ADAMS Theory in a Nutshell for Class ME543 Department of Mechanical Engineering The University of Michigan Ann Arbor March 22, 2001

Transcription

1 ADAMS heory n a Nutshell for Class ME543 Department of Mechancal Engneerng he Unversty of Mchgan Ann Arbor March, Dan Negrut dnegr@adams.com Brett Harrs bharrs@adams.com

2 Defntons, Notatons, Conventons. Generalzed Coordnates used n ADAMS In ADAMS, the poston of a rgd body s defned by 3 Cartesan coordnates x, y, and z. x p = y () z he orentaton of a rgd body s defned by a set of 3 Euler angles that correspond to the 3--3 seuence rotaton: ψ, θ, and φ, respectvely. hese 3 angles are stored n an array (not vector) n the followng form: ψ e = φ () θ he set of generalzed coordnates assocated wth rgd body n ADAMS s denoted n what follows by p = e (3) Based on ths choce of generalzed coordnates, the body longtudnal and angular velocty are obtaned as u= p& (4) where w= Be& Bz (5) snφsnθ cosφ B = cosφsnθ snφ (6) cosθ

3 and w s the body angular velocty expressed n body local coordnate system. Euaton (6) s mportant as t defnes the relatonshp between the angular velocty of the body;.e., an ntrnsc characterstc of the body, and our choce of generalzed coordnates. Fnally, note the relatonshp between the tme dervatve of the body orentaton matrx A and angular velocty w : A& = Aw% (7) where a ~ represents the skew-symmetrc operator. For an entre mechancal system models contanng nb bodes, the array nb [ ] = K = K n (8) wth n= 6 nb wll descrbe at a gven tme the poston and orentaton of each body n the system.. Jonts n ADAMS Jonts n ADAMS are regarded as constrants that act among some of the coordnates through n of E.(8). From a mathematcal perspectve, such a constrant assumes the expresson Φ ( ) = (9) For example, a revolute jont actng between bodes would nduce a set of 5 constrants lke the one n E.(9) to allow one degree of freedom between the two bodes connected by ths jont. he collecton of all constrants nduced by the jonts present n the model s denoted by F : ( ) ( ) ( ) ( ) = Φ ( ) Φ ( ) Φ ( ) F = F F K Fnj K m () where nj s the number of jonts n the model, and m s the sum of the number of constrants nduced by each jont. Note that n R, whle m F R. ypcally, m< n ;.e., the number of generalzed coordnates s larger than the number of constrants they must satsfy. 3

4 By takng one tme dervatve of the poston knematc constrant euatons of E.(), the velocty knematc constrant euatons are obtaned as F& = () By takng yet another tme dervatve of E.(), the acceleraton knematc constrant euatons are obtaned as ( ) F & = F && t () Euatons () through () can be seen as condtons that the generalzed coordnates array along wth ts frst and second tme dervatves must satsfy. hs s to ensure that the evoluton of the mechancal system makes sense;.e., the mechansm s assembled, and the parts move such that the constrants mposed by jonts are obeyed at every tme. 3. Motons n ADAMS From a mathematcal perspectve, motons ndcate that a generalzed coordnate of the system, or an expresson dependng on generalzed coordnates explctly depends on tme. As an example, consder a smple pendulum connected to ground through a revolute jont. A moton mght mpose that the angle assocated wth ts unue degree of freedom wll change n tme lke α sn( π t) =. Generally, a moton s represented as a tme dependent constrant euaton: (,t) Φ = (3) Revstng the defnton of the poston, velocty, and acceleraton knematc constrant euatons, for constrant euatons nduced by ether jonts or motons n the most general case the followng euatons must satsfed at any tme t. F ( ) F,t = (4) (, t) = F (, t) & (5) t (, t) = ( ) (, t) F & F&& F& F (6) t tt 4

5 Euatons (5) and (6) are obtaned by takng one and respectvely two tme dervatves of the poston knematc constrant euaton of E.(4). Fnally, a set of generalzed coordnates s sad to be consstent, f t satsfes the poston knematc constrant euatons. Lkewse, a set of generalzed veloctes s consdered consstent provded for a consstent poston confguraton, & satsfes the velocty knematc constrant euatons of E.(5). Fnally, n ths document a vector uantty marked wth an over-bar ndcates that the vector s expressed n a local body reference frame. Note n ths context the over-bar for the angular velocty n E.(5). 5

