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UNIT-1

Basic Concept Of Vibration

Ans. All bodies having mass and elasticity are capable of vibration. When external force is

applied on the body, the internal forces areset up in the body which tend to bring the body in theoriginal position. The internal forces which are set up are the elastic forces which tend to bring

the body in the equilibrium position. Consider an example of swinging of pendulum.At extreme

position whole of the kinetic energy of the ball is converted into elastic energy which tends to bring the ball in the equilibrium/mean position. At mean position whole of the, elastic

energy is converted into kinetic energy and body continues to move in opposite direction becauseof it. Now the whole of kinetic energy is converted into elastic energy and this elastic energy

again brings the ball to the equilibrium position. In this way, vibratory motion is repeated

indefinitely and exchange of energy takes place. This motion which repeats itself after certaininterval of time is called vibration.

Main Causes Of Vibration

Ans. The main causes of vibration are:

1. Unbalanced centrifugal force in the. system due to faulty design and poor manufacturing.

2. Elastic nature of system.

3. External excitation applied on the sysbm

4. Winds may cause vibration of cerim sv stem such as electricity lines, telephone lines etc



Disadvantages Of Effects Of Vibration

Ans. Disadvantages harmful effects vibration:

1. Vibration causes excessive and unpleasant stresses in the rotating system.

2. Vibration causes rapid wear and tear of machine parts such as gears and bearings.

3. Vibration causes loosening of parts from the machine.

4. Due to vibrations locomotive can leave the track causing accident or heavy loss.

5. Earthquakes are the cause of vibration because of which buildings and other

structures (like bridges) may collapse.

6. Proper readings of instruments cannot be taken because of heavy vibrations.

7. Resonance may take place if the frequency of excitation matches with the natural

frequency of system causing large amplitudes of vibration thereby resulting in

failure of systems e.g. — Bridges

How can you eliminate/reduce unnecessary vibrations?

Ans. Unwanted vibrations can be reduced by:

1. Removing external excitation if possible.

2. Using shock absorbers.

3. Dynamic absorbers.

4. Proper balancing of rotating parts.

5. Removing manufacturing defects and material inhomogeneities.

6. Resting the system on proper vibration isolators.

What are the advantages of vibration?

Ans. Advantages of vibration

1. Musical Instruments like guitar.



2. In study of earthquake for geological reasons.

3. Vibration is useful for vibration testing equipments.

4.. Propagation of sound is due to vibrations.

5. Vibratory conveyors are based on concept of vibration.

6. Pendulum clocks are based on the principle of vibration.

What is the importance of vibration study?

Ans. Importance of vibration study. The imp of vibration study is to reduce or eliminate

vibration effects over mechanical components by designing them suitably. Proper design and

manufacture of parts will reduce.unbalance in engines which causes excessive and

unpleasent stress in rotating system because of vibration, roper design of machine parts willreduce and tear due to vibration and loosening parts. The proper designing and material

distribution prevent the locomotive m leaving the track due to excessive vibration which may

result in accident or heavy loss. Proper designing of structure buildings can prevent the conditionof resonance which causes dangerously large oscillations which may result in failure of that part.

Define the following:

(i) Periodic Motion (ii) Time period (iii) Frequency (iv) Amplitude (v) Natural frequency

(vi) Fundamental mode of vibration (vii) Degree of freedom (viii) Simple Harmonic Motion

(S.H.M.) (ix) Resonance

(x) Damping (xi) Phase Difference (xi,) Spring stiffness

Ans. Definitions

(i) Periodic motion: A motion which repeats itself after certain interval of time is

called periodic motion.

(ii) Time Period : It is time taken to complete One cycle.

(iii) Frequency: No’s of cycles in one sec. Units = H

(iv) Amplitude: Maximum displacement of a vibrating body from mean position is

called Amplitude.



(v) Natural frequency: When there is no external force applied on the system and it is given a

slight displacement the body vibrates. These vibrations are called free vibrations and frequency

of free vibration is called Natural frequency.

(vi) Fundamental mode of vibration: Fundamental mode of vibr!sternis a mode

(vii) Degree of freedom:

• The minimum no’s of co-ordinates required to specify motion of a system at any instant is

called degree of freedom.

