Damping Coefficient and Damping Ratio MCQ Quiz - Objective Question with Answer for Damping Coefficient and Damping Ratio - Download Free PDF

Concept:

In a vibratory system with two springs in parallel, the equivalent stiffness is the sum of the individual stiffnesses.

The natural frequency of vibration is given by:

\( f = \frac{1}{2\pi} \sqrt{\frac{k_{eq}}{m}} \)

Calculation: 

Given:

Mass, \( m = 10~kg \)

Stiffness of each spring, \( k_1 = k_2 = 2~N/mm = 2000~N/m \)

Since the springs are in parallel:

\( k_{eq} = k_1 + k_2 = 2000 + 2000 = 4000~N/m \)

Now, apply the formula for natural frequency:

\( f = \frac{1}{2\pi} \sqrt{\frac{4000}{10}} = \frac{1}{2\pi} \sqrt{400} = \frac{20}{2\pi} = \frac{10}{\pi}~Hz \)

Concept:

Free damped Vibration: \(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

where m is mass suspended from the spring, k is the stiffness of the spring, x is a displacement of the mass from the mean position at time t and c is damping coefficient.

Since excitation force is absent in the equation hence it is the equation of free damped vibration.

Forced damped Vibration:

\(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = Fcos\omega t\)

Damping factor (ξ): The ratio of the actual damping coefficient (c) to the critical damping coefficient (cc) is known as the damping factor or damping ratio. i.e. \(\xi=\frac{c}{c_c}\)

Calculation:

Given:

\(9\ddot x + 9\dot x + 36x = 0\)

m = 9, c = 9, k = 36

\(\xi= \frac{c}{{{c_c}}} = \frac{c}{{2\sqrt {km} }}= \frac{9}{{2\sqrt {36 \times 9} }}= 0.25\)

Concept:

The damping ratio (ξ) is defined as ratio of viscous damping coefficient to critical viscous damping coefficient.

\(⇒ {{ξ }} = \frac{{{c}}}{{2\sqrt {{{km}}} }}{{\;\;\;\;\;}} \ldots \left( 1 \right)\)

Where, ξ = damping ratio, c = damping coefficient, k = spring constant, 

Calculation: 

Given: 

m = 0.1 kg, k = 25 kN/m, c = 40 Ns/m

Using equation (1),

\(⇒ {{ξ }} = \frac{{40}}{{2\sqrt {25 \times 10^3\times 0.1} }}{{}}\)

⇒ ξ = 0.4

Concept:

Free damped Vibration: \(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

where m is mass suspended from the spring, k is the stiffness of the spring, x is a displacement of the mass from the mean position at time t and c is damping coefficient.

Since excitation force is absent in the equation hence it is the equation of free damped vibration.

Forced damped Vibration:

\(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = Fcos\omega t\)

Damping factor (ξ): The ratio of the actual damping coefficient (c) to the critical damping coefficient (cc) is known as the damping factor or damping ratio. i.e. \(\xi=\frac{c}{c_c}\)

Calculation:

Given:

\(9\ddot x + 9\dot x + 36x = 0\)

m = 9, c = 9, k = 36

\(\xi= \frac{c}{{{c_c}}} = \frac{c}{{2\sqrt {km} }}= \frac{9}{{2\sqrt {36 \times 9} }}= 0.25\)

Concept:

The damping ratio (ξ) is defined as ratio of viscous damping coefficient to critical viscous damping coefficient.

