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AME 20213 Fall 2012: Lab Exercise #3 Procedure Topic: Dynamic Response

Overview: In this laboratory exercise, the students will conduct a series of experiments to explore both how measurement systems and physical systems dynamically respond to a forcing function. First Order Response – Temperature Measurement: Background A thermocouple (TC) acts as a typical first-order dynamic system due to heat transfer properties of the TC. The inner-workings of the TC consist of a pair of dissimilar wires connected via two junctions, a hot and cold junction. A difference in temperature between the junctions forms a voltage difference across them. This voltage is proportional to the temperature difference.

The material and size of the metal probe will affect the rate at which it responds to temperature, and can either shorten or elongate the time it takes to reach a steady-state temperature. The exchange of heat through the thermocouple can be described by conservation of energy, dudt= Q→ ρVc dT

dt= hAs T∞ −T (t)( ) , (1)

where ρ, V, As, and c are the density, volume, surface area, and specific heat of the TC tip, respectively. T is the temperature of the tip and T∞ is temperature of the surrounding fluid. The parameter h is the heat transfer coefficient and dictates how effectively heat is transferred from the fluid to the TC – a higher h means that the TC will warm up more quickly.

At steady state T = T∞ , and there is no heat transfer between the fluid and TC. The solution to this first order differential equation responding a step input (sudden immersion in T∞ ) is

T (t) = T∞ + T0 −T∞( )exp −hAsρVc

t#

$%

&

'( , (2)

where T0 is T(t = 0). If we consider the general form of the decaying part of a first order system exp −t τ( ) , we can see the time constant is clearly related to the physical parameters of the system. In this lab, you will measure the temperature response of the thermocouple when suddenly immersed in warm water and extrapolate the time constant as well as the heat transfer coefficient. Setup and Data Acquisition 1. Ensure that the thermocouple is connected to its controlling box. Connect the TC box to Channel 1 on

the LabQuest using a BNC cable and the provided 5 V cable. When combining the two cables, be sure that the source and GND are connected properly together.

2. Power on your LabQuest unit set-up the sensor for 0-5 V. 3. Set the acquisition time to be > 20 seconds. You can choose the acquisition rate, though the default is

likely sufficient. 4. With the TC not in the water, begin acquiring data. After a few seconds, submerge the tip of the TC

into the warm water until data acquisition is complete. 5. Save your data to a flash drive. Insert the drive in the LabQuest’s USB port, go to File ⇒ Export.

Click the flash drive icon in the dialog, enter a filename, and click ok.

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Useful Notes 1. The properties of the TC are ρ = 8.730 g/cm3, V = 1.762 x 10-3 cm3, As = 7.055 x 10-2 cm2 and c =

0.448 J/g-K. 2. A reasonable heat transfer coefficient is on the order of 500-5000 W/m2-K.

Second Order Response – RLC Circuit: Background Inside the box provided for this part of the lab is a circuit consisting of a variable resistor (R), an inductor (L), and a capacitor (C) – hence RLC circuit. These types of electrical circuits are popular and effective examples of dynamic systems. This one in particular has the ability of switching between first and second order. Said switch simply removes the inductor resulting in an RC circuit. Figure 1 shows the schematic for this system.

Figure 1. RLC circuit used in this lab. Ei is the input voltage (We call it Vin) and Eo is the output voltage (we call it Vout). We know that the voltage across the capacitor in this RLC circuit can be described by a second order differential of the form

LC d2Voutdt2

+ RC dVoutdt

+Vout =Vin t( ) , (3)

If the input voltage is sinusoidal, then the magnitude ratio for the a second-order system is defined as

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M ω in( )= 1

1− ω in ωn( )2[ ]2

+ 2ζω in ωn[ ]2% & '

( ) *

1 2

and the phase shift is

φ ω( ) =

− tan−1 2ζωin ωn

1− ωin ωn( )2

"

#$$

%

&'' for ω ωn ≤1

−π − tan−1 2ζωin ωn

1− ωin ωn( )2

"

#$$

%

&'' for ω ωn >1

)

*

+++

,

+++

In this lab, you will use a function generator to create an artificial sinusoidal input Vin t( ) = Vin sin 2π fint( ) and measure the output with an oscilloscope. From the oscilloscope, you will be able to measure the reduction in amplitude of the signal and the time lag (which can be related to phase lag) of the output.

