Vibration Analysis Using MATLAB

8/13/2019 Vibration Analysis Using MATLAB

http://slidepdf.com/reader/full/vibration-analysis-using-matlab 1/26



c1

c2

c3

c4

c5

c6

c7

k1 = 200 N/mk2 = 200 N/mk3 = 300 N/mk4 = 300 N/mk5 = 300 N/mk6 = 400 N/mk7 = 400 N/mm 2 =20kgm 3 =30kg

m 4 =20kgm 5 =40kgm 6 =60kg

m 1 =10kgx

1x

2x

6x

5x

4x

3

D YNAMICS OF VIBRATING S YST MS S !MMARY

T"#$ %&'(%) "*$ $+cc&$$,+--. &m( $)%*)& )"& * *-.)#c*- &%# *)#( (, )"&

g( &% # g & +*)#( $ ,(% )"& # %*)#( (, * + ,(%c& 6D(F *m'& m&c"* #$m

+$# g *g%* g& $ & +*)#( $ MAT AB ® 8*$ $+cc&$$,+--. #m'-&m& )& # $& &%*-

# $)* c&$ T"& $.$)&m c"*%*c)&%#$)#c$ 8&%& ,(+ *m&-. )"& $#x *)+%*-

,%& +& c#&$ * m( *- $"*'&$ A c(m'%&"& $# & m( *- * *-.$#$ 8*$ '&%,(%m&

8#)" )8( c(9(% # *)& )%* $,(%m*)#( $ * '%('(%)#( *- *m'# g A '*%)#c+-*% +$&

,(% MAT AB 8*$ '&%,(%m# g )"& #m'+-$& %&$'( $& ,(% )8( $&'*%*)& ( 9'&%#( #c&xc#)*)#( ,(%c&$ *''-#& )( )"& $.$)&m O)"&% ).'&$ (, &xc#)*)#( 8&%& #$c+$$&

* ,# *--. $(m& (, )"& ,+ *m& )*- *$$+m')#( $ (, )"& $.$)&m 8&%& *-)&%& )(

( $&% & )"& &,,&c) ( *m'# g

1 I NTROD!CTION

I )"#$ %&'(%) )"& $.$)&m # ,#g+%& 1 8#)" $#x D&g%&&$ (, F%&& (m :D(F; 8#-- &

c( $# &%& T"& g( &% # g & +*)#( 8#-- & &%# & +$# g *g%* g#* m&c"* #c$

# $&c)#( 2 T"& #) 8#-- & $+ <&c) )( * +m &% (, #,,&%& ) * *-.$#$ '%(c& +%&$

T"& m*)%#x m&)"( 8#-- &)&%m# & )"& + *m'& *)+%*- ,%& +& c#&$ * m( *-

$"*'&$ # $&c)#( 3 A k# &m*)#c * *-.$#$ 8#-- & '&%,(%m& # $&c)#( 4 8#)"

m( *- * *-.$#$ )"*) +$&$ )8( c(9(% # *)& )%* $,(%m*)#( $ * '%('(%)#( *-

*m'# g T"#$ '%(c&$$ 8#-- ,# &$)#m*)&$ ,(% *m'# g %*)#($ *m'# g ,*c)(%$

* *m'& ,%& +& c#&$ A k# &)#c * *-.$#$ 8#-- )*k& '-*c& # $&c)#( 5 8"&%&

)8( $&'*%*)& '%&9 &)&%m# & ( 9'&%#( #c &xc#)*)#( ,(%c&$ 8#-- & *''-#& )( )"&

$.$)&m N+m&%#c*- * * *-.)#c*- '%(c& +%&$ *%& )"&%& . &x*m# & ,(% )"&#%

&,,&c)# & &$$ # '%& #c)# g )"& #m'+-$& %&$'( $& F# *--. # $&c)#( $ 6 * 7

*-)&% *)# & &xc#)*)#( ,+ c)#( $ * m&)"( $ ,(% m( &--# g *m'# g 8#-- &

#$c+$$&



Figure 1 – 6DoF Mass-spring-damper model with parameters for mass and springconstants.



2 S YST M !ATION

T"& * *-.$#$ #$ '%&$& )& ,(% * '+%&-. )%* $-*)#( *- $.$)&m :8"&%& ( %()*)#( #$

# (- & ; T"& *g%* g& & +*)#( ,(% + ,(%c& ($c#--*)#( 8#)" *m'# g #$ g# &

.>

)?T? #9?T? #@?V? #@?D? # = 0 (2.1)

"&%& T #$ )"& # &)#c & &%g. : ; V #$ )"& ()& )#*- & &%g. : ; D #$ )"& & &%g.

