Damped Free Vibration MCQ Quiz in मराठी - Objective Question with Answer for Damped Free Vibration - मोफत PDF डाउनलोड करा

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 

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

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

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:

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 

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

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

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:

Vibration:  Vibration is a periodic motion of small magnitude. But for simplicity, we can assume it as a simple harmonic motion with a small amplitude.

Damped Vibration:

• When there is a reduction in amplitude over every cycle of vibration, the motion is said to be damped vibration.
  This is due to the fact that a certain amount of energy possessed by the vibrating system is always dissipated in overcoming frictional resistances to the motion.

Un-damped vibrations:

• When there is no friction or resistance present in the system to contract vibration then the body executes un-damped or damped free vibration.

Damped Free Vibration:

• Damped free vibration describes the mobility of an object without the action of any externally applied force.

​\(m\frac{{{d^2}x}}{{d{t^2}}} + c\frac{{dx}}{{dt}} + kx = 0\)

Where m is the mass suspended from the spring, s is the stiffness of the spring, x is the displacement of the mass from the mean position at time t and c is the damping coefficient. Since excitation force is absent in the equation hence it is the equation of free-damped vibration.

Damped Free Response x = Xe^-ζω_nt. sin(ω_d.t + ϕ₁)

where: x = displacement of mass from the mean position, X = Amplitude, ζ = damping factor or damping ratio, ω_d = Damped frequency, ϕ₁ = phase angle.

Decrement ratio or successive amplitudes ratio:

\(\frac{{{x_0}}}{{{x_1}}} =\frac{{{x_1}}}{{{x_2}}} =\frac{{{x_2}}}{{{x_3}}} =\frac{{{x_3}}}{{{x_4}}} ....=e^{ζω_nt}=Constant\) 

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