Finite element analysis of nonlinear structures with Newmark method

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1 Iteratioal Joural of the Physical Scieces Vol. 6(6), 95-40, 8 March, 0 Available olie at ISSN Acadeic Jourals Full Legth Research Paper Fiite eleet aalysis of oliear structures with Newark ethod H. Hashadar*, Z. Ibrahi ad M. Jaeel Departet of Civil Egieerig, Uiversity of Malaya, 5060 Kuala Lupur, Malaysia. Accepted 7 Jauary, 0 The Newark ethod is a explicit ethod ad the ost iportat aspects of this subfaily are the possibility of ucoditioal stability for oliear systes ad secod-order accuracy. The possibility of ucoditioal stability ad secod-order accuracy allows the use of a large tie step, ad the explicitess of each tie step ivolves o iterative procedure. To evaluate the uerical properties of the proposed faily ethod i the solutio of liear elastic ad oliear systes, its coputig sequece withi a sigle tie step ust be realistically reflected i the aalysis. I this paper, the cocept of Newark ethod for structure is explaied ad applied. Key words: Noliear dyaic respose optiizatio, structural, Newark, oliear dyaic aalysis. INTRODUCTION A Newark diffusive schee is preseted for the tiedoai solutio of dyaic systes cotaiig fractioal derivatives. This schee cobies a classical Newark tie-itegratio ethod used to solve secod-order echaical systes, with a diffusive represetatio based o the trasforatio of the fractioal operator ito a diagoal syste of liear differetial equatios, which ca be see as iteral eory variables. The focus is give o the algorith ipleetatio ito a fiite eleet fraework. This ethod is iplicit ad based o the assuptio of the liear chage of acceleratio durig each tie step. Other types of variatio of the acceleratio durig a tie iterval ca also be assued (Berard, 00). I geeral, these assuptios will idicate how uch of the acceleratio at the ed of the iterval eters ito the relatioships for velocity ad displaceet. I 959, Newark (Bradford, 999) preseted a ethod which perits differet types of variatio of the acceleratio to be take ito accout. The ai features of this ethod are give i the work. The Newark-beta ethod is a ethod of uerical itegratio used to solve differetial equatios. It is used i fiite eleet aalysis to odel dyaic systes, recallig the cotiuous-tie equatio of otio: u u t u t () A differetial equatio is a atheatical equatio for a ukow fuctio of oe or several variables that relates the values of the fuctio itself ad its derivatives of various orders. Differetial equatios play a proiet role i egieerig, physics, ecooics ad other disciplies (Nakahira, 985). Visualizatio of the airflow ito a duct was odelled usig the Navier-Stokes equatios, a set of partial differetial equatios. Differetial equatios arise i ay areas of sciece ad techology; wheever, a deteriistic relatioship ivolvig soe cotiuously chagig quatities ad their rates of chage (expressed as derivatives) is kow or postulated. Usig the exteded ea value theore, the Newark- ethod states that the first tie derivative (velocity i the equatio of otio) ca be solved as: u u u γ () Where *Correspodig author. E-ail: haidreza@siswa.u.edu.y. γ u () ( γ ) u γ u 0 γ

2 96 It. J. Phys. Sci. Therefore, u u ( γ ) u γ tu (4) Sice the acceleratio also varies with tie, however, the exteded ea value theore ust also be exteded to the secod tie derivative to obtai the correct displaceet. Thus, values will produce uerical dapig. Equatio (9) ca therefore be writte as (Zapieri, 006): () I additio to the expressios for the displaceet ad velocities, the coditio of dyaic equilibriu at the ed of the tie iterval was: u uu uβ Where agai (5) C K P M () The followig expressio for the acceleratio at the ed of the tie step was yielded: u β ( β) u β u 0 β (6) Newark showed that a reasoable value of is 0.5, therefore the update rules are, u u ( u u) (7) u u u β u β u Settig to various values betwee 0 ad ca give a wide rage of results. Typically, / 4, which yields the costat average acceleratio ethod, is used (Nakahira, 990). NEWMARK'S RELATIONSHIPS FOR ACCELERATION, VELOCITY AND DISPLACEMENT Newark expressed the velocities ad displaceets at the ed of a tie icreet i ters of the kow paraeters at the begiig ad the ukow acceleratio at the ed of the tie step as: ( γ ) γ (9) β β (8) (0) Where γ ad β are paraeters which ca be varied at will. The value of γ is take to be equal to as other M P C K () Equatios (), () ad () fro the basis is used for the o-liear aalysis of the structural systes by Newark ethod. I geeral, uless is take as zero, the calculatio procedure is for oe tie icreet. ANALYSIS USING THE NEWMARK METHOD Step : Assue value for the acceleratio vector at the ed of the tie step. Step : Copute the velocity ad displaceet vectors ad Equatios () ad (4). at the ed of the tie step usig Step : Update the stiffess ad dapig atrices. Step 4: Calculate the acceleratio vector usig Equatio (). Step 5: Copare the coputed acceleratio vector with the assued oe. If these are equal or withi a perissible differece, the calculatios for the step have bee copleted, if ot. Step 6: Assue the last calculated value of to be the iitial value i the ext iteratio to step. The rate or the covergece of the process towards the equality of the derived ad assued acceleratio is a fuctio of the tie icreet t ad is fully discussed i (Zapieri, 006). The criterio of the covergece

