OFFICE TtIE CORRESPONDENCE Date Printed: Aug …OFFICE OF TtIE SECRETARY CORRESPONDENCE CONTROL TICKET Date Printed: Aug 28, 2007 12:16 PAPER NUMBER: ACTION OFFICE: LTR-07-0565 EDO

OFFICE OF TtIE SECRETARY

CORRESPONDENCE CONTROL TICKET

Date Printed: Aug 28, 2007 12:16

PAPER NUMBER:

ACTION OFFICE:

LTR-07-0565

EDO

LOGGING DATE: 08/27/2007

AUTHOR:

AFFILIATION:

ADDRESSEE:

SUBJECT:

Po Kee Wong

MD

DaleKlein

TAMCE 2008 symposium by Fw: submission of five papers for ES1A9 to choose for presentation

ACTION:

DISTRIBUTION:

LETTER DATE:

ACKNOWLEI)GED

SPECIAL HANDLING:

Appropriate

Encls to: EDO

08/25/2007

No

Made publicly available in ADAMAS via EDO/DPC

NOTES:

FILE LOCATION: ADAMS

DATE DUE: DIATE SIGNED:

Billie Champ - Fw: FW: TAMCE 2008 Sympos'ium- by Fw: Submission of five papers for ESIA-0 to ch'o-ose for- -pre-s-entati onsae1

From: "Po Kee Wong" <[email protected]>To: <[email protected]>, <[email protected]>, "Susan McReynolds"<[email protected]>, <[email protected]>, "Mita Desai" <[email protected]>,<[email protected]>, <[email protected]>, <[email protected]>,<[email protected]>, <[email protected]>, <[email protected]>, <[email protected]>,<[email protected]>Date: 8/25/2007 2:40:23 PMSubject: Fw: FW: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 tochoose for presentations

Dear Referee at Mathforum ET AL:

You are cordially invited to review and evaluate the contents of the 16 attachments in this E-mail pertinentto the subject matter of" THE IMPACTS OF NEW SOLUTIONS OF OLD PROBLEMS INMATHEMATICAL AND EXPERIMENTAL SCIENCES " to be presented at TAMCE 2008 in Turkey and atESIA9 in Beijing for open discussions.

Your time and efforts spent for the subject matter with criticism and open discussions will be appreciated.

Very truly yours,

Wong, Po Kee0] ]2413 Spencer JRoad, Silver Spring, Maryland 20910-2344 USATel: [email protected]

----- Original Message -----From: Po Kee WongTo: Michelle. [email protected] ; [email protected]: Po Kee Wong; Adam WongSent: Wednesday, August 01, 2007 12:13 PMSubject: Fw: FW: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 to choose forpresentations

Dear Chancellor Rhee and Mayor Fenty:

With reference to my offer to you to solve your immediate needs of most updated text books inMathemaitcs and Sciences in my last E-mail having been sent to you, the current E-mail with 16attachments are submitted to you for your consideration:

----- Original Message -----From: Po Kee WongTo: Yonhua TzengCc: D LI] ; [email protected] ; Po Kee WongSent: Monday, April 16, 2007 9:28 PMSubject: Re: FW: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 to choose forpresentations

Billie Champ - Fw FW: TAMCEý 2008 Symposiu•m- u--by Fw: Submission- &f five Papers f-r ESIA9-to choose f6orpresentations Page 2!1

Dear Professor Yonhua Tzeng and professor dshieh:

Thank you for your message

----- Original Message -----From: Yonhua TzengTo: [email protected]: glULSent: Monday, April 16, 2007 7:27 PMSubject: Re: FW: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 to choose for

presentations

Dear Dr. Wong, Po Kee (UDO):

Would you please send me your updated and detailed CV so that I can forward it to departments atNCKU that fit your background and expertise?

Thank you very much for your willingness and enthusiasm in helping with NCKU! We do need a lot of

helps from NCKU alumni!

Sincerely,

Yonhua TzengProfessor of EEVP for Research and DevelopmentNational Cheng Kung UniversityTainan, Taiwan, R.O.C.

Wong, Po [email protected]

On 4/16/07, LL[] [ <[email protected]> wrote:Dear Prof. Tzeng:Got an email from Alumni Center as following. Any suggestion?

DB

[]DDDDD:LLLDLLDLE

-----------------------------------------------------------------------------------

Billie Champ.- Fw: FW:TAMC.E 2008 Sym po.sium by . F..w: Submissionof fi..ve papers for -ESIA9 t.o choose for presentations Page 31

----- Original Message -----From: Po Kee Wong <mailto:[email protected] >To: [email protected] ; [email protected] ; [email protected]: Po Kee Wong <mailto:[email protected] > ; Wong, Adam <mailto:[email protected]>;

Simon Tam < mailto:[email protected]>Sent: Saturday, April 14, 2007 1:55 PMSubject: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 to choose for

presentations

Dear TAMCE 2008 Symposium.org ET AL:

The attachments contained in the communications with various organizations worldwide aresubmitted to you for your consideration for open review and evaluation and to pick the topics that you wantto be presented at TAMCE 2008 Symposium:

Your time and effort spent on the subject matter will be appreciated.

Very truly yours,

Po Kee Wong, Ph.D.Te1:+301-585-3453Wong, PoKee E]LLIflL


----- Original Message -----From: Po Kee Wong < mailto:[email protected]>To: darcham [email protected] ; [email protected]: Po Kee Wong < mailto:[email protected]>Sent: Tuesday, March 20, 2007 9:19 AMSubject: Fw: Submission of five papers for ESIA9 to choose for presentations

The is the first one of five consecutive 5 E-mails of communications being forwarded to you:

----- Original Message -----From: Po Kee Wong <mailto:[email protected]>To: ??? <mailto:[email protected]> ; [email protected] ; Dong <mailto:[email protected] >Cc: Po Kee Wong <mailto:[email protected]> ;'simon tam' < mailto:[email protected]>

Wong, Adam < mailto:[email protected]>Sent: Sunday, January 28, 2007 11:39 AMSubject: Fw: Submission of five papers for ESIA9 to choose for presentations

Dear Professors Wu;Gosney and Dong:

Billie Cha mrp - Fw: FW: TAMCE 2.08 Symposium. by Fw: Submission of five papers for ESIA9 to choose for presentations Page _4.1

This to inform you that,by the instruction of Professor Wu, the paper ABSTRACT entitled "IMPACTS OFNEW SOLUTIONS OF OLD PROBLEMS IN MATHEMATICAL AND EXPERIMENTAL SCIENCES " hasbeen successfully uploaded to your website and given a paper code:171.

However, the paper ABSTRACT entitled "FORMULATION AND SOLUTION OF A SYSTEM OF NON-COPLANAR BEAMS IN FINITE DEFORMATIONS " has NOT been successfully uploaded to your websiteand NOT given a paper code number. This paper ABSTRACT has been SUCCESSFULLY forwarded toall of you in our previous communications. One of the reasons may be the Figure in the submittedABSTRACT occupies too much spaces for transmission during the submission. Please help to input thesubmitted ABSTRACT and send a paper code number of that submission to me.

Very truly yours,

Wong,PoKee 0[100EL


----- Original Message -----From: Po Kee Wong < mailto:[email protected]>To: ??? <mailto:[email protected] > ; [email protected] ; Dong < mailto:[email protected]>Cc: Po Kee Wong < mailto:[email protected]>Sent: Saturday, January 27, 2007 4:43 PMSubject: Fw: Submission of five papers for ESIA9 to choose for presentations

Dear Professors Wu; Gosney and Dong:

The abstract of the submitted 5 papers is attachedtogether with the 5 papers and other relevant documents for your review and evaluation.

Yours truly,

Wong, Po Kee LE LE ZE E] [I

Wong, Po Kee2413 Spencer Road, Silver Spring, Maryland 20910-2344 USATel:[email protected]

----- Original Message -----From: Po Kee Wong <mailto:[email protected] >To: ??? <mailto:[email protected]> ; [email protected] ; Dong < mailto:[email protected]>Cc: Po Kee Wong <mailto:[email protected] >Sent: Saturday, January 27, 2007 4:18 PMSubject: Fw: Submission of five papers for ESIA9 to choose for presentations


Billie Champ - Fw_ FW: TAMCE 2008 sympos'ium by Fw: Submission.of fiv..e .pap..e rs Ifor -ESIA9 .to _.choose. for pre sent.-ations Page.5

When you open the Fwmathforrumorg Po Kee Wong Angles-Google Search... you should beable to see 26 topics of mathematical discussions. However, some commercial companies intentionallyblock some of the websites for their own advertisements without my permission. If you click the topics thatyou want to see for two or more times you can open the website that you want to see and read !!! None ofall those illegal pop-ups belong to me!!!. Both you and I can take them to the Courts in USA; UK andChina for their illegally blocking legitimate communications

Very truly yours,

Wong, Po Kee 00000

Po Kee Wong2413 Spencer Road,Silver Spring, Maryland 20910-2344 USATel:[email protected]

----- Original Message -----From: Po Kee Wong <mailto:[email protected]>To: ??? < mailto:[email protected]> ; [email protected] ; Dong <mailto:[email protected] >Cc: Po Kee Wong <mailto:[email protected]>Sent: Saturday, January 27, 2007 12:46 PMSubject: Submission of five papers for ESIA9 to choose for presentations


Being forwarded to you in the attachments are 5 papers and one abstract for your selection of theappropriate one or more than one of the papers for the presentations at ESIA9 according to your needs.

Your time and effort spent on the review and evaluation of the submitted papers for your needs willbe appreciated. I look forward to hearing from you about your decision will be appreciated.

Sincerely yours,

WONG,PO KEE DOE]

Po Kee Wong, Ph.D.pokwong@verizon. net------ Forwarded MessageFrom: Po Kee Wong < [email protected]>Date: Sun, 15 Apr 2007 22:05:02 +0800To: <[email protected] >Cc: Po Kee Wong <[email protected]>Subject: Fw: TAMCE 2008 Symposium by Fw: Submission of five papers for ESIA9 to choose for

presentations

------ End of Forwarded Message

Billie Champ - Fw:FV.. TAMCE 2008 Sym po-siu.m 'by Fw: Su brnission of five -p-pers for ES!.9 to choose for .presentatio•ns Page6

CC: "Po Kee Wong" <[email protected]>

I c:\temp\GW)0-0001.TMP Pa~ge 1 ,1I. . . .-- ...... . .... ... I- ... - -. . . - I

Mail Envelope Properties (46D0777B.ACE : 10 : 6862)

Subject: Fw: FW: TAMCE 2008 Symposium by Fw: Submission of five papersfor ESIA9 to choose for presentationsCreation Date 8/25/2007 2:27:58 PMFrom: "To Kee Wong" <pokwong Ocverizon.net>

Created By: pokwong(Rverizon.net

Recipientsnrc.gov

OWGWPOO2.HQGWDOO1CHAIRMAN

nsf.govabement

mcpsmd.orgJerryD Weast

dc.govMayorMichelle.Rhee

kva.sersas

nobel.semichael.sohlman

tsplescopycenter.comcc0026

gmail.commita.9955 (Mita Desai)

yahoo.cornanisohedral

sonic.netmooresu (Susan McReynolds)

snet.netDomrosa

I c:\tempýGWý06601.T-M-P ,Page 2.1c :. . .. ... . . ......... . ... . .. . . . . ....... P a ge... .. . .. . .. .. ... 2.

mathforum.orgrichard

Post Office RouteOWGWPO02.HQGWDO01 nrc.gov

nsf.govmcpsmd.orgdc.govkva.senobel.setsplescopycenter.comgmail.comyahoo.comsonic.netsnet.netmathforum.org

Files Size Date & TimeMESSAGE 10474 8/25/2007 2:27:58 PMTEXT.htm 26394Po Kee Wong Ip resume.doc 28160Dr Wong photo.doc 33792ECQ Po Wong.doc 31744image002.JPG 115945image004.JPG 174355Fw_ mathforum org Po Kee Wong Angles - Google

Search(http wwwgoogle comrsearch_.htm 1227ICONE1 3-50509.pdf 216582COVER of COLLECTION OF TRUTH ARTICLES.pdf 2085161IMPACTS FROM NEW SOLUTIONS OF OLD PROBLEMS IN MATHEMATICAL AND

EXPERIMENTAL SCIENCES.doc 23552IMECE 2001.doc 65536IMECE2003-43540 (2).doc 51712IMECE2003-43586.doc 51200IMECE2003-43536.doc 320512COMPARISON OF TRAJECTORY SOLID ANGLE WITH GEOMETRIC SOLID ANGLE IN

SCATTERING THEORY(FULL PAPER).doc 53760Stanford Thesis.pdf 1174515FW 11 websites where you can obtain my patents.htm; international conference papers and

relevant engineering and scientifi 17684Mime.822 6124411

OptionsExpiration Date: None

c:\&temD\GWOVVOO1 .TMP .Page 3:1

Priority:ReplyRequested:Return Notification:

Concealed Subject:Security:

StandardNoNone

NoStandard

Dr. Po Kee Wong2413 Spencer Road

Silver Spring, MD 20910Tel. 301-585-3453

Email: pokwong(cverizon.net

Work History

Systems Research Company, Silver Spring, MD 1976-Present

Owner and founder of SRC specializing in science and technology. Responsible for theplanning, marketing, managing, engineering and producing proposals, reports, technicalpapers and data for national and international meetings. Presented more than 20 papers.Attended more than 100 conferences. USPTO has granted 3 patents, 1 is pending.

Boston Public Schools, Boston, MA 1979-2001

High school and Middle School teacher. Taught regular and Chinese Bilingual math andscience courses. Licensed in the state of Massachusetts. My students have received thetop awards in math from BPS. Had taught all math courses using the TI calculators.

Engineer 1971-1975

Worked for Stone & Webster, MA and General Electric, CA performing review andevaluation of computer codes, industrial and governmental technical reports. Used in.-house codes to check and to sign out specific areas in structural mechanics.

Education

1970 PHD Aeronautics and Astronautics STANFORD1966 ENG Applied Mechanics CALTECH1961 MS Mechanical Engineering UTAH1956 BS Mechanical Engineering CHENG-KUNG U.

Professional Organizations:ASME, AIAA, IAF, NYAS, and AMS

Languages:Mandarin, Cantonese

Certification:National Board for Professional TeacherMassachusetts Department of Education, Professional Development Provider

Dr. Po Kee Wong2413 Spencer Road

Silver Spring, Maryland 20910-2344 USATel. and Fax.:301-585-3453

E-mail: [email protected]

EXECUTIVE CORE QUALIFICATIONS:

(1) Leading Change: Established the sole proprietary SYSTEMS RESEARCH COMPANY (SRC) in1976. The company has been cleared and recognized by the US Federal Government Small BusinessAdministration (SBA) and the Department of Defense (DOD), Defense Logistic Agency (DLA) whichgranted SRC a Commercial And Government Enterprise (CAGE) Code: 5R583 for Mechanization OfContract Administration Services (MOCAS) since February 20, 1979.He plans, directs, and implements all operations of SYSTEMS RESEARCH COMPANY from 1976 topresent.

(2) Leading People: Recruits, selects, supervises and develops professional and support staff involved indiverse endeavors having been evidenced by the contents of his technical proposals written andproduced solely by himself and submitted to various federal governmental organizations thatinclude DOD,DOE,NRC,NSF,DOT and NASA. Evidences of this ability can be seen from thefollowing proposals:

A. SRC-DOE unsolicited proposal No. P7900450 entitled" INITIATION OF THE DEFINITION OFTRAJECTORY SOLID ANGLE AND ITS INFLUENCE ON CLASSICAL, QUANTUM ANDSTATISTICAL MECHNICS " January 17, 1979.

B. SRC-NSF SCREAMS proposal No. 98-12001 entitled " IMPACTS OF HIGH POWER FUNCTIONS;TRAJECTORY SOLID ANGLE; THE WONG'S ANGLES AND THE PHYSICAL ECONOMICTHEORY FOR SCIENTIFIC COMPUTING RESEARCH ENVIROMENTS " February 25, 1998.

C. SRC-NSF SBIR Proposal No. DMI-9760362 entitled " ESSENCE OF HIGH SCHOOLMATHEMATICS: AP CALCULUS &AP STATISTICS " June 10, 1997

D. SRC- BMDO/TOI proposal entitled " APPLICATIONS OF THE TRAJECTORY SOLID ANGLEAND THE WONG'S ANGLES TO SOLVE PROBLEMS OF FAILURE OF (THAAD)-Theater HighAltitude Area Defense Missile System " November 1, 1997.

E. SRC-NASA proposal No. NRA-96-HEDS-03-076 entitled " APPLICATIONS OF THETRAJECTORY SOLID ANGLE AND THE WONG'S ANGLES TO SOLVE FUNDAMENTALPROBLEMS IN PHYSICS " March 21, 1997.

F. SRC-NASA proposal No. TRIANA-0003-0006 entitled " APPLICATIONS OF THE TRAJECTORYSOLID ANGLE AND THE WONG'S ANGLES FOR TRIANA " July 22, 1998.

G. SRC-NASA PROPOSAL No. AIST-0042-0006 entitled " APPLICATIONS OF THE TRAJECTORYSOLID ANGLE AND THE WONG'S ANGLES TO SUPPORT ESTO PROGRAMS: AIST;ATI;IIPAND HPCC/ESS." January 22, 2000.

(3) Result Driven: Obtained the first Federal Governmental Contract for SRC from DOT-TSC underOrder No. TS-15054 in 1978 responding to the solicitation of a program in collisions of structures and as atest case of SRC' s unsolicited proposal NO. TSC-UP-77-27.Established stature in the profession as members of ASME, AIAA, MAA,AMS,AAAS, New YorkAcademy of Sciences; having been cited in 12 published " who's who" biographies.Presents and publishes high qualitative technical papers in AIAA, ASME, MAA ..etc. professionalconferences and meetings at regional, national and international meetings.Contributes in the invention of several granted and pending US Basic Patents with international impacts inphysics, mathematics, engineering and high technologies.

(4) Business Acumen: The ability of Po Kee Wong to acquire and administer human, financial, material

and information resources can be seen from the contents of the SRC-US Governmental various agenciesproposals from A,B,C ........ to G. In all these proposals that provide new technology to the U.S.Government to enhance decision making, they include the key characteristics as being prescribed in thisjob requirements for candidates.

(5) Building Coalitions/Communications: The ability of Po Kee Wong to explain, advocate, and expressfacts and ideas in a convincing manner, and negotiate with individuals and groups internally andexternally has been repeatedly demonstrated in his U.S. Patents, presentations and publications oftechnical papers at regional, national and international conferences as well as inside the contents of theabove proposals from A,B,C, to G.

(6) TECHNICAL QUALIFICATIONS.

PART I.TECHINCAL PAPERS RELEVANT TO BOTH US PATENTS HAVING BEEN REVIEWED,PRESENTED AND PUBLISHED AT INTERNATIONAL CONFERENCES: 68pages.

1. 1." BASIC NEEDS OF HUMAN BEINGS AS THE PURPOSES AND FOUNDATIONS FOR THEEXISTENCE OF GOVERNING INSTITUTIONS AND THE ADVANCEMENT OF SCIENCE ANDTECHNOLOGY ".A paper presented at 2001 ASME International Mechanical Engineering Congress &Exposition on Tuesday, November 13, 2001, 9:30AM at Technical Session # E&TM- 11 at Hilton NewYork/Sheraton New York, New York City, NY.USA.8 pages.

1.2. " APPLICATIONS OF THE TRAJECTORY SOLID ANGLE (TSA) AND THE WONG'S ANGLES(WA) TO SOLVE PROBLEMS OF THAAD FOR BMDO AND FOR FUTURE MISSIONS OF NASA"US Copyright Registration Number TX5-375-549, April 19, 2001, presented at the Proceedings of theFifteenth SSI/Princeton Conference on Space Manufacturing, May 7-9, 2001.14 pages.

1.3. " APPLICATIONS OF THE TRAJECTORY SOLID ANGLE (TSA) AND THE WONG'S ANGLES(WA) TO SOLVE PROBLEMS OF THAAD FOR BMDO AND FOR FURTURE MISSIONS OF NASA"Excerpts of document No. 1.2 published in " SPACE MANUFACTURING 13 SETTLINGCIRCUMSOLAR SPACE" Proceeding of the Fifteenth SSI/Princeton Conference on Space ManufacturingMay 7-9,2001. Page 98 to page 101. 4pages.

1.4. " APPLICATIONS OF TRAJECTORY SOLID ANGLE (TSA) AND WONG'S ANGLE (WA) TOSOLVE FUNDAMENTAL PROBLEMS IN PHYSICS AND ASTRONOMY " IAF-00-J. 1.10 paperpresented and published at 51 s". International Astronautical Congress, 2-6 Oct 2000/Rio de Janeiro, Brazil.5pages.

1.5. " APPLICATIONS OF TRAJECTORY SOLID ANGLE (TSA) AND WONG'S ANGLES (WA) FORLAUNCHING OF SPACE VEHICLES " IAF-00-S.6.03 paper presented and published at the 51StInternational Astronautical Congress, 2-6 Oct 2000/Rio de Janeiro, Brazil. 4 pages

1.6. " NUMERICAL DATA FOR SATELLITE ALTITUDE CONTROL BY MEANS OF WONG'SANGLES " AIAA-96-1047-CP paper presented and published at the 16'h. International CommunicationsSatellite Systems Conference, February 25-29,1996 Washington DC. page 517 to page 523, 7pages.

1.7. " ON THE FORMULATION AND SOLUTION OF A CLASS OF MAGNETO-VISCOELASTO-DYNAMICS (MVD) GOVERNING EQUATIONS OF MOTION" presented and published at the 1995ASME Design Engineering Technical Conferences-The 15 th. Biennial Conference on Mechanical Vibrationand Noise, September 17-20, 1995 Boston, Massachusetts, DE-Vol. 84-2 Volume 3- Part B page 1451 to1456. 7pages.

1.8. " ON THE IRROTATIONAL-FLOW VELOCITY POTENTIAL FUNCTION AND A NEWSTREAM FUNCTION OF FLUID MECHANICS " paper No. 80-C2/Aero-3 presented and published atthe ASME Century 2 Aerospace Conference, San Francisco, California, August 13-15, 1980. 10pages.

1.9. " ON THE UNIFIED GENERAL SOLUTIONS OF LINEAR WAVE MOTIONS OFTHERMOELASTODYNAMICS AND HYDRODYNAMICS WITH PRACTICAL EXAMPLES "paperNo. 67-APM-32 presented and published at the ASME Applied Mechanics Conference, Pasadena,California, June 26-28, 1967. 9pages.

PART II.3 (SRC) SYSTEMS RESEARCH COMPANY'S TECHNICAL PROPOSALS HAVING BEENSUBMITTED TO NASA FOR SUPPORT: 11 3pages.

II. 1. SRC-NASA proposal No.NRA-96-HEDS-03-076 entitled " APPLICATIONS OF THETRAJECTORY SOLID ANGLE AND THE WONG'S ANGLES TO SOLVE FUNDAMENTALPROBLEMS IN PHYSICS " submitted on March 21, 1997. 26pages.

II. 2. SRC-NASA proposal No. TRIANA-0003-0006 entitled " APPLIACTIONS OF THETRAJECTORY SOLID ANGLE AND THE WONG'S ANGLE FOR TRIANA " submitted on July 22,1998. 36 pages

II. 3. SRC-NASA proposal No.AIST-0042-0006 entitled " APPLICATIONS OF THE TRAJECTORYSOLID ANGLE AND THE WONG'S ANGLE TO SUPPORT ESTO PROGRAMS:AIST;ATI;IIP ANDHPCC/ESS." Submitted on January 22, 2000. 5 Ipages.

Eý11ýý qýomp -- imagý002JPGý Page 1I B illie....C h .. m p.. ............. .... . . . . .. .. ... . . . . . .. .. .... ... .. P a ge.... . . . .. . .. . .. . . . . . . . . . . .. 1

Ai t~9 ii i~ . tU USS-951K)~ ~~I...........-'~'- or lhl zilual pltbicaii p-;e (s oUW I,(K n:£950

slertin" 'cac PLU v, EM'11 st.--ncIL p SrtAro S ~ clý py~.- In :IdtHioii~n. f

~fl'iC~ sr''-rii~ ii" e ~c~nt~** -kcdllrcnv Order Rim~-'

I if,-.Ik io~A 10 e.rmrg firom, vw- and, ;rjIthm ow. m.11,: inclUdeo-nuyitobmgiph i 2000D

Sara Rainx

flcIisr ,-rz rr tiv~n o crle wLm -hr. hi-,r'Mnp ,rrmfvd&. JiarThc' corrc-ft*n fi:n; mbt *,ý rift't.~a~edi~ieI~ hcv ---z? tno ulm*,n.. .v~tt No Chui,'O .33ii0 ihr 1 '117 . j~:lludw cvur vun,n.-

LDCATJICI

Pt t1jR( AT I ON

EICONOI11s ".. WARDS.~

AlBESHIS O ASSO(2A'IfNS, INS-r~TES. t rc

OTHER~ WHO'S,~ WHOS IN WHICH VA." A.RE LISTED)

SN I E

a

I i~ntify thr,. thtý CA),-:'ew y'mni i.,. couvw.i an,,: innv ý-: Pu~Iishcd.S 16N.-V \TL4U ___ ________

... ..... ........ ..... .I Billie ChamD - imaae004.JPG Pagel II - ..... t-. . . --.. -...-..- ... ..-.....- ..-- -. ~.-..-...... . . . -. .-..-..-----------.----. ,-.---.-.~.-.. --.-..-....--.-.-.----..---------..-.-.-.----..-.----.---- ~.~..-..,---..

N NI i RNA IV 1., 10( IiJ I iI CA.* .U NI!C. 1 Wii. )Gk; NkN M B R iD

2-413SeR4-iad

L ASn A IBC

R*-t. 71J\-4 -5 LP

Oc:i? N-1r Won.

O~f~ir C)UTSi.-O-D)ENG ISr2L .ITUI.iO TIPE 2] ~IS-CýE,,.R-

\O~kCflN~i10 I'IAITC O: 1.068.Ck IllCLENR

.l ( nu .I1p."-t*' I ~ i iE

21 CM R1- V I t.111-. IL c

-. czit~r umi'rrc~icij ti~' Lo d (h,: F ifth -01c

2C104I Ohuffi tun'. Ifi~t-lnt of tLii 21

zIiopl~:ta~, np. ac ~and ukear, r'ierl'cma

a; thii i~iIVC

It i'Lic'l Thi to Im 2 by~c ! ( dimc Ihvn \V, w~~illR

piek~ wv-ci , '~ 41h4 -NO- I

As I nm a1:('i vii. an, n-wilr c- Ia'!(

Iit rlr L' R i' r t 't: p'ts. bm iv-441

nvifý v a Zrai m~ns flU'CE t'2 tt o 11

NowrzI)i,l 'k 31~V~ df .ll -~t

Ct-..'iti r p.it -, in

k4t ~ .ite"L± C lar? .4 ýLeii' ed 1

QVij4JI.j AIAA tH i~~ AMF-71:tJ.t-. t

I l.a.

II .' lIt II

Bi e hamp -.Fw_ mathforum-org Po Kee Wong AngTes '-Google Search(http w-vv--__googlecor-nsearch-.htm •Page• •1

---- Original Message -----From: pokwong <pokwonpq(.rcn.com>To: pokwonpq(verizon.net <pokwoncqiverizon.net>; pokwong <pokwonaq(rcn.com>Date: Wednesday, September 06, 2006 5:32 AMSubject: mathforum.org Po Kee Wong Angles - Google Search (http://www.qoo-qle.com/search?

http://www.google.com/search?q=+site:mathforum.org+Po+Kee+Wong+Angles&hl=en&lr=&ie=UTF-8&filterz-0

Billie Champ -'ICONE1 3-60660_.pdf PageB i e . .Ch a m p .. ....... ....... ... . . .. . ... . . .. .. .. . .... . .. . ... . . . . . ... ... 1. .

13 th International Conference on Nuclear EngineeringBeijing, China, May 16-20, 2005

ICONE13-50509

Fundamental Challenging Problems for Developing New NuclearSafety Standard Computer Codes

Po Kee WongSystems Research

Company (SRC)Silver Spring, Maryland

USAPhone: 301-585-3453

Fax: 301-585-3453Email:

pokw [email protected]

Adam E. WongSystems Research


USAPhone: 301-585-3453

Fax: 301-585-3453Email:


Anita WongSystems Research


USAPhone: 301-585-3453

Fax: 301-585-3453Email:


KEYWORDS: TRAJECTORY SOLID ANGLE,GEOMETRIC SOLID ANGLE, WONG'S ANGLE

ABSTRACT

Based on the claims of the US Basic patents number5,084,232; 5,848,377 and 6,430,516 that can beobtained from typing the Patent Numbers into the Box ofthe Websitehttp://164 195.1 00 11/netahtml/srchnum ht mand their associated published technical papers havingbeen presented and published at InternationalConferences in the last three years and that all thesehad been sent into US-NRC by E-mail on March 26,2003 at 2:46 PM., three fundamental challengingproblems for developing new nuclear safety standardcomputer codes had been presented at the US-NRCRIC2003 Session W4. 2:15-3:15 PM. at the WashingtonD.C. Capital Hilton Hotel, Presidential Ballroom on April16, 2003 in front of more than 800 nuclear professionalsfrom many countries worldwide. The objective and

scope of this paper is to invite all nuclear professionalsto examine and evaluate all the current computer codesbeing used in their own countries by means ofcomparison of numerical data from these three specificopenly challenging fundament al problems in order to setup a global safety standard for all nuclear power plant sin the worid.

INTRODUCTION

Problem Number (1) provides impacts enumeratedin the followings:1. Provides a new Statistical Mechanics in Physics.2. The calculation of collision cross-sections in particlephysics, based on by utilization of Geometric Solid Angle(as that was used in the well known Sir Rutherford'sAlpha Scattering), must be re-examined again incomparison with that by utilization of Trajectory SolidAngle (TSA) (US Patent 5,084,232). Please also readthe IAC-02-J.P.02 paper.

1 ICONE13-50509 Copyright ©2005 by CNS

]

I Billie Cham- P -11COND 3'-_60609.pidf Page 22 ý1Billie CharnD - ICONE13:50509.Ddf Page 2 ]

3. A new Hydrogen Model is being proposed bysubstituting the trajectory equations of the completeTwo-Body solution by Max Born into the definition ofTrajectory Solid Angle (US Patent 5,084,232).4. Application of TSA in a cold-atom atomic fountainclock as described by Bigelow (Ref 10 of paper IAF-00-J.1.10) This is the JPL project trying to improve theaccuracy of the Norman Ramsey' Atomic Clock.5. All the proposed tasks in the 1979 UnsolicitedDepartment of Energy Proposal number P7900450should be re-examined again.

Problem Number (2) Wong's Angles (WA) Patentprovides:1. A precise method to measure and determine the realtrajectories of object s under the actions of many-forcefields in various environment s andto guide the using of appropriate instrument s formeasurement s in experiment s (for examples: How to setup stations around the Nuclear Power Rant to shootdown in-coming missiles to hit the Nuclear Power Plant;How to set up instrument s around a nuclear reactor coreto track the internal flow conditions inside the nuclearreactor core... etc.)2. The (WA) provides a unique and precise method todesign the Digital Sensing Processor (DSP). The (DSP)can be used directly in all instrument s or to be integrated

into the Central Processor Unit (CPU) in all calculatorsand computers attached to regulate and control allscientific instrument s including but not limiting to: DigitalCameras; Digital Telescopes; Digital Microscopes;Digital Theodolites; High Definition TV; CAD-CAMSystem Design in Mechanical Engineering; SurveyingSystem Design in Civil and Architectural Engineering;Aircraft and Airport Landing System Design and all otherrelevant instrument s for measuring; tracking andcontrolling of object s.Problem number (3) provides a unique correction of thecalculating procedures that have been prevailingly usedin all computers and calculators for several decades.This correction must be made, because of the errors incomputers and calculators have been extensively usedin various fields of sciences, engineering, technologiesand mathematics in education. The corrections thatinvolve in functions of complex variables are veryimport ant to aerosp ace re-entry vehicles and nuclearreactor cores both include the coupling motions ofdynamics and heat transfer. Therefore, the correctionsmust be made.

