Damped Free Vibration MCQ Quiz in मल्याळम - Objective Question with Answer for Damped Free Vibration - സൗജന്യ PDF ഡൗൺലോഡ് ചെയ്യുക

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:

Viscous Damping or Damped Vibration:

In an elastic body, the vibrations die out after some time due to internal molecular friction of the mass of the body which is represented in the figure as a spring and mass system.  

The damping provided by fluid resistance is called viscous damping and represented as a dashpot in the figure.

Considering forces on mass when displaced below though a distance x from equilibrium position -

Where,

B-B equilibrium position

\(x\) = displacement of the mass from the mean position at time t

\({\dot x}\) = velocity of the mass at time t

\(\ddot x\) = acceleration of the mass at time t

When the mass moves downward resisting forces acting on mass are -

• Inertia = \(m\ddot x\) ⇒ upward direction
• Damping Force = \(c\dot x\) ⇒ upward direction
• Spring force = \(sx\) ⇒ upward direction

Conclusion

• Displacement is in the downward direction and spring force is acting in the upward direction
• Velocity is in the downward direction and damping force is a resisting force acting in the upward direction.
• Acceleration is in the downward direction and inertia force is resisting force which acts in the upward direction

Explanation:

If the system is underdamped it will swing back and forth with decreasing size of the swing until it comes to a stop. Its amplitude will decrease exponentially.

\(x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)\)

Overdamped System: ζ > 1

\(x(t) = A{e^{( - \xi + \sqrt {{\xi ^2} - 1} ){ω _n}t}} + B{e^{( - \xi - \sqrt {{\xi ^2} - 1} ){ω _n}t}}\)

This is the equation of aperiodic motion i.e. the system cannot vibrate due to over-damping. The magnitude of the resultant displacement approaches zero with time.

Underdamped: ζ < 1

\(x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)\)

\(x(t) = A{e^{ - \xi {ω _n}t}}\sin ({ω _d} + \phi )\)

This resultant motion is oscillatory with decreasing amplitudes having a frequency of ω_d. Ultimately, the motion dies down with time.

Critical Damping: ζ = 1

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

The displacement will be approaching zero with the shortest possible time.

Explanation:

If the system is underdamped it will swing back and forth with decreasing size of the swing until it comes to a stop. Its amplitude will decrease exponentially.

\(x\left( t \right) = {e^{ - \xi {\omega _n}t}}\left( {A{e^{i{\omega _d}t}} + B{e^{ - i{\omega _d}t}}} \right)\)

Overdamped System: ζ > 1

\(x(t) = A{e^{( - \xi + \sqrt {{\xi ^2} - 1} ){ω _n}t}} + B{e^{( - \xi - \sqrt {{\xi ^2} - 1} ){ω _n}t}}\)

This is the equation of aperiodic motion i.e. the system cannot vibrate due to over-damping. The magnitude of the resultant displacement approaches zero with time.

Underdamped: ζ < 1

\(x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)\)

\(x(t) = A{e^{ - \xi {ω _n}t}}\sin ({ω _d} + \phi )\)

This resultant motion is oscillatory with decreasing amplitudes having a frequency of ωd. Ultimately, the motion dies down with time.

Critical Damping: ζ = 1

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

The displacement will be approaching to zero with shortest possible time.

Concept:

Logarithmic decrement is the log of successive decrement ratio:

Logarithmic decrement 

\(\delta =\ln\left| {\frac{{{X_{n}}}}{{{X_{n+1 }}}}} \right|= \frac{1}{n}\ln\left| {\frac{{{X_{1}}}}{{{X_{n+1 }}}}} \right|\)

Where n is the number of oscillations made to reduce the amplitude from X1 to X(n+1)

Calculation:

Given X4 = 0.25 X0

\(\frac{{{X_4}}}{{{X_0}}} = 0.25\;\)

\(\frac{{{X_4}}}{{{X_0}}} = \frac{X_4}{X_3}\times\frac{X_3}{X_2}\times\frac{X_2}{X_1}\times\frac{X_1}{X_0}\)

Logarithmic decrement 

\(\delta = \frac{1}{4}\ln\left| {\frac{{{X_0}}}{{{X_4}}}} \right|\)

\(\delta = \frac{1}{2}\ln\left| 2 \right|=0.3465\)                                                                

Concept:

Logarithmic decrement is the log of successive decrement ratio:

Logarithmic decrement 

\(\delta =\ln\left| {\frac{{{X_{n}}}}{{{X_{n+1 }}}}} \right|= \frac{1}{n}\ln\left| {\frac{{{X_{1}}}}{{{X_{n+1 }}}}} \right|\)

