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Higher order asymptotic homogenization for dynamical problems
(Wydawnictwo Politechniki Łódzkiej, 2021) Awrejcewicz, J.; Andrianov, I.I.; Diskovsky, A.A.
In general, asymptotic homogenization methods are based on the hypothesis of perfect scale separation. In practice, this is not always the case. The problem arises of improving the solution in such a way that it becomes applicable if inhomogeneity parameter is not small. Our study focuses on the higher order asymptotic homogenization for dynamical problems. Systems with continuous and piecewise continuous parameters, discrete systems, and also continuous systems with discrete elements are considered. Both low-frequency and highfrequency vibrations are analyzed. For low-frequency vibrations, several approximations of the asymptotic homogenization method are constructed. The influence of the boundary conditions, the system parameters is investigated.
Modelling and Analysing of a Spring Pendulum Motion in the Presence of Energy Harvesting Devices
(Wydawnictwo Politechniki Łódzkiej, 2021) Abohamer, M. K.; Awrejcewicz, J.; Starosta, R.; Amer, T.S.; Bek, M.A.
Energy harvesting will become more and more essential mechanical vibration applications of many devices. The vibrations can be converted by appropriate devices into electrical energy, which can be used as a power supply instead of ordinary ones. This paper investigates a dynamical system associated with two devices: a piezoelectric device and an electromagnetic one. These devices are connected with a nonlinear damping spring pendulum with 2DOF, in which its supported point moves in a circular path. The equations of motion are obtained using Lagrange’s equations of the second kind. The asymptotic solutions of these equations are obtained up to the third approximation utilizing the perturbation approach of multiple scales. The comparison between these solutions and the numerical ones reveals high consistency between them. The steady-state solutions are obtained, and their stabilities are tested. The influences of the excitation amplitudes, the damping coefficients, and the different frequencies on energy harvesting devices outputs are examined and discussed. The work is essential due to its significance in real-life applications. The developed methodology and obtained results can be useful in various applications like power supply of sensors and charging electronic devices.
Thermal waves in composite membrane with circular inclusions in hexagonal lattice structures
(Wydawnictwo Politechniki Łódzkiej, 2021) Andrianov, I.V.; Awrejcewicz, J.; Starushenko, G.A.; Kvitka, S.A.
Non-stationary heat conduction in the fibre composite materials is studied. 2D сomposite media consists of matrix with circular inclusions in hexagonal lattice structures. The perfect contact between different materials is assumed on the boundary of the fibres. The local temperature field in the unit cell is modeled by the heat equation. Time variable excludes from the original boundary value problem using of Laplace transform. Asymptotic homogenization method allows to reduce original problem for multiply-connected domain to the sequence of boundary value problems in simply-connected domains. Composite material is supposed densely-packed with a high-contrast. In this case, the local boundary value problem includes a small parameter equal to the ratio of the distance between the inclusions to the characteristic cell size. Using of additional small parameter and thin layer asymptotics allows to solve local problem analytically. Closed form expression for effective heat conductivity is obtained.