UNIT-1 Basic Concept Of Vibration PDF

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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 the original position. The internal forces which are set up are the elastic for...

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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 the original 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 because of 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 certain interval 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 will reduce 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 condition of 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 directed towards the centre.

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

(zx) Resonance : When the frequency of external force is equal to the natural frequency of a vibrating 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 order to avoid resonance.

(x) Damping: It is resistance provided to the vibrating body and vibrations related to it are called damped 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 no external 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 and a2 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 determined whereas 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, the resultant 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 discrete systems 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 Further during 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 the inertia 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 body i 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 having torsional 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-Tube manometer 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 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.

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:

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 viscous medium. 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. Such type 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 of this 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 a direction perpendicular lines of magnetic flux is produced by current is induced in the object.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.

Ans. 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 undamped system

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 start and 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.