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 .[1] 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.

Examples

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Two-body unidimensional system

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System of masses in one spatial dimension.

For example, consider a system consisting of two point-like masses confined to a straight track. The state of that system 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 m1, m2, the kinetic energy of the system is

 

This formula can also be written as

 

where

 

N-body system

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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 ni free Cartesian coordinates (where ni = 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:[2]

 

where Ini is the ni × ni identity matrix, or more fully:

 

Rotating dumbbell

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Rotating dumbbell.

For a less trivial example, consider two point-like objects with masses m1, m2, attached to the ends of a rigid massless bar with length 2R, the assembly being free to rotate and slide over a fixed plane. The state of the system can be described by the generalized coordinate vector

 

where x, y are the Cartesian coordinates of the bar's midpoint and α is the angle of the bar from some arbitrary reference direction. The positions and velocities of the two particles are

 

and their total kinetic energy is

 

where   and  . This formula can be written in matrix form as

 

where

 

Note that the matrix depends on the current angle α of the bar.

Continuum mechanics (consistent mass v lumped mass)

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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 accuracy and performance. For example, lumped mass and consistent mass are two different ways of modelling the mass matrix in structural dynamics and finite element analysis. Lumped mass concentrates the entire mass of an element at its nodes, resulting in a simple, diagonal mass matrix that is computationally efficient but less accurate. Consistent mass distributes the mass evenly along the element's length, using the same interpolation methods as the stiffness matrix to create a more accurate, but more complex, fully populated off-diagonal mass matrix.

Consistent Mass Matrix

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The consistent mass matrix is based on an integral of the material density over the element volume ___domain, typically evaluated numerically (e.g. using an approximate Gaussian integration scheme). The term "consistent" refers to the fact that the shape functions used to calculate the element mass matrix are the same as those used to calculate the element stiffness matrix.

For example, the consistent mass matrix Me for a two-dimensional truss element with two degrees of freedom at each end is as follows:

 

where m is the total mass of the element.

The consistent mass matrix provides a more accurate representation of the mass distribution in a continuum. It includes the effect of mass coupling between different degrees of freedom and different nodes, including the effects of the rotational inertia for elements with rotational degrees of freedom.

Lumped Mass Matrix

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Lumped mass approximation is a simpler and often more efficient approach for the representation of the distributed mass within a structure. In this intuitive approach, the mass within each element is assumed to be lumped equally onto the nodes such that the sum of the nodal masses associated with the translational degrees of freedom for each global direction equals the total mass of the element. Usually, only the translational mass is included, even for elements with rotational degrees of freedom, and all off-diagonal terms are excluded (i.e., there is no mass coupling between the different degrees of freedom). Without rotational mass terms, overall rotational inertia comes from the relative translation of the nodes.

For the two-dimensional truss above, the lumped mass matrix is as follows:

 

Intuitively, one can generate this mass matrix by simply lumping half of the element mass onto each of its nodes.

Although the consistent mass matrix is a better approximation of the element mass distribution, the lumped mass matrix is often preferred for efficiency reasons, particularly for reasonably well refined meshes. As the lumped mass matrix is diagonal, the storage requirement is much lower and the amount of numerical effort to solve the model can be greatly reduced.

The lumped mass matrix will theoretically produce a lower level of accuracy than the consistent mass matrix. But in practice, the accuracy of the lumped mass approximation is comparable to the accuracy of the consistent mass when the mesh is sufficiently refined. This is particularly true for the lower vibration modes of a structure. Of course there are situations that cannot be modelled with a lumped diagonal mass matrix that considers only translational terms. An example is the torsional vibration of a beam modelled with beam elements. Unless a rotational mass term corresponding to the twist rotation of the beam is included in the mass matrix, the torsional vibration modes cannot be calculated. Lumped mass matrices usually yield natural frequencies that are lower than the exact values, particularly for coarse meshes.

See also

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References

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  1. ^ Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3
  2. ^ Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978 0 521 57572 0