Handou # 6 (MEEN 67) Numercal Inegraon o Fnd Tme Response of SDOF mechancal sysem Sae Space Mehod The EOM for a lnear sysem s M X DX K X F() () X X X X V wh nal condons, a 0 0 ; 0 Defne he followng varables, and wre EOM () as wo frs-order Eqs. o o o Y X; Y X () M Y DY KY F() & Y Y (3) whch can be wren n marx form as Y 0 Y 0 Y M K M D Y M F (4) Or, Y AY b (5) Y X 0 0 wh Y ; ; Y X A M K M D b M F Ths s known as he sae-space formulaon. Eq. (5) s o be negraed numercally wh nal condon vecor Y X V T o o o MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03
If he appled load s NOT a funcon of me, hen an equlbrum sae s defned afer a very long me as Y 0 YE A bo (6) Compuaonal sofware such as Mahcad, Mapple, Mahemaca, Malab, ec has bul-n funcons or commands o perform he numercal negraon of equaons se n he form Y AYb, even when sysem s nonlnear,.e. A=A(Y). A few words abou numercal negraon mehods Typcal numercal negraon mehods nclude a) Euler (smple) mehod b) Fourh and Ffh-Order Runge-Kua negraors, c) Rosenbrock Mehod see references on page, d) Adams Predcor Correcor Mehods e) Average Acceleraon and Wlson-θ (Implc) Mehods In mos mehods, he selecon of an adequae me sep s crucal for numercally sable and accurae resuls. (a)-(b) are favored by he young naes no numercal compung and because of her ready avalably n modern compuaonal sofware. (c) (d) are more modern (mplc) mehods wh auomaed nermedae reszng of he me sep whle performng he negraon. Mehods (e) have long been favored by srucural mechanc analyss when negrang Mulple DOF (lnear) sysems All mehods suffer from defcences when nonlneares are apparen hus forcng exremely small me seps and he ensung MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03
cos wh los of numercal compung (me). (Memory) Sorage appears no o be an ssue anymore. Sae-space mehod for MDOF sysems. Recall he EOMS for a lnear sysem are where MU+DU+KU () =F() (7) U,U, and U are he vecors of generalzed dsplacemen, F s he vecor of velocy and acceleraon, respecvely; and () generalzed (exernal forces) acng on he sysem. M,D,K represen he marces of nera, vscous dampng and sffness coeffcens, respecvely. Defne he followng varables, Y=U ; Y=U and wre EOM (7) as a se of n-frs-order Eqs. () (8) MY +DY +KY =F & Y =Y (9) whch can be wren n marx form as Y 0 I Y 0 = - - + - Y -M K -M DY -M F (0) Or, Y AY b () U 0 I 0 wh Y ; A ; b - - - U -M K -M D M F () The marces are square wh n-rows = n columns, whle he vecors are n- rows. MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 3
and nal condons Y( 0) Uo U o. A s a n x n marx. I s he nxn deny marx, and 0 s a nxn marx full of zeroes. Condons for a good numercal negraor In general a numercal negraon scheme should a) reproduce EOM as me sep 0 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. The numercal negraon reles n represenng me dervaves of a funcon wh an algebrac approxmaon, for example T dx x x lm d 0 x x x x x ~ Eq. above s exac only f 0 (3) Numercal negraon mehods are usually dvded no wo caegores, mplc and explc. Consder he ODE x fx, (4) In an explc numercal scheme, he ODE s represened n erms of known values a a pror me sep,.e. x x f, (5) x, MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 4
whle n an mplc numercal scheme x x f (6) x, Explc numercal schemes are condonally sable. Tha s, hey provde bounded numercal soluons for (very) small me seps. For example, T where Tn and n n K M are he naural perod and naural frequency of he sysem, respecvely. The resrcon on he me sep s oo severe when analyzng sff sysems,.e. hose wh large naural frequences. n cr (7) Implc numercal schemes are uncondonally sable,.e. do no mpose a resrcon on he sze of he me sep. (However, accuracy may be compromsed f oo large me seps are used). ANALOGY beween numercal schemes and a fler A few words of wsdom released n class MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 5
