Sinusoidal Response of a 2 nd Order Torsional Mass-Spring ... · PDF fileSinusoidal Response of a 2 nd Order Torsional Mass-Spring Damper System Important: Bring your short form from

ME 451: Control Systems Laboratory

Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 1

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



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, ωn, and the damping ratio, ζ. Both ωn and ζ are functions of system parameters, both physical and control parameters. In this experiment we will alter the ωn and ζ values by changing the feedback control gains. The objective of this experiment is to investigate the relationship between ωn and ζ and the frequency response of the system, as well as the relationship between the feedback gains and ωn and ζ. 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

22

2

2)(

)()(

nn

n

ss

K

sU

sYsG

ωςωω

++== (1)

where Y (s) and U(s) are the Laplace transforms of the output and input variables, respectively, ωn is the natural frequency, and ζ is the damping ratio. For a sinusoidal input

)*sin()( tAtu ω= , 22

)(ω

ω+

=s

AsU

the response of the system, in Laplace domain, can be written as

)2)(()(

2222

2

nn

n

sss

KAsY

ωςωωωω

+++=

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

)*sin()()( φωω += tjGAty , )( ωφ jG∠≡ (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 )( ωjG and the phase lags behind the input by )( ωjG∠ .



For the standard second-order system in Eq.(1), given the values of ωn and ζ, the “gain" )( ωjG

and the “phase" )( ωjG∠ can be expressed as a function of ω , as follows

2222 4)1()(

rr

KjG

ςω

+−= ; )(

n

rωω≡ (3)

−−=∠=

21

*2arctan)(

r

rjG

ςωφ (gives units of radians)

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 )(n

rωω≡ , for different values of ζ in Fig.1.

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 θ angular position of rotor ω angular velocity of rotor

The parameters of the system, shown in Fig.2, include



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 θ as output, can be written as

++=

++==

)/()/(

)/(11

)(

)()(

22 JksJbs

Jk

kkbsJssT

ssG

θ (4)

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

J

kn =ω ,

J

bn =ζω2 ,

Jk

b

2=ζ (5)

It is clear from Eq.(5) that ωn and ζ 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

)( 21 θθθ &KKKKkT dpfhe −−=

** If you had calculated the gain ek from earlier experiment you would remember that it was found to be

close to 25



Figure 3. Block Diagram of Programmable Torsional mass-spring-damper system

where hK is the hardware gain of the motor; T is the input torque; the feedback generated torque θ1K−

is the type of torque generated by a spring; and the feedback generated torque .

2 θK 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

TkbJ =++ θθθ.

&& ⇒ θθθθθθ 12 KkKKkKKkKkbJ ehehdpfeh −−=++ &&&&

and then ( ) ( ) ( ) dpfeheheh KkKKkKkKkKbJ θθθθ =−+−+ 12&&&

The block diagram of the programmed system is shown in Fig.3. The transfer function of this system, with

dθ as input and θ as output, can then be expressed as

( )[ ] ( )[ ]

++++==

JKkKksJKkKbs

JKkK

s

ssG

eheh

pfeh

d //

)/(

)(

)()(

122θ

θ (6)

which has the same structure as that in Eq.(4). The DC gain of the programmed system is

)(1KkKk

KkKK

eh

pfeh

+= , (7)

and the natural frequency and damping ratio are given by the relations

J

KkKk ehn

)( 1+=ω ;

JKkKk

KkKb

eh

eh

)(2 1

2

++=ζ (8)

One of the objectives of this experiment is to study the effects of varying the natural frequency ωn and the damping ratio ζ on the sinusoidal response of the torsional mass. We will vary ωn and ζ by varying the

gains 1K and 2K in software. The hardware gain, ehkK , 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.



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



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



5) At which frequency (A, B, or C) does the response below correspond on the bode plot above?

Answer: A



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. 5. Matlab script “tsfreq.m" 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.



Figure 4 – Equipment Setup

Torsional Spring (Rod)

ECP

Servomotor

Mass

Encoder #3

Encoder #1

Encoder #2



4. Experimental Procedures Procedure: In this experiment you will analyze the frequency response of the open-loop and the closed-loop 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. Run Ecp32 from start menu 2. Download me451lab5.cfg from lab website and save to desktop 3. File -> Load Settings - select this file

Setting Feedback Gains

1. Setup -> Control Algorithm 2. Select continuous time and state feedback 3. Click setup algorithm 4. Set Kpf, K1, K2… 5. Click OK 6. Click Implement Algorithm 7. Click OK

Setting Input Signal

1. Command -> Trajectory 2. Select sinusoidal 3. Click Setup 4. Set amplitude, frequency, number of reps, and open v. closed loop 5. Click OK 6. Click OK

Running System

1. Command -> Execute 2. Check Normal Data Sampling 3. Click Run – you should notice the disk turning 4. Once the data is uploaded, click OK.

Plotting Data

1. Plotting -> Setup Plot 2. Make sure only the information you wish to plot is listed in the left axis display 3. Click Plot Data.



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-spring-damper 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-spring-damper 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



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.



Table 1. Experimental results for Kpf = K1 = K2 = K3 = K4 = K5 = K6 = 0

Run# Ac Frequency (Hz) Number of Reps Ar/Ac φ (degrees)

1 0.5 V 0.5 5 2 0.5 V 1 10 3 0.5 V 1.5 20 4 **0.1 V** 2 50 5 **0.1 V** 2.5 20 6 0.5 V 3 25 7 0.5 V 5 30 Table 2. Experimental results for Kpf = 0.1, K1 = 0.1, and K2 = 0.002


1 0.1 rad 0.5 5 2 0.1 rad 1 10 3 0.1 rad 1.5 15 4 0.1 rad 2 20 5 0.1 rad 3 25 6 0.1 rad 5 30 Table 3. Experimental results for Kpf = 0.1, K1 = 0.1, and K2 = 0.01


1 0.1 rad 1 15 2 0.1 rad 2 20 3 0.1 rad 3 25 Table 4. Experimental results for Kpf = 0.2, K1 = 0.2, and K2 = 0.002


1 0.1 rad 1 15 2 0.1 rad 2 20 3 0.1 rad 5 30



Laboratory Report

Name: Section: Date: A.1 Estimate the system’s natural frequency, fn (cycles/s) and damping ratio, ζ. 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, ωn, and the damping ratio, ζ. Equation (3) will be helpful when determining ζ.



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.



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. 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.

Sinusoidal Response of a 2 nd Order Torsional Mass-Spring ... · PDF fileSinusoidal Response of a 2 nd Order Torsional Mass-Spring Damper System Important: Bring your short form from

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