Modelling of Ring-Shaped Compound Ultrasonic Waveguides by Means of Finite Elements Method

The paper describes a technique for modelling and optimization of ring-shaped compound ultrasonic waveguides consisting of two sequentially joined segments of different materials by means of finite elements method. The possibility of using such waveguides for amplifying vibrations in amplitude has been justified in the paper. The advantage of the developed technique consists in possibility of its realization by means of standard engineering software, particularly COMSOL Multiphysics. The correctness and efficiency of the technique is proved by comparing the numerical data with the simulation results by means of transfer matrix method using equations of vibration of Euler – Bernoulli and Timoshenko type. It is shown that in compound ring-shaped waveguides two kinds of vibration modes are possible – variable-sign and constant-sign, moreover only constant-sign modes are of practical interest for amplification of vibration amplitude. Recommendations for selection of optimal geometric parameters of the waveguides are given, particularly it is shown that for ensuring maximum vibration amplification factor it is necessary to choose central angles of the waveguide segments with account for calculated dependence between amplification factor and angle, characterized by presence of several local maxima of the amplification factor. It is noted that the high accuracy of the existing semi-analytical methods for calculating and designing ring-shaped waveguides is achieved using methods based on the application of Timoshenko-type equations of vibration.

MATHEMATICAL MODELLING OF VIBRATIONS OF NON-UNIFORM RING-SHAPED ULTRASONIC WAVEGUIDES

The article describes technique for modelling of ultrasonic vibrations amplifiers, which are implemented in the form of non-uniform ring-shaped waveguides, based on application of harmonic balance method. Bending vibrations of the waveguide are described by means of non-uniform integral and differential equations equivalent to Euler–Bernoulli equations in order to simplify calculation of amplitude-frequency characteristics of vibrations, particularly, to exclude the need of working with singular matrices. Using harmonic balance method, equations of vibrations are reduced to overdetermined non-uniform linear system of algebraic equations, which least-squares solution is determined by means of pseudo-inverse matrix. On the basis of analysis of numerical example possibility of existence of variable-sign and constant-sign vibration modes of the waveguide is shown and it is determined that for realization of amplifying function it is necessary to use waveguide at constant-sign vibration mode. The constant-sign vibration modes are combinations of bending defor-mation and extensional deformation of central line of the waveguide and they are detected due to accounting extensibility of the central line in equations of vibrations. Validity of the obtained results is confirmed by comparing them to the results of modelling by means of finite element method.

Periodic oscillations for a graphene-based electrostatic micro actuator

We study the mechanical oscillations for a novel model of a graphene-based electrostatic parallel plates micro actuator introduced by Wei et al.(2017), considering damping effects when a periodic voltage with alternating current is applied. Our analysis starts from recent results about this MEMS model with constant voltage, and provides new insights on the periodic mechanical responses for a variable input voltage. We derive sufficient conditions on the system physical components for which periodic oscillations with constant sign exist together with their stability properties. Specifically, under some conditions, the existence of three periodic solutions is established, one of them is negative and the others are positive in sign. The positive one nearby the origin is asymptotically locally stable, whilst the other two are unstable. Additionally, we prove that no further constant sign periodic solutions can be found. The existence of periodic solutions is approached from direct and reverse order Lower and Upper Solutions Method, and the stability assertions are derived from the Liapounoff-Zukovskii criteria for Hill's equations and the linearization principle. Theoretical results are complemented by numerical simulations and numerical continuation results. Furthermore, these numerical simulations evidence the robustness of the graphene-based MEMS model over the traditional ones.

Existence and multiplicity for Hamilton-Jacobi-Bellman equation

<p style='text-indent:20px;'>This paper is concerned with the existence and multiplicity of constant sign solutions for the following fully nonlinear equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} -\mathcal{M}_\mathcal{C}^{\pm}(D^2u) = \mu f(u) \ \ \ \ \text{in} \ \ \Omega,\\ u = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on}\ \partial\Omega, \end{array} \right. \end{equation*} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded regular domain with <inline-formula><tex-math id="M4">\begin{document}$ N\geq3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{M}_\mathcal{C}^{\pm} $\end{document}</tex-math></inline-formula> are general Hamilton-Jacobi-Bellman operators, <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> is a real parameter. By using bifurcation theory, we determine the range of parameter <inline-formula><tex-math id="M7">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the above problem which has one or multiple constant sign solutions according to the behaviors of <inline-formula><tex-math id="M8">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M9">\begin{document}$ 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \infty $\end{document}</tex-math></inline-formula>, and whether <inline-formula><tex-math id="M11">\begin{document}$ f $\end{document}</tex-math></inline-formula> satisfies the signum condition <inline-formula><tex-math id="M12">\begin{document}$ f(s)s&gt;0 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M13">\begin{document}$ s\neq0 $\end{document}</tex-math></inline-formula>.</p>

CONSTANT SIGN AND NODAL SOLUTIONS FOR NONLINEAR ROBIN EQUATIONS WITH LOCALLY DEFINED SOURCE TERM

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).