Response of MDOF syses Degree of freedo DOF: he nu nuber of ndependen coordnaes requred o deerne copleely he posons of all pars of a syse a any nsan of e. wo DOF syses hree DOF syses
he noral ode analyss EOM- Eaple: Response of DOF syse FBD - EOM & & & & In ar for, EOM s
EOM - eaple ẍ EOM M K F F K C M & In general for M s he nera of ass ar n n C s he dapng ar n n K s he sffness ar n n F s he eernal force vecor n s he poson vecor n
Synchronous oon Fro observaons, free vbraon of undaped MDOF syse s a synchronous oon. All coordnaes pass he equlbru pons a he sae e All coordnaes reach eree posons a he sae e Relave shape does no change wh e e consan A sn φ No phase dff. beween and A sn φ or or A e A e j φ j φ
Response of DOF syse eaple- EOM Synchronous oon Sub. no EOM A A K M M K de M K Characersc equaon CHE sn φ A sn φ A φ j A e φ j A e or or
Response of DOF syse eaple- 3 3 4.634.366 CHE Solve he CHE Naural frequences of he syse ; A A A A.73.634 A A.73.366 A A Fro
Response of DOF syse eaple-3 Ap. rao A A.73 Ap. rao A A.73 he frs ode shape he second ode shape φ.73 φ.73.73 sae drecon -.73 Oppose drecon
Response of DOF syse eaple-4 In general, he free vbraon conans boh odes sulaneously vbrae a boh frequences sulaneously.73 c sn ψ c.73 sn ψ c, ψ ψ c,, are consans depended on nal condons
Inal condons sn.73 sn.73 ψ ψ c c 4 & & Inal condons and cos.73 cos.73 ψ ψ c c & & Velocy response 4 sn.73 sn.73 4 ψ ψ c c & & cos.73 cos.73 ψ ψ c c
Inal condons.73.73 c sn ψ c sn ψ 4.73.73 c cos ψ c cos ψ 4 Eqs., 4 unnowns Solve for four unnowns c 3.73, c.68, ψ ψ π / he response s π.73 π.73 3.73 sn.68 sn.73 cos 3.73.73 cos.68
Inal condons 3 sn.73 sn.73 ψ ψ c c.464 & & a Inal condons and.73 & & b Inal condons and ry o do
Suary Free-undaped EOM M K Drec Mehod he oon s synchronous: consan and φ j φ Asn φ or Ae 3 Egen value proble M K K M Characerscs equaon 4 5 de K M K M,,, n n n nn Egen value K N naural freq. n,,, K N Egen vecor N ode shapes
Suary Free-undaped 6 Free-undaped response Drec Mehod A sn φ A sn φ K N AN sn N φn N A sn φ where A and φ are fro nal condon and v
Eaple l l θ Deerne he noral odes of vbraon of an auooble sulaed by splfed -dof syse wh he followng nuercal values W 3 lb l 4.5 f J C W g r 4 lb/f r 4 f l 5.5 f 6 lb/f
Forced haronc vbraon Eaple EOM F sn Syse s undaped, he soluon can be assued as X X sn Sub. no EOM F X X [ ] F X X Z Spler noaon, [ ] F Z X X
Forced haronc vbraon [ ] [ ] adj F Z Z F Z X X Z Where and are naural frequences F Z X X he apludes are F X F X
Forced haronc vbraon 3 EOM F sn 3, F sn X F 5 4 3 - - -3-4 Force response of a DOF syse X F Sae drecon X F Oppose drecon X F -5 3 X F
Solvng ehods
Modal analyss Inroducon s a ehod for solvng for boh ransen and seady sae responses of free and forced MDOF syses hrough analycal approaches. Uses he orhogonaly propery of he odes o decouple he EOM breang EOM no ndependen SDOF equaons, whch can be solved for response separaely.
Coordnae couplng l l g θ Ref. θ l θ l l g θ Ref. θ l θ θ l l l l l l J θ θ l l l J l l
Concep of odal analyss θ θ F l l l l l l J F K M N N r r r r n n N Λr r EOM n odal coordnae Independen SDOF equaons EOM n physcal coordnae Coordnaes are coupled Solve for r ransfor r bac o
Orhogonaly egen vecor vecor of ode shape If M and K are syerc and hen and j are sad o be orhogonal o each oher. n nj j M, j j K, j M M K K
Noralzaon u noralzed egen vecor respec o ass ar u j Mu, j u Mu u C, C s consan Fro egen value proble or K M K M u Ku Mu u Ku u Mu
Modal ar Modal ar s he ar ha s coluns are he ode shape of he syse [ ] u n u u U K hen K M O M M K K I U MU nn n n K M O M M K K KU U Λ Specral ar
Modal analyss undaped syses- Procedures. Draw FBD, apply Newon s law o oban EOM. Solve for naural frequences hrough CHE 3. Deerne ode shapes hrough EVP 4. Consruc odal ar noralzed U [ u u K ] U MU U KU I Λ u n 5. Perfor a coordnae ransforaon Ur M K F MU r KUr F U MUr U KUr U F r Λr U F M K F de K M K M
Modal analyss undaped syses- U F Λr r 4 3 4 3 N N N N F F F F u u u u u u u u u r r r r r r NN N N N N N nn n n N K M O M M K K M K M O M M K K M Independen SDOF equaons, can be solve for r 6. ransfor he nal condons o odal coordnaes Ur I MU U Ur MUr U M U M U r Fro and M U r & &
Modal analyss undaped syses-3 7. Fnd he response n odal coordnaes r K 8. ransfor he response n odal coordnae r bac o ha n orgnal coordnae Ur
Eaple Modal analyss - 3 3 3 7 9 EOM v Inal condons
Eaple Modal analyss - dof srng-bead syse a EOM v Inal condons
Eaple Modal analyss -_
Rgd body ode Rgd body ode s he ode ha he syse oves as a rgd body. he syse oves as a whole whou any relave oon aong asses. here s no oscllaon. n
Rgd-body odes Copue he soluon of he syse. Le g, 4 g and 4 N/. Inal condon. v
More han wo degrees of freedo Calculae he soluon of he n-degree-of-freedo syse n he fgure for n 3 by odal analyss. Use he values 3 4 g and 4 4 N/, and he nal condon wh all oher nal dsplaceens and veloces zero.
Modal analyss on daped syses EOM M C& K F he orgnal odal analyss can be appled o MDOF daped syse f and only f CM K KM C Necessary and suffcen condon Such syse s called classcally daped. However, here are subses of he above syses where C can be wren as a lnear cobnaon of M and K. C αm βk Suffcen bu no necessary condon α and β are consans. Such syse s called proporonally daped.
Modal analyss on daped syses For proporonally daped M C& K F M αm βk & K F Le Ur MU r αm βk Ur& KUr F Preulply by U U MUr U αm βk Ur& U KUr U F r αi βλ r& Λr U F N hus, he syse when s wren n odal coordnaes r can be decoupled no ses of SDOF equaons where ζ n α β n r ζ r& n n r N M
Modal analyss on daped syses E. A bel-drven lahe bearngs are odeled as provdng vscous dapng shafs provde sffness bel drve provdes and appled orque. J c J J3 3 N..s/rad g. N./rad /rad Zero nal condons Appled oen M s a un pulse funcon
Modal analyss on daped syses E.