Some nonlinear mechanical problems solved with analytical solutions

In this paper the analytical solution of nonlinear ordinary differential systems is addressed. Some of the problems are classical in the related literature and exhibit chaotic behavior in certain ranges of the involved parameters despite being simple-looking deterministic systems. The solutions are approached by means of the old technique of power series to solve ordinary differential equations. The independent variable is time in all the illustrations and elementary recurrence algorithms are obtained. This is an alternative to the standard numerical techniques and ensures the theoretical exactness of the response. Several examples are included and trajectories diagrams, phase plots, etc. are shown. The desired numerical precision is attained using time steps several times larger than the usual ones. The availability of an analytical solution may be an additional tool within a standard qualitative analysis. The solution of higher order problems and governed by partial differential equations is under study.Fil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Universidad Tecnológica Nacional; ArgentinaFil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Unidad de Direccion; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería. Área Estabilidad; ArgentinaFil: Buezas, Fernando Salvador. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Unidad de Direccion; Argentina. Universidad Nacional del Sur. Departamento de Física; Argentin

Coupled Free Vibrations Of Tapered Box-Beams Made Of Composite Materials

In this paper, analytical solutions are developed for the free vibration analysis of tapered thin-walled laminated-composite beams with closed cross-sections. The present approach is based in a recently developed model that incorporates in a full form the shear flexibility. The model considers shear flexibility due to bending as well as non-uniform torsional warping. The model is briefly reviewed with the aim to present the equilibrium equations and the related boundary conditions and constitutive equations. The lamination can be selected in order to manifest different types of elastic couplings. The typical laminations for a box-beam, like Circumferentially Uniform Stiffness and Circumferentially Asymmetric Stiffness stacking sequences, are analyzed. The exact values (i.e. with arbitrary precision) of frequencies are obtained by means of power series schemes. A parametric analysis is performed for different taper ratios, stacking sequences and materials.Fil: Piovan, Marcelo Tulio. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; ArgentinaFil: Filipich, Carlos Pedro. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; ArgentinaFil: Cortínez, Víctor Hugo. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentin

The power series method in the effectiveness factor calculations

In the present paper, exact analytical solutions are obtained for nonlinear ordinary differential equations which appear in complex diffusionreaction processes. A technique based on the power series method is used. Numerical results were computed for a number of cases which correspond to boundary value problems available in the literature. Additionally, new numerical results were generated for several important cases.Fil: Filipich, Carlos Pedro. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; ArgentinaFil: Villa Saravia, Luis Tadeo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Salta. Instituto de Investigaciones para la Industria Química. Universidad Nacional de Salta. Facultad de Ingeniería. Instituto de Investigaciones para la Industria Química; ArgentinaFil: Grossi, Ricardo Oscar. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Salta. Instituto de Investigaciones para la Industria Química. Universidad Nacional de Salta. Facultad de Ingeniería. Instituto de Investigaciones para la Industria Química; Argentin

Vibration of non-homogeneous rectangular membranes with arbitrary interfaces

The study deals with the generalized solution of the title problem. The free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled. Previous studies by other researchers include straight parallel to the borders and oblique interfaces, and bent ones. The solution is found by means of a direct variational method with a series composed with a complete set of functions. Two alternative sets are explored: trigonometric and power series. Such series are uniformly convergent to the exact solution. The approach is straightforward and very efficient from the computational viewpoint. A determinant-factorization method is employed to automatically eliminate eventual spurious frequencies. The well-known analogy between plates and membranes does not hold in this problem and a demonstration is included. Diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented. Also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included. Comparison between the two alternatives and with other authors' results show excellent agreement. In all cases the computational cost is very low.Fil: Filipich, Carlos Pedro. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería; ArgentinaFil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería; Argentin

Vibration of orthotropic plates: Discussion on the completeness of the solutions used in direct methods

The importance of the completeness of a set of solutions used in a direct variational method is herein discussed regarding a plate vibration application. The property is relevant to two issues: the claim of an “exact” solution and the avoidance of the loss of eigenfrequencies.Fil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería; ArgentinaFil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; Argentin

