On the motion of a triple pendulum system under the influence of excitation force and torque


  • T. S. Amer Faculty of Science, Tanta University, Egypt
  • A. Galal Faculty of Engineering, Tanta University
  • A. F. Abolila Faculty of Engineering, Tanta University




Triple pendulum, Nonlinear dynamics, Multiple scales technique, Resonance, Stability.


In this article, a nonlinear dynamical system with three degrees of freedom (DOF) consisting of multiple pendulum (MP) is investigated. The motion of this system is restricted to be in a vertical plane, in which its pivot point moves in a circular path with constant angular velocity, under the action of an external harmonic force and a moment acting perpendicular to the direction of last arm of MP and at the suspension point respectively. Multiple scales technique (MST) among others perturbation methods is used to obtain the approximate solutions of the equations of motion up to the third approximation because it is authorizing to execute a specific analysis of the system behaviour and to realize the solvability conditions in view of the resonance cases. The stability of the considered dynamical model utilizing nonlinear stability analysis approach is examined. The solutions diagrams and resonance curves are drawn to illustrate the extent of the effect of various parameters on the solutions. The importance of this work is due to its uses in human or robotic walking analysis.

Author Biography

T. S. Amer, Faculty of Science, Tanta University, Egypt

Prof. of Applied Mathematics (Theoretical Mechanics)


M. V. Plissi, C. I. Torrie, M. E. Husman, N. A. Robertson, K. A. Strain, H. Ward, H. Lück, J. Hough, GEO 600 triple pendulum suspension system: seismic isolation and control, Rev. Sci. Instrum. 71(6), 2539-2545 (2000)

J. Awrejcewicz, G. Kudra, The piston-connecting rod-crankshaft system as a triple physical pendulum with impacts, Int. J. Bifurc. Chaos 15, 2207-2226 (2005)

H. P. Raymond, L. N. Virgin, Pendulum models of ponytail motion during walking and running, J Sound Vib 332, 3768-3780 (2013)

J. Awrejcewicz, G. Kudra, G. Wasilewski, Experimental and numerical investigation of chaotic regions in the triple physical pendulum, Nonlinear Dyn 50, 755-776 (2007).

M. K. Gupta, N. Sinha, K. Bansal, A. K. Singh, Natural frequencies of multiple pendulum systems under free condition, Arch Appl Mech 86,1049-1061 (2016).

M. Braun, On some properties of the multiple pendulum, Arch. Appl. Mech. 72, 899-910 (2003).

M. K. Gupta, P. Sharma, A. Mondal, A. Kumar, Visual recurrence analysis of chaotic and regular motion of a multiple pendulum system, Arab J Sci Eng 42, 2711-2716 (2017).

J. Awrejcewicz, G. Kudra, Stability analysis and Lyapunov exponents of a multi-body mechanical system with rigid unilateral constraints, Nonlinear Analysis 63, e909 – e918 (2005).

J. Awrejcewicz, G. Kudra, C. H. Lamarque, Investigation of triple physical pendulum with impacts using fundamental solution matrices, Int J Bifur Chaos 14, 12, 4191-4213 (2004)

J. Awrejcewicz, R. Starosta, G. S. Kamińska, Stationary and transient resonant response of a spring pendulum, Procedia IUTAM 19, 201-208 (2016).

R. Starosta, G. S. Kamińska, J. Awrejcewicz, Parametric and external resonances in kinematically and externally excited nonlinear spring pendulum, Int J Bifurcat Chaos 21, 10, 3013-3021 (2011).

R. Starosta, G. S. Kamińska, J. Awrejcewicz, Asymptotic analysis of kinematically excited dynamical systems near resonances, Nonlinear Dyn 68, 459-469 (2012).

T. S. Amer, M. A. Bek, I. S. Hamada, On the motion of harmonically excited spring pendulum in elliptic path near resonances, Adv Math Phys, Volume 2016, 15 pages (2016).

A. H. Nayfeh, Perturbations methods, Wiley, Weinheim (2004).

J. Awrejcewicz, R. Starosta, G. Kamińska, Asymptotic analysis of resonances in nonlinear vibrations of the 3-dof pendulum, Differ Equ Dyn Syst 21, 1&2, 123-140 (2013).

T. S. Amer, M. A. Bek, M. K. Abouhmr, On the vibrational analysis for the motion of a harmonically damped rigid body pendulum, Nonlinear Dyn 91, 2485–2502 (2018)

T. S. Amer, M. A. Bek, M. K. Abouhmr, On the motion of a harmonically excited damped spring pendulum in an elliptic path, Mech Res Commu 95, 23-34 (2019).

H. I. Jaafar, Z. Mohamed, M. A. Shamsudin, N. A. Mohd Subha, L. Ramli, A. M. Abdullahi, Model reference command shaping for vibration control of multimode flexible systems with application to a double-pendulum overhead crane, Mech Syst Signal Process 115, 677-695 (2019).

R. Kwiatkowski, T. J. Hoffmann, A. Kołodziej, Dynamics of a double mathematical pendulum with variable mass in dimensionless coordinates, Procedia Engineering 177, 439-443 (2017).

S. Wolfram, An elementary introduction to the Wolfram language, Wolfram Media; 2nd. edition (2017).

G. Kamińska, R. Starosta, J. Awrejcewicz, Two approaches in the analytical investigation of the spring pendulum, Vib Phys Syst 29, 2018005 (2018).

J. Awrejcewicz, Classical mechanics: Dynamics, Springer, Berlin (2012).

S. Rajasekar, M. A. Sanjuan, Nonlinear resonances, Springer, Berlin (2016).