Meccanica (2012) 47:835–844 DOI 10.1007/s11012-011-9452-y Dynamics of the shift in resonance frequency in a triple pendulum Ivan Rivas-Cambero · Jose M. Sausedo-Solorio Received: 31 July 2009 / Accepted: 15 June 2011 / Published online: 12 July 2011 © Springer Science+Business Media B.V. 2011 Abstract We propose a general model for pendular systems with an arbitrary number of links arranged sequentially. The form of this model is easily adapt- able to different settings and operating conditions. The main subject of analysis is a system obtained as a spe- cific case taken from the general analysis, a three-links pendulum with damping subject to periodic perturba- tion. We performed a theoretical analysis of the fre- quency response and compared it with results from temporal integration. Moreover, a law was obtained explaining the behavior of the shift of the resonant fre- quencies due to a change in a parameter. Keywords Triple pendulum · Resonance frequency · Frequency shift · Friction 1 Introduction In general and whenever possible, at the starting point of any analysis of a system, a linear mathematical I. Rivas-Cambero División de Ingenierías, Universidad Politécnica de Tulancingo, Tulancingo Hidalgo, Mexico e-mail: irivas@upt.edu.mx J.M. Sausedo-Solorio ( ) Centro de Investigación Avanzada en Ingeniería Industrial, Universidad Autónoma del Estado de Hidalgo, Pachuca Hidalgo, Mexico e-mail: jmsaucedos@yahoo.com model is used. If the model is represented through dif- ferential equations, a further step is taken to propose a more detailed model, increasing the complexity of its equations and solution by using non-linear differen- tial equations. Due to non-linearities in a system, some conditions might generate solutions with co-existing attractors or even chaotic behavior [1]. The interest in the study of pendular systems is con- tinually increasing, based on the very rich dynamic be- havior they display and the corresponding mathemat- ical models that can be used to analyze very complex systems. Even a simple pendulum can be used as a standard benchmark to test another systems. Despite general models for an arbitrary number of links being used to control issues of the inverted pendulum [2], the lack of general models ready to apply to an n-links regular pendulum is well known. In [3] a model is pro- posed that uses equations of motion with a n-links pen- dulum fixed at its upper extremity using frictionless elements. Caertmell [4] developed a complete model of a plane pendulum for two links taking into account friction and compliance as well as links with variable lengths and masses. Instead of using previously derived equations for a fixed number of links [5–8], we propose a general matrix representation for the n-links pendulum with damping. With this model, it is straightforward to de- compose and separate the first derivatives of the dy- namic variables as well as to solve the system equa- tions using simulation language or a software pack- age. Another kind of analysis tool is by means of a