Vibration with more (than one) degrees of freedom (DOF) a) longitudinal vibration with 3 DOF. b) rotational (torsional) vibration with 3 DOF
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1 irtion with ore (thn one) degrees of freedo (DOF) ) longitudinl irtion with DOF ) rottionl (torsionl) irtion with DOF ) ending irtion with DOF d) the D (plnr) irtion of the fleil supported rigid od with DOF
2 The free longitudinl undped irtion wit DOF - the two sses the two rigid odies - the two independent oordintes the two DOF - the two eloities derities of orresponding oordintes - the two elertions derities of orresponding eloities - the single stiffnesses. The free-od digr : F S F S F S F S F S F S - the single spring fores : The spring defortions re : F F F S S S lengthening the tension spring fore lengthening the tension spring fore shortening the pressure spring fore. The equtions of otion re : F F i i S F F S F S F S
3 or : with sustitutions : finll in tri for : or K M where : M - is the squre ss tri - is the olun tri (etor) of elertions K - is the squre stiffness tri - is the olun tri (etor) of displeents - is the olun tri (etor) of zeros. Both M nd K re setril with respet to the in digonl ( = = =) nd he positie lues on the in digonl ( > ). Further M is digonl (nonzero lues onl on the in digonl zeros out of the in digonl).
4 The supposed solution is : further : t C sin t t C sin t t C sin t t C sin t where C C - plitudes of the orresponding oordintes - nturl irulr frequen. Put into the equtions of otion : C C sin sin t C sin t C sin t t C sin t C sin t The sin( t+) eer n e neled the C nd C unnowns n e ftorized : C C C C The sste of two lgeri equtions with two unnowns is hoogenous (zeros on the right side). The triil solution is : C = C = The triil solution represents no otion. We serh the non-triil solution representing irting odies : C C The first eqution gies the rtio etween plitudes C nd C s : C C while the seond eqution gies the se s : This sipl ss tht : C C
5 or : This result n e epressed s : This deterinnt is lled frequentionl deterinnt. The sustitution n e used : where is lled eigenlue. Then : or : wht is lled frequentionl polno. The roots re two eigenlues : 4 where nd susequentl : re two nturl irulr frequenies nd finll : f re two nturl frequenies or eigenfrequenies. The two eigenlues two nturl irulr frequenies nd two nturl frequenies f re sorted in the sending order < < f < f.
6 One the eigenlue is lulted the plitudes C nd C n e lulted. But the equtions : re linerl dependent. C C C C Sipl seond eqution is onl ertin ultiple of the first one. It does not gie the seond infortion. (The ultiplier is ) The equtions gie onl the rtio C /C : C C not the finl plitudes C =? C =? For this reson the oe presented results re not red s plitudes C C ut. The infinite nuer of solutions eists for eple : or or or or or or n other ultiple of this results. The rtio / is lws the se.
7 These nd re not the finl plitudes. The finl plitudes re their ertin ultiples (the ultiplier see elow). Beuse the refer to the oordintes (displeents) the will e orgnized in olun tri : C C C C sin t sin t The lue of the ultiplier is deterined fro the initil onditions. The olun tri is lled ode shpe or eigenetor. As eplined oe is onl one fro infinite nuer of solutions n other solution is n it s ultiple. The nueril lues re hnged ut their rtio is ept. This ft leds to the proess of norliztion of ode shpe. To norlize ode shpe ens to ultipl it ertin nuer. The nueril lues of the ode shpe re hnged ut their rtio is ept. Often used ind of norliztion is norliztion to unit. This ens tht the highest lue in the ode shpe is others re less in the proper rtio. The lrgest lue in the ode shpe is found nd whole ode shpe is diided this lue. The lrgest lue diided itself is = others diided the lrgest re < in proper rtio. Eple : Mode shpe efore norliztion : the lrgest lue is = ode shpe fter norliztion :
8 Other often used ind of norliztion is norliztion to ss tri. The denointor is lulted s T M where M is the ss tri is ode shpe (efore norliztion) T is trnsponed ode shpe (written s row tri). All eers in the ode shpe re diided this denointor. For the ode shpe fter norliztion T M Beuse two lues of eist two sets of results eist. The re orgnized in so lled odl tri or tri of ode shpes : The first row orresponds to the first oordinte the seond row orresponds to the seond oordinte. The first olun orresponds to the first nturl frequen the seond olun orresponds to the seond nturl frequen. The single oluns re single ode shpes. The nueril eple : = g = g = N/ = N/ = N/ M g K 5 N 5 N K M
9 g g N 6 N s s 6 74 f 5 8 Hz K M s 55 s the seond eqution is sipl first eqution diiding - the seond eqution is sipl first eqution ties sipl hoie sipl hoie 4 5 5
10 the first ode shpe referring to the first nturl irulr frequen = 6 s - norliztion to unit : the iu lue in is the seond ode shpe referring to the seond nturl irulr frequen = 74 s The ode shpes re orgnized in the odl tri (the tri of ode shpes). 5 5 The rows in odl tri refer to the oordintes the oluns - the ode shpes refer to the nturl frequenies. 5 5 The ode shpe interprettion. The first ode shpe irtion with the nturl irulr = 6 s -. 5 t 5 sin t sin t t Both odies irte in the se diretion. The plitude of the od is twie the plitude of the od.
