Chapter 4 General Force Vibration

Dr.-Ing. Nantawatana Weerayuth Page 163

4

1. 1-DOF

2. 1-DOF

3. Convolution Duhamel

4.


4.1 (Periodic Function)

( )F t Periodic

( ) ( )F t k F t (4.1)

2

1,2,...,k

(4.1) ( )F t

4.1 4.2

( ) ( 2 ) ( 3 ) .... ( )F t F t F t F t (4.2)

4.1

4.2


( )F t

( Fundamental Period ) ( Fundamental Frequency)

2

(4.3)

Periodic: ( )F t

0

1 1

( ) cos( ) sin( )2

n n

n n

aF t a n t b n t

(4.4)

0a na nb

4.2

(Periodic Force) 2 (4.4)

0

1 1

( ) cos( ) sin( )2

n n

n n

aF t a n t b n t

0a na nb

0

2( )cos( ) ; 0,1,2,...na F t n t dt n

(4.5)

0

2( )sin( ) ; 1,2,3,...nb F t n t dt n

(4.6)

4.2.1

( )F t (Even Function)

(Odd Function) 0a na nb


Index: n

1. (Even Function) :

( ) ( ),f x f x x (4.7)

2( ) ,f x x 2( ) 1f x x ( ) cos( )f x nx [ y ]

4.3


2. (Odd Function) :

( ) ( ),f x f x x (4.8)

( )f x x , 3( )f x x , ( ) sinh( )f x x ( ) sin( )f x nx [

180 ]

4.4


1. () () = () 2. () () = ()

3. () () = () 4. () () = ()

5. () () = ()

6. ( )f x ( ) 0c

c

f x dx

7. ( )h x 0

0 0

( ) ( ) ( ) 2 ( )

c c c

c c

h x dx h x dx h x dx h x dx

8. ( )f x : [ 0na ]

0

2 1( )cos( ) ( )cos( ) 0 ; 0,1,2,...n

t Odd function Odd function

a f x n x dx f x n x dx n

0

2( )sin( ) 0 ; 0,1,2,...n

t even function

b f x n x dx n

9. ( )f x : [ 0nb ]

0

2( )cos( ) 0; 0,1,2,...neven function

a f x n x dx n

0

2 1( )sin( ) ( )sin( ) 0; 0,1,2,...n

t odd function odd function

b f x n x dx f x n x dx n

(Even Function)

(Odd Function)



4.5

4.5

0

0 02

( )

02

for t

f t

F for t

( )f t

na nb (4.5) (4.6)

0

2( )cos( ) ; 0,1,2,...na F t n t dt n

0

2( )sin( ) ; 1,2,3,...nb F t n t dt n

1. na : 0n 0

02 2

0

002 2

2 2( ) ( ) ( )

F

a F t dt F t dt F t dt

2

00 0 0

0

22( )( )

2

Fa F dt F


1,2,...n

0

02 2

0 002 2

2 2 2( )cos( ) ( )cos( ) ( )cos( ) ( )cos( )n

F

a F t n t dt F t n t dt F t n t dt F t n t dt

2

0 20 0

0

22cos( ) sin( )n

Fa F n t dt n t

n

1 2 3, , ,...a a a

20 0

0

0 ( 1,2,3,..)2 2( ) sin( ) sin( )2

n

for all nF Fa n t n

n n

2. nb : 1,2,3,...n

0

02 2

0 002 2

2 2 2( )sin( ) ( )sin( ) ( )sin( ) ( )sin( )n

F

b F t n t dt F t n t dt F t n t dt F t n t dt

2

0 20 0

0

22sin( ) cos( )n

Fa F n t dt n t

n

1 2 3, , ,...b b b

2

0 0

0

0

0 (2,4,6,..)2 2

( ) cos( ) 1 cos( ) 22 (1,3,5,...)

n

if n is evenF F

b n t n Fn n if n is odd

n

(4.4) ( )f t

0 0

1,3,5,...

