Free Vibrations

Calculate the natural frequency vibration for an undamped free vibration.

Understand what do you mean by damping and its effect in a springdamper mass system vibrations.

Identify whether an damped free vibration is underdamped, critically damped or overdamped.
Types of Vibrations:
There are three important types of vibrations from subject point of view:

Free or natural vibrations : Natural vibrations take place at a frequency called fundamental or natural frequency, which is characteristics of the given component based on its material and geometrical properties. Free vibrations lack damping, dissipating or viscous forces. Applied force is applied only at the beginning of the motion and it is not continued further.

Damped vibrations : Here, the amplitude of the vibrations keeps on reducing every cycle of vibration. This is because energy is lost to the surroundings in terms of friction, viscous resistance or hysteresis. These losses are irreversible, the system energy reduces every oscillation and hence the amplitude of vibrations goes on reducing.

Forced Vibrations : Here the body is periodically subjected to disturbing forces, and it continues vibrating with the frequency other than the natural frequency of the component.
Undamped Free Vibration
\[m \ddot{x} = mg  k( x + \delta_{st})\]
since \[mg = k \delta_{st}\] \[m \ddot{x} =  kx\] or \[m \ddot{x} + kx = 0\] \[\ddot{x} + \frac{k}{m} x = 0\] The solution of this differential equation is:
where \[\omega_n = \sqrt{\frac{k}{m}}= \mathrm{natural\ frequency}\]
\(C_1\) and \(C_2\) are determined from the boundary (initial) conditions.
Damped Free Vibrations:
A damper generally consists of a plunger inside an oil filled cylinder, which dissipates energy by churning the oil.
When damping is present, the equation of motion is: \[m\ddot{x} + c\dot{x} + kx = 0\]
Case I. Two distinct (negative) real roots
If \(c^2 > 4mk\), the system is overdamped.
\[y = C_1 e^{r_1t} + C_2 e^{r_2t}\] Case II. One repeated (negative) real root
If \(c^2 = 4mk\), the system is critically damped.
\[y = C_1 e^{rt} + C_2 t e^{rt}\] Case III. Two complex conjugate roots
If \(c^2 < 4mk\), the system is underdamped. \[y = C_1 e^{At} \cos Bt + C_2 e^{At} \sin Bt\] where A is the real part of the root of characteristics equation, and B is its imaginary part.