Chapter 3 _ Free Damped Vibrations _ Mechanical Vibrations

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2/28/12 Chapter 3 : Free Damped Vibrations _ Mechanical Vibrations 1/21 ptumech.loremate.com/mv/node/3 Mechanical VibUaWionV 976 Recommend Press Ctrl & '+' To enlarge te[t and pics! SeaUch SeaUch WhiV ViWe: SeaUch ChapWeUV Home Topics Chapter 1 : Elements Of Vibrations Chapter 2 : Undamped Free Vibrations Chapter 3 : Free Damped Vibrations Chapter 4 : Force Vibrations Chapter 5 : Two Degree of Freedom S\stem Chapter 6 : Several Degree of Freedom S\stem Chapter 7 : Continuous S\stems Home ChapWeU 3 : FUee Damped VibUaWionV Q. 1. What is damping? Ans. Damping is the resistance offered by a body to the motion of a vibratory system. The resistance may be applied to liquid or solid internally or externally At the start of the vibratory motion the amplitude of vibration is maximum wkij6es on decreasing with time. The rate of decreasing amplitude depends upon the amount of damping. Q. 2. Classify different types of damping. Ans. Types of Damping 1. Viscous 2. Coulomb 3. Structural 4. Non-linear, Slip or interfacial damping 5. Eddy current-damping 1. Viscous damping: When the system is allowed to vibrate in viscous medium the damping is called viscous Viscosity is the property of the fluid by virtue of which it offers resistance to moment of one over the other. The force F required . to maintain the velocity x of plate is given by:

Transcript of Chapter 3 _ Free Damped Vibrations _ Mechanical Vibrations

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Chapter 1 : Elements Of

Vibrations

Chapter 2 : Undamped Free

Vibrations

Chapter 3 : Free Damped

Vibrations

Chapter 4 : Force Vibrations

Chapter 5 : Two Degree of

Freedom System

Chapter 6 : Several Degree of

Freedom System

Chapter 7 : Continuous Systems

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Chapter 3 : Free Damped Vibrations

Q. 1. What is damping? Ans. Damping is the resistance offered by a body to themotion of a vibratory system.The resistance may be applied to liquid or solid internally orexternally At the start of the vibratory motion the amplitudeof vibration is maximum wkij6es on decreasing with time.The rate of decreasing amplitude depends upon the amountof damping.

Q. 2. Classify different types of damping. Ans. Types of Damping1. Viscous2. Coulomb 3. Structural 4. Non-linear, Slip or interfacial damping 5. Eddy current-damping 1. Viscous damping: When the system is allowed to vibratein viscous medium the damping is called viscous Viscosity isthe property of the fluid by virtue of which it offers resistanceto moment of one over the other.The force F required . to maintain the velocityx of plate is given by:

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The force F can also be written as:

where c is called viscous damping coefficientFrom (1) and (2),

The main components of viscous damper are cylinder, pistonand viscous fluid.

The damping resistance depends upon pressure difference onboth sides of piston in viscous medium. The clearance is leftbetween piston and cylinder walls. More the clearance, morewill be the velocity of piston and less will be the value ofviscous damping coefficient.

Equation of Motion

and B = specific damping capacity2. Coulomb Damping: When a body is allowed to slide overthe other body the surface of o offers resistance to themovement of 9Lod over it. This resisting force is called forceof friction.

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coefficient of friction Some of the energy is wasted in friction and amplitude ofvibrations goes on decreasing. Such type of damping is calledcoulomb damping. 3. Structural damping : This type of damping arises becauseof intermolecularfriction beti- the molecules of structure which opposes itsmovement. The magnitudeof this damping is very small as compared to other damping.

Elastic materials duringloading and unloading from a loop or stress strain curveknown as_hysteresis loop. The area of this loop gives theamount of energy dissipated in one cycle during vibrations.This is also called hysteresis damping.The energy loss per cycle is given as;

If energy dissipated is treated equal to energy dissipated byviscous damping then;

The damping’force, F =

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The amplitude decay is of exponential nature.

4. &on-linear, p or Interfacial damping : Machine elementsare connected through various joints and microscopic slipoccurs over the joints of machine elements which usdisspoint of energy when machine elements are in contact withfluctuating load. The energy dissipated per cycle dependsupon coefficient of friction, pressure at contacting surface andamplitude of vibration. There is an optimum value of contactpressure at which energy dissipated is maximum for differentamplitudes.