#$$#'*)#( +& )( *m'# g : ; # #$ )"& c(9(% # *)& $.$)&m # #$ )"& ,#%$)

&%# *)# & (, )"& c(9(% # *)& # 8#)" %&$'&c) )( )#m& :m$ 91;

T"& # &)#c & &%g. (, )"& $.$)&m g# & # ,#g+%& 1 c* & &x'%&$$&

T=12m1x12@12m2x22@12m3x32@12m4x42@12m5x52@12m6x62 (2.2)

"&%& m # #$ )"& %&$'&c)# & m*$$ ,(% &*c" (, )"& m*$$&$ :kg;

T"& '()& )#*- & &%g. c* *-$( & &x'%&$$& )"+$>

V=12k1x12@12k2:x29x1;2@12k3:x39x2;2@12k4:x49x3;2@12k5:x59

x4;2@12k6:x69x5;2 (2.3)

"&%& k # #$ )"& %&$'&c)# & $)#,, &$$ (, &*c" (, )"& $'%# g$ :Nm 91;

&%g. #$$#'*)#( +& )( *m'# g #$ &x'%&$$& >

D=12c1x12@12c2:x29x1;2@12c3:x39x2;2@12c4:x49x3;2@12c5:x59

x4;2@12c6:x69x5;2 (2. )

"&%& c # #$ )"& %&$'&c)# & *m'# g c(9&,,#c#& ) ,(% &*c" *m'&%

B. $+ $)#)+)# g # )( )"& *g%* g& & +*)#( ,(% # = 1

?T? 1=m1x1 )?T? 1=m1x1 ?V? 1=k1x19k2:x29x1; ?D? 1=c1x19c2:x29

x1;

!$# g )"& *g%* g& & +*)#( g# & # (2.1) )"& ,#%$) & +*)#( :,(% # = 1; #$

)"&%&,(%&>

m1x1@k1@k2x19k2x2@c1@c2x19c2x2=0 (2.!)



T"& $*m& '%(c& +%& m*. & c( )# +& ,(% # = 2>

?T? 2=m2x2 )?T? 2=m2x2 ?V? 2=k2:x29x1;9k3:x39x2;

?D? 1=c2:x29x1;9c3:x39x2;

A "& c& )"& & +*)#( ,(% # = 2 #$ g# & >

m2x29k2x1@k2@k3x29k3x39c2x1@c2@c3x29c3x3=0 (2.6)

S# c& )"& $.$)&m "*$ 6 D(F #) ,(--(8$ )"& $*m& '*))&% )"%(+g" )( # = 6 )"& $&)

(, *g%* g& & +*)#( $ c* (8 & '%&$& )& # m*)%#x ,(%m*)

m1000000m2000000m3000000m4000000m5000000m6x1x2x3x4x5x6@

:c1@c2;9c200009c2:c2@c3;9c300009c3:c3@c4;9c400009c4:c4@c5;9c500009c5:c5@c6;9c600009c6:c6@c7;x1x2x3x4x5x6@:k1@k2;9k200009k2:k2@k3;9

k300009k3:k3@k4;9k400009k4:k4@k5;9k500009k5:k5@k6;9k600009

k6:k6@k7;x1x2x3x4x5x6=0

(2.")

B. $+ $)#)+)# g )"& *''%('%#*)& *-+&$ ,%(m ,#g+%& 1 *g%* g& $ & +*)#( $ c*

& ,(+ ,(% )"#$ '*%)#c+-*% $.$)&m

100000002000000030000000200000004000000060x1x2x3x4x5x6@:c1@c2;9

c200009c2:c2@c3;9c300009c3:c3@c4;9c400009c4:c4@c5;9c500009c5:c5@c6;9

c600009c6:c6@c7;x1x2x3x4x5x6@4009200000092005009300000093006009

30000009300600930000009300700940000009400E00x1x2x3x4x5x6=0

(2.#)

+*)#( :2 E; c( c-+ &$ )"& &%# *)#( (, )"& g( &% # g & +*)#( ,(% )"#$

'*%)#c+-*% $.$)&m # m*)%#x ,(%m

3 S YST M C ARACT RISTICS

I (% &% )( ,# )"& *)+%*- ,%& +& c#&$ * m( *- $"*'&$ (, )"& $.$)&m )"&M*)%#x m&)"( 8*$ +$& c"($& ( &% )"& (- &% m&)"( &c*+$& #) #$ m(%&*''%('%#*)& ,(% c(m'+)*)#( *- * *-.$#$

A9H 2Ix=0 (3.1)

"&%&

A=JmK91JkK (3.2)



S(- # g )"& c"*%*c)&%#$)#c & +*)#( ,(% )"& &#g& *-+&$ (, A * )"& '&%,(%m# g* $ +*%& %(() ( )"& %&$+-)* ) m*)%#x g# &$ )"& *)+%*- ,%& +& c#&$ ,(% )"&$.$)&m T"& MAT AB c( & # ,#g+%& 2 8*$ +$& )( ( )"#$>

K = [400,-200,0,0,0,0;-200,500,-300,0,0,0;0,-300,600,-300,0,0;0,0,-

300,600,-300,0;0,0,0,-300,700,-400;0,0,0,0,-400,800]M =[10,0,0,0,0,0;0,20,0,0,0,0;0,0,30,0,0,0;0,0,0,20,0,0;0,0,0,0,40,0;0,0,0,0,0,60]% Mass and stiffness matrices are set u !" = in#$M &K% " is defined and t'en t'e ei(en#a)ues * ei(en#ect+rs are f+und in +rder% t+ s+)#e acc+rdin( t+ t'e matri met'+d![#,d] = ei($"nat fre. = s.rt$d% " )+t +f a)) t'e m+da) s'a es is f+rmed)+t$#ause;

% /ac' time t'e user resses t'e return e , t'e m+da) s'a es i)) a ear% in ascendin( +rder +f natura) fre.uenc !)+t$#$ ,5ause;)+t$#$ ,6ause;)+t$#$ ,4ause;)+t$#$ ,3ause;)+t$#$ ,2ause;)+t$#$ ,1

Figure 2 – M$%&$'®

code to sol e for the natural fre uencies and modal shapes



T"& c( & # ,#g+%& 2 '%( +c& )"& ,(--(8# g %&$+-)$ ,(% )"& *)+%*- ,%& +& c#&$

H =7 11670000006 45410000005 0L760000004 2E5E0000001 45140000002 6

5L1

T"& m( *- $"*'&$ m*. & '-())& ,%(m )"& c(%%&$'( # g &#g& &c)(%$ T"&.

