FREQUENCY RESPONSE
Modal method uses mode shapes to reduce the size, uncouple the equations and make the mathematical solution more efficient.
Usually mode shapes are computed in the initial stages of a dynamic analysis, so modal frequency response is a extension of normal mode analysis.
As a first step, damping is not included.
Mode shapes [Φ] are used to transform the problem from physical variables to modal co-ordinates . This equation is exact if all modes are used, but really only a few are used, to reduce the size of the problem, so it is an approximation.
Substituting the equation of motion in modal co-ordinates is obtained, this is still a coupled problem.
Pre-multiplying by [Φ]T, and using the orthogonal property of mode shapes, the equation of motion is formulated in terms of the generalised mass and stiffness, which are diagonal matrices, and is equivalent to a set of uncoupled single-degree of freedom (dof) system.
If damping is present, the modes in general do not diagonalise [B] matrix. In this case, the coupled system is solved in a manner similar to direct method, but the size of the system is smaller, as it is described in modal co-ordinates.
Modal damping is applied to each mode separately, the uncoupled equations can be maintained.