Topology optimization of plate structures subject to initial excitations for minimum dynamic performance index

Transcription

1 th World Congress on Structural and Multdsclnary Otmsaton 7 th -2 th, June 25, Sydney Australa oology otmzaton of late structures subject to ntal exctatons for mnmum dynamc erformance ndex Kun Yan, Gengdong Cheng State Key Laboratory of Structural Analyss for Industral Equment, Faculty of Vehcle Engneerng and Mechancs, Dalan Unversty of echnology, Dalan 623, P. R. Chna. Abstract hs aer studes otmal toology desgn of damed vbratng late structures subject to ntal exctaton. he desgn objectve s to mnmze an ntegrated square erformance measure, whch s often used n otmal control theory. he artfcal densty of the late element s the toology desgn varable. he Lyaunov s second method s aled to reduce the calculaton of erformance measure to the soluton of the Lyaunov equaton. An adjont varable method s develoed n our study, whch only needs to solve the Lyaunov equaton twce. However, when the roblem has a large number of degrees of freedom, the soluton rocess of Lyaunov equaton s comutatonal costly. hus, the full model s transform to a reduced sace by mode reducton method. And we roose a selecton method to decrease the number of egenmodes to further reduce the scale of reduced model. Numercal examle of otmum toology desgn of bendng lates s resented for llustratng valdty and effcency of our new algorthm. 2. Keywords: Adjont method, vbraton control, toology otmzaton 3. Introducton Structural toology otmzaton and structural vbraton control have called attenton both n theoretcal research and ractcal alcatons n engneerng. Structural toology otmzaton rovdes a owerful automated tool for mrovng the structural erformance n the ntal concetual desgn stage. Usually, otmzaton roblems are formulated to mnmze the materal usage or to otmze the structural erformance. Structural vbraton control s a artcularly mortant consderaton n dynamc system desgn. Many control algorthms have been develoed for assve and actve control. Passve control systems that do not requre any external ower are wdely used to reduce the resonse of structures. In engneerng alcatons, shell structures are wdely used. Structural toology otmzaton and structural vbraton control of shell structures has receved an ever ncreasng attenton. Several researchers have aled structural toology otmzaton technques to structural vbraton control roblems. In most of exstng works, structural toology otmzaton technques are used to obtan the layout of ezoelectrc or damng materal on a man structure. Kang et al. nvestgate the otmal dstrbuton of damng materal n vbratng structures subject to harmonc exctatons by usng toology otmzaton method []. Cha et al. ntroduced cellular automata algorthms nto the layout otmzaton of damng layers [2]. Zheng et al. dealt wth toology otmzaton of lates wth constraned layer damng treatment for maxmzng the sum of the modal damng ratos, whch are aroxmated wth the modal stran energy method [3]. In ths aer, the roblem of a late or shell just contans damng materal wll be consdered. Many erformance ndces have been consdered n vbraton control otmzaton roblems, lke H or norms. In tme doman, there s a classc roblem formulaton of assve structural vbraton control that deals wth the dynamc system dsturbed by ntal condtons. he objectve s to fnd desgn arameters of the damed vbraton system that mnmze the erformance ndex n the form of tme ntegral of the quadratc functon of state varables (dslacement and veloctes, e.g. see equaton (5)). hs erformance ndex can be evaluated by Lyaunov s second method [4]. Based on the Lyaunov equaton, the evaluaton of erformance ndces are smlfed nto matrx quadratc forms and do not requre the tme doman ntegraton. Parameter otmzaton roblems wth a quadratc erformance ndex have been solved by ths method [5]. Wang et al. aled the Lyaunov equaton to solve the transent resonse otmzaton roblem of lnear vbratng systems excted by ntal condtons [6]. Du aled the Lyaunov equaton to obtan the otmum confguraton of dynamc vbraton absorber (.e., DVA) attached to an undamed or damed rmary structure [7]. A well-known effcent soluton technque for calculatng the dynamc resonse of structures s to transform the model nto a reduced sace. Varous methods for ths requrement are avalable now, such as the Guyan reducton, mode sueroston, modal acceleraton and Rtz vector methods [8]. Among others, the mode sueroston method s generally recognzed as an effcent aroach for dealng wth large-scale roortonally damed structures. Generally, the structural resonse of reduced model s exressed as a lnear combnaton of ther frst 2 H

