MAE 321 Project: Vibration Isolation System Analysis Due on May 6th, 2020 at 5 pm. Introduction Professor Glauser is designing an off road course at his Adirondack lake house to be used to test out his John Deere Gator RSX 860i suspension system. For this project, the Gator suspension system will be simplified to a one degree of freedom system modeled by a linear spring in parallel with a viscous damper. A good reference for understanding this modeling approach is articulated in Example 2.4.2 and homework problems 2.56 and 2.57 in the text (Inman 2014). In these problems, the road surface input is assumed to be a simple sinusoid in space as shown in Figure 1 (Figure 2.17 from Inman 2014). Figure 1. A simple model of a vehicle traveling with a constant velocity on a wavy surface that is approximated as a sinusoid (Inman 2014). Professor Glauser feels that this simple wavy surface is not sufficiently aggressive for testing his Gator so he plans to design a much more difficult course. His first approach will be to design a periodic (but not simple harmonic) ground surface as shown in Figure 2 with a much larger amplitude when compared to the wave shown in Figure 1. Y(x) A= 15 cm x L Figure 2. The periodic ground surface. L = 12 m and each section is L/3 and amplitude A = 15 cm. Note that the function shown in Figure 2 is periodic with spatial period L. Professor Glauser has decided that L = 12 meters seems like a good number to start with. He has also selected the initial amplitude of 15 cm (peak to peak of 30 cm). The L domain can be assumed to be broken up into 3 equal length segments equal to L/3 = 4 meters. Objective Professor Glauser is interested in the displacement(s) to the chassis and subsequently transmitted to the driver and other occupants as the Gator traverses across the periodic off road course for different speeds. You can assume that the Gator unloaded weighs approximately 700 kq. The Gator is a side by side vehicle so you can assume two occupants including the driver with a combined weight of 150 kg. Also of interest is the displacement(s) when the Gator is loaded with the two occupants and its max carrying capacity in the dump bed of 200 kg. Professor Glauser would like you to calculate these displacements using the stock shocks and springs which can be assumed to provide an equivalent stiffness of 20,000 N/m and damping of 1000 N sec/m. Professor Glauser would also like your recommendations on how he might change these stock values for stiffness and damping to reduce the displacements by approximately 20 percent. Approach 1. Your first task is to perform a spatial Fourier series decomposition of the periodic off road course shown in Figure 2. You will do this using Matlabs symbolic solver to get the Fourier coefficients and reconstruct the function including several of the Fourier modes. Plot the individual modes and their sum in Matlab and superimpose the reconstruction for various Fourier modes on the function shown in Figure 2. This will guide you in how many modes you need to keep in your follow-on analysis. Refer to Ch 3 section 3, text (Inman 2014) and notes to help you with this. Keep in mind that this function is periodic in space with period L. Hence the Fourier modes you extract will be waves with wavelengths that are fractions of L. Hint: the function shown in Figure 2 is odd in the x direction and its amplitude symmetric around 0 which significantly simplifies the Fourier series analysis. 2. Your second task is to now combine the analysis from Ch. 2, Section 4 in the text (Inman 2014) on base excitation with Ch. 3 Section 3 (Inman 2014) periodic but not harmonic inputs to come up with frequency and displacement information for different speeds over the course. Since you will be programing all of this in Matlab you will easily be able to change speeds in your program to get this information. Refer to Example 2.4.2 in the text (Inman 2014) specifically as a starting point to help you understand how to approach getting the results Professor Glauser would like. Note that the y(t) used for input in Example 2.4.2 (Inman 2014) has one frequency ωb as input which is calculated as shown on the top of page 158 (Inman 2014). For Example 2.4.2 there is only one wavelength equal to 6 meters as shown in Figure 1 above and hence only one value for ωb. From the Fourier analysis in task 1 you know that you have multiple values for the wavelength corresponding to the various spatial Fourier modes from the periodic off-road course. Hence, you will now have multiple values of ωb. and correspondingly, multiple functions for y(t) with different frequencies and amplitudes. The number of values will depend on how many Fourier series modes you think are necessary to adequately approximately as determined from task 1. From your Matlab program results you can compute the necessary information to develop a Table like Table 2.1 in the text (Inman 2014) for each of your individual values for ωb and corresponding y(t) for the various Fourier spatial modes you think you need. Use the Gator with the two occupants for r1 and x1 and for r2 and x2 add the 200 kg in the dump bed. Use velocities of 20 km/h, 40 km/h, 60 km/h and 80 km/h. Note, you will want to program into your Matlab code Equation 2.70 (Inman 2014) so you can easily extract the values for x for each of the y inputs from the various Fourier modes. 3. Your third task is to now reconstruct x(t), the displacement the chassis and occupants are feeling as a function of time using several of the Fourier modes as they traverse the course for the 4 speeds used in task 2. Refer to Ch. 3, Section 3 from the text (Inman 2014) to help with this. The principle of superposition is key to this approach. You will have 4 plots, one for each speed. If you choose to perform multiple truncations for the Fourier series, you will have 4 plots for each truncation. 4. Your 4th and final task is to provide to Professor Glauser, recommendations on how the stiffness and/or damping can be changed to provide approximately 20 percent reduction in the displacement felt by the chassis and occupants as they traverse the off-road course, both with and without the dump bed loaded. Presentation of Results The report will be limited to no more than 5 pages and will include an Introduction which sets the stage for the project, including a nice free body diagram, derivations of the governing equations, providing relevant theoretical background (you can refer to the text here). A summary of your Fourier series results for the off-road course from your Matlab/symbolic results is included here but derivations are not required. This will be followed by a Results and Discussion section where you clearly present your results from tasks 1 – 4. Make sure to number and provide useful captions for your tables and figures. Topics to include would be how you made decisions on the number of Fourier modes to keep and why, the process you used to decide on recommendations for the 20 percent reductions and etc. You can then provide a short Conclusion. Finally, an Appendix that includes your published Matlab codes is required. You can include any additional material you don’t include in the main body of the report in the appendix if you think it is relevant. References Inman, D. J. Engineering Vibration, 4th ed. (2014).
