ME617 - Handout 7 (Undamped) Modal Analysis of MDOF Systems

Transcription

1 ME67 - Hadout 7 (Udamped) Modal Aalysis of MDOF Systems he goverig equatios of motio for a -DOF liear mechaical system with viscous dampig are: MU+DU+KU F () () t () t where U,U, ad U are the vectors of geeralized displacemet, velocity ad acceleratio, respectively; ad F () t is the vector of geeralized (exteral forces) actig o the system. M,D,K represet the matrices of iertia, viscous dampig ad stiffess coefficiets, respectively. he solutio of Eq. () is uiquely determied oce iitial coditios are specified. hat is, at t U U, U U () () o () o I most cases, i.e. coservative systems, the iertia ad stiffess matrices are SYMMERIC, i.e., eergy () ad potetial eergy (V) i a coservative system are M M K K. he kietic U MU, V U KU (3) he matrices are square with -rows colums, while the vectors are - rows. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

2 I additio, sice >, the M is a positive defiite matrix. If V >, the K is a positive defiite matrix. V deotes the existece of a rigid body mode, ad makes K a semi-positive matrix. I MDOF systems, a atural state implies a certai cofiguratio of shape take by the system durig motio. Moreover a MDOF system does ot possess oly ONE atural state but a fiite umber of states kow as atural modes of vibratio. Depedig o the iitial coditios or exteral forcig excitatio, the system ca vibrate i ay of these modes or a combiatio of them. o each mode correspods a uique frequecy kows as a atural frequecy. here are as may atural frequecies as atural modes. he modelig of a -DOF mechaical system leads to a set of - coupled d order ODEs, Hece the motio i the directio of oe DOF, say k, depeds o or it is coupled to the motio i the other degrees of freedom,,. I the aalysis below, for a proper choice of geeralized coordiates, kow as pricipal or atural coordiates, the system of -ODE describig the system motio is idepedet of each other, i.e. ucoupled. he atural coordiates are liear combiatios of the (actual) physical coordiates, ad coversely. Hece, the motio i physical coordiates ca be costrued or iterpreted as the superpositio or combiatio of the motios i each atural coordiate. Positive defiite meas that the determiat of the matrix is greater tha zero. More importatly, it also meas that all the matrix eigevalues will be positive. A semi-positive matrix has a zero determiat, with at least a eigevalues equalig zero. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

3 For simplicity, begi the aalysis of the system by eglectig dampig, D. Hece, Eq.() reduces to MU+KU () t F() t (4) at t U U, U U ad () o () Presetly, set the exteral force F, ad let s fid the free vibratios respose of the system. o MU+KU (5) he solutio to the homogeous Eq. (5) is simply U φ cos( t θ ) (6) which deotes a periodic respose with a typical frequecy. From Eq. (6), U φ cos( t θ) (7) Note that Eq. (6) is a simplificatio of the more geeral solutio st U φ e with s i ad where i (8) Substitutio of Eqs. (6) ad (7) ito the EOM (5) gives: MU+KU Mφ cos( t θ) +Kφ cos( t θ) cos( t ) M +K φ θ ad sice cos( t θ ) for most times, the MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 3

4 or M +K φ (9) Mφ K φ () Eq. () is usually referred as the stadard eigevalue problem (mathematical argo): Aφ λ φ where Α M K adλ () Eq.(9) is a set of -homogeous algebraic equatios. A otrivial solutio, φ exists if ad oly if the determiat Δ of the system of equatios is zero, i.e. Δ M +K () Eq. () is kow as the characteristic equatio of the system. It is a polyomial i λ, i.e. 4 6 Δ a+ a + a + a a (3) i Δ a + ( ai λ ) i his polyomial or characteristic equatios has -roots, i.e. the λ k k set { },,... ± sice ± λ. k k or { },,... he s are kow as the atural frequecies of the system. I the MAH argo, the λ s are kow as the eigevalues (of matrix A) MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 4

