ME 451: Control Systems Laboratory
Department of Mechanical Engineering
Michigan State University
East Lansing, MI 48824-1226
ME451 Laboratory
Experiment #5
Sinusoidal Response of a 2nd Order
Torsional Mass-Spring Damper System
Important: Bring your short form from lab #2 to the lab.
You will need them for comparison.
__________________
ME451 Laboratory Manual Pages, Last Revised: October 31, 2007
Send comments to: Dr. Clark Radcliffe, Professor
Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System
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Reference:
C.L. Phillips and R.D. Harbor, Feedback Control Systems, Prentice Hall, 4th Ed.
Section 4.2, pp. 121-124:
Time Response of Second-Order Systems
Section 4.4, pp. 129-132:
Frequency Response of Systems
Appendix B, pp. 635-650:
Laplace Transform
1. Objective
The response of a linear system to a sinusoidal input is useful for predicting its behavior for arbitrary
periodic inputs, but more importantly, for compensator design. For second-order systems, the sinusoidal
response depends primarily on the natural frequency, n, and the damping ratio, . Both n and  are
functions of system parameters, both physical and control parameters. In this experiment we will alter the
n and  values by changing the feedback control gains. The objective of this experiment is to investigate
the relationship between n and  and the frequency response of the system, as well as the relationship
between the feedback gains and n and . The second-order system we choose for this experiment is a
torsional mass-spring-damper system, with torque as input and angular displacement as output. We obtain
the transfer function of the system and identify specific parameters of the system that affect sinusoidal
response. Specifically, we identify parameters that affect the natural frequency and the damping ratio. We
vary these parameters to experimentally verify the change in sinusoidal response.
2. Background
2.1. Second-order systems
The standard form of transfer function of a second-order system is
Y ( s)
K n2
G( s) 

U ( s) s 2  2 n s   n2
(1)
where Y (s) and U(s) are the Laplace transforms of the output and input variables, respectively, n is the
natural frequency, and  is the damping ratio. For a sinusoidal input
u (t )  A sin(  * t ) ,
U ( s) 
A
s 2
2
the response of the system, in Laplace domain, can be written as
Y ( s) 
KA n2
( s 2   2 )( s 2  2 n s   n2 )
Assuming poles of G(s) are in the left-half plane, the steady state response of the system (after transients
have decayed) can be written as
y(t )  A G( j ) sin(  * t   ) ,
  G ( j )
(2)
It is clear from Eq.(2) that a sinusoidal input produces a sinusoidal output. The amplitude of the output is
scaled by a factor of
G( j ) and the phase lags behind the input by G ( j ) .
Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System
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For the standard second-order system in Eq.(1), given the values of n and , the “gain"
and the “phase" G ( j ) can be expressed as a function of
G ( j ) 
K
(1  r 2 ) 2  4 2 r 2
;
G( j )
 , as follows
r(

)
n
(3)
 2 * r 
(gives units of radians)
2 
1  r 
  G ( j )   arctan 
On a logarithmic scale, they can be plotted to generate what are known as gain and phase plots, or
Frequency Response diagrams. The Frequency Response diagrams for a standard second-order system are
plotted as a function of the frequency ratio r  (

) , for different values of  in Fig.1.
n
Figure 1. Frequency Response diagram for standard second-order system
2.2. Torsional mass-spring-damper system
Recall the torsional mass-spring-damper system in laboratory experiment #2, shown here again in
Fig.2. The system variables are
T external torque applied on rotor
 angular position of rotor
 angular velocity of rotor
The parameters of the system, shown in Fig.2, include
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J moment of inertia of rotor
b coefficient of viscous friction
k spring constant
The transfer function of the mass-spring-damper system, with T as input and
G( s) 
 ( s)
T ( s)


as output, can be written as

1
1
(k / J )
  2

Js  bs  k k  s  (b / J ) s  (k / J ) 
(4)
2
Figure 2. Torsional mass-spring-damper system
The transfer function above bears close resemblance with the standard second-order transfer function in
Eq.(1). The only difference is the DC gain of (1/k), which appears in Eq.(4). By comparing Eqs.(1) and (4),
the expressions for natural frequency and damping ratio can be obtained as
n 
k
,
J
2 n 
b
,
J
 
