Chaotic motion of a parametrically excited microbeam
Introduction
Microscale continuous elements (Dehrouyeh-Semnani, 2014, Dehrouyeh-Semnani, 2015, Farokhi and Ghayesh, 2015, Ghayesh and Farokhi, 2015, Kahrobaiyan et al., 2012, Karparvarfard et al., 2015, Mohammad-Abadi and Daneshmehr, 2014, Mohammadabadi et al., 2015, Sahmani et al., 2013, Tang et al., 2014b), such as microbeams, are present in many microelectromechanical systems, such as in micro energy harvesters, microswitches, airbag accelerometers, vibration and shock sensors, biosensors, and microactuators, just to name a few. In some of these applications, microbeams are subject to longitudinal forces; these forces, under dynamical working conditions, vary with time. The unsteady axial load is of a harmonic type which is superimposed to a constant (mean) value; in this paper, the axial force is assumed to be in the form of , with as the mean value (or mean axial load), as the amplitude of the force variations, and as the frequency of the axial load (or the excitation frequency).
The dynamical behaviour of microbeams subject to a time-dependent axial load is mainly studied in two mean axial load regimes. These regimes are known as the sub and supercritical, where the threshold is called critical mean axial load. When the mean axial load reaches the critical value, the stability of the system is lost by divergence, loading to buckling. Another interesting feature in the dynamical behaviour of microbeams is size effects (Akgöz and Civalek, 2014, Dai et al., 2015, Şimşek and Reddy, 2013); this effect may change the qualitative as well as quantitative behaviours of the system. The modified couple stress theory is employed in this paper in order to take into account small-size effects.
Many contributions on the dynamical behaviour and stability of microbeams can be found in the literature (Baghani, 2012, Ghayesh et al., 2013a, Ghayesh and Farokhi, 2013, Şimşek, 2010, Şimşek and Reddy, 2013, Tang et al., 2014a); the first class of the literature analysed the free dynamics, while the second class examined the forced statics and dynamical behaviour of microbeams subject to a time-varying transverse force or a constant axial force, mainly with the aim of obtaining frequency–response curves for the former (of the second class) and post-buckling/bending amplitude for the latter case (of the second class). This paper is the first which analyses the sub and supercritical complex global dynamics of microbeams subject to a time-dependent axial load by constructing the bifurcation diagrams of Poincaré sections, with special consideration to period-n, quasiperiodic, and chaotic motions.
Starting the literature review with a fundamental work by Kong, Zhou, Nie, and Wang (2008), who obtained the size-dependent linear natural frequencies of a microbeam based on the Euler–Bernoulli theory. There are other linear studies in the literature, for instance, by Wang, Xu, and Ni (2013), Asghari, Kahrobaiyan, and Ahmadian (2010), and Ma, Gao, and Reddy (2008) which examined the size-dependent dynamics of microbeams on the basis of the Timoshenko theory. Salamat-talab, Nateghi, and Torabi (2012) examined the static and dynamic behaviours of a third-order shear-deformable functionally graded microbeam. Asghari, Ahmadian, Kahrobaiyan, and Rahaeifard (2010) analysed the size-dependent motion characteristics of a functionally graded microbeam via use of the modified couple stress theory. Ramezani (2012) contributed to the field by deriving the equations of motion of a Timoshenko microbeam taking into account small-size effects and geometric nonlinearities. Ansari, Gholami, Faghih Shojaei, Mohammadi, and Sahmani (2013) obtained the buckling response of a functionally graded microbeam based on a strain gradient theory. Farokhi and Ghayesh (2015) analysed the coupled dynamics and statics of a microbeam subject to a thermal loading; the mechanical properties of the microbeam were considered temperature-dependent. Medina, Gilat, and Krylov (2012) analysed the buckling response of an electrically actuated initially curved microbeams. Ghayesh, Farokhi, and Amabili (2013b) examined the pull-in characteristics and nonlinear response of an electrically actuated microbeam taking into account small-size effects.
This paper, for the first time, examines the complex nonlinear dynamics of a microbeam subject to a time-dependent longitudinal load for both the sub and supercritical regimes; this is accomplished by constructing the bifurcation diagrams of Poincaré sections when the amplitude of the axial load variations is varied as the bifurcation parameter. The microbeam is modelled via the modified couple stress theory. The potential and kinetic energies are constructed and inserted into Hamilton’s principle; this operation results in the continuous model of the system. A model reduction procedure is performed using an assumed-mode technique and the Galerkin procedure, which yields the reduced-order model of the system. This is solved via direct time integration through use of the variable step-size modified Rosenbrock method, yielding the motion amplitude as a function of time. The amplitude of the time-dependent component of the axial load is varied and the bifurcation diagrams of Poincaré sections are constructed; the analyses include different cases near the critical value for the constant component of the axial load, for both sub and supercritical regimes. For a selected system parameters, more detailed motion characteristics, including time traces, phase-plane portraits, fast Fourier transforms (FFTs), and Poincaré maps are displayed.
Section snippets
System energy and Hamilton’s principle
The system under consideration, shown in Fig. 1, consists of an Euler–Bernoulli microbeam of length L, Young’s modulus E, thickness h, cross-sectional area A, second moment of area I, mass density and length-scale parameter l. A planar Cartesian coordinate system with x and z as the longitudinal and transverse axes, respectively, is considered. The displacements in the x and z directions are denoted by u(x, t), and w(x, t), respectively; t is time. The microbeam is subject to a time-dependent
Model discretisation and solution technique
Eq. (15) forms a nonlinear continuous expression for the transverse motion of the system involving a time-dependent coefficient. In order to be able to solve this continuous model, a model reduction, based on the Galerkin scheme, is conducted by means of an assumed-mode method. To this end, the following series expansion is introducedwhere denotes the kth eigenfunction of a linear hinged-hinged beam (Ghayesh, 2011) and represents the kth generalised coordinate.
Bifurcation diagrams of Poincaré sections
In this section, the bifurcation diagrams of Poincaré sections are constructed with special consideration to chaotic motions and different routes to chaos. The mean value of the longitudinal load is set near the critical value, both in the sub and supercritical regimes, and the amplitude of the time-dependent component of the axial load is varied as the bifurcation parameter.
In general, there are three main routes to chaos, namely quasiperiodic, period-doubling, and direct transition; these
Conclusions
In this paper, the nonlinear sub and supercritical complex dynamics of a parametrically excited microbeam has been examined numerically. The continuous models for the kinetic and potential energies were developed taking into account the small-size effects via the modified couple stress theory. A model reduction was conducted through use of Galerkin’s discretisation method. The resultant model was solved by means of a direct time-integration method and the bifurcation diagrams of Poincaré
Acknowledgment
The financial support to this research by the start-up grant of the University of Wollongong is gratefully acknowledged.
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