# Analytical Method for Determination of Internal Forces of Mechanisms and Manipulators

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Distributed Dynamic Loads and Approximation Matrix

## 3. Discrete Models for Elastic Calculation of Elements and Entire Construction of Mechanisms and Manipulators

## 4. Dynamic Equilibrium Equations of the Discrete Models of the Elements and Joints

## 5. Decisive Equations for Determining Internal Forces

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Tschiras, A.A. Structural Mechanics; Stroyizdat: Moscow, Russia, 1989; ISBN 5-274-00555-1. [Google Scholar]
- Darkov, A.V.; Shaposhnikov, N.N. Structural Mechanics; Lan’: St. Petersburg, Russia, 2010; ISBN 978-5-8114-0576-3. [Google Scholar]
- Shakirzyanov, R.A.; Shakirzyanov, F.R. Lecture Notes in Structural Mechanics; KGASU: Kazan, Russia, 2014. [Google Scholar]
- Neubert, V.H.; Hahn, H.T.; Lee, H. Lumped parameter beams based on impedance methods. J. Eng. Mech. Div.
**1970**, 96, 69–82. [Google Scholar] [CrossRef] - Golebiewski, E.P.; Sadler, J.P. Analytical and experimental investigation of elastic slidercrank mechanisms. J. Eng. Ind.
**1976**, 98, 1266–1271. [Google Scholar] [CrossRef] - Sadler, J.P. A Lumped Parameter Approach in the Kineto-Elastodynamic Analysis of Mechanisms. Ph.D. Thesis, Rensselaer Polytechnic Institute, New York, NY, USA, 1972. [Google Scholar]
- Sadler, J.P.; Sandor, G.N. A lumped parameter approach to vibration and stress analysis of elastic linkages. J. Eng. Ind.
**1973**, 95, 549–557. [Google Scholar] [CrossRef] - Sadler, J.P.; Sandor, G.N. Nonlinear vibration analysis of elastic four-bar linkages. J. Eng. Ind.
**1974**, 96, 411–419. [Google Scholar] [CrossRef] - Zienkiewicz, O.; Taylor, R.; Zhu, J. The Finite Element Method: Its Basis and Fundamentals; Butterworth-Heinemann: Oxford, UK, 2013; ISBN 978-0-0809-5135-5. [Google Scholar]
- Strang, G.; Fix, G. An Analysis of the Finite Element Method; Wellesley-Cambridge Press: Wellesley, MA, USA, 2008; ISBN 978-0-9802-3270-7. [Google Scholar]
- Tokhi, M.O.; Mohamed, Z.; Amin, S.H.M.; Mamat, R. Dynamic Characterizaion of a Flexible Manipulator System: Theory and Experiments. Available online: https://zapdf.com/dynamic-characterisation-of-a-flexible-manipulator-system-th.html (accessed on 4 September 2018).
- Tokhi, M.O.; Mohamed, Z.; Shaheed, M.H. Dynamic characterization of a flexible manipulator system. Robotica
**2001**, 19, 571–580. [Google Scholar] [CrossRef] - Chung, J.; Yoo, H.H. Dynamic analysis of a rotating cantilever beam by using the finite element method. J. Sound Vib.
**2002**, 249, 147–164. [Google Scholar] [CrossRef] - Shaker, M.C.; Ghosal, A. Nonlinear modeling of flexible manipulators using non-dimensional variables. J. Comput. Nonlinear Dyn.
**2006**, 1, 123–134. [Google Scholar] [CrossRef] - Yue, S.; Tso, S.K.; Xu, W.L. Maximum dynamic payload trajectory for flexible robot manipulators with kinematic redundancy. Mech. Mach. Theory
**2001**, 36, 785–800. [Google Scholar] [CrossRef] - Korayem, M.H.; Haghpanahi, M.; Heidari, H.R. Maximum allowable dynamic load of flexible manipulators undergoing large deformation. Sci. Iran.
**2010**, 17. Available online: http://scientiairanica.sharif.edu/article_3246.html (accessed on 04 September 2018). - Korayem, M.H.; Haghpanahi, M.; Heidari, H.R. Maximum allowable load of very flexible manipulators by using absolute nodal coordinate. Aerosp. Sci. Tech.
**2015**, 45, 67–77. [Google Scholar] [CrossRef] - Du, Z.; Yu, Y.; Yang, J. Analysis of the Dynamic Stress of Planar Flexible-Links Parallel Robots. Front. Mech. Eng. China
**2007**, 2, 152–158. [Google Scholar] [CrossRef] - Dwivedy, S.K.; Eberhard, P. Dynamic analysis of flexible manipulators: A literature review. Mech. Mach. Theory
**2006**, 41, 749–777. [Google Scholar] [CrossRef] - Li, H.; Hao, G. Constraint-Force-Based Approach of Modelling Compliant Mechanisms: Principle and Application. Precis. Eng.
**2016**, 47, 158–181. [Google Scholar] [CrossRef] - Utenov, M.U. Investigations of forces, arising from the self mass of links with constant and variable cross-sections in their plane-parallel motion. In Proceedings of the First International Scientific and Practical Conference, Almaty, Kazakhstan, 18–19 October 2000. [Google Scholar]
- Utenov, M.U. The matrix of approximation of element force under the action of distributed load with parabolic intensity. In Proceedings of the First International Scientific and Practical Conference, Almaty, Kazakhstan, 18–19 October 2000. [Google Scholar]
- Utenov, M.U. Construction of discrete models of planar rod mechanisms in the elastic calculation. Bull. Kazakh Acad. Transp. Commun.
**2001**, 6, 61–64. [Google Scholar]

**Figure 4.**The discrete model of the first type of beam under the action of distributed trapezoidal loads.

**Figure 7.**Rigid joint with constant cross-sections of the element, where the external concentrated loads are attached.

**Figure 9.**The mechanism under discussion of the links showing the constructed transverse dynamic distributed loads.

**Figure 10.**The mechanism of the links illustrating the constructed longitudinal dynamic distributed loads.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Utenov, M.; Sobh, T.; Baigunchekov, Z.; Zhilkibayeva, S.; Patel, S.
Analytical Method for Determination of Internal Forces of Mechanisms and Manipulators. *Robotics* **2018**, *7*, 53.
https://doi.org/10.3390/robotics7030053

**AMA Style**

Utenov M, Sobh T, Baigunchekov Z, Zhilkibayeva S, Patel S.
Analytical Method for Determination of Internal Forces of Mechanisms and Manipulators. *Robotics*. 2018; 7(3):53.
https://doi.org/10.3390/robotics7030053

**Chicago/Turabian Style**

Utenov, Muratulla, Tarek Sobh, Zhumadil Baigunchekov, Saltanat Zhilkibayeva, and Sarosh Patel.
2018. "Analytical Method for Determination of Internal Forces of Mechanisms and Manipulators" *Robotics* 7, no. 3: 53.
https://doi.org/10.3390/robotics7030053