Vibration Analysis for Piezoceramic Circular Plates with V-Notches. Part 1: Theory

In this paper the transverse vibration characteristics of piezoceramic circular plates with V-notches are investigated theoretically through use of the Ritz's method incorporated with the defined equivalent constants. The Ritz's method is employed with two sets of admissible displacement functions, algebraic-trigonometric polynomials and corner functions, to guarantee convergence sufficiently and represent the stress singularity, respectively. Moreover, the equivalent constants derived by comparing the characteristic equations of transverse vibration between isotropic and piezoceramic disks are applied to suspend the electrical field consideration regarding the piezoelectricity. With the aid of theoretical analysis, the non-dimensional frequency parameters of transverse vibration modes for completely free V-notching circular plates are exhibited; in addition, the frequency variations depending on various notch angles and depths are explored. Numerical calculations using the finite element method (FEM) are performed and the results are compared with the theoretical analysis. It is shown that the resonant frequencies predicted by theoretical analysis and calculated by FEM are in good agreement.

The Brachistochrone With a Movable End-Point and the Nonsimultaneous Variations

In this paper, the necessary and sufficient conditions for minima plane path with a movable end-point are developed. Using the calculus of variations the considered conditions are based on the zero first-order nonsimultaneous variation and on the positive second-order variation in the functional of integral type corresponding to mechanical systems. The applied procedure is the coordinate parametric method. The obtained solutions are tested on a brachistochrone with one end-point constrained to lie on a circle. The exact solution is compared with the approximate one obtained with Ritz’s method.

Variational solution of the steady magnetic field distribution in ferromagnetic conductor

A distribution of the steady magnetic filed in a very large ferromagnetic conductor of arbitrary, but known cross-section is determined using variational approach with Ritz's method for approximate solving of the problem. Then an axial component of the magnetic vector potential is presented as finite functional series, automatically satisfying boundary condition that the normal component of the magnetic induction on the conductor's surface vanishes. The convergence and the accuracy of the proposed method are investigated. Several results for maps of the force lines and of the inductivity per unit conductor length are compared with corresponding numerical results obtained by Finite Element Method (program package FEMM) and excellent agreement s observed.

Multiple objective optimization of hydro-thermal systems using Ritz's method

This paper examines the applicability of the Ritz method to multi-objective optimization of hydro-thermal systems. The algorithm proposed is aimed to minimize an objective functional that incorporates the cost of energy losses, the conventional fuel cost and the production of atmospheric emissions such asNOxandSO2caused by the operation of fossil-fueled thermal generation. The formulation includes a general layout of hydro-plants that may form multi-chains of reservoir network.Time-delays are included and the electric network is considered by using the active power balance equation. The volume of water discharge for each hydro-plant is a given constant amount from the optimization interval. The generic minimization algorithm, which is not difficult to construct on the basis of the Ritz method, has certain advantages in comparison with the conventional methods.