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ME 354A Vibration and Control Assignment 2 For problems 1 to 6, consider corresponding forcing functions applied at given locations, to determine the response of system. You may need to derive the equations of motion again (if it is not straightforward as to how the forcing would enter the equation of motion) keeping in mind the external forcing. Assume appropriate initial conditions in symbolic form as and when required. 1. A double pendulum consists of two bobs of mass m1 & m2 suspended by inextensible massless strings of length L1 & L2 .Use Newton’s second law to derive four equations of motion for the rectangular coordinates X1, x2, y1, y2. Then, express X1, X2, y1, y2 in terms of the angles θ1 and θ2, eliminate the tensions T1 & T2 in the strings and obtain two equations of motion for θ1 and θ2. Consider a torque T = T0sin(ωt), applied at 0.

2. A cylinder of mass m and mass moment of inertia Jo is free to roll without slipping, but is restrained by spring k, as shown in figure. Derive the equation of motion and determine natural frequency of oscillation. Also obtain the full response for a torque T = T0sin(ωt) applied at 0.

3. Consider the figure below of a ship floating in sea. Here, M is the metacentre of the ship which depends on the shape of the ship’s hull and is independent of the inclination θ. G is the center of gravity and h is the metacentric height. Derive the equation of motion and determine natural frequency of oscillation. Assume that the ship has mass m and mass moment of inertia J about the rolling axis. What will be the final response of the ship under sea wave excitation.

[Can assume that oscillating waves lead to an effective forcing of F = F0sin(ωt), applied at a distance d from M along the direction M to G] 4. Write the differential equation of motion for the system shown in figure and determine the natural frequency of damped oscillations of the system. Furthermore a torque T = T0sin(ωt)+ T0sin(3ωt) is applied at the hinge. Find the response and comment on the values of ω which should be avoided.

5. A plate of area A and weight W is attached to the end of a spring and is allowed to oscillate in a viscous fluid. If ꞇ1 is the natural period of undamped oscillation (i.e. with system oscillating in air) and ꞇ2 the damped period with plate immersed in fluid, show that,

Here μ is the coefficient of viscosity of the fluid. For this system, now consider a force F = F0sin(ωt)+ F0sin(5ωt), applied on the plate along the spring length. What is its overall response?

6. The figure below shows a uniform bar hinged at point O with springs of equal stiffness at each end, the bar is horizontal in the equilibrium position with spring forces k1 and k2. Determine the equation of motion and its natural frequency. Finally apply a force F = F0sin(ωt) at the right end of rod and obtain the response of the system.

7. A counter rotating eccentric mass exciter shown in figure below is used to determine vibration characteristics of a structure of mass 181.4 kg. At a speed of 900 rpm, a stroboscope shows the eccentric mass to be at the top at the instant the structure is moving upward through its static equilibrium position, & the corresponding amplitude is 21.6 mm. If the unbalance of each wheel of the exciter is 0.0921 kg m, determine

(a) Natural frequency of structure (b) Damping factor (c) Amplitude at 1200 rpm. 8. Figure below represents a simplified diagram of a spring- supported vehicle travelling over a rough road. Determine the equation for the amplitude of W as a function of the speed & determine the most unfavourable speed.

9. An electric motor of mass 68 kg is mounted on an isolator block of mass 1200 kg & the natural frequency of the total assembly is 160 cycle per minute, with a damping factor of 0.10. If there is an unbalance in the motor that results in a harmonic force of F = 100sin3.14t, determine the amplitude of vibration of the block and the force transmitted to the floor.

10.

Show that the energy dissipated per cycle for viscous friction can be expressed as

11. Determine the steady state response of the spring mass damper system subjected to the excitation as shown in the figure.