2 dozens or hundreds egenmodes. However, for some cases, the egenmodes of low order may have no effect on the structural resonse of reduced model. In ths aer, a selecton method s used to fnd these egenmodes to decrease the number of bass vectors to further reduce the scale of the reduced model. In ths aer, an aroach s develoed for toology otmzaton nvolvng a quadratc erformance ndex of lnear elastc shell structure subject to ntal exctatons. Mode reducton method and egenmode selecton method are used to decrease the comutng tme of otmzaton rocess. At last, a cantlever late examle and several llustratve results are resented. 4. oology otmzaton roblem formulaton 4. Governng equatons Consder a vscously damed lnear vbraton system governed by the equaton: M u& + Cu& + Ku = () where M(N N) s the mass matrx, C(N N) s the damng matrx, K(N N) s the stffness matrx, and u(n ) s dslacement vector. N s the structural degree of freedoms. Assume the system s excted by ntal dslacements or veloctes. And the desgn roblem s to fnd n M, K and C matrces to mnmze a erformance matrx n the form & where, q( u, u) u Q u + u Q u (,u) J = q u & dt (2) = & & s a quadratc functon of u and u&. ransent dynamc resonses have to be u &u erformed to evaluate the objectve functon. Drect or adjont methods can be aled to evaluate the resonse senstvty requred for evaluaton senstvty of the erformance. Alternatve, f we relace the uer bound of ntegraton to nfnte, we can use Lyaunov s second method to evaluate the erformance wthout erformng transent dynamc resonse analyss. o aly Lyaunov s second method to ths system, t s necessary to rewrte Eq.() n the state sace form X & = AX (3) Where O I u X = M K M C u & A = he matrx A s (2N 2N). he vector X s (2N ). Structural desgn arameters such as mass densty, damng rato and srng stffness are contaned n the matrx A. he otmzaton roblem s to choose these arameters to mnmze the erformance measure J defned by J X QX dt = (5) for a gven ntal exctaton X(). In Eq.(5), Q(2N 2N) s a ostve sem-defnte symmetrc weghtng matrx and t denotes tme. Accordng to Lyaunov theory of stablty, for an asymtotcally stable system, there exst a symmetrc ostve sem-defnte matrx P(2N 2N) satsfyng (6) A P + PA = Q Eq.(6) s the well-known Lyaunov equaton. Based on the Lyaunov s second equaton, the Eq.(5) can be further smlfed as J = X PX (7) ( ) ( ) hat s to say, to mnmze J n Eq.(5) s equvalent to mnmze X ( ) PX( ), where X( ) vector and the unknown symmetrc matrx P can be obtaned by solvng Eq.(6). (4) s the ntal state 4.2 Mathematcal formulaton of toology otmzaton roblem In ths aer, the toology otmzaton roblem for fndng the otmal dstrbuton of gven materal to mnmze the quadratc ntegral form structural erformance ndex of a vbratng structure excted by ntal exctaton s consdered. he mathematcal formulaton of toology otmzaton roblem s exressed as ( ρ, ρ2,..., ) fnd ρ N e mn s.t. J = X X & = AX QXdt 2

