{"id":247968,"date":"2026-09-11T12:56:15","date_gmt":"2026-09-11T09:56:15","guid":{"rendered":"https:\/\/azbuki.bg\/?p=247968"},"modified":"2026-09-11T12:56:15","modified_gmt":"2026-09-11T09:56:15","slug":"differential-method-for-synthesis-of-cam-lever-mechanisms-for-generation-of-non-strict-monotonous-position-functions","status":"publish","type":"post","link":"https:\/\/newspaper.azbuki.bg\/en\/xxvii-international-scientific-conference-transport-2025\/differential-method-for-synthesis-of-cam-lever-mechanisms-for-generation-of-non-strict-monotonous-position-functions\/","title":{"rendered":"Differential Method for Synthesis of Cam-lever Mechanisms for Generation of Non-strict Monotonous Position Functions"},"content":{"rendered":"

Vitan Galabov, Roumen Roussev, Blagoyka Paleva-Kadiyska
\n<\/strong>Technical University, Sofia, Bulgaria, Faculty of Technics and Technology – Yambol of Trakia University,
\n<\/em>Bulgaria, Todor Kableshkov University of Transport, Sofia, Bulgaria<\/em><\/p>\n

https:\/\/doi.org\/10.53656\/isct-2025.18<\/a><\/p>\n

\"\"
\nPDF<\/a><\/p>\n

Pages 225-234<\/em><\/p>\n

Abstract.<\/strong>\u00a0<\/strong>Mechanisms generating periodically variable rotation of the output link (monotonous displacement functions) can be divided into two groups. The mechanisms of the first group generate only strict monotonic functions (without prolonged and even instantaneous dwell of the output link). The second group of mechanisms generates non-strict monotonic functions (with a prolonged or instantaneous dwell of the output link).<\/p>\n

This study focuses on functional generator mechanisms with a six-bar topological structure, which are a combination of a cam and lever mechanism.<\/p>\n

An original structure of a jointed cam-lever mechanism is proposed, designed to generate strict and non\u2013strict monotonous displacement functions.<\/p>\n

The mechanism’s kinematic and dynamic characteristics depend on the displacement function type and parameters, or its derivative transfer functions, i.e., on the given law of motion. Laws without a finite or infinite jump (acceleration break) in the second transfer function, or the output acceleration, are preferred. For this purpose, known functions can be used, or functions suitable for the target proposition can be composed.<\/p>\n

A law of motion was chosen that is per the requirement for resetting the output velocity, acceleration, and its derivative (jerk) for the boundaries of the output link dwell interval. This is recommended for the synthesis of high-speed cam-lever mechanisms on many technological machines.<\/p>\n

A rational mathematical model has been developed, based on elements of differential geometry, for the synthesis of planetary cam-lever mechanisms with a six-bar topological structure and a common geometric axis of rotation of the input and output links.<\/p>\n

An example is presented that illustrates the overall process by which a mechanism with the indicated structure was synthesized and which verifies the derived mathematical model.<\/p>\n

Keywords:<\/em> planetary cam-lever mechanisms; law of motion; synthesis<\/p>\n

 <\/p>\n

    \n
  1. Introduction<\/strong><\/li>\n<\/ol>\n

    When designing machines and technical devices with a continuous production process, it is increasingly necessary to synthesize mechanisms for generating monotonic functions for converting uniform rotation into periodical unidirectional uneven rotation when solving several groups of technical problems. One of them is a constant angular velocity drive at the input and temporary dwell of conveyors and conveyor belts, according to the requirements of a certain technological process.<\/p>\n

    Mechanisms generating rotational periodic rotation of the output link (monotonous displacement functions) can be divided into two groups. The mechanisms of the first group generate only strict monotonic functions (without prolonged and even instantaneous dwell of the output link). Double-crank four-bar linkages and slotted link mechanisms with full rotation of the flat-face (also known as slotted-link) generate approximately specified, strict monotonic displacement functions [1, 2, 3], unlike mechanisms with non-circular gears [4, 5].<\/p>\n

    The second group of mechanisms generates non-strict monotonic functions (with a prolonged or instantaneous dwell of the output link). This group includes mechanisms with periodical output rotations of the output link in a finite interval of the input rotation (Maltese cross, ratchet, anchor, and other types of mechanisms) [6]. Changes in accelerations with jumps, or inertial load, are avoided by some combined mechanisms involving a cam mechanism [7, 8].<\/p>\n

    This study focuses on functional generator mechanisms with a six-bar topological structure, which are a combination of a cam and lever mechanism.<\/p>\n

    The aim is to compile a rational mathematical model, based on elements of differential geometry, for the synthesis of cam-lever mechanisms with a general geometric axis of rotation of the input and output links.<\/em><\/p>\n

