Abstract: In order to reveal the mechanism of water wheel chaotic rotation, the Mechanical Mechanism and energy conversion of water wheel chaotic rotation are studied by the method of moment analysis. The mathematical model of the Malkus water wheel rotation is transformed into Kolmogorov system. Based on the different coupling modes of inertia moment, internal moment, dissipation moment and external moment, the main influencing factors and internal mechanical mechanism of the Malkus water wheel chaotic rotation are analyzed and discussed by using the method of theoretical analysis and numerical simulation. The conversion among Hamiltonian energy, kinetic energy and potential energy is investigated. The relationship between the energies and the Rayleigh number is discussed. The main factors affecting chaotic rotation are internal energy, kinetic energy and Hamiltonian energy. Through analysis and simulation, it is found that the lack of torque mode can not make the system generate chaos, but the full torque mode can make the system produce chaos, at the same time, the system can produce chaos only when the dissipation and external force match, at this time the water wheel is in a chaotic rotating state. The Casimir function is introduced to analyze the system dynamics. The bound of chaotic attractor is obtained by the Casimir function. The Casimir function reflects the energy conversion and the distance between the orbit and the equilibria. These relationships are illustrated by numerical simulations.
Abstract: Functionally graded piezoelectric materials (FGPMs), combining the properties of functionally graded materials and piezoelectric materials, provides a new idea for multi-functional and intelligent lightweight components, and it has broad application prospects in electronic devices. Based on the elasticity and electric equilibrium equations, the Mian and Spencer functionally graded plate theory is extended from elastic materials to piezoelectric materials to study the cylindrical bending of FGPM plates, where the material elastic constants, piezoelectric and dielectric parameters are assumed to vary continuously and arbitrarily along thickness direction. Accordingly, the elasticity solutions for cylindrical bending of FGPMs plates under the uniform transverse loading are obtained. Numerical examples are given to demonstrate the influence of piezoelectric effects on the static responses of the presented FGPMs plates.