"Never regard study as a duty, but as the enviable opportunity to learn to know the liberating influence of beauty in the realm of the spirit for your own personal joy and to the profit of the community to which your later work belongs"- Albert Einstein Introduction to ibrationIntroduction to ibrationVibrationVibrationAny motion that repeats itself after an Any motion that repeats itself after an interval of time is called vibration or interval of time is called vibration or oscillation.oscillation.The swinging of a pendulum is a typical The swinging of a pendulum is a typical example of vibration.example of vibration. IntroductionIntroduction Soil Dynamics is defned as that part of geotechnical engineering which deals with the behaviour of soils under dynamic loading.Soil dynamics has applications in a variety of felds of Civil Engineering.Soil Dynamics IN!"#$%&!I#NIN!"#$%&!I#NFoundations may be subjected to either static loads or a combination of static and dynamic loads;the latter lead to motion in the soil and mutual dynamic interaction of the foundation and the soil. odes of !ibration "asic Defnitions Vibration (or Oscillation): It is a time-dependent,repeating motion which may be translational orrotational. Periodic motion: It is a motion which repeats itselfperiodically in e!al time inter"als. Period: #he time in which the motion repeats itself iscalled the $Period%. &ycle: #he motion completed in a period is called a$&ycle%. 're!ency: #he n!mber of cycles in a !nit of time is(nown as the $fre!ency%.#he period and fre!ency are in"ersely related )egree of 'reedom: #he n!mber of independent co-ordinates re!ired to describe the motion of a systemcompletely is called the $)egree of 'reedom%. $ynamical 'ystems(echanical vibrations #scillations in mechanical systems ELEMENTA! "AT# $% V&'AT&N(ELEMENTA! "AT# $% V&'AT&N( #!#TEM##!#TEM#A vibrating system, in general, A vibrating system, in general, includesincludes

a mean of storing potential energya mean of storing potential energy

)spring or elasticity*, )spring or elasticity*, a mean of storing kinetic energy a mean of storing kinetic energy

)mass or inertia* and)mass or inertia* and

a means by which energy is gradually lost a means by which energy is gradually lost )damper*+ )damper*+!he vibration of a system involves the !he vibration of a system involves the transfer of its potential energy to transfer of its potential energy to kinetic energy and kinetic energy to kinetic energy and kinetic energy to potential energy, alternativelypotential energy, alternatively++ $ynamical 'ystems Mass element iscous damper $ynamical 'ystems 'teady 'tate ibration'teady 'tate ibration If the system is damped, some energy is If the system is damped, some energy is dissipated in each cycle of vibrationdissipated in each cycle of vibration and and must be replaced by an e,ternal source if must be replaced by an e,ternal source if a state ofsteady vibration is to be a state ofsteady vibration is to be maintainedmaintained++ ibration of the simple pendulumibration of the simple pendulum ibration of the simple pendulumibration of the simple pendulumLet the bob of mass m be released after giving an angularLet the bob of mass m be released after giving an angular displacementdisplacement At position )* the velocity of the bob is +ero and hence itsAt position )* the velocity of the bob is +ero and hence its ,inetic energy is +ero,inetic energy is +ero 'ut it has potential energy of magnitude 'ut it has potential energy of magnitudemgl -).cosmgl -).cos / /0 0w.r.t the datum position 1 w.r.t the datum position 1#ince the gravitational force mg induces a tor2ue mgl sin#ince the gravitational force mg induces a tor2ue mgl sin about point $* the bob starts swinging to the left fromabout point $* the bob starts swinging to the left from position ). position ).This gives the bob certain angular acceleration in theThis gives the bob certain angular acceleration in the cloc,wise direction and by the time it reaches position 1*cloc,wise direction and by the time it reaches position 1* all of its potential energy will be converted into ,ineticall of its potential energy will be converted into ,inetic energy.energy. ibration of the simple pendulumibration of the simple pendulum3ence the bob will not stop in position 1* but 3ence the bob will not stop in position 1* but will swing to position 4.will swing to position 4.3owever* as it passes the mean position 1* a 3owever* as it passes the mean position 1* a counter cloc,wise tor2ue starts acting on the counter cloc,wise tor2ue starts acting on the bob and causes the bob to decelerate. The bob and causes the bob to decelerate. The velocity of the bob reduces to +ero at the left velocity of the bob reduces to +ero at the left extreme position. 'y this time* all the ,inetic extreme position. 'y this time* all the ,inetic energy of the bob will be converted into energy of the bob will be converted into potential energy.potential energy. Again due to the gravity tor-ue, the bob continues Again due to the gravity tor-ue, the bob continues to attain a counter clockwise velocityto attain a counter clockwise velocity.. .ence the bob starts swinging back with .ence the bob starts swinging back with progressivelyincreasing velocity and passes the progressivelyincreasing velocity and passes the mean position again+mean position again+!his process keeps repeating, and the pendulum !his process keeps repeating, and the pendulum will have oscillatory motionwill have oscillatory motion.owever, in fraction, the magnitude of oscillation .owever, in fraction, the magnitude of oscillation ))* gradually decreases and the pendulum * gradually decreases and the pendulum ultimately steps due to the resistance )damping* ultimately steps due to the resistance )damping* offered by the surrounding medium )air*+offered by the surrounding medium )air*+!his means that some energy is dissipated in each !his means that some energy is dissipated in each cycle of vibration due to damping by the aircycle of vibration due to damping by the air.. $E/"EE #0 0"EE$#($E/"EE #0 0"EE$#(!he minimum number of independent !he minimum number of independent coordinates re-uired to determine completely coordinates re-uired to determine completely the positions of all parts of a system at any the positions of all parts of a system at any instant of time defines the degrees of freedom instant of time defines the degrees of freedom of the system+of the system+The simple pendulum represents a single degree The simple pendulum represents a single degree of freedom -#5%0 system.of freedom -#5%0 system.Motion can be stated in terms of the Motion can be stated in terms of the or in terms or in terms of the cartesian coordinates xof the cartesian coordinates x and y.and y. 'ut x and y are not independent6 they 'ut x and y are not independent6 they are related through relationare related through relationxx1 1 7 y7 y1 1 8 l8 l1 1 where l 8 the constant length ofthe where l 8 the constant length ofthe pendulum.pendulum.Thus any one coordinate can describe Thus any one coordinate can describe the motion of the pendulum.the motion of the pendulum.The choice of The choice of as the independent as the independent coordinate will be more convenient than coordinate will be more convenient than the choice of x or y.the choice of x or y. $I'&"E!E AN$ N!IN%#%' '1'!E('$I'&"E!E AN$ N!IN%#%' '1'!E('AA large no. of practical systems can be described large no. of practical systems can be described using a finite no. of degrees of freedom.using a finite no. of degrees of freedom.#ome systems especially those involving #ome systems especially those involving continuous elastic members* have an infinite no. continuous elastic members* have an infinite no. of degrees of freedom.of degrees of freedom.E,ample2 &antilever beamE,ample2 &antilever beam 9e have an infinite no. of coordinates to specify its 9e have an infinite no. of coordinates to specify its deflected configuration.deflected configuration.The infinite no. of coordinates define its elastic The infinite no. of coordinates define its elastic deflection curve.deflection curve.

