Load Dependent Ritz Vector Algorithm and Error Analysis

Transcription

1 Load Depedet Ritz Vector Algorithm ad Error Aalysis Writte by Ed Wilso i 006. he Complete eigealue subspace I the aalysis of structures subected to three base acceleratios there is a requiremet that oe must iclude eough modes to accout for 90 percet of the mass i the three global directios. Howeer, for other types of loadig, such as base displacemet loads ad poit loads, there are o guidelies as to how may modes are to be used i the aalysis. I may cases it has bee ecessary to add static correctio ectors to the trucated modal solutio i order to obtai accurate results. Oe of the reasos for these problems is that umber of eigeectors required to obtai a accurate solutio is a fuctio of the type of loadig that is applied to the structure. Howeer, the maor reaso for the existece of these umerical problems is that all the LDR ectors of the structural system are ot icluded i the aalysis. I order to illustrate the physical sigificace of the complete set of LDR ectors for a structure cosider the usupported beam show i Figure a. he two-dimesioal structure has six displacemet DOF, three rotatios (each with o rotatioal mass) ad three ertical displacemets (each with a ertical lumped mass). he six stiffess ad mass orthogoal eigeectors with frequecy, i radias per secod, ad period, i secods, are show i Figure b to g. he maximum umber of atural eigeectors that are possible is always equal to the umber of displacemet DOF. he static ectors (modes) hae ifiite frequecies; therefore, it is ot possible to use the classical defiitio that the eigealues are equal to if the eigealues are to be umerically

2 ealuated. A ew defiitio of the atural eigealues ad the ew algorithm used to umerically ealuate the complete set of atural eigeectors will be preseted i detail later i the paper. M0.05 M0.0 M (a) Beam Model I.0 E0,000 (b) Rigid Body Mode 0 (c) Rigid Body Mode 0 (d) Dyamic Mode (e) Static Mode (f) Static Mode (g) Static Mode Figure. Rigid-Body, Dyamic ad Static Modes for Simple Beam ote that the rigid-body modes oly hae kietic eergy ad the static modes oly hae strai eergy. Whereas, the free ibratio dyamic modes cotai both kietic ad strai eergy; ad, the sum of kietic ad strai eergy at ay time is a costat. Also, the eigeectors with idetical frequecies are ot uique ectors. Ay liear combiatio of eigeectors, with the same frequecy, will satisfy the orthogoally requiremets.. SRUCURAL EQUILIBRIUM EQUAIOS he static ad dyamic ode-poit equilibrium equatios for ay structural system, with d displacemet degrees-of-freedom (DOF), ca be writte i the followig geeral form: & () M u(t) + Ku(t) R(t) + R D( u, u&, t) Fg( t)

3 At time t the ode acceleratio, elocity, displacemet ad exteral applied load ectors are defied by u &(t), u&( t), u(t) ad R(t), respectiely. he ukow force ectors, R ( u,u& t), are the forces associated with iteral eergy D, dissipatio such as dampig ad oliear forces. I most cases, these forces are selfequilibratig ad do ot cotribute to the global equilibrium of the total structure. he sum of R ad R ca always be represeted by the product F g(t), where F is a d by L D matrix of L liearly idepedet spatial load ectors associated with both liear ad oliear behaior, ad g (t) is a ector of L time fuctios. hese time fuctios are directly specified for liear aalysis, ad are ealuated by iteratio for oliear elemets. For may problems, oliear forces may be restricted to a subset of all DOF, so that required i what follows. L < d, although this is ot he ode-poit lumped mass matrix, M, eed ot hae mass associated with all degrees-offreedom; therefore, it may be sigular ad mathematically positie semi-defiite. Also, exteral loads may be applied to displacemet DOF that do ot hae mass ad produce oly static displacemets. he liear elastic stiffess matrix K may cotai rigid-body displacemets, as is the case for ship ad aerospace structures; therefore, it eed ot be positie-defiite. I order to oercome this potetial sigularity the term ρ Mu(t) may be added to both sides of the equilibrium equatios, where ρ is a arbitrary positie umber. Or, Equatio () ca be writte as M u& (t) + Ku(t) Fg(t)+ ρ Mu(t) R(t) () While K ad M may be sigular, it is assumed here that the effectie-stiffess matrix, K K + ρ M, is osigular. herefore, the effectie-stiffess matrix represets a real structure with the additio of exteral sprigs to all mass DOF; these sprigs hae stiffess proportioal to the mass matrix. he purpose of this paper is to preset a geeral solutio method for the umerical calculatio of displacemet ad member forces. he proposed method ca be used for both static ad dyamic loads ad has the ability to iclude arbitrary dampig ad oliear eergy dissipatio. he deriatio of the ector-geeratio algorithm preseted i this paper is self-