2( ) ( )sin( )

2 n

F Ff t n t

n



4.6

4.6

Periodic

0

02 4

( )4 4

04 2

for t

f t F for t

for t

4.6 y

( ) ( )f t f t Even Function

0nb 1,2,3,...n

na ( 1,2,3,...n )

0

2( )cos( )na F t n t dt


0

4 4 2

0 0 02 4 4

2 2( )cos( ) ( )cos( ) ( )cos( ) ( )cos( )n

F

a F t n t dt F t n t dt F t n t dt F t n t dt

4

0 040

4

4

2 22 2 2cos( ) sin( ) ( ) sin( ( )) sin( ( ))

2 4 4n

F Fa F n t dt n t n n

n n

0 0 022 2sin( ( )) sin( ( )) sin( ) sin( ) sin( )4 4 2 2

n

F F Fa n n n n n

n n n

0 ( 1)

02

0 ( 2,4,6,...)2

sin( ) 22 ( 1) ( ) ( 1,3,5,...)

nn

if n is even nF

a n Fn if n is odd n

n

( )f t

( 1)

0 02

1,3,5,...

2( ) ( 1) ( )cos( )

2

n

n

F Ff t n t

n

4.2.

- 2

2 3 2

( )mx cx kx f t (4.9)


(4.4)

0

1 1

( ) cos( ) sin( )2

n n

n n

af t a n t b n t

(4.9)

0

1 1

( )

cos( ) sin( )2

n n

n n

f t

amx cx kx a n t b n t

(4.10)

(4.10) 0( 2)a

(1 1

cos( ) sin( )n nn n

a n t b n t

)

(Linear System)

(Principle of Super Position)

( (4.10))

0

2

amx cx kx (4.11)

1

cos( )nn

mx cx kx a n t

(4.12)

1

sin( )nn

mx cx kx b n t

(4.13)

(4.11) (4.12) (4.13)

3

1. (Steady State Solution) (4.11)

1( )px t C (4.11) 02a

Ck

01( )

2p

ax t

k (4.14)



3

22 2 2

( ) cos( ); 0,1,2,...(1 ( ) ) (2 )

np n

a kx t n t n

nr nr

(4.15)


2

32 2 2

( ) sin( ); 1,2,3,...(1 ( ) ) (2 )

np n

b kx t n t n

nr nr

(4.16)

n

r

n

1

2

2tan

1 ( )n

nr

nr

(4.17)

(4.10)

0

2 2 20

2 2 21

( ) cos( )..2 (1 ( ) ) (2 )

sin( )(1 ( ) ) (2 )

np n

n

nn

n

a a kx t n t

k nr nr

b kn t

nr nr

(4.18)

(4.18)

( 1,2,..,n )

3 n 1nr

Amplitude nr

(4.18)

2n 3

Periodic


4.3 (Steady State Solution) -

1 DOF

(Periodic Function) 4.7

4.7

( )mx cx kx f t

4.2 ( )f t

( 1)

0 02

1,3,5,...

2( ) ( 1) ( )cos( )

2

n

n

F Ff t n t

n

1,3,5,...n

(4.10)

( 1)

0 02

2 2 21,3,5,..

2( ) ( 1) cos( )

2 (1 ( ) ) (2 )

n

p n

n

F F knx t n t

k nr nr


4.4 (Steady State Solution) -

4.8

( )f t

4.8

4.3

( )mx cx kx f t

4.1 ( )f t

0 0

1,3,5,...

2( ) ( )sin( )

2 n

F Ff t n t

n

1,3,5,...n

(4.10)

0 0

2 2 21,3,5,..

2( ) sin( )

2 (1 ( ) ) (2 )p n

n

F F knx t n t

k nr nr


4.5 (Steady State Solution)

Hydraulic Valve -

4.9 (a) ( )p t 4.9

(b) 2500k N/m c=10 N-s/m m=0.25 kg

4.9

( )mx cx kx f t

m ( ) ( )f t Ap t A

( )p t ( m )

2 26 2(50) 625 10

4 4

dA m

4.9 (b) ( )p t 2 ( )f t

m


(Fundamental Frequency) 2 2

2

rad/s ( )p t 4.9 (b)

y ( )p t (Even Function) ( ) ( )f t Ap t

0nb

n

( )f t

50,000 02

( )

50,000 ( )2

At for t

f t

A t for t

na ( 0,1,2,3,...n )

0

2( )cos( )na F t n t dt

0n

2 2

0

0 0 0

2 2

2 2 2( ) ( ) ( ) ( ) ( )a F t dt F t dt F t dt F t dt F t dt

2

0

0

2

250,000 50,000 ( ) 50,000a At dt A t dt A

1,2,3,...n

2

0 0

2

2 2( )cos( ) ( )cos( ) ( )cos( )na F t n t dt F t n t dt F t n t dt


2

0

2

250,000 cos( ) 50,000 ( )cos( )na At n t dt A t n t dt

By Part Integration : 1cos( ) sin( ) sin( )tt n t dt n t n t dtn n

2

0

2

2

0

250,000 sin( ) sin( ) 50,000 cos( )

