|
Vibrations
Introduction
Vibration is an oscillation wherein the quantity is a parameter defining the motion of a mechanical system.
Oscillation is the variation, usually with
time, of the magnitude of a quantity with
respect to a specified reference when the
magnitude is alternately greater and
smaller than the reference.
Vibration is mechanical oscillation about a reference position. Vibration is an
everyday phenomenon, we meet it in our homes, during transport and at
work. Vibration is often a destructive and annoying side effect of a useful
process, but is sometimes generated intentionally to perform a task.
Free vibration
When a free undamped mass-spring system is set into oscillation the added
energy is constant, but changes form from kinetic to potential during the
motion.
At maximum displacement the velocity and therefore also the kinetic energy
is zero, while the potential energy is 1/2kD². At the equilibrium position the
potential energy is zero and the kinetic energy is maximum at 1/2mV².
For the sinusoidal motion
d = D sin wn t
we can also find the velocity:
V = 2*PI*fnD.
Using energy conservation laws we then get the natural resonance
frequency:
fn=squareroot(K/m) / (2PI)
Forced vibration
If an external sinusoidal force is applied to the system, the system will follow
the force, which means that the movement of the system will have the same
frequency as the external force. There might, however, be a difference in
amplitude (and phase).
For frequencies below its natural frequency, the amplitude of the vibrating
system will increase as the frequency is increased, a maximum being
reached at the natural frequency. If there was no damping in the system
(c = 0), the amplitude would approach infinity.
If the frequency of the external force is increased the frequency of the
spring/mass/damper system will increase to the same value, but the
amplitude (and the phase) will change in accordance.
|