1. Develop and draw the MDOF system for the vertical vibrations of the un-damped three dimensional model of a typical three-wheeled solar or energy efficient car (include wheels, car body, driver, minimum 7 DOF).
2. Derive the linearized equations of motions and write them in matrix notation (horizontal vibrations can be neglected). Show and explain your derivations and reasoning.
3. Assume reasonable values for the parameters (give your reasoning/source) and find the undamped natural frequencies of the MDOF system.
4. Calculate, display graphically and describe the vibrations characterized by the mode shapes of the system.
5. Assume small, proportional damping of the system, i.e., dampers acting parallel to the springs with the damping level of each damper proportional to the matching spring stiffness (the damping matrix can be found as the stiffness matrix multiplied by a constant factor).
6. Calculate the system response if the car is driving with a speed of 40 km/h on an uneven road (displacement excitation of the ground connected to wheels via spring & damper). The unevenness of the road can be assumed to have a sinusoidal shape with a vertical amplitude of 0.025 m and a horizontal wavelength (distance between peaks) of 10 m. Comment on the excited vibrations, illustrate and describe your results. Discuss the two cases when only one side of the car (e.g. the left wheel) or both sides of the car drive over the uneven part of the road.
7. Calculate the vertical acceleration of the car and driver and comment on your results. What would be the optimum driving speed in order to minimize the acceleration felt by the driver? Calculate the transient response of the system when driving through a pothole in the road. The pothole can be modelled as a sudden step in the road surface applied to one of the 3 wheels. Typically the car can be modelled as a system of ordinary differential equations and solved (e.g., in Matlab using ode45).
9. Comment on how realistic and accurate your modelling of the car and road unevenness is,and on possible improvements.