5 Kowledge summary a) A -DOF system has -atural frequecies. b) If M ad K are positive defiite, the <.... c) If K is semi-positive defiite, the..., i.e. at least oe atural frequecy is zero, i.e. motio with ifiite period. his is kow as rigid body mode. Note that each of the atural frequecies satisfies Eq. (9). Hece, associated to each atural frequecy (or eigevalues) there is a correspodig atural mode vector (eigevector) such that [ ] (),,... M +K φ (4) λ i i i he -elemets of a eigevector are real umbers (for udamped system), with all etries defied except for a costat. he eigevectors are uique i the sese that the ratio betwee two elemets is costat, i.e. ϕ( k ) costat for ay i,,... ϕ ( k ) i he actual value of the elemets i the vector is etirely arbitrary. Sice Eq. (4) is homogeous, if φ is a solutio, so it is αφ for ay arbitrary costat α. Hece, oe ca say that the SHAPE of a atural mode is UNIQUE but ot its amplitude. For MDOF systems with a large umber of degrees of freedom, >>3, the eigevalue problem, Eq. (), is solved umerically. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 5

6 Nowadays, PCs ad mathematical computatio software allow, with a sigle (simple) commad, the evaluatio of all (or some) eigevalues ad its correspodig eigevectors i real time, eve for systems with thousads of DOFs. Log goe are the days whe the graduate studet or practicig egieer had to develop his/her ow efficiet computatioal routies to calculate eigevalues. Hadout # 9 discusses briefly some of the most popular umerical methods to solve the eigevalue problem. A this time, however, let s assume the set of eigepairs, φ is kow. { i () i } i,... Properties of atural modes he atural modes (or eigevectors) satisfy importat orthogoality properties. Recall that each eigepair, φ satisfies the equatio { i () i } i,... M i +K φ () i, i,.... (5) Cosider two differet modes, say mode- ad mode-k, each satisfyig Mφ K φ ad Mφ K φ (6) k ( k) ( k) Pre-multiply the equatios above by φ( k ) ad φ to obtai MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 6

7 ad φ Mφ φ K φ ( k) ( k) φ Mφ φ K φ k ( k) ( k) (7) Now, perform some matrix maipulatios. he products φ Mφ ad φ Kφ are scalars, i.e. ot a matrix or a vector. he traspose of a scalar is the umber itself. Hece, ( φ K φ( k) ) ( K φ( k) ) ( φ ) ad φ ( k) φ K φ sice KK k K φ ( k) ( k) sice φ Mφ φ Mφ MM for symmetric systems. hus, Eqs. (7) are rewritte as ad φ Mφ φ K φ ( k) ( k) ( a) (8) φ Mφ φ K φ k ( k) ( k) ( b) Subtract (b) from (a) above to obtai if k follows that ( ) k ( k) φ Mφ (9), i.e. for WO differet atural frequecies; the it MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 7

8 for k φ Mφ ad φ K φ () k k for k φ Mφ M ad φ K φ K M () where K ad M are kow as the -modal stiffess ad -modal mass, respectively. Defie a modal matrix Φ has as its colums each of the eigevectors, i.e. [ ] Φ φ φ φ ().. ad the modal properties are writte as [ M ]; [ K ] Φ MΦ Φ K Φ () where [M] ad [K] are diagoal matrices cotaiig the modal mass ad stiffesses, respectively. he eigevector set φ k, is liearly idepedet. Hece, ay vector (v) i -dimesioal space ca be described as a liear combiatio of the atural modes, i.e. v a φ Φa (3) a a v φa+ φa φa [ φ φ.. φ] Φa.. a MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 8

9 System Respose i Modal Coordiates he orthogoality property of the atural modes (eigevectors) permits the simplificatio of the aalysis for predictio of system respose. Recall that the equatios of motio for the udamped system are MU+KU () t F() t (4) at t U U, U U ad () () U Φ q (4) 3 Cosider the modal trasformatio () t () t Ad with U () t Φ q () t, the EOM (4) becomes: M Φq+KΦqF () t which offers o advatage i the aalysis. However, premultiply the equatio above by Φ to obtai ( Φ MΦ ) q+ ( Φ K Φ ) qφ F () t (5) ad usig the properties of the atural modes, Φ M Φ M ; Φ K Φ K, the Eq. (5) becomes [ ] [ ] [ M] q+ [ K ] qq Φ F() t (6) 3 Eq. (4) sets the physical displacemets U as a fuctio of the modal coordiates q. his trasformatio merely uses the property of liear idepedece of the atural modes. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 9

