Laboratory Exercise 3: Dynamic System Response Laboratory Handout AME 250: Fundamentals of Measurements and Data Analysis

Prepared by: Matthew Bennington Date exercises to be performed:

Deliverables:

Part I

1) Using the output from the DO find τ

2) From τ and C calculate R for the Variable Resistor

Part II 3)Complete the RLC response table. Using these results and a program or M-file, construct two plots on of M(ω) and other of Φ(ω) versus the normalized frequency ratio. These plots must contain the data along with the theoretical curves.

4) Does the data support the conclusion the RLC circuit behaves as a second-order system in both cases? Finally, compare the values of the damping ratio found with the value of R/Rc. Do this by comparing the data with the corresponding values determined using various values of R/Rc in the given equations. How well do the experimental and theoretical values of ζ compare?

Part III 5) What is the natural frequency (ωn) and damping ratio( ζ) of the bat? Where you able to change it? Why or why not?

6) Plot the experimental and theoretical response of the bat

Introduction:

The main objective of this laboratory exercise is to investigate the dynamic response characteristics of first-order and second order measurement systems. The first week, the dynamic Response of a first order RC circuit and second order RLC circuit will be studied. The second week you will examine the impulse response of an aluminum bat. Data will be acquired using a digital oscilloscope (DO).

First Order System Response:

For this part of the exercise a DO will be used to determine the time constant of an RC circuit. A function generator will be used to generate a square wave of given magnitude and frequency. This circuit consists of a resistor (R) and a capacitor (C) and is shown in Figure 1 below.

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Variable Resistor: R= 1 Ω to 910 Ω

Ei

C=0.68 µF

Figure 1. First Order Circuit

We know from Kirchhoff’s voltage law

dV RC +V = E (t) (1) dt i

Therefore τ=RC, τ being the time constant and R being the value of the variable resistor.

Make sure the output cable from the FG is attached to the input of the RLC box and in parallel to channel 1 on the DO. The output of the RLC box should be connected to channel 2 of the DO. In this way, both the input and output signals of the RLC box can be viewed on the DO. Make sure that the toggle switch is set to “1st Order.” Now turn the R knob on the RLC box to that which you chose before running the lab. Set the FG and DO to their initial settings, 100 Hz frequency and 3.64 V peak to peak amplitude and no DC offset. On the DO, for both channels AC with divisional settings of 500 mV and 2.50 ms. (You can change these values in order to view a particular part of the graph better.) Using the cursors at 63.2% of the input magnitude find the time. This will correspond to the time constant τ and is illustrated in Figure 2.

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Figure 2. First Order System Response

Second Order System:

In this part a FG and DO will be used to determine the response characteristics (the magnitude ratio and the phase lag as functions of the input frequency) of an electrical RLC circuit. This circuit consists of a resistor (R), an inductor (L) and a capacitor (C) and has the response characteristics of a second-order system. The circuit will be characterized by providing an input sinusoidal wave of known amplitude and frequency from the FG to the RLC circuit and measuring the circuit’s output amplitude and time delay using the DO. The electrical diagram of the RLC circuit is shown below

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Variable Resistor: R= 1 Ω to 910 Ω L= 5 mH (R= 9.0 Ω)

Eo Ei

Figure 3. RLC Circuit

The voltage differences, V, across each component in an AC circuit are V = RI for the resistor, V=L(dI/dt) for the inductor and V=Q/C for the capacitor, where I = dQ/dt. In this circuit, all three components are in series. Thus application of Kirchhoff’s voltage law for the circuit gives

 2   d Q  + dQ + Q = ω L 2  R Ei sin( t) (5)  dt  dt C

This second order-linear differential equation can be solved for Q to yield the steady-state output voltage amplitude

