Baseline Validation
Determine the natural frequencies of the system-For the baseline system
Design Modification
Notice that the baseline design has a large amplitude response when the engine speed is around 435 or 2026 rpm. As these are very common operating speeds (idle and cruise), you would like to alter the baseline design to move these resonances as far away from their current values as possible. Due to other constraints in the system you are limited in the alterations you can make.
The modulus of the belt can be altered by as much as 10% higher and lower. Use MATLAB to plot the lowest two engine speeds that excite resonance as a function of belt modulus for the full range of modulus values. The tensioner spring constant can also be changed by as much as 10% higher and lower. Plot the lowest two engine speeds that excite resonance as a function of tensioner spring constant for the full range of values.
The geometry of the system can also be altered within some bounds. As illustrated in the figure, you can change the position and radius of each except the crankshaft pulley so that it fits in a square box centered at the original pulley position. The sides of the box are two centimeters larger than the pulley diameter. Thus, for example, the water pump pulley must fit in a box with center at (0, 167.5) mm and sides 155 mm. The radii of the pulleys cannot be reduced by more than 1 cm from the baseline design. As you change a pulley radius, you should consider changing the inertia of that pulley. However, all but the tensioner and idler pulleys are connected to devices with inertias much larger than the pulley. Thus, you should only alter the tensioner and idler inertias when you change their radii.
The possible design space is very large. To get a feel for how system changes alter the natural frequencies you can make plots as you change only a couple parameters at a time. Keeping the baseline pulley locations plot the engine speed of the lowest two resonances as a function of each pulley radius. Keeping the baseline radii, for all but the crankshaft pulley, plot the engine speed of the first two resonances as a function of x and y positions.
Include all plots in a report that includes physical arguments to explain all your results.
You should now have a good understanding of how changing individual system parameters alters the natural frequencies of the system. Try to find a design that changes the first two natural frequencies more than any of the previous slight modifications. You will need to construct some benchmarking (objective) function that measures the improvement.
Once you have chosen a final design, you should draw it to scale using whatever drawing program you like. Include this as well as all the specifications of your final design in your report.
Steady State Analysis
A more thorough analysis of your final design must be done. In this analysis, you add a tensioner damper in parallel with the tensioner spring to your model. What is the system damping matrix if the damper has constant 750 N-s/m? Remembering that the third engine harmonic is dominant, plot the steady state response amplitude of each pulley as a function of the engine speed for speeds ranging from 0 to 7000 rpm. How does the damping you added alter this plot? Scale your plots so that the resonance peaks do not mask the response at other frequencies.
Discussion
You have optimized your design only considering two factors (the first two natural frequencies). There are, however, many other factors that serpentine belt designers must consider. From the analysis of your final design, can you see any possible problems with it? Discuss some of the issues that may not make this the best design. Discuss also the benchmarking function you decided to use. Did you weight moving the first or second natural frequency more? Why? Discuss how your optimal design depends on the specific objective function you used.