Optimization synthesis of flat lever mechanisms by pressure angle

Authors: Kotov A.V., Krol D. G., Ph. D. in Phys. And Math., Assoc. Prof.

This article is a translation of the original work of the same name, which was written in Russian and published in a peer-reviewed journal. I decided to prepare and publish its English version for several reasons. First, science and engineering thinking have no language barriers. Publishing a translation is a step towards drawing attention to my research from a wider audience, including foreign colleagues, engineers and researchers who may find the proposed method useful. Second, publishing the article in English helps increase the visibility of the blog itself in foreign search engines. This means that my developments and findings are more likely to reach those who truly need them. I am open to discussion, feedback and professional dialogue with anyone who finds the topic of my research relevant. I will be glad if this material proves useful beyond the Russian-speaking audience.

Introduction. The problem of ensuring high efficiency and reliability of lever mechanisms is directly related to minimizing losses in their kinematic pairs, which, in turn, depend on the pressure angle [1, 2]. Exceeding the pressure angle beyond the recommended optimal value leads to an increase in friction losses and radial loads in the joints of the lever mechanism, which results in increased wear, jamming, and a decrease in the overall efficiency of the mechanism [3].

Traditionally, the synthesis of mechanisms based on the pressure angle is classified as a kinematic design task and is solved using graphical or analytical methods, which include constructing position and velocity plans [4 – 6]. Such synthesis can precede force analysis and provide additional qualitative and quantitative assessment of the potential force loading of the mechanism under study. However, formulating and solving this problem often requires cumbersome calculations and graphical constructions, which complicates the possibility of algorithmization and optimization [7].

The application of vector analysis or complex number theory for describing the kinematics of planar lever mechanisms is a well-established and effective approach [8, 9]. It allows for uniform, compact, and minimal analytical expressions to calculate the positions and velocities of all characteristic points and links of the mechanism. At the same time, the potential of vector analysis and complex numbers for solving kinematic synthesis problems, in particular optimization of pressure angles in joints, has not been fully explored. In this regard, the development of a methodology for the rapid assessment of pressure angles at the stage of mathematical modeling, based on this apparatus, retains its scientific and practical relevance.

The aim of this study is to develop a methodology for the optimization synthesis of a planar lever mechanism based on pressure angle constraints. To achieve this goal, original analytical dependencies for calculating the pressure angle were obtained, as well as numerical optimization algorithms.

Materials and Methods of Research. To build a mathematical model for the kinematic analysis of a planar lever mechanism, the coordinate transformation method in a fixed basis was used, implemented using both vector analysis and complex number theory. In the vector model, pressure angles in joints were calculated through scalar and vector products, while in the model using complex numbers – through the analogy of these products and the argument of a complex number. For each mathematical model, several methods for calculating pressure angles in joints were applied.

Figure 1 shows the structural group of the lever mechanism, where link 1 is taken as the input link and link 2 as the output link. To simplify calculations when solving the problem, we neglect the masses of the links and friction in the kinematic pairs. Figure 1 shows the pressure angle θB in joint B, the reaction force vector R21 and the velocity vector vB (taking into account the assumptions made, the direction of the reaction force R21 will coincide with the line of link AB, which ultimately makes it possible to determine the pressure angle based solely on kinematic parameters – the radius vector of the link and the velocity vector of the point).

Along with the pressure angle, the transmission angle γB is often considered, which complements the angle θB to 90° (see Figure 1). Therefore, the optimization synthesis of planar lever mechanisms can be performed both by the pressure angle and by the transmission angle [10, 11].

1 – input link; 2 – output link
Figure 1. – Design structural group of the lever mechanism

The proposed calculation of pressure angles in the mechanism's joints will be based on the geometric property of the scalar and vector products or their analogs as applied to complex number theory. Therefore, all initial kinematic parameters of the lever mechanism (depending on the basis of the original mathematical model) must be presented in the following general form:

where ri – is the radius vector of the link along which the reaction force acts; vi – is the velocity vector of the actual point of application of the reaction on the driven link; j – is the imaginary unit.

In this work, starting from expression (1), the following notations are adopted: the upper underscore corresponds to a geometric vector, and the lower underscore – to a complex number vector.