    The choice of the law of motion that pursuant with the requirement for resetting the output velocity, acceleration, and its derivative (jerk) for the dwell of the output link dwell interval, which is recommended for the synthesis of high-speed cam-lever mechanisms on many technological machines [9, 10, 11, 12, 13].<\/p>\n

      \n
    1. Structure and mathematical model for mechanism synthesis<\/strong><\/li>\n<\/ol>\n

      An original structure of a jointed cam-lever mechanism is proposed, designed to generate strict and non\u2013strict monotonous displacement functions (Fig.1). The mechanism consists of a jointed parallelogram OABC <\/em>with a double base joint O<\/em>, connecting the input link of length a<\/em>, and the output link of length b<\/em>. Diagonally to the base center is situated the center B<\/em> of a roller with radius r<\/em>, which contacts a fixed cam. For each position, a conditional link can be introduced connecting the center B<\/em> with the corresponding center of curvature Bo<\/sub><\/em> of the cam profile cG<\/sub><\/em>, resulting in a six-bar topological structure formed by two sequentially connected four-bar linkages OABB0<\/sub><\/em> and B0<\/sub>BCO<\/em>.<\/p>\n

      \"\"<\/p>\n

      Figure 1. <\/strong>Cam mechanism combined with a jointed parallelogram<\/p>\n

      The input and output coordinates are the angles \u03c6a<\/sub><\/em> and \u03c6b<\/sub><\/em> relative to the x<\/em>-axis, respectively. In its initial position, the configuration of the jointed parallelogram is rectangular, with the input link OA<\/em> making an angle 90\u00b0 with the x<\/em> axis. So , where the initial value of \u00a0is . The dependence \u00a0represents the displacement function of the mechanism, and its derivatives \u00a0and \u00a0are the first and second transfer functions of the mechanism, respectively. These functions determine the pitch curve cB<\/sub><\/em> of the cam, its profile cG<\/sub><\/em>, and their general evolute, which is the problem of the mechanism synthesis.<\/p>\n

      The synthesis of the cam profile is performed with a predefined or accepted function of the mechanism displacement , its derivative transfer functions , \u00a0and the lengths a<\/em> and b<\/em> of the levers of the jointed parallelogram OABC<\/em>.<\/p>\n

      The pitch curve cB<\/sub><\/em> of the cam is determined by the coordinates of the center B<\/em> as a function of the entrance angle \u03c6a<\/sub><\/em>:<\/p>\n

      \"\"<\/p>\n

      The first derivatives of the coordinates from equation (1) to \u03c6a<\/sub><\/em><\/strong> are necessary to determine the angles of force transmission in the mechanism, and together with their second derivatives, to determine the evolute of the cam profile. For the firsts and seconds derivatives of the coordinates, we get:<\/p>\n

      \"\"<\/p>\n

      The angles of motion transmission \u03b3B<\/sub><\/em> and \u03b3C<\/sub><\/em>, to the centers B<\/em> and C<\/em>, respectively, largely determine the force conditions under which the cam-lever mechanism operates. Using well-known formulas from differential geometry, the angular coefficient of the tangent to the pitch curve cB<\/sub><\/em> and the angles of motion transmission are determined:<\/p>\n

      \"\"<\/p>\n

      where<\/p>\n

      \"\"<\/p>\n

      The longer the dwell interval, the smaller the values of \u03b3B<\/sub><\/em> and \u03b3C<\/sub><\/em>. This is especially true for \u03b3B<\/sub><\/em>, which can reach unallowable small values.<\/p>\n

      Therefore, from a hinged parallelogram one can move to a four-bar linkage (Fig. 2), in which the sides \u041e\u0410<\/em> and OC<\/em> retain their lengths a<\/em> and b<\/em>, respectively, while the side AB<\/em> accept a value of lAB<\/sub> > b<\/em> (for example with 30% to 50%) and for the side BC<\/em> the length lBC<\/sub> \u2248 a<\/em> is accept. Thus, the mathematical model for determining the coordinates of the center B<\/em> will take the form:<\/p>\n

      \"\"<\/p>\n

      where<\/p>\n

      \"\"<\/p>\n

      \"\"<\/p>\n

      Figure 2. <\/strong>Cam mechanism combined with a four-bar linkage<\/p>\n

      The cam profile cG<\/sub><\/em> is the equidistant of the pitch curve cB<\/sub><\/em>, located at a distance r<\/em> (the radius of the roller) along the normal n<\/em> with an angular coefficient \"\":<\/p>\n

      \"\"<\/p>\n

      The \u201c+\u201d and \u201c-\u201d signs in equations (8) correspond to the outer and inner cam profiles relative to the roller. The curve of center of a machining tool can be determined from equations (8) by replacing r<\/em> with r – r<\/sub><\/em>, where r<\/sub><\/em> is the radius of the machining tool.<\/p>\n

