ME 563-Fall 2020

Homework No. 1

Due: September 4, 2020 11:59 pm on Gradescope

ME 563 -Fall 2020 Homework Problem 1.1 Consider the one-degree-of-freedom system shown below attached to springs of stiffnesses k1, k2, k3, k4, k5, and k6, respectively.

a) Draw a FBD of the mass and each spring.

b) Determine the equivalent spring stiffness of the six spring system.

c) Determine the equilibrium position of the system, xst.

F t bY p3

Kids In

Ftp k3

ka EgMar Iv FzF f I t

b FyA

Sgk4

E I t Fp p

b

Ks ke

E t trp p

Springs K and K2 are in series

4k z 4k t 4k 142 14122141 Kc

Springs Ks and Koo are in parallelKs6 Ks T Ks

Springs 142 and Ks are in parallel1423 Kcet Kz Kike 1Kz Kitko

kit K2 Kike

1423 K Ke t K kg t Kzk314 1 K2

Springs Ksa are Ky are in series

4456 4kg6 t YK4 1456 Ks6K4_Kg t Ks6

Kygo kicks t Kakoka t Ks tko

Springs 1423 and Kisco are in parallel1423456 1423 t K4S6

KcKz 1 14kg tKells K4KS 1K4K6t

Kit ka ka t Ks KG

key kikztkckztkzkdlkytkstkd l.lkLgtk4kdlkctKzK 1Ke Ky 1 Kst Ko

kegox

gA EFX mg c Request O

tix

Umg

NST Mfg Mg Kitty ka tKs tkokikztkikztkzkdlkxtkstkd ckykstkqkc.lk1Kz

ME 563 -Fall 2020 Homework Problem 1.2 Consider the three-degree-of-freedom system shown below made up of three particles A, B, and C, each of mass m, with system moving within a horizontal plane. Let x1, describe the absolute motion of particle A, x2 describe the motion of paticle B relative to A and x3 describe the motion of particle C relative to B. All the springs are unstretched when x1=x2=x3=0. Assume all surfaces to be smooth.

a) Draw individual free body diagrams of each particle.

b) Use the Newton- Euler formulation to derive three differential equations of motion for the system. Your final equations should not include any forces of reaction.

c) Write the equations of motion derived in b) in matrix form. Identify the mass, damping, and stiffness matrices in these equations.

cixf fCHL Kya MX

yKI NAD

TNceCalag gKala

NAB Kx o

cis INK MX

mn

kHzc smoniitxF atcrib Tapa moxa

Block A Fox Cd KX NAB mix DBlock B EFX NAB 1 KX t cis ME 2

b Efg caYz kHz 1Nrsc MIZ 3

Block C Sfx Kas cab F MCI Ij 4

I SFy Nrsc mn z s

1 T2

2mn49 Kastens Kx CNY

3 c Sn

2mn12 coke Rx

Rearranging

2mn40 t Ctx 432 t Ktx 43 O2m42 t Cox's 1kHz O

moi this cab c Kaz F

In matrix form

i.si isK O K 4

3

ME 563 -Fall 2020 Homework Problem 1.3 A double pendulum consists of two bobs of mass m1 and m2, suspended by inextensible, massless strings of length L1 and L2.

a) Draw individual free body diagrams of each particle.

b) Use Newton’s 2nd Law to derive four equations of motion for the rectangular coordinates x1, x2, y1,and y2.

c) Express x1, x2, y1,and y2 in terms of angles q1 and q2, eliminate tensions T1 and T2 in the strings and obtain two equations of motion for q1 and q2.