(viii) Simple Harmonic Motion (S.H.M..) : The motion of a body “to” and “fro” about a fixed

point is called S.H.M. S.FLM.. is a periodic motion and it is function of “Sine” or “Cosine”.

Acceleration of S.H.M. is proportional to displacement from the mean position and is directedtowards the centre.



In S.H.M. acceleration is directly proportional to the displacement from the mean position and isdirected towards the centre.

(zx) Resonance : When the frequency of external force is equal to the natural frequency of avibrating body, the amplitude of vibration becomes excessively large. This is known as

“Resonance” . At resonance there are chances of machine part or structure to fail due to

excessively large amplitude. It is thus important to find natural freuqencies of the system in orderto avoid resonance.

(x) Damping: It is resistance provided to the vibrating body and vibrations related to it are calleddamped vibration.

(xi) Phase difference : Suppose there are two vectors

(xii) Spring stiffness : It is defined as unit deflection. Units : N/m.



What are the various parts of a vibrating system?

Ans. Various parts of the mechanical system (vibratory system) are : —

(A) Spring

(B) Damper

(C) Mass

Damping force c ± acting upwards Accelerating force m i acting downwards

Spring force kx acting upwards

Explain different methods of vibration analysis ?

Ans. Different methods of vibration analysis are:

Energy method : According to this method total energy of the system remains constant i.e. sum

of kinetic energy and potential energy always remains constant.



Rayleigh Method : This method is based on the principle that maximum kinetic energy of the

term is equal to the maximum potential energy of the system.



According to this method the sum of forces and moments acting on the system is zero if noexternal force is applied on the system.

Consider fig. I

. Classify different types of vibrations.

Ans. Types of Vibrations

I. Free and Forced



To and fro motion of the system when disturbed initially without any extornal force acting on it

are called free vibrations. e.g. pendulum. To and fro motion of the system under the influence of

external force are called forced vibrations. e.g. Bell, Earthquake.

II. Linear and Non-linear vibrations

Linear vibrations : The linear vibrations are those which obey law of superimposition. If a1 anda2 are the solutions of a differential equation, then a1 + a2 should also be the solution.

Non-linear vibrations : When amplitude of vibrations tends towards large value, then vibrations become non-linear in nature. They do not obey law of superimposition. III. Damped and

Undamped vibrations

Damped vibrations are those in which amplitude of vibration decreases with time. These

vibrations are real and are also called transient vibrations.



Undamped vibrations are those in which amplitude of vibration remains constant. In ideal system

there would be no damping and so amplitude of vibration is steady.

1V. Deterministic and Random vibrations (Non-Deterministic).

Deterministic vibrations are those whose external excitation is known or can be determinedwhereas Random vibrations are those whose external excitation cannot be determined. e.g.

Earthquake

V. Longitudinal, Transverse and Torsional vibrations

What are beats? Ans. When two harmonic motions pass through some point in a medium simultaneously, theresultant is the sum of two motions. This superimposition of harmonics is called interference.

When two harmonics are in phase then their resultant amplitude is maximum and the resultant

amplitude is minimum when two harmonics are out of phase. This phenomenon continuously

occurs i.e. amplitude becomes maximum and minimum repeatedly. This is called “beats”. For beats to occur, the difference in frequencies of two waves should be very less.



Maximum amplitude = 2A

Minimum amplitude 0.

Derive the relation for the work done by the harmonic force

Ans. Let harmonic force F = F0 sin cot is acting on a vibrating body having

motion



Add the following harmonic motions analytically and check the solution graphically.



Graphically:



Draw BC parallel to OD and DC parallel to OB

Measure OC, OC = 9.6 cm at an angle 76°

Hence by graphically we get the same result as by analytical method.

Split the harmonic motion x = 10 sin (wt + 2r/6) into two harmonic motions one having

phase angle of 00 and other having 45° phase angle.

Ans. Let the equations are:



Show that the resultant motion of three harmonic motions given below is zero.



Hence proved.



How do you add two harmonic motions having different frequencies?

Ans. Let two harmonic motions with slightly different frequencies be: -



How can we make a system vibrate in one of its natural mode?

Ans. When a system is displaced slightly from its equilibrium position and allowed to vibrate

then these are called free vibrations and the system is said to vibrate m its natural mode without

any external force impressed on it.

