Tointroducethefundamentalconceptofvibrationof1DOF systemTodevelopthegoverningequationof2DOFsystemfor vibration analysis.TodevelopthegoverningequationofMultiDOFsystemfor vibration analysis.Tounderstandtheroleofconditionmonitoringinvibration analysis. Objective:11M012 VIBRATION AND CONDITIONS MONITORINGICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition MonitoringUnit I Introduction&elevanceofandneedforvibrationanalysis'Mathematical modeling of vibrating systems ( Discrete and continuous systems ' revie%ofsingle'degreeoffreedomsystems'freeandforced vibrations) *arious damping models $round vibration testing.S!!"bu#: ICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition MonitoringUnit II T$o De%ree&o'&(reedo) S#te)#$eneral solution to free vibration problem ' damped free vibration ' Forced vibration of undamped system 'dynamic vibration absorbers ' Technical applications *ibration test on torsion pendulumUnit III Mu!ti De%ree&o'&(reedo) S#te)#Freeandforcedvibrationsofmulti'degreeoffreedomsystemsin longitudinal torsional and lateral modes 'Matri+ methods of solution'normal modes ' Orthogonally principle',nergy methods "ntroduction to vibrations of plates Dynamics of rotating machineryUnit IV Vibr"tion Contro!"ntroduction(&eductionof*ibrationattheource'#ontrolof *ibration(bytructuraldesign(Materialelection(-ocali.ed additions(/rtificialdamping(&esilientisolation)*ibration isolation *ibrations measurement on lathe ICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition MonitoringUnit V Condition b"#ed M"inten"nce *rinci+!e# , A++!ic"tion#"ntroduction'#onditionMonitoringMethods'TheDesignof "nformationsystem)selectingmethodsofmonitoring)Machine conditionmonitoringanddiagnosis(*ibrationseveritycriteria( MachineMaintenancetechniques(Machineconditionmonitoring techniques(*ibrationmonitoringtechniques("nstrumentation systems(#hoiceofmonitoringparameter /coustictesting) /ctive noise and vibration controlTe-tboo.#1. ingaresu . &ao) Mechanical *ibrations) Prentice 0all Publish) 1e% Delhi)2212.2.3..&ao)*ibratory#onditionMonitoringofMachines)1arosa Publishing 0ouse) 1e% Delhi)2222.To introduce the fundamental concept of vibration for engineering students.To develop the governing equation for Free 4 forced vibration system.To analy.e the influence of vibration in mechanical system.Objective:Vibration of Sin%!e De%ree o' (reedo) ##te)ICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition MonitoringIntroduction5 /nymotionthatrepeatitselfafteranintervaloftimeis called vibration or oscillation.5 Theoryofvibrationdeals%ithstudyofoscillatorymotion of body 4the forces associated %ith them."ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring( ) sinnx t A t .( ) cosn nx t A t ..2 2( ) sin ( )n n nx t A t x t MAT/AB code"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com%% Fundamental of vibration-P.senthil kumart=0:0.01:5;w=10;%% Disla!ement of s"stem#= $.%sin&w.%t';lot&t(#()b)'hold on%% *elo!it" of s"stemd#=$.%w.%!os&w.%t';lot&t(d#()r)'%% a!!eleration of s"stemd$# = -$.%w.%w.%sin&w.