The average acceleraon mehod for numercal negraon of SDOF equaon: M X DX K X F() () Consder a change of hnkng frame by defnng Arhmec ~ Connuum (funcon) X X, F F,, X X F F (7) Wre Eq. () a wo mes, M X DX K X F (8a) a M X DX K X F (8b) a Subrac (b) from (a) o oban: where M X DX KX F (9) X X X FF F, X X X, ec Noe ha known quanes a = are X, X, X. (0) Ths numercal mehod s exremely popular among he srucural dynamcs communy. Is exenson o MDOF sysems wll be shown laer. The oher favore mehod, he Wlson-θ scheme, wll also be gven n laer lecures (MDOF sysems). MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 6
Now, assume he acceleraon s consan whn he me nerval for 0 () X a se as an average value a X X. The velocy and dsplacemen follow from negraon of Eq. (6) whn he me nerval,.e. X X a (a) X X X a (b) Acceleraon velocy X X X a X X X X a + + dsplacemen X X X X a X + MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 7
A he end of he me nerval, he velocy and dsplacemen equal X X X a (3a) X X X X a (3b) And he dfferences n velocy and dsplacemen re X X X X X X X X a X X X X X X a X X X X X 4 X X X 4 (4a) (5b) 4 (6a) from (5b), X X X X and no (4a) X X X 4 X X X X MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 8
X X X (6b) Noe ha n Eqs (6), X, X, depend on he known values obaned a he pror me sep,.e. X, X and he unknown X. Thus, replace X, X no he dfference equaon (9), M X DX KX F 4 M X X X D X X KX F Rearrangng erms leads o where K X F (7) * * * 4 K K D M (8a) * 4 F FM X D M X (8b) are known as pseudo dynamc sffness and dynamc force, respecvely The recpe for he numercal negraon o fnd he sysem me response s MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 9
A me, known varables are X, X (curren sae) X M F DX K X () fnd from EOM: () form pseudo sffness and forcng funcons, K *, F Eqs. (8), (3) Calculae * X K F, and X X X ; (4) X XX, X X X a + (5) Increase me o + and reurn o sep () from The average acceleraon mehod s an mplc mehod,.e. numercally sable and conssen. The dsadvanage s ha requres memory 3 o sore X, X, X, X, F. 3 A non-ssue n he s cenury MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 0
Average acceleraon mehods for numercal negraon of a nonlnear sysem Consder he sysem wh EOM where gx, X 3 o o 3 M X g X, X F( ) (30) s a nonlnear funcon, for example g X, X g k X k X F sgn( X ) As wh he lnear sysem, evaluae Eq. (30) a wo mes (closely spaced): a M X g X, X F (3a) a M X g X, X F (3b) Subrac (b) from (a) o oban: where (3) M X g g F X X X F F F,,,, g g X X g g X X (33) A Taylor seres expanson of he nonlnear funcon a gves g g g g X X O X, X X X X, X 0 X (34) defne local lnearzed sffness and dampng coeffcens as MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03
Hence, K g X ; D g X, X X X, X g g K X D X (35) and he dfference Eq. (3) becomes lnear M X D X K X F (36) Eq. (35) s formally dencal o he one devsed for a lnear sysem. Thus, he numercal reamen s smlar, excep ha a each me sep, lnearzed sffness and dampng coeffcens need be calculaed. The recpe s hus dencal; however wh he apparen nonlneary, he mehod does no guaranee sably for (oo) large me seps. The recpe for he numercal negraon o fnd he sysem me response s A me, known varables are X, X (curren sae) () fnd from EOM: X M F g X, X (a) fnd local (lnearzed) sffness and dampng coeffcens, (K, D ) from eq. (35) (b) form pseudo sffness and forcng funcons, K *, F from MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03