How does a continuous system differ from a discrete system in the nature of its equation of

motion?

Ans. Continuous systems have infinite degree of freedom and so the no. of solutions are infinite.

The equation of motion for continuous system involve both displacement (x) as well as time (t).

Discrete systems have finite degree of freedom and so the no. of solutions are finite. The discretesystems may be single degree of freedom system, 2 degree freedom system and so on. The

number of equations depends upon the degree of freedom of discrete system. Further equation of

motion of discrete systems involve only position (x) and not time (t).

What do you mean by undamped free vibrations?

Ans. If the body vibrates with internal forces and no external force is included, it is Furtherduring vibrations if there is no loss of energy due to friction or resistance, it is known as

undamped free vibration.



Consider the relation for the frequency of spring mass system in vertical position.

Ans.



What is D’Alembert’s Principle?

Ans. D’Alembert’s principle states that if the resultant force acting on a body along with theinertia force is zero, then the body will be in static equilibrium.

Inertia force acting on the body is given by

Assuming that the resultant force acting on body is F, then the body will be in static equilibrium

if

Consider fig. 2.2., the spring force of the body Kx is acting upwards and acceleration of the bodyi is acting in downward direction. The accelerating force is acting downward so inertia force is

acting upwards, so the body is

M static equilibrium under the action of the two forces Kx

and mi. Mathematically it can be written as



Derive the relation for natural frequency of torsional vibrations.

Ans. Consider a rotor having mass moment of inertia I connected at end of the shaft havingtorsional stiffness KT and is rotated by an angle 0 as shown in fig. 2.3.

According to Newton’s law equation of motion can be written as:



Derive the relation for natural frequency of the compound pendulum.



Ans. The system which is suspended vertically and oscillates with small amplitude under the

action of force of gravity is known as compound pendulum (Fig. 2.4)

Let W = Weight of rigid body

o = Point of suspension

k = Radius of gyration about an axis passing

through centre of gravity G.

h = Height of point of suspension frQm G.

I = Moment of Inertia of the body about 0.

I = + mh2 (Parallel axis theorem)



Find the natural frequency of the column of liquid contained in a simple a-Tubemanometer as shown in figure 2.5. Length of tube is 0.2 m.

Ans. Let p mass density of liquid

A cross-sectional area of tube.

length of the column of liquid

or monometer tube.

Let at any moment liquid is displaced by a

distance x from its mean position.

Applying Energy method



Determine the effect of mass of spring on natural frequency of the system as shown in Fig.

2.6.

Ans. Let x be the displacement of mass m and so velocity

will be x. The velocity of spring element at distance y from the fixed end is given

by where 1 is the total length of spring.

Let p be the mass per unit length of spring element, the



Differentiating the above equation

Determine the natural frequency of spring mass pulley system as shown in Fig. 2.7.



A cylinder of diameter D and mass in floats vertically in a liquid of mass density p as

shown in Fig. 2.8. Find the period of oscillation if it is depressed slightly and released. .

Ans. Let us assume x be the displacement of the cylinder,

Determine the frequency of oscillation of the system shown in Fig. 2.9 .



Determine the natural frequency of spring controlled simple pendulum as shown in Fig.

2.10.

Ans. Let us say the system is displaced by an angle 0 to the right. Equation of motion can be

written as;



Determine the natural frequency of the system shown in Fig. 2.11.

Ans. Let m be the mass of circular cylinder and r be the radius of the cylinder.



Differentiating the above equation

The natural frequency of a. spring-mass system is 20 Hz and when extra 3 kg mass is



attached to its mass the natural frequency reduces by 4 Hz. Determine the mass and

stiffness of the system.

A spring-mass system has a time period of 0.25 sec. What will be the new

time period if the spring constant is increased by 30%?

Ans. We know



A car is having a mass of 1000 kg and its spring gets deflected 3 cm under its own load..

Find the natural frequency of car in vertical direction.

Ans. Stiffness of spring is given by

Natural frequency of spring-mass system in vertical position is given by



A torsional pendulum has a rod of 5 mm diameter. Find out its length for natural

frequency of 10 Hz. The inertia of the mass fixed at the free end is 0.0120 kg rn2., Take G =

0.84 x 1011 N/m2.

