- 1. 8/31/2021 1 Hareesha N G, Asst. Prof, DSCE, BLore-78
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- 3. BASIC CONCEPTS OF VIBRATION • All bodies having mass and elasticity are capable of producing vibration. • The mass is inherent of the body and elasticity causes relative motion among its parts. • When body particles are displaced by the application of external force, the internal forces in the form of elastic energy are present in the body. • These forces try to bring the body to its original position. • At equilibrium position, the whole of the elastic energy is converted into kinetic energy and body continues to move in the opposite direction because of it. • The whole of the kinetic energy is again converted into elastic or strain energy due to which the body again returns to the equilibrium position. • In this way, vibratory motion is repeated indefinitely and exchange of energy takes place. • Thus, any motion which repeats itself after an interval of time is called vibration or oscillation. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 3
- 4. BASIC CONCEPTS OF VIBRATION (Cntd..) • The swinging of simple pendulum as shown in figure 1 is an example of vibration or oscillation as the motion of ball is to and fro from its mean position repeatedly. • The main reasons of vibration are as follows : – Unbalanced centrifugal force in the system. This is caused because of non- uniform material distribution in a rotating machine element. – Elastic nature of the system. – External excitation applied on the system. – Winds may cause vibrations of certain systems such as electricity lines, telephone lines, etc. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 4
- 5. IMPORTANCE OF VIBRATION STUDY IN ENGINEERING • The structures designed to support the high speed engines and turbines are subjected to vibration. • Due to faulty design and poor manufacture, there is unbalance in the engines which causes excessive and unpleasant stresses in the rotating system because of vibration. • The vibration causes rapid wear of machine parts such as bearings and gears. • Unwanted vibrations may cause loosening of parts from the machine. • Because of improper design or material distribution, the wheels of locomotive can leave the track due to excessive vibration which results in accident or heavy loss. • Many buildings, structures and bridges fall because of vibration. • If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is reached, and dangerously large oscillations may occur which may result in the mechanical failure of the system. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 5
- 6. IMPORTANCE OF VIBRATION STUDY IN ENGINEERING (Contd…) • Vibration can be used for useful purposes such as vibration testing equipments, vibratory conveyors, hoppers, sieves and compactors. • Vibration is found very fruitful in mechanical workshops such as in improving the efficiency of machining, casting, forging and welding techniques, musical instruments and earthquakes for geological research. • It is useful for the propagation of sound. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 6
- 7. IMPORTANCE OF VIBRATION STUDY IN ENGINEERING (Contd…) Thus undesirable vibrations should be eliminated or reduced upto certain extent by the following methods : – Removing external excitation, if possible – Using shock absorbers. – Dynamic absorbers. – Resting the system on proper vibration isolators. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 7
- 8. DEFINITIONS • Periodic motion: A motion which repeats itself after equal intervals of time. • Time period: Time taken to complete one cycle. • Frequency: Number of cycles per unit time. • Amplitude: The maximum displacement of a vibrating body from its equilibrium position. • Natural frequency: When no external force acting on the system after giving it an initial displacement, the body vibrates. These vibrations are called free vibrations and their frequency as natural frequency. It is expressed in rad/sec or Hertz. • Fundamental Mode of Vibration: The fundamental mode of vibration of a system is the mode having the lowest natural frequency. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 8
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- 10. SHM Demonstration 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 10
- 11. DEFINITIONS (Contd..) Simple Harmonic Motion: • The motion of a body to and fro about a fixed point is called simple harmonic motion. • The motion is periodic . • The motion of a simple pendulum is simple harmonic in nature. • A body having simple harmonic motion is represented by the equation 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 11
- 12. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 12 x Asin( ) A sin( t ) 2 2 2 2 dx A cos( t ) dt and d x A sin( t ) x dt
- 13. DEFINITIONS (Contd..) Damping: – It is the resistance to the motion of a vibrating body. – The vibrations associated with this resistance are known as damped vibrations. Phase difference: – Suppose there are two vectors x1 and x2 having frequencies ω rad/sec, each The vibrating motions can be expressed as x1=A1.sin (ωt) x2=A2.sin (ωt+φ) – In the above equation the term φ is known an the phase difference. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 13
