It is a diagonal matrix obtained by either row or column. Shape functions are given by 3nade bar element get more help from chegg. Solve eigenvalue problem for a number of eigenmodes. The consistent mass matrix can be obtained naturally through finite element. In previous papers1,2, the author studied the dynamic finite elements with diagonalized consistent mass matrix, but all of them are incompatible elements. There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. The mass contributions therefore interact and can not naturally be lumped to a single value at a point.
Structural dynamics department of civil and environmental engineering duke university henri p. The central difference method is based on finite difference. Dynamic analysis in finite element methods consistent. Solve eigenvalue problem for a number of eigenmodes 5. The program could be used to determine the consistent mass matrix, one. On semianalytical integration specified for mass matrix of finite. The finite element method roughly, the history of the finite element starts from the early 1900s, when it was used for elastic bars continua using discrete equivalent formulation 1.
Introduction to dynamics using finite elements analysis fea. However, the resulting consistent mass matrix is not in diagonalized form, which is inconvenient for numerical computation. To do dynamic and vibration finite element analysis, you need at least a mass matrix to pair with the stiffness matrix. The mass matrix formed in this way is called as lumped mass matrix. In the previous section, two integration formulas for a consistent mass matrix mij. So, to circumvent this difficulty, people use a lumped mass matrix. However, it is a general belief that consistent mass matrix. Modal testing has traditionally been used to confirm the validity of finite element models of structures. Consistent matrix formulations for structural analysis using finite.
The lumped mass representation is a heuristic discretization of mass. Consider the rectangular domain and the finite element mesh composed of five nodes and three elements as shown in the topleft part of the results the five piecewise linear basis functions associated to each of the nodes are shown in the results the mass matrix is defined as. The preceding section shows how this eigenvalue problem can be transformed into standard form. Find such that, where is the stiffness matrix and is the mass matrix. Vibration analysis using finite element method part 1dynamic analysis.
Dynamic analysis in finite element methods consistent mass matrix. Finite element discretization replace continuum formulation by a discrete representation for unknowns and geometry unknown. The finite element method fem is the most widely used method for solving problems of engineering and mathematical models. The advantage of having diagonal mass matrix in an eigenvalue analysis is the ease and reduction of computational effort as well as requirement of lesser storage space. Rao, in the finite element method in engineering sixth edition, 2018. What do you mean by consistent and lumped mass matrices. T v mnndv structural dynamics direct derivation of the bar element step 4 derive the element stiffness matrix and equations. Perform further analysis timehistory or response spectra. Lumped mass matrix in explicit finite element method for. When solving time dependent pdes using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. Derive the element stiffness matrix and equationsdefine the stiffness matrix for an element and then consider the derivation of the stiffness matrix for a linearelastic spring element. Their documentation should have some information, but it will be specific to their software and will probably not include specifics about the stiffness matrix. For engineering, a consistent set of imperial units is slug mass, foot length and seconds time. Microsoft powerpoint me 478 introduction to finite element analysis.
The consistent mass matrix is formed by using the actual mass distribution within the residual evaluations. The mass matrix is called the consistent mass matrix because it is derived using the same shape functions use to obtain the stiffness matrix. Consistent engineering units in finite element analysis. Lets derive the finite element equations for a timedependent. From this aspect, a diagonalized mass matrix has a computational advantage.
Consistent mass matrix an overview sciencedirect topics. How to find natural frequency of simply supported beam. An iterative procedure to calculate the inverse of the consistent and the mixed mass matrices in the framework of explicit finite element method is presented. In this paper, the compatible dynamic finite elements with diagonalized consistent mass matrix are studied. Also, we have used the same shape function for computing both mass and stiffness matrices. The lumped mass matrix for a beam element is given as follows for the beam element below. The components of the mass matrix are usually of the form. Chapter 2 introduction to the stiffness displacement. Nonsap a general finite element program for nonlinear dynamic analysis of. The lumpedmass matrix will be used in this analysis with 2. Effect of mass matrix formulation schemes on dynamics of.