A Trigonometric Recurrence Algorithm for Solving Nonlinear Problems in Structural Dynamics

An analytical-numerical methodology for solving nonlinear problems governed by partial differential equations (PDE) is presented. The authors have previously used a method named WEM that consists essentially of the statement of extended trigonometric series of uniform convergence (UC). Theorems, demonstrated previously, ensure the UC of the essential functions and the exactness of the eigenvalues. WEM has been applied to nonlinear dynamic problems in two different ways: As a direct variational method and as a solution in the classical sense. The present tool is applied in the second fashion and starts from the statement of the extended trigonometric series for all the unknown functions and derivatives involved in the PDE. The nonlinearities are treated in the same way. The application of consistence conditions leads to recurrence relationships among the coefficients of the extended series. For the sake of comparison an initial conditions problem (the well-known Duffing oscillator) is numerically solved. Then a beam example is solved in detail: A linear (both material and geometrical) supported beam, with end springs of nonlinear analytical response, under the action of a dynamic distributed load and/or prescribed initial conditions, is studied. Two numerical examples are included.Fil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería; ArgentinaFil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Universidad Tecnológica Nacional. Facultad Regional Bahía Blanca; Argentin

Uniform convergence series to solve nonlinear partial differential equations: Application to beam dynamic

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problems governed by partial differential equations. In particular, the method is applied to the solution of a uniform beam supported at its ends with nonlinear rotational springs and subjected to dynamic loads. The beam is assumed to be both material and geometrically linear and the end springs are the Duffing type. The action may be a continuous load q = q(x) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, and the nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yields arbitrary precision results. An illustration example shows that efficiency of the method.Fil: Filipich, Carlos Pedro. Universidad Tecnológica Nacional; Argentina. Universidad Nacional del Sur; ArgentinaFil: Rosales, Marta Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional del Sur; Argentin

Arbitrary precision frequencies of a free rectangular thin plate

A variational method developed by the authors and named whole element method (WEM) is used to find the arbitrary precision frequencies of a rectangular thin plate (within the Germain}Lagrange theory) having its four borders free of constraint. WEM consists is proposing an adequate functional and a sequence representing the plate transversal displacement w(x, y). Such a sequence is made of a linear combination of functions belonging to a complete set in L2. The sequence, and not each co-ordinate function, is required to satisfy the essential or geometric conditions. The sequence generation is systematic and no analysis of the classical natural modes of the plate is needed. In particular, trigonometric functions which a priori belong to a complete set in the domain are used in the present analysis. The solving equations involving very simple sums arise from the minimization of the functional. WEM is based on theorems which show the ultimate exactness of the eigenvalues and the uniform convergence of the essential functions of the problem. To the authors knowledge this problem has no classical solution.Fil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; ArgentinaFil: Rosales, Marta Beatriz. Universidad Tecnológica Nacional; Argentina. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Full modeling of the mooring non-linearity in a two-dimensional floating structure

The dynamic behavior of a two-dimensional model of a small floating structure anchored by chains is analyzed. The structure is first modeled as a two-degrees-of-freedom oscillator with a strongly non-linear stiffness and subjected to a harmonic wave force. This type of structure is sometimes named Catenary Anchor Leg Mooring (CALM) system. The prescription of the vertical displacement leads to a simplified SDOF equation. An algebraic recurrence algorithm is employed to obtain a non-truncated differential equation that may be solved with the desired accuracy. Other authors have solved similar problems with approximate formulations of the geometric non-linearities. A numerical example is presented as an illustration. The time integration is carried out with a standard integration scheme and a power series approach. It is found that the response obtained after considering the strong non-linearity without previous truncations is qualitative different from the one found with a few terms of the expansions.Fil: Rosales, Marta Beatriz. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; ArgentinaFil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Universidad Tecnologica Nacional. Facultad Regional Bahía Blanca; Argentin

Time integration of non-linear dynamic equations by means of a direct variational method

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by taking n partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.Fil: Rosales, Marta Beatriz. Universidad Nacional del Sur. Departamento de Ingeniería; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Filipich, Carlos Pedro. Universidad Nacional del Sur. Departamento de Ingeniería; Argentin