250,000 sin( ) sin( )

n

ta A n t n t dt A n dt

n

tA n t n t dt

n

2

02

2

2 1 150,000 sin( ) cos( ) 50,000 sin( )

2 150,000 sin( ) cos( )

n

ta A n t n t A n t

n n n

tA n t n t

n n

5

2 22 2

2 10(1,3,5,...)50,000

2cos( ) cos( 2 ) 1)

0 (2,4,6,..)

n

A if n is oddAa n n n

nif n is even

5

0

2 21,3,5,...

2 10( ) cos( )

2 n

Ff t A n t

n

1,3,5,...n

(4.10)

5 2 2

2 2 21,3,5,..

25,000 2 10( ) cos( )

(1 ( ) ) (2 )p n

n

A A knx t n t

k nr nr


3 0,1,n 3

5 2 5 2

1 32 2 2 2 2 2

25,000 2 10 2 10 9( ) cos( ) cos(3 )

(1 ( ) ) (2 ) (1 (3 ) ) (6 )p

A A k A kx t t t

k r r r r

2500 1000.25

n

k

m rad/s

2

rad/s

0.031416100n

r

100.2

2 2(0.25)(100)c n

c c

c m

1

1 2

2tan 0.0125664

1 ( )

r

r

rad

1

2 2

6tan 0.0380483

1 (3 )

r

r

rad

( )px t

( ) 0.019635 0.015930cos( 0.0125664) 0.0017828cos(3 0.0380483)px t t t m


4.5

4.10

3

4.10

4.3

( )f t

(Constant Sampling Time)

1 2, ,..., Nt t t


1 1 2 2( ), ( ),..., ( )N Nf f t f f t f f t N (

2,4,6,...) 1 ( ) N tN

N t 4.11

4.11

Trapezoidal Rule

0

1

2 N

i

i

a fN

(4.19)

1

2 2cos( ); 1,2,...

N

n i i

i

a f n t nN

(4.20)

1

2 2sin( ); 1,2,...

N

n i i

i

b f n t nN

(4.21)

(4.18) 0

2 2 20

2 2 21

( ) cos( )..2 (1 ( ) ) (2 )

sin( )(1 ( ) ) (2 )

np n

n

nn

n

a a kx t n t

k nr nr

b kn t

nr nr


(4.17)

1

2

2tan

1 ( )n

nr

nr

2n

r

4.6 Hydraulic Valve

4.1 0.01

1

0.12 ( )p t

0.12 s 2 2 52.360.12

rad/s

0.01t 0.12 120.01

Nt


(4.19)-(4.21)

12

0

1

268166.7

12i

i

a p

12

1

2 2cos( )

12 0.12n i i

i

a p n t

12

1

2 2sin( )

12 0.12n i i

i

b p n t

4.2

2

( ) 34083.3 26996.0cos(52.36 ) 8307.7sin(52.36 )...

1416.7cos(104.72 ) 3608.3sin(104.72 )

5833.3cos(157.08 ) 2333.3sin(157.08 ) ...

p t t t

t t

t t N m


4.4

- Periodic

(Impact Force)

1. (Convolution Integral) Duhamel Integral

2. (Laplace Transform)

3. (Numerical Methods)

(Convolution Integral)

Periodic

(Impulsive Force)

0t

(Impulse)

2 1F t mx mx (4.22)

(Impulse) F t F

t t

tFdt

F (4.23)

1 Unit Impulse

0lim 1

t t

ttFdt Fdt

f (4.24)


(Unit Impulse) 1f 0t

Dirac Delta Function

( ) ( )t t f f (4.25)

F 0t

( )tF F (4.26)

t ( )t

1. ( ) 0t t

2. 0

( ) 1t dt

3. 0

( ) ( ) ( )F t t dt F

4.4.1 (Unit Impulse Response) :

- 4.12(a) 0t

4.12 (b)

4.12


0mx cx kx (4.27)

0 00( ) cos( ) sin( )

nt nd d

d

x xx t e x t t

(4.28)

(4.28)