Compatible dynamic finite element with diagonalized. The consistent mass matrix can be obtained naturally through finite element formulation, which is consistent to the assumed form functions. Note that the entries associated with the two rotational dofs. An alternative to the consistent mass matrix is the lumped mass matrix. Dynamic finite element with diagonalized consistent mass matrix. This video explains how to find natural frequency of simply supported beam using lumped mass matrix approach used in finite element analysis. The entire consistent mass matrix for a 2node beam element is listed in eq. Correct determination of natural frequencies is extremely. Eulerbernoulli beam finite element deriving the mass and stiffness matrices duration. Spatial numerical integration is essential for finite element analysis. Considering the subject of the question, as a general case, one can define lumped mass as concentrated mass in a much simpler way. Whereas, a consistent mass has its mass distributed over the entire span. On semianalytical integration specified for mass matrix.
Theory, implementation, and practice november 9, 2010 springer. Mass, stiffness, and damping matrix estimates from structural measurements. Consistent and lumped mass matrix approximations the major complication of the standard eigenvalue problem introduced by the finite element method is the mass matrix m. Consistent matrix formulations for structural analysis.
The consistent mass matrix is determined by using the shape. The convergence of the iterative procedure is proved. By closer inspection, it turns out that the row sum of the matrix elements in m is 1 one, people just lump together the matrix elements of m on the diagonal, which then becomes an identity matrix. A new strategy for the mass matrix lumping of enriched elements for explicit. From the results, it is evident that force based finite element formulation gives better results efficiently with a single element and the discretised displacement based model results converge to that of forcebased approach. In this paper, the compatible form functions are obtained not only for the tetrahedron elements, but also for the. Vibration analysis using finite element method maheshgadwantikar. Note that a pound is not a unit of mass for engineers or scientists and should never be used as a unit of mass. Dvco37 describe the terms lumped mass matrix and consistent mass matrix and identify which formulation is appropriate to elements being used. Discuss the integral equation for element mass, highlighting the variables which it is dependent upon. Dynamic finite element with diagonalized consistent mass. Modal analysis of a tapered timoshenko beam using force.
What is lumped mass modification of finite element method. The matrices are obtained as a linear combination of lumped and consistent mass matrices. Derive the consistent mass matrix for a uniformly spaced 3node bar element as shown, and check the conservation of mass note. Alternatively, using diagonal or lumped mass formulation not only reduces the computational cost but can also simplify the program coding. As time passed, the finite element method has progressed to become the most powerful and complex tool for engineering analysis. Consistent matrix formulations for structural analysis using finiteelement techniques. Lets derive the consistentmass matrix for a bar element. Lumped mass matrix an overview sciencedirect topics. Method of finite elements i modal analysis workflow of computer program 1. What is the difference between lumped mass and consistent. For consistent fe methods, you use galerkin orthogonality. Introduction to finite element analysis fea or finite. Reactor analysis and safety division, argonne national laboratory, argonne, illinois, u. As a special example, a consistent mass matrix of standard eight node brick element is.
Spyrakos put out a book called, finite element modeling in the mid90s, i think. Gavin fall 2018 1 preliminaries this document describes the formulation of sti. Mass matrix computation in the finite element method. For discrete approximations of continuum mechanics as in the finite element method, there may be more than one way to construct the mass matrix, depending on desired computational and accuracy performance. Consistent mass method for frames and finite elements free vibration characteristics of a 3d mixed formulation beam element with forcebased consistent mass matrix 9 december 2015 journal of vibration and control, vol. Science required the invention of a slug as a real mass unit in the imperial system. A general mass lumping scheme for the variants of the extended. It is important to perform comparative accuracy study based on coarse mesh. In general, the consistent mass matrix of an element in the global coordinate system is. The purpose of fea analytical solution stress analysis for trusses, beams, and other simple structures are carried out based on dramatic simplification and idealization. Nonlinear transient dynamic analysis by explicit finite. Assemble the element equations to obtain the global or total equations and introduce boundary conditionswe then show how the total stiffness matrix.
Perform further analysis time history or response spectra. The fem is a particular numerical method for solving. We distribute the complete mass of the element at the two nodes. Lumped mass matrix in explicit finite element method for transient. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. This is obtained by using the shape functions 2 for the elements and is given by 3 this involves off diagonal entries and also referred in the cae community as full or coupled mass matrix in fea softwares. To derive the consistent mass matrix for a 2node, 1d beam element of length l. Consistent diagonal mass matrices and finite element. Consistent diagonal mass matrices and finite element equations for one. Hence, these mass matrices are called as consistent matrices.
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