# What are the types of mass matrix?

## What are the types of mass matrix?

The construction of the mass matrix of individual elements can be carried out through several methods. These can be categorized into three groups: direct mass lumping, variational mass lumping, and template mass lumping.

What is element mass matrix?

In analytical mechanics, the mass matrix is a symmetric matrix M that expresses the connection between the time derivative of the generalized coordinate vector q of a system and the kinetic energy T of that system, by the equation.

### What is a lumped mass matrix?

Lumped Mass Matrix: It is a diagonal matrix obtained by either row or column sum lumping schemes commonly used in literature [3] . It presents a computational advantage especially in the problems of impact /crash analysis the procedure as involves then a mass matrix inversion.

Instead, the actual load distributions are converted into consistent nodal loads which are statically equivalent loads that does the same work as the actual load set over the finite element displacements.

## What is element stiffness matrix?

The stiffness matrix is the n-element square matrix A defined by. By defining the vector F with components. , the coefficients ui are determined by the linear system Au = F. The stiffness matrix is symmetric, i.e. Aij = Aji, so all its eigenvalues are real.

Are the element stiffness matrix and element flexibility matrix always non singular?

All element stiffness matrices are singular.

### What is elemental stiffness matrix?

What is meant by element stiffness matrix?

## Is element stiffness matrix singular?

All element stiffness matrices are singular. Element stiffness matrices can not be inverted. For element stiffness matrices, there is no unique solution to {q} = [k]{u}. For element stiffness matrices, there is at least one non-trivial (non-zero) vector {u} for which [k]{u} = {0}.

What is a consistent mass matrix?

The mass matrix resulting in this way is referred to as the consistent mass matrix. Lumped mass matrix According to this assumption, the mass of the element is concentrated at its nodes by static considerations, that is, they are obtained as the reactions of a simply supported beam under the load m(x) (see Fig. 11.2.3 ).

### What is the lumped mass matrix of the one dimensional element?

Then the consistent mass matrix can be written as According to the diagonalization method introduced in Chapter 4.6, the lumped mass matrix of the one-dimensional element is John T. Katsikadelis, in Dynamic Analysis of Structures, 2020

How to find the lumped mass matrix of a space truss element?

To find the lumped mass matrix of the element, as in the case of the bar element, the total element mass in each direction is distributed equally to the nodes of the element and the masses are associated with translational degrees of freedom in each of the X, Y, and Z directions. Thus, the lumped mass matrix of a space truss element is given by

## What is the mass formulation for Timchenko beam elements in Abaqus?

The quadratic Timoshenko beam elements in Abaqus/Standard use a consistent mass formulation, except in dynamic procedures in which a lumped mass formulation with a 1/6, 2/3, 1/6 distribution is used. For details, see Mass and inertia for Timoshenko beams .