Possibility of application of the theory of complex numbers to the solution of engineering problems of kinetic analysis of flat lever mechanisms

Authors: Kotov A.V.

This article is an English translation of a similar article posted on the blog in Russian. The translation was done using Google Translate, so do not judge strictly. There will certainly be literary errors, but I think the general meaning of the information that I tried to put into this work will be clear to the English-speaking audience. The purpose of this article is to expand the circle of readers of the blog among the English-speaking audience, to tell about the original analytical method of calculating flat lever mechanisms by the method of transforming coordinates in an invariable basis using the theory of complex numbers.

Despite the fact that the foundations of the course «Theory of Mechanisms and Machines» can be said to go back centuries, but if we have to turn to the analytical calculation of lever mechanisms, we will find that no significant shifts in helping to solve these problems have appeared in educational literature so far. It seems that the methods of calculating lever mechanisms have remained at the same level, and science has exhausted itself in this direction and cannot offer anything new, modern and effective.

But what should mechanical engineering do in this case, in which real lever mechanisms have found wide application, and for their rational projection it is necessary to carry out the corresponding calculations? Unfortunately, the well-known principle works here: the drowning man will save himself. Therefore, if science cannot come to the aid of mechanical engineering in terms of developing modern methods for calculating lever mechanisms, then engineers have to solve this problem on their own, by developing their own methods and algorithms.

The main criteria by which engineers evaluate proposed and developed new calculation methods, including lever mechanisms, are: accessibility; clarity; efficiency; universality; possibility of formalization and algorithmization; ability to be implemented in modern mathematical packages and programming languages. If the proposed or developed method satisfies all the requirements as much as possible, this allows the engineer to use his working time more rationally and usefully, and also to save his strength and nerves.

During my work at three industrial enterprises, including the flagships of the Belarusian and Russian agricultural engineering «Gomselmash» and «Rostselmash», as well as constant study of relevant thematic materials, I have encountered only one method of calculating lever mechanisms, which has been effectively used in the Scientific and Technical Center of JSC «Gomselmash» for more than 20 years and to this day. This method of studying lever mechanisms is based on the vector method of transforming coordinates in a constant basis, a detailed description of which is devoted to several dozen scientific papers, including my authorship. For a long time, I believed that I would not meet a worthy competitor to this method in my practice. And if there is no competitor, then, as a rule, any method has no further development, no competition, which ultimately leads to stagnation and oblivion.

And so, starting to write the first chapter of the dissertation, I approached the study of materials of foreign authors on the current state of methods of synthesis and analysis of lever mechanisms and noticed that many of them use the theory of complex numbers. This approach seemed quite unusual to me, given that in Russian-language scientific works it is presented very poorly, literally 2-3 works, essentially duplicating the calculations of foreign authors. And I decided to understand this method a little, with the hope of learning something new for myself.

As is known, a complex number is a two-dimensional number that consists of a real and imaginary part. But physically, it is impossible to represent this number in real life - it is the same as trying to represent the fourth dimension in our three-dimensional space.

Any two-dimensional geometric vector in the complex plane can be represented as a complex number vector using one of the following three notations (see Figure 1):

where j - imaginary unit; rx and ry are real numbers, respectively, of the real and imaginary parts of the complex number; r - modulus of the complex number; q - argument of the complex number, which is the angle of inclination to the real axis of the complex plane, rad.

Figure 1 – Representation of a complex number vector on the complex plane

The first algebraic form of representation of a complex number in expression (1) corresponds to a rectangular (Cartesian) coordinate system, and the second and third to a polar coordinate system. The second form of notation is called trigonometric, the third is exponential, and the transition from one form of representation to another is carried out using the well-known Euler formula. The transition between rectangular (cartesian) and polar coordinate systems is carried out using the simplest mathematical functions:

The use of complex numbers in various fields has been known for quite a long time, particularly in electrical engineering, where the theory of complex numbers is used to represent voltage, current, resistance, and all quantities dependent on them, changing according to the law of a sinusoidal function, when calculating electrical circuits. It is from electrical engineering that I borrowed the underscore symbol to denote the vector of a complex number in expression (1), which will be used further on. This allows us to distinguish the vector of a complex number from its modulus, and also does not create a discrepancy with the designation of ordinary vector quantities, which I am accustomed to using in my works.

