A novel approach to study effects of asymmetric stiffness on parametric instabilities of multi-rotor-system Anuj Kumar Jain a, b, * , Vikas Rastogi a , Atul Kumar Agrawal a a Department of Mechanical, Production, Industrial and Automobile Engineering, Delhi Technological University, Delhi,110042, India b Department of Mechanical Engineering, The Northcap University, Gurgram,122017, India article info Article history: Received 1 February 2017 Received in revised form 23 August 2017 Accepted 15 October 2017 Keywords: Asymmetric stiffness Extended lagrangian Hamiltonian Amplitude Multi-rotor Bond graph modeling abstract The main focus of this paper is to study effects of asymmetric stiffness on parametric instabilities of multi-rotor-system through extended Lagrangian formalism, where sym- metries are broken in terms of the rotor stiffness. The complete insight of dynamic behaviour of multi-rotor-system with asymmetries is evaluated through extension of Lagrangian equation with a case study. In this work, a dynamic mathematical model of a multi-rotor-system through a novel approach of extension of Lagrangian mechanics is developed, where the system is having asymmetries due to varying stiffness. The ampli- tude and the natural frequency of the rotor are obtained analytically through the proposed methodology. The bond graph modeling technique is used for modeling the asymmetric rotor. Symbol-shakti ® software is used for the simulation of the model. The effects of the stiffness of multi-rotor-system on amplitude and frequencies are studied using numerical simulation. Simulation results show a considerable agreement with the theoretical results obtained through extended Lagrangian formalism. It is further shown that amplitude of the rotor increases inversely the stiffness of the rotor up to a certain limit, which is also affirmed theoretically. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction In recent decade, the dynamic behaviour and diagnostics of cracked rotor have gained momentum. At present, many researchers [1,2] are working on cracked rotor system, however very few researchers have focused on the issue of multi- cracked rotor system [3]. In literature, few authors have investigated the issue of crack evaluation for multi-rotor shaft. The investigations and studies on such crack are available in various papers [4,5] for beam structures and [6] for rotor. However, model based identification of cracks in rotor system can be found in paper of Sekhar [7]. Nevertheless, no such paper is available to evaluate the frequencyand amplitude of multi-rotor-system, which is being attempted in this paper. Rotating machinery is generally used in diverse engineering applications, such as industrial turbo-machinery, automo- biles, machine tools, household accessories, power plant's turbines and futuristic micro and nano-machines etc. Various studies exposed that the vibration brings a change in the physical properties like mass, stiffness and damping capacity that forces a change in modal parameters like natural frequencies. The existence of any type of asymmetries, transverse cracks, * Corresponding author. Department of Mechanical, Production, Industrial and Automobile Engineering, Delhi Technological University, Delhi,110042, India. E-mail addresses: anujjainkanbay@gmail.com, anujkjain@ncuindia.edu (A.K. Jain). Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi https://doi.org/10.1016/j.jsv.2017.10.027 0022-460X/© 2017 Elsevier Ltd. All rights reserved. Journal of Sound and Vibration 413 (2018) 159e172