SPECIFIC EXAMPLES OF PROBLEMS

Initial ejection velocity of a particleV sub.zero = 9.8 m/sec at point 0

The gravitation in the direction perpendicular to theplane XOY g = 9.8 m/sec squared a = square of v sub.zero / g = 9.8 m = Max. range of the particle can hit rl=9m r2= 10n r2- a = 10 - 9.8 = 0.2 m will notbe hit sector angle =pi/6 shaded targeted area =4.974 square m. Find the cumulative probability for theparticle ejected from point 0 randomly to hit theprescribed targeted area.

Problem No. (2):Given all reliable sensing instruments used for

detection of particles or objects moving in variousenvironment s, how to arrange and set them up instations to track; to measure; the position, velocity andacceleration vectors of the moving particle or object withthe least amount of parameters to be experiment allymeasured to obtain the accurate data?

Given

2 by CNS

I Billie' Charn- p - lC0N'E13'----505'-0'9.'p-'df _Pag~e..3_ jB li C h m - ICNL .... .. .... .. . .. .. . ... . . . . .... . . . . . .. . ...... .. .. . . ......... a ge.. . .. . . .. .. . . . . . .. . . . . .. . . . .3

Specific example: three ground stations identified asstations A B, C. Stations A is at the West of Station B.Stations C is at the north of Station B. Segment ABwhich is underneath the earth surface =0.4R(R=Radiusof the earth at sea level=OR, where 0 is the center ofthe earth). Segment BC which is also underneath theearth surface = 0.3 R. A high altitude object is observedwith two sets of Wong' s Angles from Stations A and Bas Alpha 1 = 30 degrees; Beta 1 = 60 degrees; Gamma1 = 60 degrees at time t = tl at point P1. Alpha 2 = 60degrees; Beta 2 = 30 degrees; Gamma 2 = 90 degreesat time t = t2 at point P2Find: The position, velocity and acceleration vectors ofthe object moving from point P1 to point P2.

Problem No. (3):Given:Zl=xl +1 yl, Z2 = x2 + I y2, where xl, yl, x2, y2 areunknown real numbers to be determined from solvingthe two following simultaneous equations involved inHigh Power Functions of Complex Numbers. where i = (

1 ) A (1/2) is the unit imaginary number.Find:The Principal Solution of Z1 and Z2 from the followingtwo simultaneous equations (1) and (2):

for the first time in October 1974. Comparing the (TSA)method with all other methods at that time, all othermethods became approximate. For examples: the MonteCarol Methods; the Geometric Solid Angle (GSA)method are all conditionally accurate in some givenranges of parameters but not precise in all ranges of thegiven para mete rs.

2. The definition of (TSA) is explicitly defined with allparameters implicitly contained within the definition whileall the other methods do not.

3. Due to the precise definition of (TSA), it is applicablefor macroscopic bodies as well as for microscopicparticles of mathematically defined infinitely small sizeunder the actions of any force and moment fieldsbetween and among the bodies and particles. Therefore,the (TSA) provides great impacts to the entire range ofphysics; from the calculation of the collision crosssections of sub-nucleus particles in high-energy physicsand to that of galaxies in astronomy. The applications ofall other methods are relatively limited.force fields (which include: the hydrogen model; Alphascattering; moon-earth model; Comet

4. The Geometric Solid Angle (GSA) of any targetedarea, being finite or infinitesimal ly small, is unchangedwith respect to the location of the source where theparticle is ejected. The (GSA) is not related to theparameters of ejection of the particle at all. It is a puremathematical quantity. The TRAJECTORY SOLIDANGLE (TSA) is a term containing all the parameters ofgenerating the particle and the targeted area to be hit.Thus the (GSA) of any targeted area is always finite andunchanged while that of (TSA) can be zero. Thisexplains why the (TSA) can be and should be used tosolve the P.sub.2 targeting problem for particles andbodies under the action of any force and moment fieldsand that the (GSA) cannot and should not be consideredas the correct solution for the P.sub.2 problem. There willbe errors comparing the use of (TSA) between the usesof (GSA) to solve the same problem. The errors willrange from 0% to more than 100%.

5. Since the collision cross sections of many problems incentral Halley scatters around the solar system etc.)have been based on the use of (GSA) for calculationand have been published in textbooks around the world,the future assertion the truth of (TSA) will provide a greatimpact to all those results in the past.

6. The (TSA) concept and its definition not only confirmsthe well known Heisenberg's principle of uncertainty inphysics, but also provides the precise definition andprocedures to calculate the uncertainty in term ofnumbers as precise as we want.

Arc Sin( Z1 + Z2 )=(3^(1/2)-i)I(i(1 +i3^A(1/2))A(-1+ i)) Eq.(1)

Arc Sin (Z1 - Z2)=((3A(1/2)-i)A(1+ i3A(1/2)))A(-1 + i) ......... Eq.(2)

The above specific problem is considered the simplestproblem in comp arison with other problems involved infunctions of complex variables of many Element aryTranscendent al Functions in the general solutions of aset governing equations of Thermo-V isco-Elastodynamics appeared in many references shown inthe U.S. Patent No.: 6,640,516 that have been used forLOCA, Fuel Pin Design, and Thermal HydraulicTransient Analysis in Nuclear Power Plants andAerospace Industries for years since 1968.

SUMMARY

1. The invention of TRAJECTORY SOLID ANGLEprovides the most precise definition to solve the problem

3 ICONE13-50509 Copyright @©2005 by CNS

I 0ý11_ie P"mp - CONE! 3-60500.pdf B li C h m. . . . . .. .. . . .. . . ........... .P age 4.

7. The most important concept of (TSA) is that thedefinition can be applied to discover new laws and newparticles by comparisons and matches of the unknownresults with the already confirmed and proved results. Ifthere are new laws of physics that describe the particlemotions other than those of Newton's classicalmechanics and Einstein's narrow and general relativity,the present (TSA) concept is still applicable to obtain theprecise P.sub.2 function for the problem.

8. Four examples are selected to illustrate how to obtainthe probability distribution functions by means of (TSA).They are: Ajpha scattering; A particle in uniform,isotropic linear motion; A particle under assumedconstant-gravity pull from a plane surface; A particle in amedium where the resistance force is 'linearlyproportional to the velocity of the particle and under auniform gravitational field. These examples are selectedon the basis that they are well known and can be foundfrom the open literatures. They were selected with theintention to show that even with such simple well-knownexamples, the correct probability functions andcumulative distribution functions of these problems havenever been obtained before. Whether exact solutionscan be obtained from the equations of motion thatgovern other problems will not be the issue because theequation of motions can always be solved by means ofnumerical analysis together with computer programming.The key issue is that through the definition of (TSA), theP.sub.2 functions can be precisely defined and obtained.The (TSA) can be applied to solve the most fundament alproblems in physics that include all the subjects listed ascited references in this application.

NOMENCLATURE

BRIEF DESCRIPTION OF THE DRAWINGSFIGURE No.1:FIGURE No. 1 shows a finite surface area ABCDA thatcan be described with respect to a fixed coordinatesystem designated to be hit by a particle generated andejected from the origin of the coordinate system. Thefinite area ABCDA to be hit can be subdivided intoalmost infinite number of infinitesimal surface areas,thus the total surface area in vector form can beexpressed as

Ir itrT •AA

47r R'S

Lr U" S-o JLl"A= A, =YAA5

S=1

U0LN UiI t.u VI 1Td A Acos2dA_z. 4,R J2=.A4 /t2 .A• 4

Each infinitesimal surface area.delt a.Asub.s isconnected by the position vector R.sub.s that defines theequations of a surface. The spherical surface formed bythe position vector passing through a particularinfinitesimal surface area (which is as small as almostlike a point) is 4.pi.R.sub.s. sup.2, where O<R.sub.s<Infinity.

The probability for the particle to hit a infinitesimalsurface area delta.A.sub.s depends on the particle's unittangent vector T.

IfT is parallel (direct common sense, since the surfacearea vector has a unit vector perpendicular to thesurface area, thus mathematically it should have beensaid perpendicular) to the surface area delta.Asub.s,the particle will miss the surface.

IfT is perpendicular (it should have been said parallelmathematically) to.delta.A.sub. s the particle will hit thesurface at a right angle.The probability for the particle to hit on.delt a.Asub.s isP. sub. s which can be expressed in the following.

The probability for the particle to hit the entire surfacearea ABCDA is therefore P. sub.2 that equals to thesummation of P. sub.s, where s is summed from s=1 tos= to infinity.

These complete the proof that the (TSA) can be used tosolve the P.sub.2 targeting problem. (The unit tangentvector T contains all the parameters of generation andejection of the particle and satisfies the govemingequations of the laws of physics).

FIGURE No. 2 shows that particles under assumedconstant gravitational pull on a plane are generated andejected from the origin of the coordinates. The particlescan be electrons, ions or unchanged particles. Thegravitation g can also be simulated by an electric and/ormagnetic field for the charged particles. The plane is thetarget surface to collect the particles. The figurerepresent s the schematic diagram of a massspectrometer. Given the initial velocity v. sub .o of anejected particle, find the probab ility of the particle thatwould hit the predetermined area bound by r.sub. 1 lessthan or equal to .r that is also less than or equal tor.sub.2; phi. sub. 1 less than or equal to phi that is alsoless than or equal to phi.sub.2.

4 ICONE13-50509 Copyright @2005 byCNS

[Billie Cqhamp- ICONE1 3-50_509:pRdf ... . . . . . . . . . . . . . . . . . . . . . . . .. [

FIGURE No. 3 shows the probability density function ofthe particle hitting on the plane surface. The cumulativeprobability to hit on any area on the plane surface canbe calculated by carrying out the integration precisely.For example:Given v sub.o =9.8m/sec g=9.8m/secA2

a= (v su.o)A2/g =9.8m = Max. Range r sub.1 = 9mr sub.2 =10 m r sub.2-a =10-9.8 = 0.2 m will not be hitphi sub.2 -phi sub. 1 = pi/6 =0.5235987756 radian

The targeted area =4.974 mA2

The cumulative probability for the particle to hit thetarget area=.OMEGA/(4. pi.)=0.0035846.

FIG. No. 4 shows the classification of regions that can bereached only by the high; only by the low, and by bothhigh and low trajectories of the particles.

Region H bounded by OPBT.sub.h AO can be reachedby high trajectory only.

Region L bounded by DBCD can be reached by lowtrajectory only.

Region HL bounded by OPBDIO can be reached by bothhigh and low trajectories.

Region outside of OAT.sub.h BCIO cannot be reachedby either high or low trajectories.

The TRAJECTORY SOLID ANGLE (TSA), ProbabilityDensity Function (pdf), and Cumulative DistributionFunction (cdf) are all different in each region. They areall zero outside of the region bounded byOAT.sub.hBCIO.

FIGURE No. 1 is applicable for the general targetingproblems in mass spectrometers particle accelerators,super-colliders; actual missiles and rockets targetingproblems scattering and collision of astronomical bodies,chaos of classical dynamics and quantum mechanics,fluid dynamics and the weather prediction . .. etc.

FIGS. No. 2, 3, 4 are demonstrated in great details howto apply the invention of (TSA) to solve a specific well-known-simple problem but its Probability DensityFunction (pdf) and Cumulative Distribution Function (cdf)have never been precisely obtained by all other methodsbefore the invention of (TSA).

The procedures to find the distribution function P.sub.2for a particle striking a predetermined area, given all itsparameters of generation and ejection can be

systematically summarized in the following steps:

1. Solve the set of govemi ng equations that govem thetrajectory of the particle and obtain the position vector,the velocity vector and the trajectory equation in terms ofinitial conditions and all other parameters in thegoveming equations.

2. Find the unit tangent vector from the velocity vector orfrom the trajectory equation.

3. Find the unit normal vector and the differential surfacearea from the goveming equation of the surface to bestruck.

4. Find the intersection of the trajectory on the surfaceand set the intersection coordinates in terms of the twoindependent variables that define the surface.

5. The incident angle of the particle on the trajectorystriking at the surface and be defined from the innerproduct of the unit tangent vector to the unit normalsurface vector expressed in terms of the twoindependent variables at the intersection.

6. The trajectory solid angle for the problem can beobtained from integration over the cosine of the incidentangle multiplying the differential surface area divided bythe square of the position vector of the surface.

7. The probability distribution function can be defined asthe ratio of the trajectory solid angles (TSA).

There are 11 separate tasks proposed tasks to be donein the SYSTEMS RESEARCH COMPANY'S 143 pagestechnical proposal DOE No. P7900450 that was sent forsupports in 1979 to the High Energy Physics Division ofthe US Department of Energy.

There are also at least 3 technical proposals havingbeen submitted to NASA for funding and support.

The values of the invention depend on whether thesolution of the P.sub.2 targeting problem by means ofthe (TSA) is TRUE. If it is, it will provide all the impacts topractitioners, public and private decision makers, andthe general public especially involved in education:

It will affect many previous Nobel Laureates' work inscattering and collision crossections of particles; inStatistical Mechanics and Quantum Mechanics. Specificwork of interests include: Rutherford's Alpha Scattering;Hofstadter's electron scattering; Yang's p-p collision andits scattering and the geometric picture; Fermi-Dirac,Bose-Einstein, Maxwell-Bo Itzmann statistics; quantummechanics based on Schrodinger's equation; Schwinger

5 ICONE13-50509 Copyright @2005 by CNS

I Billie ChamD - IC- ONE13MW509.pdf Page 6 ,I

and Feynman's quantum electrodynamics andHeisenberg's uncertainty principle... etc.

All these topics are in the current text books of physicsfor graduate and undergraduate levels in all universitiesin the world. It follows that will influence the selection ofmaterials for the secondary curriculum planning anddevelopment according to the impacts.

ACKNOWLEDGEMENT

This paper is an original excerpt from the U.S. PatentNo. 5,084,232 entitled 'Trajectory Solid Angle's Impactsto Physics and High Technologies "Published by theU.S. Patent Office on January 28, 1992. The previouswork contributed by many others leading to the inventionof Trajectory Solid Angle (TSA) to solve the P sub. 2targeting problem and that also become the foundationof the new statistical mechanics is gratefullyappreciated. The informat ion by many others can beread from the REFERENCES CITED and theBACKGROUND OF THE INVENTION from the patentspecification, which can be obtained from:

http://164 1Q5.1 00.11/netahtml/srchnum ht m

The U.S. Patent 5,848,377 entitled 'Wong' s Angles (WA)toDetermine Trajectories of Objects " published by the

U.S. Patent Office on December 8, 1998 can also beobtained from the above site.

Technical papers recently produced by means of theclaims of both the (TSA) and the (WA) patents havingbeen reviewed, presented and published at internationalConferences are provided in the following to ourcolleagues for review and evaluation of this submittedpaper for ICONE10:

REFERENCES

PART I.TECHINCAL PAPERS RELEVANT TO BOTH USPATENTS HAVING BEEN REVIEWED, PRESENTEDAND PUBLISHED AT INTERNATIONALCONFERENCES: 68p ages.

I.1."BASIC NEEDS OF HUMAN BEINGS AS THEPURPOSES AND FOUNDATIONS FOR THEEXISTENCE OF GOVERNING INSTITUTIONS ANDTHE ADVANCEMENT OF SCIENCE ANDTECHNOLOGY ". A paper presented at 2001 ASMEInternational Mechanical Engineering Congress &

Exposition on Tuesday, November 13, 2001, 9:30AM atTechnical Session # E&TM-1 1 at Hilton NewYork/Sheraton New York, New York City, NY.USA8pages.

1.2. " APPLICATIONS OF THE TRAJECTORY SOLIDANGLE (TSA) AND THE WONG'S ANGLES (WA) TOSOLVE PROBLEMS OF THAAD FOR BMDO AND FORFUTURE MISSIONS OF NASA" US CopyrightRegistration Number TX5-375-549, April 19, 2001,presented at the Proceedings of the FifteenthSSI/Princeton Conference on Space Manufacturing, May7-9, 2001.14 pages.

1.3. "APPLICATIONS OF THE TRAJECTORY SOLIDANGLE (TSA) AND THE WONG'S ANGLES (WA) TOSOLVE PROBLEMS OF THAAD FOR BMDO AND FORFURTURE MISSIONS OF NASA" Excerpts of documentNo. 1.2 published in "SPACE MANUFACTURING 13SETTLING CIRCUMSOLAR SPACE' Proceeding of theFifteenth SSI/Princeton Conference on SpaceManufacturing May 7-9, 2001.Page 98 to page 101.4pages.

1.4. "APPLICATIONS OF TRAJECTORY SOLID ANGLE(TSA) AND WONG'S ANGLE (WA) TO SOLVEFUNDAMENTAL PROBLEMS IN PHYSICS ANDASTRONOMY " IAF-00-J.1.10 paper presented andpublished at 515'. International Astronautical Congress,2-6 Oct 2000/Rio de Janeiro, Brazil. 5pages.

1.5. "APPLICATIONS OF TRAJECTORY SOLID ANGLE(TSA) AND WONG'S ANGLES (WA) FOR LAUNCHINGOF SPACE VEHICLES " IAF-00-S.6.03 paper presentedand published at the 51"•. International AstronauticalCongress, 2-6 Oct 2000/Rio de Janeiro, Brazil. 4 pages

1.6. "NUMERICAL DATA FOR SATELLITE ALTITUDECONTROL BY MEANS OF WONG'S ANGLES" AIAA-96-1047-CP paper presented and published at the 16 1h.

International Communications Satellite SystemsConference, February 25-29, .1996 Washington DC.page 517 to page 523, 7 pages.

1.7. "ON THE FORMULATION AND SOLUTION OF ACLASS OF MAGNETO-VISCOELAST O-DYNAMICS(MVD) GOVERNING EQUATIONS OF MOTION"presented and published at the 1995 ASME DesignEngineering Technical Conferences-The 15 h. BiennialConference on Mechanical Vibration and Noise,September 17-20, 1995 Boston, Massachusetts, DE-Vol.84-2 Volume 3- Part B page 1451 to 1456. 7pages.

6 ICONE13-50509 Copyright @2005 by CNS

BillONieCh•amp-ICONE3-5050_9-pdf P age 7

1.8. "ON THE IRROTATIONAL-FLOW VELOCITYPOTENTIAL FUNCTION AND A NEW STREAMFUNCTION OF FLUID MECHANICS" paper No. 80-C2/Aero-3 presented and published at the ASME 7Century 2 Aerospace Conference, San Francisco,California, August 13-15, 1980. 10pages.

1.9. "ON THE UNIFIED GENERAL SOLUTIONS OFLINEAR WAVE MOTIONSOFTHERMOELAST ODYNAMICS ANDHYDRODYNAMICS WITH PRACTICAL EXAMPLES ./"paper No. 67-APM-32 presented and published at the _ASME Applied Mechanics Conference, Pasadena, rCalifornia, June 26-28, 1967. 9pages.

PART II.THREE (SRC) SYSTEMS RESEARCH COMPANY'STECHNICAL PROPOSALS HAVING BEEN SUBMITTED ,TO NASA FOR SUPPORT: 113p ages. '

II. 1. SRC-NASA proposal No.NRA-96-HEDS-0 3-076entitled "APPLICATIONS OF THE TRAJECTORY SOLIDANGLE AND THE WONG'S ANGLES TO SOLVEFUNDAMENTAL PROBLEMS IN PHYSICS " submittedon March 21,1997.26 pages.

II. 2.SRC-NASA proposal No. TRIANA-0003-0006entitled "APPLIACTIONS OF THE TRAJECTORYSOLID ANGLE AND THE WONG'S ANGLE FORTRIANA" submitted on July 22, 1998. 36 pages

I1. 3. SRC-NASA proposal No.AJST-0042-0006 entitled" APPLIACTIONS OF THE TRAJECTORY SOLIDANGLE AND THE WONG'S ANGLES TO SUPPORT 1ESTO PROGRAMS: AIST; ATI; lIP AND HPCC/ESS."Submitted on January 22, 2000. 51 pages.

-- ----------- ,--' 0. 0

t .'.u hh0

ICONE13-50509 Copyright ©2005 by CNS

:BillieýCha mp _IGONE1 3-50509.pdf ........ ........ . -... .. - a ge -8

T'[

FL#,r R. %

B~t..t. Li.Cd~ .e rn .11 010 sai L-14 -ea1.rtr

--

8 ICONE13-50509 Copyrght@2005 by CNS

Billie Champ - COVER of CO LL.ECTION OF TRUTH ARTICLES._pdf Page 1

Sn

+z

M--lIs

BilieChamp- MPACTS .-FROM NEW SO-LUT-[ON-SO F O•L•D PROBBLEM-S IN M AT H EMATI-CAL ANID EXPERI M ENTAL SCIE bSi

IMPACTS FROM NEW SOLUTIONS OF OLD PROBLEMS IN MATHEMATICALAND EXPERIMENTAL SCIENCES

ABSTRACT

Submitted to

NINTH INTERNATIONAL CONFERENCE ON ENGINEERING STRUCTURALINTEGRITY ASSESSMENT

ESIA9"Engineering Structural Integrity - Research, Development and Application"

15-19 October 2007, Beijing, China

ByPo Kee Wong, Ph.D.(LI 0 0)

SYSTEMS RESEARCH COMPANY, USA

E-mail:[email protected]

In response to the Second Call-for-Papers from ESIA9, five papers are beingsubmitted to all participants of our colleagues worldwide for open review and evaluationand to assess their impacts and values in mathematical and experimental sciences withtheir applications in Engineering Structural Integrity:

(1) IMECE 2001/T&S-23408 paper, 7 pages with partial section translation inChinese.

(2) IMECE 2003-43540 paper, 3 pages.(3) IMECE 2003-43536 paper, 5 pages.(4) IAC-02-J.P.02 paper, 7 pages(5) IMECE 2003-43586 paper, 3 pages.

The above 5 papers together with this abstract are being submitted electronicallyto Professor Sujun Wu (DO LI1)

Biliie Oarnp - IMECE2oO3-43540-(2).doc B illie. .Ch am p. .. ... . . . ... . . . . .. ....... .. P a ge .1 .I

2003 ASME INTERNATIONAL MECHANICAL ENGINEERING CONGRESS AND R&D EXPO ProceedingsIMECE2003

November 15-21, 2003, Washington DC USA

IMECE2003-43540

the uniquely-corrected method to compute high power functions

Po Kee Wong/SYSTEMSRESEARCH COMPANY (SRC)

U.S. Federal Supply Code:5R583

Adam Wong/SYSTEMSRESEARCH COMPANY(SRC)


Anita Wong/SYSTEMSRESEARCH COMPANY (SRC)


AbstractThe present paper provides a unique correction of the

calculating procedures that have been prevailingly used in allcomputers and calculators for several decades. This correctionmust be made, based on' the impacts of the computers andcalculators have been extensively used in various fields ofsciences, engineering, technologies and mathematics ineducation.

introductionIn recent years, several large companies that produce hand-

held calculators have been competing continuously with oneand other to break into the market for educators of all levels tolearn and to use their calculators for teaching in the classrooms.During the summer months of June-August, 1995, manyseminars were conducted in the metropolitan Boston Areas inthe State of Massachusetts. CASIO offered workshops for theuse of CFX-9800G; Texas Instruments offered workshops forthe use of TI-82; Hewlett Packard offered for the use of HP38G. Educators from the Boston Public Schools of the City ofBoston, Massachusetts were invited and assigned to attend theworkshops offered by CASIO and TI. Educators fromeverywhere were invited to attend the HP 38G workshops. Itwas undoubtedly that all the participants in these workshopswere benefited from utilizing the calculators to implement theirmathematics and science curricula in one way or the other. Inparticular, educators from the State of Massachusetts came forthe workshops enthusiastically because they were givenProfessional Development Points (PDP) to fulfill therequirements for their re-certification to teach in the State of

Massachusetts. Others came to seek for the choices of theappropriate calculator in order to implement their curriculaeffectively in their own classes. In addition to the abovereasons, the first author of this paper also participated all theworkshops in order to select the appropriate calculator for theAdvanced Placement Calculus that was being offered at theCharlestown High School of the Boston Public Schools underthe grant funded by the EAGLE program of the Boston Plan ForExcellence in The Public Schools in academic year 1995-96.The author learned a lot from attending those workshops andalso fed back his opinions that could and should be updated andto be built-in into the calculators for wider applications not onlyfor teaching but also for research in Physics and Mathematics.For examples: special functions like circular cylindrical andspherical Bessel's Functions; Legendre Functions and ErrorFunction should be built-in into the calculator to solve manyproblems in Physics and in Engineering; likewise the LargrangeInterpolation Formula should also be built-in for curvefitting... etc. After finishing the participation of all fourworkshops in July-August 1995, the first author was asked byMr. Richard Stutman, a BPS mathematics teacher and colleagueworking in the Boston Teacher's Union (BTU), to solve a fun-and- game problem that was involved in high power functionsof infinite orders. Responding to his request, the author soughtto solve the problem by means of the CFX-9800G; TI-82 andHP-38G. As a result of this effort, a major error in theprocedures of calculating the high power functions was foundsimultaneously in all three calculators CFX-9800G; TI-82 andHP-38G. The major error had been corrected and filed forexamination with the U.S. Patent Office in order to clear the

I Copyright ©0 #### by ASME

Billie Champ-lMECE2003-43540(?).doc Page 2

Billie Champ - IMECE2008'48640 (2).doc Paýge 2]

legal liability problems from the companies.

SUMMARY OF THE CORRECTIONMathematical procedures of calculation of a mathematical

function in symbolic form can be defined in many ways almostat our own wills. However, there are examples that proceduresand the symbolic expression of the mathematical functions willnot be unique if one changes its standard calculatingprocedures. The power functions are some of these examples.The errors to calculate the high power functions contribute fromCFX-9800G; TI-82; and HP-38G are that they all start from thebase upward to the higher exponential power, while the correctway should be started downward from the top exponentialpower to the base. These can be cleared from the followingexamples A and B:

A. Errors in Numerical Computations

23 2

4 = 64 = 4096 is not correct

DETAIL DESCRIPTION OF THE CORRECTIONThe above examples A and B in errors can be corrected as

the followings:

A. Correct Numerical Computation.

23 9

4 = 4 = 262,144

32 8

3 = 3 = 6,561

23 9

2 = 2 =512

32 3

3 = 9 = 729 is not correct

23 2

2= 8 = 64 is not correct

B. Correct Symbolic Representation Uniquely Involved inSolving Equations of High Power Functions.

xx

x - 2=0 x = 1.476684337

xX

xx - 2=0 x= 1.446601432B. Errors in Symbolic Representation uniquely Involved

in Solving Equations of High Power Functions.

xx

x - 2 = 0 meansanswer x= 1.1414213562

xx

xx - 2 = 0 means

answer x= 1.336709735

2x

x - 2 = 0 leads to wrong

3x

x -2 = 0 leads to wrong

(3x)(2x)

-2=0 x=1.064146805

x(2x)

(3x) -2 = 0 x=.6140723908

(3x) 2 C. Examples of Correct Solutions of more ComplicatedEquations of High Power Functions.(2x) (6x )

x -2 = 0 means xwrong answer x = 1.100152079

-2 = 0 leads to2 2

2 Copyright © #### by ASME

BiWe d'*rnjp'_- IME .CE .2 .0 .0 1 3-4 -3 .540 -2 doc B i.....l. ... e...Ch a m p. ..... ................. ....... .. ... .. . . .. . .. .. ..... .. ...... ... ... P a ge....p 3e

2 (x )x

x= 1.41421356

(x )-5x +6=0

and x= 1.565552276

y40(x) - 6 = 0 x = 1.449978187

y40(x) - 7 = 0 x = 1.450087526

y40(x) - 8 = 0 x = 1.4501607

y40(x) - 9 = 0 x 1.450213659

y40(x)- 10 = 0 x= 1.450254088

(x)2(x)

(x)

(x)(x)

- 5 (x) +6 =0

x = 1.476684337 and x= 1.635078475 CONCLUSION

xx x

Denote yl (x) = x ; y2(x) = x y3(x)=x -..... etc.

The solutions of the following equations of High PowerFunctions can be obtained:

y2(x) - 2 0

.y3(x) - 2 = 0

y4(x) - 2 =0

y5(x) - 2 0

y6(x)- 2 0

y7(x) - 2= 0

y8(x) - 2 0

y9(x) - 2 0

ylO(x)- 2 = 0

yl15(x) - 2 = 0

y20(x) - 2 = 0

y30(x) - 2 = 0

y40(x) - 2 = 0

y40(x) - 3 = 0

y40(x) - 4 = 0

y40(x) - 5 = 0

x = 1.559610469

x = 1.476684337

x = 1.446601432

x = 1.432694806

x = 1.425385621

x = 1.421227912

x= 1.418734462

x= 1.417182504

x= 1.416190183

x= 1.414502086

x= 1.414258764

x = 1.414214713

x= 1.414213592

x= 1.447839583

x= 1.449395757

x= 1.44979292

What is claimed is:

1. A unique method of calculating and solving equationsinvolved with High Power Functions has been madefor all current and future computers and calculatorsthat are built-in with the wrong procedures to calculatethe High Power Functions.

AcknowledgmentsThe first author of this paper thanks to his friend and

colleague Mr. Richard Stutman of Boston Public Schools forthe fun-and-game problem which led to the discovery of theerrors for the calculation of High Power Functions in computersand calculators back to 1995.

ReferencesAll documents that have been filed with the U.S. Patent

Applications No. 08/980,657 by Po Kee Wong since 1995.


......... .. ------ II Biillie Clharnp - IMECE2003-43586.doc Page -1 1I Billie Chamo - IMECE2003-43586.doc l-'agel I

2003 ASME INTERNATIONAL MECHANICAL ENGINEERING CONGRESS AND R&D EXPO ProceedingsIMECE2003

November 15-21, 2003, Washington DC USA

IMECE2003-43586

AN ALTERNATE COMPLETE SOLUTION OF THE "COCONUTS" PROBLEM INNUMBER THEORY

Po Kee Wong/SYSTEMSRESERCH COMPANY (SRC)


Adam Wong/SYSTEMSRESEARCH COMPANY (SRC)


Anita Wong/SYSTEMSRESEARCH COMPANY (SRC)


AbstractThe " Coconuts " problem was described in 1926 by Ben

Ames Williams in a short story that appeared in The SaturdayEvening Post. Solutions have been given by Dirac and byWhitehead, among others. Their solutions involve sixindeterminate equations that have been reduced to a singleDiophantine equation with two unknowns. We give a newalternate solution obtained by means of a geometric series.

introductionAs can be seen from the article in reference No. I by

Gardner, the history of the "Coconuts" problem had beendescribed in the October 9, 1926 issue of the Saturday EveningPost appeared as a short story by Ben Ames Williams. Theproposed solutions of the problem had been previously involvedwith Noble Laureate Physicist Professor P. A. M. Dirac of theCambridge University and by Mathematician Professor J. H. C.Whitehead of the University of Oxford in England. The olderversion of the problem can be expressed in six indeterminateequations that can further be reduced into a single Diophantineequation with two unknowns. Up to the present, thisDiophantine equation can be solved by means of computerprogramming. The modern version of the older version of theWilliams' "Coconuts" problem was re-stated by Herda inreference No. 2 as following:

" A pile of coconuts was collected by four woman, assistedby a monkey. During the night, one of the woman arose whilethe others were asleep and divided the nuts into four equalshares, with one nut left over which she gave to the monkey.She hid one share, put the rest of the nuts into a single pile and

went back to sleep. In turn, each of the other three women wentthrough the same procedure, and in each case there was alwaysone nut to give to the monkey! In the morning, the four womendivided the remaining nuts into four equal shares and again hadone nut left over for the monkey. What is the smallest numberof nuts could have been in the original pile? "

The problem was re-stated by Professor Herda of theUniversity of Massachusetts at Boston to his students as amidterm take-home examination question and to write acomputer program to solve the problem. The authors, withoutprevious knowledge in Number Theory and knowing the paperby Gardner, provided an independent complete solution of theproblem. The method of obtaining the complete closed-formsolution is by means of a Geometric Series that appears in theformulation.

