MassMatrix

Mass matrix is a fundamental object in mechanical and dynamical systems that encodes how mass is distributed in a system and how this distribution translates accelerations into generalized forces. It appears in equations of motion and vibration analyses, where it maps generalized accelerations to inertial forces.

In the finite element method and other discretization techniques, the mass matrix M arises from integrating mass density against shape functions over the domain. A common form is M_ij = ∫_Ω ρ(x) N_i(x) N_j(x) dΩ, where ρ is the density and N_i are the shape functions. Such a mass matrix is symmetric and positive definite. Two main variants are used: the consistent mass matrix, which exactly represents the mass distribution as derived from the discretization, and the lumped mass matrix, a diagonal approximation obtained by methods such as row-sum, which simplifies storage and accelerates time integration.

In structural dynamics and vibrational analysis, the mass matrix combines with a stiffness matrix K to form the generalized equations of motion M q'' + C q' + K q = F, where q are generalized coordinates. The mass matrix is typically sparse for locally supported elements, and its properties influence stability and accuracy of numerical solvers. For eigenfrequency analyses, the natural modes satisfy det(K − ω^2 M) = 0.

Beyond finite elements, mass matrices also appear in rigid body dynamics, where spatial formulations yield a block matrix that includes translational mass and rotational inertia. In practice, the choice between a consistent or lumped mass matrix depends on accuracy needs and computational constraints.