Ans. The natural frequency of pendulum is given as



. Find the natural frequency of the system shown in Figure 2.12.

Ans. Since the three springs are in parallel1 their equivalent sfess can be calculated



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 thevibratory motion the amplitude of vibration is maximum wkij6es on decreasing with time. The

rate of decreasing amplitude depends upon the amount of damping.

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 dampingis 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:

The force F can also be written as:

where c is called viscous damping coefficient

From (1) and (2),



The main components of viscous damper are cylinder, piston and viscous fluid.

The damping resistance depends upon pressure difference on both sides of piston in viscousmedium. The clearance is left between piston and cylinder walls. More the clearance, more will

be the velocity of piston and less will be the value of viscous damping coefficient.

Equation of Motion



and B = specific damping capacity

2. Coulomb Damping: When a body is allowed to slide over the other body the surface of o

offers resistance to the movement of 9Lod over it. This resisting force is called force of friction.

coefficient of friction

Some of the energy is wasted in friction and amplitude of vibrations goes on decreasing. Suchtype of damping is called coulomb damping.

3. Structural damping : This type of damping arises because of intermolecular

friction beti- the molecules of structure which opposes its movement. The magnitude



of this damping is very small as compared to other damping. Elastic materials during

loading and unloading from a loop or stress strain curve known as_hysteresis loop. The area ofthis loop gives the amount 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 by viscous damping then;



The damping’force, F =

The amplitude decay is of exponential nature.

4. &on-linear, p or Interfacial damping : Machine elements are connected through various joints

and microscopic slip occurs over the joints of machine elements which usdisspoint of energy

when machine elements are in contact with fluctuating load. The energy dissipated per cycle

depends upon coefficient of friction, pressure at contacting surface and amplitude of vibration.There is an optimum value of contact pressure at which energy dissipated is maximum for

different amplitudes.



5. Eddy current damping : If a non-ferrous conducting object (e.g. plater d etc.) moves in adirection perpendicular lines of magnetic flux is produced by current is induced in theobject.1iiiiIrent is proportional to vlocity of the object. The current induced is called eddy current

which set up its own magnetic field opposite to original magnetic field that has induced it. This

provides resistance to motion object It forms magnetic field . This type of damping produced by

eddy currents is

called eddy current damping. it is used in vibrometers and in some vibration control

systems.

Derive the relation for energy dissipated in viscous damping per cycle.



Prove that frequency of vibration of system having coulomb damping is same as that of

undamped system.

Ans. Frequency of damped oscillations

• Free vibrations with dry friction or coulomb damping

(b) Mass displaced towards rigit & moving towards right



The frequency of vibration of system having coulomb damping is same as that of undampedsystem



Prove that amplitude loss per cycle in c4 damping is :

Ans. Rate of Decay of oscillation: Let 1A be the amplitude of body from mean position to startand after half cycle, let x be its amplitude. The velocity of mass =0 at two extreme positions.

(Refer Fig. 3.9)

Therefore, total energy of the system at two extreme positions be

The difference between the two energies must be equal to energy dissipated or work done

against friction.



Differentiate between Coloumb and Viscous damping.

Ans. Differences between Viscous damping & Coulomb damping

1. In case of viscous damping ratio of any two successive amplitudes is constant whereas in

coulomb damping difference between two successive amplitudes is constant.

2. In viscous damping envelope of the maximas in displacement-time plot is an

exponential curve here as in coulomb damping envelope of maximise of displacement-time plot

is a straight line.

3. In case of viscous damping the body once disturbed and from equilibrium position will come

to rest in equilibrium position although it make theoretically infinite time to do so Whereas incase of coulomb damping the body may finally come to rest in equilibrium position or in

displaced position depending upon initial amplitude and amount of friction present.

. What is the response of single degree of freedom system with viscous damping when it is:

Ans. Differential equation of damped free vibrations



Solution of equation (1) can be written as

where A1, A2 = Arbitrary constants

Critical damping constant and damping ratio

The critical damping c is defined as value of damping coefficient c for which



Depending upon the value of damping ratio e, the damped systems 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 once the system is disturbed, it will take infinite time to come back to

the equilibrium condition.