*%& g# & ( ,#g+%& 3 c(m # & # ( & '-() * )"&. *%& $# g-& (+) # *$c& # g(% &% ,%(m )"& -(8&$) )( "#g"&$) ,%& +& c. # )"& $+ $& +& ) ,#g+%&$ 4 )( L#$'-*.& ( '*g& E

T"& . *m#c c"*%*c)&%#$)#c$ (, )"& $.$)&m *%& )"& '%('&%)#&$ +%# g #)$ *c)# &

* c"* g# g $)*)&$ T"&%&,(%& #$c+$$# g )"&$& # (- &$ ()# g )"& (% &% #

8"#c" )"& m*$$&$ * $)#,, &$$ &-&m& )$ ,(% )"*) c( $)#)+)& &*c" &g%&& (,

,%&& (m "#) )"&#% *)+%*- ,%& +& c. T"& c*+$& (, )"#$ #$ +& )( )"& $+%%(+ # g

'%('&%)#&$ (, )"& $.$)&m T"#$ # ,(%m*)#( #$ (% &%& * $+mm*%#$& # )* -& 1

N*)+%*- F%& +& c. :H J%* /$K;

M*$$ :m JkgK;

S+%%(+ # g $)#,, &$$ &-&m& )$ &,) )( %#g") :k JNm 91K;

1 4514 m 5 40 k 5 300 k 6 400

2 65L1 m 6 60 k 6 400 k 7 400

1. !1rad*s

2.6!+1rad*s

.2#!#rad*s

!.,+"6ra d*s

".116"rad*s

6. ! 1rad*s

F /0 3 - M D$& 4$5 . %4 $%/0$& F0 7/ 8 $0 8 & /0 8 D D $ 4 9 %4 & F% F %4 F /0 .



4 2E5E m 4 20 k 4 300 k 5 300

5 0L76 m 3 30 k 3 300 k 4 300

6 4541 m 2 20 k 2 200 k 3 3007 1167 m 1 10 k 1 200 k 2 200

%$'& 1 – $%/0$& F0 7/ 8 $ 8 D 0D 0: $&& 8$% D % $84 M$ $ D% FF & M %

T"& "&* #&% m*$$ * $)#,,&% $'%# g *%%* g&m& )$ )& )( "* & -(8&% *-+&$ ,(%

H (8& &% #) m*. & ( $&% & ,%(m )* -& 1 )"*) )"& -(8&$) *)+%*- ,%& +& c.

*c)+*--. (cc+%$ ,(% m 5 %*)"&% )"* m 6 ($$# -& c*+$&$ ,(% )"#$ *%& )"& *%#* c& #

$+%%(+ # g $)#,, &$$ &-&m& )$ ,(% m 5 8"&%&*$ m 6 "*$ )"& $*m& k *-+& ( ()"

$# &$ * *-$( )"& ,*c) )"*) m 5 "*$ $+%%(+ # g m*$$&$ ( ()" $# &$ 8"&%&*$

m 6 #$ *g*# $) * 8*-- ( ( & $# & C( &%$&-. -#g")&% m*$$&$ 8#)" -(8&% k *-+&$

"* & "#g"&% *)+%*- ,%& +& c#&$ T"#$ #$ * )%& )"*) -(($&-. m*)c"&$ ,%(m m 4 )(

m 1 T"& &xc&')#( (cc+%$ *) m 3 8"#c" #$ -(c*)& # )"& m# -& (, )"& $.$)&m

'%(m()# g * -(8&% *)+%*- ,%& +& c. F%(m )"& )* +-*)#( (, )"&$& %&$+-)$ #) #$

'($$# -& )( $&& )"*) )"& *)+%*- ,%& +& c. ,(% &*c" (, )"& $#x &g%&&$ (, ,%&& (m

#$ "#g"-. &'& & ) ( & &%. ()"&% &-&m& ) # )"& $.$)&m



F /0 – M D$& 4$5 F 0 1. !1 0$D* F /0 ! – M D$& 4$5 F 0 2.6!+1 0$D*

F /0 6 – M D$& 4$5 F 0 .2#!# 0$D* F /0 " – M D$& 4$5 F 0 !.,+"6 0$D*

F /0 # – M D$& 4$5 F 0 6. ! 1 0$D* F /0 + – M D$& 4$5 F 0 ".116" 0$D*



4 M ODA A NA YSIS

T"#$ #$ * k# &m*)#c * *-.$#$ 8"#c" *''%(*c"&$ )"& 6D(F $.$)&m *$$+m# g )"*)

,%&& *m'& # %*)#( (cc+%$ T"& *m'# g 8#-- & '%('(%)#( *- # (% &% ,(% )"&* *-.$#$ )( 8(%k 8#)" = 0 1 * = 0 2