3 N = where, ρ s the artfcal densty of th element, mn element, V s the secfc volume ton, and e e N e ρ V V = V e = mn ρ, =, 2,..., ρ (8) N e ρ s lower bound of artfcal densty, N s the number of elements n desgn doman. e V s the volume of th An artfcal damng materal model that has a smlar form as the SIMP aroach s used and the artfcal denstes of elements are taken as desgn varables. he elemental mass matrx and stffness matrx are exressed by where, ~ ρmn ρ mn ~ M = ρ M, K = ( ρ ) + ρ K ρmn M ~ and K ~ are the elemental mass matrx and stffness matrx of th element wth ρ = (9), resectvely; s the enalty arameter and t s set to be =3 n ths aer. he Raylegh damng theory s emloyed, and the elemental damng matrx s obtaned by ρ ρ C where, α and β are the damng arameters. mn mn ρmn ~ ~ ( ρ ) + ρ ( αm + βk ) = (9) 5. Senstvty analyss scheme he toology otmzaton roblems always are solved by gradent-based mathematcal rogrammng algorthms, whch need the senstvty analyss of the objectve functon wth resect to desgn varables. In ths aer, a senstvty analyss scheme derved by adjont varable method s aled, whch s more effcent than drect varable method n the roblems nvolvng a large number of desgn arameters. For the case, ntal condton ndeendent of desgn arameters, the senstvty analyss scheme can be exressed as J = 2N 2N P λkl D ( ) X( ) = where, λ s the adjont matrx. λ and D can be obtaned by + λa + S = where, S = X( ) X( ) = k = l = X () kl Aλ () Q A A D = + P + P A where, ( M ) K ( M ) K M C M C 6. ransformaton of equatons to reduced sace When the analyss model has a large numbers of DOFs, the soluton of Lyaunov matrx equaton s comutatonal costly, whch wll makes the comutng tme of otmzaton rocess ncreased sgnfcantly. For examle, for a,-dof system, the number of unknowns n P s 2,,. hus model reducton s necessary to mlement the roosed aroach. he mode reducton method and egenmode selecton method are used to decrease the comutng tme of otmzaton rocess. 6. mode reducton method o use mode reducton method, a lnear transformaton s emloyed, whch can be exressed as (2) u = u m (3) where, u and u m are the dslacement vectors of full model and reduced model, resectvely; s the transformaton matrx. Generally, matrx contans the frst several egenmodes of full model. However, for some cases, the egenmodes of lower order may have no effect on the structural resonse. A selecton method s aled 3

4 to fnd these egenmodes to decrease the number of bass vectors n transformaton matrx to further reduce the scale of reduced model, and wll be ntroduced n next secton. he transformaton matrx s exressed as = φ φ,..., φ (4) { } c, c2 where, c, c 2, c m are the number of st, 2nd, mth reserved egenmodes. he mass, damng, and stffness matrces of reduced model are resectvely obtaned by he ntal condtons of reduced model are obtaned by Mre = M, Cre = C, K = K u re, = Mre Mu, v c m re (5) re, = Mre Mv Include the senstvty of matrx wth resect to desgn arameters n senstvty analyss scheme wll make the analyss much comlcated. hus, n ths aer, the senstvty of matrx wth resect to desgn arameters s gnored. 6.2 Egenmode selecton method We use the model artcaton factor (MPF) to evaluate whch egenmode n frst several egenmodes of full model have no effect on the structural resonse. Hgh value of MPF of th egenmode means that ths egenmode has large effect on structural resonse. Low MPF value means that ths egenmode has a lttle effect on structural resonse. he MPF value s obtaned by 2 2 ( uφ ) ( vφ ) MPF =, ( u u )( φ φ ) ( v v )( φ φ ) (6) MPF = (7) where, u and v are the ntal dslacement and velocty vector, resectvely, and φ s the egenvector of th egenmode. he MPF values of all egenmodes are located between and. For the case an otmzaton roblem has both ntal velocty and dslacement, the MPF values for ntal velocty and dslacement need to be calculated searately and weghted summed. he weghted coeffcents are the objectve functon values from usng the ntal velocty and dslacement as ntal condton searately. 7. Numercal examle o avod the checkerboard henomenon, the senstvty flter method s used, the flter radus s.5. For some cases, drastc change of the desgn may cause that the Lyaunov equaton cannot be solved. hus, the move lmt of desgn arameter s set to be.2. In ths secton, a numercal examle s resented to verfy the senstvty analyss scheme and the roosed aroach. Y m X 2m Fgure : Geometry model We consder a 2m m.m rectangular late. he left edge of the late s clamed and other three edges are free as shown n fgure. he materal arameters s E=69GPa, v=.3, ρ = 27 kg/m 3. A concentrate mass element locates at the mddle of the rght edge of the late, and m=5kg. he ntal condton s that the Z-drecton velocty of mass element s m/s. he late s unform meshed by 4-nodes square element, 4 2, as shown n Fgure 2. he objectve functon s = 2 u where, u mass s the Z drecton dslacement of mass element. J mass dt (8) 4

