# sigularity of the mass matrix

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When analyzing trusses, one can compute the natural frequencies of the truss, using matrix methods, i.c. solve the eigenvalue problem associated with the construction. One needs to assemble the **mass matrix** for that. This mass matrix is singular and can therefor not be inverted. What does it physically mean, that the mass matrix is singular ?

(The corresponding stiffness matrix is also singular, meaning the truss cannot deform, without being supported in some manner.).

Would the singularuity of the mass matrix mean, that the truss cannot

**vibrate** without being supported ?
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My linear algebra is a bit rusty, but I don't believe a singular matrix necessarily has any bearing on the stability of the truss. A singular matrix just means the equations describing the motion/forces on the individual bodies are not independent. This is believable considering truss systems often have redundant members. For example, if the forces on member A can be fully realized if one knows the forces on members B, C and D, then the equations for member A can be omitted. Leaving them in your calculations will lead to a system matrix that is singular. The rank of your matrix (number of linearly independent columns) will be less than the actual dimension of the matrix. If you want the a nonsingular version of your matrix, you'll need to do some row reduction techniques. Hope this helps.

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That is an answer to another question, I think. again: what does it **mean** that a mass matrix is singular ?

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"A singular matrix just means the equations describing the motion/ forces on the individual bodies are not independent."

So if you have a 10x10 mass matrix that is singular (det=0), then this matrix can likely be reduced to a 9x9 or less.

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that is the mathematical meaning. now for the physical meaning.

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