1. (35 points) Modeling a vehicle suspension: This problem is related to Project 1. The simplified schematic for a vehicle suspension is given below. a. You can solve this part by closely following along the discussion we had in class. For this vertical mass-spring-damper system, use the method described in class to obtain the mathematical model as an ordinary differential equation in terms of the output y and the input u. Direction of vehicle's motion Damper Mass Displacement, y www Spring Road input, u 2 cm b. For this part you will have to leverage your prior knowledge of solving ordinary differential equations. Please review that material in order to solve this part. The vehicle hits a speed bump whose height is 2 cm, representing a step input to the system. When this displacement u(t) acts on the vehicle (via the wheel), what is the displacement y(t) experienced by the vehicle mass (chassis)? You are given the following vehicle suspension parameters: The mass of the vehicle is 1600 kg, the spring constant is 40 kN/m, and the damping constant is 12,800 kg/s. You may use the Laplace transform table and your prior knowledge of ordinary differential equations to solve the problem. Assume zero initial conditions, i.e., y(t = 0) = 0 m and y(t = 0) = 0m/s. You might need the following (and other) Laplace transforms: b (s+a)+b a + b } } s[(sa)+b] = eat sin(bt) and = 1 eat {cos(bt) + sin(bt)}