NomenclatureAssuming that there are N nuts in the pile and that there are

P women in the game. Denoting:W I---the amount of nuts hidden by the first woman with

RI nuts left over.W2---the amount of nuts hidden by the second woman with

R2 nuts left over.W3---the amount of nuts hidden by the third woman with

R3 nuts left over.ThenWr---the amount of nuts hidden by the r th. Woman with Rr

nuts left over.Wp---the amount of nuts hidden by the p th. Woman with

I Copyright © #### by ASME

Billi.e.Cham.p.7 IMIECIE2003-4M8.6.doc B ilie. . . .. . .. . . .. . . . . . ... . ... . .P a g e 2 ;

Rp nuts left over.Where r are positive integers between I and p.Nuts hidden by each woman can be expressed in general

Wr = (R sub(r-I) - 1) / P for r equal to and greater than 2 whileWI= (N -1)/PNuts left over in the pile Rr after each woman took her

share can be expressed in generalRr = (P-I) Wr

These two general expressions can be understood better bythe followings:

WI = (N-1)/P

W2 = (RI-1)/P =(N-I)(P-l) / PA2 - I/P

W3 = (R2-1)/P= (N-I) (P-I)A2/PA3 -(P-1)/PA2 -I/P

And

RI = (P-I) WI=(N-l)(p-l)/P

R2 =(P-l) W2=(N-I)(P-I)A2/PA2 - (P-I)/P

R3 =(p-1) W3=(N-I)(P-I)A3/PA3 - (P- I)2/PA2 - (P-I)/P

Based on the above results, both Wr and Rr can beexpressed, in general as:

Wr=(N- I)(P-I)A(r- I )/PAr-(P-I)A(r-2)/PA(r- I).. .(P-I)/PA2-I/P.

Rr=(P- I )Wr=(N- I )(P- I )"r/PAr - ( (P- I )P^r -P(P- I )Ar)/PAr

And when r=P

Rp=(N- I )(P- I )AP/PAP_ ((P_ I )PAPP(P_ I )AP)/PAP

= (P-1) W p ................................................. E q. (l)

Equation (1) represents the nuts left over in the nextmorning.

The P th. Woman hid the amount of nuts Wp is represented

by Equation (2) expressed in the following:

Wp=(N-I )(P-I )A(P-1 )/P'P_(PAP-P(P_ I)^(P. I ))/P^P.....(2)

Again in the next morning, the left over amount of nuts Rpin Equation (1) is to be divided by the P th. Woman and gaveone to the monkey, then

(R p - 1)/ P = 1 ............................................... (3)

The net gain of nuts by the r th. woman is denoted by Tr.

Tr= Wr + I -Wr +(Rp-l)/P= Wr +((P-I) Wp-l)/P ...... (4)

Substituting Rp from Equation (1) into Equation (3) withfurther simplification and obtaining

(N+P- 1)(P - l)AP = (I + I) pA (P+I) ..................... (5)

The problem is now reduced to find N and I as integers thatsatisfy Equation (5) for any given P as total number of womenin the game.

It is when (N+P - 1) = PA (P+I) and (1+1) =(P - I)AP theleft-hand side and the right-hand side of Equation (5) becomesidentical to each other for any given P values, thus

N=PA(P+ 1) -P + I ................ (6)

I= ( P -- I ) A p . I ............................................. (7)

Are obtained as the general solution of the problem.

The solution represented by equations (6) and (7) is thesmallest solution because the numbers on both sides ofEquation (5) are divisible by (P - I)AP x P A(P+I)

The complete general solution of the problem can now besummarized in the following:

P = Total number of women in the game

The total number of nuts in the pile is N

N=PA(P+1) -P +1

The amount of nuts that the r th. woman hid is Wr

Wr-=(P- I)A(r- I) PA(P+ I)/PAr -

The amount of nuts that the P th. woman hid is Wp

Wp=P(P- I)A(P- 1)-I

The total number of nuts that the r th. woman got is Tr

Tr = Wr +1 =(P- I)A(r- I) P^(P+ 1) / PAr +(P- I)^P - 2

The total number of nuts that the P th. woman got is Tp

Tp = Wp + I =P (P-I)A(P-I) + (P-I)AP - 2

The last pile of nuts that each woman got is I

I=(P- 1)AP- I

The monkey got (P + 1) coconuts

2 Copyright © ##" by ASME

Billie Champ I.MIE-,CIE2003-4.58•6.OPCae --IPa.ge. ...3......:

All these results can also be counter-checked from the totalnumber of nuts in the pile that should be equal to the sum of allnuts that each woman got plus the nuts that the monkey got,that is

N = Summation of Tr (from r=l to r=P ) + ( P + I)=Summation of Wr (from r= I to r=P) + PI + (P+ I)=Summation of Wr (from r=l to r=P) + P (P-I)^P+l=Summation of ((P-l)^(r-I) P^(P-r+I) -1) (from r-l to

r=P) + P (P-1)^P +1= P^(P+I)- P +1

The above general and complete solution of the problemcan be used to make numerical check for the older version ofthe Williams' " Coconuts" problem as indicated by the paper byGardner by setting P = 5 sailors then N = 15,621 nuts. When Pbecomes very large, there will be as many indeterminateequations to be set up according to the paper by Gardner. Thereduction of these indeterminate equations into the singleDiophantine equation and then seeking the solution of theDiophantine equation is extremely tedious and time consumingeven with the help of a powerful computer.

Brooks and Professor Richard Sampson of Bates College tohave revised the abstract of the paper and making it concise andprecise to have aroused the attentions - from all themathematicians in the world. The authors also thank Mr. L.Rizzo of Bates College for producing the' video record of thepresentation of this paper at Bates College on August 9, 1995.

ReferencesI. Gardner, M.: Mathematical Games. Scientific

American, page 118-122, April, 19582. Herda, Hans-Heinrich. : Lecture Notes for Number

Theory. University of Massachusetts at Boston, March,2003.

Based on the presentation of this paper, the advantages ofthe complete closed-form solution over the previous solution bymeans of the Diophantine equation is therefore distinctive.

CONCLUSIONS

I. The current complete solution is in closed -formwithout solving the simultaneous linear equations ofcongruence.

2. Its advantages over all other previous solutions bymeans of the reduced Diophantine equations are:

a. Saving a great deal of computing time to obtain thesolutions especially when the number of women in thegame is a very large number.

b. Providing the complete information of the solution ofthe problem that has never been obtained before : suchas Wr, Wp, Tr, and Tp terms as described in thispaper.

3. Equation No. (6) N = P^(P+I) - P + I can be integratedinto the current computer programs worldwide to search for thelargest prime numbers continudusly.

AcknowledgmentsThe authors would like to thank Professor Herda for his

comments about the authors' independent complete solution ofthe "coconuts" problem and his reference to search Gardner'spaper. The authors are also thankful to Adele Premice ofScientific America for providing information to obtainGardner's papers. The authors appreciate Professor Robin

3 Copyright © ####4 by ASME

. .... .. .. .. ... .

Billie Champ - IMECE203.-43536.doc -~ - - - - -- - - age1.

2003 ASME INTERNATIONAL MECHANICAL ENGINEERING CONGRESS AND R&D EXPO Proceedings IMECE2003

November 15-21,2003, Washington DC USA

IMECE2003-43536

APPLICATION OF THE WONG'S FORMULA FOR GEAR DESIGN AND ANALYSIS

Po Kee Wong/Systems ResearchCompany (SRC)

Adam Wong/Systems ResearchCompany (SRC)

Anita Wong/Systems ResearchCompany (SRC)

AbstractEven though the famous ancient Greek Appollonius

problem had been solved by many others by means of theCoordinate Geometry (Analytical Geometry) as indicated, forexample, from page 56 to page 59 in the book " CRC ConciseEncyclopedia of Mathematics" by Eric W. Weisstein in(Reference No. 1), the complete closed-form solution of theproblem by means of Trigonometry and Plane Geometry that isindependent from the coordinate systems has never beenobtained until it was first derived, obtained and filed for a U.S.Patent Application by the first author of this paper on October31, 1994 ( Reference No. 2).

The closed-form formula can be applied for CAD/CAMpackaging of circular cylindrical objects. They are simple to beused, cost effective and that they also provide the highestprecision and accuracy in closed-form equations containingonly six parameters. With all advantages over all other andprevious methods, the Wong's Formula are highly competitivein worldwide market for automobile, ship and airplanemanufacturing industries.

introductionThe Wong's Formulas are simple to be used almost by

anyone who understands basic mathematics and/or algebra I.The inventor of the Wong's Formulas had already submitted thespecification of the invention containing the Formulas in asealed envelope to the U.S. Patent Office because of thesimplicity of the Wong's Formulas can be easily infringed. Thesubmitted Wong's Formulas should remain to be sealed beforeand even after the application is to be granted until when theU.S. Patent Law fails to protect this patent.

The procedures of using the Wong's Formulas can best bedescribed in the following with reference to the notations shownunder each figure:

(1) Measure or design to set the values of a, b, c, r sub.A, rsub.B, r subC. by lengths that fulfill the conditions fornon-penetrating gears.

(2) Substitute these values into the Wong's Formulasr=r(a,b,c,rsub.A,rsub.B,rsub.C), then r can becalculatedprecisely and accurately.

(3) The center of the constructed fourth gear W sub.l canbe determined from points A, B, C, with:

(4) AWsub.l=Rsub.A=r+rsub.ABwsub. l=Rsub.B=r+rsub.BCwsub.l=Rsub.C=r+rsub.C wherersub.A,rsub.B,rsub.C are positive values when thefourth gear is tangented externally to the three fixedgiven gears A,B,C respectively as being shown byFigure No.1. They should be set as negative valueswhen the fourth gear is tengented internally to the threefixed given gears A,B,C respectively as being shownby Figure No.2. Figures No.3 to No.8 show clearlythe meaning of the positive and negative values ofrsub.A, rsub.B, rsub.C relevant to the configurations ofthe arrangement of the designed gears.

NomenclatureFigure No. I to No. 8 provide a specific numerical example

of how to use the Wong's Formulas to determine the locationsof the centers of the fourth gears and their radii tangented to theother three given gears. As can be seen from these 8 figures,there are 8 different solutions for the problem.

Figures No.9 to No. 10 provide degenerated cases of theproblem represented by Figures No.1 to No. 8. The three givengears are initially tengented to one and others. No movementscan be transmitted from one to the other. Two solutions as

I Copyright © ####: by ASME

Billie.Champ'- IMIECE208485M.doc'.- ___ . ..... .. .... .. ............. ...... ..Bilie Ch m p M. .. _ . _. . . .. . . . . . . . . .. . .Page 2,

indicated by the Figures are possible. These two Figures showhow to apply the Wong's Fonnulas for static packaging ofobjects of circular cylindrical shapes. The Wong's Formulascan be repeatedly applied to calculate the centers and new radiiof other circular cylindrical objects to fill in the emty spacesbound by the three initially tangented circles.

Figure No. II provides a specific example of how to usethe Wong's Formulas to find the center of the fourth gear suchthat the sum of its distances to the centers of the other threegiven gears is a minimum. That is to find (CWsub.S +AWsub.S + BWsub.S) = 6.7664362 = minimum

Figure No. 12 is the general figure used to derive theWong's Formulas that should be described in the followings:

A, B, C, are the center of rotation of each gear A, B, Crespectively.

rsub.A, rsub.B, rsub.C are the radii of each gear A, B, Crespectively.

Segment BC=a, Segment CA=b, Segment AB=c are thedistances separated the rotation axis of gears A, B, Crespectively.

The radius of the fourth gear to be constructed andtangented to the three given gears is denoted as r.

Rsub.A, Rsub.B, Rsub.C are to be used to locate thecenters Wsub.l of the fourth gears. They are measured fromcenters A, B, C respectively with the calculated values of r thatcan be obtained from the Wong's Formulas asr=r(a,b,c,rsub.A,rsub.B,rsub.C).

Figure No.1

The Wong's Formulas holds true for the followingconditions for non-penetrating gears:

(a+b) is greater than or equal to c, (b+c) is greater than orequal to a, (c+a) is greater than or equal to a and that a isgreater than or equal to ( rsub.B +rsub.C) , b is greater than orequal to ( rsub.C + rsub.A ), c is greater than or equal to (rsub.A+ rsub.B ).

AcknowledgmentsThe author Po Kee Wong appreciates the administrators

and examiners of the U.S. Patent Office for their time and theirefforts to have examined all the Patents that have beensubmitted to them for a fair examination according to the U.S.Patent Law.

ReferencesI. Eric W. Weisstein, ". CRC Concise Encyclopedia of

Mathematics " ISBN 0-8493-9640-9 Copyright 1999published by CRC Press LLC, 2000 Corporate Blvd.,N.W., Boca Raton, Florida 33431 pp. 56 to 59.

2. Po Kee Wong, " WONG'S FORMULA FORCAD/CAM PACKAGING OF OBJECTS " U.S.PatentApplication Number 08/331,980 filed on 10/31/94

Figure No.2

a-3 b-4 c-5

; rA.2

rB.1 rc-I ; radius r =1.0298221

RA= AW Ir+~rA.3

.0298221 R B=20298221 =RC centered at W1a-3 b=4 c-5 ; rA.-

2 r,.-I rcl_1

RA-AW2 .2.029822 1

R BBW2 .3.0298221

centered at W2

radius r =4.0298221

R cc=C 2 3.0298221

2 Copyright © ###:# by ASME

Billie Champ - IMECE2003-43536.doc Pag 31

Billie Champp- .IMECE2003-43536.doc Page 4.i

Figure No.10

Figure No.7

a=3 b-4 c-5 ; rA.-2

rB.1 rC=-I ; radius r -3.7513535

RA.AW7.I .7513535 rB-BW7 =4

.7513535 RC--'CW7 =2

.7513535

centered at W7

a=3 b-4 c=5

; rA.3 r,.2

rC=l ; radius r - 6/23

36 6 W .2-- R.C

A. AW 10 , 7 3- B6 1W 23 ~ 10- C 'io -23

centered at W 1

Figure No. 11

A

Figure No.8

b-CA'4PA.AW8.351.39652

9 Rf, =CW 8-348.39652

RB=BW =350.39652 C

BaBC=3

~W8a-3 b=4 c-5 ; rA=-

2 r Bf-1 r0 =l radius r -- 349.39652

RA=AW8 - 351.39652 R8 BW8 . -350.39652 RC=CW8 -348.39652

centered at WB

CWs-- 1.0239078 AVS= 3,3885249 B6W=2,3540035


I Billie Champ - IMECE.2003-435•36doc - ------ P.g!.. ..

Figure No. 12

a-BC b.CA c-AB ; r A, rB, and rC each is the radius

of circle A,B,C respectivelyis the radius of the circle co be constructed and tangented

with the given circles AB.C respectively

RA AWl r+r +rA RB= BWI- r B RC=CWI- r + r c

WI is the center of the circle to be constructed

Figure No. 9

a-3 b-4 c-5 ; r.- -3

rB -2 rC- -1 ; radius r-6

RA AAWgr+rA-3 RB.BW9r+rB-4 RcBCWg.r+rCr5 centered at W.


Billie Champ - COMPARISON OTRAJECTORY SOLID ANGLE WITH GEOMETRIC SOLID ANGLE IN SCATTERING.T.IQOV

IAC-02-J.P.02

COMPARISON OF TRAJECTORYSOLID ANGLE WITH GEOMETRIC

SOLID ANGLE IN SCATTERINGTHEORY

Po Kee Wong, Adam Wong, Anita Wong

E-mail: pokwong(&rcn.corn

SYSTEMS RESEARCH COMPANY

U.S.A.

53rd International AstronauticalCongress

The World Space Congress - 200210-19 Oct 2002/Houston, Texas

Bie .Champ -. cOMPARISON OF TAECPT SOLID ANGL WITH GEOMETIC SOI ANGL IN S.ATTERING .T.I- (

IAC-02-J.P.02

COMPARISON OF TRAJECTORY SOLID ANGLE WITHGEOMETRIC SOLID ANGLE IN SCATTERING THEORY

Po Kee Wong, Adam Wong, Anita Wong

E-mail: [email protected]

SYSTEMS RESEARCH COMPANY (SRC)

U.S.A

ABSTRACT

The objective of the paper is openly toinvite all physicists, mathematicians andengineers in the world to re-examine andto confirm the ultimate truth and theworldwide impacts of two U.S. BasicPatents No. 5,084,232 and No.5,848,377 which can be obtained from:http://164.195.100.11/netahtmnl/srchnum.htmThe application of Trajectory SolidAngle (TSA) to obtain the correctcollision cross-sections in NuclearPhysics and in Astronomy by theexample of obtaining the correctscattering cross-section of the well-known Alpha Scattering was shown in apaper IAF-00-J.1.10. entitled "Applications of Trajectory Solid Angle(TSA) and Wong's Angles (WA) SolvingFundamental Problems in Physics andAstronomy " presented and published atthe 5 1 St. International AstronauticalCongress, 2-6 Oct 2000/Rio de Janeiro,Brazil.

The Alpha Scattering was done intheory and in experiment by SirRutherford. The differential scatteringcross section derived from using thegeometric solid angle can be seen fromall the textbooks of physics in the world.However, the differential scattering crosssection derived from using the TSA hasnot been known by most of ourcolleagues in the world and it is differentfrom the previous results. The presentand the previous theoretical resultsconverge to be the same only when theAlpha particle is far away from thestationary heavy nucleus. That waswhere Sir Rutherford made hismeasurement and therefore the oldtheory and the experiment wereconfirmed. The Alpha Scattering isreally similar to the scattering of theComet Halley by our solar system eventhough they are under the actions ofdifferent force fields. In 1976-79, thesenior author of this paper

-Bi.lie Champ - C0MPARiSON OF TAjECTOkRY .SOLI ANGLE-WiTH-GEOMEETRIC .O._ON_.fGLEE SCA-T•ERIN•G T _-f~q

communicated with JPL of NASA andurged JPL to conduct an experiment toconfirm the curvature effects of thetrajectory of the Comet Halley comingcloser to our solar system in those years.It is unfortunate that the communicationshave never been answered even up tonow. Without repeating the analysis, thetrajectory equation can be expressed bymeans of the spherical coordinate systemthat was shown in the paper IAF-00-J.1.10 Ref. 2 Figure No.1 and in Ref.3Figure No. 1 for axially symmetricmotion. The scattered particle isrestricted in a plane surface which isperpendicular to the XOY plane; theparticle is coming from the negative Zaxis going upward as the scattering angle@ that is measured from the XOY planeas zero degree. After going through allthe process as indicated in the aboveexample A. of the paper IAF-00-J.1.10,the differential trajectory solid angle canbe obtained as

(2 pi (e)(Sin @)A2 d @)/((l+eCos @)A2+(e Sin @)A2) A(l/2) (1)

that is obviously different from theusual differential geometric solid angleobtained as 2pi Sin @ d@ (2)

where e is the eccentricity of the orbitand pi = 3.14159 .......

This paper will be concentrated in thepresentation and publication of thegraphical results of numerical dataobtained from the above two differentialscattering cross-sections in greater detailin order to distinguish the differences byComparison of the Trajectory SolidAngle (TSA) with the Geometric SolidAngle (GSA) in Scattering Theory forthe Central Force Fields.

INTRODCUTION

As mentioned in the above Abstract, inorder to fulfill the objectives of thispaper, the following direct technicalcommunications with the 2001 NobelLaureate Professor Ketterle of MIT aredirectly quoted without excerptions andmodifications in the following for openinvitation of other physicists;mathematicians; engineers and otherrelevant scientists to join our openreviews and evaluations either toconfirm or to deny the truth of this paperin Scattering Theory for the CentralForce Fields in Physics:

Dear Professor Ketterle:Thank you for your E-mail replying withreference to the subject matters and giving meyour web site to search for your technical paperscontributing your success and achievements inBEC.As a result of searching for your technicalpapers, I found that two of your papers are highlyeducational to me with possibility of creating ourmutual technical interests. They are:(I) " Collective enhancement and suppression inBose-Einstein condensates " by WolfgangKetterle and Shin Inouye, January 23, 2001,Lecture notes of the Cargese Summer School2000. 38 pages.(2) " Does Matter Wave Amplification Work forFermions?" by Wolfgang Ketterle and ShinInouye, PHYSICAL REVIEW LETTERSVolume 86, Number 19, 7 May 2001. page 4203to 4206. 4 pages.The particular technical areas in these two papers(I) and (2) that we may have mutual interests todiscuss and share are that:In your paper No.(] ) page 2 under the title of 1.SCATTERING OF LIGHT AND MASSIVEPARTICLES Equation (1), the Hamiltoniancontains QUOTED " The strength of thecoupling is parametrized by the coefficient Cwhich in general may depend on the momentumtransfer) " UNQUOTED. The term C appears

IBillie champ.-COMPARIS•ON•OFT-RAiEcTORY-SOLID ANGLE WITHGEOMETRIC SOLID ANGLE IN SCATTERINGTF•'4.(

again in Equations (11), (12) on page 5; Equation(16) on page 6; Equations (19), (21) of page 8;Equations (26), (27) on pages 9 and 10.In particular, Equation (26) C = ( 4pi *h^2/MV)awhere "a" is now the scattering length forcollisions between condensate and impurityatoms. 4pi a^2 = collision cross-section.In your paper No. (2), page 4203 under the titleof "Scattering theory.---""Scattering Theory" relevant to the calculation of" Collision Cross-section" have been repeatedlydemonstrated in my patents and the IAF -00--J. 1.10 papei" entitled " APPLICATIONS OFTRAJECTORY SOLID ANGLE (TSA) ANDTHE WONG'S ANGLES (WA) SOLVINGFUNDAMENTAL PROBLEMS IN PHYSICSAND ASTRONOMY "It is my humble opinion that the example B.Application of TSA to obtain the correctcollision cross-sections in Nuclear Physics and inAstronomy by the example of obtaining thecorrect scattering cross-section of the well-known Alpha Scattering in the paper IAF-00-J. I. 10 should be related to our mutual interests.Based on the above in-depth technicaldiscussions, I hope that you are willing toprovide your expertise to examine a copy of mnySRC-NASA proposal No. NRA-96-HEDS-03-076 entitled " APPLICATIONS OF THETRAJECTORY SOLID ANGLE AND THEWONG'S ANGLES TO SOLVEFUNDAMENTAL PROBLEMS IN PHYSICS"March 21, 1997 which is also the reference No.6in the IAF-00-J. 1.10 paper.Your are cordially invited to participate many ofthe proposed tasks listed in the SRC-NASAproposal NRA-96-HEDS-03-076 of yourchoices. The proposal ( 29 pages ) is being sentto you by fax to you at No. 1-617-253-4876your fax. number).With my best regards, I am,Sincerely yours,Po Kee Wong, Ph.D. & P.ME.CEO, SYSTEMS RESEARCH COMPANY(SRC)U.S. Federal Supply Code: 5R583 forMechanization Of Contract AdministrationServices (MOCAS)50 Bradley Street, Somerville, Mass. 02145-2930 USATel. and Fax.: 1-617-628-8157E-mail :pokwong(amassed.net

----- Original Message -----From: Wolfgang Ketterle

<ketterle(@MIT.EDU>To: pokwong<pokwonoqdmassed.net>Date: Sunday, October 14, 20012:45 PMSubject: Re: (1) Congratulation toyou for winning the Nobel Prize (2)Sending you two US basic Patentsand 5 international technical papersto assess their truth relevant to theBose-Einstein Statistics that is thefoundation of your BEC

At 11:45 PM 10/13/2001 -0400,you wrote:

Dear Professor Ketterle:

The original E-mail to you wasfailed to be executed by the E-mail server.Please excuse the separate E-mails being sent to you inpiecewise condition.

Your time and effort to helpme out will be gratefullyappreciated. I look forward tohearing from you about thesubject matter No. (2) again.

Very truly yours,

Po Kee Wong

Thanks for your congratulations.I am not an expert with regard toyour other request.RegardsWolfgang Ketterle

Wolfgang KetterleJohn D. MacArthur Professor of PhysicsResearch Laboratory for Electronics,MIT-Harvard Center for UltracoldAtoms,and Department of Physics

IB.iqi Champi- CoMPARISON OF TRAJECTORY 860LID ANGLE WITRGEOMETRIC SOLID ANGLE IN sCATTERINGTI-R' _•(s

Massachusetts Institute of Technology,Room 26-24377 Massachusetts Ave., Cambridge, MA02139-4307, USATel 1-617-253-6815Fax 1-617-253-4876Email [email protected] http://cua.init.edu/ketterle group/

Based on the above data, the orbitalequations can be described in thefollowings:NUMERCIAL DATA AND

GRAPHICAL PRESENTATION

The most recent data about the CometHalley can be obtained from:

t, // .4 1 /11 _: .4 L' iLUZ: WV.1w.ss .1p -,-.gov o zons uoc.litm

Go to Appendices/Examples and Clickon Comet data screen then thefollowings will be shown:

JPL/DASTCOM3 Halley 1997-Apr-02 11:13:57Rec #: 20181 soln data arc: 1835-1989 # obs:n.a. FK5/J2000.0 osculating elements (AU,DAYS, DEG, period in Julian years): EPOCH=2446480.5 != 1986-Feb- 19.0000000 (TDB) EC=.967276875 QR= .587103582 TP=2446470.9589491 OM= 58.8601271 W=111.8656638 IN= 162.2421694 A= 17.94154996MA=. 1237401 ADIST= 35.295996338 PER=75.9973 N= .012969228 ANGMOM=.018487217 DAN= 1.80537 DDN=.84911 L=191.5461888 B= -56.6792985 TP= 1986-Feb-09.4589491 Physical & non-grav parameters(KM, SEC; AI & A2 in AU/d^2): GM= n.a.RAD= 5.6 AI= 3.88D-10 A2= 1.55D-10 MI=5.5 M2= 13. kl= 8. k2= 5. PHCOF= .030COMET comments 1: soln ref.= IHW 61, radiusref. is Belton,M (1991) 2: kl=8, k2=5, phasecoef.=0.03; ref. for magnitude laws is ICQ 1994Handbook

Select ... [E]phemeris, [F]tp, [K]ermit, [M]ail,[Riedisplay, ?,:

GIVEN: (l)R(MIN)=0.587103582 AU ;(2) ANGLE OF INCLINATION OFCOMET HALLEY'S OBITS TO THEECLITIC PLANE = 18 DEGREES; (3)THE PERIOD OF THE COMETHALLEY REVOLVING AROUNDTHE SOLAR SYSTEM = T = 75.9973YEARS ; (4) THE AVERAGEDISTANCE FROM THE CENTER OFTHE SUN TO THE CENTER OF THEEARTH IS DEFINED AS ONEASTRONAUTICAL UNIT (AU)

FIND: ALL OTHER OBITALELEMENTS AND RELEVANTPHYSICAL QUANTITIES OF THECOMET HALLEY.

RESULTS: R(MAX)=35.295996338AU ; SEMI-MAJOR AXIS = a=17.94154996 AU ;SEMI-MINORAXIS=b=4.552186934 AU; DISTANCEBETWEEN THE CENTER OF THERELLIPTICAL ORBIT OF THE COMETHALLEY TO THE FOCUS (CENTEROF THE SUN) = c =17.35423961 AU;ECCENTRICITY OF THE ORBIT

Billie Champ- COMPAFRON OF TRAjECTORY SOLID ANGLE WITHGE-.METI!CSOLID- ANGLE IN-SCATTERING T-ffR,6

e=0.967276875 ; DISTANCE FROMTHE POINT OF INTERSECTION OFTHE ORBIT WITH THE ECLITICPLANE TO THE CENTER OF THESUN = 1.154967099 AU

Then, the orbital equation of the CometHalley in plane motion about the Sun is

scattering angle @.where E=e=0.967276875. in the TI-83Equations (4), (5) and (6) is theEccentricity of the orbit of Comet Halleyabottfian.

Ploti Plot0 Plo03\rlBE(sin(e))A2/((1+Ecos(e))^2+(Esin(e) )A2)^. 5\raflsin(O)\r3EEsin(e)/( (1+Ecos(e) )^2+(Esin(e )A^2)A. 5

r =a (I1- e^2)/(l +eCos@) (3)

Based on Equation (3), the followingEquations (4), (5) and (6) can be derivedfrom using the Trajectory Solid AnglePatent No. 5,084,232.The differential Trajectory Solid Anglewith respect to the differential Scatteringangle d@ and divided by 2pi is

rl= e( Sin @ ) A 2 / ((l+eCos @ ) ^2 +(e Sin @ )A2)A(.5) (4)

The differential Geometric Solid Anglewith respect to the differential scatteringangle d@ and divided by 2pi is

r l:E(lirI(O))AZ/(Ci+E~oSCOL.

X:.4i3D9312 Y:.371?'153S

x=.'i33Oia7 :2Sq.?r2 = Sin @. (5)

The RATIO of the DifferentialTrajectory Solid Angle (TSA) to theDifferential Geometric Solid Angle(GSA) is

r3= rl/r2 = e Sin @ /(( + e Cos @ ) A

2 + (e Sin @ ) A2)A(.5) (6)

Equations (4), (5) and (6) were firstshown on page No. 12 in the 1979 SRC-DOE proposal No.P7900450 that is thereference No.7 in the U.S. Patent No.5,084,232.

Equations (4), (5) and (6) can be typedinto a TI-83 calculator. Each equationcan be graphed as a function of the

r3=Esin(8)/C((+Ec05C8))AL_

8560X:.r15030211 Y=.'CSP0'i5

CONCLUSIONS

The claims and impacts of the BasicPatents No. 5,084,232 and No.5,848,377 can be seen and traced fromthe Website given in the abstract. Theclaims are also summarized in the

Billie Champ - COMPARISON OF TRAJECTORY .SOLID ANGLE WITH .IGEO.METRIC.SCOLID-,ANGLE IN SCATTERING.T.HIRV(ý

IMECE 2001 paper recently presented inNew York City on November 13, 2001.

However, it is very amusing to providethe old conclusions written more than 23years ago in the SRC-DOE 1979proposal No. P7900450 on page No. 22:

1. The probability distribution functionfor a particle striking a designatedarea, given the parameters of itsgeneration and ejection is presented.It is being initiated and defined as theratio of the trajectory solid angle,being subtended to the point fromwhere the particle is released, over4pi. The trajectory solid angledeviates but includes the geometricsolid angle as a special case. It isdefined from the definition of theincident angle of a trajectory thatintersects the surface to be struck bythe particle. The trajectory of theparticle is governed by thefundamental laws in classical andmodem physics.

2. The present result, in contrast withthe technique presented by Maxwell-Boltzmann statistics, is derived fromthe basic principles of classicalmechanics and differential geometry.If the presentation is reviewed andconfirmed to be correct, the subjectmay provide an impact on the theoryof statistical mechanics and quantummechanics of modem physics thusaffecting the theory and itsinterpretation of experimentalresults.

3. Important applications of the subjectare widely open in numerous areasfrom problems in physics toastronomy, from aerospace

engineering, nuclearprecision design ofinstruments.

safety toelectronic

REFERENCES

1. Wolfgang Ketterle and Shin Inouye,"Collective Enhancement andSuppression in Bose-EinsteinCondensates " January 23, 2001,Lecture notes of the CargeseSummer School 2000. 38 pages.

2. Wolfgang Ketterle and Shin Inouye,"Does Matter Wave Ampliflication

Work for Fermions ? " PHYSICALREVIEW LETTERS Volume 86,Number 19, 7 May 2001. Page 4203 to4206. 4 pages.

3." Horizons User Manual " from:httq://www.ssd.epl.nasa.gov/horizons doc.hml

4. " BASIC NEEDS OF HUMANBEINGS AS THE PURPOSES ANDFOUNDATIONS FOR THEEXISTENCE OF GOVERNINGINSTITIONS AND THEADVANCEDMENT OF SCIENCEAND TECHNOLOGY " A paperpresented at 2001 ASMEInternational MechanicalEngineering Congress & Expositionon Tuesday, November 13, 2001New York City, NY. USA. 8 pages.

5. " APPLICATIONS OFTRAJECTORY SOLID ANGLE(TSA) AND THE WONG'SANGLES (WA) TO SOLVEFUNDAMENTAL PROBLEMS INPHYSICS AND ASTRONOMY"IAF-00-J.1.10 paper presented andpublished at 5 S'. InternationalAstronautical Congress, 2-6 Oct

Billie Champ.- COMPARISON OF TRAJECTORY SO-LID ANGLEWITH GEOMETIC SOLID ANGLE IN SCATTERING TIEQRY(:

2000/Rio de Janeiro, Brazil.

6. " APPLICATIONS OF THETRAJECTORY SOLID ANGLE (TSA)AND THE WONG'S ANGLES (WA)TO SOLVE PROBLEMS OF THAADFOR BMDO AND FOR FUTUREMISSIONS OF NASA " published in "SPACE MANUFACTURING 13SETTLING CIRCUMSOLAR SPACE "Proceeding of the 5'. SSI/PrincetonConference on Space Manufacturing,May 7-9,2001. Page 98 to page I01; 4pages.