\(⇒ {{ξ }} = \frac{{{c}}}{{2\sqrt {{{km}}} }}{{\;\;\;\;\;}} \ldots \left( 1 \right)\)

Where, ξ = damping ratio, c = damping coefficient, k = spring constant, 

Calculation: 

Given: 

m = 0.1 kg, k = 25 kN/m, c = 40 Ns/m

Using equation (1),

\(⇒ {{ξ }} = \frac{{40}}{{2\sqrt {25 \times 10^3\times 0.1} }}{{}}\)

⇒ ξ = 0.4

Concept:

Damping ratio:

\(\xi = \frac{C}{{{C_c}}}\)

ξ > 1: Overdamped system

ξ = 1: Critically damped system

ξ < 1: Under-damped system

Differential equations of damped free vibration

\(m\ddot x + c\dot x + kx = 0\)

Calculation:

2ẍ + 8ẋ + 32x = 0

On comparing with: mẍ + cẋ + kx = 0

M = 2 kg, c = 8 Ns/m, k = 32 N/m

Damping factor,\(\xi = {\rm{\;}}\frac{{\rm{c}}}{{2\sqrt {{\rm{km}}} }} = {\rm{\;}}\frac{8}{{2\sqrt {32 \times 2} }} = \frac{8}{{2{\rm{\;}} \times {\rm{\;}}8}} = 0.5\)

∵ ξ < 1, the system is underdamped.

Concept:

Peak amplitude \(A=\frac{{{F}_{0}}/s}{\sqrt{{{\left[ 1-{{\left( \frac{w}{{{w}_{n}}} \right)}^{2}} \right]}^{2}}+{{\left( \frac{2\xi w}{{{w}_{n}}} \right)}^{2}}}}\)

Where, ω_n = Natural frequency, ω = forced frequency

ξ = Damping factor or damping ratio

Calculation:

\(A=\frac{({{F}_{0}}/s)}{\sqrt{{{\left[ 1-{{\left( \frac{w}{{{w}_{n}}} \right)}^{2}} \right]}^{2}}+{{\left( \frac{2\xi w}{{{w}_{n}}} \right)}^{2}}}}\)

 ξ = 0 given

\(A=\frac{\frac{{{F}_{0}}}{m\times \frac{s}{m}}}{\sqrt{{{\left[ 1-{{\left( \frac{w}{{{w}_{n}}} \right)}^{2}} \right]}^{2}}}}\)

\(\Rightarrow A=\frac{\frac{{{F}_{0}}}{m\cdot w_{n}^{2}}}{1-{{\left( \frac{w}{{{w}_{n}}} \right)}^{2}}}\)

F₀/m = a

\(\Rightarrow A=\frac{\frac{a}{w_{n}^{2}}}{1-{{\left( \frac{w}{{{w}_{n}}} \right)}^{2}}}\)

\(\Rightarrow A=\frac{\frac{3}{{{20}^{2}}}}{1-{{\left( \frac{10}{20} \right)}^{2}}}\)

\(\Rightarrow A=~\frac{3}{400\times \frac{3}{4}}\)

\(\Rightarrow A=\frac{1}{100}m\)

Explanation:

Damping coefficient: The measure of the effectiveness of damper, reflects the ability of the damper to which it can resist the motion is called the damping coefficient. It is denoted by c.

Damping force is given by:

\(Damping\;force\;\left( F \right) = \; - c\frac{{dx}}{{dt}}--cv\)

⇒\(c=-\frac{F}{v}\)

where c is the damping coefficient.

Additional Information

Damping ratio:

The ratio of the actual damping coefficient (c) to the critical damping coefficient (cc) is known as the damping factor or damping ratio.

When the damping factor is more than equal to 1, it leads to aperiodic motion.

Logarithmic decrement 

Logarithmic decrement is defined as the natural logarithm of the ratio of any two successive amplitudes on the same side of the mean line.

\(\delta = \ln \frac{{{x_0}}}{{{x_1}}} = \ln \frac{{{x_1}}}{{{x_2}}}\)

Concept:

Critical damping provides the quickest approach to zero amplitude for a damped oscillator.

Damping ratio \(\zeta = \frac{{Actual\;damping}}{{Critical\;damping}} = \frac{c}{{{c_c}}}\)

Where CC is the critical damping coefficient

\({C_C} = 2\sqrt {km} = 2m{\omega _n}\)

Calculation:

Explanation:

Damped vibration:

When the energy of a vibrating system is gradually dissipated by friction and other resistance, the vibrations are said to be damped vibration.