10 Ad sice [M] ad [K] are diagoal matrices. Eq. (6) is ust a set of -ucoupled ODEs. hat is, Mq+ Kq Q M q + K q Q... M q + K q Q (7) (8) M q + K q Q with, K Or M,... he set of q s are kow as modal or atural coordiates (caoical or pricipal, too). he vector Q Φ F() t is kow as the modal force vector. hus, the maor advatage of the modal trasformatio (4) is that i modal space the EOMS are ucoupled. Each equatio describes a mode as a SDOF system. he uique solutio of Eqs. (8) eeds of iitial coditios qo, q o. U Φq U Φq specified i modal space, i.e. { } Usig the modal trasformatio, o o; o o, it follows q Φ U ; q Φ U (8) o o o o However, Eq. (8) requires of the iverse of modal matrix Φ, i.e. Φ Φ I. For systems with a large umber of DOF, >>, fidig the matrix Φ is computatioally expesive. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

11 A more efficiet to determie the iitial state { q, q } o o i modal coordiates follows. Start with the fudametal trasformatio, Uo Φq o, ad premultiply this relatioship by Φ M to obtai, Φ MUo Φ MΦqo [ M ] q, o sice [ M ] Φ MΦ, hece or q q q o o [ M ] [ M ] Φ MUo Φ MU o φ MU φ MU, q ok ( k) o ok ( k) o Mk Mk, (9a) (9b) Eqs. (9) are much easier to calculate efficietly whe -DOF is large. Note that fidig the iverse of the modal mass matrix [M] - is trivial, sice this matrix is diagoal. Comparig eqs. (8) ad (9a) it follows that [ M ] Φ Φ M (3) he solutio of ODEs M q + K q Q with iitial coditios { o, } o q q follows a idetical procedure as i the solutio of the SDOF respose. hat is, each modal respose adds the homogeeous solutio ad the particular solutio. he particular solutio clearly depeds o the time form of the modal MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

12 force Q(t), i.e step-load, ramp-load, pulse-load, periodic load, or arbitrary time form. Free respose i modal coordiates Without modal forces, Q, the modal equatios are (3a) M q + K q Q H H with solutios, for a elastic mode qo qh qo cos( ) si t + t if (3b) ; ad for a rigid body mode qh qo + q o t if (3c),,. Forced respose i modal coordiates For step-loads, Q S, the modal equatios are (3a) M q + K q Q S ad; for a elastic mode,, q Q q qo cos si cos t + t + t K o S ( ) ( ) ( ) (3a) MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

13 ; ad for a rigid body mode,, QS q qo + q o t+ t M (3c) For periodic loads,, the modal equatios are with solutios M q + K q Q cos( Ωt) P for a elastic mode,, ad,,. (33a) Ω QP q Ccos( t) + S si cos t + Ωt K ( Ω ) (33b) Note that if destructio. Ω, a resoace appears that will lead to system For a rigid body mode,, Q P q cos qo + q o t Ωt (33c) M Ω For arbitrary-loads Q, the modal respose is q q q cos( t) + si( t) + Q si ( t τ) dτ o t o ( τ ) M (34) for a elastic mode,. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 3

14 System Respose i Physical Coordiates Oce the respose i modal coordiates is fully determied, the system respose i physical coordiates follows usig the modal trasformatio U Φ q () t () t q t q U [ φ φ.. φ ] φ q + φ q φ q.. q () t () t q U φ (35) ( t ) Oe importat questio follows: are all the modal resposes importat ad eed be accouted for to obtai the respose i physical coordiates? If ot, savigs i computatio time are evidet. Hece, the physical respose becomes m U φ q, m< (36) () t ( t ) If m<, the how may modes are to be icluded to esure the physical respose is accurate? hat is, which modes are importat ad which others are ot? MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 4

15 Example: Cosider the case of force excitatio with frequecy Ω ad actig for very log times. he EOMs i physical space are MU+KUF P cos ( Ωt) Let s assume there is a little dampig; hece, the steady state periodic respose i modal coordiates is (see eq. (33b)): Ad thus, q Q K P Ω ( ) cos ( Ωt) (37a) Q P U U cos( t) P Ω Φq φ cos Ω K ( Ω ) (38) he physical respose is also periodic with same frequecy as the force excitatio. ( t) Recall that K M φ K φ ad Q P φ F P However, owadays the egieer i a hurry prefers to dump the problem ito a super computer; ad for UU P cos( Ωt), fids the solutio U P K Ω M F (39) P at a fixed excitatio frequecy Ω. Brute force substitutes beauty ad elegace, time savigs i lieu of uderstadig! MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 5

