Handout # 13 (MEEN 617) Numerical Integration to Find Time Response of MDOF mechanical system. The EOMS for a linear mechanical system are

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1 Handou # 3 (MEEN 67) Numercal Inegraon o Fnd Tme Response of MDOF mechancal sysem The EOMS for a lnear mechancal sysem are MU+DU+KU =F () () () where U,U, and U are he vecors of generalzed dsplacemen, velocy and acceleraon, respecvely; and F () s he vecor of generalzed (exernal forces) acng on he sysem. M,D,K represen he marces of nera, vscous dampng and sffness coeffcens, respecvely. The soluon of Eq. () s unque wh specfed nal condons U, U. o o Numercal soluon mehods drec numercal negraon modal analyss (mode dsplacemen or mode acceleraon) (Undamped and Damped) Mode superposon mehods are dscussed earler, see Lecure noes 7,8 and. The accuracy of hese mehods s deermned by he number of modes seleced. I s also mporan he me spen n performng he egenvecor analyss. Modal superposon mehods are mos general. elegan, and very powerful. The marces are square wh n-rows = n columns, whle he vecors are n- rows. MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8

2 Drec numercal negraon mples a marchng n me, sep-by-sep procedure, solvng he algebrac form of Eq. () a specfc me values. The drec qualfcaon mples ha, pror o he numercal negraon; no ransformaon of he EOM () s carred ou. In a numercal negraon mehod, EOM () s sasfed a dscree me nervals, Δ apar. In essence, a sor of quas-sac equlbrum s sough a each me sep. Recall ha a relable numercal negraon scheme should a) reproduce EOM as me sep Δ b) provde, as wh physcal model, bounded soluons for any sze of me sep,.e. mehod should be sable c) reproduce he physcal response wh fdely and accuracy. Numercal negraon mehods are usually dvded no wo caegores, mplc and explc. x = f x, () Consder he ODEs ( ) In an explc numercal scheme, he ODEs are represened n erms of known values a a pror me sep,.e. ( ) x =x + Δf x,, (3) + whle n an mplc numercal scheme ( ) x =x + Δf x, (4) + + Explc numercal schemes are condonally sable. Tha s, hey provde bounded numercal soluons only for (very) small me seps. For example, MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8

3 Tn Δ τ cr = (5) π π where T and K n= ωn = M are he naural perod and naural ωn frequency of he sysem, respecvely. The resrcon on he me sep s oo severe when analyzng sff sysems,.e. hose wh large naural frequences. Noe ha n a MDOF sysem, he condon defned n Eq. (5) s oo resrcve snce T n refers o he smalles perod of naural moon (larges naural frequency). More ofen han no, hs crcal me s NOT known unless he egenanalyss s compleed. In nonlnear sysems, he crcal me sep lm worsens. Implc numercal schemes are uncondonally sable,.e. do no mpose a resrcon on he sze of he me sep Δ. (However, accuracy may be compromsed f oo large me seps are used). An negraon mehod s uncondonally sable f he soluon for any nal condon does no grow whou bound for any me sep Δ, n parcular when s large The mehod s only condonally sable f he above holds Δ provded ha s smaller han a ceran value, usually T called he sably lm. n Δ T. n MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 3

4 The Wlson-θ mehod I s an exenson of he lnear-acceleraon mehod. Tha s, whn a me sep, as shown n Fgure, he acceleraon vecor s proporonal o me. Over he me nerval τ θ Δ, U τ + τ = U + + θδ θ Δ U U (6) wh θ. Wlson-θ lnear acceleraon mehod X + θ X + X Acceleraon ( τ ) X = a+ bτ X : componen of vecor U θ > τ ι+ = ι +Δτ X + θ +θδτ dsplacemen X + X τ + velocy X + ( τ) X X + θ X ( τ) = X + aτ + b τ τ + +θδτ 3 τ τ X = X + X τ + a + b 6 +θδτ Fg. Depcon of componens of acceleraon, velocy and dsplacemen for numercal negraon Wlson-θ mehod Inegraon of Eq. (6) gves he vecor of velocy as Ths numercal mehod s exremely popular among he srucural dynamcs communy. MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 4

