ME56300 Mechanical Vibration

1. 20% Consider the system shown below. Write down the equation of motion, and calculate the response assuming a) the system is initially at rest, and b) the system has an initial displacement of 0.05 m.
   
   
   
   
2. 20% Compute the initial conditions such that the response of

oscillates at only one frequency ( w ).

3. 20% Consider a vibration system described below
   
   x+ax+64x = 3cost6t

with initial conditions x (0) = x₀  and x (0) = x₀ .  Here a represents the damping constant that may take any positive value. Analyze the undamped and damped natural frequencies. Write down the expression of the complete solution to the system. You do not need to solve the unknown coefficients, but you need to explain clearly the solution steps. (Hint: here you need to take into account the role of damping, and consider all possible values of a that can yield under-damped, critically damped, and over-damped systems).

4. 20% Consider the base excitation problem for the configuration shown in the figure. In this case the base motion is a displacement transmitted through a damper. Derive an expression for the force transmitted to the support in steady-state. (Hint: the force transmitted to the support is F=kx)