DYNAMIC MODELLING OF N-CARDAN TRANSMISSIONS WITH SHAFTS IN SPATIAL CONFIGURATION. Part II. THE ALGORITHM OF DYNAMIC MODELLING

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1 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 DYNAMIC MODELLING OF N-CARDAN TRANSMISSIONS WITH SHAFTS IN SPATIAL CONFIGURATION. Part II. THE ALGORITHM OF DYNAMIC MODELLING Cdrua JALIU, Radu S ULESCU, Livia HUIDAN Transilvania University f Bv, cjaliu@unitbv.r Keywrds: general algrithm, cmputatinal dynamic mdelling, -Cardan transmissin Abstract: A general algrithm fr the dynamic mdelling f Cardan transmissins is presented in this paper. This algrithm is based n the assembling f analytical mdels (f mdular type), previusly established, and it is exemplified n a representative -Cardan transmissin, with values frm cmpanies catalgues. The mdelling is made in the premise that the intermediary Cardan shafts are elastic; it is als analyzed the case in which the intermediary shaft is rigid.. INTRODUCTION A library f mdular crrelatins regarding the frces frm the typical mdules f Cardan transmissins was elabrated in the first part f this paper; the establishment f an algrithm fr n-cardan transmissins and the simplificatin f the mdelling f frces and mments becme pssible by assembling these crrelatins. A general algrithm fr the establishment f frces and mments frm the classical - Cardan transmissins, statically determined it is further presented in the premise that the intermediary shaft is elastic, by assembling the kinematical crrelatins with the mdular nes. The technical data that are used in mdelling this transmissin are taken frm cmpanies catalgues [3]. Firstly, fr a distinct highlighting f the elastic intermediary shaft influence, the inertial effects are neglected. The prpsed generalized algrithm is applied n the example f the -Cardan transmissin frm Fig.. c M M Fig.. Variant f -Cardan statically determined transmissin, with an elastic intermediary shaft 77

2 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7. NOTATIONS AND PREMISES The ntatins and premises used in the previus paper are further valid. The fllwing tw premises are added: The intermediary shaft is cnsidered elastic and the inertia mment is neglected. The elasticity f the intermediary shafts is described thrugh the rigidity cefficient: cij = T j ij [Nm/ in which T j represents the mment f rtatin fr the j frk, and ij designates the angular defrmatin, meaning the angular difference between the rtatin angles f the ij intermediary shaft frks: ij = i j () Fr the intermediary shafts made f steel tube, the rigidity cefficient and the inertia mment have a well-knwn expressin [7]: 4 4 G ( D d ) c = [Nm/ in which: 3l l ij [mm] represents the length f the intermediary Cardan shaft, G = 8 [N/mm ] represents the mdule f transversal elasticity fr steel, D, d [mm] represents the external diameter and internal diameter, respectively, f the intermediary Cardan shaft. 3. ADAPTATION OF KINEMATICAL CORRELATIONS TO THE PREMISE OF ELASTIC SHAFTS Unlike the kinematical mdelling f the n-cardan transmissins with rigid elements, in the mdelling f the transmissins with elastic intermediary shafts, the equalities i = j are transfrmed in equalities f j = i ij type; the ther crrelatins remain unchanged. Fr exemplificatin, the kinematical crrelatins necessary in mdelling the 3- Cardan transmissins with elastic intermediary shafts are further presented: =, tg i = arctg, = + 9 csi j ij i ij ; () tg ( + ) tg i = arctg i cs, j = i ij ; (3) i sin cs = tg i sin( + ) ( + i ), tg i = sin cs ( + ) ; (4) i Taking int accunt the fact that the changes f the mments f rtatin fr an intermediary shaft is usually reduced in relatin t its medium value, the simplificatin cnstant can be admitted; therefre, the angular speeds and acceleratins can be deduced with the fllwing relatins: Ai i =, Ai =, j i ; (5) + A sin + csi ( i ) ( ) + i ( Ai ) sin ( + ) + ( A ) sin ( + ) i A i i =, j i. (6) i 773

