Frequency-Dependent Element Mass Matrices

1992 ◽  
Vol 59 (1) ◽  
pp. 136-139 ◽  
Author(s):  
N. J. Fergusson ◽  
W. D. Pilkey
Keyword(s):  
Stiffness Matrix ◽  
Shape Function ◽  
Mass Matrix ◽  
Power Series Expansion ◽  
Shape Functions ◽  
Frequency Dependent ◽  
Mass Matrices ◽  
The Matrix ◽  
Matrix Expansion ◽  
Consistent Mass Matrix

This paper considers some of the theoretical aspects of the formulation of frequency-dependent structural matrices. Two types of mass matrices are examined, the consistent mass matrix found by integrating frequency-dependent shape functions, and the mixed mass matrix found by integrating a frequency-dependent shape function against a static shape function. The coefficients in the power series expansion for the consistent mass matrix are found to be determinable from those in the expansion for the mixed mass matrix by multiplication by the appropriate constant. Both of the mass matrices are related in a similar manner to the coefficients in the frequency-dependent stiffness matrix expansion. A formulation is derived which allows one to calculate, using a shape function truncated at a given order, the mass matrix expansion truncated at twice that order. That is the terms for either of the two mass matrix expansions of order 2n are shown to be expressible using shape functions terms of order n. Finally, the terms in the matrix expansions are given by formulas which depend only on the values of the shape function terms at the boundary.

Finite Element Modeling of Dynamic Behavior of Some Basic Structural Members

1992 ◽  
Vol 114 (1) ◽  
pp. 3-9 ◽  
Author(s):  
R. C. Engels
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Modeling ◽  
Mass Matrix ◽  
Matrix Approach ◽  
Structural Members ◽  
Shape Functions ◽  
Generalized Coordinates ◽  
Stiffness Matrices ◽  
Consistent Mass Matrix

A method is described to model the dynamics of finite elements. The assumed modes method is used to show how static shape functions approximate the element mass distribution. The deterioration of the modal content of a model can be linked to the neglect of interface restrained assumed modes. Restoration of a few of these modes leads to higher accuracy with fewer generalized coordinates compared to the standard consistent mass matrix approach. Also, no need exists for subdivision of basic elements such as rods and beams. The mass and stiffness matrices for several basic elements are derived and used in demonstration problems.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

2018 ◽  
Vol 18 (10) ◽  
pp. 1850121 ◽  
Author(s):  
Junchao Wu ◽  
Dongdong Wang ◽  
Zeng Lin
Keyword(s):  
Vibration Analysis ◽  
Mass Matrix ◽  
Meshfree Method ◽  
Higher Order ◽  
Structural Vibration ◽  
Frequency Error ◽  
Shape Functions ◽  
Matrix Formulation ◽  
Order Accuracy ◽  
Mass Matrices

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

Conception of FEM Analysis Using Special Multi-Area Plane State of Stress Elements

2015 ◽  
Vol 797 ◽  
pp. 115-124 ◽  
Author(s):  
Monika Mackiewicz ◽  
Tadeusz Chyży ◽  
Sandra Matulewicz
Keyword(s):  
Explicit Form ◽  
Stiffness Matrix ◽  
Shape Function ◽  
Vertical Direction ◽  
Fem Analysis ◽  
Geometrical Parameters ◽  
Single Element ◽  
Shape Functions ◽  
Plane State ◽  
State Of Stress

The idea of the special finite elements conception has been presented in the paper. The conception is based on the assumption that area of the structure with different stiffness and geometrical parameters is described by a single element. The idea of the special so-called “multi-area” plane state of stress (shield) elements is the application of modified shape functions. This modification is the use of broken shape functions in both horizontal and vertical direction of plane state of stress finite element. It is proposed to determine explicit form of stiffness matrix for the elements. This increases the computational efficiency compared to numerical integration. To simplify the final form of stiffness matrix, method of broken shape function “expanding” has been applied. The method of simplification of stiffness matrix explicit form has been tested in calculation examples and results have been presented in the paper.

