MADE BY –Chirayu Olkar Regular 110550119048Darshil Shah – D TO D 08Parth Bhatth – D TO D 10

Forced Damped Vibrations

1. INTRODUCTION 2. ANYLASIS -Complementary function -Particular Integral - Phase angle 3.Magnification Ratio -Frequency response curve 4.Plot of phase angle verses frequency ratio .

INDEX

INTRODUCTION

Vibrations

Free Forced

Undamped Damped Undamped Damped

In free un-damped vibrations a system once disturbed from its initial position executes vibrations because of its elastic properties. There is no damping in these systems and hence no dissipation of energy and hence it executes vibrations which do not die down. These systems give natural frequency of the system.

In free damped vibrations a system once disturbed from its position will execute vibrations which will ultimately die down due to presence of damping. That is there is dissipation of energy through damping. Here one can find the damped natural frequency of the system.

In forced Damped vibration there is an external force acts on the system. This external force which acts on the system executes the vibration of the system.The external force may be harmonic and periodic, non-harmonic and periodic or non periodic.

Examples of forced vibrations are air compressors, I.C. engines, turbines, machine tools etc,.

1. Forced vibration with constant harmonic excitation

2. Forced vibration with rotating and reciprocating unbalance

3. Forced vibration due to excitation of the support

A: Absolute amplitude B: Relative amplitude4. Force and motion transmissibility

Analysis of forced vibrations

Forced vibration with constant harmonic excitation

Total Forces acting on the body are 1. External harmonic force ( Downwards)2.Inertia force ( Upwards)3. Damping force ( Upwards)4.Spring force (Kx upwards)

From the figure it is evident that spring force and damping force oppose the motion of the mass. An external excitation force of constant magnitude acts on the mass with a frequency . Using Newton’s second law of motion an equation can be written in the following manner.Equation 1 is a linear non homogeneous second order differential equation. The solution to eq. 1 1.complimentary function

2. particular Integral

The complimentary function part of eq, 1 is obtained by setting the equation to zero.

The complementary function solution is given by the following equation

Equation 3 has two constants which will have to be determined from theinitial conditions. But initial conditions cannot be applied to part of thesolution of eq. 1 as given by eq. 3. The complete response must bedetermined before applying the initial conditions. For complete response theparticular integral of eq. 1 must be determined. This particular solution willbe determined by vector method as this will give more insight into theanalysis.

Particular solution to be

Differentiating the above assumed solution and substituting it in eq. 1

Following points are observed from the vector diagram

1. The displacement lags behind the impressed force by an angle .

2. Spring force is always opposite in direction to displacement.

3. The damping force always lags the displacement by 90°. Damping force is always opposite in direction to velocity.

4. Inertia force is in phase with the displacement. The relative positions of vectors and heir magnitudes do not change with time.

From the vector diagram one can obtain the steady state amplitude and phase angle as follows

The above equations are made non-dimensional by dividing the numerator and denominator by K.

Therefore the complete solution is given by

The two constants A2 and 2 have to be determined from the initial Conditions.The first part of the complete solution that is the complementary function decays with time and vanishes completely. This part is called transientvibrations. The second part of the complete solution that is the particular integral is seen to be sinusoidal vibration with constant amplitude and is called as steady state vibrations. Transient vibrations take place at dampednatural frequency of the system, where as the steady state vibrations take place at frequency of excitation. After transients die out the complete solution consists of only steady state vibrations.

MAGNIFICATION FACTOR

The ratio of the amplitude of the steady-state response to the static deflection under the action of force F0 is known as magnification factor (MF).

Thus, the magnification factor depends upon:(a) the ratio of frequencies,W/Wn , and(b) the damping factor.The plot of magnification factor against the ratio of frequencies (W/Wn) fordifferent values of z is shown . The curves show that as thedamping increases or z increases, the maximum value of the magnificationfactor decreases and vice-versa. When there is no damping (z = 0), it reachesinfinity at W/Wn = 1, i.e. when the frequency of the forced vibrations is equal tothe frequency of the free vibration. This condition is known as resonance.

In practice, the magnification factor cannot reach infinity owing to friction which tends to dampen the vibration. However, the amplitude can reach very high values.Below shows the plots of phase angle vs. frequency ratio (w/wn) for different values of z. Observe that, Irrespective of the amount of damping, the maximum amplitude of vibration occurs before the ratio w/wn reaches unity or when the frequency of the forced vibration is less than that of the undamped vibrations. Phase angle varies from zero at low frequencies to 180° at very high frequencies. It changes very rapidly near the resonance and is 90° atresonance irrespective of damping. In the absence of any damping, phase angle suddenly changes from zero to 180° at resonance

Plot of phase angle versus Frequency ratio

1.The phase angle varies from 0 at low frequency ratio to 180 at very high frequency ratio .

2.At the resonance frequency (that is at W= Wn) the phase angle is 90 and damping does not have any effect on phase angle .

3.At frequency ratio less then 1,higher the damping factor higher is the phase angle ,whereas at frequency greater then unity ,higher the damping factor –lower is the phase angle .

4.The variation in phase angle is because of damping .If there is no damping ,the phase angle is either 0 or less then 180 and at resonance the phase angle suddendly changes from 0 to 180.

Observations

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