T"& $.$)&m & +*)#( $ (, 2.# 8&%& g# & )"+$>

100000002000000030000000200000004000000060x1x2x3x4x5x6@:c1@c2;9c200009c2:c2@c3;9c300009c3:c3@c4;9c400009c4:c4@c5;9c500009c5:c5@c6;9c600009c6:c6@c7;x1x2x3x4x5x6@400920000009200500930000009300600930000009300600930000009300700940000009400E00x1x2x3x4x5x6=0

(2.# - repeated) T"& '%( -&m 8*$ $(- & # MAT AB +$# g )"& c( & g# & # ,#g+%& 10



% tiffness and mass matrices defined, as ef+reK = [400,-200,0,0,0,0;-200,500,-300,0,0,0;0,-300,600,-300,0,0;0,0,-300,600,-300,0;0,0,0,-300,700,-400;0,0,0,0,-400,800]M =[10,0,0,0,0,0;0,20,0,0,0,0;0,0,30,0,0,0;0,0,0,20,0,0;0,0,0,0,40,0;0,0,0,0,0,60]% r+ +rti+na) dam in( a )ied t+ t'e s stem t+ f+rm t'e dam in( matri = 0!1&M 0!2&K % 9e)+ - :sed t+ c'ec t'e +rder +f undam ed natura) fre.uencies, i))e needed )ater f+r t'e matri !" = in#$M &K[#,d] = ei($"nat fre. = s.rt$d % irst c+-+rdinate transf+rmati+n!!!% Mass-<+rma)ised tiffnessK ti)da = $M -0!5 &$K &$M -0!5 % Mass-<+rma)ised >am in(ti)da = $M -0!5 &$ &$M -0!5

% indin( ei(en#a)ues and ei(en#ect+rs f+r t'e n+rma)ised stiffnessmatri[#1,d1] = ei($K ti)da % ec+nd c+-+rdinate transf+rmati+n!!!% "rran(in( t'e ei(en#ect+rs int+ ascendin( +rder acc+rdin( t+ natura)% fre.uenc ! = [#1$ ,5 ,#1$ ,6 ,#1$ ,4 ,#1$ ,3 ,#1$ ,2 ,#1$ ,1 ] % >ia(+na) matrices f+r use it' t'e ?r? c+-+rdinates after t'e sec+nd% transf+rmati+n9i( @am da K = [ ]A&[K ti)da]&[ ]9i( @am da = [ ]A&[ ti)da]&[ ] % :sed f+r c+n#ersi+n fr+m r$t c+-+rdinates ac t+ $t c+-+rdinates[ ] = M -0!5&[ ]

F /0 1, – M$%&$' ® 8 D % 5 0F 0M ; < 5 0$% 0 7/ 0 D F 0 M D$&$ $&< .

T"& '%(c&$$ (, c(9(% # *)& )%* $,(%m*)#( * & & )+*--. ,# # g )"& &c&$$*%.'*%*m&)&%$ 8#-- & &)*#-& )"%(+g" &*c" $)&'

F#%$)-. *''%('%#*)& *-+&$ $"(+- & $+ $)#)+)& ,(% C ,(% )"& *m'# g m*)%#x

T"& '*%*m&)&%$ = 0 1 * = 0 2 *%& g# & * #) #$ k (8 )"*)>

C= M@ J K

T"&%&,(%&>

Mx@ M@ x@ x=0



S( )"& & +*)#( (, m()#( # )&%m$ (, Px:);Q ,(% 6D(F m*. & $)*)& )"+$>

100000002000000030000000200000004000000060x1x2x3x4x5x6@E194000009401029600000960123960000096012696000009601449E000009E0166x1x2x3x4x5x6@4009200000092005009300000093006009300000093006009

30000009300700940000009400E00x1x2x3x4x5x6=0

( .1)

I (% &% )( ,(%m * $.mm&)%#c &#g& *-+& '%( -&m )"& m*$$9 (%m*-#$& $)#,, &$$

* *m'# g * C m+$) & ,(+ >

=M912 JMK912

C=M912CJMK912

T"&$& m*)%#c&$ 8&%& c*-c+-*)& )"%(+g" )"& MAT AB c*-c+-*)#( $ $"(8 # ,#g+%&10>

=4009141 4214000092E2 E4275009244 L4L000009367 42356009367 423500009244 L4L06009212 132000009424 26417009326 5LE6000094EL EL7LE00

C=E392L 6LE5000095L 3L70105951 43L30000977 15EL125975 L3420000950 622E126945 254E00009L0 50L7150970 21E700009105 32E1166

T( ,(%m )"& ,#%$) c(9(% # *)& )%* $,(%m*)#( * C c* & &x'%&$$& # )&%m$

(, )"& &8 c(9(% # *)& $.$)&m P :);Q

I #@:0 1I@0 2 ; #@ #=0

A )"&%&,(%&>

I #@C #@ #=0

S+ $)#)+)# g )"& m*$$9 (%m*-#$& * $)#,, &$$ * *m'# g m*)%#c&$ & +*)#(( .2) #$ ,(%m& >

100000010000001000000100000010000001 1 2 3 4 5 6@E392L 6LE5000095L 3L70105951 43L30000977 15EL125975 L3420000950 622E126945 254E00009L0 50L7150970 21E700009105 32E1166 1 2 3 4 5 6@4009141 4214000092E2 E4275009244 L4L000009367 42356009367 423500009244 L4L06009212 132000009424 26417009326 5LE6000094EL EL7LE00 1 2 3 4 5 6=0

( .2)

T"#$ c(m'-&)&$ )"& ,#%$) c(9(% # *)& )%* $,(%m*)#(



T"& &#g& *-+&$ * &#g& &c)(%$ ,(% 8&%& ,(+ 8#)" MAT AB J K #$ &,# &

*$ )"& (%m*-#$& m*)%#x (, $)#,, &$$ &#g& &c)(%$ # *$c& # g (% &% (, *)+%*-

,%& +& c.

