Computer aided cam design of roller-follower cam mechanism considering kinematic and dynamic requirements

Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 COMPUTER AIDED CAM DESIGN OF ROLLER-FOLLOWER CAM MECHANISM CONSIDERING KINEMATIC AND DYNAMIC REQUIREMENTS Tran Minh Quang*, Nguyen Quang Hung College of Technology - TNU SUMMARY In this paper, a new method to design cam mechanisms with translating roller follower using Matlab and Inventor software is presented. The minimization of the cam size and the contact stress can be determined by controlling design parameters, such as the cam base circle radius, the follower face width and the follower offset. During the design procedure, a number of constraints regarding the pressure angle and the contact stress are taken into account. The finite element aproach is used to perform the analysis. A specific design case has been done to prove the practical applicability of this new approach. Keywords: Cam profile, Cam design, Synthesis, Cam follower-mechanism, FEA INTRODUCTION* basic principle of designing a cam profile is A cam may be defined as a machine element still used to determine the coordinates of having a curved outline or a curved groove, points at the center of the follower, then which, by its oscillation or rotation motion, coordinates of points at the cam profile. gives a predetermined specified motion to NURBS is used to construct a smooth curve another element called the follower. Cam from these separate points. Once the function mechanism with roller follower is used of cam profile is absolutely constructed by widely in many classes of machines because using NURBS curve, a MATLAB program due to the cam and follower it is possible to will be done to demonstrate the curve and to obtain an unlimited variety of motions. control the kinematic conditions. The cam However, when a motion of follower is given profile then can be exported into a CAD arbitrarily, designers are sometime very software, such as Inventor. Finally, a difficult to build the cam profile to satisfy modeling of cam is expressed to simulate the kinematic and dynamic characteristics, such cam mechanism on the Inventor software. as displacement, velocity, acceleration of Also a process of finite element analysis will follower, pressure angle, and even contact be executed to control dynamic requirements. stress. There are a number of documents Furthermore, the results of the design process presenting about how to construct these using this method can be used to manufacture profiles by using normal functions, such as cam on CNC machines. An example cam Cycloid function, Sin functionetc[1.2]. design with translating roller follower from a There are many new approaches executed to given displacement of follower has been done improve smooth of the cam profile, but these to prove the practical applicability of this new methods are still limited to kinematic approach. requirements [3]. The paper presents a new CAM PROFILE DESIGN PRINCIPLE method to design cam mechanisms with FROM A GIVEN DISPLACEMENT OF translating roller follower concerning both THE FOLLOWER kinematic and dynamic requirements, using The family of pitch profile can be determined Matlab and Inventor software. Firstly, the from the parametric equation [1]. 2 2 2 * F(x,y,) = (x - xp) + (y - yp) – rf = 0 (1) Tel: 01697 165805, Email: minhquangclc06m@gmail.com 125 Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 3/2 Where (x , y ) are coordinates of points at the 2 2 p p ()r r s center of the follower, as functions of the bf  (5)  p  2 22 variable  ; rf is radius of roller. (rb r f )  s  2 v  a ( r b  r f )  s xp = (k+s)sin + e.cos Where: a is acceleration; of the follower 3) The offset e must satisfy the constraints: yp = (k+s)sin - e.sin ; 0< e < s. k() r  r22  e (2) bf 4) The contact stress of any point at the cam k is the distance parallel to the direction of curve is always smaller than permitted contact motion between center of the Cam and center stress for the cam: σmax ≤ [σ]. of the follower. CCAM PROFILE SMOOTHING BY USING The coordinates of points at the cam profile NURBS CURVESAM PROFILE can be calculated with the following SMOOTHING BY USING NURBS CURVES equations [2]. In the section, NURBS curve is used to 1/2 construct complex and smooth Cam profile 22 dyp   dx p   dy p  with many