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SELF EXCITED VIBRATIONS

Mechanical vibrations Seminar on

Ramakanth P Joshi

100922003

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OBJECTIVE

The theory of self excited vibration has important application in mechanicalsystems.

Shortly after the official opening of the Tacomas narrows bridge at pugetsound,washigton, the main span of bridge underwent self excitation oscillationwhich resulted in the destruction of bridge.

steady transversely blowing wind was held responsible

FOCUS ON

Definition: self excited vibration

Mathematical representationDynamic stability analysis for self excited vibration

Friction induced vibration

Flow induced vibration

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DEFINITION: SELF EXCITED VIBRATION

There are systems for which the exciting force is a function of the motion parametersof the system, such as displacement, velocity, or acceleration . Such systems are

called self-excited vibrating systems since the motion itself produces the exciting

force.

In self excited vibration the alternating force that sustains the motion is created or

controlled by the motion itself; when the motion stops the alternating force

disappears,

In a forced vibration the sustaining alternating force exists independently of the

motion and persists even when the vibratory motion is stopped

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MATHEMATICAL REPRESENTATIONself-excited vibration can be considered as a free vibration with negative damping.

Solution for this equation can be written as

Which is clearly a vibration with exponentially increasing amplitude

A mechanical system is statically stable if a displacement from the equilibrium position sets

up a force (or couple) tending to drive the system back to the equilibrium position. It is

statically unstable if the force thus set up tends to increase the displacement. Therefore

static instability means a negative spring constant k

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DYNAMIC STABILITY ANALYSIS FOR SELF EXCITED VIBRATION

A system is dynamically stable if the motion (or displacement)converges or

remains steady with time. On the other hand, if the amplitude of displacement

increases continuously (diverges) with time, it is said to be dynamically

unstable. The motion diverges and the system becomes unstable if energy is fed

into the system through self excitation. To see the circumstances that lead to

instability, we consider the equation of a

single degree of freedom system:

If a solution of the formx(t) = Cest, where C is a constant and Ieads to the characteristicequation

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The roots of the equations are

Roots of equation x(t) = Cest

the motion will be diverging and a periodic if the roots S and S are real and positive.

This situation can be avoided if c/m and k/m are positive.

The motion will also diverge if the roots S and S are complex conjugates with positivereal parts. To analyze the situation, let the roots S and S of Eq. be expressed as

The above equation shows that for negative p, c/m should be positive and for positive p2+q2

,K/m should be positive.

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FRICTION INDUCED VIBRATIONS AND ITS FREQUENCY

Small amplitude behavior

The equation of motion of the system is

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Since the base is moving at constant velocity v1, and the

corresponding coefficient of friction at that velocity is 1,

the system reaches static equilibrium with:

If now a disturbance applies a small positive increment of

velocity, , to the mass, as shown in Fig

But, from the fig

On substitution

Upon subtracting

We now see that gives the change in force for a unit

change in velocity, and is therefore equivalent to a

viscous damping coefficient

It is usually more convenient to write

in the non-dimensional formWhere,

1

2

3

4

5

6

7

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The solution for the equation can be written as

If then the total non-dimensional damping coefficient, , is negative, it can be seen

from the above equation that the exponent becomes positive, and x will always consist

of an exponentially growing oscillation, until limited in some way.

Large amplitude behavior

The amplitude will limit if the energy derived from the friction force over a complete cycle

becomes equal to the energy dissipated by the damper.

Limiting will always occur when the peak velocity due to the oscillation, , exceeds the

mean rubbing velocity, v1

Assuming that the waveform remains sinusoidal, this corresponds to a displacement limit,

xmax, given by:

CONTINUEDu..

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FRICTION-INDUCED VIBRATION IN AIRCRAFT LANDING GEAR

The conventional cantilever type of aircraft landing gear, when fitted with brakes, canexhibit two main forms of friction-induced vibration:

1) When the brakes are working normally, if the coefficient of friction between therubbing surfaces in the brakes tends to decrease as the velocity increases, negative

damping may result, producing the vibration known as brake judder or brakechatter, at low freq and break squeal at high frequencies

2) When the brakes lock completely, perhaps because the anti-skid device, if fitted,does not always operate down to very low speeds, the friction characteristicsbetween the tires and the runway can then provide the negative damping instead.

Measurements of tire friction show that the negative slope of coefficient of frictionversus v, necessary for instability, tends to occur when the runway is wet, and thespeed is low.

In both the cases it will lead to fatigue failure

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DYNAMIC INSTABILITY CAUSED BY FLUID FLOWH

igh-tension electric transmission lines have been observed under certain weatherconditions to vibrate with great amplitudes and at a very slow frequency.

A rough calculation shows that the natural frequency of the span is of the same order

as the observed frequency.

We have a case of self-excited vibration caused by the wind acting on the wire which,

on account of the accumulated sleet, has taken a noncircular cross section

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STABILITY CRITERIA

In aerodynamic work it is customary to resolve the total air force on an object into twocomponents:

a. In the direction of the wind (the drag or resistance D).

b. Perpendicular to the wind (the lift L).

These two forces can be measured easily with the standard windtunnel apparatus.

= tan-1(v/V)

The lift and drag forces Land D have vertical upward component (i.e., components

opposite to the direction of the motion) of Lcos and D sin . The total upward damping

force F of the wind is

The criterion for dynamic stability is

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CONTINUED..

Thus the system will be unstable when

The values of the lift and drag of an arbitrary cross section cannot be calculated from

theory but can be found from a wind-tunnel test. The results of such tests are usually

plotted in the form of a diagram such as

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KARMAN VORTICES

When a fluid flows by a cylindrical obstacle, the wake behind the obstacle is no

longer regular but in it will be found distinct vortices of the pattern shown in Fig. This phenomenon has been studied experimentally

The cylinder moves forward by about 41/2 diameters during one period of the

vibration.

The eddy shedding on alternate sides of the cylinder causes a harmonically varying

force on the cylinder in a direction perpendicular to that of the 'stream.

The maximum intensity of this force can be written in the form usual for most

aerodynamic forces (such as lift and drag) as follows

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REDUCTION OF FLOW INDUCED VIBRATION

In high speed cars the flow induced lift

forces can unload the tires thereby causing

problems with steering control and stabilityof the vehicle. Although lift forces can be

countered partly by adding spoilers ,the

drag forces will increase. In recent years

movable inverted airfoils are being used to

develop a downward aerodynamic force

with improved stability characterstics

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REFERENCES

Mechanical vibrations, J.P. DEN HARTOG, Fourth edition, McGraw-Hill book company

Mechanical vibrations,Singiresu S Rao,fourth edition, ISBN 81 7758-874-

5,Pearson sducation.

Shock and vibration handbook,Cyril M. Harris and Charles E. Crede, volume

1,McGRaw hill company.