For the first method of calculating the pressure angle, we use the geometric property of the scalar product and its analog as applied to complex number theory:

where v*i – is the vector of the complex conjugate number of the actual velocity of the point of application of the reaction on the driven link; Re – is the function for extracting the real part of a complex number.

For the second method of calculating the pressure angle, we use the geometric property of the vector product and its analog as applied to complex number theory:

where Im – is the function for extracting the imaginary part of a complex number.

The third method of calculating the pressure angle consists of dividing expression (3) by (2) and then applying the arctangent to the result:

Note that in expressions (2) – (4), the numerator uses the modulus of the number, which follows from the need to determine the pressure angle as the acute angle between the direction of the reaction force transmission and the direction of the velocity of the force application (the calculated angle must be in the first quadrant).

The fourth method for determining the pressure angle is applicable exclusively to complex numbers. As noted in the study [9], mathematical models of planar lever mechanisms built using the apparatus of complex numbers effectively determine all angular characteristics of vector quantities through the argument of a complex number (e.g., the orientation angle of a vector). Therefore, for the direct calculation of the pressure angle, one can use the argument of a complex number, which is expressed by the following relationship:

where arg – is the function for extracting the argument of a complex number.

The argument of a complex number determines the angle in the range [0…π], but since the pressure angle by definition is acute, expression (5) contains a condition that guarantees obtaining precisely an acute angle by choosing the minimum value between the calculated angle and its complement to 180°.

Since the velocity vector of the actual point and its kinematic analog are collinear, both quantities can be used in formulas (2) – (5). Moreover, using the velocity analog vector in these expressions is preferable, as it simplifies calculations by reducing intermediate computation steps.

Expressions (2) – (5) are identical and lead to the same calculation result. Therefore, the choice of one of them for further research is not fundamental. The key criterion is the mathematical apparatus chosen for constructing the mathematical model of the planar lever mechanism: geometric vectors or complex number theory.

Let us consider the algorithm for conducting the optimization synthesis of a planar lever mechanism taking into account the constraint on the maximum permissible pressure angle in its joints. As the object of study, we take a six-link mechanism, the kinematic diagram of which is shown in Figure 2, a. The mechanism has the following parameters [12]: LOA=0.15 m; LAB=0.97 m; LO1B=0.60 m; LO1С=0.45 m; LCD=0.86 m; a=0.5 m; b=0.37 m.

a)                                                                 б)
a – original mechanism; b – synthesized mechanism
Figure 2. – Kinematic diagram of a planar six-link mechanism

We assume that for this mechanism, kinematic analysis has already been performed using the coordinate transformation method in a fixed basis using vector analysis [8] or using complex number theory [9], i.e., the radius vectors of all its characteristic points, as well as the analogs of their velocity vectors, have been obtained (see expressions (1)). Then the algorithm for the synthesis of a planar lever mechanism based on the pressure angle can be represented as the following block diagram shown in Figure 3.

Figure 3. – Block diagram of the mechanism synthesis algorithm based on pressure angle

For planar lever mechanisms, the pressure angle θ < 90° [1]. When the pressure angle reaches θ = 90°, the mechanism is in so-called "dead" positions, which in statics leads to jamming, and in dynamics, the mechanism overcomes such positions due to inertia [13]. This phenomenon is observed in improperly designed mechanisms. To eliminate such situations during the synthesis of mechanisms, permissible values of pressure angles [θ] are specified. Currently, there are no uniform standards for permissible pressure angles. In practice, based on accumulated experience, it is recommended for preliminary calculations in mechanisms with only rotational pairs to take [θ] = 45…60°, and in mechanisms with rotational and translational pairs [θ] = 30…45 [14]. In this work, for the study, we will accept the constraint on the maximum pressure angle [θ] = 45°.

Results and Discussion. Figure 4, a shows a graph (hodograph) of the change in pressure angles in joints B, C, and D of the original mechanism. As can be seen from the graphs, for joints B and C of the mechanism, there is a significant excess of the accepted permissible level of the pressure angle (shaded area), which requires targeted optimization synthesis.