4 cotaied ad oly uses the fudametal laws of physics ad mathematics. ear the ed of the paper, it will be poited out that each step i the solutio algorithm is othig more tha the applicatio of well-kow umerical techiques that hae existed for oer fifty years. It is a extesio of Load Depedet Ritz ectors that hae bee preiously described Wilso, CHAGE OF VARIABLE Equatio () is a exact equilibrium statemet for the structure at all poits i time. he first step i the static or dyamic solutio of this fudametal equilibrium equatio is to itroduce the followig chage of ariable: u (t) ΦY(t) ad u (t) ΦY(t) & & (3) he d by matrix Φ of spatial ectors are calculated ad ormalized to satisfy the followig orthogoality equatios: Φ MΦ (4) Φ KΦ I or, Φ KΦ I ρ (5) he by diagoal matrices are I for the uit matrix ad for the geeralized mass matrix associated with each ector. herefore, Equatio () ca be writte as a set of ucoupled equatios of the followig form: Y & (t) + IY(t) Φ R( t) (6) If equals d, the itroductio of this simple chage of ariables ito Equatio () does ot itroduce ay additioal approximatios. he umber of ozero terms i the diagoal matrix idicates the maximum umber of dyamic ectors ad is equal to the umber of lumped masses i the system (or, mathematically, the rak of the mass matrix). If a ector has zero geeralized mass it idicates that it is a static respose ector. It is ot practical to calculate all d static ad dyamic shape fuctios for a large structure. First, it would require a large amout of computer time ad storage. Secod, a large umber of ectors that are ot excited by the loadig may be calculated. herefore, a trucated set of

5 atural eigeectors will be calculated that will produce a accurate solutio for a optimum umber of LDR ectors. I order to miimize the umber of shape fuctios required to obtai a accurate solutio the static displacemet ectors produced by L liearly idepedet spatial fuctios () F associated with the loadig (t) R will be used to geerate the first set of ectors. he liear () idepedet spatial load fuctios F ca be automatically extracted from R (t) based o the type of exteral global loadig ad the locatio of the oliear elemets... CALCULAIO OF SIFFESS ORHOGOAL VECORS he first step i the calculatio of the orthogoal ectors defied by Equatio (4) ad (5) is to calculate a set of stiffess orthogoal ectors V where each ector satisfies the followig equatio: for m mk mf (7) 0 for m hese stiffess orthogoal displacemet ad load ectors are calculated ad stored i the followig arrays: V F [ 3 ] d [ f f f f f ] 3 d (8) All ectors are geerated i sequece,, After each ector is made stiffess orthogoal ad ormalized it is iserted ito positio. For example, cosider a ew displacemet cadidate ector (produced by the load ector f ) that is ot stiffess orthogoal as defied by Equatio (7). his ector ca be modified to be stiffess orthogoal by coductig the followig umerical operatios: ormalizatio ector by the applicatio of the followig equatios: ˆ ˆ ad fˆ ˆ β f where ˆ β f ; therefor e ˆ fˆ β (9)

