transient vibration
TRANSCRIPT
-
7/29/2019 Transient Vibration
1/3
ME 535 - Vibrations
Transient Vibration
Impulse: time integral of force: $ ( )F F t dt =
If and unit impulse or function$F = 1 0
is a unit impulse at and has the following( ) t t = properties:
( )
( )
t tt
t dt
= > =
= < <
0
1 00
for allanyassumed value for
for
If is multiplied by any time function , as shown in the following figure,( ) t f t( )the product will be zero everywhere except where , and its time integral will be:t =
( ) ( )f t t dt f( ) = < <
0 0
Since the impulse acting on a mass will result in a sudden change in itsF dt m dv= $Fvelocity (without an appreciable change in its displacement) given by:
$F
m
dv
dt=
-
7/29/2019 Transient Vibration
2/3
ME 535 - Vibrations
Transient Vibration
Page 2
Since, under free vibration, an undamped spring-mass system will have the following response:
,xx
t x tn
n n= +&( )
sin ( ) cos0
0
the response of an undamped spring-mass system initially at rest, and excited by an impulse
is:$F
xF
mt Fh t
n
n= =$
sin $ ( )
where: is the response to a unit impulse.h tm
tn
n( ) sin=1
For a damped spring-mass system, the response to an impulse is:
xF
me t Fh t
h tm
e t
n
t
n
n
t
n
n
n
=
=
=
$
sin $ ( )
( ) sin
11
1
11
2
2
2
2
Arbitrary Excitation
We can deal with any arbitrary excitation as a series of
impulses:
Strength of impulse: $ ( )F f=
Contribution to response at t is dependent on :( )t
( ) ( )f h t
-
7/29/2019 Transient Vibration
3/3
ME 535 - Vibrations
Transient Vibration
Page 3
where: is the response to an impulse.( )h t
If the system is linear, then superposition holds and:
convolution integral( ) ( ) ( )x t f h t dt
= 0
Example (4.2.1 pg. 92):
Determine the response of s single DOF system to
the step excitation shown.
h tm
tn
n( ) sin=1
( ) ( )
( )
x tF
mt d
F
mt d
x t Fk
t
n
t
n
n
t
n
n
( ) sin sin
( ) cos
= =
=
000
0
01
or, using Laplace transforms:
( )
mx kx F u t sk
mX s
F
m s
x s
F
m
s s
k
m
x tF
ktn
&& ( ) ( )
( )( ) cos
+ = +
=
=
+
=
0
2 0
0 2
0
1
1