16 Example: Fid atural frequecies ad atural mode shapes of UNDAMPED system. Give EOMs for a DOF - udamped- system: ORIGIN : M M d dt X X + K K K K + K X X () K Z where Mmo, M5 mo, Kko; K5 ko m o 5 m o d dt X X + k o k o k o k o + 5 k o X X K Z (a) PROCEDURE O FIND NAURAL FREQUENCIES AND NAURAL MODES: Assume the motios are periodic with frequecy, ie X a cos t X a cos t () Set the RHS of Eq. () equal to. Substitutio of () ito () gives k o m o k o 7 k o k o 5 m o he homogeeous system of eqs a a cos t cacel cos(t) sice it is NO zero for all times has a o-trivial solutio if the determiat of the system of equatios equals zero, i.e. if Δ ( 7 k o 5 m o ) ( k o m o ) 4 k o Let λ k o m o k o 7 k o, ad expadig the products i the determiat k o 5 m o λ 5m o λ 7 k o m o + k o m o + 4 k o 4 k o a a (3) Let m o λ λ k o Leads to: with: a λ a : 5 + b λ + c b : 7 (4) c : he roots (eigevalues) of the characteristic equatio are

17 λ.5.5 b b 4 a c b + b 4 a c : λ a : a λ k o m o ad the atural frequecies are: (b) Explaatio: DOF (X) ad DOF (X) move i phase, with X>X for φ : k o m o ( k o.643 k o ) φ (.643) k o k φ o : : φ φ is the d eigevector (atural mode).3 φ ( λ ).5 Fid the eigevectors: : : λ (b) Explaatio:DOF (X) ad DOF (X) move 8 deg OU of phase, with X > X (c) fid the umerical value for each atural frequecy: lb.5 m o : Sice.87 k o g :.66 m o rad sec f : π 7.7 f Hz m o he two equatios i (3) are liearly depedet. hus, oe caot solve for a ad a. Set φ : arbitrarily; ad from the first equatio for φ ( k o m o ) k o φ : φ φ.6 k o.757 k o k o φ : k o m o φ.6 is the first eigevector (atural mode) k o : 5 lb i Note that mass must be expressed i physical uits cosistet with the problem, i.e. lb sec i.5 (.757)

18 Perform same work usig a calculator M : 5 m o K : 7 k o Use BUIL IN fuctios - Not much learig Let Z : M K λ : sort( eigevals( Z) ) : ( λ ).5 : ( λ ).5 λ rad sec sec π Hz atural modes: φ : eigevec Z, λ φ : eigevec Z, λ.849 φ φ.36 ( φ ).6 φ ( φ ).3 φ which are the same ratios as for the vectors foud earlier

19 Example: Udamped Modal Aalysis m o Mode K m : lb sec K lb M m m 3.95 i i (b) Fid iitial moddal displacemets ad velocities ad modal force vector (Q) At time ts, the system is at RES at its static equilibrium positio, hece the iitial coditios are ull displacemets ad ull velocities. Of course, the same applies to modal space, i.e. ull iitial displacemets ad velocities X for geerality, defie: X o : X ft V ft o : Calculate iverse of A matrix sec ad i modal coordiates q o : Φ iv X o q o_dot : Φ iv V o velocity (disp & velocities) Defie modal force he EOMs i modal space are ucoupled ad equal to where q () t : δ m cos t.8 3 δ m ft (c) Modal EOMs ad modal resposes q o q o_dot Q : Usig the cheat sheet, ad sice the Iitial coditios are ull, the respose i modal coordiates are Φ F i lb : k g o : 5 lb g 3.74 ft i sec ORIGIN : Equatios of motio: atural frequecies, modal matrix (eigevectors) m o 5m o d dt Defie matrices: M M X X + M : k o k o m o k o 7k o X X (a) FIND modal masses ad stiffesses modal masses ad stiffesses: 5m o lb sec i K : k o k o k o Z k o 7k o Mode Q 6 3 d M mi q i d t lb Q δ m : K m 7.95 : Φ : 39.5 sec ft sec q () t : δ m cos t rad + K mi q Q i i Q δ m : K m Φ iv : Φ No eed for actual calculatio - a kowledge statemet suffices Both atural modes will be excited i,.6 where: are the "static" deflectios i modal space. δ << δ, thus first modal respose is MORE importat.3 give: Z o :. i provides a F o : k o Z o costat force M M : Φ M Φ F : M m : M m : lb F o K M : Φ K Φ M M, M M, at ts, Iitial coditios: system is at RES o-diagoal elemets are very small o zero b/c of roudoff i umerical calculator K m : MM, MM, rad sec