6 Remoe from ˆ all preiously calculated stiffess orthogoal ectors by the applicatio of the followig equatios: ~ ˆ ~ α ad ˆ f f α f (0a ad b) Multiplicatio of Equatio (0a) by K yields the followig equatio: K ~ Kˆ α K to () If the ew ector ~ is to be stiffess orthogoal K~ must equal zero. herefore, α Kˆ () After Equatios (0a) ad (0b) are ealuated they must be ormalized by the applicatio of the equatios: ~ β ~ ad f β f where β therefore f ~ ~ f ; (3) It is ow possible to check if the cadidate ector was liearly idepedet of the preiously calculated ectors by checkig if the proposed ew ector umerical roud-off. herefore, is othig more tha If β < tol reect as a ew stiffess orthogoal ector (4) he alue for tol is selected to be approximately 0-7. he first block of cadidate ectors is obtaied by solig the followig set of equatios, () () where the static loads F ad displacemets V are d by L matrices: KV () () () LDL V F (5) ote that the effectie stiffess matrix eed be triagularized, K LDL, oly oce. Additioal blocks of cadidate ectors ca be geerated from the solutio of the followig recursie equatio:

7 ( i) ( i ) K V MV F ( i) (6) If, durig the orthogoality calculatio, a ew displacemet or load ector i the block is idetified as the same (parallel) as a preiously calculated ector it ca be discarded from the block ad the algorithm is cotiued with a reduced block size. If the block size is reduced to zero, prior to the productio of d ectors, it idicates that all of the static ad dyamic ectors, excited by the iitial load patters, hae bee foud..3. MASS ORHOGOAL VECORS After all blocks of the stiffess orthogoal ectors are calculated they ca be made orthogoal to the mass matrix by the itroductio of the followig trasformatio: Φ VZ (7) Substitutio of Equatio (7) ito Equatio (4) produces the followig by eigealue problem: where M Z (8) M V MV. he stiffess ad mass orthogoal ectors are the calculated from Equatio (7). he static modes hae zero periods, or 0. herefore, i order to aoid all potetial umerical problems, it is recommeded that the classical Jacobi rotatio method be used to extract the eigealues ad ectors of this relatiely small eigealue problem. Equatio () ca ow be rewritte as & (9) M u(t) + Ku(t) ρ Mu(t) Fg(t) he trasformatio to modal coordiates produces the followig ucoupled model equatios: Y & (t) + [ I ρ ] Y(t) Φ Fg( t) (0) herefore, a typical modal equatio,, ca be writte as Y & (t) + [ ρ ] Y (t) φ Fg( t) () he umber of static shape fuctios is equal to the umber of zero diagoal terms i the matrix. For the static modes is equal to zero ad the solutio is writte as

8 Y (t) Fg( t) φ () For the dyamic elastic modes the geeralized mass for each mode is ad the classical free-ibratio frequecies (radias per secod) ad the periods of ibratios (secods) ca be calculated from π ρ ad (3) ote that the eigealue always has a fiite umerical alue; howeer, the frequecy ad period ca hae ifiite umerical alues ad caot be umerically calculated directly for all modes. For example, able summarizes the eigealues, frequecies ad periods for the simple beam show i Figure. able. Eigealues for Simple Beam for ρ 0. 0 Mode umbe r Eigealue Frequecy ρ Period π he geeralized stiffess ad mass for the ormalized ectors are as follows: φ Kφ ρ 0 for rigid - body modes for dyamic modes for static modes (5)

9 φ Mφ ρ 0 for rigid - body modes for dyamic modes for static modes (6) herefore, it is ecessary to sae both the geeralized stiffess ad geeralized mass, for each mode, i order to determie the static, dyamic or rigid-body respose aalysis of the mode. he solutio for the dyamic modes ca be obtaied usig the piece-wise exact algorithm []. For all rigid-body modes will equal by direct, umerical or exact, itegratio from / ρ. herefore, their respose ca be calculated Y& (t) φ R( t), Y& (t) Y& ( t) dt, ad Y ( t) Y& ( t) dt ρ (4) he sum of the static, dyamic ad rigid-body resposes produces a uified method for the static ad dyamic aalysis of all types of structural systems..4. MAHEMAICAL COSIDERAIOS Except for referece to the Jacobi ad piece-wise itegratio methods Wilso, 003, the umerical method for the geeratio of stiffess ad mass orthogoal ectors preseted is based o the fudametals of mechaics ad requires o additioal refereces to completely uderstad. Howeer, it is ery iterestig to ote that the method is othig more tha the applicatio of seeral well-kow umerical techiques: First, the chage of ariables itroduced by Equatio (3) is a applicatio of the stadard method of solig differetial equatio ad is also kow as the separatio of ariables i which the solutio the solutio is expressed i terms of the product of space fuctios ad time fuctios. A special applicatio of this approach i classical structural dyamics is called the mode superpositio method i which the static mode respose is eglected. Secod, the additio of the term ρ Mu(t) to the stiffess matrix is called a eigealue shift i mathematics. Howeer, it is worth otig the zero eigealues associated with the static modes are ot shifted.

