1 A mathematical model for the forces on a car suspension system The goal of this project is to describe one of the ways in which resonance phenomena. appear in the "real world\" and to use second order linear DEs to study this phenomenon mathematically. we consider the forces on the suspension system of a car traveling on a road that consists of poured concrete sections. Unevenness in the road can cause a bumpy ride and, in the worst case, lead to dangerous resonance. To describe this situation, we use a mathematical model based on the second order linear differential equation mfi'+hri'+k.r =by}+lcy. (1) We use dot-not ation for the first and second time-derivatives of the functions 1' = TH) and y = y(t), as common in physics and engineering. Figure 1: Single degree of freedom model of a car To get a simple model, we lump the mass m of the car into a single point at the center of mass of the car. This is kowu as a magic degree off'reedom model. The vertical deection of the \"car\" (or mass m) from its equilibrium position is described by the function Mt), where we have assumed upward direction is positive, see Figure 1. For the car traveling on an uneven road, the function y(t) represents the elevation of the road under the car at time t, which can also be viewed as the vertical displacement of the wheels. The constant. k is the spring constant representing the stiffness of the springs. For our model we can look at one spring of stiffness k replacing four springs (one at each wheel) of stiffness k/4. The constant b describes the viscosity of the fluid used in the hydraulic suspension units. We will use a sine function to model the profile of the concrete sections of the road. Thus, using the z-axis for the direction of the road, the elevation of the road is given by h(z) = asin 2TZ (2) 0.10- 0.05 o - 40 0.05 - Figure 2: The function h(z) for a = 0.02 m and L = 19 m Note that the period L of h(z) represents the length of the sections, while 2a, the maximal change of elevation, measures how much the sections are distorted. The axes in Figure 2, where the profile of the road is given for a = 2 cm and L = 19 m, are not to scale (the amplitude is much smaller relative to the length of the sections, meaning that the actual distortions of the road are quite small). Problem 1: The car travels at constant velocity v in the z-direction. Find its position z(t) after time t (assuming z(0) = 0) and insert this into (2) to find the function y = y(t) = h(z(t)) in (1). Also find y and insert both into (1). What is the resulting differential equation? 32 Resonance for an nndamped suspension sys- tern The result of Problem 1 is an inhomogeneous second order linear differential equation with constant coefcients. We will consider the special case where b : 0, i.e. there are no hydraulic units in the suspension, or the driver hasn't noticed that the hydraulic liquid has leaked out. In this case the DE found in Problem 1 describes driven motion without damping. W'hen solving this DE we will work with the initial values x(0) : 0 and 1(0) : 0, i.e. the car initially rests in its equilibrium position. This initial value problem can be solved by the method of undetermined coefcients, separately for the resonant and non-resonant cases. Problem 2: Look up Sections 5.4 and 5.18 of the class notes for the deni- tions of the frequencies of free vibrations e; and driving frequency 1/ as well as Section 5.19 for the denition of pure resonance. (a) Express w/Qwr and the 7/271' in terms of m. k, L and L'. (h) Express the condition for pure resonance in terms of an equation for m, k, L and v. Solve this for u and call the resulting value can. This is the critical velocity at which the care will experience resonant vibrations. (c) From Sections 5.18 and 5.19 of the Notes, nd the formulas for the solution T(t) of the D3 found in Problem 1 with h = 0 and initial conditions 33(0) : 0 and 3'7(0) : 0, separately for the resonant and non-resonant cases. Use the abbreviations to and 7 from part (a) as well as F" : ka/m to simplify the formulas. (We will insert specic numerical values for the constants later.) So far our general considerations did not require a choice of units. All further work should be done in the mkssystem, using meters (in) for distance, kilograms (kg) for mass and seconds (5) for time. In particular. this means that the unit for k becomes kgfs2 (as made necessary by the fact that the terms mi and Inc in (1) must have the same unit). For the suspension system of a :reaHife' car one could nd out the con- stant k by an experiment (putting some extra weight m0 into the car and observing the amount so by which the body of the car is lowered, one can then nd k from may = ksu). Let's assume this has already been done and the result is k : 120000 kg/s2. Problem 3: Assume that the length of the concrete sections is L : 19 meters long, that the mass of the car is 1250 kg and the mass of the driver 80 kg. (a) Find vmt for this car and driver. (b) At what velocity does a series of speed bumps at 4 meter spacings get the car into resonance, still assuming broken hydraulic units and the same masses and spring constant as above\"7 Comment on what is achieved by putting speed bumps in a residential neighborhood. (c) Plot the solution curve for 58(25) where L, m, k and 11 : Um: are as in part (a). Use a : 2 cm, i.e. the street level varies by 4 cm within 19 meters. (d) Assume that the bottom of the car is 10 inches above the ground when the car is at rest. If you drive at resonance speed, how long does it take for the bottom of the car to hit. the road? Problem 4: Suppose you drive at a speed 7.: which is 5 0/9 lower thzm the resonance speed from Problem 3(a). Plot the solution of the initial value problem 37(0) : 0, 35(0) : O for the DE found in Problem 1 for this vi Comment! How would this ride end? 3 Suppressing resonance by damping One lesson to be learned from this project is that one should always keep the hydraulic liquid in a car at a proper level. To see this more explicitly: let us look at a mathematical example where the numbers are kept relativer simple, so the DEs can be solved without too much effort. Problem 5: (a) Using the formula from Section 5.19 in the Notes, nd the unique solution of the initial value problem i+z=sint 37(0) =0, i(0):0r Does one get resonance? (b) Using the methods of the characteristic equation and undetermined coeicients. nd the general solution of :i + 01$ +1: = sint. (c) Without determining explicit values of the constants c and c2 in your answer to (b), what can you say about the behavior of the solution for large times? (d) Comment on the effect of damping (e.g. by fixing the hydraulic damp- ing system of a car) on the risk of catastrophic resonance. Use the language of Section 5.17 of the Notes to describe what happens. 6