b
2 Jk
(5)
It is clear from Eq.(5) that n and  are functions of system parameters J, b, and k, which are typically
fixed. Hence, if we build the system in Fig.2, we will have a specific second-order response. For studying
the effect of system parameters on the response, we must be able to change J, b, and k. We achieve this by
removing the spring and damper in Fig.2 and programming a motor to generate the torques generated by
the spring and the damper. Specifically, the motor is programmed to generate the torque given by the
relation
T  ke Kh ( K pf  d  K1  K2)
** If you had calculated the gain
k e from earlier experiment you would remember that it was found to be
close to 25
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Figure 3. Block Diagram of Programmable Torsional mass-spring-damper system
where
K h is the hardware gain of the motor; T is the input torque; the feedback generated torque  K1
.
is the type of torque generated by a spring; and the feedback generated torque K 2  is the type of torque
generated by a damper. When this torque is applied on the rotor, the dynamic equation of the programmed
system becomes
.
J  b  k  T
and then
 J  b  k  Kh ke K pf  d  Kh ke K2  Kh ke K1
J  b  K k K   k  K k K   K k K 
h e
2
h e
1
h e
pf
d
The block diagram of the programmed system is shown in Fig.3. The transfer function of this system, with
 d as input and  as output, can then be expressed as
G( s) 
( K h k e K pf / J )

 ( s) 
 2

 d (s)  s  b  K h k e K 2  / J s  k  K h k e K1  / J 
(6)
which has the same structure as that in Eq.(4). The DC gain of the programmed system is
K (
K h k e K pf
k  K h k e K1
),
(7)
and the natural frequency and damping ratio are given by the relations
n 
( k  K h k e K1 )
;
J