6 INIIAL CONDIION ANALYSIS Intal Condton (IC) Analyss s concerned wth determnng a consstent confguraton of the mechancal system model at the begnnng of the smulaton at tme t. Durng IC analyss, the mechansm must be assembled and the veloctes of the parts n the mechansm must be consstent. o be assembled, the generalzed coordnates must satsfy all constrant euatons;. e., ( ) F,t = (7) whle for the generalzed veloctes to be consstent, they must satsfy the velocty knematc constrant euaton F (, t) = & Ft (, t) (8). Poston Intal Condton Analyss Durng IC analyss, the user mght want sometmes for certan reasons to fx the value of some of the generalzed coordnates,, etc., of the generalzed coordnate array n E.(8). In other words, f the user prescrbes the poston of body 3 n the system to be 3 =.9, 4 =., 6 =, the solver should assemble the mechansm and n the same tme do ts best to satsfy the prescrbed condtons. hs IC analyss s solved n ADAMS va an optmzaton approach. he constraned optmzaton problem solved mnmzes the cost functon f(, K, n) = w( ) + K + wn( n n ) (9) subject to the constrant euatons ( ) factors, whle F,t =. In E.(9), the values w are weght can be regarded as generalzed coordnates nducng an ntal confguraton of the system. Notce that ths confguraton = K need n not be consstent. he user may prescrbe that some of the entres n the array;.e.,,, etc., are to be regarded "exact". As t wll be justfed shortly, the correspondng weghts w, 6

7 w, etc., wll be gven large values;.e., values lke the ones correspondng to generalzed coordnates. he remanng weghts, namely that the user dd not specfy are gven values of. Wth ths, the constraned optmzaton problem wll manage n ts soluton seuence to keep the user mposed IC values almost unchanged, whle focusng on adjustng the "free-to-change" generalzed coordnates to fnd the soluton of the problem. Notce that the soluton of the problem means mnmzng the cost functon whle satsfyng the constrants. In matrx notaton, the constraned optmzaton problem reads for n R, mnmze subject to f ( ) = ( ) W ( ) () ( ) In E.(), W s a dagonal matrx of weghts, F,t = () ( ) W = dag w, w, K, wn () whle the constrants of E.() that must be satsfed n the optmzaton problem are exactly the poston knematc constrants of E.(4). ADAMS approxmates the non-convex optmzaton problem of Es.() and () by a successon of convex problem that are guaranteed to have a soluton, whch can be found n one teraton. In ths context, the set of non-lnear constrant euatons of E.() s lnearzed n the vcnty of (, t ) ( ) ( )( =, t +, t ) F F F (3) Euaton (3) replaces E.() and wth the notaton d the now convex optmzaton problem reads Mnmze f ( d) = dwd (4) Subject to (, t ) (, t) the Lagrangan F + F d= (5) o solve the convex constraned optmzaton problem of Es.(4) and (5) defne 7

8 F ( ) (, ) = f ( ) + ( ) + ( ) d l d l F F d (6) he optmalty condtons for ths problem are F = d F = l whch lead to the followng lnear system of euatons: W F ( ) d ( ) l F( ) = F (7) (8) Based on E.(8), ADAMS computes the value of d, and gven soluton of the convex optmzaton problem as computes the = + d (9) he confguraton nduced by the new set of generalzed coordnates obtaned as n E.(9) corresponds to the lnearzed problem of Es.(4) and (5). herefore, whle the soluton satsfes the condtons of E.(5), t mght be that t does not satsfy the orgnal non-lnear system constrant euatons nduced by the system s jonts as n E.(). If ths s the case, then the confguraton just obtaned s set to be the new another teraton startng wth the lnearzaton of E.(3) s carred out. ypcally, after a couple of teratons the soluton of the lnear convex, and optmzaton problem wll satsfy the non-lnear constrant euatons nduced by the jonts of the mechancal system. he approach fals when the only assumpton made by the numercal method nvolved ceases to hold;.e., when the lnearzaton n E.(3) s far from approxmatng the non-lnear manfold nduced by the orgnal constrant euatons. It s worth notng here that even when the manfold s hghly non-lnear, the soluton seuence wll converge provded the startng pont s close enough to the fnal soluton. hs s the reason for whch t s essental for the algorthm to have a good startng pont. 8

