Mass matrix

In computational mechanics, a mass matrix is a generalization of the concept of mass to generalized coordinates. For example, consider a two-body particle system in one dimension. The position of such a system has two degrees of freedom, the position of each particle, which can be described by the generalized position vector

$\mathbf{x}=[x_1\, x_2]^\top$.

Supposing the particles have masses, m₁ and m₂. We can write Newton's second law for each particle as

$F_i = m_i \ddot{x}_i$

and the kinetic energy of the system as

$E = \sum_{i=1}^{2} \frac{1}{2} m_i \dot{x}_i{}^2$

Putting the masses into the matrix

$M=\begin{bmatrix}m_1&0\\0 & m_2\end{bmatrix}$,

the same equations of motion for the two-particle system become

$\mathbf{F}=M \ddot{\mathbf{x}}$

and the total kinetic energy is given by

$E=\frac{1}{2} \dot{\mathbf{x}}^\top M \dot{\mathbf{x}}$.

The structure of the mass matrix becomes more complicated in more dimensions. For example, in two dimensions, there are two degrees of freedom for a given particle, so if the ith particle corresponds to degree of freedom j and j+1, then

M_j,j = M_{j + 1,j + 1} = m_i

For example, in two dimensions the two-body system has 4 degrees of freedom (in Cartesian coordinates these would be [x₁,y₁],[x₂,y₂]). Then, the generalized position vector would be

$\mathbf{x}=[x_1\, x_2\, x_3\, x_4]^\top$

and the mass matrix would be

$M=\begin{bmatrix}m_1&0&0&0\\0&m_1&0&0\\0&0&m_2&0\\0&0&0&m_2\end{bmatrix}$

For applications in which mass is distributed such as rigid-body dynamics, there may be off-diagonal terms. For example, in one dimension if two particles with mass are connected by an ideal spring with a uniformly distributed mass, the effective acceleration of all points along the spring would correspond to differential mass elements, the acceleration of which would interpolate between the velocities of the two particles.^{[citation needed]}

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. For example, a lumped-mass method, in which the deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element.

See also

• Stiffness matrix