7. Modal methods for structural problems#
\[\mathbf{M} \ddot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{f} \ .\]
7.1. No free rigid motion#
If a structure has no free rigid motion, the stiffness matrix of mechanical systems is symmetric definite positive.
Spectral decomposition of the problem.
\[\left[ s_i^2 \mathbf{M} + \mathbf{K} \right] \hat{\mathbf{u}}_i = \mathbf{0} \ ,\]
or in index and matrix form
\[\begin{split}\begin{aligned}
0 & = s_i^2 M_{jk} U_{ki} + K_{jk} U_{ki} = \\
& = \mathbf{M} \mathbf{U} \mathbf{S}^2 + \mathbf{K} \mathbf{U} \ ,
\end{aligned}\end{split}\]
with the diagonal matrix \(\mathbf{S}\) collecting the eigenvalues,
\[\mathbf{S} = \text{diag} \left\{ s_i \right\} \ .\]
Properties. For eigenvectors with different eigenvalues,
\[\begin{aligned}
\hat{\mathbf{u}}_j^* \mathbf{M} \hat{\mathbf{u}}_i = 0 \qquad , \qquad
\hat{\mathbf{u}}_j^* \mathbf{K} \hat{\mathbf{u}}_i = 0 \ .
\end{aligned}\]
Nodal and modal unknowns. The nodal vector can be written as a combination of modes, being \(\mathbf{q}\) the vector of modal amplitudes,
\[\mathbf{u} = \mathbf{U} \mathbf{q} = \left[ \hat{\mathbf{u}}_1 | \dots | \hat{\mathbf{u}}_N \right] \mathbf{q} \ .\]
Laplace domain. In Laplace domain
\[\begin{split}\begin{aligned}
\left[ s^2 \mathbf{U}^* \mathbf{M} \mathbf{U} + \mathbf{U}^* \mathbf{K} \mathbf{U}\right] \mathbf{q}(s) & = \mathbf{U}^* \mathbf{f}(s) \\
\text{diag}\left[ s^2 m_i + k_i \right] \mathbf{q}(s) & = \mathbf{U}^* \mathbf{f}(s) \ .
\end{aligned}\end{split}\]
Modal damping Adding modal damping, with simultaneous diagonalization with mass and stiffness matrices,
\[\begin{split}
\text{diag}\left[ s^2 m_i + s c_i + k_i \right] \mathbf{q}(s) & = \mathbf{U}^* \mathbf{f}(s) \\
\mathbf{q}(s) & = \text{diag}\left[ \frac{1}{m_i ( s^2 + 2 \xi_i \omega_i s + \omega^2_i )} \right]\mathbf{U}^* \mathbf{f}(s) \ .
\end{split}\]
The original equation becomes
\[\left[s^2 \mathbf{M} + s \mathbf{C} + \mathbf{K} \right] \mathbf{u} = \mathbf{f} \ ,\]
with
\[\begin{split}\begin{aligned}
\mathbf{M} & = \mathbf{U} \text{diag}\left\{ m_i \right\} \mathbf{U}^* \\
\mathbf{C} & = \mathbf{U} \text{diag}\left\{ c_i \right\} \mathbf{U}^* \\
\mathbf{K} & = \mathbf{U} \text{diag}\left\{ k_i \right\} \mathbf{U}^* \\
\end{aligned}\end{split}\]
and the eigenproblem reads
\[\begin{split}\begin{aligned}
\mathbf{0} & = \left[ s_i^2 \mathbf{M} + s_i \mathbf{C} + \mathbf{K} \right] \hat{\mathbf{u}}_i \\
\mathbf{0} & = \mathbf{M} \mathbf{U} \mathbf{S}^2 + \mathbf{C} \mathbf{U} \mathbf{S} + \mathbf{K} \mathbf{U} \\
\end{aligned}\end{split}\]
Nodal vector. Nodal vector thus reads
\[\mathbf{u}(s) = \mathbf{U} \mathbf{q}(s) = \mathbf{U} \text{diag}\left[ \frac{1}{m_i ( s^2 + 2 \xi_i \omega_i s + \omega^2_i )} \right]\mathbf{U}^* \mathbf{f}(s) \]
and the internal forces usually derived from a manipulation of the term \(\mathbf{K} \mathbf{u}(s)\),
\[\mathbf{K} \mathbf{u}(s) = \mathbf{K} \mathbf{U} \text{diag}\left[ \frac{1}{m_i ( s^2 + 2 \xi_i \omega_i s + \omega^2_i )} \right]\mathbf{U}^* \mathbf{f}(s) \ .\]
7.1.1. Dimension reduction#