=0 143690 1EE00 2E260 40L290 3L4390 73570 3E4E90 437L0 43230 40550 04610 553L0 553490 423E90 0L2290 50510 3L2590 31040 423E90 00L690 444690 15E790 74020 222E0 473E0 46E790 3E110 52320 360190 07540 34570 61110 61E090 337790 103E0 0165

* C 8#-- & +$& # )"& & +*)#( # )&%m$ (, P%:);Q )"& %&$+-) (, )"& $&c(

c(9(% # *)& )%* $,(%m*)#( T"&. '%( +c& (% &%& *-+&$ )"*) %&-*)& )( *)+%*-

,%& +& c. * *m'# g *$ $"(8 &-(8

=J KT =H 12000000H 22000000H 32000000H 42000000H 52000000H62

C=J KTC =2 H10000002 H20000002 H30000002 H40000002 H50000002 H6

T"& MAT AB c*-c+-*)#( $ '%( +c& )"& ,(--(8# g %&$+-)$ ,(% * C>

=2 10670000007 07100000001E 367700000025 LE5400000041 655200000050 6474

C=0 52130000001 51420000003 77350000005 2L71000000E 43100000001022L5

T( c(m'-&)& )"& $&c( c(9(% # *)& )%* $,(%m*)#( * C *%& $+ $)#)+)&# )( )"& $.$)&m & +*)#( &x'%&$$& # )&%m$ (, P%:);Q T"#$ '%( # &$ )"& ,(%m*)

,(% &c(+'-& & +*)#( $ )( & ,(%m& R*)"&% )"* &x'%&$$# g )"& $.$)&m *$ *

$# g-& 6D(F & +*)#( *$ &,(%& #) m*. (8 & $)*)& # )&%m$ (, 6 $&'*%*)&

$# g-& D(F :$D(F; & +*)#( $

I%#@ C%#@ %#=0



100000010000001000000100000010000001%1%2%3%4%5%6@2 H10000002 H20000002 H30000002 H40000002 H50000002 H6%1%2%3%4%5%6@H 12000000H 22000000H 32000000H 42000000H 52000000H 62%1%2%3%4%5%6=0

S+ $)#)+)# g # *-+&$ ,(% * C g# &$ ( .3) >

100000010000001000000100000010000001%1%2%3%4%5%6@0 52130000001 51420000003 77350000005 2L71000000E 431000000010 22L5%1%2%3%4%5%6@2 10670000007 07100000001E 367700000025 LE5400000041 655200000050 6474%1%2%3%4%5%6=0

( .3)

T"& & +# *-& ) 6 $D(F & +*)#( $ *%&>

%1@0 5213%1@2 1067%1=0

%2@1 5142%2@7 0710%2=0

%3@3 7735%3@1E 3677%3=0

%4@5 2L71%4@25 LE54%4=0

%5@E 4310%5@41 6552%5=0

%6@10 22L5%6@50 6474%6=0

T( ( )*# )"& ,# *- $(-+)#( ,(% xP)Q $#m'-.>

x)=JMK912 P%)Q=JSKP%)Q ( . )

"&%&>

S=JMK912



I (% &% )( )* +-*)& )"& %&$+-)$ ,(% &*c" D(F &*c" (, )"& $D(F & +*)#( $ ,(+ #

)&%m$ (, P%:);Q m+$) & * %&$$& :& +*)#( ( .3) ; T"&%&,(%& )"& *)+%*-

,%& +& c#&$ *%& c*-c+-*)& ,%(m )"& *-+&$ (, 8"&%& &*c" )&%m #$ & +*- )(H #2 >

H 1=2 1067=1 4514 %* /$

H 2=7 071=2 65L1 %* /$

H 3=1E 3677=4 2E5E %* /$

H 4=25 LE54=5 0L76 %* /$

H 5=41 6552=6 4541 %* /$

H 6=50 6474=7 1167 %* /$

T"& *m'# g ,*c)(%$ *%& ,(+ #%&c)-. ,%(m &*c" *-+& (, C T"& *m'# g

%*)#( $ *%& ,(+ ,%(m )"& &*c" *-+& (, C 8"&%& &*c" )&%m #$ & +*- )( 2 H#>

1=0 5213:2 2 1067;=0 17L6 N$/m

2=1 5142:2 7 071;=0 2E47 N$/m

3=3 7735:2 1E 3677;=0 4402 N$/m4=5 2L71:2 25 LE54;=0 51L6 N$/m

5=E 4310:2 41 6552;=0 6532 N$/m

6=10 22L5:2 50 6474;=0 71E7 N$/m

F# *--. *m'& ,%& +& c#&$ :H ; *%& ,(+ ,%(m & +*)#( ( .!).

H =H 19 #2 ( .!)