      The radius of curvature of the curve of center \u03c1B<\/sub><\/em> and the cam profile \u03c1G<\/sub><\/em> are determined by a well-known formula from differential geometry:<\/p>\n

      \"\"<\/p>\n

      Their common center of curvature<\/p>\n

      \"\"<\/p>\n

      determines their common evolute \u03b5<\/em> when xA<\/sub><\/em>, yB<\/sub><\/em> and their derivatives are functions of the angle \u03c6a<\/sub><\/em>.<\/p>\n

      \u00a0<\/strong><\/p>\n

        \n
      1. Choice of the law of motion of the output link<\/strong><\/li>\n<\/ol>\n

        The mechanism’s kinematic and dynamic characteristics depend on the displacement function type and parameters, or its derivative transfer functions, i.e., on the given law of motion. Laws without a finite or infinite jump (acceleration break) in the second transfer function, or the output acceleration, are preferred. For this purpose, known functions can be used, or functions suitable for the target proposition can be composed.<\/p>\n

        According to the technological process, the intervals of uniform rotation of the input link and dwell of the output link 3 are determined. In the range of motion of the output link, the engineer must select an appropriate output rotation law. It is possible to choose an output rotation function with high-order osculation towards the x<\/em>-axis in the entire interval of the input coordinate \u03c6a<\/sub><\/em>, which practically leads to a prolonged dwell of the output link. In the article, the authors present this possibility.<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        \"\"<\/p>\n

        Figure 3. <\/strong>Displacement function and derived transfer functions<\/p>\n

        \u00a0<\/strong>Example<\/em><\/strong>. To synthesize a mechanism of the considered type for driving a conveyor with appropriately chosen values a<\/em> = 200 mm, b<\/em> = 100 mm, r<\/em> = 30 mm.<\/p>\n

        From equations (11) and a choice normalized polynomial and its derivatives (14), the displacement function of the output link \"\" and its derivative transfer functions \"\" are determined (Fig. 2).<\/p>\n

        \"\"<\/p>\n

        The kinematic diagram of the synthesized mechanism is shown in Fig. 1.<\/p>\n

        \"\"<\/p>\n

        Graphs of the motion transmission angles determined by equations from (4) to (7) are presented in Fig. 4.<\/p>\n

        \"\"<\/p>\n

        Figure 4. <\/strong>The angles of motion transmission \u03b3B<\/sub><\/em> and \u03b3C<\/sub><\/em><\/p>\n

         <\/p>\n

        Conclusion<\/strong><\/p>\n

        This study focuses on functional generator mechanisms with a six-bar topological structure, which are a combination of a cam and lever mechanism. The lengths of its links are determined by requirements for maximum sizes, according to the purpose of the mechanism, as well as by the influence of the b\/a<\/em> ratio on the curvature the curve of center of the cam.\u00a0 The smaller the ratio b\/a, the smaller the variation in the curvature the curve of center. But the reduction in length b<\/em> is limited by design considerations related to the sizing of the bearing assemblies of the mechanism and the radius r<\/em> of the roller.<\/p>\n

        A law of motion was chosen that is per the requirement for resetting the output velocity, acceleration, and its derivative (jerk) for the boundaries of the output link dwell interval. A rational mathematical model has been developed, based on elements of differential geometry, for the synthesis of planetary cam-lever mechanisms with a six-bar topological structure and a common geometric axis of rotation of the input and output links. This makes it possible to fully reveal the geometry of the cam mechanism, the pitch curve of the roller and of various machining tools, the angles of motion transmission, and the common evolute of the cam profile.<\/p>\n

        The presented example illustrates the overall process by which mechanisms with the indicated structure are synthesized and proves the correctness of the developed mathematical model.<\/p>\n

        \u00a0<\/strong><\/p>\n

        \u00a0<\/strong><\/p>\n

        REFERENCES<\/strong><\/p>\n

        [1] SUH C. H., RADCLIFFE C. W., Kinematics and Mechanisms Design. New York: John Wiley & Sons, 1978, 434 p. ISBN 978-0-471-01461-4.<\/p>\n

        [2] PEISAKH E. E., NESTOROV V. A., System of designing flat lever mechanisms. Moscow, Mashinostroenie, 1988, p. 232. (in Russian).<\/p>\n

        [3] ERDMAN A. G., SANDOR G. N., Mechanism Design: Analysis and Synthesis. 2nd Edition. New Jersey: Prentice-Hall Inc., 1991, vol. 1. 650 p. ISBN 978-0-13-573536-7.<\/p>\n

        [4] LITVIN F. L., Theory of Gearing. Moscow, Nauka, 1968, p. 584. (in Russian).<\/p>\n

