DEVELOPMENT AND COMPARISON OF HIGH ORDER TOROIDAL FINITE ELEMENTS FOR CALCULATING DISC SPRINGS. Christoph Wehmann, Frank Rieg

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1 URN (Pape: un:nbn:de:gbv:ilm-20iw-052:3 56 TH INTERNATIONAL SCIENTIFIC COLLOQUIUM Ilmenau Univesity of Technology, 2 6 Septembe 20 URN: un:nbn:gbv:ilm-20iw:5 DEVELOPMENT AND COMPARISON OF HIGH ORDER TOROIDAL FINITE ELEMENTS FOR CALCULATING DISC SPRINGS Chistoph Wehmann, Fan Rieg Univesity of Bayeuth Depatment of Engineeing Design and CAD Univesitätsst. 30, Bayeuth ABSTRACT The pesent manuscipt deals with the development and the application of tooidal finite elements. These finite elements belong to the class of cuvilinea elements and, theefoe, a special element fomulation is equied. This means to expess stess, stain, and displacement in cuvilinea coodinates. Hee, an 8-node quadilateal element is developed which is descibed by an isopaametic intepolation. The development of the govening equations is one main topic of this contibution. Anothe main topic is to evaluate the quality and efficiency of these elements with espect to the calculation of disc spings. In this context, calculations with tooidal elements ae checed against expeimental data which consists of load-deflection-cuves. In addition, the effects of fiction and diffeences to the exact ectangula coss section ae consideed. With the help of Ris pocedue, chaacteistic cuves with negative sping ates ae obtained. Index Tems Tooidal finite elements, cuvilinea coodinates, covaiant deivatives, disc spings, geometical nonlineaities. INTRODUCTION Tooidal finite elements ae vey inteesting in many applications of mechanical engineeing. They tae advantage of a special symmety, the otational symmety. Machine elements lie disc spings o complete aggegates lie pessue vessels featue this symmety and, theefoe, they ae pedestinated to be calculated by tooidal finite elements. The pesent contibution deals with high ode tooidal finite elements which ae used fo calculating disc spings. Disc spings ae widely used in constuction of machines, vehicles, and appaatus and thei special mechanical behaviou equies numeical methods in ode to calculate displacements and stesses. 2. BASIC MECHANICAL EQUATIONS The basis fo deiving a finite element fomulation is epesented by the equilibium equations. Expessed by a vaiational pinciple, these equations can have the fom of equation (, compae [2]. This fom esults as the undefomed shape is used fo efeencing and the Lagangian stain λ is choosen fo descibing the mateial behaviou. Lie it is shown hee, it is valid fo static analysis without body foces. S δλ dv = t δu da ( V Heein S name the second Piola-Kichhoff s stesses, δλ ae the vitual Lagangian stains, and δu ae the vitual displacements. The stess vecto is symbolized by t. Choosing the undefomed shape as the efeenceconfiguation means the Lagangian way of efeencing is applied. It follows that integation in left pat of equation ( has to be adopted to the volume V of the undefomed finite element espectively disc sping. In the same way, the suface integation in ight pat of equation ( opeates on the suface of the undefomed disc sping. This suface aea A is the suounding suface of the whole finite element model (hee the model of the disc sping, excluding pats of this aea with inematic bounday conditions. The Lagangian stain λ can be obtained by consideing a mateial line element and its change of quadat of length. This change of length is shown in equation (2. l 2 L 2 = dx dx dx dx (2 By intoducing the Lagangian stain λ as a esult of a Taylo s ow (see (3, ( T x x 2 λ = (3 the change of length can be expessed as follows: dx dx dx dx =2dX λdx (4 A c 20-TU Ilmenau

