To answer your first post, I suggest you read a book on the Finite Element Method. The book I read in my Masters course was Reddy, An Introduction to the Finite Element Method and it starts with the governing equations, derives the numerical approximations, and ends with computer code to implement it.
To answer your second post, a rigid body has six degrees of freedom, three translations and three rotations. In a static structural model, you need to constrain all six degrees of freedom or there is no solution. If one of the DOF is unconstrained, that means there is zero stiffness in that direction. A natural frequency is sqrt(k/m). If k is zero the frequency is zero. If the rigid body has no constraints, there are six zero frequency modes.
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