E = Q = i Eo (6) C C (1/ C − Lω 2 ) 2 + (Rω) 2

From Equation (3) and the solution equation for Q, the magnitude ratio is

E 1 M (ω) = o = (7) 2 2 2 Ei []− ω ω + []ω ω 1 ( / n ) 2(R / Rc )( / n )

And the phase lag is

− 2(R / R )(ω /ω ) (8) φ(ω) = tan 1  c n   1− (ω /ω )2  n

4 This equation yields positive values of φ(ω). By convention, because φ(ω) is a phase lag, it is plotted as having negative values. Further for ω > ωn, the phase shift must be referenced correctly. Thus the conventional plot of φ(ω) ( ino) versus ω would actually o be -φ(ω) for ω ≤ ωn and -180 φ(ω) for ω > ωn. Also note that in Equations 4 and 5, the = resonant frequency is given by Rc 2 L / C . One other note, the damping ratio, ζ, equals R/Rc for this circuit.

Make sure the output cable from the FG is attached to the input of the RLC box and in parallel to channel 1 on the DO. The output of the RLC box should be connected to channel 2 of the DO. In this way, both the input and output signals of the RLC box can be viewed on the DO. Make sure that the toggle switch is set to “2nd Order.” Now turn the R knob on the RLC box to that which you chose before running the lab. Set the FG and DO to their initial settings, 100*K (K being the frequency multiplier for your group) Hz frequency and 3.64 V peak to peak amplitude sine wave and no DC offset. On the DO, set both channels AC with divisional settings of 500 mV and 2.00 ms. (You can change these values in order to view a particular part of the graph better.)

Data will be analyzed in the final form of M(ω) and φ(ω), each versus the normalized frequency ration, ω/ωn. These values will be determined from the raw data. This includes the input and output amplitudes, Ei and Eo and phase lag time, ∆t, which is the time between the peak of Ei and the corresponding peak of Eo. The phase lag in degrees o equals –(360 )( ∆t/Ti) where Ti is the inverse of the input frequency in HZ and the minus sign indicates a lag in time.

Once a satisfactory set of signals has been captured on the DO display, use the cursors to record the data. Enter all the raw data in the first three columns of Table 2. The last three columns can be filled in after the lab. When done with an input frequency, set the next one on the FG and repeat the process.

Table 2. RLC Response Data Table o Freq.(Hz) Ei (V) Eo (V) ∆t (s) ω (rad/s) M(ω ) Φ(ω) ( ) 100 * K 500 * K 1000 * K 1200 * K 1600 * K 2000 * K 2200 * K 2500 * K 2650 * K 2800 * K 3100 * K 3600 * K 4000 * K 5000 * K 7000 * K

5 10000 * K

When you are finished, using a voltmeter measure the total resistance of the RLC circuit (between the center pins of the IN and the OUT connectors). Subtract 9Ω from this value and record it. This is the value of R for this case, which will be needed later in the calculations.

Variable Resistor:

7 6 5 8 4 9 3 10 2 11 1

Max Min

This is a diagram of the variable resistor and the number values assigned to each tic mark. Turn the knob to the number which your group has selected and run part II at this resistance.

Bat Impulse Reponse:

In week two you will examine the system response of bat striking a ball. This can be characterized as an impulse response since the ball will be stationary. In your text the step response of a second order system is reviewed on page 113. The response to an impulse is the derivative of the step input response.

dy s yδ (t) = (9) dt

Therefore the response can be characterized as

  −ς + ς 2 − ω  −ς − ς 2 − ω  = 1   1 nt −  1 nt  yδ (t) e e  (10) ω ς 2 −   2 n 1 for cases when the system is underdamped (0<ζ<1) this equation simplifies to the form

6 −ςω ω e nt = n ω yδ (t) sin( d t) (11) 1−ς 2 where the ringing frequency or damped natural frequency (ωd) is defined as

ω = ω − ς 2 d n 1 (12)

The impulse response is plotted in Figure 4 for a range of damping ratios and a natural frequency of 1.2.

