Question

In Rayleigh's method, which of the following energy at the mean position is equal to the maximum potential energy (or strain energy) at the extreme position?

• A. Maximum kinetic energy
• B. Minimum kinetic energy
• C. Maximum potential energy
• D. Minimum potential energy

Hint:

Rayleigh's Method summary formula: \[ \text{K.E.}_{\max} = \text{P.E.}_{\max} \] For a lump mass system with displacement \(x = X\sin(\omega_n t)\): \[ \frac{1}{2} m (\omega_n X)^2 = \frac{1}{2} k X^2 \quad \Rightarrow \quad \omega_n = \sqrt{\frac{k}{m}} \]

Answer 1

The Correct Option is A

Concept: Rayleigh's method is a foundational technique used to approximate the fundamental natural frequency of an undamped vibrating system. The principle is rooted firmly in the law of conservation of mechanical energy for conservative systems. In a system executing simple harmonic motion (SHM) without any dissipative mechanisms (such as viscous or dry friction damping), the total mechanical energy (\(E_{\text{total}}\)) remains constant at every instant throughout the vibration cycle: \[ E_{\text{total}} = \text{Kinetic Energy (K.E.)} + \text{Potential Energy (P.E.)} = \text{Constant} \] Let us trace the energy distribution across different phases of the oscillation:

Step 1: Identifying the characteristics at the extreme positions.
At the extreme boundaries of oscillation, the moving mass momentarily stops to reverse its direction of travel. Therefore, the instantaneous velocity (\(v\)) drops to zero: \[ v = 0 \quad \Rightarrow \quad \text{K.E.} = \frac{1}{2}mv^2 = 0 \] At this exact coordinate, the spring or structural element reaches its highest state of deformation, which maximizes its strain energy. Consequently, the total energy of the system is entirely composed of potential energy: \[ E_{\text{total}} = \text{P.E.}_{\max} \]

Step 2: Identifying the characteristics at the mean position.
As the system passes through its static equilibrium (mean) position, the elastic deformation or displacement (\(x\)) relative to the equilibrium state is zero: \[ x = 0 \quad \Rightarrow \quad \text{P.E.} = \frac{1}{2}kx^2 = 0 \] At this point, the elastic potential energy is completely converted into kinetic energy, and the system reaches its maximum velocity (\(v_{\max}\)). Thus, the total energy is composed entirely of kinetic energy: \[ E_{\text{total}} = \text{K.E.}_{\max} \]

Step 3: Equating the energy bounds according to Rayleigh's Principle.
By equating the total energy expressions from Step 1 and Step 2, we establish Rayleigh's energy conservation criterion: \[ \text{Maximum Kinetic Energy}_{\text{at mean position}} = \text{Maximum Potential Energy}_{\text{at extreme position}} \] This expression allows for the direct derivation of natural frequencies by substituting harmonic profile functions. This aligns with Option (1).