What is the meaning of free vibration?

Free vibration is a type of vibration in which the external force is removed after giving the initial displacement to a body/system. In other words, in this type of vibration, force is applied at the initial displacement of the system, and after the initial displacement, force is removed, and the system vibrates at the natural frequency.

What do you mean by damped vibration?

Damped vibration is described as the kind of vibration in which resistance force/friction force acts on the vibrating system. In this type of vibration, as the system oscillates, there is loss of energy with movement takes place, which means the system vibrates with decreasing frequency. The diminishing of vibrations with time is called damping.

Let us consider the forces on the mass m when it is displaced through a distance below the equilibrium position during vibratory motion.

Let $s$ = stiffness of the spring, $c$ = damping coefficient (damping force per unit velocity), ${\omega }_n$= frequency of natural undamped vibrations, $x$ = displacement of the mass from the mean position at time t, $v =\stackrel{·}{ x}$ = velocity of the mass at time t, $f = \stackrel{··}{x}$ = acceleration of the mass at time t.

When the mass moves downwards, the friction force of the dashpot acts in the upward direction.

Now, the forces acting on the mass are inertia = $m\stackrel{··}{x}$(upwards), damping force = $c\stackrel{·}{x}$ (upwards), spring force = $sx$ (upwards).

As the sum of the inertia force and the external forces on a body in any direction is zero,

$m\stackrel{··}{x}+c\stackrel{·}{x}+sx=0 \mathrm{or} \stackrel{··}{x}+\frac{c}{m}\stackrel{·}{x}+\frac{s}{m}x=0$

It is a differential equation of the second order. Its solution will be of the form

$x=A{e}^{{\alpha }_{1}t}+B{e}^{{\alpha }_{2}t}$

Where A and B are some constants and are the roots of the auxiliary equation

$$\begin{gathered} {\alpha }^2+\frac{c}{m}\alpha +\frac{s}{m}=0 \\ {\alpha }_{1,2}=-\frac{c}{2m}\pm \sqrt{{\left(\frac{c}{2m}\right)}^2-\left(\frac{s}{m}\right)} \end{gathered}$$

The ratio of  ${\left(\frac{c}{2m}\right)}^2$ to $\left(\frac{s}{m}\right)$ represents the degree of dampness provided in the system and its square root is known as damping factor or damping ratio $\zeta$ , i.e.,

$$\begin{gathered} \zeta =\sqrt{\frac{{\left({\displaystyle \raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$2m$}\right.}\right)}^2}{{\displaystyle \raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}}}=\frac{c}{2\sqrt{sm}} \\ c=2\zeta \sqrt{sm}=2\zeta m{\omega }_n=2\zeta \frac{s}{{\omega }_n} \end{gathered}$$

damping coefficient,  $c=2\zeta \sqrt{sm}=2\zeta m{\omega }_n=2\zeta \frac{s}{{\omega }_n}$

when $\zeta =1$, the damping is known as critical damping. The corresponding value of damping coefficient $c$ is denoted by $c_c$.

Thus, under critical damping conditions, $c=2\sqrt{sm}=2m{\omega }_n=\raisebox{1ex}{$2s$}\!\left/ \!\raisebox{-1ex}{${\omega }_n$}\right.$

and, $\zeta =\frac{c}{c_c}=\frac{\mathrm{Actual} \mathrm{damping} \mathrm{coefficient}}{\mathrm{Critical} \mathrm{damping} \mathrm{coefficient}}$

Thus when $\zeta =1$, the damping is critical, $\zeta >1$, the system is over-damped, and when $\zeta <1$, the system is under-damped

In the context of a critical damping system, there would be less time required to return the displace object to its mean position without oscillation. By using this concept, guns are designed to fire bullets in a very less interval of time.

Some important points are given in the following points.

• An undamped system $\left(\zeta =0\right)$vibrates at its frequency which depends upon the static deflection under the weight of its mass ${\omega }_n=\sqrt{\raisebox{1ex}{$g$}\!\left/ \!\raisebox{-1ex}{$∆$}\right.}$
• When the system is underdamped $\left(\zeta <1\right)$, the frequency of the system decreases to ${\omega }_d=\sqrt{1-{\zeta }^2}{\omega }_n$ and the time period increases to $T_d=\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{${\omega }_d$}\right.$. The amplitudes of the vibrations decrease with time, the ratio of successive amplitudes being constant. The vibrations die down with time.
• At critical damping, $\zeta =1, {\omega }_d=0, and T_d=\infty$. The system does not vibrate and the mass m moves back slowly to the equilibrium position.
• For an overdamped system, $\zeta >1$, the system behaves in the same manner as for critical damping.
• is the ratio of the existing damping in a system to that required for critical damping, i.e., $\zeta =\raisebox{1ex}{$c$}\!\left/ \!\raisebox{-1ex}{$c_c$}\right.$ .