N*)+%*-,%& +& c#&$

:%* /$;

D*m'# g %*)#( :N$/m;

D*m'# g F*c)(%:N$/m;

D*m'&,%& +& c#&$

:%* /$;

1 4514 0 17L6 0 5213 1 427L

2 65L1 0 2E47 1 5142 2 54L1

4 2E5E 0 4402 3 7735 3 E4E1

5 0L76 0 51L6 5 2L71 4 3555

6 4541 0 6532 E 4310 4 EE72

7 1167 0 71E7 10 22L5 4 L4E4

%$'& 2= 8 M5$0 ' %9 5$0$M % 0 '%$ D %40 / 4 M D$& $ $&< .

T* -& 2 $"(8$ )"& )%& &)8&& '*%*m&)&%$ ( )*# & )"%(+g"

F%(m )* -& 2 #) m*. & ( $&% & )"*) *m'# g %*)#( # c%&*$&$ 8#)" )"& *)+%*-,%& +& c. T"#$ #$ # )+#)# & &c*+$& 8#)" * "#g"&% ($c#--*)# g ,%& +& c. #) #$ -#k&-.)"*) m(%& *m'# g (cc+%$ T"& *m'& ,%& +& c#&$ # c%&*$& 8#)" *)+%*-,%& +& c. +) () *) * -# &*% %*)& *$ $"(8 . g%*'" 1 T"& %&-*)#( $"#'&)8&& )"&m #$ *c)+*--. c( &%g& ) &c*+$& *m'# g ,*c)(% # c%&*$&$ )( * -#m#)(, 1 T"&%&,(%& *) "#g"&% *-+&$ )"& *m'& ,%& +& c#&$ *%& $#g #,#c* )-. -(8&%)"* )"& *)+%*- ,%& +& c#&$

0$54 1= 8 M5$0 ' %9 $%/0$& F0 7/ 8 $ D D$M5 D F0 7/ 8$80 D$M5 0$% .



5 N ON9 RIODIC UCITATION

5 1 I M ! S F ORC :F ORC 1 M; T"#$ $&c)#( '&%,(%m$ * k# &)#c * *-.$#$ ( )"& $.$)&m $"(8 # ,#g+%& 1 ,(% *#m'+-$& ,(%c& *c)# g $'&c#,#c*--. ( m 1 T"& #m'+-$& ,(%c& #$ &,# & *$ F

F=F) ) (!.1)

"& c( $# &%# g * + #) #m'+-$&>

F=,WX W (!.2)

T"& c( (-+)#( # )&g%*- c* & ,(%m& >

x)=0),W")9W W (!.3)

T"& %&c)* g+-*% '+-$& c* & c( $# &%& * *-.)#c*--. *$ )"& $+m (, )8( $)&',+ c)#( $ 8"&%& F:); = F 0 ,(% * *m'& $.$)&m

x1)=F0k:19c($H ); ) )1 (!. )

x2)=F0kJ19c($H )9)1K )=)1 (!.!)

T"& $+m (, 8"#c" #$>

x)=F0kJ9c($H )@c($H )9)1K )Z)1 (!.6)F(% )"#$ $.$)&m )"& *-+&$ *%&>

x)=1200J9c($:7 1167);@c($:7 1167)9)1;K (!.")

T"& (%#g# *- &xc#)*)#( ,#-&$ ,(% +$& 8#)" MAT AB *%& g# & # A''& #x 2 T(*''-. )"&$& ,(%c&$ )( )"& '*%)#c+-*% $.$)&m $"(8 # ,#g+%& 1 * #)#( *- $&c)#( $(, c( & 8&%& * & )( '&%,(%m )"& )%* $,&% ,+ c)#( #m'+-$& %&$'( $& *c( (-+)#( # )&g%*- I ,#g+%&$ 11A * 12A )"& (%#g# *- $&c)#( $ *%& $"* & (+)# g%&. * )"& * & $&c)#( $ *%& # -*ck

T"& '*%*m&)&%$ ,(% )"& )%* $,&% ,+ c)#( 8&%& ,(%m& )"+$>

G\$=U:$;F:$;=1m$2@2 H $@H 2

F(% m 1> 1m=0 1 2 H 6=10 22L5 * H 62=50 6474

T"& #m'+-$& $(-+)#( #$ g# & ( )"& &x) '*g& +$# g * -# &*% $#m+-*)#( *c( (-+)#( # )&g%*- T"& '-()$ *%& g# & # (% &% (, )"& c(%%&$'( # g $(-+)#( $()& . )"& c( &

dt=0!1;



t=0 dt 50;f=[Ber+s$50,1 ;+nes$100,1 ;Ber+s$)en(t'$t -150,1 ];su )+t$311 ;)+t$t,f ;tit)e$A .uare in utA ;a is$[0 50 0 1!5] ;

ause; %<umerica) s+)uti+n C im u)se res +nses s=tf$0!1,[1 10!22D5 50!6474] ; % usin( m1 = 10 ( and #a)ues fr+m sect 4% in t'e stiffness and dam in( matrices in terms +f r$t f+r nat fre. E6'=im u)se$s s,t ;s+) 1=)sim$s s,f,t ; %)inear simu)ati+ns+) 2=c+n#$f,' &dt; %c+n#+)uti+nsu )+t$312 ;)+t$t,s+) 1$1 )en(t'$t ,t,s+) 2$1 )en(t'$t ;tit)e$ AFm u)se res +nse +f s stemA ;)a e)$ AGimeA ;)a e)$ AHut utA ;

ause; %"ttem ted ana) tica) #a)idati+n C ste res +nse=0!005!&$$-c+s$7!1167&t $c+s$7!1167!&$t-$t-0!1 %:sin( <at! re.E6!su )+t$313 ;)+t$t,tit)e$ A stem Ies +nse t+ a ste in utA ;)a e)$ AGimeA ;)a e)$ A $t A ;

F /0 11$ – M$%&$' ® 8 D F 0 %4 -5 0 D 8 >8 %$% F 08 ?1.M $55& D% m 1 %4 6D F < % M.