20 (d) he respose i physical coordiates, X ad X, equals (from trasformatio xaq) with X () t : q () t + q () t Φ X () t δ m cos t + δ m cos t Ad equatios of motio reduce to: k o k o X d dt X ; hece > K k o 7k o X ed X ed F o X ed X ed F F o 3 lb.6 X ( t) :.6 q () t.3 q () t for graph below: X () t δ m.6 cos t + δ m (.3) cos t large : δ m i δ m (.3).57 4 i.3 Explaatio: Sice q ad q are o-zero, the physical motio, X &X, shows excitatio of the WO fudametal modes of vibratio - BU respose for secod mode is much less GRAPHs ot eeded for exam: displacemets (ich) time (sec) X X termial value Note that there is o dampig or atteuatio of motios. π Not too complicated physical respose. It shows domiace of first mode (lowest atural freq or largest period) π π.37 sec. sec ermial coditio: If dampig is preset ad sice the applied force is a costat, the system will achieve a ew steady state coditio. I the limit as t approaches very, very large values Ad solvig this system of equatios usig Cramer's rule F o k o k o F o X ed : X ed : Δ Δ Δ : 4 k o 4k o determiat of system of eqs. X ed 3 i X ed 3 i Note that the graph of udamped periodic motios Z(t) ad X(t) shows oscillatory motios abut these termial or ed values. OR K F 3 3 i recall Z o. i Z o 5 X ed

21 COMPARE actual respose with a respose eglectig q. Ideed mode does ot afffect the physical respose, except for motio X sligthly displacemets (ich) X X termial time (sec)

22 Normalizatio of eigevectors (atural modes) Recall that the compoets of a eigevector φ are ARBIRARY but for a multiplicative costat. If oe of the elemets of the eigevector is assiged a certai value, the this vector becomes uique, sice the - remaiig elemets are automatically adusted to keep costat the ratio betwee ay two elemets i the vector. I practice, the eigevectors are ormalized. he resultig vectors are called NORMAL MODES. Some typical NORMS are L orm: max( q ) q (39a) k L orm: q q q q (39b) Or makig the mass modal matrix equal to the idetity matrix, [M]I, i.e. hece φ Mφ (39c) M φ K φ K M (39d) his ormalizatio has obvious advatages sice it will reduce the umber of operatios whe coductig the modal aalysis. However, the physical sigificace of the modal equatios is lost. Note that the modal Eqs. (6) become: MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 6

23 q + q Q Your lecturer recommeds this ormalizatio procedure be coducted oly for systems with large umber of degrees of freedom, >>>. Note that the ormalizatio process is a mere coveiece, devoid of ay physical sigificace. Rayleigh s Eergy Method he method is a procedure to determie a approximate value (from above) for the fudametal atural frequecy of a MDOF system. At times, the full solutio of the eigevalue problem is of NO particular iterest ad a estimate of the system lowest atural frequecy suffices. Recall that the pairs { i, () i } φ satisfy Kφ M φ i,... with properties Φ M Φ [ M ]; Φ K Φ [ K ] () i i () i i.e. with modal stiffess ad masses calculated from: K K () (); () (), ad i i φ i Kφ i Mi φ i Mφ i i (4) M i hat is, K () i () i i i φ Kφ M i φ() i Mφ() i (4) Above, the umerator relates to the potetial or strai eergy of the system for the i-mode, ad the deomiator to the kietic eergy for the same mode. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 7

24 Cosider a arbitrary vector u ad defie Rayleigh s quotiet R(u) as R ( u) uku umu (43) R( u ) is a scalar whose value depeds ot oly o the matrices M & K, but also o the choice of the vector u. Clearly, if the arbitrary vector u coicides with (or is a multiple of) oe of the atural mode vectors, the Rayleigh s quotiet will deliver the exact atural frequecy for that particular mode. It ca also be show that the quotiet has a statioary value, i.e. a miimum, i the eighborhood of the system atural modes (eigevectors). o show this, sice u is a arbitrary vector ad the atural modes are a set of liearly idepedet vectors, the oe ca represet Where { } u φc Φc (44) c c c.. c is the vector of coefficiets i the expasio. Substitutio of the expressio above ito Rayleigh s quotiet gives ( Φc) K( Φc) ( Φc) M( Φc) R ( u) c Φ KΦ c c Φ MΦ c [ K ] [ M ] R( u) c c c c (45) MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 8

