Q1. Use Newton's 2nd law to derive the equations of motion for the system shown in Figure 1 where the solid, uniform cylinder of mass m and radius R is rolling on a rough slope without slipping. The cylinder is connected to the fixed surface through two springs with same  stiffness k and a damper with damping constant c.

Q2. A foot pedal mechanism for a machine is approximately modeled as a pendulum connected to two springs of the same stiffness (
k₁ = k₂ = k) as shown in Fig. 2 below. The purpose of the springs is to keep the pedal roughly vertical. Compute the spring stiffness needed to keep the pendulum at 1o from the horizontal and then compute the corresponding natural frequency. Assume that (i) the angular deflections are small such that the spring deflection can be approximated by the arc length; (ii) the pedal may be treated as a point mass m; and (iii) the pendulum rod is massless. The values in Fig. 2 are:

m 5k g, g 9.8m/s² ,l = 1 0..4m,l₂ = 0.7m,l₂ = 1.0m.

Q3. A small sports car can be modeled as a single-degree-freedom system. The car deflects the suspension system under its own weight 0.05 m. The suspension is designed to be critically damped. If the car has a mass of 1361 kg (a Porsche Boxster), calculate the equivalent damping and stiffness constants of the suspension system. If two passengers, a full gas tank, and luggage totaling 290 kg are in the car, how does this affect the effective damping?

Q4. The free vibration response of an electric motor of weight 500N mounted on a foundation is shown in Figure 3. Find out (i) the spring constant and damping constant of the foundation, and (ii) the undamped and damped natural frequency of the electric motor.3

Q5. A boy riding a bicycle can be modeled as a spring-mass-damper system with an equivalent weight, stiffness and damping constant of 800N, 50,000N/m, and 1,000N-s/m, respectively. Due to the different setting of the concrete blocks on the road, the level surface decreases suddenly as indicated in Fig. 4. Assume that the bicycle is free of vertical vibration before encountering the step change on the road surface, determine the vertical displacement of the boy caused by the step change in the road surface.

Q6. Derive the equation of motion and find the steady-state response of the rotational system shown in Fig. 5 when it is subjected to an applied torque M cos ωt 0 , with the following data: k = 5000N/m, l = 1m, m = 10 kg, M₀ = 100 Nm, ω = 100 rpm. The rotary inertia of the rigid bar about the hinge O can be calculated by 7ml²/ 48.

Q7. A single cylinder reciprocating compressor on spring support has the following data:

Engine mass = 600 kg,
Equivalent mass of reciprocating parts = 20 kg,
Stroke of piston = 225 mm (motion is assumed to be harmonic, stroke of the piston means the total
travel of the piston),
Static deflection of spring due to weight of compressor = 40 mm, and
Ratio of consecutive amplitudes in free vibration = 1.00 to 0.55.

Determine: (i) the amplitude (in mm) of compressor vibration at 320 rpm; (ii) the force transmissibility expressed as a decimal fraction; and (iii) the force (in N) transmitted to the base at this speed.

Q8. A mass m = 100 kg is hung on a spring of stiffness 10 kN/m. It is pulled down 50 mm below its static equilibrium position and released. There is a frictional resistance which is proportional to the velocity and is 360 N when the velocity is 1 m/s. (1) write down the equation of motion and find its solution; (2) calculate the time which elapses and the distance which the body moves from the instant release until it is again at rest at the highest part of its travel.