3 Hashadar et al. 97 Figure. Acceleratio durig ties. give by Newark is the equality of the assued ad calculated values of the acceleratio at the ed of the tie step (Laier, 00). For MDOF systes, this ivolves the copariso of two vectors. Sice it is highly ulikely that all the eleets i the calculated vector will be equal to the correspodig eleets i the assued vector, a covergece criterio ust be icluded i the process. The criterio ay be based o a copariso of the values of the or of the vectors ad/or a copariso of the idividual eleets give either as a percetage or a absolute differece. The choice of type ad agitude of the perissible differece is a fuctio of the required accuracy ad is left for the experiece ad judget of the aalyst. Iterpretatio of the paraeter It is of iterest to ote how the acceleratio durig the tie iterval varies with variatios i the values of. Although, it is ot possible to defie a relatioship for all values of, for at least four values, the variatio i the acceleratio durig the tie step ca be described. Three of these variatios are show i Figure. It appears that a choice of β correspods to assuig a uifor value of the acceleratio durig the tie iterval which is equal to the ea of the iitial ad fial value; a value of β correspods to assuig a step fuctio with a uifor value to the iitial value for the first half of the tie icreet ad a uifor value equal to the fial value for the secod half; ad a choice of β correspods to a liear chage of acceleratio 6 durig the tie iterval. The latter value of results i the basic equatios as developed i the stadard liear acceleratio ethod. The ai differece betwee the two algoriths is that i the Newark ethod, equilibriu of the dyaic is forced for the total acceleratio, velocity ad displaceet vectors at tie whilst i, the other equilibriu, it is oly esured for the icreetal acceleratio, velocity ad displaceet vectors. The latter ay result i a accuulatio of errors uless the acceleratio is recalculated fro the equatios of otio at the ed of the tie step (Ma-Chug, 005). The Newark 0 ethod The value 0 leads to a explicit algorith for the Newark ethod ad is therefore discussed separately. Whe 0 the expressio for the displaceet at t is give as: t t (4) Substitutig the expressio for give by Equatio () ito Equatio (4) ad solvig for yields:. P [ M C ] C K (5) ca be calculated fro Equatio (5). Oce