Billie Coamp - 8fýniford Thesis.pdf Billie~ Champ - tanford hes...p. Page..l

Waves in Viscous Fluids,Elastic Solids, and

Viscoelastic MaterialsA Dissertation

Submitted to the Department of Aeronautics andAstronautics and the Comnmittee on the Graduate

Division of Stanford University in Partial Fulfillment ofthe Requirements for the Degree of Doctor of Philosophy

ByPo Kee Wong

a-lea bi s oi n

ali eaf corn

1. Billie Cham'p- - Stanford -Thesis. pdf -Fia g-e 2,1Bi lli. C h a . ...an . .d.... i...... . .. . . . ... .. .. . .. . . . ..... P a ge... . ... ... .... ... .. .. . .. 2

© Copyright 2006, Po Kee Wong

All Rights Reserved.

No part of this book may be reproduced, stored in aretrieval system, or transmitted by any means,

electronic, mechanical, photocopying, recording,or otherwise, without written permission

from the author.

ISBN: 1-60002-009-7

I Billie Champ - Sta nf6rd- The s-is'."'p-d-f .Page 3.B illie.... .. .......Ch a m p ............ ....S t. . .. ..n f.... ..rd..... ..... ... .. ... .. . ..... .. . . . .. . .P ag.. .. . .. ... .. ..

I certify that I have read this thesis and that in my opinion it isfully adequate, in scope and quality, as a dissertation for the degree ofDoctor of Philosophy.



Approved for the University Committee on the Graduate Division:

Dean of the Graduate Division

iii

Billie Champ - Stanford Thesis.pdf -Page 4

1ý Billie Champ -_Stanfoýd Tý6s__Is.pdf - Page'61Billie Champ - Stanford Thesis.pdf Page 5 I

ACKNOWLEDGEMENT

The author wishes to express his appreciation to Professor MaxAnliker for the opportunity to participate in the Stanford biomechanicsresearch program. With his encouragement ample time and freedomwere provided to develop the present theory as a basis for furtherresearch.

The author is also indebted to Professors I-Dee Chang and C. Chaofor their critical comments and for reading the manuscript.

Mrs. Jane Fajardo and Miss Kathy Hunt have contributed theirhelp in editing and typing the manuscript, and the author would like toexpress his most sincere thanks to them.

Financial support from the United States Office of Naval Researchunder Contract No. NOOO 14-67-A-0112-0007 is gratefullyacknowledged.

V

Bhllie C•hamp -.Stanford Thes is.pd.. .. .iPage.6

IBi!llie Champ - Stanford Thesis.p'df Pa eV1

LIST OF SYMBOLS

ai radii of the concentric spherical shells with i=1,2,3,4,

... identifying the number of the layerrA- vector potential function with m = 1,2,3,4,... identifying the kind of

material

A l

An,' arbitrary constants

BIm

B2.m,

b geometrical length

C° sound speed of the viscous fluid

c, dilatational wave speed in an infinite viscoelastic medium of

material tn

c,,, shear wave speed in an infinite viscoelastic medium of material in

D partial differential operator with respect to time

e base of natural log e = 2.732 ...

E onedimensional strain, small number of a ratio of two

characteristic lengths, perturbation of wave number, or small

radius

E Young's modulus of the elastic material

f frequency of vibration

h thickness of a spherical shell

vii

.... ..... .. I

1, BýHH6 Cýýmp - Stanford Thesis.pdf Page 6 '1...Billie.Champ. ..Stanford. ... .... . ... P.ge.. . .

i subscript i =1,2,3,4,...

j subscriptj =1,2,3,4,...identifying the order of the vibrational mode

a, spherical Bessel function

ka, dilatational wave number

k,m shear wave number2/ 2

cm = 4k•1, ký,

k bulk modulus of an elastic solid

m subscript identifying the kind of material

n degree of Legendre's function Pn (cos 0), or the meridianal wave

number of the spheres or spherical shells, or the order of Bessel's

functions an and Yn

P instantaneous pressure

P0 equilibrium pressure

Pe excess pressure (perturbation pressure) Pe = P - Po

r spherical radial coordinate

T period of vibration

t time as an independent variable in the governing equations

td attenuation coefficient (time constant) of overdamped or damped

motionr

u displacement vector

tA velocity vectorruii displacement vector of a continuum of material m

viii

....... .. ..I Bilhý Coarnp - Stanford Thesis.pdf Page 9 1I BI llie... C h amp... S tanford. .... .. . . .... .. .. .... .. .. . . .. . .......... . . .. ... . .. ............... . P a e

ru rm radial component of the displacement vector of a continuum of

material m

Y,, spherical Bessel's function

Z mechanical resistance or impedance of a viscoelastic material

1.model; Z is equivalent to the reciprocal of Young's modulus I in

E

the case of an elastic solid, otherwise it is a linear operator

Zd dilatational mechanical resistance of a viscoelastic material model;1

Zd is equivalent to the reciprocal of Lam6's constant - in the case

of an elastic solid, otherwise it is a linear operator

Z, shear mechanical resistance of a viscoelastic material model; Z, is1

equivalent to the reciprocal of Lame's constant - in the case of anP

elastic solid, otherwise it is a linear operator

P

P" Flhigge's 41 ) viscoelastic linear operatorsQQ,

Y 1Ys Bland's 42) viscoelastic moduli

P/X Lam6's constants in elastic solids

4 1 1 Iim -

ix

I Billkaq4mp - Stanford Thesis.pdf Billie Champ - Stanford Thess.p .. .. Page 10

I-Zs

11'

0)

(,0

first shear viscosity of a viscous fluid

second shear viscosity of a viscous fluid

first kinematic viscosity of a viscous fluid

second kinematic viscosity of a viscous fluid

rotational vector of a continuum

angular frequency of vibration

roots of characteristi c equations, where subscriptjidentifies the

number of modes in the spherical radial direction, while n

identifies the number of modes in the meridianal direction

density of the continuum in equilibriumpo

VIm kalar potential functions of a continuum of material 7n

X in0 spherical coordinate in the circumferential direction

4 spherical coordinate in the meridianal direction

x

I Billie Champ - Stanford Thesis. p-df .Page1_Billie Champ - Stanford Thesis. pdf Page 11

CHAPTER 1

INTRODUCTION

The work presented here deals with the formulation of a theorythat can be effectively applied to study the interactions of fluid, solid,and viscoelastic materials. On the basis of the linearized Navier-Stokes equation of a viscous fluid and the displacement equation ofmotion of an elastic solid, the governing equations of motion for aclass of three-dimensi onal homogeneous, isotropic linearlyviscoelastic continua are systematically constructed. The generalsolutions of these equations of motion are obtained by means of Lam&Helmholtz-Stokes potentials. Each equation of motion is thentransformed into a scalar and a vector potential equation. The scalarpotential equation is separable in eleven coordinate systems, whereasthe vector potential equation is separable in only six coordinatesystems. Choosing the spherical coordinate system for illustration andintroducing the Debye potentials to resolve the vector potential for thepurpose of obtaining a separable solution, we can reduce each equationof motion into three independent scalar potential equations which areall separable in spherical coordinate s. The field quantities are obtainedin terms of these three scalar potentials from which the transient orharmonic solutions can be sought for mixed or nonmixed boundaryvalue problems. Finally, applications of the general solutions forphysical problems in spherically symmetric, axially symmetric,torsional and nontorsional motion are given in thirteen examples.These examples can be used for the modeling studies of aero andhydro space vehicles, geological wave problems and macroscopicbiomechanics.

Recent developments in this field were edited by GreensponI) in asymposium volume of eleven articles from four fields, namely, airblast loading and response, acoustic interaction, aeroelasticity andhydroelasticity. However, another field which also involves fluid-solid interactions and is rapidly expanding is biomechanics. It isactually an old subject which was dormant for many years and has

I

Billie' Champ - Stanford Thesiis.'pdf SP~agel 12i . . .....Billie ... . . ...... ...Ch a m p......... . ...... ..Sta n fo rd.. . . . . . .. .. . ...........e....... ........... ... . . . .P... . .g e........ .. . .. . .. I

Po Kee Wong

only recently regained its popularity. Typical examples of fluid-solidinteraction in biomechanics can be seen in a series of articles byAnliker et al. 2-6) on the theoretical and experimental developments ofnoninvasive methods to determine the elastic and inelastic propertiesof individual arteries, veins, eyes and hearts. Other examples can befound from articles edited and reviewed by Fung7' 8). In view of theimportance of this field to medical technology and the need forsystematic organization and efficient application of engineering andscientific principles to medicine and biology, we are presenting here amathematical analysis of the interaction of viscous fluids, elastic solidsand viscoelastic materials.

The theory is formulated starting out from the classicalconservation laws-of momentum, mass and energy for continua withdifferent constitutive equations. This approach is conventional fromthe point of view of continuum mechanics 8). Texts and articles on thissubject were given by Prager9 ), Scipio"°), Jaunzemis11 ), Eringen 2 ),Truesdell 13, and many others. It is well known that the governingequations are in general nonlinear and that their analytic solutions arequite formidable. For example, in the case of a viscous fluid thegeneral solution of the Navier-Stokes equation still remains unknownexcept in a few particular cases as indicated by Schlichtig 14). In viewof the difficulties in solving the simultaneous nonlinear partialdifferential equations and the fact that the linearized solutions of theseequations still represent a wide class of practical engineering andscientific problems, we confine our self to first seeking the linearizedsolutions of these equations and then applying the successive iterationmethod to find the nonlinear solutions. In the case of viscous fluidsthis method, known as the perturbation method, is for exampledescribed in articles by Eckart '), Truesdell16 ) and Hunt 7 ). It had beensystematically developed by Kaplun18), Van Dyke' 9), Cole2-), Chang2 1)

and others.Having confined our goal, we linearize the equations about the

equilibrium state and obtain a set of displacement equations of motionfor different materials. For simplicity these sets of equations arederived for isothermal conditions which means that the energyequation will not be coupled with the momentum equations and the

2

Biie Cain pStanford Thesis.pdfPge1 -ýa .... ...... -4-Tilge

Waves in Viscous Fluids, Elastic Solids, and Viscoelastic Materials

equation of continuity. The coupling of these equations with thethermal and electromagnetic fields can undoubtedly be included later.Examples of thermal coupling can be found in articles by Truesdell 16)

who provided the harmonic plane wave solution and in publications byWu22) who gave the transient plane wave solution for the case ofviscous, thermally conducting fluids. Examples of thermallyconducting elastic solids are described in articles by Deresiewicz 23)

who obtained the harmonic solutions of dilatational plane waves in anunbounded solid, axially symmetric nontorsional (longitudinal) motionof a circular cylinder, and the thickness-stretch motion in an infiniteplate. Wong 24) also dealt with problems involving thermal couplingand provided the harmonic solutions of multilayered cylindrical shellsin torsional, longitudinal, flexural and circumferential motion and theharmonic solutions of multilayered spherical shells in axiallysymmetric torsional and nontorsional motion. Examples ofelectromagnetic or other field couplings were given by Scipio 10)

The displacement equations of motion are linear partial differentialequations and may be solved by means of the spectral operator theorysystematically presented by Friedman 25). However, there are at leasttwo difficulties encountered. First, the separability of the generalsolutions of, these equations for a given coordinate system is inquestion; second, the boundary condition is generally in tensorialform.

In order to remove the first difficulty the Helmholtz-Stokespotentials were introduced in the field of fluid mechanics for thevelocity vector of the fluid field. The decomposition theorems of thevelocity vector are described for example by Lagerstrom et al. 26) andWu2 2 ). In elastodynamics, the identical potential functions, namedafter Lam6, were introduced for the displacement vector of the elasticsolid. A detailed review, in which Lamr's potentials were used for theintegration of the equations of motion of elasticity, was given by

28)Sternberg and Curtin8. In either case, the velocity vector or thedisplacement vector is decomposed into two parts. Thisdecomposition of a vector is now generally known as Helmholtz'stheorem. It will be shown later in this text that the use of Lam6'spotentials is sufficient for the solution of the governing displacement

3

I Billie Champ - Stan-ford The'sisi.pdf Pa ge 14:Bi llie .. ...... S t n f r . ... . . .. .... .. . ........ . P a.. ... . . .. . .... . .......... ... . .e... . . ... . . . . . .. .. . .. .1 4... . . . . . . . . .. . . .. . . ... . . . . . . . . .. . . . ... I

Po Kee Wong

equations of motion for elastic solids and viscous fluids as well asviscoelastic materials. By means of Lam6's potentials, the governingdisplacement equations of motion of isotropic materials' are nowdecomposed into two separate partial differential equations: the scalarpotential equation and the vector potential equation.

In order to remove the second difficulty, these potential equationsshould be transformed into the proper orthogonal coordinatesdescribing the boundaries. When this is done, one again, faces the firstdifficulty: the transformed potential equations may not have aseparable solution. It should be clarified here that the generalsolutions of the potential equations do not depend upon the coordinatesystem, which is arbitrarily chosen to describe the boundaryconditions. We require not only the separable solutions of the fieldequations (i.e., the governing potential equations), but also separablesolutions of those field quantities which are actually used to match theboundary conditions. The potential equations can further be reducedinto a scalar Helmholtz equation and a vector Helmholtz equation bysuppressing the time variable either by Laplace transform methods orby separation of variables. The separable solution of the scalarHelmholtz equation in eleven curvilinear orthogonal coordinatesystems was shown by Moon and Spencer29); however, he separablesolution of the vector Helmholtz equation is considerably involved. Acommon technique of achieving the vector solution is to find somescalar potentials which can be satisfied with the separable scalarHelmholtz equation. These potentials, when they are constructed torepresent the Lam6 vector potential, should then be consistent with theHelmholtz theorem. In a spherical coordinate system these scalarpotentials, known as Debye potentials, were discussed by Wilcox3 ° .The necessary conditions for the separable solution of the vectorH elmholtz equation in a general curvilinear orthogonal coordinatesystem were given by Morse and Feshbach31 ). They found thatseparation of variables can only be achieved for spherical, conical andgeneral cylindrical coordinates. Seeking the harmonic solution forgeneral cylindrical coordinates in the case of a homogeneous, isotropicelastic solid, they gave the results of the displacement and the stressfields in terms of three scalar potential functions that satisfy the

4

1. Billie Champ - Stanford Thesis-.pdf' Page 115B ill... .. .. ..Cham p.... ......Stanford............... .. .. . .. . . . ... .. a.. .e.... .5


potential equations of motion of elastodynamics. However, Morse andFeshbach 31) assert that these general harmonic solutions are usefulonly in the case of circular cylindrical coordinates. For other generalcylindrical coordinates on a given coordinate surface (such as ellipticalor rectangular cylindrical coordinates), the field quantities that shouldbe satisfied with the boundary conditions are not separable on thatcoordinate surface as they are on the radial surface in the case ofcircular cylindrical coordinates.

For other than circular cylindrical boundaries the boundary valueproblems are considerably more complicated, and very limitedinformation is available. For example, the harmonic wave propagationin elastic rods of elliptical cross section with a stress-free surface hasbeen given by Wong et al. 32). This applies only to an infinite rod. Ifthe rod is of finite length, the stresses at both ends of the rod can notbe zero. This fact was also pointed out in Pochammer's 33) andLove's 34) work on a circular cylindrical rod. The impossibility ofdevising separable solutions for problems with nonmixed boundaryconditions can also be demonstrated for rectangular cylindricalcoordinates. Mindlin and Fox 35) gave a set of discrete points of thefrequency spectrum for an infinite rectangular rod with stress-freeboundaries, for special values of the ratio of width to depth. Aspointed out by Mindlin 36), the frequency equation in this case can notbe expressed in terms of a finite number of known transcendentalfunctions, because of the complexities arising from mode conversionat the two perpendicular boundaries. Mathematically these difficultiescould be anticipated because of the presence of two characteristiclengths in the problem which arise whenever the nonmixed conditionsshould be satisfied. However, recently Miklowitz 37,38) presented ameans for solving some problems in this class, focusing on the semi-infinite plate as an example. He pointed out that for problemsinvolving nonmixed end conditions, the direct separation methods failto yield solutions, and the usual transform methods have been orientedonly to the mixed-end conditions. In view of this, to accommodate awider class of boundary value problems we should seek a generalsolution of the goyerning displacement equations of motion that can beapplied for harmonic as well as transient and for mixed as well as

5

I Billie Champ - Sta'6ford""Týesis.pdf SP~age 16 1Bl ie.. C h a m p..... . ... ..... ........ .. . . . . ... .. . . . . .. . . . ... ... ... . . . .. . . .. . .. . . . .. . . . . . . . .. . . . .. . . . . . . .. 1 6

Po Kee Wong

nonmixed boundary and initial value problems. This has been done byWong 39, 40 ) only in the case of an elastic solid for the sphericalcoordinates and the general cylindrical coordinates, and in the case ofa viscous fluid for the circular cylindrical coordinates. However, aspointed out by Anliker et al. 2-6) and by Fung7 ' 8), the materials involvedin biomechanics are far more complicated than a linearly elastic solidor a viscous fluid; thus, the research results should be generalized tocover a wider class of materials.

6

I Billie Champ - Stanford Thesis.pdf' 10age i1:1B i.ie C h m p . . . .n . . ... ............. .. 1.......

CHAPTER 2

DEVELOPMENT OF POTENTIAL EQUATIONS OFMOTION

If we neglect the effects of thermal conduction, the linearizedequations which govern the mechanical behavior of a continuum arereduced to the equation of continuity and three equations expressingequilibrium in the sense of d'Alembert:

fp + (V = 0 (2.1)at

J " V.=O(2.2)

where p and po are the local densities of the continuum in the

perturbed and unperturbed states,. is the velocity vector and q is thestress tensor. For homogeneous, isotropic, linearly elastic solids, thecomponents of the stress tensor are related to the displacement vectorru through Hooke's law

= ? (V. zu)8u+lP [e.(ej .V)u + ej. (e,.V)i] (2.3)

where X. and p. are Lam6's constants; V is the "gradient" operator; eiv

and ev are unit vectors in the direction of coordinates X i and X j.; and

680 is the Kronecker delta. If we are dealing with a homogeneous,isotropic, and linearly viscous fluid, the components of the stresstensor are related to the velocity vector A through the Newtonian law

V v VI[-P+r ' (V -4)]8 +5 e (ej')V + evj (eiV )A] (2.4)

7

Billie......Cham. .......Stanford.............. . ...... ...Page ...18I Billie Champ - Stanford Thesis.pdf •Page 18!I

Po Kee Wong

In this equation ri' and ri are the dilatational and shear viscosities; andP is the instantaneous pressure defined by the linearized equation ofstate which for isothermal conditions is given by

P0P" -I'~c(P - pO) = Q-P- PO) (2.5)

p0

where P. is the pressure in the unperturbed fluid c2 = -P

Substituting (2.3) into (2.2), one obtains the displacementequations of motion for a homogeneous, isotropic, and linearly elasticsolid:

Po2,(k,+ 2p) rV (V xu) =0 (2.6)

By combining (2.2), (2.4), and (2.5), we arrive at the equations ofmotion for a homogeneous, isotropic, and linearly viscous fluid:

02' 9 V r) C9 rp u._ - (rl'+2) .V(V,)+l -Vx (V x u)+CoV(p- p 0)=O (2.7)P t ot at

Taking the divergence of (2.7) and using (2.1) and (2.5) toeliminate A one obtains

p - P0 .

K+ (L'+ 2 u)§ 7 EE- - []= (2.8)0at Ot P-_ p_

% .

8

Billhe Champ - StanfOrd Thesis.pdf .age 19


where c. is the speed of sound in the viscous fluid, u'=:l'po=

dilatational kinematic viscosity u ril/p. = shear kinematic viscosity,

P-P. - Pe = excess density ratio, and P-P. P, = excesspo p0 P. P.

pressure ratio.

In terms of the dilatation A and the rotation vector 2L which aredefined by

A= V -i (2.9)

2Q= V xu (2.10)

equations (2.6) and (2.7) can be written respectively as

a2l'P -- (k+2p)VA+pVx2d =0 (2.11)

1-0 -- (71'+ 271) aVA + T-a-V x2H + C 2V (p-_P 0 (2.12)ot2 at

By taking the divergence and the curl of (2.11) and (2.12) and

noting from (2.10) that V.i=i0,we find

po -t2 -(k+ 21 )2A = 0 (2.13)

r80a22 V2 2 (2.14)

2at02A a8 V2 pPp- 5 -- 2(il'+ 2 T)--- A+ COV(-p) (2.15)

9

r _. i. C h-. •-am p -. _ : S t n or h i.p.. .. ._-. ...... .............. a... .... ......... . . ... . .. . . ... ...... . . . .. . ..................... .j .. .:, : 1Billie 0 hamp - Stanford Thesis.pdf.. .... ........... Page .29.il

Po Kee Wong

O2(2i") t V2 (2Q')=0 (2.16)9o0 at 2 l-

The displacement vector u can be written in terms of the Lam6's

potentials 27,3 1) and Am (see also Appendix B)

Vum =V(,, + VxA,, (2.17)

where the subscript m is introduced to identify the constitutive laws ofthe various materials which will be considered. The left-hand side ofthis expression involves three unknowns, whereas the right-hand side

contains four, namely, p, and the three components of Am. Thus

another condition must be imposed on Am, but this condition must besuch that the field quantity u is not affected, i.e., the field quantitiesmust be gauge invariant 31). The forms of this condition will bediscussed later.

The substitution of (2.17) into (2.9) and (2.10) gives

Am = V 2'P,,, (2.18)

2, = V xV xA =V (V Am) V 2,m (2.19)

Choosing m = 1 to represent the viscous fluid and m = 2 the elastic

solid, we obtain by substituting (2.18) and (2.19) into (2.13), (2.14),(2.15), and (2.16)

V 2. (, + 2 p)V 2 Tp2 ] =0 (2.20)

at 2

V XVX[Po a Jt 2 E1=O (2.21)M t rq

10

I Billie Champ - Stanford Thesis.pdf P-age 21 1.B lle .Stanfor ..... . . .. . . . . . .. . . . ....df i. 'age.. .


2 o 2 a 2 (.F22)V2P] __ (i'+2ro-V'( I+c (p-9o)[=0 (2.22)

Ot at2 a0

V xVx [Loi aM t2

a -- n0at M

(2.23)

A sufficient condition for the satisfaction of these equations is that

I a2 2] (2.24)

Li Cd a i

L 2 1 a = o (2.25)

Li C2 G0t

(v'+2L)t V 2 PIat at2

2(P- P.)CO 0

PO(2.26)

(2.27)Li2 1a •r- D]41 = 0[] ) at 0

where cd = (k + 21p) /p is the square of the dilatational wave speedand c] = p / p0 that of the shear wave speed for the elastic solid.

For m =1 we can eliminate (p- p.) /po with the aid of(2.1) and(2.17) and therefore write equation (2.26) also in the form

+ý (u'+ 2 u)2 iv2 a Li I=0 (2.28)

11

Billie Champ - StanfordThesis.pdfP Page 22]

Po Kee Wong

Equations (2.24) and (2.25) are the governing potential equationsfor a linearly elastic solid whereas (2.27) and (2.28) are the governingpotential equations for a viscous fluid. The solutions of theseequations are contingent on the gauge invariance just mentioned.

If we take the divergence of (2.19) and make use of (2.10) and(2.19) we find

V-~(2•)=o =-V2 (V 4,,)- V .(V 42)

which shows that the operators V2 and V° are commutative. Hencethe divergence of (2.25) and (2.27) leads to

U2 4'V-A,) = 0

(2.29)

(2.30)

Thus it is seen that (2.17) is automatically satisfied if weadditional conditions:

I

V 'A 2 = 0

r 1 a2 r.A2) -at -A

impose the

(2.31)

(2.32)

rIwith u2 = V'(p 2 +V x A2 in the case of a linearly elastic solid, and

V A¢= 0

rV 2 (V - ,)

(2.33)

(2.34)ia0U tA)

12

I Billie Champ - Stanford Thesis.pdf --page -n


rwith = Vq1 +VxA, in the case of a viscous fluid.

Since 4 must satisfy either (2.31) or (2.32) and 4 must satisfyeither (2.33) or (2.34), there are in effect only two independent

components of the vector potentials A2 and 4. In spherical andgeneral cylindrical coordinates it is possible to choose thesecomponents such that they satisfy the scalar diffusion equations for aviscous fluid and satisfy the scalar shear wave equation for a linearlyelastic solid. 39'40 )

It is shown in Appendix A that in spherical coordinates the vector

4,written as

r r (.54 VX(er ry,)+ V xV x(errXr,) (2.35)r

with er representing the unit vector in the radial direction, satisfies the

vector diffusion equation (2.27), provided the scalar functions y,1 and

XI satisfy the scalar diffusion equations

V 2N - (2.36)U at

v 2 _k 1 (2.37)U) at

Similarly the vector

r rS=Vx(erY 2 )+VxV X(er rX 2 ) (2.38)

satisfies the vector wave equation (2.25), if Y 2 and X 2 are solutions tothe scalar shear wave equations

13

Billie Champ - Stanford Thesis.pdf Page 24I

Po Kee Wong

S2 2 W2 (2.39)V ' ,2-c at2

2 1 a2X 2 (2.40)c, at2

So far we have obtained the linear ized Navier-Stokes equations fora viscous fluid and the equations of motion for a linearly elastic solid.We have written these equations in terms of the correspondingrelations which govern the associated Lamn and Helmholtz (Stokes)potentials. It will be shown that these equations can be used toconstruct the equations of motion for a class of linear viscoelasticcontinua.

The basic idea and method of approach can best be explainedthrough electrical circuit analogy. We begin with the simplest modelin one dimension which we then generalize to allow for more complexconstitutive laws and three dimensions.

According to Ohm's law we have

I=V (2.41)R

where I is the current flowing through an electric resistance R and Vis the difference in voltage across R.

RI ~ AV=I R V V

The corresponding mechanical model of Ohm's law is the one-dimensional Hooke's law

14

I BiIe Champ- Stanford Thesis.pdfPe Rage 2.5


DE D

where q is the stress, e the strain, E the Young's modulus, and Z themechanical resistance or impedance.

E

E

By taking advantage of elementary circuit theory we canthe total mechanical resistance Z for continua with moreconstitutive laws.

For a viscous fluid the Newtonian law specifies

ae eat 1 Z

-7-at

readily findcomplicated

(2.43)

where ; is the shearing stress, = the strain, and rq the viscosity

coefficient. a is treated as an operator.at

1-

at

15

Billie Champ - Stanford Thesi's.pdfPe _Page_.2_6

Po Kee Wong

Comparing (2.43) with (2.42) we immediately identify the mechanical1

resistance of a dashpot as z =71-

In the case of the so-called Maxwell model we have a spring and adashpot in series:

E

1 1E a

The total mechanical resistance is simply

z=z1 + z2 1 1 (2.44)E a

in analogy with the electric resistance. Substituting (2.44) into

E_ G=- (2.45)

E a

we obtain the constitutive relation defined by the Maxwell model:

ý I a a C=(2.46)

16

B il iie C h a m p - S t a n f o r d T h e sis p d f. . P -a g e 2O7p


The Kelvin solid is obtained by arranging a spring and a dashpot inparallel:

E zi now

U a

zlz -2

q-2

the total mechanical resistance is now

Z = - (2.47)

-+-- E+i--zi z2 aE

and the corresponding constitutive law is

=(E+ a c- (2.48)

at

In the ease of a four parameter fluid defined by the diagram below

E =1

E

'

z =-E, Z2 _-1[ lZ4Z=112- 1 1 1

a tE1 1

Z 3- E3 += 1-at+l~

17

Bi Ih e Champ - Stanf6rd The-sis.p-d-f Billie..Cham p...Stanf. ..d...... .. .. .. Page 28j

Po Kee Wong

the total mechanical resistance is given by

ZZI + Z2 + Z3 a a+ (2.49)E 12 E 3 +r12

Ot at

and the constitutive equation can be written as

E]l a 2 E3 l 1) + E 3 w zE3 a 2 E]+-"• ( 3 + +--2.0

EIT]3 112 3 at 1 [2T•f3 3 E -113 at at 1 2

For one-dimensional problems the constitutive equation of aviscoelastic material can be systematically constructed on the basis ofcircuit analogy. A representativ e set of examples are given in Table 1.

18

Billie Champ- Stanford Thesis.pdf Page 29

I Bi!.lie Charnp- Sta~nford Thesis.Pdf Pa~ge.29 ,I

Waves in Viscous Fluids, Elastic Solids. and Viscoelastic Materials

TABLE IConstitutive

Model Name Mechanical Resistance Eq.uation

E elasticsolid E z

viscous z efluid z

Maxwell Z=1+ efluid E l z

EKelvin Z E

a;a

solid E +1 at -

3-parameter Z- Esolid ZE +rh"

3-parameter Z =

fluid III E, +12 'Z

19

Billie Champ -'Stanford Thesis.pdf Page 30.1

Po Kee Wong

Constitutive

Model Nam, Mechanical Resistance Equation

4 I 4-parameter E a41 fluid E, III E, + 2 Z

t at 2 atz

4-parameter z + -- E : •

solid E, +11, E, +112 -Iq yiýat at

DE.,l•. 0 , +11 +/

n CT,SEI _ _____

[]IE, BE _L n3 00

/ Fliigge41) Z= 0 o at 0- E

q,~~ 0-E4 t2 D :• b[

0 01 1 0 j E, L,

1: l•... +--rtlz..... +- I +__+ t'j0 E L I l E

20

B!IIie Champ - Stanford Thesis.pdf Page 31Billie Champ - Stanford Tfiesis.pdf.. ..... ....... P~age 311


The mechanical resistance or impedance Z given in Table I can berelated to the creep compliance J(t) and the relaxation modulus 7 (t)introduced by Lee,44 '4 5 ) Flugge,41 ) and Bland.4 2 ) This relationship isgiven by

(t)= Z 1- " 0( t )=[J(+ (2.51)Eat F1

and

•t)=AY)= Y(t) (2.52)

where the mechanical resistance is written as a differential operator

[3~t D1As indicated earlier, the electrical analogy offers several

convenient features:

a) Extremely complicated material models can be dealt within a simple manner by utilizing electrical network analysis.

b) As will be shown, we can extend this analogy to threedimensions and if the continuum is isotropic,homogeneous, and linearly viscoelastic, its constitutiveequations can be systematicall y constructed.

c) Without suppressing the time variable the governing fieldequations are main-rained in their general form such that awider class of boundary value problems can beaccommodated.

21

'Bilie. Champ.- S.ta. nford Thesis.pdf ........................................................... ... Page 32

Po Kee Wong

We are now deriving the linearized Navier-Stokes equations for aviscous fluid and the equations of motion for an elastic solid and theequivalent governing potential for the materials given in Table 1.

In the one-dimensional case the linear operator which interrelatesthe stress and the strain varies with the constitutive law of the material.Likewise do the corresponding two linear operators in the three-dimensional case where one of the operators governs the dilatationalmotions of the material, the other the shearing motions. Theconstitutive law for a class of isotropic, homogeneous, linearlyviscoelastic materials is proposed in the following form:

c-mj P. [m+X (Vtr)] &i + pn [e'i (e.V)u +e e,

11 Zd n L

1 1

where Z= k,,, and Z, p, are linear operators. In particular, for theZd s'

case of a viscous fluid (m = 1), the constitutive law' is given by

U=• 1 1a ,. rp a [r. (rj.v)u +ej.(ei V)rl

9 at atq•l

and according to Table I the linear operators are

= k I a= 'I '.= 1 =Zd Z, - &l=r•

For the case of an elastic solid m = 2 the constitutive law is given by

g = (V. r6j +P[ri( rj.) rj ( )r

1 Iand the linear operators I and I are the ordinary Lam6's constants

Zd Z'

22

I Billie Champ - Stanford Thesis.pdf '- Pa-ge 33 11Billie.. ............Champ... .......... .....Stanford.... ... ... .. . . . .... . ........ . .. ............................ .... .. .. .3


1- 2Zd

1P2 = Pz,

(See Table 1).With these two examples for m = 1 and m = 2 we can construct the

three dimensional constitutive equations for viscoelastic materials.Considering for example a three dimensional Maxwell fluid (m 3),we have as the operator connected with the dilatational motion

1 1

Zd

at

and as the operator pertaining to the shearing motion

1 1

P 1at

The corresponding constitutive equation can therefore be written as=0 [- P + 1 (V -_u)],8 I r .v) r+ r .(r -V)r]

1 1 i 1 1 [ei(ej ej+ I+

For the three dimensional Kelvin solid the operator associated withthe dilatational motion is

I = d 4 + ,_Zd

23

Bilfie qhamp - -Sta46rd Thesis.p .d .f BillieChamp--Stanf.rd.......... Pa ge-341

Po Kee Wong

while that connected with the shearing motion is

- l= P +TISZ t

Thus the constitutive equation for such a material can be expressed inthe form

L9 r r rr

ýi= [-P+ (ý+ 7' )(V 0)]6 + ( + tar(ej'V)r+ej'(ei'flu]

Proceeding in a similar fashion we can construct the three dimensionalconstitutive equations for the remaining viscoelastic material modelsgiven in Table 1. The results are summarized in Table 2.