2 Fig.. Definitions of DIN 2092 [] fo descibing disc spings Recalling that x = X u, the Lagangian stain can be expessed as it is displayed in (5. Hee, it can be calculated fom the displacement u. The position of each mateial point at the undefomed configuation is symbolized by X and it yields that the displacement field u has to be fomulated as a function of X, u = u(x. λ = 2 ( ( T ( T (5 Assuming that stains emain small, although displacements might be lage, a linea elationship between stesses and stains can be applied. The coesponding constitutive equation is Hooe s law, which is shown in (6. Hee, Hooe s law is genealized to the theoy of lage displacements. S = C λ (6 The intoduced quantity C can be calculated by two mateial paametes, which ae the Young s modulus E and the Poisson atio ν. 3. FORMULATION IN CURVILINEAR COORDINATES The special symmety of tooidal elements equies to adapt the basic mechanical equations pesented in section 2 to cylindical coodinates. Theefoe, the displacement field u needs to be expessed in these cuvilinea coodinates as it is displayed in equation (7. u = u e u ϕ e ϕ u z e z (7 In this equation u,u ϕ, and u z epesent the displacements in,ϕ, and z diection, while e, e ϕ, and e z build up the coesponding coodinate basis. Next, expessions fo descibing stain by cylindical coodinates ae deived. Fo that pupose, the following identity, taen fom [3], is used. λ = i,j λ ij e ij = i,j Λ ij G ij (8 the Lagangian stains, measued in covaiant coodinates. In this manne e ij name the unit matices of the catesian basis and G ij descibe the unit matices of the contavaiant basis. Equation (9 and (0 give examples fo both, the catesian and the contavaiant basis. Note that in (8 λ epesents the matix fomulation of stain. e 2 = ( G 2 G 3 G 23 G 2 G 3 2 G 2 G 3 3 = G 2 2G 3 G 2 2G 3 2 G 2 2G 3 3 (0 G 2 3G 3 G 2 3G 3 2 G 2 3G 3 3 By eplacing the supescipts 2 and 3 in (0 though i and j, the geneal expession fo G ij is obtained. The items G i of this esulting expession symbolize the coodinates of the unit vecto G i, which belongs to the contavaiant basis. The exact definition is shown in (. G i i G i = = i ( G i 2 G i 3 2 i 3 The coesponding covaiant basis vectos G i G i = G i G 2 i i = 2 G 3 i (2 i 3 i behave invese to G i, what means that thei dot poduct become one o zeo and can be expessed by the Konece-symbol. Thus, equation (3 follows, compae [4]. G i G j = δ ij (3 3.. Stain measuement in covaiant coodinates With the help of a Taylo s ow, an infinitesimal mateial line element dx can be expessed by the covaiant basis, compae equation (4. In equation (8 λ ij denote the Lagangian stains, measued in catesian coodinates, wheeas Λ ij stands fo dx = i G i dξ i (4

3 Next, it is shown that the coefficients Λ ij ae applicable to measue stain in covaiant coodinates. Theefoe, (4 and (8 ae inseted into (4. Combining the esult with (2 yields to equation (5. 2 dx λdx =2 G i dξ i Λ j G j G l dξ l (5 i,j,,l Applying ( and the geneal fom of (0, the matix vecto poduct in equation (5 will simplify accoding to (6 if elation (3 is accounted. G j G l = G j δ l (6 Finally, afte using (3 a second time, equation (7 is obtained. 2 dx λdx =2 i,j dξ i Λ ij dξ j (7 This elation points out that the coefficients Λ ij ae suitable fo descibing stain in covaiant coodinates. In ode to get a calculation ule fo these coefficients, it is equied to expess (2 by (4. The esulting equation is pesented in (8. l 2 L 2 = ( x x dξ i dξ j ij i j i j (8 Accounting that x = X u, equation (8 yields to (9. l 2 L 2 = ( ij i j i j i j dξ i dξ j (9 Next, the definition of the covaiant basis (2 is inseted to achieve equation (20. l 2 L 2 = ( G i G j ij j i (20 dξ i dξ j i j The patial deivatives in (20 have to be expessed by components elated to the covaiant coodinate basis in ode to calculate the dot poducts efficiently. Theefoe, the geneal fom of patial deivation in cuvilinea coodinates (2 which is taen fom [4] is applied. i = l ( i Γ liu l G (2 Theein the coefficients Γ li ae commonly nown as Chistoffel-symbols. Concening cylinde coodinates, they ae listed in [4] and this souce is used heeafte. Compaing (2, (4, and (7 the fomula (22 fo calculating stain in covaiant coodinates is found. Below, the explicit expession of Λ will be developed step by step, while the othe coodinates of λ will be limited to compact pesentation. Λ ij = ( G i G j (22 2 j i i j Because the covaiant basis is othogonal when using cylinde coodinates, the coodinate Λ educes to (23, accounting equation (2. Λ = (( Γ 2 u ( l G Γ u G ( l 2 Γ l u G ll (23 The quantities G ll symbolize dot poducts as it is illustated below. G ll = G l G l (24 Thei values in case of cylinde coodinates (25 ae taen fom [4]. G =, G 22 = 2, G 33 = (25 Futhemoe the following Chistoffel-symbols ae equied. Γ =Γ 2 =Γ 3 =0 Γ 2 =0, Γ 2 2 =, Γ2 3 =0 Γ 3 =Γ 3 2 =Γ 3 3 =0 (26 Finally, it is necessay to tansfom the contavaiant coodinates of displacement and substitute the deivatives with espect to ξ i by deivatives with espect to, ϕ, and z. u = u, u 2 = u ϕ, u3 = u z =, = 2, = 3 (27 Altogethe, this yields to equation (28, which is eady fo discetization by finite elements. Λ = (( 2 ( 2 ϕ 2 ( 2 (28

4 The othe coodinates Λ ij can be developed similaly, the esulting expessions ae shown below. 2 Λ 22 = ϕ u (( 2 u ϕ ( 2 ( 2 (29 ϕ u Λ 33 = z (( 2 ( 2 ϕ 2 ( 2 (30 Λ 2 =Λ 2 = ( 2 u ϕ ϕ ( u ϕ ( ϕ ϕ u z z Λ 23 =Λ 32 = ( ϕ 2 z ( u ϕ ( ϕ ϕ u z z Λ 3 =Λ 3 = 2 ( ϕ ϕ z z 3.2. Stain measuement in physical coodinates (3 (32 (33 Concening cylinde coodinates, thee ae thee diffeent possibilities to define the coodinate basis. It can be defined by the contavaiant basis, the covaiant basis o the physical basis. The advantage of the contavaiant basis and the covaiant basis is thei common mathematical handling. Consideing the teatment of the deivatives in equation (2, it would be almost impossible to use the physical basis. Howeve, the contavaiant basis and the covaiant basis also have a disadvantage, which foces to switch to the physical basis. The lengths of the basis vectos aen t one, the basis vectos aen t unit vectos [4]. By contast, the basis vectos of the physical basis ae unit vectos. This is why coodinates of the stain λ finally have to be expessed elated to the physical basis. Othewise it would be impossible to gain a fomulation of the basic equation (, which is educed to a pue coodinate fomulation. Such a fomulation is equied fo implementation in a finite element code. The geneal tansfomation ules between covaiant and physical coodinates ae given in [4]. These ules ae applied heeafte to the Lagangian stain λ. Λ =Λ, Λ ϕϕ = 2 Λ 22, Λ zz =Λ 33 Λ ϕ = Λ 2, Λ ϕz = Λ 23, Λ z =Λ 3 (34 The supescipt in equation (34 denotes that these quantities belong to the physical coodinate basis. With the help of these tansfomation ules, the stain coodinates elated to the physical basis can be obtained as following. Λ = (( 2 ( 2 ϕ 2 ( 2 (35 Λ ϕϕ = ϕ u (( u ϕ ( 2 ( 2 (36 ϕ u Λ zz = z (( 2 ( 2 ϕ 2 ( 2 (37 Λ ϕ =Λ ϕ = ( 2 u ϕ ϕ ( u ϕ ϕ ( ϕ u z z Λ ϕz =Λ zϕ = ( ϕ 2 z ( u ϕ ϕ Λ z =Λ z = 2 ( ϕ u z z ( ϕ ϕ z z (38 (39 (40

5 3.3. Mateial behaviou and calculation of stesses Having obtained the equations fo Lagangian stain coodinates elated to the physical basis of the cylinde coodinate system, the mateial behaviou has to be specified in ode to get equations fo calculating stesses. By choosing the physical coodinate basis fo stains and stesses, the magnitudes of the seveal stain and stess components ae completely stoed in thei coodinates and this is why it is sufficient to expess the mateial equations diectly in coodinates of stains and stesses. The coesponding basis matices which ae shown fo example in (8, can be cutted out. Concening the mateial behaviou, Hooe s law is applied to connect Lagangian stain and second Piola- Kichhoff s stess. Second Piola-Kichhoff s stess has to be taen, because it is wo-conjugated with the Lagangian stain, compae [3]. In liteatue, this constitutive equation which esults fom a genealization of Hooe s law to the theoy of lage displacements is also named St. Venant s law [2]. In classical linea theoy of elasticity, the catesian coodinates of stess and stain ae simply eplaced by the coesponding physical cylinde coodinates to get the Hooe s law in cylinde coodinates. If the Lagangian stain coodinates ae elated to the physical cylinde coodinate basis, it will be