Where n is the number of oscillations made to reduce the amplitude from X₁ to X_(n+1)

Calculation:

Given X₅ = 0.25 X₀

\(\frac{{{X_5}}}{{{X_0}}} = 0.25\;\)

\(\frac{{{X_5}}}{{{X_0}}} = \frac{X_5}{X_4}\times\frac{X_4}{X_3}\times\frac{X_3}{X_2}\times\frac{X_2}{X_1}\times\frac{X_1}{X_0}\)

Logarithmic decrement 

\(\delta = \frac{1}{5}\ln\left| {\frac{{{X_0}}}{{{X_5}}}} \right|\)

\(\delta = \frac{1}{5}\ln\left| 4 \right|\)                                                                

Explanation:

Damping Force: 

• The frictional force acts in the opposite direction of the displacement of any particle and reduces the energy of the system over a time period.
• Reduction of energy leads to a decrease in amplitude of oscillation this is known as Damping.
• The equation of damping force is given by

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

Where c is the damping coefficient, v is the velocity of the oscillating object. 

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

From the above discussion, we can conclude that the damping force is proportional to Velocity.

Concept:

Damped frequency is given by:

\(\omega_d =\omega_n\sqrt {1 - {{\rm{ξ }}^2}} \)

ω_n = angular frequency

ξ = damping factor = \(\frac{C}{{{C_c}}}\)

Calculation:

Given:

m = 50 kg, k = 30 kN/m

C = 0.2C_c

∴ ξ = \(\frac{C}{{{C_c}}}\) = 0.2

For a system with damping,

ω_n = \(\sqrt {\frac{k}{m}} = \sqrt {\frac{{30 \times {{10}^3}}}{{50}}} = 24.494~rad/s\)

ω_d = ω_n \(\sqrt {1 - {{\rm{ξ }}^2}} \)

ω_d = (24.494) \(\sqrt {1 - {{0.2}^2}} \)

ω_d = 23.99 ≃ 24 rad/s

Concept:

Logarithmic decrement of spring is,

\(\delta = \frac{{2\pi\xi }}{{\sqrt {1 -\xi 2\;} }}\)

where, \(\xi = \frac{C}{{{C_c}}}\)

And C_C = 2 mω_n

The natural frequency of spring is, \({\omega _n} = \sqrt {\frac{k}{m}} \) 

When the stiffness of the spring is doubled and mass is made half,

The natural frequency of spring become \({\omega _n} = \sqrt {\frac{2k}{{\frac{m}{2}}}} = 2\sqrt {\frac{k}{m}} \) 

Now, \({\xi_1} = \frac{C}{{{C_C}}} = \frac{C}{{2m{\omega _n}}} =\xi\)

\({\xi_2} = \frac{C}{{{C_C}}} = \frac{C}{{2(m/2){2\omega _n}}} = \xi\)

\({\delta _2} = \frac{{2\pi\epsilon {_2}}}{{\sqrt {1 -\epsilon _2^2} }}\)

\( = \frac{{2\pi\epsilon }}{{\sqrt {1 - {\epsilon^2}} }}\)        ---- (1)

\(\delta = \frac{{2\pi\epsilon }}{{\sqrt {1 - {\epsilon^2}} }}\)       ---- (2)

Form (1) and (2)

\(\frac{{{\delta _2}}}{\delta } = \frac{{2\pi\epsilon }}{{2\pi\epsilon }}\frac{{\sqrt {1 - {\epsilon^2}} \;}}{{\sqrt {1 - {\epsilon^2}} }}\)

\(\frac{{{\delta _2}}}{\delta } = 1\)

⇒ δ₂ = δ

Explanation:

Depending on the system damping, it can have either an aperiodic motion or a periodic motion.

Damping ratio:

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

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

Overdamped System: ζ > 1

\(x(t) = A{e^{( - \xi + \sqrt {{\xi ^2} - 1} ){ω _n}t}} + B{e^{( - \xi - \sqrt {{\xi ^2} - 1} ){ω _n}t}}\)

This is the equation of aperiodic motion i.e. the system cannot vibrate due to over-damping. The magnitude of the resultant displacement approaches zero with time.

Underdamped: ζ < 1

\(x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)\)

\(x(t) = A{e^{ - \xi {ω _n}t}}\sin ({ω _d} + \phi )\)

This resultant motion is oscillatory with decreasing amplitudes having a frequency of ωd. Ultimately, the motion dies down with time.

Critical Damping: ζ = 1

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

The displacement will be approaching to zero with shortest possible time.