<Dl> <Dd> (╲ m r ╲) (q _̈) + (╲ k r ╲) (q) = 0 . (\ displaystyle (\ begin (bmatrix) ^ (\ diagdown) m_ (r \ diagdown) \ end (bmatrix)) (\ begin (Bmatrix) (\ ddot (q)) \ end (Bmatrix)) + (\ begin (bmatrix) ^ (\ diagdown) k_ (r \ diagdown) \ end (bmatrix)) (\ begin (Bmatrix) q \ end (Bmatrix)) = 0 .) </Dd> </Dl> <Dd> (╲ m r ╲) (q _̈) + (╲ k r ╲) (q) = 0 . (\ displaystyle (\ begin (bmatrix) ^ (\ diagdown) m_ (r \ diagdown) \ end (bmatrix)) (\ begin (Bmatrix) (\ ddot (q)) \ end (Bmatrix)) + (\ begin (bmatrix) ^ (\ diagdown) k_ (r \ diagdown) \ end (bmatrix)) (\ begin (Bmatrix) q \ end (Bmatrix)) = 0 .) </Dd> <P> This equation is the foundation of vibration analysis for multiple degree of freedom systems . A similar type of result can be derived for damped systems . The key is that the modal mass and stiffness matrices are diagonal matrices and therefore the equations have been "decoupled". In other words, the problem has been transformed from a large unwieldy multiple degree of freedom problem into many single degree of freedom problems that can be solved using the same methods outlined above . </P> <P> Solving for x is replaced by solving for q, referred to as the modal coordinates or modal participation factors . </P>

Phase characteristics of unbalance phenomenon in mechanical component vibration