MECHANICAL VIBRATIONS WORK BOOK CUM LECTURE NOTES (FOR SIXTH SEMESTER MECHANICAL STUDENTS) (FOR PRIVATE CIRCULATION ONLY ) JAGADEESHA T Associate Professor Mechanical Engineering Department Name USN Section ST. JOSEPH ENGINEERING COLLEGE VAMANJOOR, MANGALORE – 575 028, CHAPTER 1- INTRODUCTION TO VIBRATION LEARNING OBJECTIVES ] Introduction to vibration ] Terminologies used in Vibration ] Simple Harmonic Motion ] Addition of Harmonics, Principle of super position applied to SHM ] Introduction to Fourier analysis, Beats ] Problems related to SHM and Fourier analysis. Vibration is defined as a motion which repeats after equal interval of time and is also a periodic motion. The swinging of a pendulum is a simple example of vibration. Vibration occurs in all bodies which are having mass and elasticity. They are caused due to several reasons such as presence of unbalanced force in rotating machines, elastic nature of the system, external application of force or wind loads and earthquakes. Vibrations are undesirable in most engineering systems and desirable in few cases. A body is said to vibrate if it has periodic motion. Mechanical vibration is the study of oscillatory motions of a dynamic system. An oscillatory motion is a repeated motion with equal interval of time. Example for useful vibration: o General industries – crushers, jackhammer, concrete compactor, etc. o Medical and health – electric massage, high frequency vibration probe for heart disease treatment o Music – string instruments i.e. guitar etc. Example for unwanted vibration : o Poor ride comfort in vehicle due to road irregularities o Sea sickness when traveling on ships, boats, etc. o Earthquakes o Fatigue failures in machine and structures Cycle : The movement of vibrating body from the mean to its extreme position in one direction then to mean , then to another extreme position and back to mean is called as M cycle of vibration, Mean position 3 2 Time period : It is the time taken to complete one cycle. It is equal to the time for the vector to rotate through 2π radians ` ωωω rad/s Reference: Frequency: It is the number of cycles per unit time Amplitude : It is the maximum displacement of a vibrating body from the mean position Phase difference : it is angle between the two rotating vectors executing simple harmonic motion of same frequency ωωω rad/s ωωω rad/s The first vector is x1= X sin ( wt ) The second vector is x2= X sin ( wt+φ) Where φ is the phase difference between x1 and x2 Resonance : it is the frequency of the external force coincides with the natural frequency of the system , a condition known as resonance occurs. During the resonance the system undergoes dangerously large oscillations Damping : It is the resistance offered to the motion of a vibrating body. Periodic motion If the motion is repeated after equal intervals of time, it is called periodic motion, The simplest type of periodic motion is harmonic motion Aperiodic motion If the motion does not repeat after equal interval of time , it is called aperiodic motion 3 DESCRIBING MOTIONS OF VIBRATING SYSTEMS Periodic motions Described as sine or cosine functions [sin (ωt) and cos(ωt)] ω = radian frequency (rad/sec) ω = 2π f ; where f is frequency (Hz) Period = time between two adjacent peaks or valleys; P = 1/ f Simple harmonic motion Leads / Lags y(t) = Asin(ωt + φ) Displacement y& (t) = Aω cos(ωt + φ) Velocity 2 y& (t) = − Aω sin(ωt + φ) Acceleration Classification of vibrations One method of classifying mechanical vibrations is based on degrees of freedom. The number of degrees of freedom for a system is the number of kinematically independent variables necessary to completely describe the motion of every particle in the system. Based on degrees of freedom, we can classify mechanical vibrations as follows: 1.Single Degree of freedom Systems 2.Two Degrees of freedom Systems 3.Multidegree of freedom Systems 4.Continuous Systems or systems with infinite degrees of freedom Another broad classification of vibrations is: 1. Free and forced vibrations 2. Damped and undamped vibrations. 