VIBRATION

1 Undamped Free Vibration* A vibration is a periodic motion of a body or system of connected bodies displaced from a position of equilibrium Free vibration occurs when the motion is maintained by gravitational or elastic restoring forces

1 Undamped Free Vibration* Undamped vibrations can continue indefinitely as frictional effects are neglected The block has a mass m and is attached to a spring having a stiffness k Elastic restoring force F = kx is directed toward the equilibrium position Acceleration a is assumed to act in the direction of positive displacement

1 Undamped Free Vibration* We have Acceleration is proportional to displacement Motion is called simple harmonic motion Rearranging, we get ωn is the natural frequency (rad/s), is homogeneous, second-order, linear, differential equation with constant coefficients, and the general solution is

1 Undamped Free Vibration* General solution is can expressed in terms of simple sinusoidal motion. Let We get Since sin (θ + Φ) = sin θ cos Φ + cos θ sin Φ, When plotted on an x-versus-ωnt axis,

1 Undamped Free Vibration* Amplitude is C and angle Φ is the phase angle Sine curve completes one cycle in time when Length of time is call a period,

1 Undamped Free Vibration* Frequency f (hertz, Hz) is defined as the number of cycles completed per unit of time,

Example A 2 kg block is suspended from a spring, the spring is stretched a distance of 40mm. Determine the frequency and the period of vibration for a 0.5 kg block attached to the same spring.