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Objective • The main Purpose of this Presentation is to get a

knowledge of

• What are Vibrations ?• What are harmful effects of Vibrations ?

• How to get rid of these mechanical vibrations via• Undamped Vibration Absorbers• Damped Vibration Absorbers

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What are Vibrations :• Vibration is a mechanical phenomenon in which a

body oscillates about an equilibrium point.

• Vibrations may be • Periodic • Random

• Vibrations are some time desirable:• Motion of the Tuning fork• Vibrations in Mobile Phones

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Harmful Effects of Vibrations

• Most of the times, vibrations have undesirable effects,

• Vibrational motions of Engines, Electric Motors and Other Mechanical Devices cause :

• Noise• Waste Of Energy• Fatigue• Mechanical Failure of Machine Components

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Causes of Vibrations in Machines

• Vibrations can be caused by :• Imbalances in rotating parts• Uneven Friction• Meshing of Gear Teeth• Etc. …

•  Careful designs usually minimize unwanted vibrations.

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Vibration Absorbers

• The vibration absorbers are also called dynamic vibration absorber.

• This is a mechanical device used to reduce or eliminate unwanted vibration.

• It consists of another mass and stiffness attached to the main (or original) mass that needs to be protected from vibration.

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Image Of Vibration Absorber

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Cont. …

• Thus the main mass and the attached absorber mass constitute a two-degree-of-freedom system, hence the vibration absorber will have two natural frequencies.

• Commonly used in machinery that operates at constant speed, because the vibration absorber is tuned to one particular frequency and is effective only over a narrow band of frequencies.

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Cont. …• Common applications of the vibration absorber

include • reciprocating tools, • sanders• saws• compactors

• large reciprocating internal combustion engines which run at constant speed (for minimum fuel consumption)

• Without a vibration absorber, the unbalanced reciprocating forces might make the device impossible to hold or control.

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Vibration Absorbers in High Voltage Transmission Lines

• The dynamic vibration absorbers, in the form of dumbbell-shaped devices, are hung from transmission lines to mitigate the fatigue effects of wind induced vibration.

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Resonance• A machine or system may experience excessive

vibration if it is acted upon by a force whose excitation frequency nearly coincides with a natural frequency of the machine or system.

• This Phenomenon Is called Resonance.

• In such a case the vibration of the machine or system can be reduced by using a dynamic vibration absorber.

• The dynamic vibration absorber is designed such that the natural frequencies of the resulting system are away from the excitation frequency.

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Design of Undamped Vibration Absorbers• When we attach an auxiliary mass m2 to a

machine of mass m1 through a spring of stiffness k2, the resulting two-degree-of-freedom system will look as in figure below.

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Cont. …• The equations of motion of the masses m1 and m2

are

• By assuming harmonic solution,

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Cont. …

• we can obtain the steady-state amplitudes of the masses m1 and m2 as

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Cont. …• We are primarily interested in reducing the

amplitude of the machine X1. In order to make the amplitude of zero, the numerator of Eq. (9.135) should be set equal to zero.

• This gives

• If the machine, before the addition of the dynamic vibration absorber, operates near its resonance, Thus if the absorber is designed such that

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Cont. …• The amplitude of vibration of the machine, while

operating at its original resonant frequency, will be zero. • By defining

as the natural frequency of the machine or main system, and

as the natural frequency of the absorber.

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Cont. …• Eqs. (9.135) and (9.136) can be rewritten as:

• As seen before, X1 = 0 at = At this frequency, Eq. (9.141) gives

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Cont. …

• This shows that the force exerted by the auxiliary spring is opposite to the impressed force (k2X2 = -F0 )and neutralizes it, thus reducing to zero.

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The graph …• The graph between X1 / st and / 1 is shown below :

• It can be seen from the above figure that the dynamic vibration absorber, while eliminating vibration at the known impressed frequency introduces two resonant frequencies.

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Summary of Undamped Vibration Absorbers

• It can be seen, from above figure that 1 is less than and Ω2 is greater than the operating speed (which is equal to the natural frequency, ) of the machine. Thus the machine must pass through 1 during start-up and stopping. This results in large amplitudes.

• Since the dynamic absorber is tuned to one excitation frequency the steady-state amplitude of the machine is zero only at that frequency. If the machine operates at other frequencies or if the force acting on the machine has several frequencies, then the amplitude of vibration of the machine may become large.

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Damped Vibration Absorber • Undamped vibration absorber gives

Two unwanted peaksHaving large amplitude. Can cause failure.To remove these peaks Damped vibration

absorber is used.

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System With Damped Vibration Absorber

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Cont…• Mathematical Model

• Steady-state solution of above equations

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Cont…

• By defining

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• Magnitudes of X1 and X2 and can be expressed as

• Equation (9.152) shows that the amplitude of vibration of the main mass is a function of µ,f, g, ç .

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Response of the main System

Effect of damped vibration absorber on the response of the machine.

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Optimally Tuned Vibration Absorber

• By substituting the extreme cases of ç= 0 and ç= ∞ into Eq. (9.152) and equating the two. This yields

• Most efficient vibration absorber is one for which the ordinates of the points A and B are equal which gives us

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Cont…• This equation does not give value for damping

ratio.

• To find damping ratioWe need to make response of main system as flat

as possible at points A & B

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Response for tuned vibration absorber

Tuned vibration absorber

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Cont…• For finding damping ratio we set derivative of

response equal to zero at point A & B

• Solving both equations and taking average damping ratio give us

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Conclusion • Amplitude of absorber mass is greater than

amplitude of the main mass (X2 > X1).• Since the amplitudes of (m2) are expected to be

large, the absorber spring (k2) needs to be designed from a fatigue point of view.

• Damping is to be added only in situations in which the frequency band in which the absorber is effective is too narrow for operation.

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