        [5] GALABOV V., On the synthesis, analysis, and production of non-circular gears, Dissertation for Candidate Doctor of Science, Higher Mechanical and Electrical Engineering Institute-Sofia, 1975, p. 222.<\/p>\n

        [6] KOZHEVNIKOV S. N., ESIPENKO YA. I., RASKIN YA. M, Mechanisms (handbook), Moscow, Mashinostroenie, 1976, p. 784. (in Russian).<\/p>\n

        [7] KARELIN V. S, Design of lever and gear-lever mechanisms (handbook), Moscow, Mashinostroeniya, 1986, p. 181. (in Russian).<\/p>\n

        [8] VOLMER J., Getriebetechnik \u2013 Kurvengetriebe \u2013 CAD\/CAM, Verlag Technik, Berlin, 1989 Congress on TMM, pp. 1303-1306.<\/p>\n

        [9] ROTHBART, H., A., CAMES. Design, Dynamics and Accuracy, N., Y., John Wiley & Sons,1965, p. 336. [10] SADEK, K. S. H., DAADBIN, A., Improved Cam Profiles for High-Speed Machinery Using Polynomial Curve Fitting, J. of Process Mechanical Engineering, Vol.204, n.E 2, 1990, pp.127 \u2013 132.<\/p>\n

        [11] BLECHSMIDT, J. L., LEE, C. H., Design and Analysis of Cam Profiles Using Algebraic Functions, ASME, DE, vol. 32 pt. 2, New York, pp. 451 \u2013 459, 1991.<\/p>\n

        [12] NORTON R., L., Design of Machinery, McGraw-Hill Inc., New York, 306 p., 1992, ISBN 978-0-07-047808-4.<\/p>\n

        [13] CARDONA A., LENS E. and Nigro N., Optimal Design of Cams, Multibody System Dynamics, vol. 7, no. 3, pp. 285 \u2013 305, 2002, ISSN 1384-5640. DOI 10.1023\/A:1015243126749.<\/p>\n

         <\/p>\n

         <\/p>\n

        Prof. Vitan Galabov, DSc.<\/strong><\/p>\n

        ORCID iD: 0009-0001-1527-6099<\/p>\n

        Technical University-Sofia<\/p>\n

        8, Kl. Ohridski Blvd., 1000 Sofia, Bulgaria<\/p>\n

        E-mail: vgalabov@abv.bg<\/p>\n

        \n

        Dr. Roumen Roussev, Assoc. Prof.<\/strong><\/p>\n

        ORCID iD: 0009-0001-8485-1229<\/p>\n

        Faculty of Technics and Technology \u2013 Yambol, Trakia University<\/p>\n

        38, Graf Ignatiev St., 8600 Yambol, Bulgaria<\/p>\n

        E-mail: roussev_r@abv.bg<\/p>\n

        \n

        Dr. Blagoyka Paleva-Kadiyska, Assoc. Prof.<\/strong><\/p>\n

        ORCID iD: 0000-0002-8514-4542<\/p>\n

        Todor Kableshkov University of Transport<\/p>\n

        158, Geo Milev St., 1574 Sofia, Bulgaria<\/p>\n

        E-mail: paleva-kadiyska.bl@abv.bg<\/a>; bip-kadiyska@vtu.bg<\/p>","protected":false},"excerpt":{"rendered":"

        Vitan Galabov, Roumen Roussev, Blagoyka Paleva-Kadiyska Technical University, Sofia, Bulgaria, Faculty of Technics and Technology – Yambol of Trakia University, Bulgaria, Todor Kableshkov University of Transport, Sofia, Bulgaria https:\/\/doi.org\/10.53656\/isct-2025.18 PDF Pages 225-234 Abstract.\u00a0Mechanisms generating periodically variable rotation of the output link (monotonous displacement functions) can be divided into two groups. The mechanisms of the first […]<\/p>","protected":false},"author":124332423427287,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[19875],"tags":[19978,19977,19979],"yoast_head":"\nDifferential Method for Synthesis of Cam-lever Mechanisms for Generation of Non-strict Monotonous Position Functions - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/azbuki.bg\/xxvii-international-scientific-conference-transport-2025\/differential-method-for-synthesis-of-cam-lever-mechanisms-for-generation-of-non-strict-monotonous-position-functions\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Differential Method for Synthesis of Cam-lever Mechanisms for Generation of Non-strict Monotonous Position Functions - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"Vitan Galabov, Roumen Roussev, Blagoyka Paleva-Kadiyska Technical University, Sofia, Bulgaria, Faculty of Technics and Technology – Yambol of Trakia University, Bulgaria, Todor Kableshkov University of Transport, Sofia, Bulgaria https:\/\/doi.org\/10.53656\/isct-2025.18 PDF Pages 225-234 Abstract.\u00a0Mechanisms generating periodically variable rotation of the output link (monotonous displacement functions) can be divided into two groups. 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