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EN
In this paper, the Ritz method is developed for the analysis of thin rectangular orthotropic plates undergoing large deflection. The trial functions approximating the plate lateral and in-plane displacements are represented by simple polynomials. The nonlinear algebraic equations resulting from the application of the concept of minimum potential energy of the orthotropic plate are cast in a matrix form. The developed matrix form equations are then implemented in a Mathematica code that allows for the automation of the solution for an arbitrary number of the trial polynomials. The developed code is tested through several numerical examples involving rectangular plates with different aspect ratios and boundary conditions. The results of all examples demonstrate the efficiency and accuracy of the proposed method.
2
EN
Vibration phenomena in mechanical structures including conical shells are usually undesirable. In order to overcome this problem, this study investigates active vibration control of isotropic truncated conical shells containing magnetostrictive actuators. The first-order shear deformation theory and the Hamilton principle are handled to obtain vibration equations. Moreover, a negative velocity feedback control law is used to actively suppress the vibration. The Ritz and modified Galerkin methods are utilized to obtain results of shell vibration. The results are validated by comparison with the results of literature and finite element software. Finally, the effects of control gain value, magnetostrictive layers thickness, isotropic layer thickness, length and semi-vertex angle of the conical shell on vibration suppression characteristics are obtained in details.
EN
An inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Ritz method with satisfier function. The Ritz method together with the least squares approximation (Ritz-least squares method) are utilized to reduce the inverse problem to the solution of algebraic equations. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.
PL
Praca przedstawia metodę aproksymacji algorytmu jakobianu dynamicznie zgodnego algorytmem typu jakobianu rozszerzonego. Rozwiązanie zadania aproksymacji sprowadza się do znalezienia optymalnej funkcji rozszerzającej minimalizującej błąd aproksymacji. W celu wyznaczenia optymalnego przybliżenia posłużymy się metodą bezpośrednią rachunku wariacyjnego, jaką jest metoda Ritza. Wywody teoretyczne poparte są badaniami symulacyjnymi robota o czterech stopniach swobody.
EN
The paper presents the approximation problem of the inverse kinematics algorithms for the redundant manipulators. We focus on the approximation of the dynamically consistent Jacobian by the extended Jacobian. For this aim we formulate the approximation problem with suitably defined approximation error. By the minimization of this error over a certain region we can design an extended Jacobian inverse which will be close to the dynamically consistent Jacobian. To solve the approximation problem we use the Ritz method. Theoretical considerations are supported by numerical simulations.
5
Surfaces Filling Polygonal Holes with G¹ Quasi G² Continuity
EN
Two constructions of surfaces filling polygonal holes in piecewise B-spline bicubic surfaces with tangent plane continuity are described. The filling surfaces are obtained by minimization of functionals which impose penalty on curvature discontinuities. One of the functionals is a quadratic form, while the other functional is defined with a parameterization-independent formula. The resulting surfaces may be used instead of class G2 surfaces in practical applications; the penalty approach enables simplification of the construction, and reduction in the degree of patches filling the hole from (9,9) to (5,5) without any visible quality degradation. The notion of class Gn quasi Gm surfaces, i.e. Class Gn surfaces optimized to approximate class Gm surfaces, is proposed.
6
Condition numbers and Ritz type methods in unconstrained optimization
EN
Condition numbers for infinite-dimensional optimization, as defined in Zolezzi (2002, 2003), are shown to exhibit a stable behavior when employing finite-dimensional solution methods of the Ritz type, in particular finite elements. The same behavior is shown to hold in connection with the so called extended Ritz method.
PL
Cienkościenna powłoka kulista jest na jednym brzegu podparta przegubowo, Drugi brzeg powłoki jest również podparty przegubowo, ale ma możliwość obrotu wokół osi powłoki. Do tego brzegu przyłożony jest moment obrotowy. Rozpatrywany jest problem utraty stateczności tej powłoki. Do jego rozwiązania wykorzystano metodę energetyczną. Przyjęto postacie funkcji sił i funkcji ugięcia po utracie stateczności, przy czym jeden ze współczynników funkcji sił wyznaczono z rozwiązania nieliniowego równania nierozdzielności; równanie to rozwiązano metodą Bubnowa-Galerkina. Drugi ze współczynników funkcji sił uzyskano z warunku brzegowego dla normalnej siły południkowej, Wyznaczono następnie zmianę energii całkowitej powłoki wywoływaną utratą stateczności. Do uzyskania równania algebraicznego, z którego wyznacza się obciążenie krytyczne, stosuje się metodę Ritza. Praca kończy się przykładem liczbowym, zaś rozwiązania problemu mają postać wykresów we współrzędnych bezwymiarowy parametr obciążenia - bezwymiarowa amplituda ugięcia powłoki. Wyniki porównano z wynikami wcześniejszego rozwiązania, w którym funkcja sił nie spełniała warunku brzegowego dla normalnej siły południkowej.
EN
A thin-walled spherical shell is pivotal at one edge, The other edge of the shell is also pivoted but retains the ability of rotation about the shell axis. This edge is loaded with a torque. A problem of stability loss of the shell is considered. The problem is solved with energetic method. The forms of force function and deflection function after stability loss arc assumed. One of the coefficients of force function was determined from solution of a nonlinear equation of indivisibility. The equation was solved with Mulmov-Cialcrkin's method. The second coefficient of the force function was obtained from boundary condition applied to normal meridional force. Then the change of total energy of the shell was determined that was caused by stability loss. Its components include the energy of membrane and bending forces, and the work of external forces. Total change in the energy depends on the coefficients of assumed deflection function and on the number defining the form of stability loss. In order to obtain the algebraic equation serving for determining of critical load the Kit/, method is used. The work ends with a numerical example. Solutions of the problem are depicted in the form of charts drawn up in the coordinate system oftlimensioiiless load parameter - dimensionless deflection amplitude of the shell. Results were compared to a previous solution in which the force function did not satisfy the boundary condition for normal meridional force.
8
Trefftz in translation
EN
This paper reviews the important concepts presented by Trefftz in 1926 regarding bounds to solutions, error estimation, and hybrid fields for use with domain decomposition. Observations are offered from the perspective of today's relatively mature state of the art in finite element methods. The numerical examples presented by Trefftz are also reviewed with the benefit of `exact' solutions made available from current commercial finite element methods. The accuracies of the solutions given by Trefftz are quantified and compared, and the effectivity indices of Trefftz's proposed error estimates are also quantified. An English translation of the original German version of Trefftz's paper is included for reference in an Appendix.
EN
The article presents the high-performance Ritz-gradient method for the finite element (FE) dynamic response analysis. It is based on the generation of the orthogonal system of the basis vectors. The gradient approach with two-level aggregation preconditioning on the base of element-by-element technique is applied to minimize the Rayleigh quotient for the preparation of each basis vector. It ensures the evolution of the regular basis vector toward the lowest eigenmode without aggregating and decomposing the large-scale stiffness matrix. Such method often happens to be more effective for dynamic response analysis, when compared to the classical modal superposition method, especially for seismic response analysis of the large-scale sparse eigenproblems. The proposed method allows one to apply arbitrary types of finite elements due to aggregation approach, and ensures fast problem solution without considerable exigencies concerning the disk storage space required, which is due to the use of EBE technique. This solver is implemented in commercial programs RobotV6 and Robot97 (software firm RoboBAT) for the seismic analysis of large-scale sparse problems and it is particularly effective when the consistent mass matrix is used.
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