Shape optimization of a flywheel

Transcription

1 Stuct Multidisc Optim 19, Spinge-Velag 000 Shape optimization of a flywheel G.R. Kess Abstact This pape deals with finding the best thickness distibution along the adius of a centally boed flywheel unde the objective of eaching an even stess distibution. A global numeical shape optimization pocedue is used, employing a two-dimensional Finite-Element- Method (FEM) model, and a one-dimensional analytical model based on simplified mechanical consideations and inspied by Stodola s solution fo an evenly stessed tubine disk without cental boe. Both models ae pesented in detail and the esults compaed. The simplified model yields inteesting conclusions. 1 Stodola s solution Within a flywheel of constant thickness without cental boe, Fig. 1a, stesses ae distibuted as plotted in Fig. 1b. The nomal stess in adial diection σ must meet the natual bounday condition at the im. Hee the cicumfeential stess σ θ is somewhat highe than the adial stess. Both stess cuves incease with deceasing distance fom the cente. They actually meet at the cente with zeo slopes. With egad to both nomal and cicumfeential stess, the mateial of the disk of constant thickness is not economically used because of the stess value vaiations. Stodola (194) discussed an optimum-stength design fo the cental disk of a steam tubine. The cental disk connects the tubine s blades to its dive shaft. Stodola s design essentially avoids a cental hole, Fig.. The inetia foces of the blades acting on the im of the cental disk ae taken to be equivalent to an aveage adial Received Octobe 9, 1998 Revised manuscipt eceived Decembe 0, 1998 G.R. Kess Institute of Design and Constuction Methods, Swiss Fedeal Institute of Technology, Tannenst. 3, CH 809 Zuich, Switzeland kess@ikb.mavt.ethz.ch stess which entes the mathematical model as a natual bounday condition. By choosing the moe geneal fomulation allowing fo a changing thickness t of the equilibium of foces in the adial diection, (σ t) + t (σ σ θ )+tω ϱ =0, (1) and equiing both stesses to be equal and also constant eveywhee, σ = σ θ = σ, () the geneal equilibium equation is educed to a diffeential equation whee only the thickness emains as a dependent vaiable t + t ϱω =0, (3) σ and the shape of the evenly stessed flywheel is defined by the solution t = t 0 e ϱω σ. (4) It must, howeve, be kept in mind that the one-dimensional model which undelies diffeential equation (1) holds only when the efeence thickness t 0 is small compaed to the diamete of the disk. Then, the nomal stess in the axial diection σ z and the shea stess τ z can be neglected. As many flywheels possess a cental boe fo the pupose of fixing them on a otating shaft, the study of Stodola s poblem povokes the question whethe a shape can be found fo which an almost even stess distibution exists in the pesence of a cental boe and in the absence of extenal adial stess. An answe to this question is found by using a numeical shape optimization scheme whee a FEM-based stuctual model povides the system equations. Two-dimensional finite-element model The equilibium equations in the adial and axial diections ae σ + τ x x + 1 (σ σ θ )+ω ϱ =0,

2 75 Fig. 1 Coodinate convention (a) and stess distibutions (b) fo a otating disk of constant thickness without cental boe Fig. Optimum tubine disk design accoding to Stodola s fomula. The cente of otation is at the left-hand side of the model and the disk is without a cental hole τ x + σ x x + 1 τ x =0. (5) Afte applying the pinciple of vitual displacements the weak vaiational fom follows: ( δw σ + δw xτ x + δwσ θ + δu τ x + ) δu xσ x dx d = δw ω ϱ dx d. (6) Fo bevity, the bounday tems have been omitted as no extenal loading is applied to the edges. Afte consideing the kinematics and the constitutive equations fo isotopic mateials the vaiational fom may be witten in tems of displacements, [ δw ( w ) C 13 u x + C 3 + C33w + δw x (C 55 w x + C 55 u )+ ( w ) δw C 1 u x + C + C3w + δu (C 55 w x + C 55 u )+ ( w ) ] δu x C 11 u x + C 1 + C13w dx d = δw ω ϱ dx d. (7) By intoducing the opeato D and the adius-dependent stiffness matix C 0 1 x D = 0 0 x 0, 0 1 C C C 3 C 1 C C 13 C () = 0 0 C 55 C C 55 C 55 0, C 3 C C 33 the element equation may be witten as (DN) T C () (DN)dxdũ = N ω ϱ dx d, (8) whee the nodal point degees of feedom ũ in the ode ũ T = {u 1 w 1 u w...}, define the stuctue of the shape function matix N. The standad isopaametic quadatic Seendipity element with eight nodes numeically evaluates the weak fom (8). Figue 4 shows a typical mesh fo analysing the flywheel.