scholarly journals An Alternative Approach to Obtaining Stiffness and Mass Matrices for Three-Dimensional Bernoulli-Euler and Timoshenko Beam-Columns

2021 ◽  
Vol 10 (4) ◽  
pp. 253-268
Author(s):  
Ruhi Aydin
Keyword(s):  
Timoshenko Beam ◽  
Mass Matrix ◽  
Tensile Force ◽  
Three Dimensional ◽  
Beam Axis ◽  
Order Effects ◽  
Shape Functions ◽  
Stability Parameters ◽  
Beam Columns ◽  
Mass Matrices

In the static analysis of beam-column systems using matrix methods, polynomials are using as the shape functions. The transverse deflections along the beam axis, including the axial- flexural effects in the beam-column element, are not adequately described by polynomials. As an alternative method, the element stiffness matrix is modeling using stability parameters. The shape functions which are obtaining using the stability parameters are more compatible with the system’s behavior. A mass matrix used in the dynamic analysis is evaluated using the same shape functions as those used for derivations of the stiffness coefficients and is called a consistent mass matrix. In this study, the stiffness and consistent mass matrices for prismatic three-dimensional Bernoulli-Euler and Timoshenko beam-columns are proposed with consideration for the axial-flexural interactions and shear deformations associated with transverse deflections along the beam axis. The second-order effects, critical buckling loads, and eigenvalues are determined. According to the author’s knowledge, this study is the first report of the derivations of consistent mass matrices of Bernoulli-Euler and Timoshenko beam-columns under the effect of axially compressive or tensile force.

A Well-Conditioned Hierarchical Basis for Triangular H(curl)-Conforming Elements

2011 ◽  
Vol 9 (3) ◽  
pp. 780-806 ◽  
Author(s):  
Jianguo Xin ◽  
Wei Cai
Keyword(s):  
Orthogonal Polynomials ◽  
Stiffness Matrix ◽  
Jacobi Polynomials ◽  
Mass Matrix ◽  
Basis Functions ◽  
Shape Functions ◽  
Reference Element ◽  
Hierarchical Basis ◽  
Normal Functions ◽  
Bubble Functions

AbstractWe construct a well-conditioned hierarchical basis for triangular H(curl)-conforming elements with selected orthogonality. The basis functions are grouped into edge and interior functions, and the later is further grouped into normal and bubble functions. In our construction, the trace of the edge shape functions are orthonormal on the associated edge. The interior normal functions, which are perpendicular to an edge, and the bubble functions are both orthonormal among themselves over the reference element. The construction is made possible with classic orthogonal polynomials, viz., Legendre and Jacobi polynomials. For both the mass matrix and the quasi-stiffness matrix, better conditioning of the new basis is shown by a comparison with the basis previously proposed by Ainsworth and Coyle [Comput. Methods. Appl. Mech. Engrg., 190 (2001), 6709-6733].

Vibration of plate and shell structures using triangular finite elements

1967 ◽  
Vol 2 (1) ◽  
pp. 73-83 ◽  
Author(s):  
R Dungar ◽  
R T Severn ◽  
P R Taylor
Keyword(s):  
Stiffness Matrix ◽  
Numerical Solutions ◽  
Mass Matrix ◽  
Arch Dam ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Experimental Frequency ◽  
Single Plane ◽  
Stiffness Matrices ◽  
Mass Matrices

