ME 3281: Spring 2013 University of Minnesota
Transform Solutions to LTI (Linear Time Invariant) Systems Part 1
Need to study Laplace Transform and the formulae for Laplace transform of various
functions.
Why?
Using Laplace transform, the solutions to ode
Applications of Bode Plots
Rajesh Rajamani
ME 3281
Department of Mechanical Engineering
University Of Minnesota
FREQUENCY RESPONSE
Assume the transfer function G(s) is
asymptotically stable
y (t ) A sin( t )
x (t ) sin( t )
G (s )
A?
?
Answer
A  G ( j )
ME 3281, Spring 2013, Prof. R. Rajamani, University of Minnesota
May 7, 2013
With proportional control, if Ydes is constant, will y ydes as t ?
Sketch the Bode magnitude plot of
Y
Ydes
Let n 2 =
Y
Ydes
transfer function:
KpK
=
=
Ts 2 + s + K p K
Kp K
2n =
ME 3281, Spring 2013, Prof. R. Rajamani, University of Minnesota
April 25, 2013
In general, the high frequency slope in the Bode magnitude plot is
determine by the relative degree of the transfer function.
Relative degree is defined as the difference betw
ME 3281, Spring 2013, Prof. Rajamani, University of Minnesota
Frequency Response of Linear Time Invariant Systems
Complex Numbers: Recall that every complex number has a magnitude
and a phase.
Example: z = a + bj,
j = 1
a is called the real part of z, a =
ME 3281, Spring 2013, Prof. R. Rajamani, University of Minnesota
April 16, 2013
Recap 1:
It is easy to find the transfer function between input and output, even
when there is a system involving multiple DOF (multiple odes).
Example:
Find the transfer func
ME 3281, Spring 2013, Prof. Rajamani, University of Minnesota
May 2, 2013
Bode magnitude plot of
( )
( )
(
)
ME 3281, Spring 2013, Prof. Rajamani, University of Minnesota
 



Example: Sketch the approximate Bode magnitude plot of
( )
( )
ME 3281, Sp
ME 3281 Spring 2013, University of Minnesota
Transform Solutions to LTI Systems Part 4
April 2, 2013
Final Value Theorem
Given F(s), how can we find lim f(t)
t
Final Value Theorem (FVT):
lim f(t) = lim sF(s)
t
s0
When is the FVT applicable?
1). F(s) shoul
ME 3281: Spring 2013 University of Minnesota
State Space Representation
The order of an ode is the order of the highest derivative in the ode.
Example:
The order of the following ode is 2:
+ + =
The order of the following ode is 4:
(4) + + + + =
Every
ME 3281, Spring 2013, Prof. Rajamani, University of Minnesota
April 9, 2013
Step response and impulse response
Step response:
a). Initial conditions are all zero.
b). The input is a unit step.
Example: Constant force acting on a springmassdamper system.
ME 3281: Spring 2013 University of Minnesota
Recap: Example on using Laplace Transforms to solve a first order ode
F
massless plate
If the force f(t) is constant, f(t) = F0 , find x(t)
Solution:
() =
0
[1 ]
The value of t at which the exponentially conve
Transform Solutions to LTI Systems Part 3
Example of second order system solution:
Same example with increased damping:
k=5 N/m, b=6 Ns/m, F=2 N, m=1 Kg
Given (0) = 0, (0) = 0, find x(t).
The revised equation for () with b=6 Ns/m is
() =
5
( 2 + 6 + 5)
Th