An alternative method to derive the uncoupled equations governing the motion of the free vibrations of a \(n \mathrm{DOF}\) system in terms of principal coordinates is to introduce a linear transformation between the generalized coordinates \(\mathbf{x}\) and the principal coordinates \(\mathbf{p}\) as \(\mathbf{x}=\mathbf{P} \mathbf{p}\), where \(\mathbf{P}\) is the modal matrix, the matrix whose columns are the normalized mode shapes. Follow these steps to derive the equations governing the principal coordinates:

(a) Rewrite Equation (8.3) using the principal coordinates as dependent variables by introducing the linear transformation in Equation (8.3).

(b) Premultiply the resulting equation by \(\mathbf{P}^{T}\).

(c) Use the orthonormality of mode shapes to show that \(\mathbf{P}^{T} \mathbf{M P}\) and \(\mathbf{P}^{T} \mathbf{K P}\) are diagonal matrices.

(d) Write the uncoupled equations for the principal coordinates.

\(\mathbf{M} \ddot{\mathbf{x}}+\mathbf{K} \mathbf{x}=\mathbf{0} \tag{8.3}\)