3 76 3 Shape optimization The objective is to find a shape of the flywheel esulting in an even stess distibution within it. Using a flywheel of constant thickness t 0 as the initial configuation its mass and otational enegy ae also to be kept constant. Mathematically, this is expessed by using the standad deviation of the adial stess distibution σ as the objective and by intoducing the equality constaints h 1 (constant mass) and h (constant otational enegy), O = (σ σ) d =min, σ: aveage stess, h 1 =πϱ ( ) t () t 0 d =0, h = ω πϱ 3 ( ) t () t 0 d =0. (9) The thickness values at the element intefaces in the adial diection take the ole of shape paametes as outlined in Fig. 3. Thus, the numbe of optimization vaiables depends linealy on the numbe of elements in the adial diection but is independent of the numbe of elements in the axial diection. The position of the nodes on the element sides is always on the staight line connecting the espective cone nodes. The objective function is minimized using the method of feasible diections accoding to the textbook of Vandeplaats (1984) as well as exact gadient infomation fom the objective and the constaint functions. Additional coections to ensue that the equality constaints emain satisfied ae made afte each line seach. Quadatic appoximations of the objective function ae used fo quickly finding the minimum value along the cuent seach diections as descibed in the textbooks of Reklaitis et al. (1983) and Baie et al. (1994). 4 FEM sample calculation and esults Table 1 gives the sample calculation input data. stess distibutions indicate that the egion adjacent to the cental hole is significantly moe stessed than othe pats of the flywheel. The shape optimization pocedue yields the esults plotted in Figs. 5 and 6. Both the stess distibutions and the shape suggest a division of the flywheel into thee egions fo a chaacteization of the optimized geomety. Region I adjacent to the hole inceases pogessively in thickness with deceasing distance to the hole. The cicumfeential stess also inceases with deceasing distance to the hole but its maximum value is much lowe than in the initial configuation. The most stiking featue of egion II is the nea constancy of both the adial and the cicumfeential stesses. Moeove, the values of both stesses ae almost equal. The thickness of egion II deceases pogessively but at a slow ate with inceasing distance to the hole. Within egion III, the shape of which is somewhat eminiscent of a bell, the adial stess dops to zeo. The diffeence between the esults shown in Fig. 5 and in Fig. 6 is in the choice of the stess used fo the objective. Figue 5 shows the shape coesponding to the most evenly distibuted cicumfeential stess. Since the cicumfeential stess has highe values eveywhee in the initial configuation, it is identical to the fist pincipal stess and minimizing the objective epesents the engineeing aspect of seeking a spatially constant pobability of bittle failue in the flywheel. Figue 6 shows the shape coesponding to the most evenly distibuted von Mises stess. The stess distibution obtained hee comes closest to a spatially constant pobability of yield failue in the flywheel. Thus, it may be concluded that local details of the shape depend on the choice of stess fo the objective function while global featues such as the existence of egions I, II, and III emain the same. Both the shape of egion II and the almost even distibution of the stesses within it bing to mind Stodola s evenly stessed tubine disk. The deviations fom pefect constancy of the stesses ae explained by the effects of steep thickness changes on the two-dimensional stess equilibium. The hypothesis is poposed that egion I alleviates the stess petubations of the cental hole and that egion III intoduces adial stesses on the oute edge of egion II which ae used in Stodola s model to simulate the foces exeted by the blades on the otating tubine disk. Table 1 Sample calculation input data disk adius 0 [mm] 500 hole adius i [mm] 10 initial thickness t 0 [mm] 0 otational speed u [evs/min] 5000 mass density ϱ [g/cm 3 ] 7.85 Young s modulus E [N/mm ] Figue 4 shows the initial configuation, the finite element mesh, and the initial stess distibutions. These 5 Simple pediction of optimum shape featues A mechanical system is poposed, consisting of a disk modeling egion II and two discete ings located at the edges of the disk, modeling egions I and III (Fig. 7). The disk has a thickness distibution as given by Stodola (194), (4). The inne and oute ings at positions and have the coss-sectional aeas A 1 and A, espectively.