24

I B i!I ie "mh-f p - Sta nf rd -Ih-e'sisipdf .... Page 35]Billie Champ - Stanford The~is.pdf Page 35 I


TABLE 2

CONSTITUTIVE EQUATIONS OF THREE-DIMENSI ONAL HOMOGENEOUS,ISOTROPIC, LINEARLY VISCOELASTIC MATERIALS

Name Constitutive Equation

= P + X(V - ,)]5 .Vr r + rj.( r+lP111ri,(ej ur e .ei.V)ru]

I-= X-ýI= P",Zdd Z

Viscous gy P.1 -- a t" (V " U) ýij

fluid a r r r r .n=] + -re • (e' V)u + ej (ei .V)u

1 1 P/ 1 0Zd at Zs a f

elastic 2 = V - + eJ " (ej eV)'t + e' , )isolid

11

m=2 Zd 22 P

1 vMaxwell qij3 =I[-P+ (V "u)]6

fluid -+k ,_m=3 a t

+ 7 .(e j*V)r+ej.(ri it~]

at

1 1 I 1

Zd + ZL +3

t ,a t

25

Billie Champ - Stanford Thesis.pdf ..---P.a.g.e...36

Po Kee Wong


K elvin a _+.) ra)solid a P+("+fl'(V "u)

m= r r r+= (P+I-) [eri • (ei Vj ej (e Vu

aak"•4 k' +1=P4 =l-J+Zd +at Z, at7

3-parameter 1 rsolid 1;i5 = [-P + (V "u)]5 8i1 1

atm=5 k¢ 72 q2t

r r.r+ j(r v r]+ 1,

P/ P2 +112Otat1 11

Zd 1 1Z-+ ,I-+

at at_____+,_______+ at3-parameter I r

fluid ';,6 =[-P+ (V'u)15i

m=6 1 + , Ia

+ d

I Iq':

+ +

a a111- P• +2 a'1'

a Iz - 6 1-1Zd 1-1

,a ,a1'&'7 X +2 -•

11Z-7P'6 1 '1

•~+a_________+r_ 7

26

Billie Champ -Stanford The'-.sis'.pd.f - ...... 37



4-parameter / (V [r

fluid /oz =[-P+ 1 I + 1)]8i

II r rm= X qI •2 + 11'2l

1 1 1 a "(ej V) +e "(er r]

I I Iu ePI T11'a P2 + ,2 a-'

at at a

zI Iý

Zd 1 1

at at

'ilI P, +112at at

4- rparameter q,8 [P + (V .u)]i0.

solid ÷a aI I/ ?12 + i'2

m=8 tt at

+ [ei (e V u e(e e "i

P/+111t P2+ t12

Iks=Zd 1 +a aPI+ 'I P2

at at

i =lP8 =Zs1 1S +

a ap1 + iot' I P2 +T12**t

27

I Billie Champ - Staýford The-s"'is.pd-f P~ag e 5.81]B li C h m - St n o d TeIs.... .. . .. .. ... . . . .... ... .. . . ... .. ... .P........g... .. . . .38

Po Kee Wong

TABLE 3

Elastic Moduli Viscoelastic Operators Viscoelastic Moduli

(transformed operator)

Fljigge [41] the present text Bland [42]

Bulk Q'. 3 2 Yý3k +' Z ,

Shear 2 Y,2p - 2p,.

Young's Q =3 __ 9' I 3Z, + 2Z, 3.2,KE P 2P'Q"+P'Q' z = Zs(Z,, + Z) 2Y. +Y

Poisson's ration P'Q- P"Q' Z, Yý - Y.,2PVQ+P"Q' 2(Z,, + Z,) 2)', + Y,

Lamd's constant - n Za

3ujP' Po Z 3

28

Billie Champ -Stanford Thesis- -.pdf Page.9.


The constitutive equations of three-dimensional homogeneous,isotropic, linearly viscoelastic materials given in Table 2 wereobtained by direct analogy from the constitutive laws listed in Table 1for one-dimensional materials. In contrast to the one-dimensionalmaterials for which we have a single operator, the mechanicalresistance Z, we have two operators for the three-dimensionalmaterials, a mechanical resistance Z, due to dilatation, and a Z, due to

shear. z, z, and z are differential operators which after lineartransformation assume the form of the so-called viscoelastic moduli.For comparison purposes the various operators are given also in Table3 together with those obtained by Flugge41 ) and Bland42 ) on the basisof the correspondence principle.

Following the formation of Table 2 and Table 3, the generalizedlinear Navier-Stokes equations and their equivalent governingpotential equations are obtained by inspection from Table 3. Theresults are shown in Table 4.

29

Billie Champ - Stanford The's_.spd'lfP __Pý6q§ .4o0

Po Kee Wong

TABLE 4

GENERALIZED LINEAR NAVIER-STOKES EQUATIONS ANDTHEIR EQUIVALENT GOVERNING POTENTIAL EQUATIONS

Name Governing Field Equations

Materials a0u r rp•,--t- (X. + 2p,)V(V u)+p Vx (V x,•) 0

r vu. =Vyý+V x A.

jrV2 O = 0

a .0

,ýV2 _.T

Xc +2p _ 1 2

pl, P. Hz Z,H

C2 P-, = '•p, po Z.,

Viscous fluid o2 L, " r 0c'/ +oC +(i'+2,1)1 (v.r,1) +n 1ýV x(Vx ,) =o

m 1r vu, V=p, +Vx A,

II~ IRdV2- a2 I =0

LASA;=•ýfv2 2 _aS/ Ut =0

30

I Billie Champ - Stanford Thesis.pdf Page 41 I


2 21 a 2 1 2Cd2 CO + -(+ 2 )- = CO+ +

P. at PoZd fP0Zs

CsIPo at poZ,

Elastic solid 2 r

m=2 pO - ( + 2p)V(V u2 )+ PVx(Vxr 2 )=O

U2 = V(P2 +V xA 2

L2•= 22V2- - 0

L2A- 220

2 + 2V_2 1 2Cd,2 -- + 2

Po PoZd PoZs

2 P 1Cs 2 - -

PO PoZS

31

Billie Champ - Stanford Thesis.pdf Page 42.,]

Po Kee Wong

Maxwell fluid

m=3

a2 U 2 1 2 (V rPO at2 PCo + + V 1U 3 )

•+ I •+

at1 )r+ Vx(VxU 3 )0

1 1

r3 = V(P3+ V xV A3r

L p3q 3 = d32 at2_ 3 -0

Lv2A = 2 =

LA3 s3= 3 V2~~~

1 2 1 2

C 3 = o + P.+ - + + 0+Po Po+ P... PoZd PoZs

Ti ,a I 1 aat- at-

71Cs3

P. + PO

1po~

32

Billie Champ - Stanford Thesis.pdf •Page 43!


Kelvin solid

m=4

ýt4 I C 2 +Ba + I'a Hj+2(p +

LI atwF-

r4 = V4+ V xV r4

L~4 I= V2 a2~4

a 0'V(Vr

C2 =C 2 + +Cd4 0 '

P 0El a0

Ot El P. 1E

1

9oZs

CO- +P.Zd

2

P0 ZS

33

Billie Champ - Stanford Thesis.pdff

Po Kee Wong

I

...... .. .... .... ... ... . . .. ....... .. .P a g e .4 4 ..

3-parameter ,1 1solid Po -- , 1 + + 1

at - + I 1 + 1at 2 _____ _______tD1a l

+T at P4 2 +12 a

=V~+V xA5vxu)

2 2

L~p 595 = d5V - s=

1 2 I 2

Cd5 =) + +c+ +

PL- PO PO P O PoZd Poz,•'1 2 +1'2 PI p2 +T12 -

+ 11 . 12-

sPO+ PO PoZs

P2 + 112

34

1. Billie Champ - Stanford Thesis.pdf P~age 45 1Bi f i h m S tanford.. ... ..... .... . . P age.. .. . . .... . . . ...... .. . .... .... . .. . ... 45


[] []

3-parameter a2r6 0 1 2 0 rfluid Po Ot2 + + 2 1 1

m ~6 at- cJ 1 +__1__2 1_111_P2 + u12H 1 ,-D- -+at 'at , 2+batE

+ ~Vx(V xu 6 ):oI 1

-+a a

r 1 ~ 6r 6 -V(P 6 + VXA6

L r •= 26V2 _ a2---T

LA 6 A 6 - 26V2 at2 6O

Cd62+ + 1 + 2PO + PO PO Po POZd poZA

at 2 -at Illat P 12at

2 6- I _ Ic o6 - _ _ + Po poZS

a a'11 "- P"2 +11 2"t

35

I Billie Champ - Stanford Thesis.pdf Page.46 II Billie Champ - Stanford Thesis.pdf Page 46

Po Kee Wong

4-parameter a I]7 +12 1 2fluid + ___ I + I + __+

a __ a11', - k + 11'2 a 2 + at 1m=7 at at at

-+ 1-

TII•" P2+12•

I I

u7 V(P7 + V X 1u7

r0Ly7A R 7V 0 at2 •7 =0

LA7 tý

+ I 2 + +cIQ 7T = C ° + 2 -0+ P. _ .O P O 1 + P O . + / P o ~ a P o Z ,

c17 1 X 2+T2 p Il- P +1 1

Po+ Po + Po PoZP

c II/ P2 + 1 - -a2 at

~ ~1I~7

36

I Billie Champ - Stanford Tfhesis.pdf Page 47 1Billie Champ - Stanford Thesis.pdf Page 47 I


4-parameter o _ 2 + 1 +2 _V

solid 1 + 1 /

0 Xq], X,+-,- P2, +1,12 -Lm=8 t] at a atu

r+ V x(v 1= o+ 11 1 v(x 8 =+a a

P1 +f, P2 + T 2fat at

L8 = V•8 t +

L rs8 = 28 V2 -a 2 =LASAS=~V - =0a2

2 2 2 1 2C+_ + + _ 2 + P+

Xl~l~a ~~+'I2 PI +111 P2 +112~

cP P P O PO ZS

PI +I - P2 -+ 12 ,

at + L 9at "1•

37

1, 8il lie Cham'p -'-sta6f6'r'd" Th-e-s-i-sEp-df SPage 48. 1Bi llie .Cham p .-S.a.f..d.8................

CHAPTER 3

GENERAL SOLUTION OF THE FIELD QUANTITIESIN TERMS OF THE POTENTIAL FUNCTIONS IN

SPHERICAL COORDINATES

In Chapter II it was shown that the differential equations of motionfor any isotropic, homogeneous, viscoelastic solid and/or fluid can besystematically obtained. By utilizing Helmholtz's theorem, thesedifferential equations can be transformed into two governing potentialequations, a scalar potential equation, and a vector potential equation:

L'm p : = _ aO .0 (3 .1)

LAm Am= V2 a2 r,,= 0 (3.2)

where L and L A, are linear differential operators, ,p and Am arethe scalar and the vector potential functions respectively, c•,, and CS2m

define the material properties (see Table 4).If the solutions to these equations satisfy the gauge invariance

or

rV -A. = 0 (3.4)

then all field quantities can be obtained from

38

Billie Champ- Stanford Thesis.pdf Page ~ t

841iechýmp - 9tanf6rd Thesis.pdf Page 4911


ru,,,= Vg9" + V x A (3.5)

r (5ur, = V (P + V x. (3.5)

r r2Q, =Vxu= V xV x A,,, (3.6)

, [- ,, ,(V r ] r r r) r re=" ,,I ,,,• .(r V r U e e V rJ

andI 2

A,= V U, V q2, (3.8)

Equations (3.1) to (3.8) can also be applied in studies of themechanical interactions of layered viscoelastic continua. Aclassification of the type of problems that' can be solved is givenbelow.

The time dependence of the solution can be determined with theaid of linear transform techniques such as Laplace transform or Fouriertransform. Whenever we can separate the dependence of the solutionon the spatial variables we can determine the space dependence ofbasic solutions of the differential equations of motion by utilizingagain linear transform techniques. Such separation of the dependenceon the spatial coordinates is for example possible when we useCartesian, circular-cylindr ical, elliptic-cylindr ical, parabolic-cylindrical, spherical, and conical coordinates. The true complexitiesenter usually into the problem through the character of the geometricdomain. Domains defined by spheres, cones, and cylinders withcircular, elliptic, and parabolic cross sections can be studied byseparation of variables. Likewise, we can analyze spherical shells andcylindrical shells with circular, elliptic, and rectangular circumferencesin the same manner, at least for certain boundary conditions. We alsocan give separable solutions for infinite domains with cavities orinclusions in the form of spheres, and cylinders of the typesmentioned. Finally, as to the boundary conditions we may prescribe

39

Bi!lle champ -_Stanford Thesis.pdf Page 50

Po Kee Wong

zero displacements or zero stresses on the boundaries, irrespective ofwhether we are dealing with a single medium or a multilayeredmedium.

As mentioned before, it is advantageous to delineate a class ofgeneral solutions in which the dependence on the spatial variable isseparated and to determine the various geometric configurations of thecontinua and the types of boundary conditions which can beaccommodated by the solutions for certain material models. We shalldemonstrate this for the case of spherical coordinates.

Depending on whether the medium is a viscous fluid or an elasticI

solid, we use for the representation of Am that given by (2.35).

Generalizing these expressions for A,! to allow for a more complexmaterial behavior (see Appendix A) we can write the field quantities interms of the scalar potentials ,p, ) ,,,, and y which are solutions of

L (, Pm = 0 (3.9)

Lain ,Am = 0 (3.10)

LA, Y = 0 (3.11)

Accprding to (A7) and (3.5) the components of the displacementvector u are then defined by

a(f rl+ 2 - rVW (3.12)u .. arr2 "r

_0 1 c + - r-- M (3.13)r 0• r 60 r sinO rV'z.

_ =- L. + 1 2rý ,,, +I I-( 2 X (3.14)rsinO &^ rsinO O•O rN0

40

I Billie Champ - Stanford Thesi's.pdf Page 51,iBillie Champ - Stanford Thesis.pdf Page 51


rWith (A7) and (3.6) the components of the rotational vector 20.

can be expressed as

2rmn_ -V2 (, _2 X ))(3.15)

r aO r a8O9r&

M om 1 - I 2 , 22f2•"-W 1 -Or(V m) rsnl-3'r2(rV' (3.17)

On the basis of (A7), (3.5), and (3.7) we can give the components ofthe stress tensor g i,, as

=Op+X V2o +2hi• 1a]V2=ITf Pý + (P + 2 p + 2pý (r [ m (3.18)

~ [] 8 ar

EQ a%2m 2 ay,,, Elr . n. = P m [ 1 - -2 a o F

Er Oaor r 2 F]

[E2 ay 2 0 2r•ry 1 2W vz\Er fl 2 2 )

O0r r ar50 r-)

+ Imr aEsInO a• (rV2" X(3.19)

41

Billie Champ - Stanford Th-essis-.df..ge_52

Po Kee Wong

E2 a 2• .M 2 aP mI= P ,. 2 - D

IsinO 5ar r sinO a4 wS2 a,•,. 2 a

2 rt.m I a 2 D)E+ P ] - u

'sine oar 2 r 2 sinO aOar rsinO ar ' m

+ Pn, r-•Tr []I a rVX2, (3.20)

[ a"' +Lap,,qoe,=•[•m+'"~flm+br Hr 20 r r

D1 a 3 V 1 C2-rwgm 2+ 2pG-- v m 1[]2, 2 arao02 r arr2

-2 a H -1--L(rV2 "X,) (3.21)[-2 " afTO ýsinO 3d• V

P q in e c m n, a , 1 a2 :),E

or ••2 H-sin2 a r 2 sinO aOaO"8

+ Pm LinO0 a 1 :21V , m I a03rm ,r" Zr9 • in0 05 & re sinO aýr8UsinO O 1rOsinO

+i. r-nO a a 2•-)(V H X. 91 (rV2Xm (3.22)E3 oLsinO a0 rsinO D2 [

] I a2 (P+ I l+ Coto &Pm O

00m ýn[ + V2m + 2p . 2 sin 20 312 r ar r 2 ao

I 1 3 r I"a21 m CO to ar2 r[],+2p", Si~ -.--- L T M + _____ V N'W ED

[] 2 c3r , or 2 0 a.,_V2rm[

+ m2sin 2T a&2 -" r r caOar w

Pm F 2_9 ) covtO 2 (V_ - (3.23)IT sin 0 a0m r2 sinO a0 F-

42

I Billie Cham p' --Sla'nford -Th'e-s'is--.-pdf' ,.Page. 53B illie . . . . . ..Ch am p.. . .. ... .. .. ..n.. ..... . .. ..... .............. P age..... ... .


By adding (3.18), (3.21), and (3.23) we obtain the mean of the normalstresses

...rr. +t 1;0 '&n+qOon P , + a + .2 m 2(p n

3 n i 3,n

where•2• r=•-+r3.V )-pB2VpI+r 3P [w(p (3.24)

&r [] , ag(rV2 y n2 V can be proved to be

identically equal to zero. The physical and mathematical interpretationof (3.24) for the cases of a viscous fluid (m -1) and an elastic solid(m = 2) are given for example in references 9 to 13.

The terms containing V 2(f)m V2Y,,, or V2yX in (3.12) to (3.23)

can be replaced by 12 at 2 C2 , a2 , and 1 at2 , according to

Table 4 and equations (3.9) to (3.11).All of the field quantities defined in terms of the potential

functions Pm, X,, , and V, can be applied for nonsymmetric motions

as well as for axially symmetric nontorsional and torsional motions.The relative contributions of cp,,, ,,,, and y , to the various types of

motion can be assessed by a direct inspection of the components of thedisplacement vector.

In the case of axially symmetric nontorsional motion we can givethe components of the displacement vectors as

Urnm= Urmn(r,O,t)

u1 ,, = U,,, (r,O , t)

43

1. B i I I i e C h a m- -p -" 8 t an' f6 r d T fi'e-s'i-s'-. -p- d f Page, 541i .... .... .... ....a ... .. .. ... . .......r... .. ..... .. . .. .. .. . .. . .P a ge.....5 4.

Po Kee Wong

0,n =0

According to (3.12), (3,13), and (3.14) we have in such a case

(pro= T, (r,O ,t)

, (r,O,t)

Thus the corresponding field quantities are defined by

2

Urm =)m + a rWj, rV

O r - o r aO r

r 02Qr,n = 0

200M = 0

2 o,,, = Iarrf€ - (: (g• ,

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

44

I Billie Champ - Stanford Thesis.pdf Pae55:1


'rrm n1P + xV (pm + 2p1 r2 FF1 r

+ b32p i

D ~a3 nigr aoLar2

C9 (rV 2W .)E

T20 []

(3.31)

2 L98V.,2 &"0 r L90tV I'n (3.32)

(3.33)4rdm " 0

11 a 29 +, +P +I

+plj 2~~i 7~~ 2 Vw

000

Oom4i 1fi p + k V 2(p ",+ 2 m p. 2gp'+Cooa

E . a & r 2 0

+ 2pnE .2' m + Ct0L2r~If -V 2 JD

(3.34)

(3.35)

(3.36)

In the case of axially symmetric torsional motion, the componentsof the displacement vector are

45

Billie Champ - Stanford Thesis.pdf Page ~I Billie Champ - Stanford Thesi's.Pdf Pag.e.PP I

Po Kee Wong

Urnl 0

UOI =0

'uOm = un (r,Ot)

Such a displacement pattern is enforced if we require

(P,, = 0

X = x,, (r,O,t)

Accordingly the field quantities can be expressed in

X , (r,0 ,t) alone.

Um =0

Uen, 0

1 I (rV 2Xr 00

V 2 a2(rV2 X)2Qr. = rV 2XZ, M r2

1 00r22MOm =- (rV 2 0

2Q"Om = 0

terms of

(3.37)

(3.38)

(3.39)

(3.40)

(3.41)

(3.42)

46

Billie Champ.- Stanfori-dc Th-esis.pdf .Page._6717


rr. -- Pin (3.43)

ren 0 (3.44)

a D1 a V 2Xqr4•,n pmr T-• -(rV ",) (3.45)

0o e Pn (3.46)

P.o, 2 , - (3.47)

Tn= (3.48)

For radially symmetric motion, we have as the components of thedisplacement vector:

Urn, = Um (r, t)UOm -- 0

zll•, = 0

The associated scalar potentials are in this case

(p = (pin (r,t)

Ym=O

and the corresponding field quantities can be expressed in terms of9,r (r,t) :

47

Billie Champ- Sta nfordThes is•. pdf PePage 58

Po Kee Wong

U rm --

UOr

Uen = 0

2unr = 0

2r0. = 0

_ =-p. + k.,V 2q),. + 21 ,.l2'f ,,rrm (•r 2

r4~n = 0

e ,n = p , + , ,ly 2(p ,,, + 2 1 P .r ar

(3.49)

(3.50)

(3.51)

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

/I =V 2(,,, + 21 ,-- & (3.60)

From the above three special cases of motions, we infer that (P,

characterizes dilatational motion, y,, represents nontorsional shearing

motion, while Xrn represents torsional shearing motion of the medium.

48

Billie'Champ -'Stanfor .d Thesis.pd .f P-- ... 69ageB. lli .C h.m.. .. t n for.. .... . . . . . . . .. . . ..... .. .... .. . .... .. . .P..... . ... . . .. . . . ... . . . . .e.. . .. . .5 9

CHAPTER 4

APPLICATIONS OF THE GENERAL SOLUTIONSTO SPECIFIC PROBLEMS

In Chapter 3 it was shown that all field quantities of a continuumcan be obtained in terms of three independent scalar potentialfunctions mp,, X m and w,, which are solutions to the governing

potential equations (3.9), (3.10) and (3.11). In this chapter it will beshown how these results can be applied to specific cases.

For purposes of illustration we simplify the procedures and seekharmonic time solutions of the governing equations (3.9), (3.10) and(3.11) for each material model. These equations are then furtherreduced into three corresponding scalar Helmholtz's equations whichinvolve two physical quantities kan and ksm which themselves vary

with the material models. A list of k•,, and ksm, for each model is

given in Table 6.The k,,,, and ks,, in Table 6 can be interpreted physically as

dilatational wave numbers and shear wave numbers which in generalare frequency dependent. The simplest forms of these wave numbersare the cases of elastic solids and viscous fluids. For materials otherthan these two, they are quite complicated. Once a boundary valueproblem is solved for a particular kind of material, one can replace theappropriate ka,, and ks,,, from Table 6 and obtain the solution of thesame boundary value problem for another material.

Now let us explore in more detail the physical meaning of thewave numbers k,,, and ks,,m by considering the two fundamentalmaterials, namely the elastic solid and the viscous fluid. Theinterpretation of these wave numbers for other materials can begeneralized from these results.

49

Billie Champ - Stanford Ihesis.pdf -Page bU

I: Billie. Champ - Stanford T~hesis.Pdf .. Page bU I1

Po Kee Wong

TABLE 6

HARMONIC SOLUTION CHARACTERIZED BY k-,, AND k

For solutions defined by

qým n= nei

Am IV X(rzm.)+ V xV x(e

for general cylindrical coordinates

Am[V X(e", mV )+ V xV xV(erX

for spherical coordinates

U., = Vq,,, + V xA .,

provided that y ,,, ,. and X., satisfy

(V 2 + k-,,), = 0

(V 2 + k2)", )y= 0

(V2 + k2m)Xm=O

2

where. kL . k2.2

S2S")'

cn L SIP)

we obtain the following expressions k,,,, and k,,for the various material models considered

C

2 -0) 2 2___________

Viscous fluid k2l - 2Cdl c2 -r'+1)o1 +

M l P0

2 j)2 (02C• - -1

ksi

PO

50

I Billie Champ - Stanford Thesii-s.pdf -P. -66 ýa 1 :1I i lli.C am .. St nf rd. .. i.. .......... .Page.. ... ..... . .. ... ... 1


Elastic solid

m 2

2 2 2 (O-k =-=d 2 2-H+2p

Cd2

Po

2

k d2 2s2 d2 +2

Cs 2 Jp 0

-Po

4.

Maxwell fluid

m-3

22

kd 3 2

2 0) 2

20)

c02 + + 2PO + PO PO POkL 1' im CO p 1io)

2 (PO+ P0

pa ltj O

Kelvin solid 2 2

m4dbc4 =- --m = 4 C 2 4 C 2 + +/ - 1 1,, + i ( 0) + 2 - ( + , i )

p 0 p 0

k -s 2 (0 -2 (0 "2s4 2 2)s4 -- ( + r i(0)

P0

51

Billie Champ-Stanford Thesis.Ipdf Page 6:2-1

Po Kee Wong

2

2

3 parameter

m=5

7

kd 5 -2

0) 2

lr 1

c5d co ++

Po .+ Po Po + Po;ý X'2 +'1 2 iw PJ1 P 2 +'1 2j0)

k]5 = 2 2-c 2 ýP01 P 2 +fT1 2(10)

2 23 parameter k2 6 -2 0_ 2

Cd6 C5 + P + R+mn= 6 0 o 2 •. Oo ... 2o P o

f'( 10) X2 + r1'2 0) l11Co I-02 + 112 i°)

22o- C o -ý2+ Pooks6 - 2

CA6 11i P!2 + 112i)

52

1BilIie Champ- stanford Thesis.pdf _P__age 63 i


4 parameter

m=7

22 _ 0- _kd7 27

cd7 2

s7Cs7

20)

I 2-I-

P+ .+ P.

PIt T1 i0)C P 2 +112'J)

T l 1i'0) P 2 + TI2'( 0)

2)24 parameter k2d c - 01

cd28 C2+ 2I = 8 0 0 + PO Po + PO

Xl '(0o k- +Tl'2io) P I +q 1'1) P2 + 12i(o)

2 Pks8-8 = 2i+

CS8 P~1O p 2 + 112 j0

53

I WHO Cha-mp I Stanford Thesis-.pdf Bili"Cam t nf rd.... . Pag ... .. " " ".. q64 ]

Po Kee Wong

As can be seen later from the examples described, the wavenumbers k•, and ks,, will appear in the frequency equations. Theycan be cast into a dimensionless form by multiplying them with arepresentative geometrical length b of the materials b k,,,, bk,,,. Assuch they depend on the physical and geometric characteristics(frequency, viscosity, density, constitutive law, and the geometricparameters). An expansion of the solution with respect to small orlarge values of the parameters bkd, and b ks, provides two extreme

cases of the general solution. For the time being we shall restrictourselves to those cases where b is finite. The cases in which thephysical quantities are kept constant and the geometrical quantities areallowed to vary will be dealt with later.

In the case of a viscous fluid, we have according to Table 6

c, 1+ '+' ] , l [

cs),

y•2

The expressions for kdl can be approximated by series expansion for

,1ý)'L+ ~ 1u2 [

1*0 b H O bHTo

54

I Bill e Champ - Stanford Thesis.pdf Page 65 iBi llie. C h am. . . S tanf rd. T h.. i.... . . . ... . . .. .. . .. . . . .. .P age.. . . .... ...... .... .... ... .. .... . . ..... .... .......


and

U'+ 2ub2 -)<< I

Zb iE rb n

In the first case kdl is approximately equal to

ki 42dl 2 (u '+2 (1-i)

In the second expansion /d, is approximately equal to

The physical interpretations of these two expansions are listed below.

kd, Approximately PossibleEqual to Conditions Physical Statement

-42 w•• 1 - co <<tu'+ 2u Viscous effect dominant2 ( 2u)b b2 over compressible effect.

Frequency response in thesmall -+ 0 order of f-.B(long wavelength) Ub 0

2•- o'+ 2a) co Viscous effect dominant') 70:7b over compressible effect.2i2,);tl Frequency response in the

finite odro L(medium wavelength) order of [-.-2u-U___ ___ ___ ___ __ _ _ ___ ___ ___ _UE b Ul

N c2 () >> u'+ 2u co Viscous effect same order

L) +"2L) b2 b with compressible. Highlarge frequency response.

(short wavelength)

55

Billie 9..h?.rn P.-. Stanford Thesis. pdf Page ~ I

I Billie Champ - 8tanf6ýd Th6ýis.pclf .... "RP..ae. W 1

Po Kee Wong

( 00< c. u'+ 2u Viscous effectandbmall -b0 0) 2 compressible effect sameb b order of magnitude. Low

(long wavelength) frequency response.

u '+ 2L Compressible effectb 2 dominant over the viscousF b effect. Frequency response

(medium wavelength) in the order of RL H[i b Ei

mal 0u '+ 2u < c 0 Compressible effectbmall 2 b dominant over the viscoust b effect. Frequency response

(long wavelength) in the order of ' +___b 2 H

The exact expression of kd must be utilized when

u'+2u Wo _ b2 2 0 )

Co C O o H [ oHlb []b []

that is when either B -o>o,> --- or 2 2 o <--o] bE br b] wb ] 1 b1

Noticing that o/•_/.-2v His equivalent to the Reynolds number of the

viscous fluid, but it is not the usual Reynolds number since it pertainsto harmonically varying motions: It should rather be called the waveReynolds number.

The values of ksi in two extreme cases can be obtained

immediately when b or b I i.e. when the frequency is

either high or low.

56

1, Billie Champ - Stanford Thesis.pdf --Page- -67'1I Billie Champ - Stanford Thesis.pdf Page 67


For an elastic solid, Table 6 gives

0,2 CO (0 k0.+21 d2 Cd2

Here the two extreme cases correspond also to high and low

frequencies or equivalently to short and long wave solutions cob >> ICd2

and c-b«< 1 and -ob > 1 and -ob << 1. Physical interpretations ofCd C, 2 Cs2

ki and k-s, for other material models can also be obtained althoughthe situation becomes quite involved, especially with regard to thedilatational wave number k,,,, for m = 3, 4, 5, 6, 7, 8,....

Now let us proceed to solve some problems of interest beginningwith spherically symmetric motion, axially symmetric torsional motionand axially symmetric nontorsional motion. In each case we shall dealwith the interior, exterior, and interior-exterio r (shell type) problems.

4.1. Spherically Symmetric Motion

This motion is defined by equations (3.49), (3.50) and (3.51) whichindicate that ur,,= Ur,, (r,t) with Ui. U=,, =0 everywhere at all times.

An equivalent definition would be yP,=rjm(r,t) with Y,,+ X,,=0

everywhere at all times. Thus one can see that the governing potentialequations (3.9), (3.10) and (3.11) are satisfied if

pM =[AmaO(kdmr)+BmYo (4 ,r ]e'O' (4.1)

With (4.1) the displacement vector is according to (3.49) given by

Ur = :[mka,.,9'o (kdn,r)+B,,kdmY'o (/d,,r ]eJ" (4.2)

57

Billie Champ - Stanford The&is.pdf Page 681, Billie Champ --'8fa-nf o-'rd-The-s-is'.pcff Page__68.:l

Po Kee Wong

where A4, B,,, are arbitrary constants and k,,, is defined according to

Table 6.In the case of viscous fluid [m = 1] we have

u,r =[Alkdla'o (kadr)+BzkdY'o(•ir ]ei (4.3)

The excess pressure Pe =(PI - Po) can be obtained from (2.25), (2.26),(2.28) and Table 6:

Pe =(P,- Po)=c4 (),-pPo ,0 '+2C - O2

1+2L) 2P l a2 (pIl[

F1 at at 2- - poc2V 2 (pl

= Po Cok 1 [Ad Oo(kdlr)+ BiYo (ýdlr ]ei'(Ot (4.4)

With (4.1) and (4.4) the remaining nontrivial field quantities canaccording to (3.55), (3.58) and (3.60) be written as

qri + P, =Aikd 21[- oo +'iw)o(kdjr)+2ioO "o(ý4d ]ie

+ B,k2 [- (pCo + icnl')Yo(kdi-,)+2itonr(),,k/1'lJe W (4.5)

21PC rK1)flpa (d r)n .10 +o Bk, F (poc +i+of')YO'"lr+ 2i.oiikjtr)L (4.,

0 (kd~r, L+ ,;, + . Y'o (k0,,') .... (4.6)

,+, 2 +iD' o (kdl r) ] ,

+ 7 0,, pC + i~oi') Y,(kd f) + N irl Y oka, ) (4.7)11 'd rF . [

58

I Billie Champ - Stanford Thesis.pdf iPaq 6 9-]


Adding (4.5), (4.6) and (4.7), we obtain

,rr +100/ +140 / (P +P')+im04'+ + 2n1 (4.8)3 ] 3 []

The physical interpretation of equation (4.8) implies that the

dilatational pressure ico w 1 + 21 [ 1 in a viscous compressible fluid isL] 3 D

equal to the algebraic sum of the static pressure P, and the excesspressure P, plus the averaged normal viscous stresses at any point ofthe fluid at any time. It should be clarified that if the Stokes relation

2holds, i.e. "1 + --n = 0 it does not mean that the fluid is incompressibl e

or the flow irrotational; the fluid is still viscous and compressible andthe dilatation A, may not vanish identically at all.