9 Fnally, a word on how the weghts w of E.(9) factor n to keep the user defned ntal condtons to ther prescrbed value. o better see how these weghts kck n, consder a case wth only one constrant that needs to be satsfed. Lkewse, assume that the system has generalzed coordnates x and y, and that the constrant euaton they must satsfy s Φ (, xy) = x y =. Obvously ths constrant euaton s satsfed by an nfnte number of pars lke (,4), (.5, 6.5), etc., but n ths smple case the user would lke to specfy that the value of x s x =, whle the value y s free to change. As the value for y s not prescrbed, assume our ntal guess takes y = 6. Snce the user prescrbed a value for x, the weght assocated wth ths generalzed coordnates s large;.e., w =. On the other hand, snce there s no condton mposed on the second generalzed coordnate, w =. Notce that prescrbed n the dscusson above refers to coordnates specfed to be exact n the ADAMS modelng language (the adm fles). Wth ths, the matrx of E.(8) assumes the form d d = λ 5 hen d 9 =, 5 d =, λ = 5, and x y IC IC = + 9 = 6 5= 9 8 As t can be easly verfed, ( x, y ) Φ = +, that s, the non-lnear constrant IC IC euaton s very well satsfed, and the correcton appled n the user prescrbed ntal condton x s of the order 9 whch for most practcal purposes s neglgble. hus, the weghts w s the means through whch the algorthm s determned to be based n fndng a soluton by changng a certan subset of generalzed coordnates rather than another set whose values were prescrbed by the user. On a fnal note, t s worth notng that durng poston IC analyss ADAMS checks for the santy of the constrant euatons nduced by the jonts and motons present n the 9

10 model. Durng redundant constrant analyss some of the constrant euatons mght turn out to be redundant. In a bengn stuaton a redundant constrant s consstent. A uck example of such a case s when ADAMS s presented wth one constrant euaton that looks lke x y = and then wth a dfferent constrant euaton that reads lke x y = he latter clearly does not add anythng to the pcture, snce when the frst euaton s satsfed the second one s automatcally satsfed too. hus, the second euaton s redundant, but t s consstent and throughout the smulaton ADAMS wll montor ths euaton to make sure that the redundant constrant contnues to be consstent. he malgn case s when the second euaton would read lke x y = If ths s the case, the two constrant euatons can not be smultaneously satsfed no matter what values the generalzed coordnates x and y assume. hs s the case of ncompatble redundant constrant euatons. ADAMS wll do a LU factorzaton wth full pvotng and nform the user about encounterng ths stuaton. ADAMS wll stop smulaton upon fndng ncompatble redundant constrants because from a modelng perspectve there s somethng ualtatvely wrong wth the system beng smulated. Note that redundant constrants are mostly encountered when too many jonts are used to model the mechancal system and therefore the number of constrant euatons nduced by these jonts exceeds the number of generalzed coordnates of the model. In what follows, two or more constrant euatons wll be called ndependent f they are not redundant.. Velocty Intal Condton Analyss he velocty IC analyss s a drect and smple applcaton of the algorthm employed for the poston IC analyss. It s drect because t s appled exactly as presented before, and t s smple because the constrant euatons that need to be satsfed are already n a lnear form. herefore, there s no need to lnearze them as was

11 the case wth the poston constrant euatons (see E.(3)) and the soluton s guaranteed to be found n one teraton. hus, the convex constraned optmzaton problem solved to retreve the ntal veloctes & mnmzes the cost functon f( &&, K, n ) = ( && ) W( && ) (3) subject to the always lnear velocty knematc constrant euatons of E.(5): F (, t ) F (, t ) & (3) + t = As for IC analyss, the weght dagonal matrx W has some very large postve entres that ensure ths tme around that the correspondng user prescrbed ntal veloctes are not changed by the optmzaton algorthm. From here on, the same procedure prevously used for poston IC analyss s employed. Note that because of the lnearty of the velocty knematc constrant euatons the algorthm s guaranteed to converge n one teraton. 3. Force and Acceleraton Intal Condton Analyss In the absence of frcton forces, the acceleraton IC analyss reures the soluton of the lnear system assembled from the euatons of moton (EOM) and the acceleraton knematc constrant euatons of E.(). he resultng system has the form ( ) M F & F = F ( ) l t Note that ths s a lnear system, and the teratve process nvolved typcally converges n one teraton. he user does not drectly prescrbe any ntal acceleraton for the force/acceleraton IC analyss. he reacton force and ntal acceleratons are evaluated based on the computed ntal poston, ntal velocty, and the appled force actng on the system at the ntal tme. For a more detaled explanaton of how the EOM are obtaned (the frst row n the E.(3)), see the Secton on Dynamc Analyss n ADAMS. Besdes &, the soluton of the lnear system above also provdes the Lagrange multplers l. he constrant reacton force and torue nduced by jont j that connects body to ground or other part n the system are computed as (3)

12 F C F& = v ( j) l ( j) (33) C F& = w ( j) l In Es.(33) and (34) the superscrpt C ndcates that the uanttes are expressed n a Cartesan coordnate system; v s the Cartesan velocty of body ; angular velocty; ( j) (34) w s the global ( ) j F represent the set of constrant euatons nduced by jont j.

