Comparison of two linearization schemes for the nonlinear bending problem of a beam pinned at both ends

The nonlinear bending problem of a constant cross-section simply supported beam pinned at both ends and subject to a uniformly distributed load qðxÞ is analyzed in detail. The numerical integration of the two-point boundary value problem (BVP) derived for the nonlinear Timoshenko beam is tackled through two different linearization schemes, the multi-step transversal linearization (MTrL) and the multi-step tangential linearization (MTnL), proposed by Viswanath and Roy (2007). The fundamentals of these linearization techniques are to replace the nonlinear part of the governing ODEs through a set of conditionally linearized ODE systems at the nodal grid points along the neutral axis, ensuring the intersection between the solution manifolds (transversally in the MTrL and tangentially in the MTnL). In this paper, the solution values are determined at grid points by means of a centered finite differences method with multipoint linear constraints (Keller, 1969), and a simple iterative strategy. The analytical solution for this kind of bending problem, including the extensional effects, can be worked out by integration of the governing two-point BVP equations (Monleón et al., 2008). Finally, the comparison of analytical and numerical results shows the better ability of MTnL with the proposed iterative strategy to reproduce the theoretical behavior of the beam for each load step, because the restraint of equating derivatives in MTnL leads to further closeness between solution paths of the governing ODEs and the linearized ones, in comparison with MTrL. This result is opposed to the conclusion reached in Viswanath and Roy (2007), where the relative errors produced by MTrL are said to be smaller than the MTnL ones for the simply supported beam and the tip-loaded cantilever beam problems. 2009 Elsevier Ltd. All rights reserved.Merli Gisbert, R.; Lazaro, C.; Monleón Cremades, S.; Domingo Cabo, A. (2010). Comparison of two linearization schemes for the nonlinear bending problem of a beam pinned at both ends. International Journal of Solids and Structures. 47(6):865-874. doi:10.1016/j.ijsolstr.2009.12.001S86587447

Random Vibrations of Nonlinear Continua Endowed with Fractional Derivative Elements

In this paper, two techniques are proposed for determining the large displacement statistics of random exciting continua endowed with fractional derivative elements: Boundary Element Method (BEM) based Monte Carlo simulation; and Statistical Linearization (SL). The techniques are applied to the problem of nonlinear beam and plate random response determination in the case of colored random external load. The BEM is implemented in conjunction with a Newmark scheme for estimating the system response in the time domain in conjunction with repeated simulations, while SL is used for estimating efficiently and directly, albeit iteratively, the response statistics

Time domain prediction of first- and second-order wave forces on rigid and elastic floating bodies

The application and development of a transient three-dimensional numerical code ITU-WAVE which is based on panel method, potential theory and Neumann-Kelvin linearization is presented for the prediction of hydrodynamics characteristics of mono-hull and multi-hull floating bodies. The time histories of unsteady motions in ambient incident waves are directly presented with regards to impulse response functions (IRFs) in time. The first order steady forces of wave-resistance, sinkage force and trim moment are solved as the steady state limit of surge radiation IRFs. The numerical prediction of the second order mean force which can be computed from quadratic product of first-order quantities is presented using near-field method based on the direct pressure integration over floating body in time domain. The hydrodynamic and structural parts are fully coupled through modal analysis for the solution of hydroelastic problem in which Euler-Bernoulli beam is used for the structural analysis. A stiff structure is then studied assuming that contributions of rigid body modes are much bigger than elastic modes. A discrete control of latching is used to increase the bandwidth of the efficiency of Wave Energy Converters (WEC). ITU-WAVE numerical results for different floating

Linearization of dynamic equations of flexible mechanisms - a finite element approach

A finite element based method is presented for evaluation of linearized dynamic equations of flexible mechanisms about a nominal trajectory. The coefficient matrices of the linearized equations of motion are evaluated as explicit analytical expressions involving mixed sets of generalized co-ordinates of the mechanism with rigid links and deformation mode co-ordinates that characterize deformation of flexible link elements. This task is accomplished by employing the general framework of the geometric transfer function formalism. The proposed method is general in nature and can be applied to spatial mechanisms and manipulators having revolute and prismatic joints. The method also permits investigation of the dynamics of flexible rotors and spinning shafts. Application of the theory is illustrated through a detailed model development of a four-bar mechanism and the analysis of bending vibrations of two single link mechanisms in which the link is considered as a rotating flexible arm or as an unsymmetrical rotating shaft, respectively. The algorithm for the calculation of the matrix coefficients is directly emenable to numerical computation and has been incorporated into the linearization module of the computer program SPACAR

The linearized three-dimensional beam theory of naturally curved and twisted beams: the strain vectors formulation

This paper presents the equations of the linearized geometrically exact three-dimensional beam theory of naturally curved and twisted beams. A new finite-element formulation for the linearized theory is proposed in which the strain vectors are the only unknown functions. The linear form of the consistency condition that the equilibrium and the constitutive internal force and moment vectors are equal, is enforced to be satisfied at chosen points. An arbitrary curved and twisted axis of the beam is taken into account which demands proper consideration of the non-linearity of spatial rotations. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by comparing present numerical results with various analytical and numerical results. (c) 2005 Elsevier B.V. All rights reserved

Constrained finite rotations in dynamic of shells and Newmark implicit time-stepping schemes

Purpose – Aims to address the issues pertaining to dynamics of constrained finite rotations as a follow-up from previous considerations in statics. \ud \ud Design/methodology/approach – A conceptual approach is taken. \ud \ud Findings – In this work the corresponding version of the Newmark time-stepping schemes for the dynamics of smooth shells employing constrained finite rotations is developed. Different possibilities to choose the constrained rotation parameters are discussed, with the special attention given to the preferred choice of the incremental rotation vector. \ud \ud Originality/value – The pertinent details of consistent linearization, rotation updates and illustrative numerical simulations are supplied.\u

Kinematically exact curved and twisted strain-based beam

The paper presents a formulation of the geometrically exact three-dimensional beam theory where the shape functions of three-dimensional rotations are obtained from strains by the analytical solution of kinematic equations. In general it is very demanding to obtain rotations from known rotational strains. In the paper we limit our studies to the constant strain field along the element. The relation between the total three-dimensional rotations and the rotational strains is complicated even when a constant strain field is assumed. The analytical solution for the rotation matrix is for constant rotational strains expressed by the matrix exponential. Despite the analytical relationship between rotations and rotational strains, the governing equations of the beam are in general too demanding to be solved analytically. A finite-element strain-based formulation is presented in which numerical integration in governing equations and their variations is completely omitted and replaced by analytical integrals. Some interesting connections between quantities and non-linear expressions of the beam are revealed. These relations can also serve as useful guidelines in the development of new finite elements, especially in the choice of suitable shape functions