In the case of the elastic solid, m = 2, the radial displacementaccording to (4.2) is

Ur2 = [A2kd29'o (kd2r)+ B 2 kd2Y'o ( 4 d2r ]eio° (4.9)

Other nonzero field quantities are

€rr2 = A2 k22 [- M9o (kd2 r)+ 2P3 ". ( d2r ]e

+ B 2k22 [- U.Y (kd 2 r)+ 2pY'o (4d2r ]ei'°t

2d2 0 4 ~ Iio (4.10)

2 El '9'o (kd2r)[] io)t

0o2 = A2k 1 M9 ,r (kd2r)+ 22p2] 1 .(k-d2r)[ .

+ B 2k1,2 0.-XYo (kd2,.)+ 2p Y d tie (4.11)PI (kd2r) N

59

_Billie Champ -_ Stanford Thesi..s. pd• Paef_Page._7.0

Po Kee Wong

~~2E1 &- I d .

LT XaO(kddrr)l 2P DLi (kd~r) 0

2 E] Y' , kd rl I-W•+B,k , k2r)+p

Adding (4.10), (4.11) and (4.13), we obtain

rr2 + ¢002 + ý 02 _ +

3 3i 3m

Applications of the harmonic solutions (4.1) toillustrated with the following problems.

(4.12)

(4.13)

(4.13) can be

4.1.1. Spherically symmetric vibration and attenuation of a viscous,compressible fluid enclosed by a rigid spherical boundary andsubjected to an initial finite excess pressure applied at thecenter of the fluid.

The boundary conditions require that the field quantities be finite atr=0 for any t> 0,. thus B1 =0 and at r=a we have ur,(a,t)=O,

which gives with (4.3)

60

1. Billie Champ - Stanford -rhesis.'Pdf .. 7ag-- 1 .1i .. . . . . . . ..llie. ...... .. ....ha m. .. . . . . .. . . . . .. .. . . . .......f. ... ....d... . ..... ...... . .... .. . ... .. . .. ... . . . .. . .. . .. . .. . . . .. . .. . .. . . .


19'o (kdl a) d E-kn (kd a )ED'd--kd7 in (kd/ a)E'-

i.e.

tan (kda )= (ýdja (4.14)

This is the characteristic equation of the problem where (kdl) is the

wave number and as such related to the wavelength dI by kdI=2

The root (kdla) of equation (4.14) is well known [see for example,Lamb 46)]:

(kda)=l1.4303Tc , 2.450nt, 3.47097r, 4.4774n , 5.4818nt (4.15)

d'l =-0.6992 0.4067 0.2881 0.2233 0.1824 (4.16)2a

Since kdl is related to o) as shown in Table 6 and is so defined inorder to obtain the solution from the field quantities given above, onecan solve for c in terms of (kdla )c-(yI Thus

j _ H__'+2 F(2 (u'+2uj (4.17)

ia w 2ac o 2 ac,

where j=1, 2,3,4,....For each ct, given by (4.15) there is a wo. such that the boundary

condition u,, (a,k)- 0 is satisfied.With the initial condition

61

I Bilhe Champ' -"Sta n--fo'-rd- Thesis I p df Page 72!I B illie . . ... '.. . . . ... ... .. . . . ." .. . . . .. . .. . . . .. .. .. . . .. . . .. ..-. ... . ..... ... ..nfo rd..................... ... . .. . . .. . ... . ... ...p d P a e 7 '

Po Kee Wong

Pe(" p)(r,o) (4.18)

and (4.4), we obtain

Pe(r,o)= 0 , AIjj=1

P oC5x L 1 [ r []- '0 Ejxj -- 02 [ aa a (4.19)

Since the ca 's are the roots of j { )=o we can determine the A1j 'sfrom

A1j =a2 foarPe(r,O)io (kdlr)dr2 2 or42

a2 ra'P,(r,O)&o ýd Ir)dr

2

(4.20)

In particular, when therepresented by a function

Pe(r,o ) AP= const.

=0

initial excess pressure is distributed and

OZ rZc

(4.21)

ELrZr a

then

sin2 FU

Poe a 2 asAt-2 2t2P°C~jsin2 •2 •o2•J (4.22)

62

Billiý qoamp -Stanford The--sis-.pd.f Billie Champ -Stanford The pf - / - I i Pa ge 13;


Thus all field quantities of the viscous compressible fluid inside therigid inclusion are uniquely determined:

ur,(r,t)= • a ______________• L,[]x rj- (423j=,po•tj sin2 •.•os2 •.• JaiL

u2 a][2[

siin2

Pe (rIt")=AP t]2a -f F. ,iE'i (4.24)

j=,~ si sinZL2J0s22 .[3•a[

sin jj=1pc si 'O 2aF

Eli- ýpc( 2 ~t~

sin 2ý

CO AP2a

CI+ PO = I -,J=,9oC5 sin2 TJ 4.os2 POi

El all rI l

+. ,I. ,2 r[ [] a Eb w

(pjc + -- aO 2iw ill (4.25)i[] a[ F r 11[]1

l O a El

AP 2rj E[]

J I P°Co sin 2 Li 1..9 1:os - [ ..2

1 - [] r [I]

D[- 2o~ +tr 1. O-- 2iwj 0 L L iw (4.26)[] al rill

[] a []

63

Billie Champ _' Stanford Thesis.p'df Bil.ie.Ch. .S a.o. .. . .. . .. ... .. Page .74

Po Kee Wong

sin2 1xJ 0• 2[7, jH

+ o AP 0- ao

c~ 1 +P r -1A, ,+ JF1 I 2 B+ [] 0 a

(P-C ER +i- &E~±.+ 11~ ~~~~ (4.27)U U E] ro 0

0] 0 all] 0

where coj is given by (4.17) and cz. represents the roots of (4.14).

The solutions are valid in the region c< r• a for 0• t• oi,where e is chosen such that grrl, 1 ,0 and ý00, will be finite at r = E

and remain finite as we let e approach zero.A detailed physical interpretation of these results is of interest. We

observe from (4.17) that the motion of the viscous fluid in theinclusion can be classified as underdamped if

2Lb' + 2u at. []

I - DE L > 0 (4.28)El aco 2 E]

as critical-damp ed if

16'+ 2u atj dI - 1 -] = 0 (4.29)

E] aco 2 []

and as overdamped if

1- j [I< 0 (4.30)] aco 2 -

64

I Billie Champ - Sta'nfo'rd Th-e*sis.p'df B..llie.Champ'.Stanford. ". . ' " " . " .Page 75


The ratio of the ai values represents the positions of the spherical

nodes. As shown by Lamb 46) for the inviscid fluid, in the second modethere are two spherical nodes which are located at1.4303 a=0.5816592 a and a. Thus one can further predict that the2.4590

third mode should have three nodes, located at 1.4303 a= 0.4120833 a,3.4709

2.450a =0.7058679 a and a and that the fourth mode should have four3.4709

nodes, located at 1.4303•a= 0.3194487 a, 2.4590 =0.5492026 a,4.4774 4.4774

3 47 09 a =0.775204 a and a, etc.4.4774

omaj•~ : $

a2 =2.4590rc a3 3.4709n cc 4.4774ic

From (4.28), (4.29) and (4.30) one can see that as ac increases, i.e. as

we increase the order of the mode, the motion of the viscous fluidbegins with an underdamped wave motion, becomes a critically-damped one and finally reaches an overdamped character. The causeof the damping is, of course, the u' and u. If (u'+ 2u )= 0 the motionof the fluid is undamped, thus the wave motion of an inviscid fluid in aperfectly rigid inclusion never decays.

Equation (4.17) can be rewritten in the form

27e 2,!(0i 2=n-+ i- (4.31)

T il

65

I

I Billie Champ - Stanford Thesis.pdf _Page_76 I.B e . .S ta n fo r... . . . . .. . . . ... ...I.. ... .. . . .. . . . . . . .. .. . . .

Po Kee Wong

Then the time factor e'wj becomes

2I7 271-"-•t +i-'•t

e (4.32)

where

2ff=- - __. 2L •• _ 2.. (4.33)T [I a El aco []

27r _ v'+ 2u g (4.34)2

T1 a 2

f is the frequency of the underdamped wave motion for a given mode

caj and T is the corresponding period, 2 is the decayed factor of all11

field quantities at any given position for a given mode (xi for the case

of underdamped motion. From (4.32) one can see that the magnitudesof all field quantities at a particular position are attenuated by a factor

of e-' = 1 at a time interval of2.7321

r a 2 2 2 dltd 27r u'+ 2u -(ut'+ 2u)kd2t 2(+2)

td is called "modulus of decay" or "time constant" and (4.35) holds for

underdamped and critical-damped motion.In the case of overdamped motion, i.e. when

66

I Billie Champ - Stanford Thesis.pdf P. a .g eBi llie. C ham p. .. Stan. ord.T.... ... ... . . .. . . . . . .. . . . ..... ... . .P age.. . .. . .. . . ... . ... .. .. .. .. 7


'i)+ 2u cxT1- • - <0

waco 2

harmonic wave motions cease and the "modulus of decay" becomes

T=i a Eb'+a 2L) x 41 2u _X j L LI27r c,(x Haco 2 LIo2

(4.36)

4.1.2. Viscous effects in fluid cones

The theory of rigid conical tubes filled with an inviscid fluid wasoriginally treated by Rayleigh 47). One can see that it is a part of thesolution of spherically symmetric motion. Since u, = u. = 0 and

uir = ur (r, t), and the fluid motion is in the radial direction only, one

can form a tube bound by two spherical radii a, and a 2 as shownbelow.

z

x

/

Y

I-

67

Billie Champ - Stanford Thesis.pdf Pa et.8

Po Kee Wong

Rayleigh treated the problem with two particular boundary conditions,namely: (1) both ends of the tube are open; (2) both ends of the tubeare closed. The same problem is extended here to account forviscosity and compressibility of the fluid. Also we allow for one moreset of boundary conditions by considering one end closed and the otherend opened.

a) Both ends opened:

In this case we require that Pe(a,t) 0 and Pe(a2, t)= 0. From (4.4)we have

Al,• o(kdlal)+ BIYo(kdlal)=0

A,qo(kdla 2) + BYo(kda 2)= 0

(4.37)

(4.38)

Since A • 0 and B, •0 we obtain as frequency equation

,9O(kdja,)YO(kdja2) -a ~9(kdla2)YO(kd,,a,) =0 (4.39)

which can be reduced to

sin kdl(a 2 - a,)=0

or

kdl(a, - a,) =.27r-(a - al)= jfkdl2

j = 1.2.3.4L - (4.40)

Thus for each kd= - al , there is a corresponding cowa 2 - a 1

satisfying the boundary conditions for opened ends:

from (4.17)

68

Billie Champ - Stanford Thesis.pdf ....P.a.ge,79


01) = j-t. a- u '+2u Ca2 c. a

0u'+2u J-r E(

(a, - a,) 2co

where j1,2,3,4,K.

The nontrivial solution of (4.37) and (4.38) isA1 _ B1 (

ra2j

(4

ýP]2 - a, [] [•2 - a, [

This means that all field quantities can be obtained in terms of c j

2 in r (4Do j 9°c2 sin• •'t•~,(

j=l ar 4a

0] pin.jr jt(a,-r) c a2-a, L,,, EP Elt "-- E? (,4LIOt

j= Cu jp c +ý ic 1 n2- J a,aa~r

j= LI [] aýr ! Fa, - a, Ej ( a2 - a,) jnr []

Elj] . ](a 2 - r)U[] sing. a a

J=• []a77 -r

9ý +Fý C c 2+ (j~j'2

.42)

.43)

.44)

1.45)

2i .11 LE (a2 -a1 .)+ ýýE 2 a j

a2r' i

2oj2i(a2 - r) [f]a, - a,

ap

(4.46)

69

I Billie Champ - Stanford Thesis.pdf .Page 80u,Billie Champ - Stanford Thesis.pdf I-'age ~iu

Po Kee Wong

Ssin j n (ar r)(a,-r),c 0 0 1+tPo= c o)C~flo +qi]

j=l J El a,r

__ j_(a_ - r)arin+IL 2 i(0j1 a, - a,

a2r ' = n[] jj---

io n (a, - r) 1

a 2 - a ,

(4.47)

D sin J(a2 - r)

S, 1 =T 1c + l'ioJ a,- a,j=l ar

i jnr(a, - r).j a -a

ar]jn(a, - r) E

Cosa, - a, Li,

(4.48)

Taking the sum of (4.46), (4.47) and (4.48), we obtain

0 . jn(a. - r) D[] sin ý D]a, -a,

rr r U, "q 1; +33P. =Yc1 0-(3poc 0 " + ih(3 q'+2rl))i= Ealr Li

H H-- 3P +ioj (31'+2i)V:p

or

ýrrl + ý 001 + •' #-

3P,+ io) (n'+ 2 )A, (4.49)

This is in agreement with (4.8).

70

np - Stanford Thesis Ddf yage OI J1i lli - .. .. .Sta n...f . .. ... .. .... . . . . . . . .


b) Both ends closed:

In this case we have u,, (a,t)" 0 and u12 (a2 ,t) 0, and with (4.3)

A ,kd. 0 (kda,) + B,kdYo'(kda,) = 0

Alkdl9l(kdla2 ) + BlkdlYol(kdla 2 ) 0

(4.50)

(4.5)

Again we must have A1 • 0, B, # 0 for a nontrivial solution, or

,4',(kdla,) + Y'(kdl 2 - YO'(kd~,~,(kl 2 (4.52)

After some manipulation, the last equation can be simplified andwritten as

tankdl(a 2 - a1) _

kdl(a 2 - a1)1

+ a2a,1)2 [kdl(a 2 - al)]2

(a 2 -a1 )

(4.53)

If we take a, = 0 in this form of the characteristic equation it reducesexactly to (4.14). For given values of a, and a2, kdl(a 2 -a ,) can bedetermined from (4.53). If kdl(a 2 - a,)- j, then for each a 2 , the

corresponding frequency cow is defined by (4.17):

Lia2 -a(W

1ib +2u a. 0 u'+2Lu a.[2.E +i i (4.54)

0 2 co(a,-a,)r]D 2 c.(a 2 -a,)

Since (4.54) represents a complex number, it is convenient to representit graphically together with the expression

71

I Billie Champ - Stanford -The s--is. pdf SPage 821B illie .. a mp. . S t.n.. r.. . .. i..... . . . . . . . .. .. .. . . ... . . . . .. .a g....... . . .... . . .... .......... . . .. ... .. . . .

Po Kee Wong

-Pe wj Otj-- e - e (4.55)

Y§7 Pe 1C c ~ j (da,-a,)

which is obtained from (4.4) and (2.1). We note that poA/pe lags/ Co

behind 0cj -- 0 (j with a phase angle /2 and that the amplitudea 2 - al

shrinks with a factor cx I/Co (a2 - a,) with respect to the expression

(4.54). When a, = 0, (4.54) reduces to (4.17).The physical interpretation of these results is as follows. After

obtaining cxi = kdl(a 2 - a, ) from (4.53) for given values of a, and a2,the mode shape of the vibrating fluid between two concentric rigidspherical boundaries can be determined immediately by taking theratio among the (xi values as demonstrated in example 4.1.1.

Corresponding to each ac the magnitude of (oi can be determined

from (o / C°CLj = 1. If (u'+2u) is known, it follows that 0j can bea 2 - a1

determined and thus the magnitude and direction of p pe Pe. The

projection of coj along the real axis gives the value of 27c and that

along the imaginary axis gives the value of 2-. 2. is the frequency ofTi

vibration for the particular mode aj and 271 gives the "decay time"

for such a mode. In particular, when the Stokes relation u'+ 2L = 0holds, then 0j = 0, the fluid becomes purely elastic, and there will be

no decay of the motion at all.According to the preceding physical interpretation, one can

perform an experiment to determine (u'+2u), provided that the mode

72

I Billie Champ - Stanford Thesis.pdf Paige -93!I ... . . . . . ... . . . .. . . . . . ..C h.. . .. . . ..p. .. . .. .. . .. . .. ... .. . ... .. ..... . ... .. ... .. . ... ... .... . ..... . . . .. . .. . .P... . . . .. . .. . . . .. . .. . ..


shape corresponding to an (xi value can be induced. Since (xi is

known theoretically from (4.53) for given a2 and a, and with Tj

determined experimentally, we can evaluate (u'+2u) from

a,'+2u 2(it 1

2 co(a 2 -a 2 )E] T.

The quantities 0j7, rs, and Pe can thus be determined from thePeP

expressions given in the graph above.The values of cxs = kdl(a, - a,) can be discussed in some detail for

two extreme cases: a1 •0; and, a>> (a 2 - a1 ), a2 >> (a 2 - a,) and

a2 -a1 = h =finite. For a1 0 the values of a.i are the same as in

example 4.1.1. In the second case the fluid cone degenerated into aviscous fluid vibrating between two almost parallel, plane rigidboundaries. Denoting a2 + a, = b = 2a, equation (4.53) can be rewrittenas

tan kdlh I 1kdlh I + (kdlz)2 2 (4.56)

4 R I 0

if )---oo and kd,/h is finite, then tankd h--I o. Therefore,

kdlh= j]2. If h >> 1, the first approximate solution of (4.56) can be

obtained by setting kdlhjPr+ E where E-< land tan E-_. Then2

(4.53) is reduced to (jrt+ E) 4E - and we haveobF

73

I Billie Champ - Stanford Thiesis.p'df -Page 84 1I Billie Champ - Stanford Thesis.pdf Page 84 I

Po Kee Wong

E n 4.J__ ]b -- + LL

jr7tcýb Ei (17r) 3ýb Ei

Thus for >> 1, k 1h can be given as

ci= kdah =j + E= jn+- - 42- lb (9- =~ + L (4.57)

j7D (j7r)3ZlbD

and the corresponding wavelength kdl is approximately

_ 27r ~_ 27th 27i E Fkd I j Jr+ E J7r j 7=hL 4 EYh'I lb ,Eli ]

17- +----- -LE (4.58)j (j7)2 Tb [] (j7t)4 obL

Since we consider the problem of standing waves (vibrations) whichare attenuated with time, the results of (4.57) and (4.58) are identicalwith those obtained by Rayleigh 47) for the case of an inviscid fluid.When (4.57) kdlh=Izj is substituted into (4.54), the frequency is

clearly a complex number except when u'+ 2u = 0 or when u'= u = 0(Rayleigh's solution)47).

c) One end closed and the other opened:

Here the boundary conditions are ur(al, t)= 0 and Pe(a 2 ,t)= 0, or with

(4.3) and (4.4)

Akdla,'. (kdala)+ BkdIY'o (kdlal) = 0 (4.59)

A4Oo(kdla 2 )+ BIYo(kdla,) = 0 (4.60)

74

Billie Champ - Stanford Thes.ispdf. a I Page85


In this case we also have 0 i 0, B1 # 0, and thus obtain as frequencyequation the relation

at . (kd Ial I Y.(kd I a )-Y'. (k dla / ),a o(k d Ia2 )0 (4.61)

which can be simplified to

tan kdl(a 2 - a,) - a - a2 _ (4.62)kdl(a 2 -a,) (a,- a,) a2 -a Ua 2l

For given values of a and a2 with a2 >a,, we can determinekdl(a 2 - a,)from (4.62). If kdl(a 2 - a1)c= a then-for each ac value,

the corresponding frequency o)j can be determined from (4.54). When

the end opened is at r- a and the closed end at r=a2, the

characteristic equation (4.62) is

tan kd, (a 2 - a,) = a 2 (4.63)kdl(a 2 - a,) a, - a,

The solutions of (4.62) and (4.63) can be evaluated by plotting

tan kdl (a 2 - a6), a 2 kdl(a 2 -a,) and - al kdl(a 2 - al) versusa) -s a a2 - ab

kdl (a2 - a,) as shown below.

75

P-age bb I1I B illi C h am. -. - .Sta n fo rd T h e s is .o d f ... ,

Po Kee Wong

i tan kdl(a, - a,)

I II I II I I

a openedI

I a2 closed

I I

III I

I I I

kd, (a 2 - a)

IIIIIII

76

Billie Champ - Stanford S :P ..... Pa~ge___:B illie... ... .. . .. . ... ..Ch am p.... .... ... .. . ......Stanford.. . . . .. . . . .... .. . . ..... .. . ... . .. . . . .. ...p 7 i


One can observe that the intersections of tan k-d(a, - a,) with(-a 1) a, and kd(a,- a provide two distinct

solutions corresponding to the two alternatives. For eachoj = kd• (a2 - a, ) the decay factor is less when the cone is open at r =-0

and closed at r a, (loudspeaker analogy). From the figure, one cansee that the maximum of the decay factor ratio for the two alternativesis e' . The decay factor ratio depends on the values of a 2 and al and

ranges from e' and e°43°3 . However with increasing this ratioreaches e' independent of a2 and a,.

In concluding this discussion of the viscous effects in fluid cones itshould be noted that the present solutions are valid only insofar as theyallow for slip in the radial direction along the wall of the tube. Strictlyspeaking, once the viscous effect is considered the solutions are exactonly for viscous fluid contained within two concentric sphericalboundaries. A formal solution for the conical pipe containing viscousfluid is rather involved and the present example serves to providesome insight into the dynamic behavior of such bodies. The fieldquantities are not written out for this case since they can be easilyobtained as for examples a) and b).

4.1.3. Spherically symmetric free Vibration of a spherical elasticsolid, a spherical viscous fluid drop and a viscoelastic sphere.

The boundary conditions considered are: grrm(r = a, t)- 0,ý,,mr= a, t)=O, and ýr,,,(r= a, t)= 0. Since =,Om r,,, = 0 isautomatically satisfied for spherically symmetric motion,rr.(r = a, t) = 0 is the only condition to be enforced.

For an elastic solid (m = 2) we have according to (4.10) and withB2 = 0 (to achieve finiteness of the field quantities at the origin r =0the characteristic equation

77

1, Bil.ie Champ:- Stanford Thesis.pdf Page 98'1

Po Kee Wong

- ~Xqo(kd2a) + 2p1•9(kd2a)= 0 (4.64)

which can be simplified to

E 1 + 2P.+2"

1- 4 (kd2a)[]tan(kda)= 0 (4.65)

This result was also obtained in another way by Love34 ). He expressed4p ~~~~2(1- 2q).-1 ý<-w4+p in terms of Poisson's ratio ; as I- With - 1< g < 4we•,+ 2 1- g

have 3> 4p > 0. Love gave the six lowest roots of (4.65) for aX + 2p

Poisson's ratio ; = 0.25: kd2a = 0.8160nt, 1.92857t , 2.93597r

3.96587r , 4.97287t and 5.9774rt.In the case of a viscous fluid (m = 1) we set B1 = 0 in the harmonic

solution (4.5) to ascertain finiteness of the field quantities at the originr = 0. The boundary condition ,rr,(r=a,t)=O leads to thecharacteristic equation for a spherical viscous fluid drop:

D 1 pc, + io(ij'+21)k (kda)jtan(kdla) = 0 (4.66)

Note that (4.66) can also be obtained by inspection from (4.65) andTable 6.

Similarly we have in the case of a viscoelastic material

[1- (_ k",a) •an(k•,,a) = 0 (4.67)s a)

78

.. ..... .....I Billie Champ - Stanford Thesis.pdf Page 89,1II Billie Chamo - Stanford Thesis.odf ~ag~ ~,i


where k,,,, and k,,, are listed in Table 6 for a given material model m.Equation (4.67) reduces to that obtained by Bland 42) when wesubstitute k~n, and k,,,, from Tables 3 and 4.

4.1.4. Spherically symmetric free vibration of an elastic shell, aviscous fluid shell (bubble) and a viscoelastic shell.

We consider here shells which are geometrically defined by twoconcentric spheres with radii r = a, and r a 2 (a 2 > a, ). For traction-free boundaries we have

n, (r= a,,,t) = 0m = a,,t ro = a,, t 0

'rr, (r a 2 ,t)=rem 0 " a 2,t - a 2 ,t 0

As indicated before, r,,, .,=O is automatically satisfied for

spherically symmetric motion and we therefore have to require only

9 rr,,(r= a,,t) 0 and ,.r. (,-= a t,)= 0.

In the case of an elastic shell (m = 2) the above boundaryconditions imply according to (4.10)

A2 k, [-X&, (kda,)+2p8&(',2,, ]+42 kd [- XY,,(k 2 ,a)+2p •,•," , ]=o 0 (4.68)

and

A2 kd2 [- X&o(kd1, 2)+2P&"(9,,a 2 ]+Bk [- X} (kdaa)+2pY0 (O4,a2 1=0 (4.69)

Seeking the nontrivial solutions we must have A 2 • 0, B 2 # 0, ora0 ) •"(ýd _ X Y.(k-d2a, )+ 2 p YO , O4V0,o (Y kd2a 2 )+2p; ( 2a2 - -Yo k.,)+2p; ( , (4.70)

-XO k~ 2a2 +2p&( Fd2a2 - k&O kd2a I )+2 p,9 d2a,

79

. Billie Champ - Stanford Thesis.pdf Page 91Bille Chmp Sta n..r ....e....... ... ... 90

Po Kee Wong

(4.70) is the characteristic equation for spherically symmetric freevibration of a spherical shell with an inner radius a, and an outer

radius a2. This equation can be simplified and written as

(kd2a2 )+ arc tan 'a2 1 . + 2p )kd2a2Jd~a2 4p D

(kd2 arcta "-[ 1 - k + 2J P kd2alI(kd 2a,)" a (4.71)

Td2a, 4p []

A numerical example of the solution of (4.71) is given below.

a, = 1.81 a2 = 2.54 Poisson's ratio = 0.34

The roots kd2a 2 are

°La- =k2a, = 7.8791995, 15.62335, 23.397896, 31.179907,

Cd2

38.964892, 46.751358, 54.538672, 62.326515,

70.114712, 77.903153, ...

For a geometrical parametric study of the characteristic equation(4.71) we consider the following two cases.

(a) a2[-a,=h, <

This means the shell is very thin and a2 z a, = a.

80

I Billie Champ - Stanford -rhesis.pdf Page 911B illie.. .. .. . .... . . . . . . ..Ch a m p...... ... ........... ..S t.... . ...nfo rd..... . .. . . ..... . ............ . . ... ... . ... .. . ..... . . ... .P ag...... .. . . .. . .91... . . . . . .. .


(b) a2 -a, = h where h is finite and a2 -->oo, a,--->oo, and

a 2 +a 1=2a. We again have H-h <1 but now the shell isEj2a [

approximately a plate of finite thickness h.

In the case (a) we arrive at a first approximation for the roots of(4.71) by expanding the expressions containing a2 into a Taylor serieswith respect to a,:

c(kAd2a+2c2O)d2a 2_2(3c C2

ElaWE (kd~a Y+c -2c)(k-d2ay +C2

+ OH-.+ ...... •=0 (4.72)

wherec 41 2(l-2q)ý +2p 1-ý

Equation (4.72) is the characteristic equation for the sphericallysymmetric free vibration of a very thin shell of radius a and thicknessIh. The first order solution of this equation gives

L ka =F V4p3 2. (4.73)

'cd2 a; = 0+ 2p)

The period of vibration of the thin shell is

_I 27r _ a,3 PX+2p (4.74)f r) p +23p+ 2p

which is identical to the result obtained by Love34 ).For the case (b) we rearrange (4.71)

81

1, Billie Champ - Stanford Thesis.pdf- 'Pa .g-.e 92 '

Po Kee Wong

1 1 +l(kd 2a2 kd2a,)

tan kd2 (a 2 - a,) kd2al kd2a2 c (4.75)LiiI 1 D•i 1 Li1+ daka

L d 2al C El d2a2 c

With a2 - a, = h, a2 + a, -b = 2a, this can be further simplified:

(4-1)2 h)+ I + ý-Ic(k,,2 b)2-- (4,,h 2 ] (4.76)(k 4 2b y - (4• 1 2 h2 ' + I [(_ - +4 c /' 2 ]+ l [(k d~ by + { Ld h 2 1

4 2c

As Yh---oo with (kd2h) finite, the right-hand side of (4.76)

approaches zero, thus tan(kd2h)--*O. Therefore kd27r=jf where

j= 1,2,3,4,.... If 1h>> I the first approximation of (4.76) can be

obtained by setting (kd2 z)=j1r+e where e<< I and tan= e.

Equation (4.76) is then reduced to

2 2lI C 2hL 2c2LL4_ _- E_ or G

For 1h>> I or Ib<< I we can approximate kd2h as

2) 4(k-d2h ) = j •r + " ' -C- '] 2C.. ( .7c +c (4.77)

The corresponding wavelength ;d2 and the corresponding period of

vibration T are given by

.82

Billie Champ - Stanford Tfte'is.pdfS. _Page_ 93IBl le C a p -S a f r ..... ...... .......... .. .... .... .Page.. . . . .. 3


_ 2rt_ 2._.,/ =2.t/, e 11 2h 0 _ c R• .o + 22c' E]== 7, -1+.• ( .8

Xd2 ir~r j n = 1i-7 H7 ý~ (4.78)

and

2nt 2rch 1 2h L c 2c2 • (4.79)T = -- + =-7- F] - 7-7 +.. +n ý

0)Cd2 ~J+CJd 2 0 ill1

A comparison of the two geometrical parametric studies of thecharacteristic equation (4.71) is listed in the chart below.

'>>(ialh--0, a finite h finite, a--+oo

43c - 1 c E[Ih 0 2c2 Qh Elk a - kd, =--rt~r +" U-0 -T- -----) +...

C .2-2) 4p 4p 2(1-22)X+2p l-q =X2 -

X++21a 1-g

j = 1, 2, 3, 4,... integerst27rt 1 T 2h 1 c [•H +¾ 2c- hjcd W70j2al Fjt 7"~a [I

j = 1, 2, 3, 4,... integers

Long period, low Short period, high frequencyfrequency

Thin spherical shell Plate of finite thickness

(Physically this represents (Physically this represents the thicknessthe breathing of the stretch of the shell in the radialspherical shell in the direction. In particular, as the radiusradial direction.) approaches infimity, it becomes the

thickness stretch vibration of a plate.)

83

I Billie Champ - Stanford Thesis.pdf Page 94:1Billie Champ - Stanford Thesis.pdf Page 94

Po Kee Wong

This geometric study of the characteristic equation (4.71) can also beformalized in the following way. Since a2 - a, = h, a2 + a, = b = 2a

and E- H we obtainD21a [I ibr

a 2 a(l+ ) hL+ I H2E DE

a, a(1+ h [)h--1 H

Substituting a2 (a1,E), a, (a,E) and a2 (h,e), a, (h,E) into (4.71), wecan then write the characteristic equation (4.71) as a function of (a,E)or as a function of (h,E). Using Taylor's series expansion for bothexpressions of (4.71) in terms of ascending powers of E and notingthat the coefficients of 1, E, ( 2 , 3, ... must be zero we obtain thesame solutions shown in the chart.

In view of the fact that the characteristic equation (4.71) is thecorrect exact solution for the free vibration of a spherical shell of innerradius a, and outer radius a2 , the two degenerated solutions justobtained can be regarded as "singular perturbation" solutions of thecharacteristic equation. It is called "singular" or "irregular" becausethere is only one limit E-->0 but two distinct solutions are produced.Other parametric physical quantities are not affected, as they appear inthe degenerated solutions. Systematic techniques and detailed reviewsof the method of perturbation can be found in books and articles byLagerstrom et al. 18), Van Dyke'9), Cole2 °) and Chang21 .

