Chapter 4 General Force Vibration
Post on 14-Nov-2015
23 Views
Preview:
DESCRIPTION
Transcript
Dr.-Ing. Nantawatana Weerayuth Page 163
4
1. 1-DOF
2. 1-DOF
3. Convolution Duhamel
4.
Dr.-Ing. Nantawatana Weerayuth Page 164
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
Dr.-Ing. Nantawatana Weerayuth Page 165
( )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
Dr.-Ing. Nantawatana Weerayuth Page 166
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
Dr.-Ing. Nantawatana Weerayuth Page 167
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
Dr.-Ing. Nantawatana Weerayuth Page 168
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)
Dr.-Ing. Nantawatana Weerayuth Page 169
4.1 (Periodic 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
Dr.-Ing. Nantawatana Weerayuth Page 170
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
Dr.-Ing. Nantawatana Weerayuth Page 171
4.2 (Periodic Function)
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
Dr.-Ing. Nantawatana Weerayuth Page 172
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)
Dr.-Ing. Nantawatana Weerayuth Page 173
(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)
Dr.-Ing. Nantawatana Weerayuth Page 174
2. (Steady State Solution) (4.12)
3
22 2 2
( ) cos( ); 0,1,2,...(1 ( ) ) (2 )
np n
a kx t n t n
nr nr
(4.15)
3. (Steady State Solution) (4.13)
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
Dr.-Ing. Nantawatana Weerayuth Page 175
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
Dr.-Ing. Nantawatana Weerayuth Page 176
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
Dr.-Ing. Nantawatana Weerayuth Page 177
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
Dr.-Ing. Nantawatana Weerayuth Page 178
(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
Dr.-Ing. Nantawatana Weerayuth Page 179
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
Dr.-Ing. Nantawatana Weerayuth Page 180
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
Dr.-Ing. Nantawatana Weerayuth Page 181
4.5
4.10
3
4.10
4.3
( )f t
(Constant Sampling Time)
1 2, ,..., Nt t t
Dr.-Ing. Nantawatana Weerayuth Page 182
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
Dr.-Ing. Nantawatana Weerayuth Page 183
(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
Dr.-Ing. Nantawatana Weerayuth Page 184
(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
Dr.-Ing. Nantawatana Weerayuth Page 185
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)
Dr.-Ing. Nantawatana Weerayuth Page 186
(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
Dr.-Ing. Nantawatana Weerayuth Page 187
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)
Dr.-Ing. Nantawatana Weerayuth Page 188
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
Dr.-Ing. Nantawatana Weerayuth Page 189
( ( ) 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
Dr.-Ing. Nantawatana Weerayuth Page 190
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
Dr.-Ing. Nantawatana Weerayuth Page 191
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
Dr.-Ing. Nantawatana Weerayuth Page 192
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)
Dr.-Ing. Nantawatana Weerayuth Page 193
(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
Dr.-Ing. Nantawatana Weerayuth Page 194
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
Dr.-Ing. Nantawatana Weerayuth Page 195
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
Dr.-Ing. Nantawatana Weerayuth Page 196
4.10 -
0F 00 t t 4.20 (a)
4.20
Dr.-Ing. Nantawatana Weerayuth Page 197
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
Dr.-Ing. Nantawatana Weerayuth Page 198
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
Dr.-Ing. Nantawatana Weerayuth Page 199
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
Dr.-Ing. Nantawatana Weerayuth Page 200
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
Dr.-Ing. Nantawatana Weerayuth Page 201
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
Dr.-Ing. Nantawatana Weerayuth Page 202
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
Dr.-Ing. Nantawatana Weerayuth Page 203
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
Dr.-Ing. Nantawatana Weerayuth Page 204
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
Dr.-Ing. Nantawatana Weerayuth Page 205
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
top related