1. ME375 Handouts - Spring 2002
MESB System Modeling and Analysis FREQUENCY DOMAIN ANALYSIS Frequency Response

2. Frequency Response Forced Responses to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Frequency Response
Forced Responses to Sinusoidal Inputs
Transient and Steady-State Response
Frequency Response
Steady-State Response to sinusoidal inputs at various input frequencies
Bode Plots
A convenient graphic display of frequency response at all input frequencies

3. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
Ex: Let’s find the forced response of a stable first order system:
to a sinusoidal input:
Forced response in s-domain:
PFE:
Use ILT to obtain forced response in time-domain:

4. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
Input is sin(2t) -2
-1.5 -1
-0.5
0.5 1
1.5 2
Response
Output
Input 2 4 6 8 10 12 14
Time (sec)

5. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
In-class Ex: Given the same system as in the previous example, find the forced response to u(t) = sin(10 t).

6. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
Input is sin(10t)
0.5 1
1.5 2
2.5 3
3.5 4
4.5 5
Time (sec) -1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8
Response
Output
Input
Transient

7. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
Ex: Let’s revisit the same example where
and the input is a general sinusoidal input: sin(w t).
Use the residue formula to find Ai’s:

8. Forced Response to Sinusoidal Inputs
ME375 Handouts - Spring 2002
Forced Response to Sinusoidal Inputs
Ex: Forced response in time-domain:
The steady state sinusoidal response in time-domain:
Stable
LTI System
Sinusoidal input
Sinusoidal output
Phase shift
Changed Magnitude
What happens to frequency?
No Change!
where

9. Forced Response to Sinusoidal Inputs

10. ME375 Handouts - Spring 2002
Frequency Response
Frequency response is used to study the steady state output ySS(t) of a stable LTI system to sinusoidal inputs at different frequencies.
In general, given a stable system:
If the input is a sinusoidal signal with frequency w , i.e.
then the steady state output ySS(t) is also a sinusoidal signal with the same frequency as the input signal but with different magnitude and phase:
where G(jw) is the complex number obtained by substitute jw for s in G(s) , i.e.

11. Frequency Response    LTI System G(s)
ME375 Handouts - Spring 2002
Frequency Response
Input u(t)
U(s)
LTI System
G(s)
Output y(t)
Y(s)
ySS u t
2p/w
2p/w t   
A different perspective of the role of the transfer function:

12. Steady-state Sinusoidal Output y(t)
Frequency Response G
Sinusoidal Input u(t)
Steady-state Sinusoidal Output y(t) G

13. In Class Exercise Ex: 1st Order System
ME375 Handouts - Spring 2002
In Class Exercise
Ex: 1st Order System
The motion of a piston in a cylinder can be modeled by a 1st order system with force as input and piston velocity as output:
The EOM is:
(1) Let M = 0.1 kg and B = 0.5 N/(m/s), find the transfer function of the system:
(2) Calculate the steady state output of the system when the input is
f(t) 2
0)=0
0.8944
( )) v
0.4851
( ))
0.3288
( ))
0.2481
( ))
0.1990
( ))
0.1661
( ))

14. In Class Exercise (3) Plot the frequency response plot
ME375 Handouts - Spring 2002
In Class Exercise
(3) Plot the frequency response plot

15. Example - Vibration Absorber (I)
ME375 Handouts - Spring 2002
Example - Vibration Absorber (I)
Let M1 = 10 kg, K1 = 1000 N/m, B1 = 4 N/(m/s). Find the steady state response of the system for f(t) = (a) sin(8.5t) (b) sin(10t) (c) sin(11.7t).
Without vibration absorber:
EOM:
TF (from f(t) to z1): K1 M1 B1 z1
f(t)
0.0036
( ))
0.025
( ))
0.0027
( ))

16. Example - Vibration Absorber (I)
ME375 Handouts - Spring 2002
Example - Vibration Absorber (I)
-0.01
-0.005
0.005
0.01
-0.04
-0.02
0.02
0.04 5 10 15 20 25 30 35 40 45 50 z1
(m)
f(t) = sin(8.5 t)
f(t) = sin(10 t)
f(t) = sin(11.7 t)
Time (sec)
Poles:

17. Example - Vibration Absorber (II)
ME375 Handouts - Spring 2002
Example - Vibration Absorber (II)
With vibration absorber:
EOM:
TF (from f(t) to z1):
Let M1 = 10 kg, K1 = 1000 N/m, B1 = 4 N/(m/s), M2 = 1 kg, K2 = 100 N/m, and B2 = 0.1 N/(m/s). Find the steady state response of the system for f(t) = (a) sin(8.5t) (b) sin(10t) (c) sin(11.7t). K1 M1 B1 z1
f(t) M2 K2 z2 B2
0.023
(-66.5))
0.001
(-90))
0.016
(-88.5))

18. Example - Vibration Absorber (II)
ME375 Handouts - Spring 2002
Example - Vibration Absorber (II)
-0.04
-0.02
0.02
0.04
-0.004
-0.002
0.002
0.004 5 10 15 20 25 30 35 40 45 50
-0.01
0.01 z1
(m)
f(t) = sin(8.5 t)
f(t) = sin(10 t)
f(t) = sin(11.7 t)
Time (sec)

19. Example - Vibration Absorber (II)
ME375 Handouts - Spring 2002
Example - Vibration Absorber (II)
Characterizing Transient Response:
The characteristic equation
Q1: Can you make a good guess of the duration of the transient period?
Q2: Can you explain the observed steady-state sinusoidal responses?
If the frequency of input is near the imaginary part of one of poles,
resonance will possibly happen.
If the frequency of input is near the imaginary part of one of zeros,
the effect of input will probably be “absorbed”.

20. Example - Vibration Absorbers
ME375 Handouts - Spring 2002
Example - Vibration Absorbers
Frequency Response Plot
No absorber added
Frequency Response Plot
Absorber tuned at 10 rad/sec 2 4 6 8 10 12 14 16 18 20
0.005
0.015
0.025
0.01
0.02
Magnitude (m/N)
Frequency (rad/sec)
-180
-135
-90
-45
Phase (deg) 2 4 6 8 10 12 14 16 18 20
0.005
0.01
0.015
0.02
0.025
Magnitude (m/N)
Frequency (rad/sec)
-180
-135
-90
-45
Phase (deg)

21. Example - Vibration Absorbers
ME375 Handouts - Spring 2002
Example - Vibration Absorbers
Bode Plot
No absorber added
Bode Plot
Absorber tuned at 10 rad/sec
Frequency (rad/sec)
Phase (deg); Magnitude (dB)
-100
-90
-80
-70
-60
-50
-40
-30 10 1 2
-180
-135
-45
Phase (deg); Magnitude (dB)
-100
-90
-80
-70
-60
-50
-40
-30
Frequency (rad/sec) 10 1 2
-180
-135
-45
A better way to graphically display Frequency Response !

22. Example -- SDOF Suspension
ME375 Handouts - Spring 2002
Example -- SDOF Suspension
Simplified Schematic (neglecting tire model)
Suspension System
How does the traveling speed influence the magnitude of joggling?
Car body M z K B
Suspension v
Wheel Ap xp
Reference L

23. Example -- SDOF Suspension
ME375 Handouts - Spring 2002
Example -- SDOF Suspension
Given:
Find:
How does the traveling speed influence the magnitude of joggling
All parameters are normalized