But let us return to foreign works on kinematic analysis using complex numbers. When solving the problem of the positions of a flat lever mechanism, foreign authors represent each of its links on the plane by the corresponding relative vector of a complex number in exponential form. Thus, a system of equations is formed by analogy with the method of closed vector contours, which is then differentiated twice by the adopted generalized coordinate when solving the problem of velocities and accelerations. The general form of such equations obtained using vectors of complex numbers in exponential form can be represented as:

To solve the equations compiled, each term of the expressions is multiplied by a complex exponent with a negative argument for one of the sought quantities (angles, analogs of velocities or accelerations), ensuring that only one unknown remains in the real or only in the imaginary part of the expression. After that, the Euler formula is applied and the real and complex parts are identified separately, thereby obtaining analytical equations that completely coincide with the known projection equations presented in classical educational literature. Therefore, this approach to applying the theory of complex numbers is interesting only from the point of view of the initial formulation of the problem being solved and partially with the methods for solving the resulting systems of equations. However, from a practical point of view, it does not satisfy the engineering criteria voiced above for the developed methods for calculating lever mechanisms.

As I have already mentioned, there is an original vector method for studying lever mechanisms, based on the method of transforming coordinates in a constant basis. This method uses such basic mathematical operations as rotation and displacement of vectors. The implementation of these operations is also possible with complex numbers, for which the operations of rotation and displacement are equivalent to simple algebraic operations - addition (subtraction) and multiplication (division):

Expression (3) can be used for the case when there are no changes in the length (modulus) of the new complex vector during rotation, and expression (4) – when there is a change in the length of the new complex vector (increase or decrease). In this case, the coefficient r is the coefficient of stretching (compression) of the length of the new vector of the complex number. A graphical explanation of the operation of rotating the vector of a complex number with and without changing its length is shown in Figure 2.

a)                                                                            b)
Figure 2 – Graphic explanation of the operation of rotating a complex number vector: a – without changing the length; b – with changing the length

Expressions (2) - (4) are basic for describing the kinematics of any flat lever mechanism by the method of transforming coordinates in a constant basis using the theory of complex numbers. We will demonstrate the application of the theory of complex numbers when conducting a kinematic analysis of a flat lever mechanism using this method, the kinematic diagram of which is shown in Figure 3.

Figure 3 – Kinematic diagram of a flat lever mechanism

The solution to the position problem can be represented as the following simple algorithm using complex numbers:

The solution to the problem of analogs of linear velocities and accelerations of characteristic points or links of a lever mechanism is solved by double differentiation of the corresponding radius vector of a complex number:

To solve the problem of analogs of angular velocities and accelerations of characteristic links of a lever mechanism, it is necessary to use the property of vector and scalar products, realized using vectors of complex numbers. If in mechanics the vector product of two vectors is found using certain mathematical calculations, then in the theory of complex numbers the analog for these mathematical operations is the algebraic operation of multiplication with a complex conjugate number. Thus, if the first of two multiplied complex numbers is presented as a complex conjugate number, then the real part of the resulting product, taking into account the algebraic sign, will represent the value of the scalar product of two vectors specified by these numbers. The imaginary part, also taking into account the magnitude of the algebraic sign, will represent the value of the vector product of these same two vectors.

Using the above-described property of vector complex numbers, the analogs of angular velocities and accelerations of characteristic links of a flat lever mechanism can be defined as:

where ri and ri* - vectors of complex conjugate numbers, i.e. a pair of complex numbers with identical real parts and imaginary parts equal in absolute value but opposite in sign.

If the specified forms of recording a vector of a complex number are used correctly, and all mathematical rules for working with them are observed, then any real number that does not contain an imaginary unit j will be a projection onto the X axis of a right Cartesian coordinate system, and the presence of a complex unit will indicate a projection onto the Y axis. Therefore, to display the results of calculations with vectors of complex numbers, the following universal form of recording can be used:

Thus, using expressions (5) - (13), the kinematics of the considered flat lever mechanism is completely described, i.e. the problems of positions, speeds and accelerations are solved.

In the future, the application of the theory of complex numbers is not limited to kinematic analysis. The ability to find the results of scalar and vector products using vectors of complex numbers opens up new possibilities for conducting force analysis and analysis of the equation of motion of a flat lever mechanism during its dynamic study.

To date, using the theory of complex numbers together with the method of transforming coordinates in a constant basis, the kinematic parameters of various flat lever agricultural machines, such as the lifting mechanism of the inclined chamber of a grain harvester, have been tested. The obtained calculation results completely coincided with the projections obtained earlier using other research methods, which once again confirms the viability of applying the theory of complex numbers to the study of the kinematic parameters of flat lever mechanisms.