25 Assume the modes have bee ormalized with respect to the mass matrix, i.e. ci i c i cic ci i c R( u) (46a) Next, cosider that the arbitrary vector u (which at this time ca be regarded as a assumed mode vector) differs very little from the atural mode (eigevector) φ ( r). his meas that i the expasio of vector u, the coefficiets c << c ; for i,,... adi r Or c ς c ; ς << for i,,... adi r i i r i he, Rayleigh s quotiet is expressed as i r R( u ) c r + c r r ς i i i, i r cr + cr ς i i, i r ς i i + ςi + i, i r i, i r r + ςi + ςi i, i r i, i r r i r R( u ) (46b) he quatities { ς i } are small, of secod order, hece R(u) differs from the atural frequecy by a small quatity of secod order. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 9

26 his implies that R(u)has a statioary value i the viciity of the modal vector φ ( r). he most importat property of Rayleigh s quotiet is that it shows a miimum value i the eighborhood of the fudametal mode, i.e. whe r. R ς i + i u (47) i, sice i > + ς i i he each term i the umerator is greater tha the correspodig oe i the deomiator. Hece, it follows that R u (48) i.e., Rayleigh s quotiet provides a upper boud to the first (lowest) atural frequecy of the udamped MDOF system. Clearly, the equality holds above if oe selects u c φ (); c. Closure Rayleigh s eergy method is geerally used whe oe is iterested i a quick (but particularly accurate) estimate of the fudametal atural frequecy of a cotiuous system, ad for which a solutio to the whole eigevalue problem caot be readily obtaied. he method is based o the fact that the atural frequecies have statioary values i the eighborhood of the atural modes. I additio, Rayleigh s quotiet provides a upper boud to the first (lowest) atural frequecy. he egieerig value of this approximatio ca hardly be overstated. Rayleigh s eergy method is the basis for the umerical computig of eigevectors ad eigevalues as will be see later. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

27 N 5 umber of DOFS mass stiff 3 M idetity( N) mass Example for estimatio of first atural frequecy ORIGIN d M U d t KU K stiff K U M K U M K U3 M K U4 M K U5 M K

28 Estimate for first atural frequecy - Use Rayleigh-Ritz method: ASSUME u ( ) calculate Rayleigh's quotiet Ru u Ku u Mu _ap Ru _ap 6.93 Estimate Actual rad/s % differece _ap M u (approx)

29 Estimate for secod atural frequecy - Use Rayleigh-Ritz method: ASSUME u ( ) calculate Rayleigh's quotiet Ru u Ku u Mu _ap Ru _ap 3.63 Estimate Actual 3.63 rad/s % differece _ap M u (approx)

30 N 5 umber of DOFS mass stiff 3 M idetity( N) mass Example for estimatio of first atural frequecy ORIGIN d M U d t KU K stiff K U M K U M K U3 M K U4 M K U5 M

31 Estimate for first atural frequecy - Use Rayleigh-Ritz method: ASSUME u ( ) calculate Rayleigh's quotiet Ru u Ku u Mu _ap Ru _ap.935 Estimate Actual 9. rad/s % differece _ap M u (approx)

32 Mode Acceleratio Method Recall that the respose i physical coordiates is m U φ q, m< (36) () t ( t ) where m<. he procedure is kow as the mode displacemet method. his method, however, fails to give a accurate solutio eve whe a static load is applied (See Structural Dyamics, by R. Craig, J. Wiley Pubs, NY, 98.). he difficulty is overcome by usig the procedure detailed below. Recall that the system motio is govered by the set of equatios (4) MU+KU F () t () t Ad, if there are o rigid body modes, i.e. all atural frequecies are greater tha zero, the U K F MU (5) () t () t where K is a flexibility matrix. From Eq. (36), m U φ q, m< (5) ( t ) Hece, Eq, (5) ca be writte as MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

33 m () t () t q U K F K M φ (53) ( t ) Usig the fudametal idetity, Kφ Mφ φ K Mφ Write Eq. (53) as () i i () i () i () i i m U() t K F() t φ q ( t ) (54) Note that US K F () t (55) is the displacemet respose vector due to a pseudo-static force F(t), i.e. without the system iertia accouted for. Hece write Eq. (54), as m () t s() t φ q U U ( t ) ; m< (56) he secod term above ca be thought as the iertia iduced respose. Example: Cosider the case of force excitatio with frequecy Ω ad actig for very log times. he EOMs i physical space are: MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8

