JNTU B.Tech IV MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS Previous Question Paper (October/November - 2020 - R16)

To determine the natural frequency of a semicircular disc of radius R and mass m pivoted at its center, we apply the energy method. For small oscillations, the total energy of the system remains constant.

First, identify the moment of inertia of the semicircular disc about its center O. For a full disc, the moment of inertia about the center is 1/2 mR². Since the semicircular disc is half of the full disc, its mass is m and its moment of inertia about the center O is:

Let θ be the small angular displacement of the disc. The kinetic energy (T) of the system is given by:

The potential energy (V) is determined by the change in height of the center of gravity (G) of the semicircular disc. The distance of the center of gravity from the center O is given by:

When the disc is rotated by an angle θ, the vertical displacement of the center of gravity is h = OG(1 - cosθ). For small angles, cosθ ≈ 1 - θ²/2, so h ≈ OG(θ²/2). The potential energy is:

According to the energy method, the total energy E = T + V is constant, so dE/dt = 0:

Simplifying the equation of motion:

Comparing this to the standard form of a simple harmonic oscillator equation, θ̈ + ω_n²θ = 0, the natural frequency ω_n is:

To derive the governing differential equation of motion for a general mechanical system, we apply Newton's Second Law or the Energy Method. Assuming a standard spring-mass-damper system consisting of a mass m, a spring with stiffness k, and a viscous damper with coefficient c, the forces acting on the mass during displacement x are the restoring force, the damping force, and the inertia force.

According to D'Alembert's principle or Newton's Second Law, the sum of all forces acting on the mass must equal the product of mass and acceleration:

Where:

• m(d²x/dt²) represents the inertial force opposing motion.
• c(dx/dt) represents the viscous damping force, proportional to velocity.
• kx represents the elastic restoring force of the spring, proportional to displacement.
• F(t) represents any external excitation force applied to the system.

For a free vibration system where no external force is present, F(t) = 0, the equation simplifies to the homogeneous second-order linear differential equation:

This equation governs the dynamic response of the system, where the roots of the characteristic equation determine the nature of the vibration (underdamped, critically damped, or overdamped) based on the damping ratio ζ.