1. Problem 19.169 Transient Response By: Matt Lausmann ME 114 Granda Assignment #3

2. The Problem

3. Possible Solutions Two very fast and efficient methods to analyze the transient response of this set up would be to employ the use of: 1)Camp-G and it’s MATLAB interface 2)SIMULINK Both methods produce the same results. Since we are trying to find the amplitude of the displacement of the block, and the amplitude of the force of the spring/damper combination on the ground, certain modifications and parameters need to be attended to for both methods of solution.

4. Camp-G – Bond Graph The following is the bond graph model used to create the differential equations with the Camp-G-MATLAB interface tool. In this bond graph, the SF entities are initialized to zero. You can see there are 2 springs and one damper, and one Source of Force (from the motor).

5. Camp-G – Campgequ.m The first modification done to campgequ.m was to define the equation used to calculate the Input function SE1. This is a sinusoidal input. The second modification done to the campgequ.m file was to activate the lines that define the output vectors for efforts and flows. The efforts vector was defined as the sum of the efforts of the 2 springs and the damper. This is so the plot of the spring and damper force on the ground could be generated.

6. Camp-G – Campgmod.m The campgmod.m file that was generated from Camp-G also needed slight modifications. The first was modification was the initialization of input variables (spring constants, etc). The second modification needed in campgmod.m dealt with the plots. The first figure is the plot of Q6, which is the first row of the p_q array and represents the vertical displacement of the motor. The second figure is the plot of the force on the ground from the spring/damper combination.

7. Camp-G – Vertical Displacement This is a plot of the vertical displacement of the motor. The plot seems to level out around +- 7.1x10 -6 meters. This corresponds exactly with the solution given in problem 19.169

8. Camp-G – Force of Spring and Damper on Ground This is a plot of the force of the spring and damper combination on the ground. The maximum force of the combination appears to be roughly 5.7 Newtons, which also agrees with the solution given for 19.169.

9. Simulink – Block Model 1 This is a copy of the block model created in Simulink. To create the Sinusoidal forcing function, a clock and sine function were employed. The top left corner of the block function represents the input forcing sinusoidal function in the differential equation of motion shown above. A scope was placed on this in order to measure its magnitude.

10. Simulink – Block Model 2 The rest of the block model is exactly like previous simple spring/damper problems. On the right side of the sum block, the differential equation is divided by the mass as shown in the “Gain2” block. Then the differential equation of motion is integrated twice. A scope is placed after the 2 nd integration in order to graph the vertical displacement of the motor. A summation block also joins the gain blocks containing the spring and damper constants b and k. This is the summation of –(b/m)xdot and –(k/m)x (which both have units of Force). The scope joined to this summation enables the plotting of the force of the spring and damper combination.

11. Simulink – Block Initialization During the construction of the block model, certain blocks were given variables, such as the gain blocks containing the variables k and b, and the gain box containing the 1/m multiplier that divides the equation of motion by the mass. For the simulation to be run, these variables needed initializations. This can be done directly in the Simulink, or can be done via the use of an external.m file. I chose to use an external input file “input.m.” This appears to the right.

12. Simulink – Force of Spring and Damper on Ground The plot of the Spring/Damper force exerted on the ground is shown to the right. This is exactly the same plot generated by the Camp- G-MATLAB method. The maximum force exerted on the ground appears to be about 5.7 Newtons.

13. Simulink – Vertical Displacement This is a plot of the Vertical Displacement of the Motor. This value varies slightly from the Camp-G value and the value from the solution to 19.169. Simulink generates a maximum displacement of a little over 5x10 -6 meters, instead of 7.1x10-6 meters. I used the same initial conditions for the model, and cannot find a solution to why there is this discrepancy. However, the variance in result is somewhat insignificant, varying by only 2 micro meters.

14. Conclusion In conclusion, both methods to analyze the transient response of this system are efficient and accurate. With accurate bond graphs and Simulink block models, plots displaying the same results calculated by hand in the solution to 19.169 can be obtained readily and easily. Although there was a slight discrepancy in values with the Simulink model, I still feel the answers generated are within a reasonable margin. Both Simulink and Camp-G are extremely valuable tools when used in conjunction with MATLAB to solve transient spring/damper problems!