4 77 Fig. 3 Relationship between nodal point coodinates and shape paamete t Fig. 4 Initial shape and stess distibutions. The adial stess satisfies the natual bounday conditions at the cental hole and the im of the flywheel. The cicumfeential stess inceases steeply with deceasing distance to the hole Fig. 5 Optimized shape and stess distibutions unde the objective of minimum standad deviation of the adial distibution of the cicumfeential stess The line load N esulting fom the integation of the adial stess σ ove the thickness is assumed to be positive, and the cicumfeential stesses σ θ in the ings ae σ θ1 = ϱω 1 + N 1, A 1 σ θ = ϱω N, (10) A whee the fist tems on the ight-hand side give the cicumfeential stess in the independently otating ing and the second tems make coections fo the tensile adial line load which tends to pull the inne ing apat and the oute ing togethe. The otational symmety of the poblem automatically satisfies the condition of foce equilibium in the cicumfeential diection. The condition of foce equilibium in the adial diection equies that the adial line loads be continuous at the intefaces between the disk and the ings. If the mateial of the flywheel is homogeneous, the kinematics ae satisfied if the cicumfeential stess is also continuous. The paticula shape of

5 78 Fig. 6 Optimized shape and stess distibutions unde the objective of minimum standad deviation of the adial distibution of the von Mises stess Fig. 7 Mechanical model consisting of Stodola s disk and two discete ings epesenting egions I and III the disk as defined by Stodola s equation guaantees the constant stess distibution indicated in () as long as the conditions at the discete ings N i = σt i, i=1,, (11) apply. This detemines the coss-sectional aeas of the ings, A 1 and A, σ A 1 = σ ϱω 1 t 0 e ϱω 1 σ, A = σ σ ϱω t 0 e ϱω σ. (1) The coss-sectional aeas of the ings ae positive as long as σ is highe than the cicumfeential stess due to the inetia body foces that would aise in the otating inne ing as an isolated system, and also lowe than the cicumfeential stess that would aise in the otating oute ing: ϱω 1 σ ϱω. (13) Then the stess σ, acting in the disk, tends to pull the inne ing apat in addition to its own inetia effect and to estain the oute ing against the inetia effect acting on it. Thus, the stess σ can be chosen feely as long as it lies between the bounds defined by (13). In accodance with the finite-element shape optimization descibed above it was decided to fix the mass M as well as the otational enegy U of the disk to the espective values of the initial shape (IS), M = t IS πϱ ( ) = πϱ ( A 1 + A )+πϱt 0 U = 1 4 t ISπϱω ( 4 4 1) = πϱω ( 3 1 A A ) + πϱω t 0 e ϱω σ d, (14) 3 e ϱω σ d. (15) This can be achieved by adjusting the efeence thickness t 0 and the constant stess value σ. This stess value will always be lowe than the maximum cicumfeential stess