3 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 4. ADAPTATION OF THE GENERAL ALGORITHM TO THE PREMISE OF ELASTIC INTERMEDIARY SHAFTS On the base f the mdular crrelatins established in the first part, the fllwing general algrithm fr the mdelling f the frces and mments frm the statically determined -Cardan transmissins is further prpsed (Fig. ): There are knwn: the kinematical scheme f the n-cardan transmissin and its state quantities:,, (independent mtin), = ;, ; ; ; l ; J ; r; a, b, J ; ; a ij, b ij, J n ; F µij ; M n (independent mment). The quantities, ;, i ;, i, ij and ij fr the cmpnent cuplings are established i [8]. Starting frm the utput shaft n and based n rel. ( part - I), the mments f rtatin (T, ), bending mments (B, ) and resultant mments (K, ) that lad each Cardan shaft frk are successively established in terms f the resistant mment M n. The resultant bending mment f an intermediary shaft ij is btained thrugh a vectrial summing f the bending mments frm the ij shaft frks: Bi and B j (see Fig. 5 part I). The axial (A, ) and radial (H, >i R, ) frces f each shaft s frks are established, establishing, at the beginning, the axial and radial frces fr the frks i and j f the unsupprted telescpic shaft ij, n the base f rel. (5, part I); The reactins frm the extremal shafts bearings (input and utput) are calculated with rel. (9, part I). On the base f the frces and mments frm the Cardan frks (T, B, K; A, R, H) and based n rel. (3 and 4, part I), the frces that are specific t the cntact znes between the frks and the crsses are established. In the premises f elastic intermediary shafts, the relatins in which interferes the influence f rigidity c are mdified. Thus, the mment f rtatin n a Cardan shaft, cnsidered elastic, becmes: Tj = ij cij (7) Fr exemplificatin, the algrithm fr -Cardan transmissins like thse frm Fig., in a mdular variant, is presented synthetically in Fig., as a flw chart. In the case in which the intermediary shaft is cnsidered rigid, its elasticity becmes null, the presented algrithm being applicable in this kind f situatins, as well. Calculus prgrams can be elabrated n the base f the prpsed algrithm, using different cmmercial sftware. This kind f prgram, elabrated in Excel, was run fr a representative set f numerical data fr the cnsidered -Cardan transmissin; this data are written abve the Figures. As an example f numerical simulatin, diagrams fr the bending and rtatin mments frm this transmissin are further presented (see Fig and ). 5. NUMERICAL SIMULATIONS After running the prgram which was elabrated n the base f the previus algrithm and the Excel sftware, a set f representative diagrams was btained, part f which being illustrated in Fig. 3 and 4. These diagrams illustrate the variatins f the angular differences (Fig. 3, a and 4, a) and rtatin mments (T ij ) (Fig. 3,a, a and 4,a, a ) fr the analyzed example, bth in dynamic cnditins (cnsidering the inertia mments f the Cardan shafts) and in static cnditins (neglecting the inertial effects). The cnsidered state parameters are written abve each Figure. 774

4 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 MOMENTS and FORCES fr the transmissin with universal jints (Fig. ). Input data:,, ; = ;, ; ; ; c ; l, J ; r; a, b, J ; a, b, J ; F µ ; M. = ; = ; =,, =, ;,i i, i =, (vezi []) T, B, K ; T, B, K ; T, B, K, T, B, K ; M. (with rel. () frm Part I) with rel. (7) = ;,,, ;,i i, i =, ; (see []). T, B, K ; T k, B, K ; T, B, K, T, B ; M. (with rel. () frm Part I) i = i + NO ( i) ( ) = e YES R, H, A ; R, H, A (with rel. (5) frm Part I) R, H, A ; R, H, A (with rel. () frm Part I) P, U, V, S ; P, U, V, S (with rel. (6) frm Part I) F', F", N', N", Q; F' i, F" i, N' i, N" i, Qi ; i =, A', A", R', H', H" ; A' i, A" i, R' i, H' i, H" i ; i =, (frm rel. () frm Part I) Fig.. The flw chart f the algrithm fr the mdular mdelling f a -Cardan transmissin, in the premise f elastic intermediary shaft 775

5 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 = = 35, = [ rad / s], = [ rad / s =, =, = 8, ], M = 5[ Nm] c = [ Nm/ J = [ kgm ], 4 c = [ Nm/ J = 3,4 [ kgm ],4,,8,6,4 fi -fi fi -fi, f [grd] ,4,,8,6,4 fi -fi fi -fi, f [grd] a. a T, f T, f T T T T T T T T -9-9 b. b. 5, 5,5 M, -T, -T 5, 5,5 M, -T, -T 5, 5, 5,5 5, ,99995 M 49, ,9999 T 49,9999 M 49,99985 T T 49,99985 T f f 49, , c. c. Fig. 3. The variatins f the angular differences between the intermediary shaft s frks ( ) and the rtatin mments (T ij ), in the premise f cnsidering a rigid intermediary shaft: withut inertial effects (a...c) and with inertial effects (a...c ). 776

