# Analytical method of definition of internal forces taking into account the distributed dynamical loads in links of robotic systems and mechanisms with statically indeterminate structures

### Abstract

In this paper the technique of analytical determination of internal forces in links of planar mechanisms and manipulators with statically indeterminate structures taking into account the distributed dynamical loads, a dead weight and the operating external loads is designed. The dynamic equilibrium equations for the discrete model of the element under the action of cross and axial inertial trapezoidal loads are derived. Also, the dynamic equilibrium equations for elements and joints that expressed in terms of the unknown parameters of the internal forces of elements under the action of the distributed trapezoidal loads are obtained. The compliance matrix of an element is received from the expressions of energy for rods, so as the replacement of construction by a set of discrete elements is based on the equality of the energies of the real structures and its discrete model. The programs in the MAPLE18 system are made on the given algorithm and animations of the motion of mechanisms with construction on links the intensity of cross and axial distributed inertial loads, the bending moments, cross and axial forces, depending on kinematic characteristics of links are obtained.### References

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[2] Zienkiewicz O.C., Taylor R.L. The Finite Element Method// fifth edition. — London: Butterworth-Heinemann. — 2000. — Vol. 1: The basis. — P. 663.

[3] The Finite Element Method in Structural Mechanics and Solid Mechanics// A bibliography. Foreign Literature. — 1973.

[4] Ciarlet P.G. The Finite Element Method for Elliptic problems// Amsterdam: North-Holland. — 1978. — P. 529.

[5] Rozin L.A. The Finite Element Method in the annex to the elastic system // М: Stroyizdat.— 1977. — P. 178.

[6] Korneev V.G. Schemes of the finite element method of high orders of accuracy // L: University of Leningrad. — 1977. — P. 208.

[7] Аlexandrov A.V., Smirnov V.A., Shaposhnikov N.N. Methods of calculation of rod systems, plates and shells using a computer // М: Stroyizdat. — 1976. — P.1. — P. 248.

[8] Segerlind Larry J. Applied Finite Element Analysis// second edition. — the USA: Wily&Sons, Inc. — 1937. — P. 427.

[9] Chiras А.А. Structural mechanics// М: Stroyizdat. — 1989. — P. 255.

[10] Utenov М.U. The report on research work «Development of analytical prediction of the theory of strength and stiffness of robotic systems and mechanisms» (an interim report on the project GF4/2019 МОN RК)// Аlmaty. — 2016. — P. 120.

Published

2018-07-18

How to Cite

ZHILKIBAYEVA, S.; UTENOV, M. U.; UTENOVA, K..
Analytical method of definition of internal forces taking into account the distributed dynamical loads in links of robotic systems and mechanisms with statically indeterminate structures.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 92, n. 4, p. 55-69, july 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/454>. Date accessed: 20 feb. 2019.
Section

Mechanics

Keywords
mechanisms, manipulators, distributed inertial forces, internal forces, dynamic equilibrium, linkage compliance, kinematic parameters, statically indeterminate mechanisms, CAD, animation