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- 15. DEFINITIONS (Contd..) Resonance: – When the frequency of external excitation is equal to the natural frequency of a vibrating body, the amplitude of vibration becomes excessively large. – This concept is known as resonance. Mechanical systems: – The systems consisting of mass, stiffness and damping are known as mechanical systems. Continuous and Discrete Systems: – Most of the mechanical systems include elastic members which have infinite number of degree of freedom. – Such systems are called continuous systems. – Continuous systems are also known as distributed systems. – Cantilever, simply supported beam etc. are the examples of such systems. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 15
- 16. DEGREE OF FREEDOM: • The minimum number of independent coordinates required to specify the motion of a system at any instant is known as degrees of freedom of the system. • In general, it is equal to the number of independent displacements that are possible. • This number varies from zero to infinity. • The one, two and three degrees of freedom systems are shown in figure 2. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 16 In single degree of freedom there is only one independent coordinate x1 to specify the configuration as shown in figure (a). Similarly, there are two (x1, x2). and three coordinates (x1, x2 and x3) for two and three degrees of freedom systems as shown in figure (b) and (c)
- 17. PARTS OF A VIBRATING SYSTEM • A vibratory system basically consists of three elements, namely the mass, the spring and damper. • In a vibrating body, there is exchange of energy from one form to another. • Energy is stored by mass in the form of kinetic energy (1/2 mv2), in the spring in the form of potential energy (1/2 kx ) and dissipated in the damper in the form of heat energy which opposes the motion of the system. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 17 2 3 1 1. Inertia (stores kinetic energy) –Mass (m) 2. Elasticity (stores potential energy) – Spring (k) 3. Energy Dissipation- Damper (C)
- 18. PARTS OF A VIBRATING SYSTEM • Energy enters the system with the application of external force known as excitation. • The excitation disturbs the mass from its mean position and the mass goes up and down from the mean position. • The kinetic energy is converted into potential energy and potential energy into kinetic energy. This sequence goes on repeating and the system continues to vibrate. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 18
- 19. PARTS OF A VIBRATING SYSTEM (contd..) • At the same time damping force (cv) acts on the mass and opposes its motion. Thus some energy is dissipated in each cycle of vibration due to damping. • The free vibrations die out and the system remains at its static equilibrium position. • A basic vibratory system is shown in figure . 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 19
- 20. F = cv Velocity Displacement Acceleration x v a k c m m F = kx F = ma Mechanical Parameters and Components
- 21. TYPES OF VIBRATION Some of the important types of vibration are as follows 1) Free and Forced Vibration • After disturbing the system, the external excitation is removed. then the system vibrates on its own. This type of vibration is known as free vibration. – Simple pendulum is one of the examples. • The vibration which is under the influence of external force is called forced vibration. – Machine tools, electric bells etc. are the suitable examples. 2) Linear and Non-linear Vibration • In a system., if mass, spring and damper behave in a linear manner, the vibrations caused are known as linear in nature. – Linear vibrations are governed by linear differential equations. – They follow the law of superposition. • On the other hand, if any of the basic components of a vibratory system behaves non-linearly, the vibration is called non-linear. – Linear vibration becomes, non-linear for very large amplitude of vibration. – It does not follow the law of superposition. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 21
- 22. TYPES OF VIBRATION (Contd..) 3) Damped and Un-damped Vibration • If the vibratory system has a damper, the motion of the system will be opposed by it and the energy of the system will be dissipated in friction. • This type of vibration is called damped vibration. • On the contrary, the system having no damper is known as un- damped vibration. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 22 n 2 d 2 Overdamped ( 1) Underdamped ( 1) Underdamped ( 0 ) Critically damped ( 1)
- 23. TYPES OF VIBRATION (Contd..) 4) Deterministic and Random Vibration • If in the vibratory system, the amount of external excitation is known in magnitude, it causes deterministic vibration. • Contrary to it, the non-deterministic vibrations are known as random vibrations. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 23
- 24. TYPES OF VIBRATION (Contd..) 5) Longitudinal, Transverse and Torsional Vibrations • Figure represents a body of mass m carried on one end of a weightless spindle, the other end being fixed. If the mass m moves up and down parallel to the spindle axis, it is said to execute longitudinal vibrations as shown in figure (a). • When the particles of the body or shaft move approximately perpendicular to the axis of the shaft, as shown in figure (b), the vibration so caused are known as transverse. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 24 If the spindle gets alternately twisted and untwisted on account of vibratory motion of the suspended disc, it is called to be undergoing torsional vibrations as shown in figure (c).