%t';lot&t(d$#()+)'title&)motion of s"stem)'#label&)time&se!')'"label&,-mlitude)' 1 DOF system2 DOF systemMulti DOF system#ondition MonitoringMAT/AB code"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition MonitoringVibr"tion Ter)ino!o%5 Thema+imumdisplacementofavibratingbodyfromits equilibriumpositioniscalledtheamplitudeofvibration 6mm7.5 The number of cycles per unit time is called the frequency of oscillation.60.75 Thetimeta8entocompleteonecycleofmotionis8no%n as the time period of oscillation.6sec75 "nthepreviousfigure)thevelocityvectorleads displacementvectorby8no%nangleof92degree.This angle is 8no%n asphase angle.5 "f a system) after an initial disturbance) is left to vibrate on itso%n)thefrequency%ith%hichitoscillates%ithout e+ternal forces is 8no%n as its natural frequency."ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring12ft 2t5 The number of coordinates that are required to specify the motion of system in space is called degree of freedom.5 :hen the ma+imum value of a range of frequency is t%ice itsminimumvalue)itis8no%nasanoctaveband.For e+ample) each of the ranges ;< 1ase ,+citation&otating FnbalanceP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.comICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.comD A!e)bert *rinci+!e4ner% *rinci+!eR"!ei%0 )et0od1 DOF system2 DOF systemMulti DOF system#ondition Monitoring( )+ 0d T Udt _+ ,.2 21 102 2dmx kxdt+ ..0 m x kx 0 F ma..0 mx kx + ..0 kx mx . .mean extremeK E P E sin x X t .cos x X t X 2 2 21 12 2KE mv m x 212PE kx 2 2 21 12 2m x kx km #o!ution"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring1 2n ni t i tx Ae A e +..0 mx kx + stx Ae 2( ) 0stms k e + /ns k m iw t tni tx Ae t0(0) x x . .0(0) x x ( ) ( )( )2n ni t i tXx t e e 1 + ]2.200nxX x _ + ,.00tannxx _ ,( ) cos( )nx t X t 12iXA e 22iXA e0(0) cos x x X . .0(0) sinnx x X Inter+ret"tion"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring0 0sin( )nx X t +2.200 0nxX x _ + ,00 .0tannxx _ ,0( ) cos( )nx t x t .0( ) sin( )n nx t x t .20( ) cos( )n nx t x t 10sin(0.9977 / 2) 10cos(0.9977 ) x t t + 4-")+!e: Vibr"tion in $"ter t"n.The column of the %ater tan8 sho%n in Fig.is =22 ft high and is made of reinforced concrete %ith a tubular cross section of inner diameter B ft and outer diameter 12 ft. The tan8 %eighs A22222lb%henfilled%ith%ater.>yneglectingthemassof the column and assuming the Goung s modulus of reinforced concrete as HeA psi) Determine thevibration responseof the %ater tan8 due to an initial transverse displacement of 12 in."ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring05603008106 104 1010i ft! ft! ft" #bE $six inch "ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring0 0sin( )nx X t +331545.6672 /E#k #b inch

( )4 4 4 40600.9551064i# d d in 0.9977 / secnkradm 2.200 0 010nxX x x in _ + ,00 .0tan2nxx _ ,10sin(0.9977 / 2) 10cos(0.9977 ) x t t + .10 0.9977 cos 0.99772x t _ + ,..210 0.9977 sin 0.99772x t _ + ,..2max9.9540 / sec x in Revie$:So!ution o' di''erenti"! e5u"tion"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring20 ms cs k + + .. .0 mx c x kx + + , s a b , s a a , s a iba ib +( ) ( )atx t A Bt e +( ) sin( )atx t Xe bt +1 1( )at btx t C e C e +242b b acsa t (ree D")+ed vibr"tion"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com5 "n the study of vibration the process of energy dissipation isgenerallyreferredtoasdamping.Themostcommon typeofenergydissipatingelementisviscousdamper. *iscous damping force is proportional to velocity of mass and act in the direction opposite to velocity of mass.1 DOF system2 DOF systemMulti DOF system#ondition Monitoring.. .