A Design for the Pitch Curve of Noncircular Gears with Function Generation

Transcription

1 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong A Design for the Pitch Curve of Noncircular Gears with Function Generation Jen-Yu Liu and Yen-Chuan Chen Astract In this paper, we employ the Fourier series to design the pitch curve of noncircular gears with function generation. With the speed ratio is represented as Fourier series, the nonlinear function of angular displacements can e acquired through integration. And then, ased on known finite separated angular displacements of two gears, the order of Fourier series is decided and a set of simultaneous equation is otained. By solving the simultaneous equations, the Fourier series function regarding angular displacement and speed ratio can e otained. Finally, synthesizing these two formulas creates the pitch curves for the noncircular pinion and gear, respectively. In this research, we demonstrate two designs examples to interpret the design process of the pitch curves of noncircular gears. In addition, the results derived from this paper can e taken as a reference to design the pitch curves of noncircular gears with function generation. Index Terms pitch curve, noncircular gears, function generation. I. INTRODUCTION The applications of noncircular gears are often applied in the transmission systems which may need variale speed ratio, nonlinear motion function, or variale input and output loadings, respectively. Basically, most of those applications are using elliptic gears therefore the elliptical pitch curve is more arresting [-]. In the field of research, most studies focus on the elliptic and the modified elliptic gears [3-]. For instance, Emura and Arakawa [-5] employed the elliptic and the modified elliptic gears at a design of the steering mechanism for automoiles and ships. Taking the linkage mechanism with a sliding slot as an example, Danieli [6] concluded a relationship of the speed ratio etween drive and driven links, and further constructed the pitch curves of the equivalent noncircular gears according to this relationship. Dooner [7] used the noncircular gears to eliminate unnecessary loading in transmission and analyzed the relationship of the loading and the speed ratio of gears with input and output on the asis of the law of conservation of energy. With this relationship, he designed the outline of two pitch curves of gears and calculated the reduced torque on noncircular gears. Usually, the pitch curves of the noncircular gears have to match a specific input/output relationship. The common procedure is to employ a known functional relationship, tooth numer, and modules to synthesize the pitch curves of the noncircular gears which fit this relationship. Sometimes, as anuscript received Novemer 7, 007. This work was supported in part y the National Science Council of Repulic of China for their financially supporting this research under Contract No. NSC 96--E Jen-Yu Liu is with the Power echanical Engineering Department, National Formosa University, 64 Wen-Hwa Road, Hu Wei 6308, Yunlin, TAIWAN. (corresponding author, phone: fax: ; davidliu@ nfu.edu.tw. Yen-Chuan Chen is with the Engineer, Accton Technology Corporation, TAIWAN. ( t4395.chung@msa.hinet.net. regards a design for the pitch curves of noncircular gears, the requirements are to satisfy finite separated angular positions, i.e., input and output angular displacement regarding the finite separated positions. The relevant literatures for designing noncircular gears with such function generation are rare. Owing to the periodicity in the Fourier series, its first order derivative, its second order derivative or aove [8-9], the demand on periodical features for angular displacement, angular velocity, and angular acceleration of the noncircular gear, the Fourier series is an adequate tool to generate the finite separated angular displacement function of the noncircular gears. In this paper, ased on the known finite separated angular displacement position, we construct angular displacement function using Fourier series and deduce the pitch curve equations of noncircular gears meeting this finite separated position according to this function. In addition, some examples are created to interpret the method for designing the pitch curves of noncircular gears with function generation. II. SYNTHESIZING PRINCIPLES OF NONCIRCULAR PITCH CURVES In Figure, the noncircular gears and with angular velocity ω and ω and angular displacement and, respectively, reversely rotate around the fixed axes O and O. Coordinate systems x y and x y are the moving coordinate attached to gears and, respectively. Figure Pitch curves of noncircular gears The rotating-speed ratio m ( is: ω d dt d m ( = = = ω d dt d when =0 then also e zero. Integrating Equation ( with respect to the relationship of and is otained: ( = m d ( 0 The distance from the contacting point P of gear or gear to the rotating center of each gear, and the center distances for two gears are r (, r ( and D, respectively. Based on the ISBN: IECS 008

