Jawaharlal Nehru Technological University Hyderabad B.Tech IV Year MECHANICAL VIBRATIONS Previous Question Paper (September - 2022 - R18)

Energy methods are analytical techniques used to determine the natural frequency of a vibrating system by equating the maximum kinetic energy to the maximum potential energy, based on the principle of conservation of energy. Common methods include the Rayleigh method, Rayleigh-Ritz method, and Dunkerley's method.

The Rayleigh Method is based on the principle that for a conservative system undergoing simple harmonic motion, the maximum kinetic energy (T_max) equals the maximum potential energy (V_max). For a system with a single degree of freedom, the natural frequency ω_n is given by:

This method assumes a mode shape for the system. By calculating the kinetic energy of the masses and the potential energy of the elastic elements at the extreme position, the fundamental frequency can be estimated. It provides an upper bound to the fundamental frequency and is particularly useful for complex systems where exact solutions are difficult to derive.

Given: Mass m = 15 kg, Frequency range = 1600 to 2400 rpm. Isolation efficiency η = 0.95 (95%).

First, convert the engine speed (f) to frequency in Hz:

The transmissibility (TR) is defined as 1 - η. For 95% isolation, TR = 1 - 0.95 = 0.05. The formula for transmissibility without damping is:

Where r = f / f_n. Since f > f_n, r^2 - 1 = 1 / 0.05 = 20, so r^2 = 21, and r = 4.58. To ensure 95% isolation across the entire range, we must design for the lowest excitation frequency (f_min = 26.67 Hz):

The natural frequency in rad/s is ω_n = 2π * f_n = 2π * 5.82 = 36.57 rad/s. The stiffness k is calculated as:

The required rubber stiffness is approximately 20.06 kN/m.