( ) mx k x m% c x + + 20 ms cs k + + ( )( )2221,212 2n n n nc c ksm m m _ t t t ,.. .0 mx c x kx + + 2,2c kis &er' ne%ative 'r $'sitivem m _ ,202c km m _ ,2c nc m 2c nc cc m 20c ks sm m+ + "ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.comC"#e:1 Under d")+ed ##te)t-1 DOF system2 DOF systemMulti DOF system#ondition Monitoring( ) sin( )ntdx t Xe t +( )21,21ns t 1 ."ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com.2< m2ms1 DOF system2 DOF systemMulti DOF system#ondition Monitoring11.54xx 1.524xx 22.54xx "ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com/pproach? :e need to find 8 4 # value. First find coefficient ofdampingfromlogarithmicdecrement.Thenfindcritical dampingandnaturalfrequencyofasuspensionsystem. 1o% %e can find 8 4 # directly. 1 DOF system2 DOF systemMulti DOF system#ondition Monitoring12ln ln(16) 2.7726xx _ ,222.77261 0.4037 2ddt 21d n 2c nc c m 2nk m 0.4037( ) sin 0.455 sinn nt td dx t Xe t e t 2max1ntx Xe "ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.comMatlab code?1 DOF system2 DOF systemMulti DOF system#ondition Monitoring9"r)onic"!! 4-cited vibr"tion5 :henthebodyvibratesundertheinfluenceofe+ternal force) then the body is said to be under e+cited vibration."ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring( ) ( ) ( )h $x t x t x t +1 2( ) cos s n n( ) i sih n n dx t C t C X t t + + 9"r)onic"!! e-cit"tion o' und")+ed ##te)"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring..mx kx F + 0 0costF F t F e ( ) ( ) ( )h $x t x t x t +1 2( ) cos sin cos( )h n nx t C t C t X t + +0 02 2 2/11stnF F kXm k mk _ ,01 22( ) cos sin cosn nFx t C t C t tk m + +cos F X t 2( ) cos cos1st$nx t X t t _ ,20cos cos cos m X t kX t F t 1 + ]02( ) cos( ) cosFx t X t tk m + + +"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring01 02FC xk m 00 12(0) 0Fx x Ck m + +.02nxC.01 22( ) sin cos sinn n n nFx t C t C t tk m + +. .02(0)nx x C 01 22( ) cos sin cosn nFx t C t C t tk m + +.0 00 2 2( ) cos sin cos1stn nnnF xx t x t t tk m _ + + , _ ,h $x x / 1nn( ) x t"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com0armonically e+cited Force? 6for Damped system7#onsideraspringmassdampersystemsubCectedtoforced vibrationthen the equation of motion becomes1 DOF system2 DOF systemMulti DOF system#ondition Monitoring.. .mx c x kx F + + 1 2( ) sin cos sin( )$x t A t A t X t + ( ) ( ) ( )h $x t x t x t +.1 2( ) cos sin$x t A t A t ..2 2 21 2( ) sin cos$ $x t A t A t x .2$ $ $m x c x kx F + + ( ).2$ $k m x c x F + 0 sin F F t ( ) ( ) ( )21 2 1 2 0sin cos cos sin sin k m A t A t c A t A t F t + + ( ) ( )2 21 2 1 2 0sin cos sin k m A c A t c A k m A t F t 11 + + ] ]( )21 2 0k m A c A F ( )21 20 c A k m A + 22 011m c A FAk k k _ ,2121 0c A mAk k _+ ,"ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.com1 DOF system2 DOF systemMulti DOF system#ondition Monitoring22 011m c A FAk k k _ ,2121 0c A mAk k _+ ,21 2 021 2nA r A X _ ,21 222 1 0nr A A _+ ,2c nc cc m 2nc m ( )21 2 01 2 r A r A X ( )21 22 1 0 r A r A + ( )( )( )201 22211 2r XAr r +( )( )( )02 22221 2r XAr r +( ) ( )( ) ( )20 0222 222(1 ) sin 2 cos sin( )1 21 2$X Xx r t r t tr rr r 1 ] + +22 22 2nnn ncm rk k "ntroductionproblemICGP.enthil !umar"ntelligent #ontrol $roup%%%.icgindia.%eebly.comMagnification factor vs Frequency &atio1 DOF system2 DOF systemMulti DOF system#ondition Monitoring( ) ( )max0max2 222 2 20111 22n n$$xXx (FXr r 1 _1 +1

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