DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 1

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1 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 1 A schematic for a small laboratory electromechanical shaker is shown below, along with a bond graph that can be used for initial modeling studies. Our intent is to understand the performance characteristics of this shaker, which is designed to provide controlled vibrational testing of objects attached to the table (represented by mass, m t ), or to forcibly excite small test specimens to evaluate their structural response properties. Parameters: L c = H, R c = 2.4 Ω, k s = N/m, b s = 60 Ns/m, r = 5 N/A, m t = 0.4 kg Study the bond graph and applied causality indicated. The input voltage, v in, is supplied by a voltage amplifier (not shown), and the induced armature current, i a, induces an electromechanical force, F em = ri a, where r is the electromechanical (gyrator) constant. Note that this bond graph is very similar to the one seen before for a pmdc motor. For the purposes of this homework, assume all elements have linear constitutive relations. Problem 1 (Transfer Function and Frequency Response of EM Shaker Case 1): (a) Simplify the bond graph model provided above for the case where armature current, i a, is considered the input. In other words, consider the system as shown below:

2 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 2 Derive state equations (show this is a 2nd order system). Write one output equation for table acceleration; i.e., y = a t. Indicate the A, B, C, and D matrices in terms of the system parameters. Solution: Strictly speaking, the the current is not the input in the laboratory setup, but for the we can still find the transfer function treating the current as a known input. From the bond graph, there are two state equations, derived as follows (note: parameters here have subscripts not used in the original document): (b) Derive the transfer function for the system in part (a) using the G(s) formula, G(s) = C[sI A] 1 B + D. Note that G(s) is a single transfer function, G(s) = a t /i a, where a t is the table acceleration. Your final results should be in terms of system parameters. You should find that the transfer function between table acceleration and armature current (as input) is, G ati a (s) = a m i a = rs 2 ms 2 + bs + k Solution: From the defined system matrices, first find the [si A] 1, then the G(s) as follows:

3 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 3

4 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 4 (c) Use the parameter values provided to determine the magnitude and phase functions of the frequency response for this transfer function. You can plot the magnitude (in db) and phase (in degrees) functions using the bode() function either in LabVIEW or Matlab. Practice using both the state-space (from (a)) and TF (from (c)) models with these computational tools. Sample programs will be provided. Solution: not provided

5 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 5 Problem 2 (Transfer Function and Frequency Response of EM Shaker Case 2): (a) Refer to the bond graph model for the case where voltage, v in, is considered the input, and derive the state equations for this case. Write one output equation for table acceleration, y = a t, and again indicate the A, B, C, and D matrices in terms of the system parameters). Solution: The bond graph and derived equations are as follows. Note that it is not necessary to insert an amp gain in your model, since this was not provided in the lecture discussion slides. It is expected that you studied the slides and thus began with the bond graph given below.

6 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 6 (b) Derive the transfer function for this (3rd order) system using the G(s) formula, G(s) = C[sI A] 1 B + D. where now G(s) is a transfer function, G(s) = a t /v in, and you should show that, G atv in (s) = a m v in = rs 2 L c ms 3 + (R c m + L c b) s 2 + (r 2 + R c b + L c k) s + R c k Solution: From the defined system matrices, the G(s) function is (using MathCAD for symbolics):

7 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 7 (c) Use the parameter values provided to plot the magnitude and phase functions of the frequency response for this transfer function as done for Problem 1.

8 DSC HW 4: Assigned 7/9/11, Due 7/18/12 Page 8 Problem 3 (Linear Response Analysis of Electromechanical Shaker): For the electromechanical shaker studied in Problem 2, solve for key system response variables assuming the input is: (a) A step input of magnitude 10 V that turns on at 0.1 seconds. Solve from t = 0 to t = 0.5 seconds. Solution: In the graphs shown below, the state trajectories are all on the bottom graph and values for each state have been normalized by the peak value of each (shown in legend). (b) A triangular pulse input with peak voltage of 10 V that has a base period of 0.2 seconds and turns on at 0.1 seconds. Solve from t = 0 to t = 0.5 seconds.