4 98 It. J. Phys. Sci. is kow K ca be calculated to for the product K.. I soe cases, this product ca be built up as a colu vector without forulatig the global stiffess atrix (Nakahira, 990; Peterso, 008). The value of 0 correspods to double pulses of acceleratio at the begiig ad ed of the tie iterval with each double pulse cosistig of a part equal to half of the acceleratio ties the tie iterval, oe occurrig just before the ed of the precedig iterval ad the other just after the begiig of the ext iterval (Peterso, 008). Aalysis usig the Newark ( 0) ethod With the value of K, C,, ad give either at the ed of the previous tie step or fro iitial coditios, the calculatio procedure of Newark ( 0) ethod for oe tie iterval ca be sued as follows: Step : Calculate fro Equatio (4). Step : Set up K ad C. Step : Calculate usig Equatio (). Step4: Calculate usig Equatio (). These steps coplete the calculatios for oe tie iterval which ay ow be repeated for the ext step. Fro the aforeetioed, it ca be see that the calculatio of as a fuctio of the calculatio of C is oly possible for cases where the dapig atrix is ot a fuctio of the velocity. Stability ad accuracy of the Newark ethod The Newark ethod is of the secod order accuracy ad oly coditioally stable. This eas that the tie iterval t ust be less tha a certai value to esure stability. The size of t is a fuctio of the value of ad the sallest period of vibratio of a syste. Recoedatios with respect to the choice of values for ad the size of tie itervals are give i Beer ad Marti (00). A study cocerig errors i the dyaic eergy resultig fro the use of the Newark algoriths is ivestigated. 00). I this ethod, total potetial eergy is used: W U V (6) Where W the total potetial eergy; U the strai eergy of the syste ad V the potetial eergy of the loadig. The advatage of this ethod is decrease uber of iterative per tie step. There are a lot of ways for the aalysis of the structure. Aalyzig structure is divided ito direct ad idirect ethod. The ethod whe applied to structural systes first predicts the displaceet vector ad the calculates the acceleratio ad velocity vectors at the iddle of a tie step. It the predicts the displaceet vector ad fro that calculates the acceleratio ad velocity vectors at the ed of the tie step. The dapig ad stiffess atrices ay or ay ot be update at this stage, depedig upo the degree of oliearity of the syste. The equatio of otio for a ulti degree (MDOF) syste ca be writte as: M C (t) k (T) P (T) (7) Where, M ass atrix; C (t) dapig atrix; K (t) stiffess atrix; displaceet vector; velocity vector; acceleratio vector; P(t) load vector Sice is a o-zero costat value, both sides of Equatio (7) ca be divided by : P C (t) /, Q K (t) /, F P (t) / The equatio ca be writte as: P QF (8) Equatio (8) is a liear differetial equatio if P ad Q are idepedet of x. ( k ) ( k ) S ( k ) V ( k ) (9) Where the suffices (k) ad (k) deote the (k)th ad (k)th iterate, respectively ad where V the eleet of the directio vector, ad S (k ) the step legth which defies the positio alog V (k ) where the total potetial eergy is a iiu. The expressio for V is, if the Fletcher-Reeves forulatio of the cojugate gradiets ethod is used, which is give by: Potetial dyaic work This ethod is ot cooly used for aalysis because it is difficult but it is perfect i result ad reality (Liqus, V J ( k ) ( k ) j i ( k ) g ( k ) V J ( k ) j i g g ( k ) g g ( k ) (0)

5 Hashadar et al. 99 Calculatio of the step legth The required polyoial for step legth is foud by substitutig the expressio for ( k ) give i a suitable expressio for the total potetial eergy: e j j j Where; ( a a S a S ) () L e a i ( x i x (( x )( x i i x x )( x )) i x ) (a) a ( v i i v (( )) x i x x i x ) (b) Figure. Diagra of frae ade. a ( v i v ) i (c) NUMERICAL AND ANALYTICAL TESTING Ad secodly to the expressio for W i ters of the step legth S ad its derivative with respect to S as give: W C 4 C S S C 5 4 C S C S S C S 4 () W S 4C C S C () Where: EA C ( a ) L(L e) EA C ) ( a a L(L e) T EA C )) C (-a) (-b) ( a ( a a a (-c) L e L( L e) C ( a a a 4 L e L(L e) ) J j i J j i F V T EA ( a a 5 L e L(L e) ) F T EA (-f) (-d) The developet of a atheatical cotrol to esure stability whe usig larger tie steps is desirable. The atheatical odel chose is a circle flat et with 9 degrees of freedo. The circle flat et was also built as a fiite eleet odel ad tested i order to verify the proposed theory give i this paper. The diagra is give i Figure. The odel cosisted of a circle et, with the bea eleet at 00 itervals. At the poits of itersectio, the circle et is claped together with yield. Specificatios of circle et are give as: Overall diesios: 600 x 600 ; spacig of the cables: 00 ; uber of fixed boudary joits: 4; uber of liks: ; Youg s Modulus: 9.60 KN/ Size: 0 x 00 The ass desity iflueces the stability liit. Uder soe circustace, scalig the ass desity ca potetially icrease the efficiecy of a aalysis ad the explicit dyaic uses a cetral differece rule to itegrate the equatio of otio explicitly through tie.deflectios due to cocetrated load at Node 4 is preseted i Table. The dapig ratio of a practical structure depeds o ay factors, such as the structure syste, the detail of joits, the foudatio coditio ad so o. The dapig ratio obtaied fro easureet of a practical structure varies ot oly with the detail of the structure itself, but also with the vibratio aplitude, the easurig ethod ad the data processig ethod. As a result, it is difficult to fid out how the dapig ratio is iflueced by differet factors D view of odel is show i figure. I Figures 4 ad 5, we preseted the tie history of the ea value of the displaceet ad ea value of the

400 It. J. Phys. Sci. Table. Deflectios due to cocetrated load at ode 4. Load (o) 000 FU ethod (T) Fiite eleet (E) ( T E ) / T*00 Z AIS DEFLECTIONS () NODE (LVDT) 6E-0 6E-0 0.