2 n

c

m

(4.29)

n

k

m (4.30)

2

212

d n

k c

m m

(4.31)

m 0x x 0t

0t

(0) (0 ) 1unit impulse mx mx f (4.32)

0(@ 0) 0x t x (4.33)

1(@ 0)x t

m (4.34)

(4.33) (4.34) (4.28)

( Unit Impulse Response) ( )g t Unit Impulse Response Function

1( ) ( ) sin( )n

t

d

d

x t g t e tm

(4.35)

4.12(c)


F

(@ 0)x tm

F

(4.36)

( Impulse Response)

( ) sin( ) ( )nt dd

x t e t g tm

F

F (4.37)

F t 4.13(a)

t (4.36) ( )x tm

F

4.13


( ( ) 0x t ) ( Impulse Response)

( )( ) sin( ( )) ( ) ;n

t

d

d

x t e t g t for tm

FF (4.38)

(4.38)

t (4.37)

(4.38) 0t

4.13 (b)

4.7 Load Cell

(Impact Force) Impulse 4.14 (a)

4.14


5m kg, 2000k N/m 10c N-s/m

Impulse 1 20F N-s 2 10F N-s 0t 0.2t

Impulse 1 2( ) ( ) ( )F t t t F F

4.14 (b)

200020

5n

k

m rad/s

100.05

2 2(5)(20)n

c

m

2 21 20 1 (0.05) 19.975d n rad/s

Impulse 0t ( 4.37)

( ) sin( ) ( )nt d

d

x t e t g tm

F

F

0.05(20)

1

20( ) sin(19.975 )

5(19.975)

0.020025 sin(19.975 ) 0

t

t

x t e t

e t m for t

E.1

Impulse 0.2t s ( 4.38)

( )

( ) sin( ( )) ( ) ;nt

d

d

x t e t g t for tm

FF

0.05(20)( 0.2)

2

( 0.2)

10( ) sin(19.975( 0.2))

5(19.975)

0.100125 sin(19.975( 0.2)) ; 0.2

t

t

x t e t

e t for t

E.2


Super Position

( 0.2)0.020025 sin(19.975 ) 0 0.2

( )0.020025 sin(19.975 ) 0.100125 sin(19.975( 0.2)) ; 0.2

t

t t

e t for tx t

e t e t for t

E.3

E.3 4.15

4.15


4.4.2 ( Response to a General Forcing

Condition)

( )F t Periodic

Function 4.16 ( Impulse )

4.16

t ( )F

( )F t t

Impulse ( )F F

( ) ( ) ( )x t F g t (4.39)

(0, )t ( )

Impulse

( ) ( ) ( )x t F g t (4.40)


(4.40) 0t

0

( ) ( ) ( )

t

x t F g t d (4.41)

( )g t Unit Impulse Response Function (4.35)

(4.41)

( )

0

1( ) ( ) sin( ( ))n

t

t

d

d

x t F e t dm

(4.42)

(4.41) (4.42) Convolution Integral Duhamel

Integral

4.8 -

(Step Function) 0F : ( 0( )F t F ) 4.17

Duhamel Integral Convolution Integral

4.17


0F (4.42)

( )

0

1( ) ( ) sin( ( ))n

t

t

d

d

x t F e t dm

0( ) ; 0F t F t

( )0

0

( ) sin( ( ))nt

t

d

d

Fx t e t d

m

( )0

2 2

0

0

2

sin ( ) cos ( )( )

( )

11 . cos( ) ; 0

1

n

n

t

t n d d d

d n d

t

d

F t tx t e

m

Fe t t

k

4.18

1

2tan

1

4.18


4.9 -

(Step Function) 0F 0t

( 0 0( ) ( )F t F t t ) 4.19

4.19

4.8

0F 0t

( )0

2 2

0

0

2

sin ( ) cos ( )( )

( )

11 . cos( )

1

n

n

t

t n d d d

d n d

t

d

F t tx t e

m

Fe t

k

0t

t 0t t

00 (

0

)

02

1( ) 1 . c ( )os( ) ;

1

n

d

t tt

Ftx t e t t

k


4.10 -

0F 00 t t 4.20 (a)

4.20


4.20(a) Step Function 1( )F t 0F

0t Step Function 2 ( )F t 0F 0t t

4.8 4.9

0 0F F

0

0

2

0

2

( )

0

1( ) 1 . cos( ) ...