Thus* the cantilever beam has an infinite no. of Thus* the cantilever beam has an infinite no. of degrees of freedom. degrees of freedom. Most structures and machine systems have Most structures and machine systems have deformable -elastic0 members and therefore have an deformable -elastic0 members and therefore have an infinite no. of degrees of freedom.infinite no. of degrees of freedom.'ystems with a finite no+ of degrees of 'ystems with a finite no+ of degrees of freedom are called discrete and3or lumped freedom are called discrete and3or lumped parameter systemsparameter systems++'ystems with an infinite no+ of degrees of 'ystems with an infinite no+ of degrees of freedom are called continuous or freedom are called continuous or distributed systemsdistributed systems.. &4A''I0I&A!I#N &4A''I0I&A!I#N #0 #0 I5"A!I#NI5"A!I#N I+ 0ree and forced vibrationsI+ 0ree and forced vibrationsa+a+

0ree vibration0ree vibrationIf a system, after an initial disturbance, is If a system, after an initial disturbance, is left to vibrate on its own, the ensuing left to vibrate on its own, the ensuing vibration is known as free vibrationvibration is known as free vibration. . No No external forces act on the system.external forces act on the system.-The response of a system is called free -The response of a system is called free vibration when it is disturbed and the left to vibration when it is disturbed and the left to vibrate about mean position0vibrate about mean position0Example: The oscillation of a simple pendulumExample: The oscillation of a simple pendulum b+ 0orced ibrationb+ 0orced ibrationIf a system is subjected to an e,ternal force )often, If a system is subjected to an e,ternal force )often, a repeating type of force*, the resulting vibration is a repeating type of force*, the resulting vibration is known as forced vibration+known as forced vibration+ "esonance"esonance::If the fre-uency of the e,ternal force coincides If the fre-uency of the e,ternal force coincides with one of the natural fre-uencies of the system, with one of the natural fre-uencies of the system, a condition known as resonance occurs and the a condition known as resonance occurs and the system undergoes dangerously large oscillations+system undergoes dangerously large oscillations+ -%ailures of such structures as buildings* bridges* -%ailures of such structures as buildings* bridges* machine foundations* and airplane wings have been machine foundations* and airplane wings have been associated with the occurrence of resonance0associated with the occurrence of resonance0 II+%ndamped and $amped ibrationII+%ndamped and $amped ibration&f no energy is lost or dissipated in &f no energy is lost or dissipated in friction or other resistance during friction or other resistance during oscillation* the vibration is ,nown as oscillation* the vibration is ,nown as undamped vibration.undamped vibration.&f any energy is lost in this way* on the &f any energy is lost in this way* on the other hand* it is called damped vibration.other hand* it is called damped vibration.In many physical systems, the amount of In many physical systems, the amount of damping is so small that it can be disregarded damping is so small that it can be disregarded for most engineering purposes+for most engineering purposes+.owever, the consideration of damping is .owever, the consideration of damping is e,tremely important in analy6ing systems near e,tremely important in analy6ing systems near resonance+resonance+ III+ 4inear and Nonlinear ibrationIII+ 4inear and Nonlinear ibration&f all the basic components of a vibrating system . &f all the basic components of a vibrating system . the spring* the mass* and the damper . behave the spring* the mass* and the damper . behave linearly* the resulting vibration is ,nown as linearly* the resulting vibration is ,nown as linear linear vibrationvibration..$n the other hand* if any of the basic components $n the other hand* if any of the basic components behave nonlinearly* the vibration is called behave nonlinearly* the vibration is called nonlinear vibrationnonlinear vibration..The differential e2uations that govern the The differential e2uations that govern the behaviour of linear and nonlinear vibrating behaviour of linear and nonlinear vibrating systems are linear and nonlinear respectively.systems are linear and nonlinear respectively. 