10 hird, the recurrece relatioship, Equatio (0), is idetical to the ierse iteratio algorithm for a sigle ector. herefore, the approach is a power method that will always coerge to the lowest eigealues of the system. Fourth, the series of ectors geerated by the ierse iteratio method is kow as the Krylo Subspace. A.. Krylo, , was a well-kow Russia egieer ad mathematicia who first studied the dyamic respose of ship structures. Howeer, Krylo did ot iclude static modes i his work. Fifth, orthogoality is maitaied, Equatios (4), by the applicatio of the modified Gram- Schmidt algorithm. heoretically, after the iitial block of orthogoal ectors are calculated, it is oly ecessary to make each ew displacemet ector orthogoal with respect to the preious two Krylo ectors. Howeer, after may years of experiece with the dyamic aalysis of ery large structural systems, we hae foud that it is ecessary to apply the Gram-Schmidt method to all preiously calculated ectors i order that the same ectors are ot regeerated. (i) Sixth, the performace of the algorithm is improed if the load ectors F, for each block, are made orthogoal with respect to the preiously calculated displacemet ectors, Equatio (6), prior to the solutio of the equilibrium equatios. his additioal step has made the algorithm uique ad ery robust..5. LOAD PARICIPAIO RAIOS AD ERROR ESIMAIO I the aalysis of structures subected to three base acceleratios there is a requiremet that oe must iclude eough modes to accout for 90 percet of the mass i the three global directios. Howeer, for other types of loadig, such as base displacemet loads, there are o guidelies as to how may modes are to be used i the aalysis. he purpose of this sectio is to defie two ew load participatio ratios, which ca be calculated durig the geeratio of the LDR ectors, to assure that a adequate umber of ectors are used i a subsequet static or dyamic aalysis. From Equatio (4), a typical modal equatio for load patter, ca be writte as Y ( t) + Y(t) φ F g( t) to & (7)

11 he error idicators are based o the two differet types of load fuctios g(t). I oe case the loads s. time excite the low frequecies; ad, i the other case the high frequecies are excited..5.. Static Loads he first error estimator is a measure of the ability of a trucated set of mode shapes to capture the static respose of the structural system. For this case the load fuctio g(t) is applied liearly from a alue of zero at time zero to a alue of.0 at the ed of a ery large time iteral. herefore, the iertia terms ca be eglected ad Equatio (7), ealuated at the ed of the large time iteral, is Y φ F to (8) s herefore the static mode participatio ca be writte as Y φ F to (9) From Equatio (3) the approximate static displacemet respose of the structure due to modes is u φ Y (30) he approximate strai eergy associated with the displacemet defied by Equatio (30) is E s u ( φ F K u ) Y φ Kφ Y φ Y (3) he exact static displacemet due to the load patter ca be calculated from the solutio of the followig static equilibrium equatio: K u F (3) he exact strai eergy stored i the structure for the load patter is calculated from

12 E s u K u u F (33) he static load participatio ratio is defied as the ratio of the strai eergy captured by the trucated set of ectors, ratio is E, to the total strai eergy, E. For the typical case where ρ 0 the r s (φ F ) u F (34) It must be poited out that for LDR ectors, this ratio is always equal to.0. Whereas, the use of the exact dyamic eigeectors may require a large umber of ectors i order to capture the static load respose. Also, if the static mode shapes are excited it is ot possible for the exact dyamic eigeectors to coerge to the exact static solutio..5.. Dyamic Respose he dyamic load participatio ratio is based o the use of the applicatio of the static loads as a delta fuctio at time zero that produces a iitial coditio for a free ibratio respose aalysis of the total structural system. It is well kow that ay type of time fuctio ca be represeted by the sum of these impulse fuctios applied at differet poits i time. his type of loadig will produce a iitial elocity at the mass poits of kietic iput to the system, for a typical load ector, is gie by u& M F. herefore, the total Ek u Mu f M & & F (35) From Equatio (3) the relatioship betwee iitial ode elocities ad the iitial modal elocities is φy & u & (36) herefore, the kietic eergy associated with the trucated set of ectors is E k u & Mu& Y & (37)

