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. 'Ne 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 m:i%+bi'+kr1t:bgj+ky. (1) We use dotnotation for the first and second time-derivatives of the functions 1' = :I:(t) and g; = 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 kown as a single degree of freedom model. The vertical deection of the \"car" (or mass m) from its equilibrium position is described by the function 1U), 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 ls: replacing four springs (one at each wheel) of stiffness k / 4. The constant 12 describes the viscosity of the uid used in the hydraulic suspension units. We will use a sine function to model the prole 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 27m 1 z : _' 7 . 2 L( ) asin( L ) ( ) Jon ,9 m Figure 2: The function 11(2) for a 2 0.02 In and L : 19 In 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 prole of the road is given for a : 2 cm and L : 19 In. 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 u in the :direction. Find its position 2(25) after time t (assuming 2(0) : 0) and insert this into (2) to nd the function y : y(t) : h(z(t)) in (1). Also nd y and insert both into (1). What is the resulting differential equation