Let us formulate the synthesis problem as follows: for a given rotation angle of the driving link (crank), it is necessary to find a set of geometric parameters of the mechanism links such that the maximum values of the pressure angle in the joints over a full revolution of the crank do not exceed the specified permissible value [θ].

We set the vector of optimized parameters, which will include all the link lengths of the mechanism, as well as the coordinates of one fixed support:

a)                                                             б)
a – original mechanism; b – synthesized mechanism
Figure 4. – Results of calculating pressure angles in the joints of mechanisms

We calculate the pressure angles in the joints using one of the expressions (2) – (5). The objective function will be the condition of minimizing the maximum values of the pressure angles for the three joints B, C, and D:

where Ci – is the weighting factor of the pressure angle, assigned individually for each joint, which is preliminarily taken equal to 1 and, depending on the results obtained, can be "softened" or "tightened" for the objective function.

To determine the maximum pressure angle in each joint, we find the derivative of θ(φ) with respect to the generalized coordinate and set it equal to zero. We solve the resulting equation using a numerical method (for example, using the mathematical package PTC MathCAD), and find the crank angle (φ) corresponding to the maximum pressure angle, and then the numerical value of the pressure angle (see Figure 3, block 5 in the diagram).

We restrict the search for optimal values of the vector of optimized parameters (6) to the following range of numbers from their initial values:

where p*i – is the initial value of the i-th element of the vector of optimized parameters.

To ensure the condition for the existence of a crank, we additionally impose on the lever mechanism under consideration constraints in the form of Grashof's theorem [3, 15], according to which the shortest link is a crank if the sum of the lengths of the shortest and any other link is less than the sum of the lengths of the remaining two links:

The further solution of the kinematic synthesis problem is carried out using one of the known optimization algorithms [16]. To solve the stated kinematic synthesis problem, a numerical optimization method implemented using the built-in minimization function in the mathematical package PTC MathCAD was applied. The result of the calculations was the following optimal set of geometric parameters of the mechanism:

Figure 2, b shows the kinematic diagram of the synthesized mechanism, and Figure 4, b shows graphs of the dependence of the change in pressure angle in its joints. As can be seen from the results obtained, the new values of the pressure angles do not exceed the accepted permissible value over the entire range of the driving link rotation angle. For the synthesized mechanism, the calculated values of the maximum pressure angle in joints B, C, D were respectively θB=44.8°, θC=44.9° and θD=14.1°, which satisfies the specified constraint on the permissible angle [θ].

However, it should be noted that tightening the design constraints during the kinematic synthesis on the geometric parameters of the lever mechanism may make it impossible to simultaneously optimize the pressure angles in all joints. If the mechanism has working and idle (with lower loads) strokes, slightly larger values for [θ] can be adopted for the idle stroke. Therefore, the synthesis of a mechanism is always a search for a working compromise between the condition of ensuring pressure angle values within acceptable limits and the choice of geometric parameters that meet the design requirements.

Conclusion. A methodology for kinematic optimization synthesis of a planar articulated mechanism has been developed and implemented, aimed at ensuring a specified constraint on the maximum pressure angle in its joints. The proposed methodology for calculating pressure angles is based on original analytical dependencies obtained using the geometric property of scalar and vector products or their analogy as applied to complex numbers, as well as the argument of a complex number. The proposed algorithm and the computational experiment conducted according to it confirmed the efficiency of the method. The developed approach can precede the force analysis and is also recommended for use in engineering practice when designing lever mechanisms that are subject to increased requirements for efficiency, wear, and absence of jamming.

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To cite this work:

Котов, А. В. Оптимизационный синтез плоских рычажных механизмов по углу давления / А. В. Котов, Д. Г. Кроль // Барановичского государственного университета. Серия: Технические науки. – 2026. – № 1(19). – С. 33-39. – EDN USEEDK.

Kotov A. V., Krol D. G. Optimizatsionnyy sintez ploskikh rychazhnykh mekhanizmov po uglu davleniya [Optimization synthesis of flat lever mechanisms by pressure angle]. Vestnik Baranovichskogo gosudarstvennogo universiteta. Seriia: Tekhnicheskie nauki [Bulletin of Baranovichi State University. Technical Sciences], 2026, no. 1(19), pp. 33-39 (in Russ.).

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