4 Sometime vibration problems are classified as: 1. Linear vibrations 2. Non-linear vibrations 3. Random vibrations 4. Transient vibrations 5. Longitudinal vibrations 6. Transverse vibrations 7. Torsional vibrations Free vibration : If a system after initial disturbance is left to vibrate on its own , the resulting vibration is known as free vibrations. Free vibration takes when a system vibrates under the action of forces inherent in the system and when the external forces are absent. The frequency of free vibration of a system is called natural frequency. Natural frequency is a property of a dynamical system Forced vibration : Vibration that takes place under the excitation of external forces is called forced vibration. the forced vibration takes place at different forced frequencies or external frequencies Damped vibration : If any energy is lost or dissipated during oscillations then the vibration is known as damped vibration\ Undamped vibration : if no energy is lost or dissipated during oscillations ,such vibrations are known as undamped vibration Linear vibration : If all the basic component of a vibrating system behave linearly, the resulting vibration is known as linear vibration. The differential equations govern linear vibratory system are linear. If the vibration is linear , the principle of superposition holds and mathematical techniques of analysis are well developed. Non linear vibration : If any of the basic components of a vibrating system behave non linearly . the resulting vibration is known as non linear vibration. The differential equations that govern non linear vibratory system are non-linear. If the vibration is non linear the principle of superposition does not hold good and techniques of analysis is well known Deterministic vibration : If the magnitude of excitation on a vibrating system is known at any given time , the resulting vibration is known as deterministic vibration Random vibration: If the magnitude of excitation acting on a vibratory system at a given time cannot be predicted , the resulting vibration is known as non deterministic or random vibration 5 Longitudinal vibration : Consider a body of mass m carried on one end of a slender shaft and other end being fixed. If the mass vibrates parallel to the spindle axis, it is said to be execute longitudinal vibration Transverse vibration : If the mass vibrates perpendicular to the spindle axis , it is said to execute the transverse vibration Torsional vibration : If the shaft gets alternatively twisted and un twisted on account of an alternate torque on the disk, it is said to execute the torsional vibration SIMPLE HARMONIC MOTION A Vibration with acceleration proportional to the displacement and directed toward the mean position is known as SHM, ( Simple Harmonic Motion) Consider a spring mass system as shown in the figure along with the displacement time diagram m1 The equation of motion of the mass can be written as follows From the right angled triangle OAB we have Where x= displacement at any instant of time X = amplitude of vibration W= angular velocity or frequency in rad/second 6 The velocity of the mass m at an instant of time t is given by V = The acceleration of mass m is given by Hence we can conclude that in SHM the acceleration is proportional to displacement and is directed towards mean position observing the equations 2 and 3 the velocity and acceleration are harmonic with the same frequency but lead a displacement vector by π/2 and π radians respectively. t x-Displacement X-amplitude T-Periodic Time x t f-Frequency f=1/T ω=Frequency in radians per second t= time t X X= A sin ωt 7 Degrees of freedom: The minimum number of independent coordinates required to determine completely the positions of all the parts of a system at any instant of time is called degrees of freedom One degree of freedom m1 Two degree of freedom K1 m m1 F1 1 x kt1 1 kt2 Rotor with inertia J 1 K m2 Rotor with Inertia J 2 Three degree of freedom m2 x 2 Infinite degree of freedom m 1 Cantilever Beam m2 Continuous system ( consider the mass of the beam ) m 3 8 Exercise Specify the no of degree of freedom for the following 1 2 Cantilever Beam y k1 y k2 m1 m2 Continuous system ( Neglect the mass of the beam ) 3 3k k k 2m m 4m m, 4 J G m2 K1 K2 5. a b L1 M1 θ1 L2 θ2 M2 9 Addition of SIMPLE HARMONIC MOTION ( SHM) The addition of two simple harmonic motion having frequency yields a resultant which is simple harmonic having the same frequency. Consider tow simple harmonic motions of x1 and x2 having the same frequency and phase difference φ as given below x 1 === X 1 sin ( ωωω t ) x 2 = X 2 sin (ω t + φ ) Adding x = x 1 +x 2 Hence the resultant displacement is also SHM of amplitude X and phase angle θ B θθθ φφφ A ωωωt O 10 Tutorial problems on Simple Harmonic Motion. 1. Add the following harmonic motion analytically and verify the solution graphically 1) X1= 3 sin (((ωωω t +++ 30))) X2= 4 cos (((ωωω t +++ 10))) ( VTU Jan 2005) 2) X1= 2 cos (((ωωω t +++ 0.5))) X2= 5 sin (((ωωω t +++ 1 ))) ( VTU July 2006) 3) X1= 10 cos ωωω t +++ πππ X2= 8 sin ωωω t +++ πππ ( VTU Dec. 2007) ((( 4))) ((( 6))) 2. A body is subjected to two harmonic motions X1= 8 cos ωωω t +++ πππ X2= 15 sin ωωω t +++ πππ what harmonic motion ((( 6))) ((( 6))) should be given to the body to bring it to equilibrium (VTU July. 2005) 3. Split the harmonic motion x = 10 sin ω t + πππ into two harmonic motions (((ωω ++ 6))) having the phase of zero and the other of 45 o 4.