1

11 . cos( ) ;(

1)

n

n

t

d

t

d

t

Fx t e t

k

F

kte t

02

0( ) cos( ( )) cos( )1

nt

d d t tF

x t e tk

1

2tan

1


4.5

(Laplace Transform)

(Initial Conditions)

( )f t

0

( ) ( ) ( )stf t e f t dt F s

L (4.43)

1 2 1 2( ) ( ) ( ) ( )f t f t F s F s L (4.44)

(Linear Vibration System)

(Linear Ordinary Differential Equation) 2

1. Laplace Transform 1

0

00

( ) ( )( ( ))

( ) ( ) (0) ( )

st

st st

df t df tf t e dt

dt dt

e f t se f t dt f sF s

L L

(4.45)

( )f t 0t (Initial Condition) (0)f


2. Laplace Transform 2

2

2

0

0 0

2

0

( ( )/ )

( ) ( )( ( ))

( ) ( )

( )(0) (0) (0) ( )

st

st st

st

df t dt

d f t d df tf t e dt

dt dt dt

df t df te se dt

dt dt

df tf s e dt f sf s F s

dt

L

L L

(4.46)

( )f t ( )f t 0t (Initial Condition) (0)f

(0)f

3. Laplace Transform n

1 2 (1) ( 1)( ) ( ) ( ) (0) (0) ... (0)n

n n n n n

n

Initial Conditions

d f tf t s F s s f s f sf

dt

L L (4.47)

( )nf t n ( )f t

4.11 1 DOF

( )mx cx kx f t

(4.47)

( ( ))mx cx kx f t L L

) ( ) ( ( ( ))mx cx kx f t L L L L

) ( ) ( ( )m x c x k x F s L L L


2 ( ) (0) (0) ( ) (0) ( ) ( )m s s sx x c s s x k s F s

2 ( ) ( ) (0) (0)ms cs k s F s mx ms c x

(0) (0) 0x x (Zero Initial Conditions)

2 2 2( ) 1 1

( )( ) 2 n n

sT s

F s ms cs k m s s

4.3

4.3


4.3

(Laplace Transform : ( ) ( )f t F sL (S-Domain)

(Inverse Laplace Transform : 1 ( ) ( )F s f t L )

(Time Domain)

11

( ) ( ) ( )2

s i

st

s i

F s e F s ds f ti

L (4.48)

(4.48)

4.3


4.12 ( )cx kx f t

( )f t bt 4.21

4.21

Ramp Function : ( )f t bt

1 1( ) ( )

a d

kx x f t bt

c c c

x bx dt

x bx dt L L

2( ( ) (0)) ( )

ds s x a s

s

(0) 0x

2 2( )

( ) ( )

d A B Cs

s s a s a s s


Partial Fraction Expansion 2( )s s a

2 ( ) ( )As Bs s a C s a d

2( ) ( )A B s Ba C Ca d

2,

dA

a

2

dB

a dC

a

2 2 2 2 2

1 1 1 1 1 1( ) ( ) ( ) ( )

( ) ( )

d d d d ds

a s a a s a s a s a s a s

4.3

1 2( ) ( ) 1atd ds x t e t

a a

L

Time Domain

2( ) (1 )at

dx t at e

a

2

( ) (1 )kt

cc k

x t t ek c


4.13 - 4.22

( )f t 0(0)x x 0(0)x x

4.22

( )mx kx F t ; 0t

( )mx kx F t L L

( )mx kx F t L L L

2 ( ) (0) (0) ( ) ( )ms s msx mx k s F s

2 2 2 2 2 2( ) (0) (0) 1 ( ) (0) (0)

( )F s msx mx F s sx x

sk k kmms k ms k ms k

s s sm m m

2 2 2 2 2 21 ( ) (0) (0)

( )n n n

F s sx xs

m s s s


4.3

1 1 0 0

2 2 2 2 2 2

1 ( )( ) ( )

n n n

sx xF sx t s

m s s s

L L

0 0

1 1 10 0

2 2 2 2 2 2

cos( )sin( )

1 ( )( )

nn

n

n n n

Convolution x t xt

sx xF sx t

m s s s

L L L

00

0

1( ) ( )sin( ( )) cos( ) sin( )

t

n n n

n n

xx t F t d x t t

m

Chapter 4 General Force Vibration

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Transcript of Chapter 4 General Force Vibration