Chaotic Dynamics of Piezoelectric MEMS Based on Maximum Lyapunov Exponent and Smaller Alignment Index Computations

We characterize the dynamical states of a piezoelectric micrcoelectromechanical system (MEMS) using several numerical quantifiers including the maximum Lyapunov exponent, the Poincaré Surface of Section and a chaos detection method called the Smaller Alignment Index (SALI). The analysis makes use of the MEMS Hamiltonian. We start our study by considering the case of a conservative piezoelectric MEMS model and describe the behavior of some representative phase space orbits of the system. We show that the dynamics of the piezoelectric MEMS becomes considerably more complex as the natural frequency of the system’s mechanical part decreases. This refers to the reduction of the stiffness of the piezoelectric transducer. Then, taking into account the effects of damping and time-dependent forces on the piezoelectric MEMS, we derive the corresponding nonautonomous Hamiltonian and investigate its dynamical behavior. We find that the nonconservative system exhibits a rich dynamics, which is strongly influenced by the values of the parameters that govern the piezoelectric MEMS energy gain and loss. Our results provide further evidences of the ability of the SALI to efficiently characterize the chaoticity of dynamical systems.

Saddle-Node Bifurcation and Homoclinic Persistence in AFMs with Periodic Forcing

We study the dynamics of an atomic force microscope (AFM) model, under the Lennard-Jones force with nonlinear damping and harmonic forcing. We establish the bifurcation diagrams for equilibria in a conservative system. Particularly, we present conditions that guarantee the local existence of saddle-node bifurcations. By using the Melnikov method, the region in the space parameters where the homoclinic orbits persist is determined in a nonconservative system.

Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic systems in nonconservative form—the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

STABILITY OF DAMPED COLUMNS ON A WINKLER FOUNDATION UNDER SUB-TANGENTIAL FOLLOWER FORCES

This study examines the dynamic stability regions of damped columns on a Winkler foundation that are subjected to sub-tangentially distributed follower forces. A nondimensionalized equation of motion for the column subjected to linearly distributed follower forces is firstly derived based on the extended Hamilton's principle. A finite element procedure, using Hermitian interpolation functions, is employed to develop the mass matrix, Rayleigh damping matrix, Winkler foundation matrix, elastic and geometric stiffness matrices due to distributed axial forces, and a load correction stiffness matrix to account for sub-tangential follower forces. Subsequently, a time history analysis using the Newmark-β method and an evaluation method for the flutter and divergence loads of the nonconservative system are presented. Finally, the dynamic stability characteristics of the nonconservative system that display the jumping phenomenon in the second flutter load are explored through a parametric study. In particular, how the stable and unstable regions of the undamped and damped Leipholz columns translate with changes in the Winkler foundation stiffness is demonstrated and discussed.

Dynamic Stability of an Elastic Beam Subjected to Follower Forces

The stability of nonconservative system of a beam is investigated when an elastic beam is subjected to follower forces. The mathematical formulations for a conservative system and a nonconservative system are established regarding to a uniform cantilever subjected to a concentrated force and a uniform distributed force axially. The displacement of a uniform cantilever is assumed to be obtained by superposing the modal functions which are normal modes in a vacuum, and is estimated by applying the Galerkin’s method. Changing the forces, the eigenvalue analysis is performed, and the root locus is calculated for the stability analysis. And, the relationship between forces and frequencies for the undamped system and the damped system of the uniform cantilever subjected to a concentrated force and a uniform distributed force is investigated. When the system is considered to be conservative, the divergence phenomenon is confirmed to appear first. On the other hand, when the system is considered to be nonconservative, the flutter phenomenon is confirmed to appear first although the critical force becomes high. And, by changing the structural damping, the destabilized effect due to the damping is confirmed when an elastic beam is subjected to follower forces.