6 79 at the edge of the hole in the initial design. The pactical significance of the optimized shape lies in the wide choice of less expensive mateials that can be used fo the flywheel. The integals in (14) and (15) can not be solved algebaically. They can, howeve, be eliminated by using the identity e a d ea a 3 e a ϱω d, a= σ. Combining (14), (15), and (16) leads to the stess σ, (16) σ = ϱ U M = 1 4 ϱω ( + ). (17) The maximum cicumfeential stess at the edge of the hole of the initial configuation depends on the mateial s Poisson s atio ν, σ θis = 1 4 ϱω [ (3 + ν)+ (1 ν)]. (18) Its value, even when neglecting the influence of Poisson s atio, can be up to thee times highe than that of the constant stess of the optimized flywheel. Substituting the esult given in (17) into the exponent of the thickness function of the disk, t () = t 0 e ϱω σ = t 0 e Mω U = t 0 e +, (19) eveals that the diametes of the im and the cental hole uniquely detemine the shape of the optimized disk as well as the ing aeas, A 1 = t + 0 e 3 A = t e 1 +, +. (0) The condition stated by (13) can now be witten as 3. (1) Unde the constaints of having the same mass and otational enegy as a efeence flywheel of constant thickness, a shape fo constant stessing can be found only when the oute diamete is at least times lage than the diamete of a cental hole. Intoducing the atio α of the two adii α =, 0 α 1, () 3 yields moe tanspaent equations fo the ing aeas and the thickness function of the disk, 1+α A 1 = t 0 1 3α e α 1+α, A = t 0 1+α 3 α e 1+α, t () = t 0 e (/ ) 1+α. (3) The ing aeas depend on the diamete atio as well as linealy on the absolute size of the flywheel. The thickness function is independent of the diamete of the flywheel. Thus, the atio α uniquely detemines the oveall shape of the optimized flywheel. In ode to detemine the emaining model paamete t 0 it is necessay to evaluate one of the integals in (14) o (15), fo instance, by using a numeical integation scheme. Figue 8 gives the aeas of the two ings and the disk ove the full pemitted ange of α as () indicates. The consideed initial-shape flywheels have a unit adius and a thickness of one hundedth. In the case of vey small values of α, the disk and the oute ing A dominate the aea of the optimized flywheel. With inceasing α, the aeas of the inne ing A 1 and the disk quickly incease and decease, espectively. In the case of values of α highe than 0.3 the inne ing tends to take up the majo potion of the total aea. The aea of the oute ing deceases slowly with inceasing α. As α appoaches the uppe limit, the whole design degeneates in the sense that all the available aea moves into the inne ing. The shapes accoding to the simplified model ae visualized on the left-hand side of Fig. 9 whee ing aeas ae epesented by solid cicles. On the ight-hand side of Fig. 9 the esults of the numeical two-dimensional FEManalysis, based on the same input data, ae shown. It appeas that the shape esults agee well fo lowe values of α up to 0.3, which seems emakable in view of the cudeness of the simplified model. The main diffeence in shape is that the exta aeas at the inne and the oute im ae moe smoothly distibuted accoding to the FEMmodel than accoding to the simplified model. The two models agee in the exta aea at the oute im emaining faily constant, the exta aea at the inne im inceasing quickly, and the thickness of the disk deceasing, all with inceasing values of α. The eason fo the degeneation of the ageement between the two models with inceasing values of α becomes obvious when examining the stess plot in Fig. 10. The adial stess has to vanish at the boundaies causing a petubation to the even stess distibution. The petubation is modeled well by the FE method but is not consideed by the simplified one-dimensional model. Thus, the shape plotted in Fig. 10 is optimal although the stesses ae not even thoughout the domain. Equation (17) also implies that the capacity of an evenly stessed flywheel fo stoing kinematic enegy pe unit mass equals the specific stength of the mateial, U M = σ = σ ϱ. (4)

7 80 Fig. 8 Relative values of the aeas of the inne (1) and oute () ings and the disk Fig. 9 Optimum shapes fo diffeent values of the diamete atio α

8 81 objective of even stessing. A simplified analytical model yields ough shape featues of the maximum stength design and elucidates the undelying essential mechanisms. Accoding to the simplified model, an even-stess shape design does not exist fo boe adii geate than the squae oot of one-thid of the adius of the flywheel. The optimum-shape pedictions of both models agee well if the adius of the boe does not exceed appoximately one thid of the adius of the disk. The stess level in the optimized design is appoximately only one thid of the maximum cicumfeential stess at edge of the boe in a flywheel of constant thickness. The specific enegystoing capacity of the evenly stessed flywheel accoding to the closed-fom solution of the simplified model equals the specific stength of the mateial being used. Fig. 10 Stess distibutions fo α =0.5 accoding to the FEM-solution This povides an immediate estimate of the enegy stoage capacity pe unit mass at optimum shape fo any given mateial. 6 Conclusions A pocedue based on FEM-analysis fo finding a maximum stength design fo flywheels with cental boe has been coded in FORTRAN and explained. The pogam yields satisfactoy esults which appoximate the Acknowledgements The autho would like to thank Ms. Bigit Wagne fo text impovements. Refeences Baie, H.; Seesselbeg, Ch.; Specht, B. 1994: Optimieung in de Stuktumechanik. Baunschweig, Wiesbaden: Vieweg & Sohn Reklaitis, G.V.; Ravindan, A.; Ragsdell, K.M. 1983: Engineeing optimization: methods and applications. NewYok: Wiley-Intescience Stodola, A. 194: Dampf- und Gastubinen. Belin, Heidelbeg, New Yok: Spinge Vandeplaats, G.N. 1984: Numeical optimization techniques fo engineeing design: with applications. New Yok: McGaw- Hill