Draw FBD

fm'smy mix

ban

zoomurn

yea Imo gmisiHole mix

2Fox T sing Tess.noz mix 1

Efg 7T cosQ Tecos02 mg Mig 2

4 Fox Tzsin0z m iz 3

Efg fTzcosQz mg Miya 4

NowSino Nyc 0519 944

sin 5 Exc Xz

cosOz _cos 92

Al mix Tiny Teak neck2Al mij Iyya I cgz y.lk Mof3A mn TzCoxe X

z

9A Migos Telge orq mg

Constraints XF toy 2 Lfcoxa xD t ya 9,5 25

The EOM can be reduced to two dotsand the Tensions can be eliminatedHowever a better choice is to write theEom using the generalized coordinatesystems 0 02

Coordinate TransformationX L sin

ix 0,40501ix OIL cos0 01,24sin0

y L cos

Ey t 04sin

if L Sino 1104050

Hz sin 02 t G Sin Q

iz de cos e t01,4COSIOIna zlzcosQz 01zs.in0ztiQ4cos0 OfLisin0

yq LcosQ cos 02

Ep Of COSA t 0722205102

_0774cos0 t L sing EE cos z t since

Now plug into governing equations

1 mL cos10,90 mL sin 0 2 1 Tsing Tzsin z O2 ML sing mL cos 0 2 TicosQ T120502 1mg D3 ML cos0,8 sin Of tmLz cosmic sin028122 t

Tzsin 2 04 m4 sin 0 10510,02 mLz sin01201 c cos E t

Mg Tacos0 0

Recall we have the following trig identitiescos0i cos i 42cos 20 i 4,2Sinai sin i 42 cos 20 isi'nQi cos j sin i 10 TsinColiOj i j 1,2

2cos i cos j cos it j t coscoli j

2sin i sing coscoliOj cos Oi10J

2

Now 3 X cos02 t 4 x sin a

El mL cosCO 020 M 02 mL SinCQ 10102 tmgs.in0e O

This is the first equation of motionNow the second equation of motion is tricky151solve 3 for T2Te ML cscozs.inQOI2tmLzOE m4cosQcsc0zFQmLzcot0z z 3A

2nd solve 4 for T2Tz mgse.COz mLicosQsec0zOF mLz02

m4secQqsinQ tmcztan 020 4A

Plug 3A into 1 and simplifyingm 24 sin0,0 2 mlzs.in 0z Etm24cosQtMLscosQz0z t T sin 01 O 3B

Plug 38 into 2 and simplifyingm 24050,0

2m220502815 m 24sing

ofm sin01202 t 2mg TCOSIO D

Now 3A xcos t 3B x sin 0 yieldsF2 m 240 mLacos10 02702 tmglzss.nlQ QdOEcM2gsin0D

The final equations of motionEl ML cosCO 020 m ml sinCQ 0402 tongs'n e O

F2

ME 563 -Fall 2020 Homework Problem 1.4 Consider the system below: it consists of a drum of mass of m, and inner radius R, an outer radius 2R and a centroidal of inertia IO. A spring of stiffness k connects the center of the drum O to a fixed wall. Block A has a mass of m. The inner surface of the disk rolls without slipping on the ground at point C. The cable does not slip on the outer radius of the drum. A force F acts to the right at the center of the drum O. Let f be the rotation angle for the drum. The spring is unstretched when f=0.

a) Using the Newton-Euler formulation, determine the equations of motion for the system using the coordinate f. Draw the free body diagrams of the drum and block individually before writing down the Newton-Euler equations.

b) Using the power equation formulation, determin the equations of motion for the system using the coordinate f. Draw a free body diagram of the entire system before writing down the power equation.

Datum

kzIT s T

tooF

mix

pn

4T

tumor Iman

a

Drum A Emo KRK TFR TBR Ic i

Ic Iot M R2

Block the Efg T 1mg Mig 2

Kinematicsx R0 x R E RIONEWT Ifip 45 32 0 Xt

KR2 T FR T STR Io tmR7 DT 1mg 3mR 2

1 t 322KR2 d 1 FR 13mgR Io 110mW

RearrangeIo ccomic IO tkr 3mgR t Fk

Now using power

1 42 40 24 KonigU Mogg t k Kat

INE F DE FT dat Fdx

kinematics

y 3R x R dx rd0g 3RO

22Kym9h01F 42 ICO

U mg3RO tyzkR

02daIetdaIedWIde

date YzIc'd t 1km9h02 ctm9R

Io t10mn00date date f 3mgRIO tkkR0

3mgRI t kn

date C 3mg R t KR'd

dWWCTe

Fde

FRIO

otlomk.IO0F kR0 3mgR EFR07

IotlOMR tkR 3mgR the

same as before