5 U τ + τ = U + τ U + + θδ θ Δ U U (7) One more negraon delvers he vecor of dsplacemens as 3 U τ τ + τ = U + τ U θδ U 6θ Δ U U (8) Specfcaon of he vecors a me τ = θ Δ ( θδ ) U +Δ θ = U + θδ U + +Δ θ θ U U Δ = U + θδ U +Δ θ + U (9) ( θδ) ( θδ) ( θδ ) 3 U +Δ θ = U + θδ U + U + +Δ θ 6θ Δ U U = U + θδ U + +Δ θ + 6 U U () From Eqs. (9) and (), solve for he acceleraon and velocy a he end of he nerval,.e. U 6 6 θ = U U +Δ +Δ θδ U U () ( ) [ ] θ θδ 3 θδ U [ ] +Δ θ = U +Δ θ U U U θδ () Now, sae EOM () a he end of he me nerval ( + θ Δ ) (3) a + θ Δ MU + θ Δ +DU + θ Δ +KU + θ Δ =F+ θ Δ where = + θ [ ] F F F F (4) + θ Δ +Δ MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 5

6 s a (me) projecon of he exernal force vecor. Subsue Eqs. () and () no (3) o deermne an equaon from whch U+Δ θ can be obaned. Nex, subsue hs vecor no Eq () and () o oban U +Δ θ and U +Δ θ. Lasly, use hese vecors o deermne n Eqs. (8) and (9) he dsplacemen and velocy vecors a τ =Δ,.e. U +Δ and U +Δ. The resulng sysem of equaons for soluon a each me sep s KU ˆ = F ˆ (5) + θδ + θδ where Kˆ = K+ a M+ ad (6) o s he effecve sffness marx, and [ ] Fˆ = F + θ F F + + θ Δ +Δ ( ao a ) ( a + + a 3 ) M U + U + U + D U U U (7) s he effecve load vecor a + θ Δ. Above, MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 6

7 6 3 θ Δ ao ao =, a =, a= a, a3=, a4 =, θ Δ θ ( θ Δ) a 3 Δ Δ a5=, a6 =, a7 =, a8= θ θ 6 (8) The effecve sffness marx can be easly decomposed (once) as ˆ T K= LDg L If M, K, D are symmerc. ˆ Oherwse, K= LU Δ,.e. he produc of a lower rangular and upper rangular form marces. Hence, he soluon of Eq. (5) proceeds sep by sep n a process of backward and forward subsuons,.e. LY= Fˆ +Δ θ U U = Y (9) Δ + θδ U Soluon of Eq. (9) gves + θδ. Nex, calculae he +Δ usng dsplacemens, velocy, and acceleraon vecors a ( ) [ ] U = a U U + a U + a U +Δ 4 + θδ 5 6 U +Δ = U + a7 U +Δ + U U U U U U +Δ = +Δ + a8 +Δ + () Analyss shows ha θ = for uncondonal sably. MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 7

8 The Newmark-β mehod Ths mehod s also an exenson of he lnear-acceleraon mehod. The velocy and dsplacemen vecors a he end of he me nerval ( ) +Δ are, as shown n Fg : ( δ) U +Δ = U + U + δ U +Δ Δ \ ( α) +Δ = +Δ + + α +Δ Δ () U U U U U where (, ) α δ are parameers seleced o promoe numercal sably and/or gan n accuracy. Lnear acceleraon mehod Acceleraon velocy X + X ( τ ) X = a+ bτ X + X X ( τ) = X + aτ + b τ τ + τ + X : componen of vecor U dsplacemen X + ( τ) X 3 τ τ X = X + X τ + a + b 6 τ + Fg. Depcon of componens of acceleraon, velocy and dsplacemen for numercal negraon Newmark mehod MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 8