In the case of a viscous fluid shell (m =) the characteristicequation can be obtained from (4.5) or simply by invoking thecorrespondence principle and using Table 6 and (4.71). Ascharacteristic equation we obtain:

84

II Billie Charn .n - Stanford Thesis.pdf Page 95 1I, - "L . . ..k -'h"m o 2- S ta fo r T. h e sis.• LZ_. f . ..... . . .. . .. ........... .... . ..... .. .. . . . . .. . .. .. .. . ... . . .. ... .... ... ..... .. . .. .


(0 1 c2+ (X-'+ 2,L )i( d)kdla)+ arc tan -- 0(kda.,

T dla 1uo _ 0('2~o~

(kdlal )+ arc tan 1 1 (4.80)al 4uio [

The difference between (4.71) and (4.80) is the coefficient

cm=4±, which is a constant for the elastic solid c = 4P and.k;, "X+2p

frequency dependent for the viscous fluid c1= 2 T+(O, M usC,+(L'+ 2u )co'

(4.80) is more complicated to solve for the unknown co (frequency).In (4.71) co is a real number, whereas in (4.80) co is complex.

Although is frequency dependent, the geometricalc=c + (u'+ 2L)co

parametric study of (4.80) will not be not be affected by this and wecan proceed as in the case of (4.71).

If we replace kd2 in (4.73) by kd, = and c2 byFCoO +iu'+2L)i

2 4i(U ' , the characteristic equation of a thin spherical viscousco + i(p 1+ 2L))fluid shell of radius a vibrating in a spherically symmetric breathingmode is

kca ()a ] 4LuiCO 2 4uim o 3kd'a=4co~ ~ ~ ~2L )1(' u • (t'20 }m -co + u+ 2L) 0)o

From this relation we deduce

C= a2 c +4 p(3o'+2u) a c +41p(3u'+2iu_ 48 a-Co('+2u) A2(u'+2u)a2 2 u'+2ua (a2co 2+4 (3+2u))2

85

I Pillýe qoo.mp - Stanford Thesis.pdf B li Ch m -Stn o d T e s ... .. .. p.di. ..f ... Page. . .. . .. ... ... .•..i.....i....J...i ...iii.. . a ,e 9.6_

Po Kee Wong

27r 2m27c+i71 (4.81)T -q

We know that the motion is overdamped because

48va2 C(u'+ 2u)[a2c2 + 4u (3u'+ 2u) 2 > 0

Similarly it follows from (4.77) that the characteristic equation of athin spherical viscous fluid plate of thickness h undergoing thicknessstretch vibrations is

____ 1__-•_a__4uiojkdh-~t+ c-~-u~-4 +.d' L +i(u'+2 -uV-J'+20 )o +

The corresponding relation for co can be given as

03 [j22+ 1 7_ (U, + 2U )+ 41 2(u,+2u)+ 02t" Gn Yma H(•

2 [jtcc ( 2u + k1.E2 0 + C0 = 0 0 (4.82)-(u,+u)h2 (I')+ +L~ - o~

The stability of the solution can be examined without solvingequation (4.82) for o). It is well known that for a solution of the forme21 where is characterized by k3 + o02+ BA + a 0 is stable (i.e. real

part of k 's are negative) if (x > 0, 03 > 0, L> 0 and cxj3 - a > 0. Wenow let X = ic and note that our characteris tic equation (4.82) assumesthe form

86

Billie Champ - Stanford_ Thesis.pdf .Pagq 97]


co3 a -i 2 3) + iaO = 0

in which ax, j3 and a are all real and positive. Hence the thicknessstretch vibrations of a spherical viscous fluid plate are also stable.

In the case of a viscoelastic shell (m =3,4,5,6,...) the characteristicequation of spherical symmetric free vibration of the shell of innerradius al and outer radius a2 can again be obtained by means of thecorrespondence principle, using Table 6 and equation (4.71):

(kcina,)+arctan My (k 1ta 2 )•-k • -•,a2 ~ a C

711 1 (•,l•(.3=(k-,,,a,)+ arc tan-C (kna 4.3

where

Performing the same geometrical parametric study as before, weobtain as degenerate solutions of (4.83) the breathing mode solution

,,a = (3 c 7, (4.84)

and the thickness stretching mode solution[] Iq / r• 2 4

2m [3 h [] [k,6, t +h + C -.. 2c -__F]+(4.84)

(j7r Y [i (j7rW Y2a []

j= 1,2,3,4,K integers

87

I Billie Champ - StanfordThes is. pdf P aqg ..98..

Po Kee Wong

4.1.5. Spherically symmetric free vibration of a spherical elastic shellcontaining a viscous fluid.

We now consider a spherical elastic shell with an inner radiusr = a, and outer radius r = a2. The field quantities must be finite at

r=O for any t>O and therefore we must have BI=O. Thedisplacements of the viscous fluid and of the elastic shell arecontinuous at the interface, while the outer surface of the shell isconsidered stress-free. Hence, with (4.3), (4.5), (4.9) and (4.10), theboundary conditions are

U,,(alt)-Ur2 (c,,t = 0 (4.86)

PO rri (al,t)- r,-2 (0 1,t = 0 (4.87)

Srr2 (a 2,t)= 0 (4.88)

and the characteristic equation of the problem stated is

kA , 1 '9; ( k ,, , , ) - k d,2 ;a () ,a , - Y O( 4 2a ,( 4 .8 9 )

k [] (p,4, +n,,~o )&,,(kdla, ): - 2 - X9(k 2 ) ~ l- Xr'o(k,,a,k2'• 0i (kA a,) 23 - Uz+ 19j(kda,) -r2 1+2p') , X I" ' =0A

-d k, 2 A] .. -=0" 2(ka, 12P (k 2a1 8 [] 2 P,.A

Lt ~~~[ 2ivj9 ) 2 D '12 (aza

o- XMn(A. 2a2 ) 0 2E0- X Y, (k~a2 )I

To examine the effects of the geometric parameters on the roots of(4.89) we examine the three cases:

(a) a1 = 0. In this case (4.89) reduces to the characteristic equation

for the free vibrations of an elastic sphere which we discussed insection 4.1.3.

88

I Billie C"mp - Stanford Thesis.pdf Page 99 IBl lie .C h a m p. S ta n fo rd ..... ."......... .... .... ............ . .. .. ................ .... ................ .............. .. P a ge.. ....... . .. . .. .. . . .. 9 9. - . . .


(b) a, finite with a2 ->c'0. Geometrically this constitutes a viscousfluid sphere surrounded by an infinite elastic medium.

(c) a2 -a,=h and a2 -a,=b=2a, with a2-a(I+c)=-

a, = a(1- .) - 2a- and - = -=---> 0. As before, this2 Ep E 2a b

will provide us with two approximate solutions of thecharacteristic equation, one corresponding to a very thin elasticshell of radius a containing a viscous fluid, the other representingthe solution to the thickness stretch vibrations of an elastic sheet ona viscous fluid.

4.2. Axially Symmetric Torsional Motion

This motion is defined by (3.37), (3.38) and (3.39), according to

which Urm,= UOm 0, UOm = U.(r,O, t= 1(rV orr a0

X,,=r m(r,Ot) and p ym=m 0 everywhere at all times. Thus thegoverning potential equations (3.9), (3.10) and (3.11) are satisfied if

Xm A,, (k mr)+ BraY,, (,,r ] eico (4.90)

From (3.39) it follows that the displacement u, is given by

u~m= ksZ., [AmO.. (ksmr )+ BY,,,Y (}.,r ]dP"(O e eiWI (4.91)

dO

With (3.40) and (3.41) the nontrivial components of the rotationalvector become

M, =k 2 UlA +2 d +rd 2 Am A (k/ rp-BY O@s r ]Pt(cosO)ewitE-1 dr dr-2 D

89

I Billie Champ - Stanford The'si's.pdf Page 100Billie .Ch .mp...ta... rd..... .. .. .. . .. Page. .100

Po Kee Wong

2-k"n(n+I XAý,,,, ()s,,r +B. Yn (•,,r ]P,, (cos0 )e0 ' (4.92)

mom ks d [?,,r9n (k5 ,r )+B,,rY,, (@Sr dPfl(COs0 )je0 " (4.93)r dr dO

and according to (3.43), (3.45), (3.46), (3.47) and (3.48) the nontrivialcomponents of the stress tensor q, are

Srr., q 00e, =n , = P (4.94)

d I. 4 (k r) Y,(ksmr)LdP,,(coso) (4.9q 4m p,,k2r j~n ~+ Bm - H dO e (4.95)

k d pj dP, (cosO) (4.96)r de Es0n dO

k,6, ksr, can be obtained from Table 6 for any of the materials listed.Applications of the harmonic solutions (4.90) to (4.96) to torsional

motion will be illustrated in four specific problems.

4.2.1. Torsional vibration and attenuation of a viscous fluid, an elasticsolid, and a viscoelastic material in a rigid spherical inclusion.

The boundary conditions require that the field quantities be finiteat r = 0 for any time t > 0 and thus that B, = 0. At r =a we have

uN, (a,t ) 0 which yields as characteristi c equation

',9 (k,,,,a )= 0 (4.97)

The roots kS,,a-- ccj,, of (4.97) are well known and are for example

given in the mathematical handbook edited by Abramowitz andStegun 48). The first four roots for n = 0,1, 2,3 are given below:

90

I Biilie Champ - Stanford Th6slsXýf P'a.ge -.0-1 1BiI~~~~~~~~~~~~~~~~~i8 ~ ~ ~ ~ ~ .. ..am .... Stnfr ...isDd . Iy -.. . . -.. --.- .--------.--..-- - ~ -.. ~


j=1j=2

j=3j=4

n=0

3.141593

6.283185

9.424778

12.566370

n=1

4.493409

7.725252

10.904122

14.066194

n=2

5.7634599.095011

12.32294115.514603

n=3

6.98793210.41711913.69802316.923621

Using Table 6 and the field quantities . given previously, one canresolve o) in terms of ksm a = for each material in . For a viscous

fluid (m = 1)we find for example from Table 6

2 2

(k )1aY. 2 -Co asla = TIj10o

po

that is

2in 2-

po a (4.98)

When 0)j, is substituted back into the solutions with the time factor

e'°•j' it is clear that the corresponding field quantifies are exponentially

decaying with time like e- 2 2----t. The positions of nodes in the.p0 a

radial direction for each fixed meridianal mode defined by n can beobtained by taking the ratio of successive aj, as illustrated fromexample 1.1. The position of the meridianal nodes can be obtained

from dp, (co)=0 which givesdO

91

BileCh m S a. rd T e i.pdf.. .. .... ...... ............ Page............. ....... ... .... .... 102.. ...............Billie Champ - Stanford Thesis.pdf .Page 102;:

Po Kee Wong

n=O n=l n=2 n=3 n=4

L 0 0 0 0. 0

0 =0j] /U [1 58 0E 7 [1

From (4.98) we conclude that the torsional motion of a viscous fluidwithin a rigid spherical inclusion is always overdamped.

In the case of an elastic solid (m = 2) we infer from Table 6

(kcoa )= cX j" -Fo a (4.99)C ~S

P 0

For a given radial (j) mode and meridianal (n) mode the torsionalvibration of an elastic sphere in a rigid inclusion is undamped and has

a frequency f -27r 2rta

In the case of a viscoelastic material defimed by the Maxwellmodel (m 3 )we obtain from Table 6

(ks,3a 2 = , iG)2o 2

or

2

0O2-) - L P( i (4.100)ri. po a

92

Billie Champ - Stanford Thesi.pdf .. Pa.. 103


The roots of (4.100) are

co -i 2 1 El (4.101)

and

[•12 [if" jn

=1-].o~p~a 0

(4.102)

The motion of the material in the cavity is

underdamped if 4Ta, _ 1>0,POP. a2

critical damped if 1=0,p0 p a-

• 2

and, overdamped if - ) n 1>0.p0 p a2

The frequency of underdamnped vibration is

f_ ] 1 J2 Ea,f = 2-I []E (4.103)

47r il po a r,

and the corresponding attenuation factor is

93

.. .. ....... ......[ Rillie Champ - Stanford Tfiesis.pdf Page 104ilI~~~~ Bili ...... .. Stanfor .h d .....104.

Po Kee Wong

td = 2,1

Ii(4.104)

For overdamped motion the attenuation factor (time constant) is givenby

td-

1 P_ _ aI

(4.105)

In the case of a viscoealstic material of the Kelvin type (mn = 4) wededuce from Table 6

(k-4a) =a2- (o2a2

1- (p +f io))

Po

or

-2

2-- 2PO a po a

The roots of (4.106) are

(4.106)

C,)

-=I+LE4p~pa 2

2 2EI D

(4.107)

94

1: Billie Champ - Stanford Thesis-pdf Va g-e-1 0 57I

..' . .. ... .. " .... .. . . "

Billie ChamD - Stanford Thesis.pdf Page 105


CO) 2

01 % H~-p0 a 2

L~pOpa 2 E1 2 1~ []

(4.108)

and the motion of the material in the spherical cavity is

underdamped if 4p 0pa- _ 1> 0,

critical damped if 4p0 1ia2 1=0,2

and, overdamped if 4p2 2 1 < 0.Ti2 0C2

The frequency of underdamped vibration is

1 11 U.4pPa2 -127t 2p0 a' UCj,

and the corresponding attenuation factor (time constant) is

td = 2pa 2

For overdamped motion the latter assumes the form

(4.109)

(4.110)

id = 1pa

jn

(4.111)

95

PHlie Cýamp Stanford Thesis.pdf Billie.Champ.-.tanf.rd_ -...... p..._.Page 106]

Po Kee Wong

From these four cases of materials, we see that the pure viscousfluid in torsional motion is damped. Other materials of viscoelastictype in torsional motion are either underdamped, critical-damped oroverdamped according to the physical quantities of the materialmodels and the radius of the cavity as well as the vibrational modecharacterized by Cjl,,.

A comparison of the results for the Maxwell material with thosefor the Kelvin material reveals some interesting aspects. Theattenuation factor for a Maxwell material does not depend on the wavenumberLjn, while that for the Kelvin material depends on a' , in

addition to the physical parameters of the model. Using the subscriptsm = 3 for the Maxwell material and m = 4 for the Kelvin material in allphysical quantities we find from (4.100) and (4.106) that the dynamicbehavior is the same for both materials if

" _(4.112)P 3 P 4

and

H -T 4 •J-1 (4.113)

This means that the displacements, rotations, and strains will be thesame for these two material models, but the corresponding stresseswill be different. This may be referred to as the "KinematicEquivalence of Viscoelastic Models."

4.2.2. Free torsional vibration of a spherical elastic solid, a sphericalviscous fluid drop and a viscoelastic sphere.

As boundary conditions we have here P,,,+ rr,,(r~a,t) 0,

r~m (r a,t) 0 and ý r, (r -a, ) 0. Since rr° n = r°m = 0 is

96

Billie Chmp - Stanford Thes-i s-.'p'df Billie.Champ~.Stanford.T. ...... ... .g 107


automatically satisfied for axially symmetrical torsional motion

r*,, (r =a,t)-0 is the only nontrivial boundary condition. Forfiniteness of the field quantities at the origin r= 0 we again requireB, 0. The characteristic equation assumed the form

d EA,, (ks.,,a)n_-da T -a 0

By means of the recurrence relations of spherical Bessel functions itcan be simplified to

(n -1)8 . (ot j,, )- otxj,, 8+, (o j. )= 0 (4.114)

where oxj,, = k,,a. Using Rayleigh's formula

( )(aj n q H_ I d I a =(oXj" j`

(4.114) can be rewritten as

(oj,, )' [(n - I )i,, + , x j ', = 0 (4.115)

Equation (4.115) is identical with the result given by Love34).In particular, when n=l we find from (4.114)

a H3 _ in (cx1 - -- cos•a 0

3 ji jl=' i (Xjl ( jl

or

97

Billie Champ_- Stanford Thesis.pdf Page.. . ..

Po Kee Wong

tana 31 1 (4.116)3 ji

34)

This result was also obtained by Love The roots of(4.116) are

all 1.8346 X31 = 3.9225

a 21 - 2.8950 a 4, - 4.9385,

All results described in this example are essentially identical tothose of example 4.2.1 with the exception that the axj, values were

there the roots of ,, (Ccy}, 0 1while they are now the roots of(4.114).

4.2.3. Free torsional vibration of an elastic shell, a viscous fluid shell(bubble) and a viscoelastic shell of spherical shape.

Again, we let a, denote the inner radius and a2 the outer radius.The boundary conditions are

P,. + 9 rrm (r =a,,It)= qrO, • =a,,t)= q.,. ()=a,,t =0

P. + ( ,rrn (r= a 2 ,t)= 'rn a2 ,,,t = y 0=a2 ,It = 0

As indicated before, P,, + ý rr., = = re.n 0 is automatically satisfied for

axially symmetric torsional motion, leaving q 4 m(r=a,t)=0 and

g rO (r = a,,t)= 0 as the required conditions. With (4.95) and these

boundary conditions, the characteristic equation of the problem is

I[(n-!)O (ý a, -(ý.a, & (,,(•a2 [(n )1( ý,,a , -(`a (`.a, j 1 (4.117)=

98

I Billie Champ - Stanford The-sis-.pdf -Page 109 1I I ~ -. ...- --.... ~..-... .... ..- - .-.......--...-.... ~- .. ...... - - . . ----.-


For the lowest nontrivial circumferential torsional mode (= 1), (4.117)can be further simplified:

ksma1 -arc tan 3k mal a, ksmla, - 3ksma 2 (4.118)3 - (k,.a, 3- (ksma 2)

A geometrical parametric study of (4.118) similar to that of (4.71)also points out some interesting features of the motions. We considertwo limiting cases:

(a) a2 -a,=h and a 2 ->a 1 =a with H---•<1.E02a El

This means that the shell is very thin and has a finite radius a.(b) a,-a 1 =h with a2---x, a -> o, and a 2+ a = 2a = b-->oo .

In this case the shell approaches a plate of finite thickness h.

Expanding a2 with respect to a, into a Taylor series we obtain incase (a)

(kma)4 + 6+ 6(ks,,a)2 (k a,,Yay Q__9+ 3(kma Y + (s,a4 [9+ 3(ks a)2 + (ý,,,a 41 Da n

+0OE-q+ ........ =0 (4.119)

The first order solution is trivial (C o ) but the second order solutionyields

(k,,,aY =.- 6 (4.120)

99

Bil!ie Cham.p - Stanford-Thesis.pdf Page 1df1Page 110

Po Kee Wong

The motions corresponding to the roots of (4.120) are however notcompatible with physical reality, which means that only the trivialsolution may be admissible.

Rearranging (4.118), we obtain

tan 9kg (a2 - a-)+3k a'a 2 () 2 -a (4.121)2 2 a L k a - 4 2 2 '9-3k,,, (a- a' 9m 9 2a, a+k aka2

With 2• c->co and (kIh) finite, the right-hand side of (4.121)

approaches zero. Therefore (k,,,h)= jrt where j = 1,2, 3,4,.... If

»>> I the first approximation of (4.121) can be obtained by setting

(ksmh)= j7t + e, where e<<l and tane= e. Then (4.121) reduces to

12 Fz2EC_= -H 1 + + ..... Elj7r+e Ob

or

12

Thus (k,,,,h) can be obtained for >> I from

Pi+ H . I 1.4 .. (4.122)jir - b E[ (j7r) 3E + ...

Equations (4.98) to (4.111) are still valid for this particular solution forif n= 1, if a is substituted for h and orj,,,. for oxj, as expressed in the

right-hand side of (4.122).

100

Si.lie.Champ- Stanford Thems.io ..... ......


4.2.4. Free torsional vibration of a viscoelastic spherical shellcontaining a viscous fluid or other viscoelastic materials.

The field quantities of the core materials are finite at r 0 at anytime t = 0, thus B, = 0. The displacements of the shell and that of thecore are continuous at the inter-spherical surface, and the outer surfaceof the viscoelastic shell is considered stress-free. From (4.91) and(4.95) with B, = 0 the boundary conditions are obtained as

u,,(a 1,t)-u,, (4,,t = 0 (4.123)

ý rol ( ,t)- ro 01,t =0 (4.124)

C r•, (a, ,t)= 0 (4.125)

They yield the characteristic equation

- ,-, .( .. @),,a,. .. k .Y. ( .,)(4.126)

--- t (ka)k+,O a'E]-'• • k. a, .÷Qa, [ a, a-(," a)Y ,., a

(,,A-, [(n -I (k_, ) a u k 2.. [, _ I ( ,.a) 0

4.3. Axially Symmetric Nontorsional Motion

This motion is defined by equations (3.25), (3.26) and (3.27) whichrequire that lrm- u,. 1(r,O ,t), U0, = uo., (r,O ,t), and U.,= 0 everywhere

at all times, or p1 (p, ,(r,Ot), y ,,= (r,O,), and X =0

everywhere at all times. Thus one can see that the governing potentialequations (3.9), (3.10) and (3.11) are satisfied provided that

101

i.•i!lie Champ - Stanford Thesis.plf-I'

Po Kee Wong

(p = [Am,, (k.,,r)+ B,,. Y,, (},,r ]Pt(cos0)e i°

and

W= n[A2m• (kr)+ B 2 mYn (ýs,,r ]P,, (cos O)e'°'

(4.127)

(4.128)

From (3.25) and (3.26) using (4.127) and (4.128) it follows that thecomponents of the displacement vector can be written in the form

U IMEl~[ E]ElEl 2

a -U , ,, (,')] [U ,,,Y,, (r)]J" Be'D (4.129)

[B Uo,,, liltEd B2G P (cos0 Elbd, ni EJ

where

[Urn &~r~h 7 dm Yn (kd'i r)

P .mY,,(r)A]51

r D5 k,,.

k,. oi'.i(:sr +!&(4k r) E-r S

k r YLI IY k

(4.129a)

(4.129b)

According to (3.41), (3.'42), (4.127) and (4.128) the components of thepertinent stresses may N, given as:

102

I Billie Champ - Stanford Thesis.pd'f 1•~g 13'1B illie.. . ....Cham... ..... .....a.... .. .......... ....... ....... P age.... .113... .


q] rr,, + PM [] 1

L LP a,,,cs(r)]L )YD(r Di mLe'iw (4.130)] U I2

Ui d L1

"dewhereD

where

Flf Xý k 'q LI

L-0 0

[lit ~29(kbr) 0

la 0 k LD E

ElE.

n(n= 1)[rk-3'.(ý,r (- ý,

2 + n I (k"n 1.) 2 1 " (ýý. 1H-n)rk& Y( '(k-. r)

]EIPl

ElElEl

ElElH]

(4.131)

(4.132)

1 k2 k- 2 y krS2p _ ,kn

EEl

Ykr) H

-E0

2l

k,, i~sYi,;( r)

The other.ield quantities can be obtained by substituting (4.127) and(4.128) into (3.40), (3.44) and (3.46).

In the following we shall discuss applications of (4.127) through(4.132) to specific problems.)

103

Billie Champ - Stanford Thesis.pdfPe Page 1-14;1

Po Kee Wong

4.3.1. Axially symmetric nontorsional free vibration of a viscousfluid in a rigid spherical inclusion.

The boundary condition of the problem requires thatUr, (r-a,O,t)-O and Uo,(r-a,O,t)0. These conditions are met ifwe

equate the determinant of (4.129a) equal to zero for r = a. We thushave

kdIa,(4'(kdIa) n(n+ 1).(ýda'anCk da) (kaa,,kl a)+ 9,, (

This is the characteristic equation of a viscous fluid in axiallysymmetric vibration in a rigid spherical inclusion. The correspondingexpressions for (kd la) and (ksa) are according to Table 6

kdja a -- wc.a

2 + (u,+ 2u )i(O

k,a -i-.a - a

There are two degenerated solutions of (4.133) for viscous fluid:

(a) Inviscid solution: u' and u ---0

In this case

k, = a:a 0

and

kda/ a=

104

I Billie Champ - Stanford Thesis.pdf Fage 1151I B.lli.hm. S r" Pg 1


where the ý's are the roots of

(4.134)

(b) Slightly viscous solution: o' and u are both small

With this restriction we find

Y2 aC4 c0 a

or

kd~fl + EE

with Ikk, a» I1>

where E << I and the e's are, as before, the roots of a()= 0.

Now (4.133) can be approximated by

LDn(n +1) EleE [-- - -4 1:(-

El 4 0U

or

ml± i .(v + 1)- c04 a n(n+l)- 2

Solving k,, a= + for (o we obtain

oa F2 uF n(n+l) Ell - a + -0

CO ] .2 c04 a n (n+ 1)-4 2 F

105

Billie Champ - Stanf6rdThesis.pdf Billie.Champ...Stanford...., ...pdf Page 1 16

Po Kee Wong

i -2 u n(n+l)- (4.135)2- "-'•, c0 a n (n +a)-42

The upper and lower signs in (4.135) correspond to [n(n+])- <2] X o.If [n(n+I)_ 2]> 0, the upper signs hold and in this case 'theattenuation coefficient (time constant) is

3

n•(n+l)- vCa

and the frequency of the vibration is given by

f =1 c J2 n(n +1) D127"aD 2 c0•a n(n+l)-•2 (4.137)

if [n(n+ 1) -- ] < 0, the lower signs hold, for which we obtain asattenuation coefficient

td n2 n(n+I) Fa (4.138)

n(n +1)-4~ 1)cO

and as frequency

f c04 u _ 42 n(n+l) IZ (4.139)

27ra C1 2 c0•a f- n(n+1)(

where 4 are the roots of (4.134).

According to Abramowitz and Stegun48) [Table 10.7, page 468],the lowest value of 4, for a given n always yields [2 _ n(n+1)J>O. Alist of the first four roots for n= 1,2,3,4 is given below.

106

ý Billie Champ - Stanford Thesis.pdf Billie..Champ...Stanford.............Page 1171


n= 1 n= 2 n=3 n= 41 2.460536 3.632797 4.762196 5.86842

j = 2 6.029292 7.367009 8.653134 9.904306= 3 9.261402 10.663561 12.018262 13.337928

/= 4 12.44526 13.88337 15.279081 16.641787

From these observations, we conclude that the lower signs in(4.135) leads to an attenuated vibration whose attenuation coefficientis given by (4.138) and whose frequency is defined by (4.139). Theupper signs in (4.135) and the subsequeny results from (4.136) and(4.137) correspond to an unstable motion.

The time constants given by (4.138) are identical to those obtainedby Lamb 46 ). He also gives an alternate form of (4.134) in terms ofRayleigh's formula. Unfortunately, Lamb made a computational errorin the-case of n = 1. The lowest mode of vibration x should be2.460536 instead of Lamb's value of 2.081.

4.3.2. Axially symmetric nontorsional free vibration of a sphericalelastic solid, a spherical viscous fluid drop and a viscoelasticsphere.

The boundary conditions of this problem areP,,,,+..m?(r=a,O,t)=O and q,o,(r=a,O,t)=O. B,, and B2, are set

equal to zero to ensure finite field quantities at the origin r = 0. Thecharacteristic equation of the problem can be obtained by setting thedeterminant of (4.131) equal to zero:

•P.(kdma)2"(kdm[a)] n(n+ 1)[(ksma)9'n (ka)- 9n (k, a)]

+ (kd.a) 2 ,,(kdma) =0 (4. 140)

Ek,,a W. (kd.a) E L--n + n- 1 -- (k,,a)'- . (k-a)LE0 - & d ] LI 2] -8 (ka~p) [] P - ksma&',(kma)

107

BiII~e Champ - Stanford Thesis.pdf Page 118Billie C-h--amp Stan fo -rd .. Th .esi -s. .pd f.. ....... . ..P~age !1.81

Po Kee Wong

where Am and p,, are the operators defined in Table 2 with iw replacinga- and where kdm and ksm are listed in Table 6 for the various materialOt

models (m). In the case of an elastic solid (m = 2), A2 and P2 are the

usual Lam6 constants, k,,2 = and k,, = with Cd2 = 2 2 andcd2 C,, P0

C,2-=-- representing respectively the dilatational and shear wave

speeds. For the elastic medium the solutions of (4.140) are given inFigure 1. The curves shown are plots of the normalized frequencyversus the meridianal wave number. According *to this graph thefrequency appears to be a continuous function of the wave number;however, the plots are only valid for discrete values of o) for eachmeridianal wave number n, which is the degree of Legendre's functionP, (cosO). For each n there are an infinite number of W's satisfying(4.140), which means that there are an infinite number of frequencycurves.

4.3.3. Axially symmetric nontorsional free vibration of an elasticspherical shell, a viscous fluid spherical shell, and aviscoelastic spherical shell.

The spherical shell under consideration is again defined by twoconcentric spheres whose radii are al, and a2. The boundary conditionsof this problem are given by

m +g,,, (r = a, , 0, t)=O0

P rOm (r=al, O, t)=O

+ +;,,, (r=a2 , 0, t)=0(;,-6,, (r =a2, ,O, t)=O0

108

Billie Champ - Stanford Thesis.pdf Page 119

Billie Champ - Stanford Thesis.pdf .Pag~e 1191


With (4.130), (4.131) and (4.132) we obtain as characteristic equation

[km n (a2)] [,n Yn (a2) (4.141)

where the matrices [a. (a,)] and [ Y. (a,)] are defined by (4.13 1)and (4.132) for a given viscoelastic material m. Equation (4.141) is a4x4 determinant and, with Ay representing an individual element, it

takes the form

All A12 A13 AM4

A21 A22 A23 A24 =0

A31 A 32 A33 A 34 (4.142)

A 41 A 42 A 43 A 44

In the case of an elastic solid (m = 2) the elements becomeAll((X) Tn + -L J, ot3(at)_2oAt,Oa'(ox,)

1- 24;'

A12 (c•1 n(n +)I)= ( -1 H- 9n# 1 41

I -2 j I-2€ j 1-2I

D] •1 -) ~2 D [A13 (ctI) = n (n +1) [ 1 2q I (aI) - 2. 1 Y (o 1)

[] -2g • -g• 2

A H (cx, n (n + 1)L 2 (1 c Y' 2(-0 2 -I- 2ý FH12z21 (Ol1O a~l 1

109

I Billie Champ - Stanford Thesis-.Pdf Ragýý 120 1I Billie Champ - Stanford Thesis.pdf Page 120

Po Kee Wong

D2A 22 (0,t0 = [0 +n

El2A2 4 (•) 1) += +

11

2I - 2

- 2¢

I - - I

I- 21

11 41 2 F(1ý (XI F2(12c 2ý 11 - 2ý

Ely" 2 F11R 21(1 41 21 IF_

I _ 2c 2ý 1 _ 2cEl

A 23 (a1 ) O =0' -aYn .I (a1 I

andA 31 = All (( 2 )

A 32 -A 12 (X 2 )

A33 =AI3 (a 2 )A34 = A14 (6 2 )

A41 =A, ]a 2)

A42 =A22a 2)A43 =A23 (()2 )

A44 =A24 (a2)

where oc2 =a2 o-aa,

and a, - --ICd

ý is the Poisson's ratio and cd the

dilatational wave speed of the elastic solid.The frequency of a vibrational mode is normalized by first

multiplying it by the inner radius of the spherical shell and thendividing it by the dilatational wave speed of the elastic medium. Thefrequency spectra are now reviewed for two particular cases. In thecase of radial vibration of a sphere the values are identical with thosedefined by Lover's solution 34). In the case of a one-layer thin shelltheory used by Baker 4 9) and by Naghdi and Kalnins 50) as far as the twolowest modes are concerned. However, the present investigationindicates that there exists an infinite set of modes of vibration for aone-layer thin shell of finite thickness. All the higher modes, however,.tend to have a constant high frequency as the meridianal wave numbercontinues to increase. These facts are demonstrated in Figures 2 and 3

110

I Billie Cham'p -S'tanford_'Th'_e'_sis.pdf Page 121III Billie Champ - Stanford Thesis.~df Pacie 121I-- *.......~....- ...-.-.---.......-----....-----.. ~.-......--..--..-.---.-..-..--.-~--.-..-----.-----...


for a thin shell with a mean radius to thickness ratio of 63 and a thickshell with a mean radius to wall thickness ratio of about 3. When thethickness of a shell with a given radius is increased, the higher modesare lowered and approach the same order of magnitude of the twolowest modes. By substituting a particular frequency value from thespectrum into the field quantities, one obtains the displacement andstress fields as functions -of the spherical coordinates r, 0 and asfunctions of time. Here the field quantities are normalized with thefunction 0 at a given time, leaving r as the only variable. Thus theprofile of the variation of the field quantities in the radial direction canbe shown by plotting them as functions of the radial coordinate. Themode shapes can also be given as vectorial sums of the displacements.The modal analysis and the frequency spectra thus obtained are shownin Figures 4 to 8.

