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Concept
Mass matrix
Résumé
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
where denotes the transpose of the vector . This equation is analogous to the formula for the kinetic energy of a particle with mass m and velocity v, namely
and can be derived from it, by expressing the position of each particle of the system in terms of q.
In general, the mass matrix M depends on the state q, and therefore varies with time.
Lagrangian mechanics yields an ordinary differential equation (actually, a system of coupled differential equations) that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system. The kinetic energy formula above is one term of that equation, that represents the total kinetic energy of all the particles.
For example, consider a system consisting of two point-like masses confined to a straight track. The state of that systems can be described by a vector q of two generalized coordinates, namely the positions of the two particles along the track.
Supposing the particles have masses m_1, m_2, the kinetic energy of the system is
This formula can also be written as
where
More generally, consider a system of N particles labelled by an index i = 1, 2, ..., N, where the position of particle number i is defined by n_i free Cartesian coordinates (where n_i = 1, 2, 3). Let q be the column vector comprising all those coordinates. The mass matrix M is the diagonal block matrix where in each block the diagonal elements are the mass of the corresponding particle:
where I_n_i is the n_i × n_i identity matrix, or more fully:
For a less trivial example, consider two point-like objects with masses m_1, m_2, attached to the ends of a rigid massless bar with length 2R, the assembly being free to rotate and slide over a fixed plane.
Source officielle
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