Modal unknowns can usually partitioned in slow (dynamical, resolved) and fast modes (with natural frequencies well above the frequency content of the forcing, and the dynamics of the system; can be treated as static modes),
\[\begin{split}\mathbf{q} = \begin{bmatrix} \mathbf{q}_s \\ \mathbf{q}_f \end{bmatrix} \ ,\end{split}\]
and the sum of their contributions give the nodal unknown,
\[\begin{split}\mathbf{u} = \mathbf{U} \mathbf{q} = \begin{bmatrix} \mathbf{U}_s & \mathbf{U}_f \end{bmatrix} \begin{bmatrix} \mathbf{q}_s \\ \mathbf{q}_f \end{bmatrix} = \mathbf{U}_s \mathbf{q}_s + \mathbf{U}_f \mathbf{q}_f \ .\end{split}\]
7.1.1.1. Truncation and direct recovery of loads#
\[\begin{aligned}
\mathbf{u}
& = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
+ \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f ( s^2 + 2 \xi_f \omega_f s + \omega^2_f )} \right] \mathbf{U}_f^* \mathbf{f}
\end{aligned}\]
7.1.1.2. Mode acceleration and static recovery of fast modes#
Static approximation of fast modes gives
\[\begin{split}\begin{aligned}
\mathbf{u}
& = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
+ \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f ( s^2 + 2 \xi_f \omega_f s + \omega^2_f )} \right] \mathbf{U}_f^* \mathbf{f} = \\
& = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
+ \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f \omega^2_f } \right] \mathbf{U}_f^* \mathbf{f}
\end{aligned}\end{split}\]
and adding and subtracting the static response of the slow modes,
\[\begin{split}\begin{aligned}
\mathbf{u}
& = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
- \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s \omega^2_s } \right] \mathbf{U}_s^* \mathbf{f} + \\
& + \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s \omega^2_s } \right] \mathbf{U}_s^* \mathbf{f}
+ \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f \omega^2_f } \right] \mathbf{U}_f^* \mathbf{f} = \\
& = \mathbf{U}_s \text{diag}\left[ \frac{-s^2 -2 \xi_s \omega_s s}{m_s \omega^2_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
- \mathbf{U} \text{diag}\left[ \frac{1}{m_i \omega^2_i } \right] \mathbf{U}^* \mathbf{f} + \\
& = \mathbf{U}_s \text{diag}\left[ \frac{-s^2 -2 \xi_s \omega_s s}{m_s \omega^2_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f}
- \mathbf{K}^{-1} \mathbf{f} \\
& = ... \\
& = - \mathbf{U}_s \left( \mathbf{U}_s^ \mathbf{K} \mathbf{U}_s \right)^{-1} \left( \right) \text{diag} \left[ \frac{1}{m_s (s^2 + 2 \xi_s \omega_s s + \omega^2_s) } \right] \mathbf{U}_s^* \mathbf{f} - \mathbf{K}^{-1} \mathbf{f} \ .
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
& \mathbf{U}_s \text{diag}\left[ \frac{-s^2 -2 \xi_s \omega_s s}{m_s \omega^2_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f} = \\
& = \mathbf{U}_s \text{diag}\left[ \frac{1}{k_s} \right] \text{diag}\left[ m_s ( s^2 + 2 \xi_s \omega_s s ) \right]\text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s) } \right] \mathbf{U}_s^* \mathbf{f} = \\
\end{aligned}\end{split}\]
…todo…
7.2. With free rigid motion#
Free rigid degrees of freedom are associated with vectors of the kernel of the stiffness matrix. The stiffness matrix is singular…