Question 1

A competitor in a hill-climb event finds during practice that his car becomes airborne at a particular point on the hill. Shortly after landing the lowest point in the car just contacts the road. The mass of the car is 2000 kg and the suspension is designed to give a natural frequency of 1.3 Hz. The static ground clearance is 100 mm. The dampers are set to give a damping ratio of 0.3 of critical damping.

(a) What is the combined stiffness of the suspension springs?

(b) What is the damping coefficient of one of the dampers assuming they are identical?

(c) Determine the minimum ground clearance after impact which would result if the damper settings were increased by 25%.

You should assume that the car can be treated as a one degree of freedom system for the transient that occurs when it hits the ground. You should assume that the suspension springs are in their static equilibrium position when the car hits the ground. You should also assume that after becoming airborne the car always hits the ground with the same downwards velocity.

Note that the transient response for a one degree of freedom system with ??< 1 is given by

You may wish to use EXCEL to solve part (c)

Question 2 

A three degree of freedom system is shown in Figure Q3.  The three masses are each 1 kg and are constrained to move in the directions shown.   The three stiffnesses are 10 kN/m, 100kN/m, and 1000 kN/m as shown.

(a)  Use a one degree of freedom approximation to estimate the lowest natural frequencies of the system in Hz.

(b)  Write the equations of motion of the system using matrices.

(c)  Using MATLAB or otherwise, calculate the three natural frequencies of the system and the associated mode shapes normalised on the largest value.

(d)  Find the principal stiffnesses for the system using the normalised mode shapes.

Question 3

Model the three degree of freedom system shown in Figure Q5 and solve for the displacements of the three masses due to a force of 10 N applied to the bottom mass at a frequency of 20 rad/s.

(a) First establish the 3x3 mass, stiffness and damping matrices.

(b) Use the mass and stiffness matrices to evaluate the three undamped natural frequencies of the system and associated mode shapes. The mode shapes must be normalised so that the largest value in the vector is 1. (It is recommended that you use MATLAB for this analysis.)
(c) Use the mode shapes to find the principal masses, principal stiffnesses, and damping factors for the three modes.
(d) Hence find the response of the three masses by modal superposition, i.e. the displacement amplitudes.