possible to do the same in nonlinea elasticity. Using Lamé s paametes, Hooe s law becomes (4. S ij = λ ( Λ Λ ϕϕ Λ zz δij 2μ Λ ij (4 Theein λ and μ symbolize Lamé s paametes, which ae defined by (42, see [6]. They can be used instead of Young s modulus E and Poisson s atio ν. Eν λ = ( 2 ν(ν E μ = 2(ν 4. FINITE ELEMENT DISCRETIZATION (42 This chapte deals with the development of an 8-node tooidal element fo numeical solving of the cuvilinea mechanical fomulation above. The following section is based on theoy and implementation of the Z88- element tous no. 8 [5]. Fist of all, the same shape functions N i ae used to intepolate displacements. Because of the otational symmety, the integation in ϕ- diection can be done analytical and a two-dimensional discetization is sufficient fo modeling disc spings o othe machine elements with the same symmety. Accodingly, the displacement (7 simplifies to (43 and evey node has two degees of feedom. u = u e u z e z (43 Figue 2 illustates shape, node position, and coodinate system oientation of the applied Z88-element tous no. 8. Z(=Y R (=X Fig. 2. Tooidal element No. 8 of Z88 [5] It follows that the shape functions fo intepolation of displacements loo lie equation (44 displays. u = N u, u z = N u z (44 Next, stesses and stains ae expessed in Z88-notation (45 fo tooidal elements. S = S Szz Sz Sϕϕ, λ = 2 Λ Λ zz Λ z Λ ϕϕ 3 (45 Because of the symmety (46, the coodinates Λ ϕ and Λ ϕz disappea. Due to the isotopic mateial behaviou the coesponding stess coodinates S ϕ and S ϕz become zeo, too. ( =0, uϕ =0 (46 Intoducing the displacement-stain tansfomation matix B, B = λ (47 whee u stands fo the nodal displacement vecto, leads to following fom of the equilibium equations (. S T B dv δu = F T δu (48 V In addition, the nodal foce vecto F is intoduced in (48. Finally, by demanding (48 to be valid fo abitay vaiations δu, the algebaic equation (49 is obtained. B T S dv = F (49 V

6 Foce F / N 25,000 20,000 5,000 0,000 5,000 Calculation of chaacteistic cuves with tooidal element no. 8 h h = 4.5 mm h = 4.0 mm h = 3.5 mm h = 3.0 mm h = 2.5 mm h = 2.0 mm h =.5 mm h =.0 mm h = 0.5 mm h = 0.0 mm h Displacement s / mm Fig. 3. Chaacteistic cuves of disc spings with diffeent heights, calculated by tous no. 8 The next chapte deals with the numeical solution of this equation by using Newton-Raphson s and Ris pocedues. 5. CALCULATION OF DISC SPRINGS In this chapte, the tooidal element developed above is applied to seveal disc spings. These disc spings diffe fom thei height h, which is the distance in s- diection between points II and III (see figue. The othe geometical paametes ae shown in (50. D e = 00.0mm, D i =5.0mm t =2.7mm (50 Fo discetization of this stuctues the mapped meshe Z88N is used, who geneates the meshes on the left of figue 3. Each of these meshes contains 200 tooidal elements, five in axial diection and 40 in adial diection. Disc spings with small heights h 2.5 mm can be calculated by Newton-Raphson-pocedue, while fo spings with lage heights the pocedue of Ris is equied. This chaacteistic is easoned by the negative sping ate of disc spings with lage heights h. The Newton-Raphson-pocedue is not capable of ovecoming points nea to the local maximum, because it is contoled by applied foce. The same poblem is obseved by Niepage, who suggests to use a pocedue contoled by displacement [7]. In this context, the pocedue of Ris can be applied. It is contoled by a given ac length incement, which can be a foce incement, a displacement incement o a composition of foce and displacement incement. It follows that Ris pocedue enables calculating chaacteistic cuves up to high foces and displacements fo disc spings with negative sping ates, too. Concening the mateial paametes on which the calculations of figue 3 ae based, the following values fo Young s modulus E and Poisson s atio ν ae used. E = 20, 000 N/mm 2, ν =0.29 (5 On the left side of figue 3, the diffeent finite element meshes ae shown. Tooidal elements geatly simplify modeling of disc spings: only the