Question

The Smallest and largest natural frequencies of a 'n' degree freedom system are ω₁ and ω_n respectively. The approximate natural frequency estimated by Rayleigh’s and Dankerley’s methods is ω_r and ω_d respectively. Which of the following statements is true?

1. ωr < ω1 and ωd < ω1
2. ωr < ω1 and ωd > ω1
3. ωr > ω1 and ωd > ω1
4. ωr > ω1 and ωd < ω1

Answer 1

Correct Answer - Option 4 : ωr > ω1 and ωd < ω
Explanation: Degree of freedom: The number of independent coordinates required to describe a vibratory system is known as the degree of freedom. A simple spring-mass system or a simple pendulum oscillating in one plane are examples of a single degree of freedom. A two-mass, two-spring system, constrained to move in one direction, or a double pendulum belongs to two degrees of freedom. Shaft carrying multiple loads (Multiple degrees of freedom): There are two methods to find natural frequencies of the system: Dunkerley's method (ω_d) (Approximate results which are less than the actual natural frequency of the system). Rayleigh Method (ω_r) or Energy method (Gives accurate results that are slightly greater than the actual natural frequency of the system)   Given that the smallest natural frequency of the system is ω₁, therefore as per the definition of two methods mentioned, ω_r > ω₁ and ω_d < ω₁.