Critical damping provides the quickest approach to zero amplitude for a damped oscillator.

Damping ratio:

The ratio of the actual damping coefficient (c) to the critical damping coefficient (cc) is known as damping factor or damping ratio.

Damping ratio \(\zeta = \frac{{Actual\;damping}}{{Critical\;damping}} = \frac{c}{{{c_c}}}\)

where C_C is the critical damping coefficient

\({C_C} = 2\sqrt {km} = 2m{\omega _n}\)

Thus critical damping is the function of mass and stiffness only.

Explanation:

The damping factor is the ratio of actual damping to the critical damping coefficient. It shows how vibration decay after damping It is denoted as:

\(ζ = \frac{C}{C_c}\) where, C = damping coefficient, C_c = critical damping coefficient

C_c = 2mω_n , where ω_n  = natural frequency , m = mass

\(ω_n = √{\frac{k}{m}} \)

C_c = 2m\( \sqrt{\frac{k}{m}}\) = \(2\sqrt{km}\)

\(ζ = \frac{C}{C_c} = \frac{C}{2\sqrt{km}}\)

when:

ζ < 1 it is called underdmped vibration.

ζ = 1 it is called critically damped vibration.

ζ > 1 it is called over damped vibration.

Concept:

For a Damped free vibration,

\(ζ = \frac{c}{{{c_c}}} = \frac{c}{{2 \sqrt{km} }}\)

where, ζ is damping factor, C is damping coefficient and C_c is critical damping coefficient.

when ζ =1, c = c_c = 2√(km)   

Calculation:

Given, c_c1 = 0.1 kg/s, m₂ = 2m and k₂ = 8k

\({c_{c1}} = 2\sqrt {{k_1}{m_1}} = 2\sqrt {km}\)

\({c_{c2}} = 2\sqrt {{k_2}{m_2}} = 2\sqrt {8k \times 2m} = 4{c_{c1}} = 4 \times 0.1 = 0.4\) kg/s.

Concept:

The damping ratio (ξ) is defined as ratio of viscous damping coefficient to critical viscous damping coefficient.

\(⇒ {{ξ }} = \frac{{{c}}}{{2\sqrt {{{km}}} }}{{\;\;\;\;\;}} \ldots \left( 1 \right)\)

Where, ξ = damping ratio, c = damping coefficient, k = spring constant, 

Calculation: 

Given: 

m = 0.1 kg, k = 25 kN/m, c = 40 Ns/m

Using equation (1),

\(⇒ {{ξ }} = \frac{{40}}{{2\sqrt {25 \times 10^3\times 0.1} }}{{}}\)

⇒ ξ = 0.4

Concept:

Free damped Vibration: \(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

where m is mass suspended from the spring, k is the stiffness of the spring, x is a displacement of the mass from the mean position at time t and c is damping coefficient.

Since excitation force is absent in the equation hence it is the equation of free damped vibration.

Forced damped Vibration:

\(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = Fcos\omega t\)

Damping factor (ξ): The ratio of the actual damping coefficient (c) to the critical damping coefficient (cc) is known as the damping factor or damping ratio. i.e. \(\xi=\frac{c}{c_c}\)

Calculation:

Given:

\(6\ddot x + 9\dot x + 27x = 1\)

m = 6, c = 9, k = 27

\(\xi= \frac{c}{{{c_c}}} = \frac{c}{{2\sqrt {km} }}= \frac{9}{{2\sqrt {27 \times 6} }}= 0.3535\)

Concept:

Critical damping provides the quickest approach to zero amplitude for a damped oscillator.

Damping ratio,

\(\zeta = \frac{{Actual\;damping}}{{Critical\;damping}} = \frac{c}{{{c_c}}}\)

Where CC is the critical damping coefficient

\({C_C} = 2\sqrt {km} = 2m{\omega _n}\)

Calculation:

Given: m = 50 kg, k = 30 kN/m