- 25. TYPES OF VIBRATION (Contd..) 6) Transient Vibration • In ideal systems the free vibrations continue indefinitely as there is no damping. • The amplitude of vibration decays continuously because of damping (in a real system) and vanishes ultimately. • Such vibration in a real system is called transient vibration. • Systems with finite number of degrees of freedom are called discrete or lumped systems. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 25
- 26. SIMPLE HARMONIC MOTION: • The motion of a body to and fro about a fixed point is called simple harmonic motion. • The motion is periodic . • The motion of a simple pendulum is simple harmonic in nature. • A body having simple harmonic motion is represented by the equation 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 26
- 27. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 27 SIMPLE HARMONIC MOTION:
- 28. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 28 x Asin( ) A sin( t ) 2 2 2 2 dx A cos( t ) dt and d x A sin( t ) x dt SIMPLE HARMONIC MOTION:
- 29. PHENOMENON OF BEATS • When two harmonic motions pass through some point in a medium simultaneously, the resultant displacement at that point is the vector sum of the displacement due to two component motions. • This superposition of motion is called interference. • The phenomenon of beat occurs as a result of interference between two waves of slightly different frequencies moving along the same straight line in the same direction. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 29 http://historicaltuning.co m/BeatsVideo.mp4
- 30. PHENOMENON OF BEATS (contd…) • Consider that at particular time, the two wave motions are in the same phase. • At this stage the resultant amplitude of vibration will be maximum. • On the other hand, when the two motions are not in phase with each other, they produce minimum amplitude of vibration. 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 30 Again after some time the two motions are in phase and produce maximum amplitude and then minimum amplitude. This process goes on repeating and the resultant amplitude continuously keeps on changing from maximum to minimum. This phenomenon is known as beat. https://www.youtube.com/ watch?v=5hxQDAmdNWE
- 31. MATLAB Program for solving Harmonic analysis Problems clc clear all syms t w=20*pi; n=1:6; x1(t)=20*t; x2(t)=-20*t+2; x(t)=x1(t)+x2(t); a0=20*(int(x1(t),t,0,0.05)+int(x2(t),t,0.05,0.1)) an=20*(int((x1(t)*cos(w*n*t)),t,0,0.05)+(int((x2(t)*cos(w*n*t)),t, 0.05,0.10))) bn=20*(int((x1(t)*sin(w*n*t)),t,0,0.05)+(int((x2(t)*sin(w*n*t)),t, 0.05,0.10))) % a0/2 =1 % an =[ -4/pi^2, 0, -4/(9*pi^2), 0, -4/(25*pi^2), 0] %bn =[ 0, 0, 0, 0, 0, 0] 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 31
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- 34. %% % Problem number-2 clc clear all syms t w=10*pi; n=1:6; x1(t)=-20*t+2; % 0<t<0.2 a0=20*(int(x1(t),t,0,0.2)) an=10*(int((x1(t)*cos(w*n*t)),t,0,0.2)) bn=10*(int((x1(t)*sin(w*n*t)),t,0,0.2)) 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 34
- 35. %% % Problem number-3 clc clear all syms t w=100*pi; n=1:6; x1(t)=abs(sin(100*pi*t)); a0=50*(int(x1(t),t,0,0.02)) an=100*(int((x1(t)*cos(w*n*t)),t,0,0.02)) bn=100*(int((x1(t)*sin(w*n*t)),t,0,0.02)) % This is the solution % a0 = 2/pi % an = [ 0, -4/(3*pi), 0, -4/(15*pi), 0, -4/(35*pi)] % an= -4/[(n^2-1)*pi] for even vlaues of n an=0 for odd values of n. % bn = [ 0, 0, 0, 0, 0, 0] 8/31/2021 Hareesha N G, Asst. Prof, DSCE, BLore-78 35
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