Mechanical Vibrations

Mechanical Vibrations

Vibration is the periodic motion of a body or system of connected bodies displaced from a position of equilibrium.

Free Undamped Vibrations

Consider a block of mass $m$ attached to a spring of stiffness $k$.

Key Parameters

• Natural Circular Frequency: $\omega_n = \sqrt{\frac{k}{m}}$ (rad/s)
• Natural Frequency: $f_n = \frac{\omega_n}{2\pi}$ (Hz)
• Period: $\tau = \frac{1}{f_n} = \frac{2\pi}{\omega_n} = 2\pi \sqrt{\frac{m}{k}}$ (s)

Simple Pendulum

For small angles ($\sin \theta \approx \theta$):

• $\omega_n = \sqrt{\frac{g}{L}}$
• Period $\tau = 2\pi \sqrt{\frac{L}{g}}$

Damped Free Vibrations

If a viscous damper ($c$) is added: $m \ddot{x} + c \dot{x} + k x = 0$

• Critical Damping Coefficient ($c_c$): $c_c = 2m\omega_n$
• Damping Ratio ($\zeta$): $\zeta = c / c_c$

If $\zeta < 1$ (underdamped), the system oscillates with decreasing amplitude.

Example: Spring-Mass Period