Indian Institute of Space Science and Technology
AE223: Kinematics and Dynamics of Mechanisms
Tutorial 12: Vibration
Aerospace Engineering
11­04­2013
1. Consider the simple pendulum of length l and mass m, shown in figure. It is kept vertical using two springs of stiffnesses k1 and k2 on each side, at distances a and b respectively from the fulcrum, and using appropriate initial compressions. Determine
(a) the expression for natural frequency as a function of the parameters l, a, b, k1, k2, m, and initial compressions (state the assumptions which enabled you to arrive at the natural frequency), and (b) the values of the above parameters at which the natural frequency becomes zero.
(adapted from Uicker, Pennock, and Shigley) 2. A set of fans have to be hung by long rods. The designer proposes to reduce cost by making the rod very slender. His logic is that very small cross sectional area is sufficient to support the weight. The company is worried that this could make the natural frequency too low and could cause resonance. As the designer, how would you assure the company that your design will not cause resonance. Show how you would prove your point with appropriate calculations. Neglect damping.
After you prove your point with damping neglected, someone points out that with damping, the resonance frequency could decrease. Suppose you want to conduct an experiment to determine the damping coefficient by measuring the decrement in oscillations. Derive an expression to estimate the damping coefficient in this fashion, and explain how you would check for resonance with the estimated damping coefficient.