Figure 4. Second Order Impulse Response.

The purpose of this lab is determine the natural frequency and damping ratio of an aluminum bat. Strain gauges have been added to the bat to measure any vibrations as a result of the bat striking the ball. The orientation of these strain gauges in relation to where you strike the bat is important. A marker will be placed on the bat to identify the proper face to strike the ball. A schematic of the bat setup is shown in Figure 5.

7 Wheat Stone Bridge/Amplifier

Digital Oscilloscope Strain Gauges

Figure 5. Schematic of baseball bat setup

The bat will be properly connected when you arrive for the lab. On the DO, set the input channel to AC. Press auto set on the DO. A steady line with some noise that reads the output from the amplifier should appear. Adjust the voltage scale on the channel 1 to be 50 mV/div and adjust the time scale to be 10ms per division. Now set up the DO trigger correctly in response to a step-input forcing: Press the trigger menu button. Select the slope menu and choose,

source = ch1 (the trigger is looking for a signal from channel 1), mode = Normal, the trigger is waiting for a single event to occur), coupling = DC. Move the trigger level knob so that the trigger level is at 20mv. This ensures that the DO will not trigger until the signal slope falls to this level. Above the graph readout on the right hand side there will be some text. You must wait for the text to read “Trig?” before proceeding. Now hit the ball with the bat. Once the signal is acquired hit the start/stop button and save the file to a disk*. Once you have done this repeat this same procedure with your lab partner.

8 Determining Damping Ratio and Natural Frequency

In Equation 11 there is an exponential term e-ωnζt which dictates the decay of the signal as a function of the damping ratio and natural frequency. It is possible to solve for either the damping ratio or natural frequency from the time trace if one quantity is known. In Figure 5 the system response is plotted versus time for a damping ratio of 0.1 and natural frequency of 1.2. The first two peaks are identified as y1 and y2.

Figure 5. System Impulse Response for ζ= 0.1 & ωn=1.2

We could then take the ratio of y2/y1 which would result in,

y −ςω +ςω 2 = e nt1 nt2 (13) y1

Therefore if the natural frequency is known we can solve for the damping ratio,

y ln 2 y ς = 1 (14) ω ∆ n t

9 WE can find the natural frequency of the signal by using pwelch or FFTs in Matlab and plotting the power spectre of the signal. The frequency with the largest magnitude corresponds to the natural frequency of the system. Remember that ω=2πf! When plotting the theoretical and experimental responses of the bat it might be necessary to shift and scale the magnitude of the theoretical response.

Error Analysis

Due to the inaccuracy of some of the equipment used as well as the refinement of your curve fit there is error associated with both the measurements and results in this lab. It would be helpful to try and quantify these since it will affect your results.

References: [1] Dunn, P. F., Measurement and Data Analysis for Engineering and Science, First Edition, McGraw-Hill, New York, NY, ©2005

* Save the frozen trace by pressing the SAVE/RECALL button at the top of the control panel. Make sure that CH1 is selected and that the “Save Waveform CH1” is selected by pressing the corresponding button on the bottom of the screen. Press the button on the right side of the screen corresponding to “To Ref 1” (the date and time should change). - Turn CH1 off by holding down the CH1 button and pressing off. - Turn REF1 trace on by pressing the white REF button and select Ref 1 by pressing the corresponding button at the bottom of the screen. - Insert your floppy disk into the oscilloscope. - Press the SAVE/RECALL button again, but this time select the button corresponding to “To File” from the buttons to the right of the screen.Make sure that “Spreadsheet File Format” is selected in the menu on the right side of the screen, then use the knob you used to change the cursor position to highlight the TEK?????.CVS file. Select the “Save Ref1 To Selected File” option from the menu on the right side of the screen. (It will take a couple minutes to save the data to the disk and the ????? will be replaced by the next sequentially available number starting at 00000. I would suggest writing down the file name it is saved as for future reference).

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