constraints about kinematics, x x  r   ; pfd   d    d   dynamics, such as displacement, velocity,       acceleration, jerk of follower and pressure dxp  angle of the cam. NURBS is a flexible d (3) function used widely on applications of CAD y ypp  x  x  dy p to construct complex curves [4.5]. While most  d basic curves, such as the circle, are not represented by B-spline curves and other During the synthesis procedure the following parametric polynomial curves [5]. The functional constraints are imposed: NURBS can be defined by generalizing B- 1) The maximum value of the pressure angle spline curve of degree p: must be smaller than the maximum permitted: h ww α ≤ [α]. The lower the value of pressure C()() u  Ni, p u P i angle, the best the transmitted force will be i0 wx transformed into the motion of the follower. If ii  h wy the pressure angle is too high, the follower ii  Nuip, ( ) ; (0  u  1) (6) sliding or friction will be increased. i0 wz ii The pressure angle can be calculated by wi [1.3]: w ve Where: Pi = (wixi, wiyi, wizi) are control tg  22 (4) points; i = 0,1,2, h, (h+1) are number of s() rb  rf  e control points, Ni,p(u) are B-spline basis Where: s is displacement of the follower, v is functions of degree p, wi is weights, u are knots. velocity of the follower, r is the base circle; e b   U  0,...,0,u ,...,u ,1,...,1 (7) is eccentricity.  p1 e p1   p1 p1  2) The roller follower radius rf is always smaller than smallest absolute value of local is knot vector radius of curvature |1/ρ|< 1/rf, that keep the uu uu contact between the cam and roller follower. N( u )i N ( u )ip1 N ( u ); N ( u ) i, pu u i , p 1 u u i  1, p  1 i ,0 If we disregard for existence of pressure i p i i  p 11 i  (8) angle, the minimum cam size can be 1 ui1uui determined from radius of curvature for roller   0 u u u  follower [1.3].  i1i 126 Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 h Where: t is parameter of initial data’s point i N() u w P i  i, p i i h and can be calculated by the chord length i0 (9) Ri,, p( u )h  C ( u )   R i p ( u ) P i i0 method: t0 =0; tn =1.  Ni, p() u w i i0 FINITE ELEMENT ANALYSIS PROCEDURE Ri,p(u) are called B-spline basic Rational Finite element method is one of the most functions. accepted and widely used tools for the solution Suppose there is a data set of (n+1) data of linear and non linear partial differential {D0(x0, y0),, Dn(xn,yn)}, An NURBS curve equations which arises during the mathematical of degree p is constructed from the data as modeling of various processes [6]. using approximate method. It always get Once the function of cam profile is absolutely through the beginning point and the ending constructed by using NURBS curve, a point, therefore the sum of the squared errors MATLAB program has been done to between the point of initial data and its demonstrate the curve and to test the corresponding point at the NURBS curve can be detected [3]: kinematic conditions. The cam profile then can be exported into a CAD software, such as f f( P , P ,.., P , w , w ,..., w ) 0 1hh 0 1 Inventor. Finally, a modeling of cam is h 2 N() t w P expressed to simulate the cam mechanism. n  j, p i j j (10) j0 Also a process of finite element analysis will  h Di i0 be executed. The cam design flowchart was  Nj, p() t i w j j0 applied in the paper as shown in Fig.1. 127 Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 Motion of follower of the variable cam angle 14 follower, the solid model of the same is S input 12 essential. Figure 5 shows a solid model of roller follower. The accuracy of the FEM 10 depends on the density of the mesh used in 8 the analysis; the outer surface of cam loaded 6 by the follower so that to obtain the correct in 4 Displacement Displacement S(mm) the region of high stress. 2 The results of finite element analysis, as 0 0 50 100 150 200 250 300 350 shown in Fig.6, indicate that the maximum Cam angle(deg) contact stress, according to Von-mises Stress, Permitted max. pressure angle is 25deg Cam base radius range from 60 to 120mm at surface of cam is 6.4MPa with a pressure Permitted max. contact stress is 1500N/mm2 load 100N in rise cycle at angle of 19.5º, Fig.2. Kinematic and dynamic requirements respectively. It absolutely satisfies the given RESULTS AND DISCUSSION dynamic requirements. In this section, an example cam design is Cam profile 80 exhibited with translating roller follower from Pitch curve 60 Cam curve a given follower displacement S() as shown 40 in Fig.2. 