TECHNICAL PAPER Nonlinear dynamics of flexible nanopositioning systems with geometrical imperfection A. Naderi Rahnama 1 • M. Mousavi Mashhadi 1 • M. Moghimi Zand 1,2 Received: 7 November 2018 / Accepted: 18 January 2019 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract In this paper, the nonlinear dynamics of flexible beams with geometrical imperfection and concentrated end-mass is investigated. Based on the Euler–Bernoulli beam theory and nonlinear strain–displacement relations, governing equations are obtained. The Galerkin method is employed to discretize the nonlinear partial differential equations of motion. The effect of system parameters on both the linear and nonlinear dynamics of the system are studied based on the numerical solution of the set of nonlinear ODEs with coupled terms. The results show that dynamic behavior of the system is affected significantly by the geometrical imperfection. It is revealed that the geometrical imperfection results in jump instability and sub-harmonics. Since, geometrical imperfections are inevitable in manufacturing process of these equipments, the effects of imperfections should be considered in design prior to the manufacturing of nanopositioning systems. 1 Introduction Flexible beam is one of the most important members used in micro/nano electromechanical systems (Cao et al. 2018; Hosseini-Pishrobat and Keighobadi 2018c; Siahpour et al. 2018; Moghimi Zand and Moghaddam 2014; Isaac Hos- seini et al. 2017; Keighobadi et al. 2018), Atomic force microscopy (Cai et al. 2018; Hosseini-Pishrobat and Keighobadi 2018a, b), nanopositioning systems and etc. Indeed, a nanopositioning system is a motioning system composed of different mechanical and electrical compo- nents capable of producing displacements in micro and nano scale. Such equipments are used in systems to pro- duce displacements in the order of micro and nano meter. Some of the important applications of this equipment are in semiconductors, optic and laser industries, nanometrology, sweeping microscopes, precise machining, genetic manip- ulations and inter-cellular activities. In nanopositioning systems, the elastic motion is created by using flexible beams, which causes single or multi stage motion. For example, Fig. 1 represents a planar flexural positioning mechanism (Wang and Xu 2017). In order to create a wide motion range in these systems, larger deformations have to be applied to the mechanism which causes geometrical nonlinearity (Yang et al. 2017; Rad- golchin and Moeenfard 2018). These nonlinearities have considerable effects on the dynamic behavior of the system and its performance. Hence, linear models of these systems are not suitable for prediction of the dynamic behavior, and therefore, it is necessary to develop a model based on nonlinear behavior (Awtar and Parmar 2013). Taking into account the fact that the flexible beams usually contain narrow and long arms, Euler–Bernoulli beam theory can explain static and dynamic behavior of such systems (Rao 2007). In order to have an overview on the dynamical behavior of the nanopositioning system, it is desired to study the dynamics of a simple beam with concentrated mass at its end. Such investigation is the main objective of the present study. Generally, vibrations of beam have been investigated both analytically and exper- imentally from different aspects by various researchers (Eslami et al. 2016; Maleki and Mohammadi 2017; Eltaher et al. 2016; Nayfeh and Mook 2008). For instance, Nayfeh and Mook (2008) studied nonlinear vibration behavior of the beam using nonlinear Euler–Bernoulli beam theory. Anderson et al. (1996) investigated vibration behavior of slender beam under the parametric excitation theoretically and experimentally. Their results revealed that the & M. Moghimi Zand mahdimoghimi@ut.ac.ir 1 School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 2 Small Medical Devices, BioMEMS and LoC Lab, School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran 123 Microsystem Technologies https://doi.org/10.1007/s00542-019-04319-0