A new approach for numerical modal analysis using the element-free method

Abstract

A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the "nite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh}Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving leastsquares interpolant is described. In Section 4, the mass and sti!ness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655}1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the "rst application of the element-free method to modal analysis. Figure 2. Frequency parameters for SS beam, with k"2, using an exponential function 6.1.1. One-dimensional tests. (a1) Optimization of parameters k, and : In order to perform this optimization, the problem of a beam simply supported at both ends, and for which an analytical solution is forthcoming, was examined. In a "rst analysis, parameter k was set to 2, which corresponds to a Gauss distribution, and we varied parameters and , solving the Galerkin equation (45) for each ( , ) pair. Figure 2 shows the solution surfaces for the "rst 16 NUMERICAL MODAL ANALYSIS 9 Figure 2. (Continued)frequencies: we observe that for all modes, there is a region of ( , ) for which the solutions tend towards a constant approximately equal to the analytical solution, and hence an in"nity of ( , ) values are acceptable. We chose for the value 1, and taking a vertical section of the surface at "1, Figure 3 results providing the evolution versus of the "rst 16 frequencies. Analysis of these curves shows that, from "3 onwards, a good solution is obtained. The lowest possible 10

A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids

“…In static analyses, strategies have been developed for alleviating the above problem, such as using the Lagrange multiplier method, the penalty method , and the direct interpolation method (Liu and Yan, 2000). In free vibration analyses using the MLPG method, orthogonal transform techniques (Atluri et al, 1999b;Ouatouati and Johnson, 1999) are utilized in order to eliminate the independent modes. For free vibration analysis, the essential boundary conditions are always homogeneous, therefore, we have u i 0 in Eq.…”

“…Some special techniques have to be used to overcome above-mentioned problems in using MLPG to static analyses. For example, the Lagrange multiplier method, the penalty method (Atluri and Zhu, 2000a), the orthogonal transformation technique (Atluri et al, 1999b;Ouatouati and Johnson, 1999), and the direct interpolation method (Liu and Yan, 2000) have been used to deal with essential boundary conditions. MLPG formulations for free vibration and forced vibration analyses of two-dimensional solids and structures are proposed in this paper to extend the MLPG method to dynamic analyses.…”

Modal-graph 3D shape servoing of deformable objects with raw point clouds

“…Afterward, without the need to identify the object’s physical properties, we assign the graph with arbitrary material parameters (including Young’s modulus E , Poisson’s ratio v , and the total mass M ). Using the finite element method (FEM) (Zienkiewicz et al (2005)) or an element-free modeling technique (such as Belytschko et al (1994); El Ouatouati and Johnson (1999)), we can compute the stiffness matrix K(boldn)∈double-struckR3N×3N and mass matrix M(boldn)∈double-struckR3N×3N of the graph. Then, we embed linear modal analysis in the graph.…”

Element-free Galerkin method for free vibration of rectangular plates with interior elastic point supports and elastically restrained edges

“…In the numerical applications, α 1 = α 2 = 1 × 10 5 , we adopt the quadratic basis function and choose the weight function w(r) as follows [21] :…”