What is the Role of Damping Coefficient and Ratio in Vibration Theory?

[chandran]

in vibration theory we talk about the following

1)damping coefficient

2)critical damping coefficient

What is the intuitive meaning of the above two

3)damping ratio

why we talk about the ratio of damping coefficient and critical damping coefficient?

Vibration theory says that every material has an inherent damping property(any deep thought on this)?

 

[Reshma]

In practice, a body in oscillatory motion experiences resistive forces due to friction, air resistances etc. The body does work against these resistive forces and the energy required for doing this work is taken from the energy of the oscillatory motion of the body.

Hence the energy and therefore the amplitude of the oscillatory motion of the body goes on decreasing with time and the body stops oscillating, we say the motion is damped. The oscillations of the body in presence of such dissipiative forces are called damped oscillations.

Are you familiar with the damped equation of motion? Then it is possible to clearly define the damping coefficient.

 

[Gokul43201]

Equation of motion of a harmonic oscillator :

$$m \ddot{x} - kx = 0$$

Equation of motion of a damped harmonic oscillator :

$$m \ddot{x} - c \dot{x} - kx = 0$$

where c is the damping coefficient.

Rewriting this as

$$\ddot{x} - \zeta \dot{x} - \omega _0^2 x = 0$$

you have critical damping when $\zeta = 2 \omega _0$. This is that special value of damping at which the solution to the above equation is

$$x = (A+Bt)e^{- \omega _d t}$$

and hence there is no oscillatory form to the response.

The damping ratio, as you seem to know, is the ratio $\delta = \zeta /\omega _0$, and it's handy to write it that way, to enable a useful classification of the damping regimes. A system is "underdamped" if $\delta < 1$ and "overdamped" if $\delta > 1$. Naturally, the system is said to be critically damped when $\delta = 1$.

I strongly recommend you look into the harmonic oscillator problem in a standard classical mechanics text.