NONLINEAR DYNAMICS OF TOPOLOGICALLY OPTIMAL TIMOSHENKO NANOBEAM BASED ON THE COUPLE STRESS THEORY

Abstract

The relevance of research.Technological sensors (sensors of load, pressure, liquid temperature, solution yield, liquid density) are used for monitoring data of technical operations of well drilling, various types repair of well, exploration drilling in the oil and gas industry on drilling and repair of all types of wells. Modern sensors are small in size and manufactured on the basis of nanoelectromechanical systems to increase sensitivity. They include constituent elements, nano beams and nano plates. These elements operate at high temperatures and are subjected to mechanical loads of various kinds. Increasing the strength of these elements is undoubtedly an urgent task. The main aim of the research is to build a mathematical model on the basis of Timoshenko kinematic hypothesis and the modified couple stress theory of a sensing element in the form of a beam, a nano-electro-mechanical sensor under the action of mechanical and thermal fields;to develop methodology for obtaining the optimal topology of nano beams for arbitrary static and dynamic loads and different boundary conditions in order to increase its rigidity; to conduct a comparative analysis of the statics and nonlinear dynamics of optimal and nonoptimal beams. Objects: element of nanoelectromechanical systemsin the form of a beam taking into account the optimal microstructure. Methods: methods of topological optimization, variational methods, second order finite difference method, Runge–Kutta type methods, Fourier and wavelet analysis, phase portrait and Poincare section. Results. A methodology for obtaining the optimal microstructure of the nano beam based on topological optimization is developed. On the basis of the Hamilton–Ostrogradsky principle, a mathematical model of a Timoshenko nano beam inhomogeneous in two directions (in thickness and length) is constructed on the basis of a modified moment theory. A comparative analysis of static bending and nonlinear dynamics was performed for optimal and non-optimal nano beams.

Keywords

• timoshenko nanobeam
• nanoelectromechanical systems
• temperature and force fields
• modified couple stress theory of elasticity
• finite difference method
• runge–kuttatype methods
• topological optimization
• nonlinear dynamics