This document discusses identifying parameters of a dynamic system from frequency response data. It describes directly measuring the magnitude response and resonance frequency from a swept sine test. The resonance frequency and magnitude can then be used to calculate the system's stiffness, mass, and gyrator modulus. A model fit VI can also find the damping coefficient by fitting the measured data to the second order system model.

1. Identifying G, k, and ωn from Bode
• Directly measure ωr and |Mss|
10
0
10
1
10
2
-200
-150
-100
-50
Frequency [Hz]
Magnitude[dB]
|Mss|= G/m
r nω ω≈
2 2
2 2
2
( )
( ) 2 n n
A s s G m s G m
B kI s s s s s
m m
ζω ω
=
+ + + +
2
nk mω=
ssG m M=
Important:

2. Swept sine frequency response
Jonathan LeSage, TA
ME 144L – Dynamic Systems and
Controls Lab

3. Frequency Response Overview
• Frequency response vs. impulse response?
– Impulse response -> Beam setup in 144L
• No inputs, other than the initial deflection, x(0) = x0
• Second order system (homogeneous):
• Time domain (Transient plots)
– Frequency response -> Shaker (subwoofer)
• Sinusoidal input:
• Second order system (inhomogeneous):
• Frequency domain (Bode plots)
2
2 0n nx x xζω ω+ + = 
2
2 ( )n nx x x F tζω ω+ + = 
[ ]0( ) sin iF t A wt φ= +
System

4. Swept Sine Testing
• Directly measure magnitude response of system
– Input sinusoid at specified frequency
– Measure the magnitude response (output/input)
– Build magnitude frequency response chart
10
0
10
1
10
2
-200
-150
-100
-50
Frequency [Hz]
Magnitude[dB]
10
0
10
1
10
2
-280
-260
-240
-220
-200
-180
-160
-140
Frequency [Hz]
Magnitude[dB]
Yet another way to characterized/find parameters
for a dynamic system!

5. Shaker Dynamic Model
• Actuated second order system
( )mx bx kx F t+ + = 
( )mx bx kx Gi t+ + = 
( ) ( )V t Ri t Gx= +  (Apply KVL)
1
( ) ( )
G
i t V t x
R R
= − 
(Newton’s 2nd)
2
( )
G G
mx bx kx V t x
R R
+ += −  
2
( )
G G
mx b x kx V t
R R
 
+ + + = 
 
 
( )
G
mx Bx kx V t
R
+ + = 

6. Into the Laplace Domain
• Take Laplace transform
• Second order system form
• For simplicity, assume I(s) = V(s)/R where R = 2Ω
2
( )
( )
X s G mR
B kV s s s
m m
=
+ +
( )
G
mx Bx kx V t
R
+ + =  ( )
B k G
x x x V t
m m mR
+ + = 
2
( ) ( )
B k G
s s X s V s
m m mR
 
+ + =  
2 2
( )
( ) 2 n n
X s G mR
V s s sζω ω
=
+ +
2 2
( )
( ) 2 n n
X s G m
I s s sζω ω
=
+ +
2
2 2
( )
( ) 2 n n
A s s G m
I s s sζω ω
=
+ +

7. Lab Outline
• Build measurement VI
– Read two voltage signals with myDAQ. (Accelerometer voltage and shaker voltage)
– Add calibration equation to VI for accelerometer ( a(t) = 98.1*Vaccel(t) )
– Add equation to calculate I(t), ( I(t) = Vshaker(t)/2 )
– Compute RMS of both signals (“AC and DC Estimator.vi”) -> Magnitude
– Compute Gain ratio
• Swept sine testing
– Measure gain ratio from 10 Hz to 150 Hz (plot values in excel)
– Fill in peak detail if necessary
• System identification
– Using swept sine data, measure the resonance frequency
– Assume
– Compute the shaker stiffness (k), using mass of block + 240grams (armature mass)
– Compute shaker gyrator modulus
• Model fit VI to find damping coefficient
– Find viscous damping factor, B, by fitting data
n rω ω=
2
nk mω=
ssG m M=
2 2
2 2
2
( )
( ) 2 n n
A s s G m s G m
B kI s s s s s
m m
ζω ω
= =
+ + + +