Question

b) Search, understand and discuss the with the help of sketch the difference between Undamped and Damped free vibration.

Answer 1

Before we start let us assume that a mass 'm' is vibrating on a spring of spring constant 'k' and a viscous medium of damping coefficient 'c'. As this is a free vibration, we will do not have to deal with any disturbing force. Then using Newton's second law of motion we can write

më + ci + kr = 0

where x is the position from the mean point, $\dot{x}$ is the velocity at that point and $\ddot{x}$ is the acceleration at that point. Now let us analyze the 2 cases 1 by one

UNDAMPED FREE VIBRATION

The salient features of an undamped vibration are as follows

• In this case, vibration is not affected by the effect of any restricting agent, like the medium or a damper.
• If undisturbed by any external agent, the vibration tends to go on for an indefinite time.
• As evident, such a model is impossible to be replicated in the real world. Even if we try to do it in a vacuum chamber, eventually the energy will degrade causing it to stop, though it will take a long time.

For undamped vibration, the damping constant c is 0. Thus the equation of oscillation reduces to

mi + ks = 0

or, c + (k/m).x = 0

From here, we can obtain the following important equations

• the natural frequency of vibration $w = \sqrt{k/m}~~rad/s$
• the natural frequency of vibration $f = 2*\pi /w=2\pi\sqrt{k/m}~Hz$
• The time period of vibration $T = 1/f =\frac{1}{2\pi}\sqrt{m/k}~s$

Also solving the ODE for the equation of motion, we get

where A and B are constants, which are to be determined by the initial conditions for position and velocity of vibrating particle

DAMPED FREE VIBRATION

The salient features of this type of vibration are

• In this case, vibration dies out due to internal molecular friction of vibrating mass and friction of medium of vibration.
• The effect o0f decay of the vibration amplitude is called "damping"
• As evident, this is a lot more closer to the real world, and literally almost every vibration we see in our surroundings are viscous, meaning they needed a continuous supply of force to continue vibrating, in absence of which the vibration stops.

The equation given at first is the equation for damped free vibrations.

më + ci + kr = 0

$or,~\ddot{x}+(c/m)\dot{x}+(k/m)x=0$

The roots of this equation are

$x=-\frac{c}{2m}\pm \sqrt {(c/2m)^2 - (s/m)}$

From here we get the damping ratio as $\xi = \sqrt{\frac{(c/2m)^2}{(s/m)}} =\frac{c}{2 \sqrt{sm}}$

Depending on the value of the damping ratio, there are 3 cases

• Over damping. In this case, the damping ratio is more than 1. As evident, the damping force is stronger than the spring force, and hence the vibration is not possible.
• Critical damping. In this case, the value of the damping constant is 1. In this case, also vibration is not possible.
• Under damping. In this case, the value of damping constant is less than 1. vibration dies out slowly due to decay

Please note that undamped vibration is also a special case of damping in which the value of the damping ratio is 0.

The solution of the differential equation for damped vibration is given by

$x = Ae^{\alpha_1t}+Be^{\alpha_2t}$

where $\alpha_1$ and $\alpha_2$ are the roots of the equation, and A and B are constants determined by the initial conditions.

Plotting these motions on a graph, we obtain something like this (here free vibrations mean free undamped vibrations).

[critical damp! G= 1 [over damp & Znatet t I 4 = 0 [under damped (free vibrations

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