b  K h ke K 2
2 ( k  K h k e K1 ) J
(8)
One of the objectives of this experiment is to study the effects of varying the natural frequency n and the
damping ratio  on the sinusoidal response of the torsional mass. We will vary n and  by varying the
gains K1 and K 2 in software. The hardware gain, K h k e , the rotor inertia, J , and the damping constant,
b, will remain fixed during experiments. These values are the same as those found in lab #2.
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Pre-Lab Sample Questions
Use the plot below to answer questions 1-3:
1) What is the phase angle at this frequency, in degrees?
Answer:
Phase angle = -90°
2) What is the non-dimensional gain at this frequency?
Answer:
Gain = 1.5
3) What is the gain at this frequency in decibels?
Answer:
Gain = 3.5 dB
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4) All three systems above have the same natural frequency. What is their natural frequency (in rad/s)
and which system has a higher damping ratio?
Answer:
ωn = 6 rad/s
System C has the highest damping ratio
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5) At which frequency (A, B, or C) does the response below correspond on the bode plot above?
Answer:
A
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3. Description of Experimental Setup
3.1. Hardware and software
1. Electromechanical plant, ECP Model 205
The electromechanical plant is comprised of the torsion disk, a DC servo motor that
provides an external torque to the disk, and sensors for measurement of angular position
of the disk.
2. ECP input/output electronics unit, Model 205
It contains the power supply unit for the DC motor and associated electronic hardware.
3. Digital signal processor (DSP)
The DSP, installed in the PC, takes the sensor signals in digital form, provided by the
analog-to-digital convertors (ADC), and computes the torque to be generated by the
motor. The digital torque signal is sent to the motor via the digital-to-analog convertors
(DAC).
4. Software
ECP executive program in Windows NT environment.
(download from www.egr.msu.edu/classes/me451/radcliff/lab/software.html/)
5. Matlab script “tsfreq.m"
(download from www.egr.msu.edu/classes/me451/radcliff/lab/software.html/)
This script generates Frequency Response plots from the data obtained (amplitude and
phase lag) from experiments.
6. Figure 4 below shows the basic equipment setup.
3.2. Basic setup
1. Place two 500 gm masses on the lowest disk at equal distances from the center. Verify
that the masses are secured to the disk.
2. Verify that the disk can rotate freely. You can do this by turning the disks manually.
Caution: In this work, and all future work, make sure to stay clear of the mechanism. If
the system appears to react violently, you should immediately click the “Abort Control"
button on the ECP program window. If the problem persists, promptly turn off the ECP
input/output electronic unit by pressing the red button.
Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System
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Encoder #3
Torsional
Spring (Rod)
Encoder #2
Servomotor
ECP
Mass
Encoder #1
Figure 4 – Equipment Setup
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4. Experimental Procedures
Procedure: In this experiment you will analyze the frequency response of the open-loop and the closedloop system. You will also vary the control gains and predict the change in response to such variation.
The following provide instructions for operating the software. Please refer to these instructions as needed
throughout the procedure.
Starting Program
1.
2.
3.
Run Ecp32 from start menu
Download me451lab5.cfg from lab website and save to desktop
File -> Load Settings - select this file
Setting Feedback Gains
1.
2.
3.
4.
5.
6.
7.
Setup -> Control Algorithm
Select continuous time and state feedback
Click setup algorithm
Set Kpf, K1, K2…
Click OK
Click Implement Algorithm
Click OK
Setting Input Signal
1.
2.
3.
4.
5.
6.
Command -> Trajectory
Select sinusoidal
Click Setup
Set amplitude, frequency, number of reps, and open v. closed loop
Click OK
Click OK
Running System
1.
2.
3.
4.
Command -> Execute
Check Normal Data Sampling
Click Run – you should notice the disk turning
Once the data is uploaded, click OK.
Plotting Data
1.
2.
3.
Plotting -> Setup Plot
Make sure only the information you wish to plot is listed in the left axis display
Click Plot Data.
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PART A:
1. Test the open-loop system with no servo motor drive
a) Turn the ECP electronics unit OFF
b) Displace the disk and observe its behavior.
Caution: Do not displace the disk more than 20° because you can damage the long slender
vertical shaft on this system by displacing it too far
“Be nice to our shaft and it will be nice to you…”
Questions:
A.1 Estimate the system’s natural frequency, fn (cycles/s) and damping ratio, ζ.
2. Open Loop Frequency response
Procedure: In this experiment you will analyze the response of the OPEN LOOP torsional mass-springdamper system to a sinusoidal input of varying frequencies and create frequency response diagrams for the
system.
a) Turn the ECP electronics unit ON and start the ECP program
b) Make the system effectively open-loop by setting all feedback gains to 0.
Kpf = K1 = K2 = K3 = K4 = K5 = K6 = 0
c) Set input to sinusoidal with the following parameters.
Select open-loop
Amplitude = 0.5v
f = 0.5 Hz
Number of reps = 5
d) Run the system and plot control effort (E(s) from block diagram) and Encoder-1 Position
(Position of the disk).
e) Determine ratio of output amplitude to input amplitude (gain) and the phase lag (in degrees).
Measure amplitudes peak to peak.
Remember: phase lag can be calculated from the ∆t between the input and output signals
on the graph. φ=∆t*f*360.
To read the graph accurately, you may need to rescale the horizontal axis. Plotting ->
Axis Scaling
f) Repeat steps c-f for the frequencies and amplitudes listed in Table 1. Always set the
“Amplitude (Volts)" to 0.5 WITH THE EXCEPTION of the runs at 2 Hz and 2.5 Hz. In these
runs, use an amplitude of 0.1V. The input will be scaled down, in turn scaling down the
output. However, the ratio that you record will be unaffected. Do you understand why this is
done at this particular frequency? Note that the number of repetitions is increased with
