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1. CHAPTER 1 INTRODUCTION

2. 1.1 Introduction Mechanical Vibrations Defined as oscillatory motion of bodies in response to disturbance. Oscillations occur due to the presence of a restoring force Vibrations are everywhere: Human body: eardrums, vocal cords, walking and running Vehicles: residual imbalance of engines, locomotive wheels Rotating machinery: Turbines, pumps, fans, reciprocating machines Musical instruments Excessive vibrations can have detrimental effects: Noise Loosening of fasteners Tool chatter Fatigue failure Discomfort When vibration frequency coincides with natural frequency, resonance occurs.

3. Failures Caused by Mechanical Vibrations Aeolian, wind-induced or vortex-induced vibration of the Tacoma Narrows bridge on 7 November 1940 caused it to resonate resulting in catastrophic failure. Tacoma Narrows Bridge Collapse, USA (See video) Millennium Bridge , London: Pedestrians, in reaction to lateral motion of the bridge, altered their gait and started behaving in concert to induce the structure to resonate further (forced periodic excitation ).

4. 1.2 Fundamentals of Vibration In simple terms, a vibratory system involves the transfer of potential energy to kinetic energy and vice-versa in alternating fashion . When there is a mechanism for dissipating energy (damping) the oscillation gradually diminishes . In general, a vibratory system consists of three basic components : A means of storing potential energy (spring, gravity) A means of storing kinetic energy (mass, inertial component) A means to dissipate vibrational energy (damper ) Example: Pendulum

5. Fundamentals (contd.) At position 1: the kinetic energy is zero and the potential energy is At position 2: the kinetic energy is at its maximum At position 3: the kinetic energy is again zero and the potential energy is at its maximum . In reality, the oscillation will eventually stop due to aerodynamic drag and pivot friction (resulting in dissipation of energy in the form of heat).

6. Degrees of Freedom The number of degrees of freedom is the number of independent coordinates required to completely determine the motion of all parts of a system at any time . Examples of single degree of freedom systems:

7. Degrees of Freedom (contd.) Examples of two degree of freedom systems:

8. Degrees of Freedom (contd.) Examples of three degree of freedom systems:

9. Discrete and continuous systems Many practical small and large systems or structures can be described with a finite number of DoF . These are referred to as discrete or lumped systems. Some large structures (especially with continuous elastic elements) have an infinite number of DoF . These are referred to as continuous or distributed systems. In most cases, for practical reasons, continuous systems are approximated as discrete systems with sufficiently large number of lumped masses, springs and dampers. This leads to a large number of degrees of freedom system and hence better accuracy.

10. 1.3 Classification of Vibration Free and forced vibrations Free vibration : System is initial disturbed and then left to vibrate without influence of external forces. Forced vibration : Vibrating system is stimulated by external forces. If excitation frequency coincides with natural frequency, resonance occurs. Undamped and damped vibration Undamped vibration : No dissipation of energy. In many cases, damping is (negligibly) small (1 – 1.5%). However small, damping has critical importance when analyzing systems at or near resonance. Damped vibration : Dissipation of energy occurs and vibration amplitude decays. Linear and nonlinear vibration Linear vibration : Elements (mass, spring, damper) behave linearly. Superposition holds. Mathematical solutions well defined. Nonlinear vibration : One or more element behave in nonlinear fashion. Superposition does not hold, and analysis technique not clearly defined.

11. Classification of Vibration (contd.) Deterministic and random vibrations Deterministic vibration : Can be described by implicit mathematical function as a function of time. Random vibration: Cannot be predicted. Process can be described by statistical means.

12. 1.4 Vibration Analysis A vibratory system is a dynamic system for which the variables such as the excitations ( inputs) and responses (outputs) are time dependent. Responses generally depend on initial conditions as well as the external excitations. General procedure for solution of vibration problems: Mathematical modeling D erivation/statement of governing equations Solving of equations for specific boundary conditions and external forces Interpretation of solution(s) Most practical systems are very complex. Therefore mathematical modeling requires simplification.

13. Example: Mathematical Modeling of a Forging Hammer Following, the three mechanical elements of a vibratory system (namely spring, mass and damping) are discussed. Forging Hammer SDoF Model Two DoF Model

14. 1.5 Spring Elements Pure spring element are considered to have negligible mass and damping For linear springs, force is proportional to spring deflection (which is the relative motion between the two ends of the spring): For linear springs, the potential energy stored is: Actual springs sometimes behave in nonlinear fashion. It is important to recognize the presence and significance of nonlinearity. It is often desirable to generate linear estimate.

15. Spring Elements (contd.) Equivalent spring constant Example: In the cantilever beam with the lumped mass, m the mass of the beam is assumed to be negligible compared to lumped mass. The deflection at the free end is: This procedure can be applied for various geometries and boundary conditions. Stiffness, k = Force/Deflection :

16. Equivalent spring constant Springs in parallel: where In general, for n springs connected in parallel: Spring Elements (contd.)

17. Springs in series: Both springs are subjected to the same force: Combining the above equations: Spring Elements (contd.)

18. Substituting into the first equation gives: Dividing by k eq  t throughout gives: For n springs in series: Spring Elements (contd.) Note: that when springs are connected to rigid components such as pulleys and gears, the energy equivalence principle must be used.