13 KINEMAIC ANALYSIS ypcally, for a knematc analyss to be carred out a number of ndependent constrant euatons eual to the number of generalzed coordnates n the model must be prescrbed. For the mechansm to actually change ts confguraton n tme, some of these constrants must be motons;.e., they should depend on tme.. Poston Level Knematc Analyss Gven the poston of the system at tme t, the problem here s to determne the poston at tme t > t. Because of the non-lnear nature of the constrant euatons n E.(4), a Newton-Raphson teratve method s used n ADAMS to compute at tme t. o understand how ths method works and what ts lmtatons are frst note that t s obtaned from a aylor expanson based lnearzaton of the non-lnear constrant euatons: (, t ) = (, t ) + (, t )( ) F F F (35) Snce the number of constrants s eual to the number of generalzed coordnates, frst note that the matrx (,t ) F s suare. As the constrant euatons were assumed to be ndependent ths matrx s also nvertble. Based on an explct ntegrator an ntal startng confguraton ( ) s determned, and the teratve algorthm proceeds at each teraton j by fndng the correcton hen, ( j + ) ( j) ( j) ( j) D ( t ) ( ) ( j ) ( j t = ) F, D F, (36) = + D, wth the teratve process beng stopped when the correcton ( j) D and/or the resdual ( j) (, t) F become small enough. As for the poston IC analyss, ADAMS can fal to fnd f whle wth an ntal guess far enough from the consstent soluton, the lnearzaton turns out to be a poor approxmaton of the non-lnear manfold. In these stuatons, the remedy les n 3

14 decreasng the smulaton step-sze, and thus causng to le closer on the manfold to the last consstent confguraton.. Velocty Level Knematc Analyss Velocty knematc analyss s straghtforward as the velocty knematc constrant euatons are lnear n velocty. Wth already avalable after the poston knematc analyss, the non-sngular matrx (,t ) s solved for the new velocty. F s evaluated and the lnear system of E.(5) 3. Acceleraton Level Knematc Analyss Acceleraton knematc analyss s mmedate, as at tme t t s found as the soluton of the lnear system of E.(6). Notce that the same matrx that s factored for velocty knematc analyss s used for a forward/backward substtuton seuence to solve for the generalzed acceleratons &. Once & s avalable, the Lagrange multplers assocated wth the set of constrants actng on the system are computed as the soluton of the lnear system Fl= & (37) F M hs euaton s dentcal to the frst row of the lnear system of E.(3) and n fact represents precsely the euatons of moton. More nformaton on how E.(37) s obtaned s provded n the next Secton. 4

15 DYNAMIC ANALYSIS. Nomenclature, Conventons, Defntons. In addton to the defntons and notatons ntroduced at the begnnng of ths document, the followng uanttes wll be used n formulatng the rgd body euatons of moton. M - generalzed mass matrx J - generalzed nerta matrx expressed about the prncpal local reference frame K - knetc energy, defned as K = umu+ w Jw (38) m l R - array of Lagrange multplers. he number m of Lagrange multplers s gven by the number of constrant euatons nduced by jonts connectng a body to other bodes n the system. (,&,t) f n 6 F = R - the vector of appled forces 6 (,,t) R Q& - the generalzed force actng on the body. Obtaned by projectng the appled force F upon the generalzed coordnates. ypcally, Q P ( P ) R ( P ) f = n (39) where wth P v beng the velocty of the pont of applcaton P of the external force F, the projecton operators are computed lke P P v P = (4) u R w P = (4) z 5