4inear vibration4inear vibration"rinciple of superposition holds* "rinciple of superposition holds* mathematical techni2ues of analysis are mathematical techni2ues of analysis are well developed.well developed.Nonlinear vibrationNonlinear vibration#uperposition principle is not valid* #uperposition principle is not valid* techni2ues of analysis are less well ,nown.techni2ues of analysis are less well ,nown.All vibratory system tend to behave All vibratory system tend to behave nonlinearly with increasing amplitude of nonlinearly with increasing amplitude of oscillation;vibration.oscillation;vibration. I+ $eterministic and "andom ibrationI+ $eterministic and "andom ibration$eterministic $eterministic &f the value or magnitude of the excitation &f the value or magnitude of the excitation -force or motion0 acting on a vibratory -force or motion0 acting on a vibratory system is ,nown at any given time* the system is ,nown at any given time* the excitation is called excitation is called deterministicdeterministic..The resulting vibration is ,nown as The resulting vibration is ,nown as deterministic vibration.deterministic vibration. Non-deterministicNon-deterministic&n some cases* the excitation is nondeterministic or &n some cases* the excitation is nondeterministic or random* the value of the excitation at a given time random* the value of the excitation at a given time cannotcannot be predicted.be predicted.&n some cases* a large collection of records of the &n some cases* a large collection of records of the excitation may exhibit some statistical regularity. &t is excitation may exhibit some statistical regularity. &t is possible to estimate averages such as the mean possible to estimate averages such as the mean and mean s2uare values of the excitation.and mean s2uare values of the excitation.E,ample2 7ind velocity, ground motion during E,ample2 7ind velocity, ground motion during earth-uakeearth-uake..&f the excitation is random* the resulting vibration is &f the excitation is random* the resulting vibration is also random* it can be described only in terms of also random* it can be described only in terms of statistical 2uantities.statistical 2uantities. I5"A!I#N ANA41'I' 8"#&E$%"EI5"A!I#N ANA41'I' 8"#&E$%"EAA vibratory system is a dynamic system for vibratory system is a dynamic system for which the variables such as the excitations which the variables such as the excitations -inputs0 and responses -outputs0 are time.-inputs0 and responses -outputs0 are time.dependent.dependent.The response of a vibrating system generally The response of a vibrating system generally depends on the initial conditions as well as depends on the initial conditions as well as the external excitations.the external excitations.Most practical vibrating systems are very Most practical vibrating systems are very complex* and it is impossible to consider all complex* and it is impossible to consider all the details for a mathematical analysis* only the details for a mathematical analysis* only the most important features are considered in the most important features are considered in the analysis to predict the behaviour ofthe analysis to predict the behaviour of the the system under special input conditions.system under special input conditions. #ften, the overall behaviour of the #ften, the overall behaviour of the system can be determined by system can be determined by considering even a simple model of the considering even a simple model of the comple, physical system+comple, physical system+The analysis of a vibratory system The analysis of a vibratory system involves four steps:involves four steps:9+9+(athematical modelling(athematical modelling:+:+$erivation of the governing e-uations$erivation of the governing e-uations;+;+'olution of the governing e-uation'olution of the governing e-uation s principle)ii* $>Alembert> s principle)iii* !he principle of conservation of )iii* !he principle of conservation of

energyenergy.. 4. #olution of the governing E2uation4. #olution of the governing E2uationE2uation of motion must be solved to find E2uation of motion must be solved to find the response of the vibrating system.the response of the vibrating system.-i0 9e can use standard methods of solving -i0 9e can use standard methods of solving differential e2uation.differential e2uation.-ii0 Laplace transformation methods -ii0 Laplace transformation methods -iii0 Numerical methods-iii0 Numerical methods

-%5M* %EM* 'EM* etc.0-%5M* %EM* 'EM* etc.0