6 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 = = 35, = [ rad / s], = [ rad / s =, =, = 8, ], M = 5[ Nm] c = 956,896[ Nm/ J = [ kgm ], c = 956,896[ Nm/ J [ kgm ],4,,8,6,4 fi - fi fi - fi, f [grd] ,4,,8,6,4 fi - fi fi - fi, f [grd] a. a T, f T, f T T T T T T T T -9-9 b. b. 5, 5,5 M, -T, -T 5, 5,5 M, -T, -T 5, 5, 5,5 5, ,99995 M 49,99995 M 49,9999 T 49,9999 T 49,99985 T 49,99985 T f f 49, , c. c. Fig. 4. The variatins f the angular differences between the intermediary shaft s frks ( ) and the rtatin mments (T ij ), in the premise f cnsidering an elastic intermediary shaft: withut inertial effects (a...c) and with inertial effects (a...c ). Frm these numerical simulatins result, n ne side, the variatins f the angular differences between the intermediary Cardan shaft s frks and f the rtatin mments that lad the elements f the -Cardan transmissin in terms f the rtatin angle f the input shaft and, n the ther side, the influence f the Cardan shafts inertia n the rtatin mments, by nullifying the inertia mments J, J si J. 777

7 Fascicle f Management and Technlgical Engineering, Vlume VI (XVI), 7 The cmparative analysis f the diagrams frm Fig. 3 and 4 highlights the fact that the influence f the Cardan shafts inertia, in nrmal cnditins f use is practically negligible. Thus, in static cnditins (Fig. 3,c and 4,c), the input rtatin mment (T ) cincides, in mdule, with the utput rtatin mment (M =5 Nm); in dynamic cnditins (Fig. 3,c and 4,c ), the mment T has small sinusidal scillatins that are practically negligible. 6. CONCLUSIONS In the cnditins f the cnsidered numerical example, it can be bserved that the elasticity f the intermediary shaft influences mainly the transmissin s kinematics and has a reduced influence n the rtatin mments (see Fig. 4,a, a and c ). Therefre, the main cnclusins that result frm the numerical simulatins systematized in Fig. 3 and Fig. 4, can be synthetically frmulated as fllws: The analyzed transmissin is hmkinetic due t the fact that the input and utput rtatin mments cincide in mdule, in the cnditins f neglecting frictin and inertial effects (T =T see Fig. 3,c and 4,c); in the mean time, the amplitude f the rtatin mments f the intermediary shafts frks (T, T ) describes the dynamic lad f the same shaft. While the extremal Cardan shafts are laded with cnstant r apprximately cnstant rtatin mments, the intermediary shaft is laded with an scillating rtatin mment, f csinusidal type, whse amplitude is f apprx. 3 % frm the medium value. In usual applicatins, the axial inertia mments f the Cardan shafts have insignificant influence n the transmissin frces and mments (see Fig. 3,c and 4,c ). Accrding t Fig. 4,a, and a the variatins f the angular differences that describe the rtatin defrmatins f the intermediary shaft ( ), are reduced, in relatin t their medium variatins; thus, it is justified the simplifying premise ij = cnstant (particular case when the intermediary shaft is cnsidered rigid). The influences f the intermediary shafts elasticity are highlighted in Fig. 4,c, cumulated with their inertial effects. The dynamics f the n-cardan transmissins can be apprached similarly. References [] DudiGH, Fl., Diacnescu, D.V., Neage, M., SHulescu, R.: On the mdelling f hmkinetical tricardanic transmissins. nd Internatinal Cnference Research and develpment in mechanical industry RaDMI, Yugslavia, ISBN , vl II, pp [] Diacnescu, D., SHulescu, R., Brca, A. Abut kinematic analysis and synthesis f 3-Cardan transmissin. Annals f the Oradea University. Fascicle f Management and Technlgical Engineering. Vl II (XII), 3, ISSN , pag 4 (Abstract). [3] *** GWB Gelenkwellen für Industrie-Anwendung. Gelenkwellenbau GmbH-Essen, KGN Autmtive, Cardantec Divisin. 778