5 Fgure 2: he fnte model Consderng the symmetry of the fnte element model, ntal condton, and constrants, only the artfcal denstes of the elements n bottom half of the structure are consdered n the otmzaton rocess. he artfcal densty of the element n the to half of structure s set to be same wth that of the element at symmetrcal oston. Consderng the accuracy of senstvty results and effcency of otmzaton rocess, n ths examle, the transformaton matrx wll contans 6 egenmodes selected from frst 3 egenmodes of full model. o verfy the accuracy of senstvty results obtaned by the roosed senstvty analyss scheme, the fnte dfference method s also aled to obtan the senstvty results. he senstvty results of the urle element as shown n fgure 3 by the fnte dfference method and adjont method are both shown n fgure 3. he damng arameters are α =., β =, and the analyss model s a unform desgn ( V =. 5 ). Numercal results show that the relatve error of the results obtaned by two methods s small for most elements. * Fnte dfference method Adjont method Senstvty Element number Fgure 3: Senstvty results of several elements from two methods Objectve functon Iteraton ste Fgure 4: Iteraton hstory of objectve functon 5

6 Frstly, erform a toology otmzaton wth α =., β =, V =. 5, and mn =. ρ by roosed aroach. Fgure 4 shows the teraton hstory of objectve functon and. From the results, a stable decrease of the objectve functon can be observed. Next, erform another toology otmzaton wth α =, β =., V =.5, and ρ by roosed aroach. he otmzed desgns are shown n fgure 5. he results mn =. wtness that the otmzed desgns under dfferent damng arameters are such dfferent. hus, obtan the accurate damng arameters are mortant to whether the otmzed desgn s reasonable. (a) α =., β = (b) α =, β =. Fgure 5: Otmzed desgns under dfferent damng arameters 8. Conclusons he roblem of toology otmzaton wth resect to vbraton control of a shell structure subject to ntal exctaton s consdered. he desgn objectve s mnmzaton of dynamc erformance ndex n the form of tme ntegral of the quadratc functon of state varables. An aroach s develoed to handle ths toology otmzaton roblem. Mode reducton method and an egenmode selecton method are aled to decrease the scale of reduced model. he numercal examle s resented to verfy the senstvty analyss scheme and the roosed aroach for toology otmzaton roblem consdered n ths aer. he results show that the senstvty analyss scheme for reduced model can obtan accurate results, and also wtness that the damng arameters have a great effect on the otmzed desgn. 9. Acknowledgement hs work s suorted by Natonal Natural Scence Foundaton of Chna (9262 and 37262).. References [] Kang Z, Zhang X, Jang S, et al. On toology otmzaton of damng layer n shell structures under harmonc exctatons[j]. Structural and Multdsclnary Otmzaton, 46(): 5-67, 22. [2] Cha CM, Rongong JA, Woeden K, Strateges for usng cellular automata to locate constraned layer damng on vbratng structures. J Sound Vb 39:9 39, 29. [3] Zheng L, Xe RL, Wang Y, El-Sabbagh A, oology otmzaton of constraned layer damng on lates usng Method of Movng Asymtote (MMA) aroach. Shock Vb 8:22 244, 2. [4] Kalman RE, Bertram, JE,Control System Analyss and Desgn Va the Second Method of Lyaunov: I Contnuous-me Systems. Journal of Fluds Engneerng, 82(2), , 96. [5] Ogata K, Yang Y, Modern control engneerng, 97. [6] Wang BP, Kts L, Plkey WD,ransent Resonse Otmzaton of Vbratng Structures by Launov s Second Method. J. Sound Vb., 96, , 984. [7] Du D, Analytcal solutons for DVA otmzaton based on the Lyaunov equaton. Journal of Vbraton and Acoustcs, 3(5), 545, 28. [8] Bathe KJ, Fnte element rocedures. Prentce Hall, New Jersey,