The values of A1 and A2 can be found by initial conditions.



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;

1

I.



An underdamped system is an oscillatory system whose amplitude decreases with time.

Theoretically the system will never come to rest although the amplitude of vibration may be very

very small.

What is the importance of critical damping?

Ans. Out of the three modes the vibrating body which has been displaced from its

mean position would come to state of rest in smallest possible time without overshooting i.e.

without executing oscillation about mean position in critical damping mode.

So critical damping is used for practical applications in large guns so that after firing thereturning to original position in minimum time without vibrating and ready for next firingwithout delay. If damping provided is overdamped or underdamped, then there will be delay.

This property is also design of an instrument



Ans. Logarithmic Decrement (Underdamped system)

It is defined as the natural logarithmic of ratio of any two successive amplitudes on same side ofmean 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



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



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:



In Question No. 13 if m = 1.5 kg, K 4900 N/m, a 6 cm and b = 14 cm, determine the value ofc for which the system is critically damped.

Ans. The equation of motion can be written as;



The system is critically damped when radical is zero

A torsional pendulum when immersed in oil indicates its natural frequency as 200 Hz. But

when it was put to vibration in vacuum having no damping, its natural frequency was

observed as 250 Hz. Find the value of damping factor of oil.

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



A mass is suspended from a spring system as shown in figure 2.13. Determine the natural

frequency of the system.

Ans. Since spring k2 and k3 are connected in parallel1 so their equivalent k is given as k = + k3.

Again k and k1 are connected in series, so the equivalent ke is given as

The natural frequency



A cantilever beam of negligible mass is loaded with mass 4m’ at the free end. Find the

natural frequency of the mass sm’.



Ans. Deflection of cantilever beam loaded at one end can be given as

and therefore stiffness of beam can be calculated as

General equation of motion for undamped free vibration is given as



Determine the natural frequency of the system as shown in figure.

Ans. Deflection of such a system is given as



Equation of motion for undamped free vibration is given as

A simply supported beam of square cross section 5 mm x 5 mm and length 1 m, carrying a

mass of 2.3 kg at the middle, is found to have a natural frequency of transverse vibrations

of 30 rad/s. Determine the Young’s modulus of elasticity of the beam.



where u max - frequency corresponding to maximum amplitude.

(ii) Putting this value in equation (I), we get;

A -, Max Amplitude

X5 -, Zero frequency deflection



What are characteristic curves? Draw them.

Ans. Characteristic Curves

A curve between frequency ratio and magnification factor is known as frequency response curve.Similarly, a curve between phase angle and frequency ratio is called phase- frequency response

curve.



UNIT-II

Two Degree of Freedom System

What are the two degree of freedom system?

Ans. The system which requires two co-ordinates to describe its motion completelyis called two degree of freedom system. In a two degree of freedom system thereare two masses which have two natural frequencies and two co-ordinates arerequired to specify the configuration of the system completely.

Define ‘Normal mode of vibration’?

Ans. In a two degree freedom system there are two natural frequencies of thesystem. The system at its lowest or first natural frequency its first and next higheri.e. second natural frequency is called its second mode. If the two masses vibrate atsame frequency and in phase it is called principal mode of vibration. If prncipa1mode of vibration the amplitude of one of the masses is then it is known as normal

mode of vibration.

Draw the mode shapes for two rotor system.

Ans.

Torsional Vibrations : Consider fig. 5.2. A shaft AB is carrying two rotors ofmoment of inertia I and ‘2 Let and °2 be the angular displacements of rotor at any.instant from mean position. The equation of motion can be written s,



Put,

Putting these values in (II) and (IV)

Solving by determinant



From (VIII)



It shows that the angular displacements of rotors are inversely proportional to theirinertia.

The section of the shaft where angular displacement is zero Is known as node. First

Mode shape

Second Mode Shape



What is co-ordinate coupling? Determine the natural frequencies of suchsystemwith dynamic coupling? (V.V. Imp.)

Ans. Co-ordinate coupling. When we apply brakes on a automobile two motions of

car body occur simultaneously.



If

Equation III is of translator nature.

Equation IV is of oscillator nature.

These are uncoupled differential equations and when then it is calleddynamic coupling.