Mass and stiffness matrices are obtained for a general-triangle element and for a right-angled-triangle element. Both bending and in-plane actions are considered, although no coupling is assumed, and the matrices relating to bending actions are obtained independently of those relating to in-plane actions. Coupling is introduced, unless all the elements lie in a single plane, when the transformation from local to global co-ordinates is made. In deriving the stiffness matrices assumptions have been made about the form of the stress components within, and on the boundaries of, the element, together with assumptions about the form of the displacement components on the boundary of the element only. The commonly made assumptions in the derivation of stiffness matrices relate to the form of the displacement components not only on the boundary but throughout the element. In order to derive satisfactory mass matrices it is necessary to assume the form of the displacement components throughout the element. For the right-angled-triangle mass matrices these displacement components have been assumed independently of the assumed boundary displacements needed for the stiffness matrix. For the general-triangle mass matrix, however, the displacements throughout the element have been made consistent with the boundary displacements which were needed for the stiffness matrix. Numerical results are given for the first few natural frequencies of a square simply supported slab and a square encastré slab. Comparison with accepted values shows that the finite-element values are accurate, and convergent as the element size is reduced. For the same number of elements it is indicated that general triangles give a more accurate solution than right-angled triangles, probably because of the more satisfactory derivation of the mass matrix for the general triangle. This advantage is offset, however, by the greater computation time required by general triangles. Calculated and experimental frequency values are also given for a single-curvature arch dam of constant thickness. Mode shapes are not given in any of the numerical solutions although they are produced as an integral part of the computer programme.

scholarly journals Finite Dynamic Elements and Modal Analysis

1993 ◽  
Vol 1 (2) ◽  
pp. 171-176
Author(s):  
N.J. Fergusson ◽  
W.D. Pilkey
Keyword(s):  
Modal Analysis ◽  
Stiffness Matrix ◽  
Correction Term ◽  
Dynamic Stiffness ◽  
Mass Matrix ◽  
Power Series Expansion ◽  
Forced Response ◽  
Dynamic Correction ◽  
Analysis Scheme ◽  
Dynamic Element

A general modal analysis scheme is derived for forced response that makes use of high accuracy modes computed by the dynamic element method. The new procedure differs from the usual modal analysis in that the modes are obtained from a power series expansion for the dynamic stiffness matrix that includes an extra dynamic correction term in addition to the static stiffness matrix and the consistent mass matrix based on static displacement. A cantilevered beam example is used to demonstrate the relative accuracies of the dynamic element and the traditional finite element methods.

AN EFFICIENT SMOOTHED POINT INTERPOLATION METHOD FOR DYNAMIC ANALYSES

2013 ◽  
Vol 10 (01) ◽  
pp. 1340007 ◽  
Author(s):  
DELFIM SOARES ◽  
ANNE SCHÖNEWALD ◽  
OTTO VON ESTORFF
Keyword(s):  
Stiffness Matrix ◽  
Shape Function ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Dynamic Analyses ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Support Domain ◽  
Edge Based

In this work, a new procedure to compute the mass matrix in the smoothed point interpolation method is discussed. Therefore, the smoothed subdomains are employed to evaluate the mass matrix, which have already been computed for the construction of the stiffness matrix, rendering a more efficient methodology. The procedure is discussed, taking into account the edge-based, cell-based, and node-based smoothed point interpolation methods, as well as different T-schemes for the construction of the support domain of the approximating shape function, which is here formulated based on the radial point interpolation method. Numerical results of different dynamic analyses are presented, illustrating the potentialities of the proposed methodology.

scholarly journals Modeling Structural Dynamics Using FE-Meshfree QUAD4 Element with Radial-Polynomial Basis Functions

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2288
Author(s):  
Hongming Luo ◽  
Guanhua Sun
Keyword(s):  
Structural Dynamics ◽  
Interpolation Method ◽  
Mass Matrix ◽  
Time Integration ◽  
Partition Of Unity ◽  
Implicit Time ◽  
Lumped Mass ◽  
Point Interpolation Method ◽  
Point Interpolation ◽  
Consistent Mass Matrix

The PU (partition-of-unity) based FE-RPIM QUAD4 (4-node quadrilateral) element was proposed for statics problems. In this element, hybrid shape functions are constructed through multiplying QUAD4 shape function with radial point interpolation method (RPIM). In the present work, the FE-RPIM QUAD4 element is further applied for structural dynamics. Numerical examples regarding to free and forced vibration analyses are presented. The numerical results show that: (1) If CMM (consistent mass matrix) is employed, the FE-RPIM QUAD4 element has better performance than QUAD4 element under both regular and distorted meshes; (2) The DLMM (diagonally lumped mass matrix) can supersede the CMM in the context of the FE-RPIM QUAD4 element even for the scheme of implicit time integration.

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