Unilateral Problem for a Viscoelastic Beam Equation Type p-Laplacian with Strong Damping and Logarithmic Source

In this manuscript, we investigate the unilateral problem for a viscoelastic beam equation of p-Laplacian type. The competition of the strong damping versus the logarithmic source term is considered. We use the potential well theory. Taking into account the initial data is in the stability set created by the Nehari surface, we prove the existence and uniqueness of global solutions by using the penalization method and Faedo-Galerkin’s approximation.

STABILIZATION OF THE GENERALIZED RAO-NAKRA BEAM BY PARTIAL VISCOUS DAMPING

In this paper, we consider the stabilization of the generalized Rao-Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group two to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group one are left for future study.

On the use of nonlinear impact oscillators in vibrating electromagnetic based energy harvesters

This work investigates the use of impact-oscillators in nonlinear electro-magnetic cantilever based broad bandwidth frequency energy harvester. The electro-mechanical model includes the dynamical equations of the nonlinear impact-oscillators, the nonlinear frequency resonant cantilever beam equation of the harvester, and the Lorenz magnetic force resulting from the surrounding nonlinear magnetic field. The simulated results of the harvested power show a band of resonant frequencies of the vibrating cantilever beam extended beyond its fundamental natural frequency. Due to the strong nonlinearity of both the impact oscillators’ forces and the magnetic flux, the harvested power of the enhanced system shows wider bands in its respective frequency response. Moreover, the harvested power of the cantilever beam shows that considering two magnets in repulsion generates higher values mainly due to the strong nonlinear transient magnetic flux, resulting into few times more power as compared to the case of considering only one magnet in the system.

Controlling Resonator Nonlinearities and Modes through Geometry Optimization

Controlling the nonlinearities of MEMS resonators is critical for their successful implementation in a wide range of sensing, signal conditioning, and filtering applications. Here, we utilize a passive technique based on geometry optimization to control the nonlinearities and the dynamical response of MEMS resonators. Also, we explored active technique i.e., tuning the axial stress of the resonator. To achieve this, we propose a new hybrid shape combining a straight and initially curved microbeam. The Galerkin method is employed to solve the beam equation and study the effect of the different design parameters on the ratios of the frequencies and the nonlinearities of the structure. We show by adequately selecting the parameters of the structure; we can realize systems with strong quadratic or cubic effective nonlinearities. Also, we investigate the resonator shape effect on symmetry breaking and study different linear coupling phenomena: crossing, veering, and mode hybridization. We demonstrate the possibility of tuning the frequencies of the different modes of vibrations to achieve commensurate ratios necessary for activating internal resonance. The proposed method is simple in principle, easy to fabricate, and offers a wide range of controllability on the sensor nonlinearities and response.

A Cantilever Beam Problem with Small Deflections and Perturbed Boundary Data

The boundary value problem of a fourth-order beam equation u 4 = λ f x , u , u ′ , u ″ , u ′ ′ ′ , 0 ≤ x ≤ 1 is investigated. We formulate a nonclassical cantilever beam problem with perturbed ends. By determining appropriate values of λ and estimates for perturbation measurements on the boundary data, we establish an existence theorem for the problem under integral boundary conditions u 0 = u ′ 0 = ∫ 0 1 p x u x d x , u ″ 1 = u ′ ′ ′ 1 = ∫ 0 1 q x u ″ x d x , where p , q ∈ L 1 0 , 1 , and f is continuous on 0 , 1 × 0 , ∞ × 0 , ∞ × − ∞ , 0 × − ∞ , 0 .

Numerical Solution of Fourth-Order Boundary Value Problems for Euler-Bernoulli Beam Equation using FDM

Euler-Bernoulli beam equation is widely used in engineering, especially civil and mechanical engineering to determine the deflection or strength of bending beam. In physical science and engineering, to predict the deflection for beam problem, bending moment, soil settlement and modeling of viscoelastic flows, fourth-order ordinary differential equation (ODE) is widely used. The analytical solution of most of the higher order ordinary differential equations with complicated boundary condition that occur in any engineering problems is not easy way. Therefore, numerical technique based on finite difference method (FDM) is comparatively easy and important for solving the boundary value problems (BVP). In this study four boundary conditions (Neumann condition) are considered for solving BVP. Absolute error calculation, numerical stability and convergence are discussed. Two examples are considered to illustrate the finite difference method for solving fourth order BVP. The numerical results are rapidly converged with exact results. The results shows that the FDM is appropriate and reliable for such type of problems. Thus present study will enhance the mathematical understanding of engineering students along with an application in different field.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.