19. 1.6 Mass / Inertia Elements Mass or inertia elements store kinetic energy and are assumed to have no spring and damping properties. The kinetic energy of a mass is proportional to the square of its velocity . Force  mass * acceleration Work = force * displacement Work done on mass is stored as Kinetic Energy. In the example shown (a building) mass of the frame is assumed to be negligible compared to mass of the floors.

20. Equivalent mass - example: To determine the equivalent mass at position l 1 we equating the kinetic energies The velocities of the mass elements m 2 and m 3 can be written as: Mass / Inertia Elements (contd.) Substituting for the velocity terms results:

21. Absorbs energy from vibratory system resulting in vibration amplitude decays . Damping elements are considered to have no mass and elasticity. Real damping systems are very complex. Damping can be modeled as: Viscous damping: It is based on viscous fluid flowing through gap or orifice. (Example: film between sliding surfaces, flow between piston & cylinder, flow through orifice, film around journal bearing) In this model damping force is opposite in direction and proportional to the relative velocity between the ends; i.e. Coulomb (dry friction) damping: It is based on friction between unlubricated surfaces. Damping force is constant and opposite to the direction of motion. 1.7 Damping Elements

22. Hysteretic (material or solid) damping: It is based on plastic deformation of materials (energy loss due to slippage b/n grains) Energy is lost due to hysteresis loop as shown in the force-deflection (stress-strain) curve of element. Damping Elements (contd.)

23. Equivalent damping element : Combinations of damping elements can be replaced by equivalent damper using similar procedures used for spring and mass/inertia elements . Here, equivalence between energy dissipation rate (Power Dissipated) is used to determine the equivalent damping coefficient. Damping Elements (contd.)

24. 1.8 Harmonic Motion Harmonic motion is the simplest form of periodic motion. It is represented by the sinusoidal curve as shown. A scotch-yoke mechanism rotating with angular velocity  gives a simple harmonic motion to the end point S or mass m . The motion of mass m is described by: Its velocity and acceleration are:

25. Harmonic Motion (contd.) Simple harmonic motion can be represented by a vector ( ) with magnitude A and angular velocity (frequency ) . The rotating vector generates a sinusoidal and a co-sinusoidal components along mutually perpendicular axes.  

26. Harmonic Motion (contd.) It is often convenient to represent sinusoidal and co-sinusoidal components (mutually perpendicular) in complex number format as shown. a and b are real and imaginary parts of the complex number and they denote the co-sinusoidal ( x ) and sinusoidal ( y ) components respectively. .  

27. Harmonic Motion (contd.) Definition of terms: Cycle is the movement of a vibrating body from its undisturbed or equilibrium position to its extreme position in one direction, then to the equilibrium position, then to its extreme position in the other direction, and back to equilibrium position. Amplitude, A is the maximum displacement of a vibrating body from its equilibrium position. Period,  is the time taken to complete one cycle of motion. It is equal to the time required for the vector ( ) to rotate through an angle of 2  . Therefore,   Where  is in rad/s and f is in Hertz (cycles/s) where  is also referred to as circular frequency Frequency, f is the number of cycles per unit time. Therefore,

28. Phase angle : the difference in angle (lead or lag) by which two harmonic motions of the same frequency reach their corresponding value (maxima, minima, zero up-cross, zero down-cross) Harmonic Motion (contd.)

29. Natural frequency: If a system, after an initial disturbance, is left to vibrate on its own , the frequency with which it oscillates without external forces is known as its natural frequency . Beats : When two harmonic motions, with frequencies close to one another, are added, the resulting motion exhibits a phenomenon known as beats. For example, if Harmonic Motion (contd.) x(t) can be rewritten as

30. In mechanical vibratory systems, beats occur when the (harmonic) excitation (forcing) frequency is close to the natural frequency. Harmonic Motion (contd.)

31. Octave: When the maximum value of a range of frequency is twice its minimum value, it is known as an octave band. For example, each of the ranges 75-150 Hz , 150-300 Hz, and 300-600 Hz can be called an octave band. In each case, the maximum and minimum values of frequency, which have a ratio of 2:1, are said to differ by an octave . Decibel : The various quantities encountered in the field of vibration and sound ( such as displacement, velocity, acceleration, pressure, and power) are often represented using the notation of decibel . A decibel ( dB ) is originally defined as a ratio of electric powers : In electrical systems power is proportional to the voltage squared hence: Harmonic Motion (contd.) where and represent some reference values of power and voltage respectively.  

32. Harmonic Analysis using Fourier Series Many vibratory systems are not harmonic but often periodic. Any periodic function can be represented by the Fourier series which is infinite sum of sinusoids and co-sinusoids. To obtain a n and b n the series is multiplied by cos ( n t ) and sin( n t ) respectively and integrated over one period.

33. Example : A Periodic Function Harmonic Analysis using Fourier Series (contd.) b. Harmonic function approximation using Fourier series a. Triangular Wave Vibration Note that the Fourier series can also be represented in terms of complex numbers:

34. With this can be written as:   Harmonic Analysis using Fourier Series (contd.) can be expressed as:   By defining the complex Fourier coefficients becomes  

35. The amplitudes and phases are defined as: harmonics Harmonic Analysis using Fourier Series (contd.) Note: that the Fourier series expression for is made-up of harmonics .   The harmonic functions are called harmonics of order n of the periodic function . A harmonic function of order n has a period of