2 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong synthesizing principle [0], r ( and r ( are acquired as follows: Dm ( = (3 + m r r D = (4 + m Then the Equations of pitch curves of gears and, R ( and R ( respectively, are: R ( = r( cosi + r( sinj Dm ( Dm ( = cos i + m ( + m ( R ( = r ( cos i + r ( sin j Dcos + sin j (5 = + m ( i ( m Dsin j (6 + III. FOURIER SERIES FUNCTION FOR FINITE SEPARATED ANGULAR DISPLACEENTS Tale shows N finite separated angular displacement positions of and. As gear rotates with a constant speed, the angular displacement of gear displays a nonlinear variation. Thus, the rotating speed ratio is not a constant. Science gear and gear are periodical rotating, the rotating speed ratio displays a periodical variation. For an aritrary integral n, Equation should satisfy the following relationship: m ( = m (n π+ (7 Let the period time of the noncircular gear is T, Equation (7 can e represented with order Fourier series [8-9]: d nπ m ( = = A0 + An cos d T nπ + Bn sin T (8 Tale N finite separated angular displacement,,,, 3,3,3 : : : N-,Ν,Ν N,Ν,Ν Sustituting Equation (8 into Equation (, the relationship etween and is: = m ( d TAn nπ = A sin nπ T TBn nπ + cos nπ T (9 From Equation (7, the period of angular displacement of noncircular gears are π, Equations (8 and (9 can e modified as: ( = + ( + Bn sin( n m A A cos n 0 n A = A + sin n (0 n + Bn cos( n 0 n n ( Equation ( must satisfy conditions such as =0 together with =0 and =π together with =π. Sustituting these conditions into Equation (, we can otain A 0 =. In Equation (, let A n /a n, B n / n, A n =na n, and B n =n n. Equations (0 and ( can e written as: ( = + ( + ( m na cos n n sin n n n = + a sin n + cos n n n ( (3 In order to find the values of various coefficients of Fourier series, such as a,, a,,, a, and, respectively, Equation (3 can e modified as : an sin( n + n cos( n = (4 Sustituting various separated positions listed in Tale into Equation (4, we can otain the simultaneous formulas to solve various coefficients of Fourier series: aα, + β, + aα, + β, + L+ aα, + β, =,, aα, + β, + aα, + β, + L+ aα, + β, =,, aα,3 + β,3 + aα,3 + β,3 + L+ aα,3 + β,3 =,3,3 aα + β + a α + β + + α a α,n + β,n + a α,n + β,n + L + a α + β = where,n,3,n,3 L a,n+ β,n =,N,N,N,N,N,N (5 α i,j = sin i,j (6 β = cos i (7 i,j,j In Equations (6 and (7, i and j N. Further, only with the total numer for unknown coefficients of a,, a,,, a, and een equal to N will there e ISBN: IECS 008

3 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong only one set of solutions for the simultaneous formulas of Equation (5. Oviously, if N is an even numer, is N/; however, for an odd N, the situation that one of those unknown coefficients of a,, a,,, a, and must e 0 in order to match the aove condition. Through a numerical computation and analysis, we find when = 0 for the case of an odd N, the solutions of Equation (5 have a etter effect for transmission and facilitates processing of the shape of gears. Thus, N/,N is = (N + /,N is even. odd, = 0. N (8 Equation (5 can e expressed as the following matrix form: where A N N a N = V N (9,,,,,3,3 V N =,N,N,N,N For an even N: α, β, L αn /, βn /, α, β, L αn/, βn/, α,3 β,3 L αn /,3 β N /,3 A N N = O α,n β,nlαn/,nβn/,n α,n β,n L αn/,n βn/,n a a N = a N / N / For an odd N: α, β, L α β α N /, N /, ( N+ /, α, β, L α β α N /, N /, ( N+ /, α,3 β,3 L α( N β /,3 ( N α /,3 ( N+ /,3 A N N = O α,n β,n Lα( N β /,N ( N α /,N ( N+ /,N α,n β,n L α ( N /,N β ( N /,N α ( N+ /,N a N Therefore, (0 ( ( (3 a (4 = a(n/ (N/ a (N+ / a N =A N NV N (5 The solutions for coefficients of Fourier series under even and odd known conditions are displayed from Equation (9 to Equation (5. Design example (N is even For a pair of noncircular gears, the finite separated angular displacements for gears and are shown in Tale. The pitch curve of noncircular gears satisfy these specified angular displacements are designed as what follows. From Tale, owing to 6 finite separated angular displacements, N equals to 6. By Equation (8, = 3. Sustituting data shown in Tale into Equations (0 and (, we can otain: Tale Even finite separated angular displacement π/9 π/9 4π/9 5π/9 3 5π/6 π 4 π/9 5π/8 5 5π/8 3π/ 6 5π/3 3π/ V 6 = A 6 6 ( (7 = Employ Guass ethod [] to get the inverse matrix of Equation (5: A (8 = Sustitute Equations (6 and (8 into Equation (5 to acquire: a a a 6 = = a By Equation (3, it can e otained: (9 ISBN: IECS 008