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Setup and Data Acquisition 1. To learn how to set up and operate the function generator and oscilloscope, watch the video provided

on the course website. 2. Connect the output of the function generator to the input of the RLC circuit box and the output of the

RLC circuit box to the digital oscilloscope. Using a T-junction piece, also connect the output of the function generator to the other channel on the scope, so both the input signal and the output from the RLC are seen simultaneously.

3. Set the variable resistor on the circuit box to be a constant value. (It’s easiest to set it at its maximum or minimum setting, in case it gets bumped out of position.) Record this value – you will need it to determine the natural frequency and damping ratio of the RLC circuit!

4. Be sure the box is switched to the second-order mode. 5. Set the function generator to be a 10 Hz sine wave with amplitude of ~1.50 V. Record the output and

input peak-to-peak voltages and the time lag of the signal. You will use these to determine the magnitude ratio and phase lag of the system.

6. Repeat Step 3 for 100 Hz, 1 kHz, 10 kHz, and 100 kHz. Physical System Response – The Baseball Bat: Background A baseball bat acts like a typical dynamic system and when the bat strikes the ball, the impact creates a vibration in the bat displacing it from equilibrium some distance y. The striking of the ball is an impulse force because it happens ‘instantaneously’ as opposed to the step or sinusoidal inputs we discussed in class. The bat is a second order system and the resulting general solution for the displacement is

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ybat t( ) =1

2ωn ζ 2 −1e−ζ + ζ 2−1% & ' (

) * ωnt

− e−ζ − ζ 2−1% & ' (

) * ωnt+

, -

.

/ 0 , (6)

where ζ is the damping ratio and ωn is the natural frequency of the bat. The bat can be considered under-damped (0<ζ<1) and this simplifies Eq. (2) to

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ybat t( ) =ωne

−ζωnt

1−ζ 2sin ωd t( ) , (7)

where ωd is the ringing frequency given by

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ωd =ωn 1−ζ 2 . (8) Therefore, if the natural frequency ωn and damping ratio ζ of a given baseball bat are known, the vibrational response to any impulse can be found. That is, it does not matter how hard the ball is struck, the bat will respond in the same manner. The ringing frequency and damping ratio of a baseball bat can be measured directly leading to the natural frequency. However, if the frequency is measured incorrectly, the wrong response will be predicted. The bat for this laboratory exercise is outfitted with four strain gages in a full bridge configuration and they respond to the displacement of the bat ybat versus time t. The strain gages are connected to an amplifier to increase the output voltage, and the output voltage is recorded by an oscilloscope, which samples at a very high rate (GHz – which easily satisfies Shannon’s sampling theorem). The oscilloscope voltage directly relates to the magnitude of the displacement.

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Setup and Data Acquisition The baseball bat will be connected to the amplifier and oscilloscope when you as shown in Figure 2.

Figure 2. Schematic of experimental apparatus and measurement system.

1. Set the input channel to “AC” and press “AutoSet” on the oscilloscope. A steady line with some small oscillations should appear. This is a real time measurement of the signal from amplifier and is the background signal from the electrical connections corresponding to the noise of the measurement system. 2. Adjust the voltage scale on Channel 1 to be 50 mV/division and adjust the time scale to be 10 ms/division. This is the scaling required to observe the response of the baseball bat. 3. Set the oscilloscope to trigger in response to an impulse forcing function. This means that the oscilloscope is prepared to record data (trigger) at the moment of the impulse on the bat:

• press the “Trigger” menu button • select the slope menu and choose the icon shown in Figure 3 • set the source as Channel 1(the trigger is looking for a signal from Channel 1) • set the mode as “Normal”(the trigger is waiting for a single event or impulse to occur), and the

coupling as “DC” • move the trigger level knob so that the trigger level is at 20mV – this ensures that the

oscilloscope will not trigger until the signal slope falls to this level • press “Single Sequence” button – the green LED should light • using the horizontal position knob, move the trigger arrow to the extreme left of the screen

4. Above the graph readout on the right hand side there will be some text. Wait for the text to read “Trig?” before proceeding. 5. Keeping safety and those around you in mind, hit the ball with the bat. Do not swing hard as the strain gages will be broken or displaced. 6. Once the signal is acquired, press the “Start/Stop” button and save the data to a disk (see Saving Data below). 7. Repeat this procedure until a good set of data has been obtained.

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Figure 3. Selection for Slope Menu on digital oscilloscope.

Saving Data 1. Save the frozen trace by pressing the “Save/Recall” button at the top of the control panel. Make sure that Channel 1 is selected and that “Save Waveform CH1” is selected by pressing the corresponding button on the bottom of the screen. 2. Press “To Ref 1” on the right side of the screen (the date and time should change). 3. Turn CH1 off by pressing “off.” 4. Turn the REF1 trace on by pressing the white “REF” button and select Reference 1 by pressing the corresponding button at the bottom of the screen. 5. Insert a floppy disk into the oscilloscope. 6. Press the “Save/Recall” button again, but this time select the “To File” button on the right of the screen. Be sure that “Spreadsheet File Format” is selected on the right side of the screen. 7. Use the cursor position knob to highlight the “TEK?????.CVS” file. Select the “Save Ref1 To Selected File” option from the menu on the right side of the screen. The “?????” will be replaced by the next sequential number, starting at 00000 when no other files are present. 8. It will take a couple minutes to save the data to the disk. Data Analysis In order to perform a fast Fourier transform (FFT) on the data, a Matlab script title lab3_fft.m is provided on the website. The code is well commented and should be straightforward to implement. The student should simply change the input file name (and extension) for each set of their data. The script plots both the actual response (voltage versus time) as well as the frequency response (amplitude versus frequency in Hz). The ringing frequency ωd is the frequency corresponding to the largest magnitude. To do this analysis, however, it is useful to smooth the data to remove noise. When done post de facto on acquired data, this is called an artificial filter. It essentially removes the fluctuations that occur on a very short time scale, allowing the longer time scale trends to become apparent. The simplest way to smooth data is to use a moving average. In this case, each data point at time t becomes an average of the data points immediately preceding and following the data point or

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ynew t( ) =...+ y t + 2Δt( ) + y t + Δt( ) + y t( ) + y t −Δt( ) + y t − 2Δt( ) + ...

N, (9)

where N is the number of neighboring data points considered. It is up to the student to determine the best N to use for this laboratory exercise. In order to determine the natural frequency and plot the theoretical response, it is still necessary to determine the damping ratio. In Eq. (3), the natural frequency and damping ratio both appear in the exponential term

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e−ζωnt . Consider the following generic image for a damped response to a step input:

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Figure 4. Generic response to a step impulse.

Consider the peaks labeled y1 and y2. If the student takes the ratio of these peaks using Eq. (3), they should obtain

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y2y1

=e−ζωnt2

e−ζωnt1= e−ζωn t2− t1( ) . (10)

Equation (6) can then be solved for the damping ratio

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ζ = −

ln y2y1

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'

( )

ωnΔt. (11)

However, the natural frequency ωn is not yet known, only the ringing frequency ωd is given by the Fourier transform. Therefore, you will need to combine Eqs. (11) and (8) to determine ωn and ζ, respectively. Useful Notes 1. To see the effect of filtering, you will likely need to plot the Fourier transform all the way to the

maximum resolved frequency, since it is only the high frequency content that is filtered. 2. In order to plot both the theoretical and actual measured data on the same plot, the theoretical

equation will need to be scaled to approximately the same amplitude as the measured data.