Conclusions

The proposed method of transforming coordinates in a constant basis using the theory of complex numbers for performing kinematic analysis of flat lever mechanisms eliminates the need to compile and solve cumbersome systems of equations obtained by the traditional method of closed vector contours. This method can easily be formalized and algorithmized in modern mathematical packages and programming languages, allowing for a comprehensive kinematic analysis of the designed flat lever mechanisms in a short time, with the possibility of their subsequent optimization synthesis.

However, the presented application of the theory of complex numbers in the kinematic analysis of lever mechanisms is limited only to the solution of two-dimensional problems, since to solve spatial problems it is necessary to use more complex forms of recording complex numbers.

References

1. Wilson, Charles E. Sadler, J. Peter. Kinematics and Dynamics of Machinery. 3rd Edition. Pearson Education Limited, 2013. – 848 p.

2. Abhary, K. A unified analytical parametric method for kinematic analysis of planar mechanisms. International Journal of Mechanical Engineering Education. – 2022. – Vol. 50(2). – Pp. 389-431.

3. Matsyuk, I. N., Shlyakhov, E., Zyma, N. V. Study of Planar Mechanisms Kinetostatics Using the Theory of Complex Numbers with MathCAD PTC. Mechanics, Materials Science & Engineering Journal. – 2017. – Vol. 8, hal-01508541.

4. Matsyuk I.N., Shlyahov E.M., Zima N.V. Kinematika ploskikh mekhanizmov v programme MathCAD s ispol'zovaniyem teorii kompleksnykh chisel [Kinematics of planar mechanisms in MathCAD program using the theory of complex numbers] Razvitiye informatsionno-resursnogo obespecheniya obrazovaniya i nauki v gorno-metallurgicheskoy otrasli i na trans-porte 2014 : Sbornik nauchnykh trudov mezhduna-rodnoy konferentsii [The Development of the Informational and Resource Providing of Science and Education in the Mining and Metal-lurgical and the Transportation Sectors 2014: Collection of scientific papers of the internation-al conference]. Dnepropetrovsk, NGU Publ., 2005, pp 407-412. (in Russian).

5. Kotov A. V., Kroll D. G. Kinematicheskiy i silovoy analiz mekhanizma pod"yema naklonnoy kamery zernouborochnogo kombayna s primeneniyem teorii kompleksnykh chisel [Kinematic and force analysis of the inclined chamber lift mechanism of a combine harvester using complex numbers theory]. Konstruirovaniye, ispol'zovaniye i nadezhnost' mashin sel'skokhozyaystvennogo naznacheniya : sbornik nauchnykh rabot [Designing, use and reliability agricultural machines destination : collection of scientific papers], 2015, vol. 24, no. 1, pp. 40-48 (in Russ.).

6. Kotov A. V., Kroll D. G. K voprosu analiza uravnoveshennosti ploskikh rychazhnykh mekhanizmov s pomoshch'yu teorii kompleksnykh chisel [On the problem of balance analysis of flat lever mechanisms using the theory of complex numbers]. Trudy mezhdunarodnoy nauchno-prakticheskoy konferentsii "Razvitiye mashinostroitel'noy otrasli i podgotovka vysokokvalifitsirovannykh kadrov novoy formatsii" (sostoyaniye, problemy i puti ikh resheniya) [Proceedings of the international scientific and practical conference "Development of the mechanical engineering industry and training of highly qualified personnel of a new formation” (status, problems and ways of their solution)]. Astana, 2025, pp. 31-33 (in Russ.).

7. Kotov A. V. Issledovaniye kinematiki ploskikh rychazhnykh mekhanizmov s primeneniyem teorii kompleksnykh chisel [Investigation of the kinematics of flat lever mechanisms using the theory of complex numbers]. Materialy nauchno-tekhnicheskoy konferentsii aspirantov, magistrantov, studentov "Studencheskiy nauchnyy dvizh" [Materials of the scientific and technical conference of postgraduates, masters, and students “Student scientific movement”]. Gomel, 2025, pp. 18–20 (in Russ.).

I hope that the materials presented on the site will be useful for your scientific or practical activities and I would be grateful for mentioning my works in your list of references when using them.

To cite this work:

Kotov A. V. Possibility of application of the theory of complex numbers to the solution of engineering problems of kinetic analysis of flat lever mechanisms. Vectormethod.blogspot.com [Electronic resource]. – 2020. – Mode of access: https://vectormethod.blogspot.com/2025/08/possibility-of-application-of-the-theory-of-complex-numbers-to-the-solution-of-engineering-problems-of-kinetic-analysis-of-flat-lever-mechanisms.html. – Date of access: 25.08.2025.