T"& #m'+-$& %&$'( $& $"(8& * %&$&m -* c& )( )"& %&c)* g+-*% # '+) +$& )(

&xc#)& m1 T"& $.$)&m %&*c)$ . *m#c*--. ,(% )"& +%*)#( (, )"& #m'+-$& *

)"& )"& ,(%c& #&$ *8*. ( &% )#m& +& )( )"& $.$)&m *m'# g T"& * *-.)#c*-

%&$+-) ,(% $)&' # '+) # () '%( +c& * %&$+-) *$ &x'&c)& *$ #) #$ * c( )# +(+$

,+ c)#( %*)"&% )"* * $(-+)#( )( )"& $)&' # '+) T"#$ #$ +& )( * + k (8 ,-*8

# )"& & +*)#( '%&$& )& # )"& c( &

5 2 A F 9SIN ! S UCITATION F ORC :F ORC 2 M;I )"#$ c*$& * "*-,9$# & '+-$& 8*$ *''-#& )( m 1 ( )"& $.$)&m g# & # ,#g+%& 1

T"& * *-.)#c*- $(-+)#( # )"#$ c*$& #$ g# & .>

x)=F0k19&9 H )19 2$# :H )9]; (!.#)

"&%&>

]=)* 91 19 2

S)*)# g (!.#) 8#)" *-+&$ %&-& * ) )( )"#$ '%( -&m>

x)=120019&95 114E)190 71E72$# :4 L4E4)90 E02; (!.+)

F /0 11' – M5/& 0 5 5& % F 0 F 08 ?1.M $55& D % m 1 %46D F < % M



T"& $(-+)#( +$# g MAT AB c( & ,(% * -# &*% $#m+-*)#( * c( (-+)#( # )&g%*-#$ g# & ( )"& ,(--(8# g '*g&

T"& '-()$ *%& g# & # (% &% (, )"& c(%%&$'( # g $(-+)#( $ ()& . )"& c( &



>G=0!1;t=0 >G 50; %Gime $'i(' res+)uti+ns s=tf$0!1,[1 0!5213 2!1067] ;f=$sin$t !&t 0!1 !&$tJ8!1 ;su )+t$311 ;)+t$t,f,A A ;)a e)$AGimeA ;)a e)$AFn utA ;tit)e$AFn ut #s GimeA ;ause; % Fm u)se res +nse $ e)+'=im u)se$s s,t ;s+) 1=)sim$s s,f,t ; %)inear simu)ati+ns+) 2=c+n#$f,' &dt; %c+n#+)uti+nsu )+t$312 ;)+t$t,s+) 1$1 )en(t'$t ,t,s+) 2$1 )en(t'$t

tit)e$ AFm u)se res +nse +f s stemA ;)a e)$ AGimeA ;)a e)$ AHut utA ;ause; % "ttem ted "na) tica) #a)idati+n - ste res +nse $ e)+ = 0!2!&$1-$e $-5!1148&t s.rt$1-0!7187 2 !&$sin$4!D484&t-$atan$0!7187 s.rt$1-0!7187 2su )+t$313 ;)+t$t,tit)e$ A stem Ies +nse t+ a ste in utA ;)a e)$ AGimeA ;)a e)$ A $t A ;

a is$[0 5 0 0!5] ;F /0 12$ – M$%&$' ® 8 D F 0 %4 -5 0 D 8 >8 %$% F 08 ?2.M $55& D% m 1 %4 6D F < % M

F /0 12' – 0 5 5& % F 0 F 08 ?2.M $55& D % m 1 %4 6D F< % M



T"& $.$)&m %&$'( $& *''&*%$ )( & *-# ,(% )"& c( (-+)#( # )&g%*- *''%(*c"

:#m'+-$& %&$'( $& m# -& '-(); &%& )"& # %*)#( $ ,(--(8 )"& )%& (, )"& # '+)

&,(%& .# g *8*. +& )( *m'# g # )"& $.$)&m T"#$ $"(8$ )"*) * +m&%#c*-

*''%(*c" c* '%( # & *cc+%*)& %&$+-)$ 8#)" *''%('%#*)& c( #)#( $ ,(% -# &*%

$#m+-*)#( ! ,(%)+ *)&-. )"& *-# *)#( 8*$ () &%. *cc+%*)& ,(% )"& "*-,9$# &

'+-$& :* *-.)#c*- *''%(*c" -(8&% '-(); T"#$ c(+- & +& )( * &%%(% # )"&

& +*)#( (% * m#$)*k& 8#)" )"& # '+) ,(% MAT AB (8& &% #) $)#-- $"(8$ * # #)#*-

)%* $#& ) '&%#( ,(--(8& . $)* #-#).

