Figure 1.1.1: Magnitude ratio as a function of time using normalized temperature data

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1 9Student : Solutions to Technical Questions Assignment : Lab 3 Technical Questions Date : 1 November 212 Course Title : Introduction to Measurements and Data Analysis Course Number : AME 2213 Instructor : Dr. David B. Go Laboratory TA : Tamuto Takakura Question 1 (5 points) First Order Response Temperature Measurement: (1) Normalize the temperature data from the thermocouple and plot the magnitude ratio as a function of time. Using the raw data and plot, estimate the time constant of the measurement system (recall steady state takes ~5τ). Linearize the data (hint: take natural logarithms of the equation) and conduct a curve fit. Extrapolate the time constant from the curve fit and compare to the original estimate. Quantify and discuss the comparison M(t) Time (s) Figure 1.1.1: Magnitude ratio as a function of time using normalized temperature data Recall that the magnitude ratio is defined as M ( t) =1 e t τ. When t = τ, the magnitude ratio is M ( t = τ ) =1 e Using Figure 1.1.1, the red line estimates when the magnitude ratio 1

2 reaches this value. This occurs at approximately.4 seconds; therefore the time constant is τ ~.4 s. Note that other measures, such as M(t = 5τ) =.99 could also be used ln(m(t)) Time (s) By linearizing the data, one obtains Figure 1.1.2: Linearized thermocouple data ln"# 1 M ( t) $ % = t. τ One can determine the time constant from a linear curve fit by noting the slope is equivalently 1/τ. For this data, the curve fit equation is ln"# 1 M t $ % = t.25, such that the time constant is τ = 1/ =.3821 s. It is also important to note that data after 1.75 seconds is scattered about a mean value of approximately 5. This is because in a true first-order system dynamic response system, the data would infinitesimally approach a magnitude ratio of one (M = 1); however, because of noise in the DAQ system, the final magnitude ratio is near one with scatter. Hence, data greater for t > 1.75 s should be excluded in the curve fit calculation. The estimations to determine time constants compare extremely well, differing only by.179 s and less than 5 %. ( ) 2

3 (2) From the curve fit time constant, estimate the heat transfer coefficient h at the tip of the thermocouple (be sure to have the correct units). How are h and τ related? Does increased heat transfer increase or decrease the time constant? Does this make physical sense? The heat transfer coefficient can be determined using the equation, τ = ρcv = mc. ha! ha! Rearranging, we can find the heat coefficient to be h ~ 253 W/m 2 -K, with a given density of 8.73 g/cm 3, bead volume of x 1-3 cm 3, bead surface area of 7.55 x 1-2 cm 2 and specific heat of.448 J/g-K. Again, using the relation above, increasing the heat transfer increases h and will decrease the time constant, and the vice versa is true for decreased heat transfer. This makes sense, since by increasing the heat transfer, the thermocouple bead would be able to react to the temperature change much faster, yielding a faster time response. 3

4 Question 2 (5 points) Second Order Response: (1) Generate a plot of the magnitude ratio M(f in ) as a function of the signal input frequency f in (recall that ω in = 2πf in ) for the RLC circuit. Also include the theoretical magnitude ratio based on the known relationship for second order systems. (Must show calculated values of the natural frequency and damping ratio.) Discuss how the experimental data compares to theory quantitatively. What type of filter is this? Can you determine the cut-off frequency? 1.9 Theory Experimental M(f in ) Natural frequency is defined as, Figure 2.1.1: Magnitude ratio as a function of signal input ω! = 1 LC, and damping ratio is, ζ = R 4L/C. Therefore, with a maximum resistance setting of 91 Ohms, the natural frequency is ω n = rad/s (2.73 khz) and the damping ratio is ζ = The theory agrees well with the experimental results, with the greatest relative error being 1.54 %. This is a low pass filter and the cut-off frequency can be estimated graphically by when M(ω in ) =.77 or approximately ω c ~ 79 rad/s (~5 Hz). f in 4

5 (2) Similar to Question (1), generate a plot of the phase shift φ as a function of the signal input frequency f in (recall that ω in = 2πf in ). Also include the theoretical phase shift based on the known relationship for second order systems. Discuss how the experimental data compares to theory quantitatively. -.5 Theory Experimental -1 φ(f in ) Figure 2.2.1: Magnitude ratio as a function of signal input The theory agrees well with the experimental results, with the greatest relative error being 8.91%. Although higher in error compared to the magnitude ratio, this is reasonable, however, since the time lag was measured via oscilloscope cursors and require human precision to align the voltage signals to get accurate measurements. Question 3 (5 points) Physical System Response The Baseball Bat: (1) Plot the voltage response as a function of time of the baseball bat with and without artificial filtering (one plot or two plots your choice). Plot the Fourier transform of the response with and without artificial filtering (one plot or two plots your choice). Explain the artificial filtering you chose and how it affects the output. f in 5

6 .25.2 Without Filtering With Filtering y(t) Time (s) Figure 3.1.1: Voltage response as a function of time of the baseball bat with and without artificial filtering 6

7 relative amplitude frequency (Hz) Figure 3.1.2: FFT of response without filtering relative amplitude frequency (Hz) Figure 3.1.3: FFT of response with filtering 7

8 relative amplitude frequency (Hz) Figure 3.1.4: FFT of response without filtering (higher frequencies shown) relative amplitude frequency (Hz) Figure 3.1.5: FFT of response with filtering (higher frequencies shown) 8

9 The artificial filtering used is a moving average; the sized used is 35 points. By using this filtering, the high frequency noise is removed and smoothed; this can be seen in Figure The reduction in the higher frequency noise can be seen when Figures and are compared. The relative amplitude is significantly reduced for frequencies in the 35 to 5 Hz range. (2) Determine the natural frequency (in Hz) and damping ratio of the bat. Be sure to show work and note the type of bat you used. Using these parameters, plot the theoretical response of the bat along with your experimental response. (Note you will want to scale the experimental response to the theory or vice versa) How do they compare? Using both Equations (11) and (8) of the laboratory handout, a system of equations can be obtained to determine the natural frequency and damping ratio. Solving for natural frequency, ω! = ω! 1 ζ! and, substituting for natural frequency in equation (11), one can obtain ln y! y ζ =! ω! 1 ζ Δt.! Taking values from Figure 3.1.1, an y 1 value of -.25 and y 2 value of is estimated. The only unknown variables in the above equation are the natural frequency and damping ratio. Solving this simple set of two equations and two unknowns, one can find that the natural frequency is f n = Hz and the damping ratio is ζ =.668. As shown in Figure 3.2.1, the theory predicts the actual baseball bat response well. 9

10 .25.2 Experimental Theory Figure 3.2.1: Theoretical and real response of the baseball bat. 1

Lab 4 RC Circuits. Name. Partner s Name. I. Introduction/Theory

Lab 4 RC Circuits. Name. Partner s Name. I. Introduction/Theory Lab 4 RC Circuits Name Partner s Name I. Introduction/Theory Consider a circuit such as that in Figure 1, in which a potential difference is applied to the series combination of a resistor and a capacitor.