5. Eddy current damping : If a non-ferrous conducting object(e.g. plater d etc.) moves in a direction perpendicular linesof magnetic flux is produced by current is induced in theobject.1iiiiIrent is proportional to vlocity of the object. Thecurrent induced is called eddy current which set up its ownmagnetic field opposite to original magnetic field that hasinduced it. This provides resistance to motion object It forms magnetic field . This type of damping produced by eddycurrents iscalled eddy current damping. it is used in vibrometers and insome vibration controlsystems.

Q. 3. Derive the relation for energy dissipated inviscous damping per cycle.

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viscous damping per cycle. Ans. Energy dissipated in viscous damping per cycle

Q. 4. Prove that frequency of vibration of systemhaving coulomb damping is same as that of undamped

system. Ans. Frequency of damped oscillations

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• Free vibrations with dry friction or coulomb damping(b) Mass displaced towards rigit & moving towards right

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The frequency of vibration of system having coulombdamping is same as that of undamped system

Q. 5. Prove that amplitude loss per cycle in c4damping is :

Ans. Rate of Decay of oscillation: Let 1A be the amplitude ofbody from mean position to start and after half cycle, let xbe its amplitude. The velocity of mass =0 at two extremepositions. (Refer Fig. 3.9)Therefore, total energy of the system at two extremepositions be

The difference between the two energies must be equal toenergy dissipated or work done against friction.

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Q. 6. Differentiate between Coloumb and Viscousdamping.

Ans. Differences between Viscous damping & Coulombdamping1. In case of viscous damping ratio of any two successiveamplitudes is constant whereas in coulomb dampingdifference between two successive amplitudes is constant.

2. In viscous damping envelope of the maximas indisplacement-time plot is an exponential curve here as incoulomb damping envelope of maximise of displacement-timeplot is a straight line.3. In case of viscous damping the body once disturbed andfrom equilibrium position will come to rest in equilibriumposition although it make theoretically infinite time to do soWhereas in case of coulomb damping the body may finallycome to rest in equilibrium position or in displacedposition depending upon initial amplitude and amountof friction present.

Q. 7. What is the response of single degree of freedomsystem with viscous damping when it is:

Ans. Differential equation of damped free vibrations

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Solution of equation (1) can be written as

where A1, A2 = Arbitrary constants

Critical damping constant and damping ratioThe critical damping c is defined as value of dampingcoefficient c for which

Depending upon the value of damping ratio e, the dampedsystems are categorized as:

This motion is also called a periodic motion. When t =0, x =A1 + A2. This system is non-vibratory in nature. When oncethe system is disturbed, it will take infinite time to come backto the equilibrium condition.The values of A1 and A2 can be found by initial conditions.

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The value of displacement x goes on decreasing with time.

In critical damping both roots are equal and are equal to - (0.The solution of critically damped system is given as;

1I.

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The amplitude vary exponentially with time. As time increasesamplitude decreases.

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An underdamped system is an oscillatory system whoseamplitude decreases with time. Theoretically the system willnever come to rest although the amplitude of vibration maybe very very small.

Q. 8. What is the importance of critical damping? Ans. Out of the three modes the vibrating body which hasbeen displaced from itsmean position would come to state of rest in smallest possibletime without overshooting i.e. without executing oscillationabout mean position in critical damping mode.

So critical damping is used for practical applications in largeguns so that after firing the returning to original position inminimum time without vibrating and ready for next firingwithout delay. If damping provided is overdamped orunderdamped, then there will be delay. This property is alsodesign of an instrument

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Ans. Logarithmic Decrement (Underdamped system)It is defined as the natural logarithmic of ratio of any twosuccessive amplitudes on same side of mean time.Consider fig. 3.13(a).Let us take two successive amplitudes be x1 and x2.Logarithmic decrement 6 is given by;

The time period of damped oscilliations

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Q 10. If an underdamped system executes ‘n’ cyclesthen prove that logrithimic decrement

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Q 11. A damping .force having magnitude 2 cos (23rt-44) N, gives 5 cos 2t m displacement. Calculate(a) Energy dissipated during first 5 seconds and(b) Energy dissipated during the first 3/4 sec.

Ans. We know the force and displacement are given as:

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Q. 14. In Question No. 13 if m = 1.5 kg, K 4900 N/m,a 6 cm and b = 14 cm, determine the value of c forwhich the system is critically damped. Ans. The equation of motion can be written as;

The system is critically damped when radical is zero

Q. 15. A torsional pendulum when immersed in oilindicates its natural frequency as 200 Hz. But when itwas put to vibration in vacuum having no damping, itsnatural frequency was observed as 250 Hz. Find thevalue of damping factor of oil.

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Ans. The expression for torsional vibrations in vacuum (c =0) is;

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