6 O T R T Y OF UCITATION

A &xc#)*)#( #$ c( $# &%& (, )"& ,(--(8# g ,(%m # (6.1) >

F)=&9)$# H) (6.1)

T( #--+$)%*)& )"& #m'-#c*)#( $ (, )"#$ ( )"& $.$)&m )"& ,+ c)#( #$ '-())& ,(% )8(

#,,&%& ) *-+&$ (, H )"& ($c#--*)# g ,%& +& c. # ,#g+%&$ 13 * 14

T"&%&,(%& #) #$ * ,+ c)#( 8"#c" #&$ *8*. # *m'-#)+ & *) * ,#x& %*)& %&g*% -&$$(, )"& *)+%*- ,%& +& c. I) #$ c-&*% )( $&& )"*) )"#$ #$ + -#k& )"& ()"&% &xc#)*)#(

,+ c)#( $ # $&c)#( 5 8"#c" 8&%& -*%g&-. $(- & )"%(+g" +m&%#c*- * *-.$#$ *$ #)

#$ #$$#m#-*% )( $#m'-& %&c)* g+-*% "*-,9$# & '+-$& (% c( $)* ) 8#)" %#$& )#m&

,+ c)#( $

T"& &$) 8*. )( )*ck-& )"#$ '%( -&m * *-.)#c*--. #$ )"%(+g" )"& ,(%c& m( *-

* *-.$#$ m&)"(

Mx@Cx@ x=BF)=BP&9)$# H)Q (6.2)

F /0 13 – F 08 F/ 8% (6.1)$55& D 94 0 @ A ! 0$D*

F /0 1 – F 08 F/ 8% (6.1)$55& D 94 0 @ A !, 0$D*



%('(%)#( *- *m'# g 8(+- (cc+% # )"& $*m& 8*. *$ # $&c)#( 4 * )"& c(9

(% # *)& )%* $,(%m*)#( $ 8(+- *-$( & *''-#& )( )"& %#g") "* $# & (, )"&

& +*)#( JBK #$ &,# & "&%& *$ * # & )#). m*)%#x





E C ONC !SIONS

T"& 6D(F $.$)&m 8*$ )%&*)& # * +m &% (, 8*.$ ()" c(m'+)*)#( *--. *

* *-.)#c*--. T"& $.$)&m & +*)#( $ * c"*%*c)&%#$)#c$ 8&%& $+cc&$$,+--. ,(+M( *- * *-.$#$ '%( # & * +$&,+- 8*. (, )*ck-# g # %*)#( '%( -&m # c( <+ c)#(

8#)" MAT AB 8"#c" c(m'+)& )"& c(9(% # *)& )%* $,(%m*)#( $ A-- (, )"& *)+%*-

,%& +& c#&$ *m'# g %*)#($ *m'# g ,*c)(%$ * *m'& ,%& +& c#&$ 8&%&

,(+ F(%c& ( 9'&%#( #c &xc#)*)#( 8*$ * *-.$& +m&%#c*--. 8#)" )"&

c( (-+)#( # )&g%*- ,(% * %&c)* g+-*% '+-$& * * "*-,9$# & '+-$& T"& %&$+-)$

8&%& *cc+%*)& +$# g )"& #m'+-$& %&$'( $& # MAT AB (8& &% )"& * *-.)#c*-

*-# *)#( '%( & )( & * c"*--& g& * )"& %&$+-)$ 8&%& -#k&-. )( & # c(%%&c)

F(%c& m( *- * *-.$#$ 8*$ c( $# &%& )( m( &- * &xc#)*)#( ,+ c)#( (, F)=&9

)$# H) O &%*-- )"& * *-.$#$ (, )"#$ 6D(F $.$)&m '%( & )( & $+cc&$$,+- * *

g(( + &%$)* # g "*$ && ,(%m& (, #)$ ,+ *m& )*- '%('&%)#&$ # ,%&& *

,(%c& # %*)#(



L R F R NC S

J1K TONG! B 1LL6 Principles of Vibration Ox,(% ! # &%$#). %&$$

J2K T OMSON T DA M D 1LLE Theory of Vibration with Applications,

5 )" & %& )#c& *--

J3K S3E6 S)+ & ) R&$(+%c&$ A *#-* -& ( -# &>"))'>//8882 8*%8#ck *c +k/,*c/$c#/& g/&+(/m( +-&$/.&*%3/&$3E6*/%&$(+%c&$/

10 A NDIC S

A NDIU 1S3E6 A$$#g m& ) "* (+)> D% U#* # g #+ Sc"((- (, g# &&%# g *%8#ck

! # &%$#). M*%c" 2010

A NDIU 2xc#)*)#( ,#-&$ ,(% $&c)#( $ 5 1 * 5 2

F(%c& 1 m

dt=0!1;t=0 dt 50;f=[Ber+s$50,1 ;+nes$100,1 ;Ber+s$)en(t'$t -150,1 ];su )+t$211)+t$t,ftit)e$ A .uare in utAa is$[0 50 0 1!5]% ause

F(%c& 2 m

>G=0!1;t=0 >G 50; %Gime $'i(' res+)uti+nf=$sin$t !&t 0!1 !&$tJ8!1 ;)+t$t,f, A A ;)a e)$ AGimeA ;)a e)$ AFn utA ;tit)e$ AFn ut #s GimeA ;ause;

Documents

Vibration Analysis Using MATLAB