A Perturbation Approach for Lateral Excited Vibrations of a Beam-like Viscoelastic Microstructure Using the Nonlocal Theory

In this paper, we investigate the lateral vibration of fully clamped beam-like microstructures subjected to an external transverse harmonic excitation. Eringen’s nonlocal theory is applied, and the viscoelasticity of materials is considered. Hence, the small-scale effect and viscoelastic properties are adopted in the higher-order mathematical model. The classical stress and classical bending moments in mechanics of materials are unavailable when modeling a microstructure, and, accordingly, they are substituted for the corresponding effective nonlocal quantities proposed in the nonlocal stress theory. Owing to an axial elongation, the nonlinear partial differential equation that governs the lateral motion of beam-like viscoelastic microstructures is derived using a geometric, kinematical, and dynamic analysis. In the next step, the ordinary differential equations are obtained, and the time-dependent lateral displacement is determined via a perturbation method. The effects of external excitation amplitude on excited vibration are presented, and the relations between the nonlocal parameter, viscoelastic damping, detuning parameter, and the forced amplitude are discussed. Some dynamic phenomena in the excited vibration are revealed, and these have reference significance to the dynamic design and optimization of beam-like viscoelastic microstructures.

Robust Stability of Discrete-time Singularly Perturbed Systems with Nonlinear Perturbation

This paper is concerned with the robust stability and stabilization problems of discrete-time singularly perturbed systems (DTSPSs) with nonlinear perturbations. A proper sufficient condition via the fixed-point principle is proposed to guarantee that the given system is in a standard form. Then, based on the singular perturbation approach, a linear matrix inequality (LMI) based sufficient condition is presented such that the original system is standard and input-to-state stable (ISS) simultaneously. Thus, it can be easily verified for it only depends on the solution of an LMI. After that, for the case where the nominal system is unstable, the problem of designing a control law to make the resulting closed-loop system ISS is addressed. To achieve this, a sufficient condition is proposed via LMI techniques for the purpose of implementation. The criteria presented in this paper are independent of the small parameter and the stability bound can be derived effectively by solving an optimal problem. Finally, the effectiveness of the obtained theoretical results is illustrated by two numerical examples.

Oriented manifold-based error tracking control for nonholonomic wheeled autonomous vehicles on Lie group for curvilinear approach

This paper considers the trajectory tracking control of wheeled autonomous vehicles (WAV) with slipping in the wheels, i.e., when the kinematic constraints are not satisfied. Usually, the coordinates system used to represent all control problems suggest invariant subspaces mutually orthogonal, but this approach can not be enough to treat curvatures significative large at different navigation speed. In order to get a slight im- provement on this topic, there are previous works showing that the kinematic problem (commonly associated with an outer loop) can be resynthesized by using other invariant subspaces, i.e., another representation of the configuration space. For this reason, the proposal reported here uses an oriented-manifold parametrized by a coordinate system on a curve viewpoint of the trajectory to describe the kinematic problem, however, the dynamic control law remains faithful to the singular perturbation approach with invariant subspaces mutually orthogonal, thus, it is possible to include the flexibility through a small factor in the dynamic model (well-known as ε), responsible to avoid the good-performance of the kinematic constraints. Only a common curvature-transformation between orthogonal and curve coordinates will be used to couple both approaches. Finally, it will be observed that when the controller is applied to the control scheme the behavior of the tracking is meaningfully improved.

Homogenization of surface energy and elasticity for highly rough surfaces

Surface energy plays a central role in several phenomena pertaining to nearly all aspects of materials science. This includes phenomena such as self-assembly, catalysis, fracture, void growth, and microstructural evolution among others. In particular, due to the large surface-to-volume ratio, the impact of surface energy on the physical response of nanostructures is nothing short of dramatic. How does the roughness of a surface renormalize the surface energy and associated quantities such as surface stress and surface elasticity? In this work, we attempt to address this question by using a multi-scale asymptotic homogenization approach. In particular, the novelty of our work is that we consider highly rough surfaces, reminiscent of experimental observations, as opposed to gentle roughness that is often treated by using a perturbation approach. We find that softening of a rough surface is significantly underestimated by conventional approaches. In addition, our approach naturally permits the consideration of bending resistance of a surface, consistent with the Steigmann-Ogden theory, in sharp contrast to the surfaces in the Gurtin-Murdoch surface elasticity theory that do not offer flexural resistance.

Theoretical modeling on the combination resonance of size-dependent microbeams

The combination resonance of size-dependent microbeams is investigated. Two harmonic forces act on the microbeam, and combination resonance is observed while the excitation frequencies differ from the resonant frequency. Microbeams with two different sources of nonlinearities including three kinds of boundary conditions, clamped-free (nonlinearity comes from large curvature and nonlinear inertial), clamped-clamped, and hinged-hinged (nonlinearity originates from mid-plane stretching-bending coupling), are taken into consideration to have a deep understanding of this phenomenon. A traveling load acting on the microbeam is presented as a special case of combination resonance. The modal discretization technique is applied to discretize the equations of motion, and then the Lindstedt–Poincare method, a perturbation approach, is employed to solve the resultant equations. The conditions for combination resonance are presented, and frequency-response curves and time histories at the resonance point are obtained for microbeams of each boundary condition. Results reveal that different sources of nonlinearities result in different performances of combination resonance. The free vibration part constitutes a large percentage of the final response. Furthermore, the situation of coexistence of combination resonance and superharmonic (or subharmonic) resonance is determined. The special case demonstrates a higher amplitude than the common combination resonance for all the boundary conditions. Parametric studies are then carried out to discuss the effects of the length scale parameter, excitation force as well as its position, and damping on the performance of the microbeam.

Supersolid phase of a spin-orbit-coupled Bose-Einstein condensate: A perturbation approach

The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit coupling includes a stripe phase with supersolid features. In this work we develop a perturbation approach to study the ground state and the Bogoliubov modes of this phase, holding for small values of the Raman coupling. We obtain analytical predictions for the most relevant observables (including the periodicity of stripes, sound velocities, compressibility, and magnetic susceptibility) which are in excellent agreement with the exact (non perturbative) numerical results, obtained for significantly large values of the coupling. We further unveil the nature of the two gapless Bogoliubov modes in the long-wavelength limit. We find that the spin branch of the spectrum, corresponding in this limit to the dynamics of the relative phase between the two spin components, describes a translation of the fringes of the equilibrium density profile, thereby providing the crystal Goldstone mode typical of a supersolid configuration. Finally, using sum-rule arguments, we show that the superfluid density can be experimentally accessed by measuring the ratio of the sound velocities parallel and perpendicular to the direction of the spin-orbit coupling.