The natural frequencies of the system are:

What are vibration absorbers ? Prove that spring force of the absorbersystem is equal and opposite to the excitition force for main system to bestationary?

Ans.Vibration Absorber. When a structure which is excited by an external harmonicforce has undesirable vibrations, it becomes necessary to eliminate them bycoupling some vibrating system to it. The vibrating system is known as vibrationabsorber or dynamic vibration absorber. Vibration absorbers are used to control thestructural resonance (consider the main figure)

The natural frequency of this system is



Where



In order that amplitude of mass ni1 is zero

Put A1 =0 (so that mass rn1 must not vibrate)



Simlilarly



Hence when the amplitude A1 = 0 i.e. main system becomes stationary the springforce

of the absorber is equal and opposite to exciting force. The energy of the mainsystem is absorbed by vibration absorber which is also called auxiliary system.

Amplitude of the auxiliary system is inversely proportional to spring constant ‘K2’.

This equation is used for design of absorber.

Discuss the effect of mass ratio on natural frequency of the vibration

absorber.

Ans. We know that

By puffing w = W2 and equating denomination of the above equation equal to 0, we

Get



Equation (I) becomes,

Natural frequency of pendulum is proportional to speed of rotational body.

(iii) Define order No. Ans. Order No: The torsional system receives certain number of torques perrevolution. The no. of these disturbing torques per revolution Is known as order no.of the system



2-cylinder engine working on 4-stroke - Order No. = 1

4-cylinder engine working on 4-stroke - Order No. = 2

6-cylinder engine working on 4-stroke - Order No. 3

Design of centrifugal pendulum absorber

For pendulum absorber to be effective its natural frequency must be equal toexcitation frequency or frequency of disturbing torques.

where f, is Natural frequency of pendulum absorber.

For effective working of pendulum absorber

i.e. natural frequency must he equal to excitation frequency/frequency of disturbingtorque.

from (i) & (ii)



For known values of ‘R’, the length of pendulum can be calculated.

Write short notes on:

(A) Untuned Dry friction damper

(B) Untuned Viscous damper

Ans. (A) Untuned Dry friction damper or Untuned vibration Absorber(Lanchester Damper) : This type of damper is very advantageous to use fortorsional vibrations near resonance conditions. It consists of two fly wheelsmounted freely over a hub. The hub is rigidly fixed to shaft undergoing vibrations.There are friction plates attached to the extension of the hub. These friction platesapply pressure on the flywheel and are responsible for driving the flywheel.

When the pressure. between the friction plates and flywheel is zero the relativevelocity is maximum but frictional torque is zero. There is no energy dissipation insuch case. When the pressure between the friction plate and flywheel is large dueto large friction torque there is no relative velocity between flywheel and shaft andenergy dissipation is zero.

When the speed of the main system is such that torsional vibrations are present inthe system then the pressure between the friction material and flywheel is suchthat both frictional torque and relative rubbing is present then there is dissipation ofenergy in the absorber which causes, reduction in amplitude of the vibrations of the

man system.



(B) Untuned Viscous damper (Houdaille Damper) : this types of damper issimilar in principle to the Lanchester Damper except that instead of using frictionplates for dry friction damping, this system uses friction damping. It consists of afreely rotating disc enclosed in the close-fitting case which is keyed to the shaft.Normally the disc rotates at the shaft speed owing to the viscous drag of the oilbetween the disc and the case. However if the shaft vibrates torsionally, viscousaction of the oil between the disc and casing gives a damping action.



Determine the two natural frequencies of vibration and the ratio ofthe amplitudes of motion of mass m1 and m2 for the system shown in Fig.5.11.

Equations of motion can be written as

Assuming the solution of the form



Ans. The equations of motion can be written as

Assuming the solution of the form

The frequency equation is obtained as

A vibratory system performs the motions as expressed by thefollowing equations:



If the system is turned through 1.5 radians and released, find thefrequencies and mode shapes

Ans. Adding both the equations, we get

Again rewriting the given equations and substituting the values of and 0



Let us assume the solution of the form

Substituting these solutions in the above two equations, we get

The frequency equation can be written as



Ans. Let us give x vertical displacement to mass as shown. Since there is no slipbetween the cord and cylinder, so vertical displacement x causes the cylinder torotate by angle 0.