9 δ =, α =, Eq. () represens he lnear acceleraon mehod 6 (θ= n Wlson s mehod). δ =, α =, brngs uncondonal sably, and Eq. () 4 represens he average acceleraon mehod dscussed for SDOF sysems. See Lecure Noes 6. The mehod solves he algebrac form of EOM () a he end of he me nerval +Δ MU +Δ +DU +Δ +KU +Δ =F+Δ () Subsuon of Eqs. () above leads o he algebrac equaon: KU ˆ = F ˆ (3) +Δ +Δ Kˆ = K+ a M+ ad (4) where o s he effecve sffness marx, and ( a a a 3 ) ( a + a 4 + a 5 ) F = F + M U + U + U + ˆ +Δ +Δ o D U U U (5) s he effecve load vecor a + Δ. Above, MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 9

10 6 δ δ ao =, a =, a =, a =, a =, αδ αδ αδ α α 3 4 Δ δ a5=, a6 =Δ( α), a7 = δδ α (6) δ, α + δ (7) In general, s recommended ha ( ) 4 The effecve sffness marx s easly decomposed (once) as Kˆ = LUΔ for soluon of Eq. (3),.e. o deermne +Δ The vecors of acceleraon and velocy follow from U. [ ] U = a U U a U + a U +Δ +Δ 3 U = U + a U + a U +Δ 6 7 +Δ (8) Sably of Numercal Mehod In general, he numercal mehod solves KU ˆ = Fand ˆ KU ˆ = F ˆ n wo consecuve me seps. +Δ +Δ For a suffcenly small me sep, ˆ ˆ Δ ˆ +Δ... ˆ ˆ F F ˆ F F = F + Δ + + KU + Δ (9) Hence ˆ ˆ ˆ F KU+Δ = KU + Δ (3) Afer some algebrac manpulaons, = U + +Δ AU LF (3) MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8

11 Hence, assume L F = for sably analyss. A s known as he convergence marx, Then, U =AU, U =AU= A U (3)... n U =AU = A U n n The numercal soluon s bounded,.e. sable, f he specral radus of he convergence marx s less han one, ( ) = λmax < ρ A (33) Tha s, none of s egenvalues s greaer or equal o un. Numercal Soluon for Nonlnear Sysems The EOMS for a non-lnear mechancal sysem are where F NL = f NL ( UUU,, ) (34) MU+F =F NL ( ) () s a nonlnear funcon of ( ) U,U, and U,.e. he vecors of generalzed dsplacemen, velocy and acceleraon. Specfcaon of Eq. (34) a wo mes, suffcenly close o each oher, gves: MU +F =F NL ( ) () +Δ NL ( ) ( +Δ ) +Δ (35a) (35b) MU +F =F MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8

12 Noe ha n (35a) all nformaon s known. Subracng (b) from (a) gves: Now, le ΔF =F F NL( ) NL( +Δ) NL( Δ) Δ MΔU + ΔF = ΔF NL ( ) () F F F Δ U + Δ U + ΔU U U U NL NL NL (36) (37) where Δ U = ( U U ), Δ U = ( U U ), Δ U = ( U U ) +Δ +Δ +Δ (38) And defne he followng marces, evaluaed a each me sep, F NL F NL F NL K = ; D = ; M = U U U (39) 3 These marces represen lnearzed sffness (K), vscous dampng (D) and nera (M) coeffcens. Subsuon of Eq. (39) no (37) and no Eq. (36) gves: ( ) M+ M ΔU+D ΔU+K ΔU= ΔF (4) The elemens of he sffness marx are, for example, FNL K b ab, =, ab, =,,... n U 3 a MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8