13 he iitial modal elocity Y & is obtaied from the solutio of Equatio (7) as ϕ F Y& (38) Substitutio of Equatio (38) ito Equatio (37) yields E k ( ϕ F ) (39) he dyamic load participatio ratio is defied as the ratio of the kietic eergy captured by the trucated set of ectors, E k, to the total kietic eergy, E k. For the typical case where ρ 0 the ratio is r d F ( φ F ) M F (40) A dyamic load participatio ratio equal to.0 assures that all the eergy iput is captured for the dyamic load coditio. I the case of base acceleratio loadig where the three load ectors are the directioal masses, the dyamic load participatio ratios are idetical to the mass participatio ratios Automatic ermiatio of LDR Vectors Sice the LDR ector algorithm starts with a full set of static ectors the static load participatio factor will always equal.0. Equatio (40), the dyamic load participatio factor ca be ealuated after each block of ectors is geerated. herefore, this factor ca be computed as the ectors are calculated ad it ca be used as a idicator to automatically termiate the geeratio of LDR ectors. Based o experiece, a dyamic load participatio ratio of at least 0.95, for all load patters, will assure accurate results for most types of loadig. his is a ery importat user optio sice the umber of ectors requested eed ot be specified prior to the dyamic aalysis.

14 .6. USE OF HE LDR ALGORIHM O CALCULAE EIGEVECORS he LDR ector algorithm, as preseted i this paper, geerates the complete Krylo subspace for a specified set of load ectors ad errors i the resultig dyamic respose aalysis are miimized. If oe examies the frequecies associated with the LDR ectors it is foud that all the lower frequecies are idetical to the frequecies obtaied from a exact eigealue aalysis. Sice the approach is related to the power method this is to be expected. he higher modes produced by the LDR ector algorithm are liear combiatios of the exact eigeectors ad compoets of the static respose ectors. he complete set of LDR ectors is the optimum set of ectors to sole the dyamic respose problem associated with the specified static load patters. herefore, the umber of LDR ectors required will always be less tha if the exact eigeectors were used. If, for some reaso, oe wishes to calculate the exact eigealues ad ectors the same umerical method ca be used. he iitial displacemet ectors eed oly be set to radom ectors. If, durig the geeratio, ectors are geerated which are idetical to preiously calculated ectors they ca be replaced with ew radom displacemet ectors. he procedure ca be termiated at ay time; howeer, the higher frequecies will ot be exact. he itroductio of iteratio for each block ca be used to calculate the exact eigealues ad ectors. ote that if the system cotais M masses, the method will geerate M exact ( i) ( i ) eigeectors; eertheless, if radom load ectors are used directly, istead of F MV, the algorithm ca cotiue ad will produce d -M static respose ectors which hae ifiite frequecies ad zero periods..7. SUMMARY OF HE COMPLEE LDR VECOR ALGORIHM he use of exact eigeectors to reduce the umber of degrees of freedom required to coduct a dyamic respose aalysis has sigificat limitatios. he effects of the applicatio of static loads to massless DOF caot be take ito accout. I additio, for certai types of loadig a large umber of ectors are required. O the other had, a large umber of exact eigeectors may be calculated that are ot excited by the loadig o the structure.

15 he use of static ad dyamic LDR ectors, preseted i this paper, elimiates the problems associated with the use of the exact eigeectors. I additio, the LDR ector algorithm produces a uified approach to the static ad dyamic aalysis of may differet types of structural systems. I additio, it is possible to check if a adequate umber of ectors are geerated prior to the itegratio of the equatios of motio.