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Laplace Transformation: T F KX X C X M

The document discusses using Laplace transforms to analyze systems with arbitrary forcing functions. It shows that taking the Laplace transform of the equation of motion of a damped mass-spring-damper system excited by an arbitrary force F(t) results in an expression relating the Laplace transform of the response x(s) to the Laplace transform of the forcing function F(s). This can be used to find the response x(t) via inverse Laplace transform. Examples are given to determine the impulse and step responses of a single-degree-of-freedom damped system.

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Pradip Gupta
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57 views3 pages

Laplace Transformation: T F KX X C X M

The document discusses using Laplace transforms to analyze systems with arbitrary forcing functions. It shows that taking the Laplace transform of the equation of motion of a damped mass-spring-damper system excited by an arbitrary force F(t) results in an expression relating the Laplace transform of the response x(s) to the Laplace transform of the forcing function F(s). This can be used to find the response x(t) via inverse Laplace transform. Examples are given to determine the impulse and step responses of a single-degree-of-freedom damped system.

Uploaded by

Pradip Gupta
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Module: 9

Lecture: 3
Laplace Transformation
The equation of motion of the system excited by an arbitrary force F(t) is
m x c x kx F t
Taking its Laplace transformation, we find

m s 2 x s s x 0 x 0 c s x s x0 k x s F s
Solving for x s we obtain the equation
F s

ms cs k

x s

m s

c x0 m x 0
m s2 c s k

The response x(t) is found from the inverse Laplace transform of the above equation, the
first term represent the forced vibration and second term represents free vibration due to initial
conditions.
For the more general case, the above equation can be written in the form
x s

As
B s

Where A(s) and B(s) are polynomials and B(s), in general, is of higher-order than A(s).
If only the forced solution is considered, we can define the impedance transform as

F s
Z s m s 2 c s k
x s
Its reciprocal is the admittance transform
G s

1
1
1

2
2
Z s
ms cs k
m s 2 n s n 2

F s (imput) H s x s (output)

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Page 1

Ex.

Drive the impulse response of SDOF by Laplace Transform.


1

G s

ms 2 n n
2

1
1

2 i n m s n i d
s n i d

But, in general
L1

1
s

e t

Hence, we get impulse response

1
1
g t L 1 G s L 1

s n i d
s n i d
1
e n i d t e n i d t

2 i d m

1
e n t sin d t
m d
Which is precisely same as given earlier.

Ex. Determine the step response of a damped single-degree-of-freedom system by the


Laplace transform method.
1
Laplace transform of step load can be written as
s
1
G s
t L 1
L 1
2
s
m s s 2 n s n 2

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1
m n

1
1
1
n i d
n i d
L 1

2 i d
2 i d
s n i d
s n i d
s

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Therefore, the step response can be obtained as follows:

1
k
1
k

n i d n i d t n i d n i d t
e
e

2 i d
2 i d

1
1
1 2

when tan 1

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n t

cos d t

1
2

Page 3

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