coss section has to be meshed. With egad to the bounday conditions, one foce and one given displacement exist. The foce is applied to the node on point I, while the displacement of the node on point III (see figue is set to zeo. 6. EXPERIMENT The quality of the esults obtained by the pesented tooidal element will be checed against expeimental data in chapte 7. Hee, it is illustated, how this expeimental data is gained. The disc sping is loaded by a hydaulic system, while beaing and loading only is applied to a small annula suface. Thus, high deflections, which ae lage than h, can be eached. Measuement of foce is pefomed by a load cell, which is placed between hydaulic cylinde and disc sping. A dial indicato is used to measue the deflection of point I. 7. COMPARISON OF CALCULATION AND EXPERIMENT This chapte deals with the esults of the measued and calculated chaacteistic cuves of the disc sping (52. D e = 00.0mm, D i =5.0mm t =2.7mm, h =3.5mm E = 206, 000N/mm 2, ν =0.3 (52

7 Foce F / N 25,000 20,000 5,000 0,000 Chaacteistic cuves of disc sping, measued and calculated Expeiment Tous no. 8 Tous no. 8, KA = 0.2, = 0.3 Tous no. 8, KA = 0.3, = 0.3 Tous no. 8, KA = 0.4, = 0.2 5, Displacement s / mm Fig. 4. Measued chaacteistic cuve compaed to calculations with tous no. 8, taen fom [8] (modified Figue 4 illustates the coesponding chaacteistic cuves. The last thee cuves, labeled by KA and μ tae additional bounday conditions into account. One of these conditions is a displacement of the applied foce and the beaing easoned by a diffeence to the exact ectangula coss section. Actually, the cones of disc spings have small oundings and, hence, the applied foce acts futhe outwads, while the beaing is located futhe inwads. The displacement of foce and beaing position is labeled KA and measued in mm, so KA = 0.2 means a displacement of 0.2 mm. The othe additional bounday condition is a fiction foce, which acts adially on the oute diamete. The coesponding fiction paamete μ which is used fo the calculations is shown in figue 4. The esults point out that with these additional effects the chaacteistic cuve can be calculated exactly up to a deflection of s h 3.5mm. Without accounting oundings and fiction, the stiffness at s < h is less than measued. At highe deflections s>h, all cuves lay upside the measued data, whethe additional bounday conditions ae accounted o not. 8. CONCLUSION AND OUTLOOK In tems of development tooidal elements ae sophisticated and equie finite element fomulation in cuvilinea coodinates. The coesponding set up of equations is pesented in this pape. Having implemented an 8-node quadilateal element based on Z88 element type no. 8, calculation esults with high qualities ae obtained. Due to the application of finite element calculation, it is possible to account additional bounday conditions, which leads to nealy exact chaacteistic cuves up to deflections s h. At highe deflections s>hthe calculated stiffness is highe than obseved expeimentally. This pobably might be explained by inelastic mateial behaviou. With egad to applications of disc spings accounting inelastic pocesses might be of less inteest, but with egad to maufactuing pocesses, it is an impotant topic fo futhe eseach. 9. REFERENCES [] DIN 2092 Tellefeden. Beechnung, Beuth, Belin, 992 [2] Wigges, P. Nichtlineae Finite-Element- Methoden, Spinge, Belin, 200 [3] Paisch, H. Festöpeontinuumsmechani. Von den Gundgleichungen zu Lösung mit Finiten Elementen, Teubne, Stuttgat, 2003 [4] Neemann, K.; Schade, H. Tensoanalysis, de Guyte, Belin, 2006 [5] Rieg, F.; Hacenschmidt, R. Finite Elemente Analyse fü Ingenieue, Eine leicht veständliche Einfühung, Hanse, München, Wien, 2009 [6] Landau, L. D.; Lifschitz, E. M. Lehbuch de theoetischen Physi, Band VII, Elastizitätstheoie, Aademie Velag, Belin, 99 [7] Niepage, P. Vegleich veschiedene Vefahen zu Beechnung von Tellefeden, Daht 34, S , , 983 [8] Zimmemann, M. Anwendung de Finite- Elemente-Analyse zu Emittlung nichtlineae Fedeennlinien, Diplomabeit, Univesität Bayeuth, 20