Mechanics, Materials Science & Engineering, March 2017
ISSN 2412-5954
Study of Planar Mechanisms Kinetostatics Using the Theory of Complex Numbers with MathCAD PTC13 Matsyuk I.N. 1,a, Shlyakhov
1
, Zyma N.V.1
1 a
shlyahove@nmu.org.ua DOI 10.2412/mmse.40.52.685 provided by Seo4U.link
Keywords: MathCAD, planar mechanism, vector, complex number, second class Assur group, kinetostatic.
ABSTRACT. The paper describes the convenient method to study the kinematics of planar mechanisms. Very convenient to study the kinematics of planar mechanisms with the help of complex numbers. The current article proposed the kinetostatic analysis also be carried out by operating in the field of complex numbers. It is recommended to use the MathCAD PTC, having great potential for operations with complex numbers. The basic idea is presented by the example of the three second class Assur groups found most frequently in modern planar mechanisms.
Introduction. As it is well known, complex numbers are used to solve geometric problems [1] and for research purposes of the motion of planar bodies and mechanisms [2, 3, 4]. Information about complex numbers usage to solve kinetostatics problems of planar mechanisms has been described insufficiently in known literature. In case, when inertia forces of the links are taking into account for kinetostatics analysis of mechanism, to perform kinematic analysis is necessary first of all. This analysis made with representing vectors as complex numbers and has a definite advantage over other methods. In this case, it is logical to provide force analysis using complex numbers too. This is facilitated by the fact, that the popular modern mathematical software (Maple, Wolfram Mathematica, MathCAD) have appropriate calculation algorithm to operate with such numbers. Analysis of the recent research. The research of the force analysis of mechanisms does not represent any difficulties today. Historically, in the beginning it has been solved only using the graphicalanalytical methods [5]. Gradually graphical-analytical methods are substituted by purely analytical methods, that became a prior due to development of computer technology [6 - 9]. Determination of forces in kinematic pairs can be carried out separating the mechanism on the groups [10] which are kinetostatically determined [11]. Another approach is to partition mechanism on links, for each of which the equilibrium equation should be written. Solving the obtained system of equations defines all unknown reactions [12]. These reactions may be represented by two components (normal and tangential [11] or horizontal and vertical [12]), i.e. vectors with known directions, but unknown magnitudes. Representation of required reactions as unknown vectors is more compact. In this case, the number of equations is minimal [7]. The research has been provided using MathCAD PTC that is powerful program for operations with complex numbers. All calculations are performed in the international standard system (NIST).
13
-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/
MMSE Journal. Open Access www.mmse.xyz
143