Writing the equations

Substituting these values in the above equations



Parts of shaft are connected in series

Two equal masses of weight 400 N each and radius of gyration 40 cm arekeyed to the opposite end of a shaft 60 cm long. The shaft is 7.5 cmdiameter for the first 25 cm of its length, 12.5 cm diameter for next 10 cmand 8.5 cm diameter for the remaining length. Find the frequency of freetorsional vibrations of the system and position of node. Take G = 0.84 106kg/cm2

Ans. The system is shown in figure



Ans. The equations of motion for the system shown in figure can be written as

Rearranging the above equations, we can write them as

Let us assume the solution of the form



The above equations can be written as

Two rotors A and B are attached to the end of a shaft 50 cm long. Weightof rotor A is 300 N and its radius of gyration is 30 cm and thecorresponding values of B are 500 N and 45 cm respecitvely. The shaft is7cm in diameter for first 25 cm, 12 cm in diameter for next 10 cm and 10cm diameter for remaining length. Modulus of rigidity for shaft material is8 x 106 kg/cm2 Find:

(i)the position of node and

(ii) the frequency of torsional vibrations.

Ans. The configuration diagram is shown in



From (I)

Considering damping :

A vibrating body having mass 1 kg is suspended by a spring of stiffness 1000 N/rn and it is

put to harmonic excitation of 10 N. Assuming viscous damping, determine;

(a) the resonant frequency

(c) amplitude at resonance

(b) phase angle at resonance

(d) frequency corresponding to peak amplitude

(e) damped frequency.

Take c =40 N - sec/m

Ans. (a) Frequency at resonance

Damping factor E is given as



First trial

Let us assume deflections as

The inertia forces are given by



UNIT VIII

INTRODUCTION TO AEROELASTIC STABILITY

Aeroelasticity:

Aeroelasticity deals with the interaction between aerodynamic, elastic and inertial forces acting

on atmospheric flight vehicles. The aerodynamic and inertial loads deform the structure. The

deformations affect the airloads, which closes the aeroelastic loop.

Collar’s triangle

Static aeroelasticity deals with the effects of structural deformations on steady aerodynamic

load distributions and total force and moment coefficients, and with static instability

(divergence). It is assumed that:

• The 6 d.o.f. airplane maneuvers are slow compared to the structural dynamics. The structure

deforms but structural vibrations have negligible effects.

• The aerodynamic loads due to change in local angles of attack develop with no delays

Dynamic aeroelasticity deals with the interaction between structural dynamics and unsteady

aerodynamics. Delays in the development of aerodynamic loads are important. The main topics

are dynamic instability (flutter) and response to atmospheric gusts (deterministic and stochastic)

• Dynamic Response

•Buffeting



Divergent behavior can occur within a few cycles

Aeroservoelasticity (ASE) deals with the interaction between aero elastic and Control systems.

The control system reads structural vibrations and activates Aerodynamic control surfaces, which

closes the aeroservoelastic loop.The models in this lecture series assume linearity of the

aerodynamic, structural and control systems

Flutter

Flutter is a self-feeding and potentially destructive vibration where aerodynamic forces

on an object couple with a structure's natural mode of vibration to produce rapid periodic

motion.

Flutter can occur in any object within a strong fluid flow, under the conditions that

a positive feedback occurs between the structure's natural vibration and the aerodynamic

forces.

That is, the vibrational movement of the object increases an aerodynamic load, which in

turn drives the object to move further. If the energy input by the aerodynamic excitationin a cycle is larger than that dissipated by the damping in the system, the amplitude of

vibration will increase, resulting in self-exciting oscillation.

The amplitude can thus build up and is only limited when the energy dissipated by

aerodynamic and mechanical damping matches the energy input, which can result in

large amplitude vibration and potentially lead to rapid failure. Because of this, structures

http://en.wikipedia.org/wiki/Natural_frequency



http://en.wikipedia.org/wiki/Vibration



http://en.wikipedia.org/wiki/Periodic_motion




http://en.wikipedia.org/wiki/Positive_feedback



http://en.wikipedia.org/wiki/Vibration#Types_of_vibration



http://en.wikipedia.org/wiki/Self-exciting_oscillation











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