4 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong 3 3 a sin( n cos( n = + + n n = sin cos sin cos sin cos3 (30 Differentiate Equation (30 with respect to, we otain rotating speed ratio m ( as: 3 3 m ( = na n cos( n + nn sin ( n = cos sin 0.00 cos sin 0.4 cos sin 3 (3 With Equations (30 and (3 sustituted into Equations (5 and (6, the equations of pitch curve of gear and are: Dm R ( = cos + m R ( Dm ( i + m ( Dcos + sin j (3 Dsin j (33 + m = + m ( i Figure shows Fourier series satisfying the finite separated angular displacements y solving Equation (30 wherein the lack old curve is representing Equation (30, and the dotted line is the corresponding values of finite separated angular displacements. Figure 3 indicates the drawn pitch curves of gears and y Equations (3 and (33. Design example (N is odd The finite separated anular displacements for gears and are shown in Tale 3. The design of pitch curves of noncircular gears must satisfy these odd finite separated angular displacements. In Tale 3, 7 known finite separated angular displacements are listed, N equals to 7. By Equation (8, = 4. Sustituting data in Tale 3 into Equations (0, (3, and (4, respectively, we have: Tale 3 Odd finite separated angular displacement π/9 π/9 5π/8 π/3 3 π/9 5π/9 4 5π/6 π 5 π/9 5π/8 6 5π/8 3π/ 7 5π/3 3π/ V 7 = A 7 7 = The reverse matrix of Equation (35 is: A = (34 (35 (36 Sustitute Equations (34 and (36 into Equation (5, we otain: Figure Fourier series function satisfy Tale a a a 7 = = a a Thus, from Equation (3, we have: (37 Figure 3 Pitch curve of gears and y Equations (3 and ( a sin( n cos( n = + + n n = sin cos sin cos sin cos3 ISBN: IECS 008

