Dynamic Vibration Absorber.pdf

Dynamic vibration absorber

SOLVE the virtual lab@ NITK Surathkal Machine Dynamics and Vibration Lab


Theory

Learning objectives

After completing this simulation experiment one should be able to

Model a given real system to an equivalent simplified single degree of freedom

system and reducing the vibration of the main system adding an absorber system with

suitable assumptions / idealisations.

Determine the amplitude of vibration for both main and additional system.

Tune a vibration absorber by varying dynamic system parameters.

Study the response of the vibration absorber for different excitation frequency and

amplitude.


Introduction

When a structure is externally excited, it will have undesirable vibrations. The amplitude of

vibration will be maximal when the system gets excited close to its natural frequency and this

can cause rapid catastrophic failure. Hence it becomes necessary to neutralize these

vibrations. One of the methods for neutralizing these vibrations is by coupling a vibrating

system to it so that the amplitude can be brought down to zero. This kind of vibrating system

is known as vibration absorber or dynamic vibration absorber (DVA). DVA are used to

control structural vibrations where it concentrates on neutralizing the amplitude of vibration

at resonance. Some practical examples of dynamic vibration absorber showed in Fig 1 and 2.

Fig 1: Dynamic vibration absorber in steel stack



Fig 2: Dynamic vibration absorber PCB

To study the dynamic absorber system, a real system considered as main system is modelled

as an equivalent single degree of freedom system and it is excited by a harmonic excitation

force F= F0 sinωt. The steady state response of the system is given by x= X sin (ωt+ϕ).

Steady state amplitude of vibration of the proposed single degree of freedom system will be

maximum at the resonance. To neutralize the effect at resonance, the main system couples

with an absorber system. This coupling will affect (suppress) the amplitude of vibration of

the main system. By the addition of absorber system, single degree of freedom analysis

cannot hold. Hence whole system should be considered two degree of freedom system.

Fig 3: A single degree of freedom system and vibration absorber system



Fig 4: Free body diagram of vibration absorber system

The free-body diagrams of the masses and are shown in Fig. 4. By application of Newton’s

second law of motion to each of the masses gives the equations of motion as:

1 1 1 1 2 1 2 0( ) sin( )m x k x k x x F t ... (1)

2 2 2 2 1( ) 0m x k x x ... (2)

Steady state response of two degree of freedom is assumed as,

1 1 sin( )x X t And 2 2 sin( )x X t

By substituting x1 and x2 in equation 1 and 2 we get,

2

1 2 1 1 2 2 0( )Xk k m k X F ... (3)

2

2 1 2 2 2( )X 0k X k m ... (4)

By solving the equation 3 and 4, amplitude of vibration of main system and absorber system

given by,

2

2 2

1 4 2

1 2 1 2 2 1 2 1 2( )

k mX

m m m k m k k k k

... (5)

and

2 02 4 2

1 2 1 2 2 1 2 1 2( )

k FX

m m m k m k k k k

... (6)

From equation 5 it can be observed that the amplitude of vibration of main system X1 can be

zero if numerator becomes zero.

2

2 2i.e. 0k m

or 2 22

2

2

n

k

m ...(7)

From equation 7, it can be concluded that when the excitation frequency is equal to the

natural frequency of the absorber, then main system amplitude becomes zero even though it

is excited by harmonic force.



Dimensionless form of equation 5 and 6 can be written as,

2

2

21

4 2 2

2 2 2 2

1 2 1 2

1

(1 ) 1st

X

X

... (8)

2

4 2 2

2 2 2 2

1 2 1 2

1

(1 ) 1st

X

X

... (9)

Equations 8 and 9 give the amplitude response of main system and absorber system as a

function of exciting frequency.

Where,

Xst = F0/k1= Zero frequency deflection of the main system

ω1= Natural frequency of the main system

ω2= Natural frequency of the absorber system

ω= Frequency of external excitation

μ= ratio of absorber mass to the main mass

Fig 5 shows the variation of X1/Xst for different frequency ratios.

Fig 5: Frequency response curves of main system



Let’s try to understand these equations by doing a few simple simulations, go to next tab

procedure to find out how to run the simulation to EXPLORE (expR) and to EXPERIMENT

(expT). A talking tutorial or a self-running demo with narration can be seen at EXPLAIN

(expN)

Dynamic Vibration Absorber.pdf

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