* 4 K K D M ; * 4 F F M X D M X (3) Calculae * X K F, and X X X ; (4) Se X XX, X X X a + (5) Increase me o + and reurn o sep () Ofen regarded as an ar, numercal compung s n acualy a well esablshed branch of mahemacs. References The followng are a mus! 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. Bahe, J-K, 98 ( s ed), 007 or laes, Fne Elemen Procedures, Prence Hall. Pche, R., and P. Nevalanen, 999, Varable Sep Rosenbrock Algorhm for Transen Response of Damped Srucures, Proc. IMechE, Vol. 3, par C, Paper C05097. Rosenbrock, H.H., Some General Implc Processes for he Numercal Soluon of Dfferenal Equaons, MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 3
Appendx A 4 Numercal Inegraon Usng Modfed Euler s Mehod I s ofen dffcul o solve (exacly) a nonlnear dfferenal equaon. Numercal negraon s hen employed o oban resuls (predcons of moon). The Modfed Euler Mehod s one ype of numercal negraon scheme. Solve for q() governed by M qcq KqQ( q, q, ) (A.) wh nal condons q0and q 0a 0. In Eqn. (A.), Qq (, q, ) may conan erms ha are nonlnear n Le V dq d (A.) and wre Eq. (A.) as a sysem of TWO frs order dfferenal equaons,.e. C K Q( q, V, ) V V q, M M M q V (A.3) Defne K C n and (A.4) M K M 4 Ths Appendx s ncluded because mos young engneerng sudens have learned only abou Euler s mehod. Hence, he appendx complemens her educaon.. However, I encourage you o abandon he usage of hs poor mehod. MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 4
as he naural frequency and vscous dampng rao, respecvely. Wh he noed defnons Eqs. (A.3) become dv Q( q, V, ) V F ( q, V, ) nv n q d M dq q F ( q, V, ) V d (A.5) Noe ha F and F are he slopes of he (V vs ) and (q vs ) curves, respecvely. q() V() q+ V q V+ + + Le he numercal approxmaons (arhmec values) be V V,and q q (A.6) ; and s a suably small me sep for numercal negraon. where 0,,... F q+ In Euler s numercal scheme, approxmae he me dervaves (or slopes) as: q V F V+ V V f, f F q, V,, q q f, f F q, V, for =,.. (A.7) MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 5
wh nal condons q0and q 0a 0 0 Eq. (7) offer a recursve relaon o calculae he numercal (arhmec) values of he varables V and q a successve mes ). The regular Euler mehod s frs order wh a runcaon error of order ( ). A modfed Euler mehod (second order accurae) wh error order ( 3 ) follows: (a) Compue prelmnary esmaes of V, q as V V F q, V,, q q F q, V, (A8.a) (b) Use hese prelmnary esmaes o oban mproved slopes as f F q, V,, f F q, V,, (A8.b) (c) Defne average slopes as per f f f f f f (A8.c) (c) and oban new esmaons usng he averaged slopes,.e. MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 6
V V f q q f, (A8.d) (d) Repea seps (a) hrough (d) a each me sep, ; 0,,..., and sarng wh he nal condons q and q a 0 0 0 0 The sze of he me sep s very mporan o oban accurae and numercally sable resuls. If s oo large, hen numercal predcons wll be n error and very lkely show unsable (oscllang) resuls. If s oo small, hen he numercal mehod wll be slow and cosly snce he number of operaons ncreases accordngly. In pracce, a me sep of he order =T n /60, where T n s he naural perod of moon gven as (/ω n ). Euler s mehod s mos mes NOT a good choce o perform he numercal negraon of lnear or nonlnear ODES. Alas, s wdely used by rooke engneers and engneerng sudens because s easy o mplemen. Ofen enough, however, predcons can be wrong and msleadng. I call Euler s mehod a BRUTE FORCE approach, MEEN 67 HD 6 Numercal Inegraon for Tme Response: SDOF sysem L. San Andrés 03 7