34 MU+KUF P cos ( Ωt) With a little dampig, the steady state periodic respose i modal coordiates is q Q K P Ω ( ) cos ( Ωt) Recall that, usig the mode displacemet method, the respose i physical coordiates is: (37a) m QP U φ cos ( Ωt ) K (38) ( Ω ) From each of the modal resposes, Q P q ( Ω ) cos ( Ωt) ; K ( Ω ) q QP Ω cos K ( Ω ) ( Ωt) (57) Sice K respose is M ; the usig the mode acceleratio method, the m Q P Ω U U SP + φ cos ( Ωt) K (58 ( Ω ) ) MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 3

35 U K F. Now, i the where the pseudo-static respose is SP P limit, as the excitatio frequecy decreases, i.e., as Ω, the secod term i Eq. (58) above disappears, ad hece the physical respose becomes: U USP K F P (59) which is the exact respose, regardless of the umber of modes chose. Hece, the mode acceleratio method is more accurate tha the mode displacemet method. Kow disadvatages iclude more operatios. Fidig the flexibility matrix is, i actuality, desirable. I particular, if derived from measuremets, the flexibility matrix is easier to determie tha the stiffess matrix. MEEN 67 HD#7 Udamped Modal Aalysis of MDOF systems. L. Sa Adrés 8 4

36 SEP FORCED RESPONSE of Udamped -DOF mechaical system ORIGIN : Dr. Luis Sa Adres (c) MEEN 363, 67 February 8 he udamped equatios of motio are: d M dt X + KX F o () where M,K are matrices of iertia ad stiffess coefficiets, ad X, VdX/dt, d X/dt are the vectors of physical displacemet, velocity ad acceleratio, respectively. he FORCED udamped respose to the iitial coditios, at t, Xo,VodX/dt, follows: he equatios of motio are: M M M M d dt x x + K K K K x x F o F o (). Set elemets of iertia ad stiffess matrices DAA FOR problem M : K : 5 Note M ad K kg 6 6 are symmetric matrices 6 6 N m : # of DOF iitial coditios Applied force vector: m. X o : V o : F o : 5 N m sec. Fid eigevalues (udamped atural frequecies) ad eigevectors Set determiat of system of eqs ( K M ) ( K M ) Δ K M K M (a) Δ a 4 + b + c a λ + b λ + c with λ (b) where the coefficiets are: a : M, M, M, M, b : K, M, K, M, K, M, + K, M, c : K, K, K, K, (c)

37 he roots of equatio (b) are: λ :.5 b b 4 a c a λ :.5 b + b 4 a c a (3) also kow as eigevalues. he atural frequecies follow as: Note that: :.. Δ( ) Δ : ( λ ).5 f : π f rad sec Hz (4) For each eigevalue, the eigevectors (atural modes) are :.. a : MODAL matrix A : a K, M, λ ( K ), M, λ Set arbitrarily first elemet of vector a.73 a.73 A is the matrix of eigevectors (udamped modal matrix): each colum correspods to a eigevector A.73 (5).73 Plot the mode shapes: A, A, DOF mode mode

38 3. Modal trasformatio of physical equatios to (atural) modal coordiates Usig trasformatio: X A q EOMs () become ucoupled i modal space: d M m q dt + K m q Q m (6) (7) with modal force vector: Q m A F o (8) ad iitial coditios (modal displacemetq ad modal velocity dq/dts) q o M m A M X o s o M m A M V o (9) he modal resposes are of the form: k... OR q k q ok cos k t + q k q ok + s ok t + s ok k si ( k t) Q mk t M mk, k + Q mk ( cos ( K k t) ) mk, k for a elastic mode k for k for a rigid body mode (a) (b) Ad, the respose i the physical coordiates is give by the superpositio of the modal resposes, i.e. X() t Aqt () CHECK Verify the orthogoality properties of the atural mode shapes M m A 6.79 : M A M m.58 4 K m : A K A.6 6 K m (5) N m kg s -

39 4. Fid Modal ad Physical Respose for give iitial coditio ad Costat Force vector Recall the vectors of iitial coditios X o m V o m s ad Costat forces: 4.a Fid iitial coditios i modal coordiates (displacemet q, velocity s) Set iverse of modal mass matrix 4 F o 5 3 m N m A iv : M m DAA FOR problem beig aalyzed: ( A M) q o : A iv X o s o : A iv V o q o 4.b Fid Modal forces: m s o ms- Q m : A F o Q m.37 4 N 4.c Build Modal resposes: q () t : q o cos t + s o si ( t) + Q m K m ( ) cos t, q () t : q o cos t + s o si ( t) + Q m K m ( ) cos t, 4.d Build Physical resposes: X() t : a q () t + a q () t for plots: 6 plot : f 4.e Graphs of Modal ad Physical resposes:. Respose i modal coordiates