16 . Formulaton of Euaton Of Moton (EOM) n ADAMS Provded n any Dynamcs book, the Lagrange formulaton of the euatons of moton leads to the followng second order dfferental euatons d K K + dt & Fl= Q (4) Consderng the choce of generalzed coordnates n ADAMS;.e., the defnton of as n E.(3), E.(4) s rewrtten for a rgd body as K K u p Fl f p + = Fl e P ( P ) R ( P ) d dt K K n z e It s worth pontng out that when dealng wth a full system of rgd bodes connected through jonts, the system Euaton Of Motons (EOM) are obtaned by smply stackng together the EOM for the bodes n the system. Snce (43) d K = Mu& (44) dt u K p wth the angular momenta defned as = (45) G K = z BJB the EOM of E.(43) are reformulated n ADAMS as z (46) p P ( ) Mu& + Fl= P K G& + Fl= e P e R ( ) he frst order dfferental euatons above are called n what follows knetc dfferental euatons, and they ndcate how external forces determne the tme varaton of the translatonal and angular momenta. f n (47) 6

17 Fnally, the tme varaton of the generalzed coordnates s related to the translatonal and angular momenta by means of the knematc dfferental euatons. By assemblng the knetc and knematc dfferental euatons ADAMS generates a set of 5 euatons that provde the nformaton necessary to fnd a numercal soluton for the dynamc analyss of a mechancal system. hese euatons are as follows: ( P ) Mu& Fl- P f (48) + p = G BJBz= (49) K G& e R ( ) + Fle P = n (5) p& u= (5) &e-z= (5) 3. Numercal Soluton for Dynamc Analyss. Jacoban Informaton Computaton. Euatons (48) through (5) ndcate how relevant varables change n tme. What s mssng n ths pcture s the fact that the soluton of ths system of dfferental euatons must also satsfy the knematc constrant euatons of Es.(4) through (6). From a numercal standpont, ths s what makes the dynamc analyss of a mechancal system the most dffcult type of smulaton. here s a multtude of methods for solvng the assembly dfferental+constrant euatons. hs document s not concerned wth these methods and t only provdes a glmpse at what ADAMS does to address ths problem. In ths context t s worth mentonng that ths assembly: dfferental and constrant euatons forms what s called a set of Dfferental-Algebrac Euatons (DAE). A DAE has an ndex assocated wth t, and rule goes that the hgher the ndex, the more challengng the numercal soluton of the DAE becomes. In partcular, the DAE nduced by the dynamc analyss problem n mechancal system smulaton has ndex 3, whch s consdered hgh. In the ADAMS Fortran solver there are two more relable methods for soluton. he most common one s a drect ndex 3 DAE solver, n whch assocated to the dfferental euatons nduced by (48) through (5) are the poston knematc constrant 7

18 euatons of E.(4). In ths approach the velocty and acceleraton level knematc constrant euatons are perodcally enforced. hs s how the solver GSIFF n ADAMS works. A second, more refned algorthm reduces the orgnal ndex 3 problem to an analytcally but yet numercally dfferent ndex DAE problem. hus, nstead of consderng the poston, the velocty level knematc constrant euatons of E.(5) are solved for along wth the knematc dfferental euatons. In the Fortran solver, ths algorthm s called SI, and whle typcally slower than the ndex 3 approach t turns out to be more accurate and robust. In what follows the ndex 3 approach s presented n a reasonable amount of detal. In an attempt to keep the presentaton smple, the ndex 3 DAE wll be ntegrated va an order mplct ntegraton formula. hs formula s the backward Euler formula - an one step, A stable algorthm that ualtatvely captures all the relevant detals characterstc to hgher order methods. Backward Euler ntegraton formula replaces the dervatve y& at tme t wth y& = y y (53) h h Based on E.(53), an Intal Value Problem (IVP) y& = g( y,t), y( ) fndng ( ) t system t = y s solved by y at tme t > t as the soluton y of the dscretzaton algebrac non-lnear y y g( t, y) = (54) h h he system of euatons n E.(54) s called a dscretzaton system snce the dervatve n the orgnal IVP problem was dscretzed usng the ntegraton formula of E.(53). Snce almost always the functon g s non-lnear, a non-lnear algebrac system needs to be solved to retreve y. hs s done n ADAMS by usng a Newton-Raphson type teratve algorthm. Based on the mplct Euler dscretzaton formula ntroduced above, all the frst order tme dervatves that appear n the euatons of moton n Es.(48) through (5) are dscretzed to produce a set of algebrac non-lnear euatons. In the ndex 3 approach, 8

