Next Article in Journal
Algebraic Numbers as Product of Powers of Transcendental Numbers
Previous Article in Journal
On Some Formulas for Kaprekar Constants
Previous Article in Special Issue
Periodic Solution of the Strongly Nonlinear Asymmetry System with the Dynamic Frequency Method
Article Menu
Issue 7 (July) cover image

Export Article

Open AccessArticle

Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain

1
Department of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050000, China
2
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuan 050043, China
3
Department of Mechanics Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(7), 886; https://doi.org/10.3390/sym11070886
Received: 17 June 2019 / Revised: 1 July 2019 / Accepted: 3 July 2019 / Published: 6 July 2019
(This article belongs to the Special Issue Nonlinear Oscillations and Boundary Value Problems)
  |  
PDF [2715 KB, uploaded 8 July 2019]
  |  

Abstract

In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion. View Full-Text
Keywords: non-smooth high-dimensional system; asymmetric system; impact periodic solution; impact recovery matrix; non-smooth analytic solution non-smooth high-dimensional system; asymmetric system; impact periodic solution; impact recovery matrix; non-smooth analytic solution
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Guo, X.; Zhang, G.; Tian, R. Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain. Symmetry 2019, 11, 886.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top