Table of Contents

## Is mass matrix always diagonal?

The technique may vary and one must be careful when using high order elements (i.e. order greater than one). Lumped mass technique is more general and it is not necessarily diagonal. Depending on formulation, considered DOFs, etc it could be diagonal or not.

## Why mass matrix is diagonal?

Each mass matrix has along its block diagonal the scalar mass, the inertia relative to the fuselage center of mass, and the inertia of the linkage. Off-diagonal terms represent inertial couplings within the system. The total mass matrix of the system is a summation of the mass matrices of each rigid body.

**What are the two types of mass matrix?**

Lumped and.

**What is the difference between lumped mass matrix and consistent mass matrix?**

The consistent mass matrix is fully populated whereas lumped mass matrix is diagonal. Stiffness matrix: The stiffness matrix [k] is symmetric. The total stiffness matrix will be formed by assembling all the elements together to form a structure.

### Is mass matrix symmetric?

The mass matrix is also a symmetric matrix.

### Is stiffness matrix always symmetric?

It is not always symmetric; in fact for a number of problems it is decidedly unsymmetric. To understand the “what and why” of this, it is important to understand physically just what the stiffness matrix is.

**What is mass matrix in FEM?**

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 lumped mass matrix in FEM?**

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.

#### Is mass matrix positive definite?

The mass matrix is always positive definite, but the non diagonal elements can be negative. The mass matrix is also a symmetric matrix. The diagonal elements are always positive, All others can be also negative.

#### What is stiffness matrix in finite element?

In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

**What is difference between stiffness and flexibility matrix method?**

When comparing the flexibility and stiffness methods, it is seen that the flexibility method requires the solution of equations of compatibility for unknown forces whereas the stiffness method requires the solution of equations of equilibrium for unknown displacements.

**What is the lumped mass matrix?**

## What is lumped mass?

A lumped mass is a load that can be applied to a node in a static stress, natural frequency (modal) or modal superposition analysis. A lumped mass can be used to resist the translation or rotation of a node.

## What is Jacobian matrix in FEM?

In the Finite Element Method, an element’s Jacobian Matrix relates the quantities wrote in the natural coordinate space and the real space. The bigger the element is distorted in comparison with a ideal shape element, the worse will be the transformation of the quantities from the natural space to the real space.

**What are flexibility and stiffness matrices?**

Since the stiffness matrix (or the flexibility matrix) relates forces (or moments) applied anywhere on a linear structure to the displacements produced anywhere, it contains all there is to know about the stiffness properties of the structure, provided there are sufficient coordinates.

**What is the relationship between stiffness matrix and flexibility matrix?**

Flexibility is the inverse of stiffness. For example, consider a spring that has Q and q as, respectively, its force and deformation: The spring stiffness relation is Q = k q where k is the spring stiffness. Its flexibility relation is q = f Q, where f is the spring flexibility.

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

All element stiffness matrices are singular.

### What is the purpose of mass matrix?

The basic function of the element mass matrix is to represent the element KE as a quadratic form (i.e. 12vTMv where v are the nodal velocities).

**How do you get a lumped diagonal mass matrix?**

For first-order elements, one way to obtain a lumped diagonal mass matrix is to sum up all the rows of the original mass matrix and assign such sum to the diagonal. However, this does not work for second-order element as there are some nodes that have negative contributions and a simple sum may result in negative values for the diagonal.

**Is there any advantage to using a diagonal mass matrix?**

Computationally, it doesn’t give you any advantage unless you aim for explicit time stepping, in which case the diagonal mass matrix makes much easier to solve. If you’re using an implicit time stepping method, you don’t gain any sparsity in the matrix. I think you only gain error at that point by not using a consistent matrix.$\\endgroup$ – Paul♦

#### Are there non-diagonalizable matrices that aren’t rotations?

But, there are non-diagonalizable matrices that aren’t rotations – all non-zero nilpotent matrices. My intuitive view of nilpotent matrices is that they ”gradually collapse all dimensions/gradually lose all the information” (if we use them over and over again), so it’s clear to me why they can’t be diagonalizable.

#### What is the mass matrix of a rotating dumbbell?

The mass matrix M is the diagonal block matrix where in each block the diagonal elements are the mass of the corresponding particle: Rotating dumbbell.