19 the poston knematc constrant euatons are appended to these euatons, along wth the force functon defnton F and. hs appendng of the force functons s done wth the sole purpose of ncreasng the number of unknowns and thus nducng a larger but yet sparser Jacoban matrx. hus, after the mplct Euler based dscretzaton Es.(4), and (48) through (5) along wth the force/torue defnton euatons assume the followng form: P Mu Mu + Flp ( A ) f = h h G- BJBz= K + e = R G- G Fl- ( A ) n h h e p p u= h h e e z = h h p = F (, e, t ) (, z,, e,,, t ) (, z,, e,,, t ) f F u p fn = n u p fn = (55) he unknowns n ths non-lnear system are u, G, z, p, e, l, f, n. he subscrpt ndcatng the tme step was dropped for convenence. Introducng the array u G z = p y e l f n the non-lnear system of E.(55) s rewrtten as (56) Y ( y) = (57) 9

20 A Newton-Raphson type algorthm fnds the soluton of ths system. hus, frst a predcton ( ) y of the soluton s provded, typcally by usng a predctor constructed around an explct ntegrator. Once an ntal guess of the soluton s provded, teratons are carred out untl the resdual ( ) ( ) ( j ) ( j = ) Y D Y y y y y ( j+ ) ( j) ( j) = y + D ( j) ( y ) Y and/or the correcton (58) ( j) D are small enough. he last pece n the puzzle s what turns out to be also the most costly one;.e., the computaton of the Jacoban Y ( ) y y ntroduced n E.(56), the expresson of the Jacoban Y ( ) Appendx., whch s obtaned from E.(55). Wth the notaton y y s provded n the he same remarks made n conjuncton wth the teratve Newton-Raphson algorthm used for IC analyss and for Knematc analyss are applcable here. hus, f the ntal guess;.e., the predcted value of ( ) y s too far away from the soluton, the teratve process mght fal to converge. hs s more lkely to happen wth dynamc analyss than wth other types of analyss, as t s clear that the system that needs to be solved at each ntegraton step s hghly non-lnear. If the teratve process fals, the ntegraton step-sze s decreased and another step s attempted. ADAMS users are famlar n ths context wth messages nformng them that the step-sze was decreased too much, and yet the convergence was not attaned. Gettng such a message s a bad omen, as typcally the user wll have to revst the model, make modfcatons n the smulaton defnng parameters, or to try a dfferent ntegrator lke SI for example. Whle talkng about the ntegraton Jacoban, t s the rght place to menton yet another pece of nformaton that the solver supples to the user, namely the need for refactorzaton. As can be seen n the Appendx, the Jacoban has a tme-nvarant sparsty pattern. When solvng for the correctons ( j) D the Jacoban needs to be factored. he pvots n the LU factorzaton are chosen at the begnnng of the smulaton and they are re-used upon a new call for the soluton of ths system. Especally for long smulatons, t turns out that after some tme the ntal pvot seuence results n a sngular

21 matrx. hs reures a refactorzaton and possbly a decrease of the ntegraton stepsze. he user s flagged when the solver runs nto such a scenaro. Fnally, although the dscretzaton formula used to convey the dynamc analyss soluton message was backward Euler t conceptually captures the essence of the ADAMS soluton seuence. ADAMS typcally uses hgher order ntegrators whenever the sgnals sent over by the problem beng solved suggest that ths would mprove performance. he expresson of the ntegraton Jacoban s ualtatvely the same, wth very mnor and nsgnfcant changes for example the denomnator of the fracton h would become ( hβ ), where β s an ntegraton formula specfc coeffcent. What s mportant to remember here s that the use of a more sophstcated ntegraton formula serves n the end the same purpose, namely to replace a frst order tme dervatve wth a lnear combnaton of future and past values of the unknown, whch s what backward Euler formula does n a very basc way through E.(54).

22 SAICS AND QUASI-SAICS ANALYSIS Statcs and Quas-Statcs analyss n ADAMS s merely a smplfed case of Dynamc Analyss. hs s because what the solver currently does s to set the coeffcent h (or ( hβ ) for more complex mult-step methods) to zero. hs effectvely mples that the value of the tme dervatve n E.(53) becomes zero. hs s how eulbrum s perceved there s no change n tme of any of the unknowns, and therefore ther tme dervatves are all zero. As a parenthess here, n ADAMS f they wsh the users can set the coeffcent above to some very small but yet non-zero value. hs s to allow the algorthm to handle neutral eulbrum confguratons wthout runnng nto sngular Jacoban matrces.