Logarithmic decrement

The ratio of two successive oscillations is constant in an underdamped system. The natural logarithm of this ratio is called logarithmic decrement.

Some related useful definitions

Undamped vibrations

Undamped vibration is described as the kind of vibration in which no resistance force/friction force acts on the vibrating system. In this type of vibration, as the system oscillates, there is no loss of energy with movement takes place, which means the system vibrates continuously at its natural frequency.

Forced vibrations

Whenever a repeated force continuously works on a vibrating system, the vibrations refer to forced vibration. The frequency of the vibrations is that of the applied force and is independent of their natural frequency of vibrations.

Period

The term period refers as the time taken by a motion to repeat itself and is measured in seconds.

Cycle

The term cycle refers as the motion completed by a vibrating object during one time period.

Frequency

The term frequency refers to the number of cycles of motion completed in one second. The value of frequency helps to obtain the other parameters of the motion.

Resonance

When the frequency of the external force is the same as that of the natural frequency of the system, a state of resonance is said to have been reached. Resonance results in large amplitudes of vibrations and this may be dangerous.

Oscillation

Whenever an object moves forward and backward about a specific mean position with respect to time, it refers to oscillation. For example, the movement of a simple pendulum about its mean position with a specific frequency can consider as an oscillation.

Damping

The term damping effect exists in vibration when there is a mechanical device called a damper attached to the vibrating system. The damper tries to reduce the intensity of vibration to a minimum level. The damper absorbs extra energy from the vibrating system and releases it when required.

Damping ratio

The term damping ratio represents the decrement rate of a specific vibrating system's amplitude. The damping ratio's value can be obtained by taking the actual and critical damping coefficient ratios. The types of vibration can decide with the help of the value of the damping ratio.

Amplitude

The term amplitude of vibration represents the height of a vibrating wave's highest point (crest) from the mean position, either from mean to crest or from mean to trough. In other words, the amplitude of a wave is the maximum displacement of the medium particle from its equilibrium position.

Common Mistakes

• Students are often confused about damping with friction. However, damping with friction means the vibration takes place in the presence of a damper and friction. Friction works as a resisting force that decreases the vibration.
• Student some times confuse about the difference between damped and undamped vibration. However, in undamped vibration friction is absent whereas in damped vibration friction is present,

Context and Applications

Free damped vibration is significant in professional courses like graduate and postgraduate courses in the following streams.

• Bachelor of Technology in Mechanical Engineering.
• Bachelor of Science in Physics.
• Master of technology in mechanical engineering.
• Master of technology in physics.
• Diploma in mechanical engineering.

Practice Problems

Q1. Whirling of the shaft occurs when the natural frequency of transverse vibration is, ____ the rotating shaft's frequency?

1. Greater than shaft rotating frequency.
2. Less than shaft rotating frequency.
3. Equal to the shaft rotating frequency.
4. None of these

Correct Option: (c) 

Explanation: Whirling of the shaft appears when the natural frequency of the transverse vibration is equal in magnitude to the frequency of the rotating shaft.

Q2. In which of the following vibration, a reduction in the amplitude of vibrations over every cycle of vibration occur?

1. Free vibrations
2. Forced vibrations
3. Damped vibrations
4. Torsional vibrations

Correct Option: (c)

Explanation: In damped vibration, the amplitude of vibration of the body reduces over every cycle of vibration.

Q3. The damping capacity of a material is its ability to,

1. Absorb shocks
2. Absorb impact
3. Withstand creep failures
4. Absorb Vibrations

Correct option: (d)

Explanation: The damping capacity is described as a characteristic of the material that shows the material's ability to absorb vibrations.

Q4. Which of the following option is correct in the context of an under-damped system, the amplitude of vibration with reference to time,

1. Increase linearly
2. Increases exponentially
3. Decreases linearly
4. Decreases exponentially

Correct Option: (d)

Explanation: In an underdamped system, the amplitude of vibration of the system decreases exponentially with reference to time.

Q5. In the spring mass system, if the mass of the system is double with spring stiffness halved, the natural frequency of vibration is:

1. Remains unchanged
2. Doubled
3. Halved
4. Quadrupled

Correct Option: (d)

Explanation: In the spring mass system, if the mass of the system is double with spring stiffness halved, the natural frequency of vibration is quadrupled.