ree and orced Vibraions of Two Degree of Syses Inroducion: The siple single degree-of-freedo syse can be coupled o anoher of is ind, producing a echanical syse described by wo coupled differenial equaions; o each ass, here is a corresponding equaion of oion. To specify he sae of he syse a any insan, we need o now ie dependence of boh coordinaes, and, fro which follows he designaion wo degree-of-freedo syse. urher DO syses find applicaion in vibraion absorbers which are very helpful for reducing he vibraion levels of he paren syse. Eaples: orging haer and anvil on ground isolaors I engine ouned on fleible base building floor ig..4 orging Haer wih anvil [] The general equaion of oion of for a dynaic for forced vibraion is; c
If we consider ζ only, he is given e n A cos d B sin d by : To obain he PI, we us now he RHS,. We will consider one ype of eciaion only : c sin We now need o guess a PI. When a linear syse is subjeced o a haronic eciaion of he for sinw, I will respond haronically a he sae frequency. There will be a phase lag beween he force and he response. Inpu : Oupu : sin sin PI sin sin ig..5 Phase lag The he soluion for he seady sae vibraion can be found by insering PI PI sin ino he EOM c sin = + Or,
And = [ ] + [ ] an = = And he oal soluion of equaion.7 ay be obained for underdaped condiion as follows. = sin + cos + sin.9 here, he values of consans A and B ay be deerined using he iniial condiion and he forcing funcion. A closer analysis of above equaion yields ha or a very larger value of, he ransien response he firs er becoes very sall, and hence he er seady sae response is assigned o he paricular soluion he second er The value of coefficien of he seady sae response, or paricular soluion becoes large when he eciaion frequency is close o he undaped naural frequency,i.e.,ω = ω. This phenoenon is nown as resonance and plays a vial role in design,vibraion analysis, and esing. Eaple.4 : opue he response of he following syse =, +.4 + 4 = sin 3, = irs, solve for he paricular soluion by using he ore convenien for of as = sin 3 + cos 3 Differeniaing yields as follows
= 3 cos 3 3 sin 3 = 9X sin 3 9X cos 3 Subsiuion and collecion of siilar ers yields as 9. + 4 sin 3 + 9. + 4 cos 3 = Since sine and cosine are independen hence coefficien of sine and cosine should vanish. 9. + 4 = 9. + 4 = Solving hese equaion for X and X and subsiuing he values, paricular soluion yields as Given ha, =.34 sin 3.3 cos 3 ω = rad s, ζ =. ω =. <, ω = ω ζ =.99 rad s Since, he syse is underdaped, herefore, he coplee soluion of he equaion yields as = sin + cos + sin + cos Differeniaing he above epression as = cos sin + cos sin sin + cos Applying he iniial condiion, he values of he consan A and B ay be obained as = + = 3 = 3 =.89
= + = = =.8 Thus he final desired soluion is =..8.99 +.89.99.34 3.3 3 Two-Degree of reedo syse: There are various seps involved in analyzing he -DO vibraing syses o ge he naural frequencies and ode shapes. To resolve he force under spring deflecion, he free body diagra is essenial required as; Resoring force by springs K & K Resoring force by spring K Basic deflecion in springs wih wo spring siffness and Displaceens of boh asses
ig..6 ree Body Diagra for DO syses [ ] [ ]{ } = {}.3 Where he siffness ari is [ ] = + Mass ari is [ ] = Mode shape : Mass Mass: nd law : Newon's Reebering ha have : We ω ω - ω ω & ω K X Resoring force by springs K & K K X X...
Here, he Eigen value us be equal o he square of naural frequency. or wo degree of freedo syse, here us be wo naural frequencies and he corresponding wo ode shapes eis. The ass & siffness arices us be syeric. The ain diagonal eleens us be posiive. or large n, here are any nuerical soluion echniques. Use deerinan = for sall syses as; or a non-rival soluion: [K]- [M]{}= {} de [K] [M] = or {} = de [K] [M] = which gives + =.4 + = The above equaion gives quadraic in naural frequency, hence wo naural frequencies eis, as n and n n n Inser w n ino [K]-w [M]{}={} By definiion, de[k]- w n [M]= & are linearly dependen, bu we can obain / Using he previous resul : Hence : n Siilarly, for he nd ode : ω n n.5 n.6
Assue ha,insering values for,, gives : n & n The asses ove in phase. X and X ove by + uni each. The asses ove ou of phase. X oves by + uni, X oves by uni. Mode Shapes are he Relaive Displaceens of Bodies a Differen requencies as shown as; Two Degree of reedo orced Vibraing Syse: Two asses are consrained by he springs and wo differen forces are acing as shown in ig..7. 3 3 ig..7 Two Degree of freedo syse
Newon s second law for each ass gives,,,,,,,, -7.8-9 3 3.3 Equaions -7 - -3 give as; -3 3 3-3 Mari and vecor noaion can be incorporaed ino -3 and -3, which is useful for generalizing o an arbirary nuber of degrees-of-freedo. undaenals of Sound and Vibraions by KTH Sweden [], his boo is used under IITR-KTH MOU for course developen.