20 0 The input parameters are set: rb = 80mm; e = -20 20mm and rf = 15mm. The NURBS curve of -40 degree p=3, i=90points and h=80. The cam -60 -80 profile and pressure angle are shown as in -100 Fig.3. The results indicate that the maximum -100 -50 0 50 100 max Pressure angle of cam angle error f80 =0.0587. The pressure angles 12.2 range from 4.75deg [min] to 12.15deg [max], 12 the maximum radius of curvature is 24.03 11.8 mm. All of these results absolutely satisfy the 11.6 kinematic requirements. 11.4 11.2 Considering kinematic requirements the Pressure Pressure angle(deg) 11 displacement, velocity and acceleration 10.8 of the follower are determined as shown in 0 50 100 150 200 250 300 350 Figure 4. Cam angle(deg) To perform finite element analysis of roller Fig.3. Cam profile and pressure angle of the cam 15 0.6 0.4 10 0.2 S(mm) 0 5 velocity velocity (m/s) -0.2 0 -0.4 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Cam angle (deg) Cam angle (deg) 30 200 20 100 ) ) 2 10 3 (m/s 0 0 acceleration acceleration jerk (m/s -10 -100 -20 -200 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 Cam angle (deg) Cam angle (deg) Fig.4. The follower motion diagrams 128 Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 In this approach, the contact stress distribution along the contact between the follower and the cam edge which give good agreement with the numerical analysis using F.E.M have been investigated. The results show some advantage in solving non-linear programming for designing cam mechanisms with translating roller follower. The entire data of the design process can be used for manufacturing process on CNC machines. REFERENCES Fig.5. 3D model of the cam 1. Robert L. Norton, Design of Machinery, McGraw-Hill International Editions, Singapore, CONCLUSIONS 1992. For design cam mechanisms with translating 2. Chen, F. Y., Mechanics and Design of Cam roller follower using Matlab and Inventor Mechanisms, Pergamon, New York, 1982. software have been studied in this paper 3. Vũ Thị Liên, Trần Minh Quang, Cam Profile Design of Cam Roller-Follower Cam Mechanism taking in the consideration the kinematic and from A Given Displacement of The Follower dynamic requirements for solving the Using NURBS Curve. NURBS curve that proposed to satisfy the 4. Piegl, L.A. and Tiller, W., Computing offsets of curvature and minimum cam size. NURBS curves and surfaces. Computer-Aided Design. v31. 147-156. 5. Vu Thi Lien, (2010) Optimization of NURBS Fitting with Simulated Annealing Method for Non-Symmetric Objects, Master’s thesis, Department of Mechanical Engineering, National Taiwan University of Science and Technology. 6. Fathi Al-Shamma, Faiz F. Mustafa, Sahar M. Saliman, An Optimum Design of Cam Mechanisms with Roller Follower for Combined Effect of Impact and High Contact Loads, Al- Khwarizmi Engineering Journal, Vol. 6, No. 4, Fig.6. Contact stress at cam surface PP 62 - 71 (2010) 129 Trần Minh Quang và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 139(09): 125 - 130 TÓM TẮT THIẾT KẾ CƠ CẤU CAM CẦN ĐẨY ĐÁY CON LĂN VỚI SỰ TRỢ GIÚP CỦA MÁY TÍNH KHI KỂ ĐẾN CÁC YÊU CẦU VỀ ĐỘNG HỌC VÀ ĐỘNG LỰC HỌC Trần Minh Quang*, Nguyễn Quang Hưng Trường Đại học Kỹ thuật Công nghiệp - ĐH Thái Nguyên Bài báo này trình bày một phương pháp mới, sử dụng phần Matlab và Inventor để thiết kế cơ cấu Cam cần đẩy đáy con lăn. Trong đó, các giá trị kích thước cam và ứng suất tiếp xúc tối thiểu được xác định bằng việc điều khiển các thông số thiết kế như bán kính vòng cơ sở, bề rộng của cam và độ lệch tâm. Trong toàn bộ quá trình thiết kế, các ràng bộc về góc áp lực và ứng suất tiếp súc sẽ được đưa vào quá trình tính toán và được xem như các điều kiện biên. Phương pháp phần tử hữu hạn được sử dụng để mô phỏng sự phân bố ứng suất tiếp xúc trên bề mặt cam. Một ví dụ thiết kế cụ thể đã được tiến hành để chứng minh được khả năng áp dụng vào thực tế của của cách tiếp cận mới này. Từ khóa: Biên dạng cam, Thiết kế cam, Tổng hợp, Cam cần đẩy, Phân tích phần tử hữu hạn Ngày nhận bài:20/6/2015; Ngày phản biện:06/7/2015; Ngày duyệt đăng: 30/7/2015 Phản biện khoa học: PGS.TS Vũ Ngọc Pi - Trường Đại học Kỹ thuật Công nghiệp - ĐHTN * Tel: 01697 165805, Email: minhquangclc06m@gmail.com 130