13 The soluon of hs algebrac equaon proceeds n he same form as for he SDOF sysem, see Lecure Noes 6. Thus, he numercal reamen s smlar, excep ha a each me sep, lnearzed sffness, dampng and nera coeffcens need be calculaed. Alhough he recpe s dencal; however wh he apparen nonlneary, he mehod does no guaranee sably for (oo) large me seps. References Bahe, J-K, 98 ( s ed), 7 laes, Fne Elemen Procedures, Prence Hall. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Veerlng, W.T., 986 ( s edon), Numercal Recpes, The Ar of Scenfc Compung, Cambrdge Unversy Press, UK. Oher San Andrés, L, 8, Numercal Inegraon o Fnd Tme Response of SDOF mechancal sysem, MEEN 67, Lecure Noes 6, hp://phn.amu.edu/me67 Pche, R., and P. Nevalanen, 999, Varable Sep Rosenbrock Algorhm for Transen Response of Damped Srucures, Proc. IMechE, Vol. 3, par C, Paper C597. MEEN 67 HD 3 Numercal Inegraon for Tme Response: MDOF sysem L. San Andrés 8 3

14 Numercal Inegraon of MDOF Lnear sysems wh Vscous Dampng - NEWMARK METHOD Orgnal by Dr. Lus San Andres for MEEN 67 class / 8 The equaons of moon are: M d U/d + C du/d + K U = P() () where M,C,K are nxn marces of nera, vscous dampng and sffness coeffcens, and U, V=dU/d, W=d U/d,and are he nx vecors of dsplacemens, velocy and acceleraons, resp, and P() s he nx vecor of generalzed forces. Eq. () s solved numercally wh nal condons, a =, Uo,Vo=dU/d =====================================================================================. Defne elemens of nera, sffness, and dampng marces: M := kg 5 K := N/m C := N.s/m 5 5 n := 3 # of DOF Inpu vecors of nal condons: U F max := := V := N m m/s T mpulse :=.. naural frequences Se samplng me: smalles naural perod T mn =. T mn Δ := Number of me seps: N mes := Pulse load T half Δ = should be less han smalles perod N mes =.4 3 = f :=.5 T mpulse EXAMPLE.6.6. F ():= F max f < T half T half ( T mpulse ) F max T half oherwse f T half < T mpulse f = 6.64 Buld me and force vecors: :=.. N mes ( ) T mpulse Δ =.86 := Δ F mp := F Noe ha mpac mus be well capured

15 Pulse load F mp. 4 5 F( ). 4 5 T mpulse Δ =.86.. Numercal samplng vs. acual funcon Se parameers for Newmark negraon mehod: δ :=.5 ( ) α := δ a := αδ a := δ αδ a := αδ a 3 := α α =.5 a 4 := δ α a 5 := Δ δ α ( ) a 6 := Δ δ a 7 := δδ Form effecve sffness marx: K e := K + a M + a C Trangularze effecve K marx, snce n=3 (small), bes o fnd nverse, (effecve flexbly marx) B e := K e Calculae response for mpac exered a one of DOFs Force vecor g () := F ()

16 numercal procedure Fnd nal Acceleraon (W) from EOM & nclude exernal force a =,.e W := M KU + CV g ( ) nal accel. = ( ) Sep by sep numercal negraon NumIn( g) := u U v V w W for.. N mes ( ) Se nal condons P g P P M a u a v a 3 w + C a u a 4 v + + a 5 w reurn u w v u W T B e P ( ) ( ) a u u a v a 3 w v ( ) a 6 w + + a 7 w U: dsplacemen, V: velocy W: acceleraon ( ) se (new) acceleraon (W), velocy (V), and dsplacemen (U) nal me := fnd numercal response z := NumIn( g) rows() z = 3 cols( z) =.4 3 Exrac me responses obaned from numercal negraon :=.. N mes U := z, U := z, U 3 := z, numercal procedure RESULTS of numercal response: T mpulse =. UNDAMPED NATURAL FREQS. T =.6.6. T max_plo :=. f =

17 =============================== VERY LARGE TIME STEP.4 U. U U U U U3 max( ) = 5.5 Δ = T mpulse Δ =.86 =============================== LARGE TIME STEP.4 U. U U U U U3 las() =.75 max( ) =.75 Δ = T mpulse Δ = 3.7

18 =============================== SMALL TIME STEP.4 U. U U U U U3 max( ) =.75 Δ = T mpulse = Δ