11 The seond ode shpe irtion with the nturl irulr = 74 s -. 5 t sin t 5 sin t t Both odies irte in the opposite diretion. The plitude of the od is one qurter of the plitude of the od. The finl irtion the tie funtion of the oordintes is the liner ointion of the ode shpes : the eloities : t 5 sin t sin t t sin t 5 sin t t t 5 os t os t t t os t 5 os t where nd re the oeffiients of the liner ointion nd nd together with phse shifts nd re integrtionl onstnts sed on the initil onditions. The initil onditions re four : t =... (t=) = (t=) = (t=) = (t=) = In the eple : = = /s = 8 = /s sin os os sin sin 5 sin os 5 os nd the solution : = 57 = = 64 = 66 rd = 6 = 459 rd finll :
12 t C sin t C sin t t C sin t C sin t where : C = 5 = 786 C = = C = = 57 C = -5 = - t 7 86 sin t 66 sin t t t 5 7 sin t 66 sin t t The odl oordintes Let us go to the sste. The equtions of otion re : Let us nlze the speifi se in whih = = = nd = =.
13 The equtions of otion re : The oordintes nd re lled pril oordintes (defined prir) or phsil oordintes (the he diret phsil ening - displeent of the odies nd ). Add nd sutrt oth equtions. First eqution + seond eqution : First eqution - seond eqution : Now use the sustitution : The equtions of otion re then : The new oordintes nd re lled prinipl oordintes or odl oordintes. The odl oordintes re the liner ointion of phsil oordintes. In the oe desried eple ( = = = nd = = ) the odl oordintes he the following diret phsil ening : = + - twie the oordinte of the enter of ss (the oordinte of the enter off ss is ) = - - distne etween the two odies. This sitution (the odl oordintes he diret phsil ening) is er unusul. Usull the odl oordintes he no phsil ening sipl the liner ointion of phsil oordintes.
14 Modl trnsfortion The odl oordintes re defined through odl trnsfortion : where - olun tri (etor) of phsil oordintes - olun tri (etor) of odl oordintes - odl tri (the tri of ode shpes). In the oe desried eple where : = g = g = N/ = N/ = N/ nd the odl tri is : 5 5 the odl trnsfortion is : 5 5 nd : Tht is ler tht the odl oordintes nd he no diret phsil ening. The equtions of otion re : K M Use the odl trnsfortion : The equtions of otion re then : K M Now ultipl the eqution the trnsponed odl tri T (fro the left side) : K M T T T
15 Define so lled odl ss tri nd odl stiffness tri s : ~ T M M ~ T K K The equtions of otion re then : ~ ~ M K This opertion is often lled swith to odl spe euse re odl oordintes. In the oe desried eple where : = g = g = N/ = N/ = N/ nd : 5 T the odl ss tri nd odl stiffness tri re : ~ M T M ~ K T 5 K Both tries re digonl (zeroes out of the in digonl). It n e proofed tht it is not n ident ut the rule. This ss tht the two equtions of otion in the odl spe re independent. ~ ~ M K This represents the two independent ses with DOF. We n esil find the nturl irulr frequenies s : s nd s Copre with the oe presented solution. This pproh swith to the odl spe to the odl oordintes nd sole independent DOF eoes useful espeill in the re of fored irtion.
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