4.3.4. Axially symmetric nontorsional free vibration of an elasticspherical shell containing viscous fluid.

The elastic shell under consideration is uniformly thick with innerradius r = al and outer radius r = a2 . The field quantities are finite atr 0 for any t > 0, thus B1, = 0. The displacements and stresses arecontinuous at the interspherical surface; the outer surface of the shell isconsidered stress-free.

The boundary conditions of this problem are

Ur1 (r=a1,O, t)-U0r2(=ai,O,t =0uo,(r=al,O, t)-uo02(ý=a,,O, t =0

PI + ',.l(r=aI,O,t)-0,0 .2(G=al,O,t =0

C01/(r-a/,,0,t) -. g,02(ý= al,,, t =0g,,.2(r =a2 , 0, t) =

1;,02(r a2,0, t) 0 O

III

I

I Billie Champ - Stanford Thesis.pdf Page 122,1I Billie. .... ......... .. ......Cham p.......... ... ........St. .......nford....... .... T h s s p f a e 1 2

Po Kee Wong

From (4.127) to (4.132), with m = 1 for the interior of the shell and

m = 2 for the shell material, the characteristic equation of the problem

is

[u P. (a1)]

0

2[P2 2&.(a,)] 2 212 ,(j

(4.143)

The characteristic equation (4.143) is considerably more complicatedthan in previous examples. The elements of the determinant aredefined by (4.129a), (4.129b), (4.131) and (4.132). In seeking the highfrequency solution of (4.143), we can make use of the asymptoticexpansions of the spherical Bessel functions a, and Y, for large

arguments. By manipulating all the dominant terms in the determinant(4.143), we can prove that the shearing effects and dilatational effectsof a spherical shell filled with compressible viscous fluid vibrating athigh frequency are almost decoupled. These results are summarized asfollows.

(a) Shearing effects

Solution PhysicalMaterials 0) Statements

2v(n2+ +n- 1)i dampedViscous a solutionfluid

+ 2 dampedcore n i solutionw 02

____________ where = 0,_1,2, 3... _____

112

I Billýe qhýmp - Stanford Theslis.pdf Page 1231B illie ..C h a m p... ,. a n.... . d...... .... ..... .... P a ge.. . ..... . . . . . ... . . . . .. ....... . . . ... . . . . . . . . .. .. .. . . ... ...... ... . . .. . ...... . .. ..... . - . .. . I


Sphricl- n undampedS eca, oscillation

elasticshell a,+aA 1 2 2r+ undamped

a, +a2 H -- oscillation

where/ = 0, 1, 2, 3...

(b) Dilatational effects

113

I Billie' Champ - Stanford Thesis.pcff Page 124 1Bi lle. h . p.. ta fod..e i.p........ .......... . . . .P age... .. ... .. . . ............ 124... ...... . ...... .... . ..

Po Kee Mong

SOLID SPHERE

bI =2

cd = 2.805 x 105 1e

18

16

14'

12

obCd

10

8

6

4

2

00 2 4 6 8 10

MERIDIANAL WAVE NUMBER

114

[Billie Cham-p - Stanford Tlhesis.pdf .. Page 125


THIN SPHERICAL SHELL

5= 21Cd 2.805 x 10 (sec-

b =2.5-

b 2 = 2.54

t = b2-b 1=0.04

bm= 2 = 2.5210

8

b m

t =63t

cob 1

Cd

C.-.'

LZ.

6

4

2

0 2 4 6 8 10

MERIDIANAL WAVE NUMBER

115

Billýý Phamp - Stanford Thesis.pdfI

Billie........Champ...... ... .. Stanf.......r..... ...................... Page..... . ag 126

Po Kee Wong

SPHERICAL SHELL

18-

16-

14-

12-

Cd8

8-

6

4

2

0-

cd' 2.805 x 10

b = 1.81

b 2 = 2.54

t = b2 - b1 = 0.73

bm 2 = 2.175

Figure 3

II I I I

0 2 4 6 8MERIDIANAL WAVE" NUMBER.

10

116

I Billie Champ - Stanford Thesis.pdf Page__ 127.I


CN

_•.r.~j.

117

I Billie Champ - Stanford Thesis.pdf Page 128 1.1B . .... ..Champ. ..Stanford. . ... . . . . . .. 12.

Po Kee Wong

N4

q

cm.

CR"

CD Nu 00C3 0

N'

118

I Billie Champ - Stanford..Thesis.pdf .......... Page 29


C4,

A!

q

00 0

r4,

0C

0ý

119

I Billie Champ - Stanford Thesis.pdf Page 1301B illie ...... ... .. . ....Ch am p... ......... ..... ...Stan fo rd.. .. ... . ........ . . . . .. ....e.... . ... . . ............... .. P a ge......... .. 130.. ........ . .

Po Kee Wong

(P~

N"

.- I

0~

(L

120

...... ... ...I PHlie qhamp - Stanford Týesis.pdf _Page 13"1 .I Billie......Cham ." S n...... ....... ..... ..


3

- - - 2-

/ /

/ ,/ ,/('I

I i,1(1

II.'~I \ *~

' \' \

" \\ \ %..

N

- -

DISPLACEMENT MODE FOR N =1 AT SECOND FREQUENCY

Figure 8

121

I Billie Champ - Stanford Thesis.pd -f P age 132:1I Bl le C a p - S a f r .. . . .... ... ... ........ ..... . ..... . . . .. . . . ... P age.. .......... .. . .132.. ... .. .

CHAPTER 5

CONCLUSIONS, DISCUSSION AND SUGGESTIONS

Conclusions

1. The governing displacement equations of motion for a class ofthree-dimensiona 1 homogeneous, isotropic, linear viscoelasticmaterials are systematically constructed.

2. General solutions of these governing displacement equations areobtained by means of the Lam6-Helmholtz-Stoke s potentialfunctions, from which each governing displacement equation ofmotion is transformed into a scalar and a vector potential equationof motion.

3. The vector potential function is further resolved in terms of twoindependent scalar potential functions in order to secure aseparable solution. This can be done when the vector equation ofmotion is expressed in terms of spherical, conical, or generalcylindrical coordinates. Therefore, the displacement equation ofmotion for each material model can be transformed into threeindependent scalar potential equations of motion.

4. The field quantities are obtained in terms of those threeindependent scalar potential functions, and from these the transientor harmonic solutions for mixed or nonmixed boundary valueproblems can be obtained.

Discussion and Suggestions

The present theory, as mentioned above, provides a systematicmeans of constructing and solving the governing displacementequations of motion for a class of three-dimension al, homogeneous,isotropic, linear viscoelastic materials.

122

Billie Champ - Stanford Thesis.pdf Billi p. . .S.... t.a.no. T.. i.... Page.133


The application of the theory to solve some physical problems isdemonstrated in thirteen examples which are all based on sphericalcoordinate systems. However, as noted in the text, the theory can beextended with regard to the following three aspects.

1. Analytical aspect

a. To provide detailed analytical general solutions andphysical examples in other separable coordinate systems,namely, the rectangular, the parabolic, the circular and theelliptic cylindrical coordinate systems, and the conicalcoordinate system.

'b. To consider thermal and electromagnetic coupling effectsin the general solutions for all separable coordinatesystems.

c. To obtain, by means of iteration from the basic lineargeneral solutions, the higher order nonlinear solutionsincluding thermal and electromagnetic coupling effects.

d. To develop a systematic method of analyzing the physicalproblems involving nonhomogeneous, anisotropicmaterials.

e. To implement electrical and mechanical circuit analogy tosimulate complicated material models.

2,3. Numerical and experimental aspects

a. There are quite a few difficult physical problems "whichhave been analytically solved, but so far no numericalvalues or experimental data have been obtained for theseproblems. Examples are: "the circumferential harmonicwave propagation in elastic rods of elliptic cross section";"the circumferential harmonic wave motions of a viscous

123

Bilie qhamp -.Sta-nford Thesis.pdf -I

......................... .........................................................PPage 134

Po Kee Wong

fluid in a rigid elliptic inclusion"; and "flexural wavemotions in a multilayered circular cylindrical shell." Allthese problems have been analytically solved, and althoughthey are physically meaningful, with important engineeringapplications, the numerical results have so far not beenobtained.

b. To establish a long-term project for tabulation of well-known functions of complex argument in the course ofevaluating the numerical values from the analyticalproblems. For example, the harmonic solutions of physicalproblems associated with thermal coupling effects, viscousfluids or viscoelastic materials in a single- or multilayeredcontinuum involve mathematical functions with complexarguments in order to retain the separability of the solutionssatisfying the boundary conditions. Also, nonmixed typeboundary value problems involve mathematical functionsof complex arguments. These can be solved by means oflinear transform techniques.

124

I Billie Charmp - Stanford T'hes's.pdf Page 135!1

APPENDIX A

SOLUTION OF THE VECTOR DIFFUSIONEQUATION AND THE VECTOR WAVE EQUATION

IN TERMS OF SCALAR POTENTIALS

The vector diffusion equation and the vector wave equation can besolved in terms of two scalar potentials, which satisfy scalar diffusionequations and scalar wave equations respectively. This will bedemonstrated here for the case of spherical coordinates.

If y (r,O, 4, t) is a scalar function of the spherical coordinates

r, 0 ,ý, then it can be readily shown that:.

rIV x(errWl=(VrW)xe, (A 1)

andrrr V2

xV x (er, ,Y)= V V- er r (A2)]a r (A2)

where er is a unit vector in the r direction.If y (r, 0, p, t) satisfies the scalar diffusion equation and the

scalar wave equation

Y= [ (A3)

R at2

then substitution of(A3) into (A2) gives

125

I Billie Champ- Stanford Thesis.pdf iPage 13.6.1

Po Kee Wong

wlay F1rarywr -r L (A4)

V xV x (er,.r/-)= V 7 - e,.r ] (A4E r [j [I1 I a' [I

R2 t2 H

If satisfies

v =11 0 O(A 5)

[]I a'Y L

and X satisfies

[i OI t a]

D El a (A6)=[I a2x X

I- at 2"

r

then A=Vx (rrr)+ VxV x(rrX

0 1 ay, D

at 0)+JýL rFa 3xD (A 7)(er wa El5 E72---z-,D

and

rI a0A. 0

v2A IV at~ (A8wA a2r Li

SF3

126

Billie Champ - Stanford Thesi's-.-pdf........ .. Page 137 .IBillie.Champ....tanford...e...... . Page 37 .1


PROOF: Taking the divergence of (A7) gives

V A=0 (A9)

Taking the curl of(A7) and making use of (A4), we obtain

r / .T + ? 7 Hg r rV xA= Vx4 x (e.,) ~-I~r

00rm0

[I ax [-IF

[:] a2 8

wa0rY1 ElwV E--.- [I-E] cr E]

El a1,w

err;[IEw

[ý at2

01 ax wr at

Vxe rr1 2

DmI aX D (A 10)

The curl of(A 10) gives, with (A9) and (A7)

r rV XA-V.(v.;/- V2x -vAA

w ml [: L2yrE1

E2at 2 H

El I arX NN

V x eml a2x mm]

52 at2 Hgr

mI aAm0

x(-e r )]= - a E- m[]1 0A2

m 1 a0

-=-F~t E Vx(er,.)+ V xVml a2 l

5 t

127

I Billie Cha'mp' - 8tanford Thesis.pdf Page 138 1Bi llie .. . .. .. . ..... .... ... .. ..p....... ....t...................T. ...esis...... ...p....f. .. .P a ge... .. . .. 1 38....

Po Kee Wong

Hence

rV xVxA=

El IaAEF- E-

E1'U a~01 a2A 0

and

El OAE

E~l a2 A EEP' - F

Therefore, if Vt satisfies v 2~ F= and X satisfies

V 2x F H- where F H is a function of the operator only,

r v~rrthen A Vxl (ery )+ V xV x V 1r X) satisfies V2A= F 9-j.

b]t E

128

Billie Cýamp - Stanford Thesls.pdf Billie.Champ. .. Stanfo.d. ... s.."p.,. Page 139:

APPENDIX B

rThe well-known Helmholtz's theorem u = VpD + V x A indicated

by equation (2.17) where ý and' A are potential functions, can betraced back to contributions of Lame and Poisson. These functions pand A are often referred to as Lame's potentials. The traditional proofof the Helmholtz theorem is diversified in the literature of severalscientific fields. It can be found in texts of potential theory,electromagnetic theory, vector analysis, partial differential equations,as well as in those of elasticity and fluid mechanics. The traditionalproof was obtained in terms of volume integrals for p and A over aninfinite domain. These results are in fact particular integralrepresentations of the Poisson's equations V2 = V and V 2 A 0- ,I Z2.

where A=V'u and 26. Vxur V.'=0.An alternate proof of such results is shown in the following

sequence: (a) vector algebra; (b) vector identity; (e) commutativenessof divergence and curl with respect to the Laplacian operator; and (d)completion of the proof.

(a) Vector algebra

r cD 2Q as known vectors at any points in the space withtheir components lying along the coordinate basis of three

r -

orthogonal unit vectors e, e2 and e3 , and A a known

scalar at the same point, where D. C Oand b ± 2

rConsider: u as an unknown vector which satisfies

.1r = A (1)

1x rD u = 20 (2)

129

I Billie Champ - Stanford Thesis.pdf .Page.140~

Po Kee Wong

rThen u can be determined from both (1) and (2). Such a result can beobtained by taking the cross product of vector C and equation (2):

Cx it) xU)/ cr- CL) x 2ý (3)

r ir)Solve u from (3) and replace UCr) from (1), obtaining

r DA- Cx2(u = F . 4 (4)

rNotice that u cannot be determined either from (2) or (3) alone. In theformer case, it is because of insufficiency of the simultaneousequations *for the components of the vector zIh, and in the latter, thedeterminant is singular.

(b) Vector identity

Since (4) is true. for any given vectors D, C and 2 fl and any givenscalar A at any point where these four quantities are finite, thenwithout loss of generality let C= D = V which is defined as a gradient,a vector operator in the usual sense; (1) and (2) become

V -4 = A (5)

rV x U = 2L (6)

and equivalence of (4) becomes

v2ri= VA-Vx2L (7)

130

I Billie Champ - Stanford Thes s.pdf Page 14i1


Substituting A from (5) and 202 from (6) gives the well-known

vector identity

v2 = V (A.ru) - Vx (Vx u) (8)

(c) Commutativeness of divergence and curl with respect to theLaplacian V 2. Taking the divergence of (8),v .(V2U')= V2(Vzi,)-V.[V X(VX,',)], and noticing thatV -[Vx (V x u)]= 0 in vector analysis, one obtains

v .(v2')= V2(V.L) (9)

Hence V V2 and V2V operated on a vector are commutative.Taking the curl of (8), V (V2I,)= V XV(V .,)- V x[V x(V x4)],

noticing that V xV (V "u ) 0 and using (8) to expand the term

-v X[VX ×(V X 1,)]- =V2 (V X U,)_-V[V .(ý X U- V2(V XU)

one obtains

v x(V11)= V2((V X) (10)

Hence VxV2 and V2 Vx are operated on a vector are also

commutative.

(d) Completion of the proof.

Ifru = u+u 2 (11)

where u, and u, are functions to be determined satisfying the

conditions

131

I'Billie Champ - Stan-ford Thesis.pdf Page,!4z2I . ..B. ....llie. . .. ... .. . . .. . . ......... ............Sta.........fo rd........................ ... . . ...p Pe1 4

Po Kee Wong

Vxul=O

and

V "U2 =0

then substitution of(11) into (7) gives

V2(ui+ U)=VA-V x2L

(12)

(13)

(14)

Taking the divergence of (14) and noticing that from (9) V. V 2 andV 2V • are commutative, from (13) V u, = 0 and from vector analysis

V .V x2L = 0, one obtains

,2V.)= V2A (15)

A sufficient condition for the satisfaction of(15) is that

V "U, = A (16)

Taking the curl of (14) and noticing from (10) that V xV 2 and

V 2V x are commutative, from (11) that V x = 0, from vector analysisthat VxVA=0 and from (8) that

V 2X(Vx2 =V[V (7xu 2 I-V xVx(Vxu2 )-VxVx(V x(2) gives

- V XV x(Vx-= -V xV x 2 ý

A sufficient condition for the satisfaction of (17) is that

V 2=

(17)

(18)

132

[:.Billie Champ - nStanford-T..sis--- 9]


Of the two pairs of equations, (12) and (16), and (13) and (18), the first

pair contains u, as an unknown and the second pair contains u2 as anunknown. (12) is satisfied if

u, = V (19)

and it follows from (16)

(20)

Equation (13) is satisfied if

u2= V xA

and it follows from (18) that

(21)

V x(V xA)= 2L (22)

Simplifying the left-hand side of (22) by means of (8)Vx(VXA)V(VA)_V2A and choosing

V "A=0 (23)

equation (22) becomes

V 2 A= - 26i (24)

Substituting (19) and (21) back into (11) completes the proof of

ru=Vp+ VXA

satisfying (7) which represents (5) and (6) provided that V 2¢ =A andI I IV 2 A=-2f2 with V'A=0.

133

Bille Champ - Stanford Thfiesis•..df Pg.I

Page 144 ý1..... ... ..... .I I .. I

REFERENCES

1. Fluid-Solid Interaction, edited by J.E. Greenspon, ASME,' NewYork, 1967.

2. Anliker, M. and K.R. Raman, "Korotkoff Sounds at Diastole - APhenomenon of Dynamic Instability of Fluid-Filled Shells,"International Journal of Solids and Structures, Vol. 2, No. 3, pp.467-491, Pergamon Press, 1966.

3. Anliker, M. and J.A. Maxwell, "The Dispersion of Waves in BloodVessels," in Biomechanics Symposium, edited by Y.C. Fung,ASME, New York, 1966.

4. Jones, E., I-Dee Chang and M. Anliker, "Effects of Viscosity andExternal Constraints on Wave Transmission in Blood Vessels,"SUDAAR Report No. 344, Department of Aeronautics andAstronautics, Stanford University, Stanford, California, 1968.

5. Anliker, M., M.K. Wells and E. Ogden, "The TransmissionCharacteristics of Large and Small Pressure Waves in theAbdominal Vena Cava," SUDAAR Report No. 362, Department ofAeronautics and Astronautics, Stanford University, Stanford,California, 1968.

6. Tuffias, H.R. and M. Anliker, "Dynamic Behavior of Eye Globes,"SUDAAR Report No. 302, Department of Aeronautics andAstronautics, Stanford University, Stanford, California, 1967.

7. Biomechanics Symposium, edited by Y.C. Fung, ASME, NewYork, 1966.

8. Fung, Y.C., "Biomechanics, Its Scope, History and Some Problemsof Continuum Mechanics in P hysiology," Journal of AppliedMechanics Reviews, Vol. 21, No. 1, pp. 1-20, 1968.

134

I Billie Champ - Sta'nford'Theisis.pdf Pabe 145 1I i I h a p t n o d Teis... .. .. . . ..... ....... .... ... ..... ..... .... . . ........... . . . . . .P a ge.. . . .. .. .. . . . .... .. . . . . . . .. 1 5 I


9. Prager, W., Introduction to Mech anics of Continua, Ginn and Co.,1961.

10. Scipio, L.A., Principles of Continua with Applications, John Wiley& Sons, 1967.

11. Jaunzemis, W., Continuum Mechanics, The MacMillan Co., 1967.

12. Eringen, A.C., Mechanics of Continua, John Wiley & Sons, 1967.

13. Truesdell, C., "The Mechanical Foundations of Elasticity and FluidDynamics," J. Rational Mechanics and Analysis, Vol. 1, pp. 125-300, 1952.

14. Schlichting, H., Boundary Layer Theory, translated by J. Kestin,McGraw-Hill, 1960.

15. Eckart, C., "Vortices and Streams Caused by Sound Waves,"Physical Review, Vol. 73, No. 1, pp. 68-76, 1948.

16. Truesdell, C., "Precise Theory of the Absorption and Dispersion ofForced Plane Infinitesimal Waves According to the Navier-StokesEquations," Journal of Rational Mechanical Mechanics andAnalysis, Vol. 2, pp. 643-741, 1953.

17. Hunt, F.V., "Propagation of Sounds in Fluids," American Instituteof Physics Handbook, 2nd edition, McGraw-Hill, 1963.

18. Fluid Mechanics and Singular Perturbations, A Collection ofPapers by Saul Kaplun, edited by P.A. Lagerstrom, L.N. Howard,and C-S. Liu, Academic Press, 1967.

19. Van Dyke, M., Perturbation Methods in Fluid Mechanics,Academic Press, 1964.

135

1: Billie Chamip - Stanford Thes -is.pdf Page 146.1B C ham p. St n o . ... P. ... ..... ..... ... ..... ...... ...........

Po Kee Wong

20. Cole, J.D., Perturbation Methods in Applied Mathematics, Ginn-Blaisdell, 1968.

21. Chang, I-D., "Navier-Stokes Solutions at Large Distances from aFinite Body,"J. Math. Mech., Vol. 10, pp. 811-876, 1961.

22. Wu, T.Y.T., "Small Perturbation in the Unsteady Flow of aCompressible, Viscous and Heat-Conducting Fluid," Journal ofMath. and Phys., Vol. 35, pp. 13-27, 1956.

23. Deresiewicz, H., "Solution of the Equations of Thermoelastic ity,"Proceedings of the Third U.S. National Congress of AppliedMechanics, pp. 287-291, 1958.

24. Wong, P.K., "On the Unified General Solutions of Linear WaveMotions of Thermoelastod ynamics and Hydrodynamics withPractical Examples," Transactions ASME, Journal of AppliedMechanics, pp. 879-887, 1967.

25. Friedman, B., Principles and Techniques of Applied Mathematics,John Wiley & Sons, 1956.

26. Lagerstrom, P.A., J.D. Cole and L. Trilling, "Problems in theTheory of Viscous Compressible Fluids," Office of NavalResearch, GALCIT Report, 1949.

27. Sternberg, E., "On the Integration of the Equations of Motion inthe Classical Theory of Elasticity," Archiv for Rational Mech. andAnal., Vol. 6, pp. 34-50, 1960.

28. Sternberg, E. and M.E. Gurtin, "On the Completeness of CertainStress Functions in the Linear Theory of Elasticity," Proceedingsof the Fourth U.S. National Congress of Applied Mechanics, Vol.2, pp. 791-793, 1962.

136

I Billie Champ - Stanford Thesis.pdf Pag'e141]l... .. ..Billie....... .. ......Ch a m p........... .. .......Sta n fo rd.......... . ..... .. .. . . ... . .. . .. . .. ...... .. . . .......... P a... .... e..... 1... . ...7


29. Moon, P. and D.E. Spencer, Field Theory Handbook, Springer-Verlag, Berlin, 1961.

30. Wilcox, C.H., "Debye Potentials," Journal of Mathematics andMechanics, Vol. 6, No. 2, pp. 167-201, 1957.

31. Morse, P.M. and H. Feshbach, Methods of Theoretical Physics,Chapter 13, McGraw-Hill, 1953.

32. Wong, P.K., J. Miklowitz and R.A. Scott, "Harmonic WavePropagation in Elastic Rods of Elliptical Cross Section," Office ofNaval Research Tech. Report No. 2, Contract Nonr-220(57), Proj.No. NR-064-487, Nov. 1965.

33. Pochhammer, L., "Ueber die Fortpflanzungs geschwindigkeite nSchwingungen in einem Unbegrenzten Isotropen Kreiszylindcr," J.fiir Math., Vol. 81, pp. 324-336, 1876.

34. Love, A.E.H., A Treatise on the Mathematical Theory ofElasticity, Cambridge University Press, Fourth Edition, 1927.

35. Mindlin, R.D. and E.A. Fox, "Vibration and Waves in Elastic Barsof Rectangular Cross Section," Journal of Applied Mechanics, Vol.27, pp. 152-158, 1960.

36. Mindlin, R.D., "An Introduction to the Mathematical Theory ofVibrations of Elastic Plates," Monograph, U.S. Army Signal CorpsEngineering Laboratories, Ft. Monmouth, N.J., Signal CorpsContract DA-36-039, Sc.-56772, 1955.

37. Miklowitz, J., "On Solving Elastic Waveguide Problems InvolvingNon-mixed Edge Conditions," Office of Naval Research Tech.Report No. 5, Contract Nonr-220(57), Proj. No. NR-064-487,August 1967.

137

I Billie Champ - Sianfo'rd Thes-isi.ýdf .Page 148 1Bi llie .C..p.....n... .. ... ... ... ... . . .P age.. ..... ... 148.. ..

Po Kee Wong

38. Miklowitz, J., "Pulse Propagation in an Elastic Plate with a Built-in.Edge," Office of Naval Research Tech. Report No. 7, ContractNonr-220 (57), Proj. No. NR-064-487, May 1968.

39. Wong, P.K., "Propagation of Harmonic Waves in an Elastic Rod ofElliptical Cross Section." Thesis, California Institute ofTechnology, September 1965.

40. Wong, P.K., "Linear Wave Motions of Viscous Fluids in ElasticMedia of Circular Cylindrical Boundaries," Lockheed Palo AltoResearch Laboratory Report, May 1968.

41. Flhigge, W., Viscoelasticity, Blaisdell Publishing Co., 1967.

42. Bland, D.R., The Theory of Linear Viscoelasticity, PergamonPress, 1960.

43. Corneliussen, A.H. and E.H. Lee, "Stress Distribution Analysis forLinear Viscoelastic Materials," in Creep in Structures, edited byN.J. Hoff, IUTAM Colloquium, Stanford, California, 1960.

44. Lee, E.H., "Viscoelastic Stress Analysis. Structure Mechanics," inProceedings of the First Symposium on Naval StructuralMechanics, Stanford, California, 1958, edited by J.N. Goodier andN.J. Hoff, Pergamon Press, 1960, pp. 456-482.

45. Lee, E.H., "Stress Analysis in Viscoelastic Materials," J. Appl.Phys., Vol. 27, pp. 655-672, 1956.

46. Lamb, H., Hydrodynamics, Sixth Edition, Dover Publications,1945.

47. Rayleigh, J.W.S., The Theory of Sound, Second Edition, DoverPublications, 1945.

138

Bille.Camp. .S.n.od..es. .. ..Pag_14ji Billie Champ' - Stanford Th .es .. is. .pdf .Paage.1.491.;


48. Abramowitz, M., and I.A. Stegun, Handbook of MathematicalFunctions, Dover Publications, 1965.

49. Baker, W.E., "Axisymmetric Modes of Vibration of Thin SphericalShells," Journal of the Acoustical Society of America, Vol. 33, No.121, pp. 1749-1759, 1961.

50. Naghdi, P.M. and A. Kalnins, "On Vibrations of Elastic SphericalShells," Journal of Applied Mechanics, Vol. 29, pp. 65-72, 1962.

139

I Billie Champ- Stanford Thesis.pdf Page ...e 5 I

Billie Cham p -. S ta.nford hesis.pdf - p. .6..1.l 1

Billie Champ -.Stanford t-hesls.pdf Pag 152.

Billie Champ - FW 11 websites where you ca n obtain -my- pa tents.htm_;i ternati•.na conferen c.e p, ap ers and releveant e ngi nediRage.1

From: Po Kee wong [[email protected]]Sent: Saturday, September 30, 2006 2:42 PMTo: '????'Cc: '[email protected]'; '[email protected]'Subject: FW: 1I websites where you can obtain my patents; international conference papers andrelevant engineering and scientific documents for open review; evaluation and discussionworldwide

Dear Chairman Sheon:

The following websites are consolidated together to facilitate for your filing and tracing of my workthat may be of your interest to collect and edit and put them into the file for your book " History of Nan TaoHigh School ".

While typing the cc of this E-mail to you, President George Bush's E-mail address pops up. Thisgives me an idea to ask you whether you want also to invite President Bush of USA and President Hu JinTao to come to our Centennial Celebration. They can meet and talk informally about the proposedprojects of collaborations between two governments to build: (1) A 3 rd. identical observatory and; (2) Anew satellite ejection station in the "High Plateau Region of China"

Please call me at + 301-585-3453 to discuss about this idea further if you think this is feasibleand mutually beneficial to all of us being involved. Please also respond this E-mail with acknowledgementthat you have already received all the E-mails of information having sent to you so far.

With my best regards to you all, I am,

Very truly yours,

WONG, Po Kee (1946-1949 Alumni)

Tel: + 301-585-3453

pokwonpqaverizon. net

iookwonpta-rcn.com

iBl!ie Champ - FW 11 websites where you cangobtain my patents.ht m; international conference papers and relevant enginee!age.2

From: pokwong [mailto:[email protected]]Sent: Friday, September 29, 2006 10:50 PMTo: [email protected]; pokwongSubject:

(1) Search for Po Kee Wong Angles:http://www. google.com/search?q=+site: mathforum.org+Po+Kee+Wong+Angles&hl=en&lr=&ie=UTF-8&filter=0

You and all members of CSTB are cordially invited to review and evaluate the TRUTH and theIMPACTS from the following websites about the confrontation between Judges and Mathematiciansworldwide on the key issue of whether the Judges should be given the Judicial Power by anygovernmental institutions in the world to rule against the ABSOLUTE TRUTH of MATHEMATICS:

(2) The following infonnation was the first one of the several public documents to have been submitted back toCAFC for the public hearing of the CAFC case number 2006-1324:

http://math forum.org/kb/thread.J spa? forumID=206&threadID=478694& messageID= 1467123

(3) The following infonnation was the second one of the several public documents to have been submitted back tothe CAFC for the public hearing of the CAFC case number 2006-1324 IN RE PO KEE WONG:

http://mathforum.org/kb/message.ispa?messagelD=1 094424&tstart=0

(4) The following was the third of the several public documents to have been submitted back to the CAFC for thepublic hearing of the CAFC case number 2006-1324 IN RE PO KEE WONG:

http://mathforum.org/kb/thread.isspa?forumlD= 1 30&threadlD=357907&messagelD= 1094423

(5) The following was the fourth (4th.) of several public documents that had been submitted back to the CAFC forthe Judges to re-consider their current disposition of the CAFC case number 2006-1324 IN RE PO KEE WONG:

http://mathforum.org/kb/message.ispa?mnessagelD=3566959&tstart=0

(6) Comparison of Trajectory Solid with Geometric Solid in Sacttering Theory:

htto://adsabs.harvard.edu/abs/2002iaf. .confE.559W

(7) U.S. Patent 5084232:http://patft.uspto.gov/netacgri/nph-Parser? Sect 1 =PTO 1 &Sect2=HITOFF&d=PALL&p= 1 &u=%2Fnetahtml%2FPTO%2Fsrchnumn.h

SBillie Champ -_Fw- 11 elbsites" w~ere you-canobtainmy p~atents.htm; international conference papers and relevant engineeage.3.

tirn&,r- I &f--G&I =5 0&s 1 =5 0 8423 2. PN. &OS=PN/5 08 423 2&RS=PN/5 08 423 2

(8) U.S. Patent 5848377httn://oatft.ustto.Lyov/netaczi/nth-Parser?Sect 1 =PTO I &Sect2=HITOFF&d=PALL&p= I &u=%2Fnetahtml%2FPTO%2Fsrchnum.htrn&r- I &f=G&1=50&s 1 =5848377.PN.&OS=PN/5848377&,RS=PN/5848377

(9) U.S. Patent 6430516httn://natft.usnto. gov/netacgi/nnih-Parser? Sect I =PTO 1 &Sect2=HITOFF&d=PALL&p= 1 &u=%2Fnetahtrnl%2FPTO%2Fsrchnum.htmn&r 1 &f-G&1=50&s 1 =6430516.PN.&OS-PN/6430516&RS=PN/6430516

(10) Mathforum Searchhttp://mathforum.org/kb/profile.isspa?userID=47317

(11) ICONE 13 paper 50509http://www.conferencetoolbox.org/ICONE13/Author/PaperDetails.cfin

Documents

OFFICE TtIE CORRESPONDENCE Date Printed: Aug …OFFICE OF TtIE SECRETARY CORRESPONDENCE CONTROL TICKET Date Printed: Aug 28, 2007 12:16 PAPER NUMBER: ACTION OFFICE: LTR-07-0565 EDO