frequency since it takes more number of cycles to reach steady state at higher frequency. You
should take measurements accordingly.
g) Run “tsfreq.m” to plot your data and print your resulting diagrams.
Questions:
A.2. From the Frequency response diagram obtained, estimate the DC gain of the open loop system. How
does this match with the DC gain obtained from Experiment#2 (Check open loop part of experiment#2)?
Explain your result. Make sure you check units.
A.3. From the Frequency Response diagram also estimate the natural frequency, ωn, and the damping ratio,
ζ. Equation (3) will be helpful when determining ζ.
PART B: Closed Loop Frequency Response
Procedure: In this experiment you will analyze the response of the CLOSED LOOP torsional mass-springdamper system to a sinusoidal input and create new frequency response diagrams for the system.
a)
Turn the ECP electronics unit ON and start the ECP program
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b) Set the feedback gains to the following.
Kpf = K1 = 0.1; K2 = 0.002; K3 = K4 = K5 = K6 = 0
c) Set input to sinusoidal with the following parameters.
Select closed-loop
Amplitude = 0.1 radians
f = 0.5 Hz
Number of reps = 5
d) Run the system and plot Commanded Position and Encoder-1 Position (Position of the disk).
e) Determine the gain and phase lag.
f) Repeat steps c-f for the frequencies and amplitudes listed in Table 2.
g) Run “tsfreq.m” to plot your data and print the resulting diagrams.
Questions:
B.1. From the Frequency response diagram obtained, estimate the DC gain of the system. How does this
match with the DC gain obtained from Experiment#2 (Check part 4 of experiment#2)? Explain your result.
B.2. From the Frequency Response diagram also estimate the natural frequency, ωn, and the damping ratio,
ζ. Determine the theoretical values for ωn and ζ from Eq.(8) using the values of K1, K2, and the values of
Kh, k, ke, b and J from experiment#2. Compare these values to your experimental data.
Part C: Effect of damping on sinusoidal response
Procedure: In this experiment you will increase the damping ratio and observe the change in the sinusoidal
response of the system. Repeat steps b through f, described in Part B. In step b, use the values
Kpf = 0.1; K1 = 0.1; K2 = 0.01; K3 = K4 = K5 = K6 = 0
and in steps c and f, use the frequency values and number of repetitions shown in Table 3.
Questions:
C.1. Using the new parameter values, calculate the DC gain of the system, the natural frequency, ωn, and
the damping ratio, ζ.
C.2. Using your results from C.1 and Eq. (3), predict the system gain and phase lag at 1 Hz. Compare your
calculations with data obtained experimentally. Comment on discrepancies, if any.
Part D: Effect of natural frequency on sinusoidal response
Procedure: In this experiment you will increase the natural frequency and observe the change in the
sinusoidal response of the system. Repeat steps b through f, described in Part B. In step b, use the values
Kpf = 0.2; K1 = 0.2; K2 = 0.002; K3 = K4 = K5 = K6 = 0
and in steps c and f, use the frequency values and number of repetitions shown in Table 4.
Questions:
D.1. Using the new parameter values, calculate the DC gain of the system, the natural frequency, ωn, and
the damping ratio.
D.2. Using your results from D.1 and Eq. (3), predict the system gains and phase lag at 1 Hz. Compare your
calculations with data obtained experimentally. Comment on discrepancies, if any.
Conclusion
Summarize the lessons you have learned from this laboratory experience, in few sentences.
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Table 1. Experimental results for Kpf = K1 = K2 = K3 = K4 = K5 = K6 = 0
Run#
Ac
Frequency (Hz)
Number of Reps
Ar/Ac
1
2
3
4
5
6
7
0.5 V
0.5 V
0.5 V
**0.1 V**
**0.1 V**
0.5 V
0.5 V
0.5
1
1.5
2
2.5
3
5
5
10
20
50
20
25
30
Table 2. Experimental results for Kpf = 0.1, K1 = 0.1, and K2 = 0.002
Run#
Ac
Frequency (Hz)
Number of Reps
1
2
3
4
5
6
0.1 rad
0.1 rad
0.1 rad
0.1 rad
0.1 rad
0.1 rad
0.5
1
1.5
2
3
5
0.1 rad
0.1 rad
0.1 rad
1
2
3
0.1 rad
0.1 rad
0.1 rad
1
2
5
 (degrees)
Ar/Ac
 (degrees)
Ar/Ac
 (degrees)
15
20
25
Table 4. Experimental results for Kpf = 0.2, K1 = 0.2, and K2 = 0.002
Run#
Ac
Frequency (Hz)
Number of Reps
1
2
3
Ar/Ac
5
10
15
20
25
30
Table 3. Experimental results for Kpf = 0.1, K1 = 0.1, and K2 = 0.01
Run#
Ac
Frequency (Hz)
Number of Reps
1
2
3
 (degrees)
15
20
30
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Laboratory Report
Name:
Section:
Date:
A.1 Estimate the system’s natural frequency, fn (cycles/s) and damping ratio, .
A.2. From the Frequency response diagram obtained, estimate the DC gain of the open loop system. How
does this match with the DC gain obtained from Experiment#2 (Check open loop part of experiment#2,
make sure you check units)? Explain your result. (Attach Frequency Response plot)
A.3. From the Frequency Response diagram also estimate the natural frequency, n, and the damping ratio,
. Equation (3) will be helpful when determining .
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B.1. From the Frequency response diagram obtained, estimate the DC gain of the system. How does this
match with the DC gain obtained from Experiment#2 (Check part 4 of experiment#2)? Explain your result.
B.2. From the Frequency Response diagram also estimate the natural frequency, n, and the damping ratio,
. Determine the theoretical values for n and  from Eq.(8) using the values of K1, K2, and the values of
Kh, k, ke, b and J from experiment#2. Compare these values to your experimental data.
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C.1. Using the new parameter values, calculate the DC gain of the system, the natural frequency, n, and
the damping ratio, .
C.2. Using your results from C.1 and Eq. (3), predict the system gain and phase lag at 1 Hz. Compare your
calculations with data obtained experimentally. Comment on discrepancies, if any.
D.1. Using the new parameter values, calculate the DC gain of the system, the natural frequency, n, and
the damping ratio.
D.2. Using your results from D.1 and Eq. (3), predict the system gains and phase lag at 1 Hz. Compare
your calculations with data obtained experimentally. Comment on discrepancies, if any.
Conclusion
Summarize the lessons you have learned from this laboratory experience, in few sentences.
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