40 q q time (s). Respose i physical coordiates x x time (s) 5. Iterpret respose: aalyze results, provide recommedatios Recall atural frequecies & periods S-S displacemet K F o 5 3 m f Hz f s s A.73.73

41 SEP FORCED RESPONSE of Udamped -DOF mechaical system ORIGIN : Dr. Luis Sa Adres (c) MEEN 363, 67 February 8 he udamped equatios of motio are: d M dt X + KX F o () where M,K are matrices of iertia ad stiffess coefficiets, ad X, VdX/dt, d X/dt are the vectors of physical displacemet, velocity ad acceleratio, respectively. he FORCED udamped respose to the iitial coditios, at t, Xo,VodX/dt, follows: he equatios of motio are: M M M M d dt x x + K K K K x x F o F o WIH RIGID BODY MODE (). Set elemets of iertia ad stiffess matrices DAA FOR problem Note M : K : 5 M ad K kg 6 6 are symmetric matrices 6 6 N m : # of DOF iitial coditios Applied force vector: m. X o : V o : F o : 98 N m sec. Fid eigevalues (udamped atural frequecies) ad eigevectors Set determiat of system of eqs ( K M ) ( K M ) Δ K M K M (a) Δ a 4 + b + c a λ + b λ + c with λ (b) where the coefficiets are: a : M, M, M, M, b : K, M, K, M, K, M, + K, M, c : K, K, K, K, (c)

42 ,,,, he roots of equatio (b) are: λ :.5 b b 4 a c a λ :.5 b + b 4 a c a (3) also kow as eigevalues. he atural frequecies follow as: Note that: :.. Δ( ) Δ : ( λ ).5 f : π f rad sec Hz (4) For each eigevalue, the eigevectors (atural modes) are :.. a : MODAL matrix A : a K, M, λ ( K ), M, λ Set arbitrarily first elemet of vector a a A is the matrix of eigevectors (udamped modal matrix): each colum correspods to a eigevector A (5) Plot the mode shapes: A, A, DOF mode mode 3. Modal trasformatio of physical equatios to (atural) modal coordiates Usig trasformatio: X A (6)

43 Usig trasformatio: X A q EOMs () become ucoupled i modal space: d M m q dt + K m q Q m (6) (7) with modal force vector: Q m A F o (8) ad iitial coditios (modal displacemetq ad modal velocity dq/dts) q o M m A M X o s o M m A M V o (9) he modal resposes are of the form: k... OR q k q ok cos k t + q k q ok + s ok t + s ok k si ( k t) Q mk t M mk, k + Q mk ( cos ( k t) ) K mk, k for a elastic mode for k k for a rigid body mode (a) (b) Ad, the respose i the physical coordiates is give by the superpositio of the modal resposes, i.e. X() t Aqt () CHECK Verify the orthogoality properties of the atural mode shapes (5) M m A 5 : M A M m 3 kg K m : A K A N m K m s- 4. Fid Modal ad Physical Respose for give iitial coditio ad Costat Force vector Recall the vectors of iitial coditios X o m Vo m s d C t t f

44 q o : ad Costat forces: 4.a Fid iitial coditios i modal coordiates (displacemet q, velocity s) Set iverse of modal mass matrix A iv X o s o : 3 F o 98 A iv V o m N m A iv : M m DAA FOR problem beig aalyzed: ( A M).67 A iv b Fid Modal forces: q o m s o ms- Q m : A F o Q m.96 3 N 4.c Build Modal resposes: q () t : q o + s o t + Q m t respose for rigid body mode M m, q () t : q o cos t + s o si ( t) + Q m K m ( ) cos t, 4.d Build Physical resposes: X() t : a q () t + a q () t for plots: 6 plot : f 4.e Graphs of Modal ad Physical resposes:.4 Respose i modal coordiates time (s)

45 q q time (s) Respose i physical coordiates x x time (s) 5. Iterpret respose: aalyze results, provide recommedatios S-S displacemet - NONE Recall atural frequecies & periods f Hz A s -