6 400 It. J. Phys. Sci. Table. Deflectios due to cocetrated load at ode 4. Load (o) 000 FU ethod (T) Fiite eleet (E) ( T E ) / T*00 Z AIS DEFLECTIONS () NODE (LVDT) 6E-0 6E Z AIS DEFLECTIONS () NODE (LVDT) 9E-0 7E-0.55 Z AIS DEFLECTIONS () NODE 4 (LVDT) 95E-0 9E-0.6 Z AIS DEFLECTIONS () NODE 5 (LVDT) E-0 0E Z AIS DEFLECTIONS () NODE 6 (LVDT) 97E-0 95E-0.06 Figure. D view of odel. Tie (s) Figure 4. Stadard deviatio of defectio agaist the height of cetre.

7 Hashadar et al. 40 Tie (s) Figure 5. Tie history of ea value of oet. Table. Frequecy o 5 ode shape. Mode Frequecy of Fu ethod Frequecy (Hz) of fiite eleet Differetials percetage oet at the ceter of the structures, which is i the directio of the loger side of the structures, whe the shorter sides are siply supported. The differetial percetage results are give i Table. I Figures 6 ad 7, the variatios of the ea values of defectio as well as the stadard deviatios of defectio at ay two arbitrary tie istats are oliear whe the height of ceter of the structures icreases. This pheoeo is uderstadable sice the stiffess of the structures reduces as the height of ceter of the structures icreases. CONCLUSIONS This proposed faily ethod is very copetitive with other itegratio ethods for solvig geeral structural dyaic probles, where the resposes are doiated

8 40 It. J. Phys. Sci. Tie (s) Figure 6. Tie history of ea value of defectio. Figure 7. Dyaic respose i directio.

9 Hashadar et al. 40 by low frequecy odes. This is because it ca itegrate the ucoditioal stability ad the explicitess of each tie step. Ucoditioal stability will allow it to use a larger tie step to perfor step-by-step itegratio ad the explicitess of each tie step ivolves o eed of uerical iteratios. The stability aalysis idicates the presece of uerical dapig ad shows a effective stability liit of t which is the sallest atural period of a syste. These stability coditios are closely related to the istataeous degree of oliearity. The objective of this work was pricipally to develop a Newark algorith aalysis theory ad verify it by uerical ad fiite eleet testig. The proposed ethod was foud to be stable for tie steps equal to or less tha half of the sallest periodic tie of the syste. Nakahira N (985). Nuerical vibratio aalysis of plate structures by Newark's ethod. J. Soud Vib., 9: Nakahira N (990). Vibratio of plae trusses by the Newark ethod Coput. Struct., 7: Peterso CM (008). Laczos revisited. Liear Algebra Appli., 48: Zapieri E (006). Approxiatio of acoustic waves by explicit Newark's schees ad spectral eleet ethods. J. Coput. Appl. Math., 85: REFERENCES Beer HFB, Marti S (00). A Hailtoia Krylov Schur-type ethod based o the syplectic Laczos process. Liear Algebra Appl., 5: 8-4. Berard P (00). Stochastic Newark schee. Probabilistic Eg. Mech., 7: Bradford MA (999). A Newark-based ethod for the stability of colus. Coput. Struct., 7: Laier JE (00). Spectral aalysis of a high-order Heritia algorith for structural dyaics. Appl. Math. Model., 5: Liqus MC (00). Noliear elastoplastic dyaic aalysis of siglelayer reticulated shells subjected to earthquake excitatio. Coput. Struct., 8: Ma-Chug DB (005). Traspose-free ultiple Laczos ad its applicatio i Padé approxiatio. J. Coput. Appl. Math., 77: 0-7.

Exercise 8 CRITICAL SPEEDS OF THE ROTATING SHAFT

Exercise 8 CRITICAL SPEEDS OF THE ROTATING SHAFT Exercise 8 CRITICA SEEDS OF TE ROTATING SAFT. Ai of the exercise Observatio ad easureet of three cosecutive critical speeds ad correspodig odes of the actual rotatig shaft. Copariso of aalytically coputed