1. VIBRATION

2. 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

3. 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

4. 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

5. 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,

6. 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,

7. 1 Undamped Free Vibration*
Frequency f (hertz, Hz) is defined as the number of cycles completed per unit of time,

8. 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.