Differential Method for Synthesis of Cam-lever Mechanisms for Generation of Non-strict Monotonous Position Functions

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. 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).

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

An original structure of a jointed cam-lever mechanism is proposed, designed to generate strict and non–strict monotonous displacement functions.

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.

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.

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.

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.

Keywords: planetary cam-lever mechanisms; law of motion; synthesis

 

1. Introduction

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.

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].

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].

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

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.

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].

2. Structure and mathematical model for mechanism synthesis

An original structure of a jointed cam-lever mechanism is proposed, designed to generate strict and non–strict monotonous displacement functions (Fig.1). The mechanism consists of a jointed parallelogram OABC with a double base joint O, connecting the input link of length a, and the output link of length b. Diagonally to the base center is situated the center B of a roller with radius r, which contacts a fixed cam. For each position, a conditional link can be introduced connecting the center B with the corresponding center of curvature B_o of the cam profile c_G, resulting in a six-bar topological structure formed by two sequentially connected four-bar linkages OABB₀ and B₀BCO.

Figure 1. Cam mechanism combined with a jointed parallelogram

The input and output coordinates are the angles φ_a and φ_b relative to the x-axis, respectively. In its initial position, the configuration of the jointed parallelogram is rectangular, with the input link OA making an angle 90° with the x axis. So , where the initial value of  is . The dependence  represents the displacement function of the mechanism, and its derivatives  and  are the first and second transfer functions of the mechanism, respectively. These functions determine the pitch curve c_B of the cam, its profile c_G, and their general evolute, which is the problem of the mechanism synthesis.

The synthesis of the cam profile is performed with a predefined or accepted function of the mechanism displacement , its derivative transfer functions ,  and the lengths a and b of the levers of the jointed parallelogram OABC.

The pitch curve c_B of the cam is determined by the coordinates of the center B as a function of the entrance angle φ_a:

The first derivatives of the coordinates from equation (1) to φ_a 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:

The angles of motion transmission γ_B and γ_C, to the centers B and C, 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 c_B and the angles of motion transmission are determined:

where

The longer the dwell interval, the smaller the values of γ_B and γ_C. This is especially true for γ_B, which can reach unallowable small values.

Therefore, from a hinged parallelogram one can move to a four-bar linkage (Fig. 2), in which the sides ОА and OC retain their lengths a and b, respectively, while the side AB accept a value of l_AB > b (for example with 30% to 50%) and for the side BC the length l_BC ≈ a is accept. Thus, the mathematical model for determining the coordinates of the center B will take the form:

where

Figure 2. Cam mechanism combined with a four-bar linkage

The cam profile c_G is the equidistant of the pitch curve c_B, located at a distance r (the radius of the roller) along the normal n with an angular coefficient :

The “+” and “-” 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 with r – r, where r is the radius of the machining tool.

The radius of curvature of the curve of center ρ_B and the cam profile ρ_G are determined by a well-known formula from differential geometry:

Their common center of curvature

determines their common evolute ε when x_A, y_B and their derivatives are functions of the angle φ_a.

 

3. Choice of the law of motion of the output link

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.

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-axis in the entire interval of the input coordinate φ_a, which practically leads to a prolonged dwell of the output link. In the article, the authors present this possibility.

Figure 3. Displacement function and derived transfer functions

 Example. To synthesize a mechanism of the considered type for driving a conveyor with appropriately chosen values a = 200 mm, b = 100 mm, r = 30 mm.

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).

The kinematic diagram of the synthesized mechanism is shown in Fig. 1.

Graphs of the motion transmission angles determined by equations from (4) to (7) are presented in Fig. 4.

Figure 4. The angles of motion transmission γ_B and γ_C

 

Conclusion

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 ratio on the curvature the curve of center of the cam.  The smaller the ratio b/a, the smaller the variation in the curvature the curve of center. But the reduction in length b is limited by design considerations related to the sizing of the bearing assemblies of the mechanism and the radius r of the roller.

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.

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.

 

 

REFERENCES

[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.

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

[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.

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

[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.

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

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

[8] VOLMER J., Getriebetechnik – Kurvengetriebe – CAD/CAM, Verlag Technik, Berlin, 1989 Congress on TMM, pp. 1303-1306.

[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 – 132.

[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 – 459, 1991.

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

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

 

 

Prof. Vitan Galabov, DSc.

ORCID iD: 0009-0001-1527-6099

Technical University-Sofia

8, Kl. Ohridski Blvd., 1000 Sofia, Bulgaria

E-mail: vgalabov@abv.bg

Dr. Roumen Roussev, Assoc. Prof.

ORCID iD: 0009-0001-8485-1229

Faculty of Technics and Technology – Yambol, Trakia University

38, Graf Ignatiev St., 8600 Yambol, Bulgaria

E-mail: roussev_r@abv.bg

Dr. Blagoyka Paleva-Kadiyska, Assoc. Prof.

ORCID iD: 0000-0002-8514-4542

Todor Kableshkov University of Transport

158, Geo Milev St., 1574 Sofia, Bulgaria

E-mail: paleva-kadiyska.bl@abv.bg; bip-kadiyska@vtu.bg