5 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong sin 4 (38 Differentiate Equation (38 with respect to, the speed ratio m ( is: 4 4 m ( = na n cos( n + nn sin ( n = cos sin cos sin cos sin cos 4 (39 Sustituting Equations (38 and (39 into Equations (5 and (6, the equations of pitch curves of gears and are as follows: R ( = r( cosi + r( sinj Dm ( Dm ( = cos i + m ( + m ( R ( = r ( cos i r sin Dcos = + m ( i + sin j (40 + j Dsin j (4 + m Figure 4 shows Fourier series meeting the finite separated angular displacements y solving Equation (38. In the figure, the lack old curve is related to Equation (38 and the dotted line is corresponding values of finite separated angular displacements. Figure 5 shows the pitch curves of gears and according to Equations (40 and (4 when the center distance D is 00mm. Figure 4 Fourier series function satisfy Tale 3 Figure 5 Pitch curve of gears and y Equations (40 and (4 IV. CONCLUSION In this paper, we provide the angular displacement curves of the noncircular gears y employing Fourier series to synthesize finite separated angular displacements and further synthesize the pitch curves of the noncircular gears. Based on the known finite separated angular displacements, simultaneous equations for the coefficients of order Fourier series which satisfy these odd as well as even finite separated angular displacements respectively are derived. By solving the simultaneous equations, we otain order Fourier series. And the pitch curves of noncircular gears y means of the synthesizing principle for the pitch curves of noncircular gears. Two design examples in this study interpret the method to synthesize the functions and the pitch curves of the noncircular gears with finite separated angular displacements through Fourier series. As a reference for a research of designing the pitch curve of the noncircular gear, this study result can contriute to design a noncircular gear, which meets the function of finite separated angular displacement. ACKNOWLEDGENT The authors would like to thank the National Science Council of Repulic of China for their financially supporting this research under Contract No. NSC 96--E REFERENCES [] Chironis, N.P. (Ed., 967, Gear Design and Application, cgraw-hill, New York. [] Chang, S. L., 99, Design of the Involute Elliptical Gears, aster Thesis, National Tsing Hua University, Taiwan. [3] Kuczewski,., 988, Designing Elliptical Gears, achine Design, April, pp.6-8. [4] Litvin, F.L., 989, Theory of Gearing, National Aeronautics and Space Administration, D.C. [5] Litvin, F.L., 994, Gear Geometry and Applied Theory, Prentice Hall, New Jersey. [6] Chang, S.L., and Tsay, C.B., 995, athematical odel of Elliptical Gear Generated y Shapers, Journal of the Chinese Society of echanical Engineers, Vol.6, No.5, pp [7] Chang, S.L., Tsay, C.B., and Wu, L.I., 996, athematical odel and Undercutting Analysis of Elliptical Gears Generated y Rack Cutters, echanism and achine Theory, Vol.3, No.7, pp [8] Chang, S.L., and Tsay, C.B., 998, Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears anufactured with Shaper Cutters, Transactions of the ASE, Journal of echanical Design, Vol.0, pp [9] Bair, B.W., 00, Computer Aided Design of Elliptical Gears, Transactions of the ASE, Journal of echanical Design, Vol.4, pp [0] Bair, B.W., 00, Computerized Tooth Profile Generation of Elliptical Gears anufactured y Shaper Cutters, Journal of aterials Processing Technology, Vol., pp ISBN: IECS 008

6 Proceedings of the International ulticonference of Engineers and Computer Scientists 008 Vol II IECS 008, 9- arch, 008, Hong Kong [] Bair, B.W., 004, Computer Aided Design of Elliptical Gears with Circular-arc Teeth, echanism and achine Theory, Vol.39, No., pp [] Emura, T., and Arkawa, A., 99, A New Steering echanism Using Noncircular Gears ( st Report, A Proposal and Analysis of echanism, Transactions of the Japan Society of echanical Engineers Series C, Vol.57, No.533, pp [3] Emura, T., Arkawa, A., and Suzuki,., 99, A New Steering echanism Using Noncircular Gears ( st Report, Trail anufacture and Experiments, Transactions of the Japan Society of echanical Engineers Series C, Vol.57, No.533, pp [4] Emura, T., and Arkawa, A., 99, A New Steering echanism Using Noncircular Gears, JSE International Journal, Series III, Vol.35, No.4, pp [5] Arkawa, A., and Emura, T., 994, otion and Power Transmission echanism for Nonuniform otion, Journal of the Japan Society for Precision Engineering, Vol.60 No.0, pp.4~ 47. [6] Danieli, G.A., 000, Analytical Description of eshing of Constant Pressure Angle Teeth Profile on Variale Radius Gear and its Applications Transactions of the ASE, Journal of echanical Design, Vol., pp.3-9. [7] Dooner, D.B., 997, Use of Noncircular Gears to Reduce Torque and Speed Fluctutations in Rotating Shafts, Transactions of the ASE, Journal of echanical Design, Vol.9, pp [8] orrison, N., 994, Introduction to Fourier Analysis, Wiley, New York. [9] Haerman, R., 998, Elementary Applied Partial Differential Equation with Fourier Series and Boundary Value Prolems, Prentice Hall, New Jersey. [0] Tong, S.H., and Yang, D. C.H., 998, Generation of Identical Noncircular Pitch Curves, Transactions of the ASE, Journal of echanical Design, Vol.0, No., pp [] Fraleigh, J.B., and Beauregard, R.A., 995, Linear Algera, 3 rd edition, Wesley, New York. ISBN: IECS 008