8. Modal methods for structural problems#

\[\mathbf{M} \ddot{\mathbf{u}} + \mathbf{K} \mathbf{u} = \mathbf{f} \ .\]

8.1. 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 \(\mathbf{K} \mathbf{u}(s)\),

\[\mathbf{K} \mathbf{u}(s) = \mathbf{K} \mathbf{U} \text{diag}\left[ \frac{1}{m_i ( s^2 + 2 \xi_i \omega_i s + \omega^2_i )} \right]\mathbf{U}^* \mathbf{f}(s) \ .\]

8.1.1. 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,

\[\begin{split}\mathbf{u} = \mathbf{U} \mathbf{q} = \begin{bmatrix} \mathbf{U}_s & \mathbf{U}_f \end{bmatrix} \begin{bmatrix} \mathbf{q}_s \\ \mathbf{q}_f \end{bmatrix} = \mathbf{U}_s \mathbf{q}_s + \mathbf{U}_f \mathbf{q}_f \ .\end{split}\]

8.1.1.1. Truncation and direct recovery of loads#

\[\begin{aligned} \mathbf{u} & = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f} + \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f ( s^2 + 2 \xi_f \omega_f s + \omega^2_f )} \right] \mathbf{U}_f^* \mathbf{f} \end{aligned}\]

8.1.1.2. 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,

\[\begin{split}\begin{aligned} \mathbf{u} & = \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f} - \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s \omega^2_s } \right] \mathbf{U}_s^* \mathbf{f} + \\ & + \mathbf{U}_s \text{diag}\left[ \frac{1}{m_s \omega^2_s } \right] \mathbf{U}_s^* \mathbf{f} + \mathbf{U}_f \text{diag}\left[ \frac{1}{m_f \omega^2_f } \right] \mathbf{U}_f^* \mathbf{f} = \\ & = \mathbf{U}_s \text{diag}\left[ \frac{-s^2 -2 \xi_s \omega_s s}{m_s \omega^2_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f} - \mathbf{U} \text{diag}\left[ \frac{1}{m_i \omega^2_i } \right] \mathbf{U}^* \mathbf{f} + \\ & = \mathbf{U}_s \text{diag}\left[ \frac{-s^2 -2 \xi_s \omega_s s}{m_s \omega^2_s ( s^2 + 2 \xi_s \omega_s s + \omega^2_s )} \right] \mathbf{U}_s^* \mathbf{f} - \mathbf{K}^{-1} \mathbf{f} \\ & = ... \\ & = - \mathbf{U}_s \left( \mathbf{U}_s^ \mathbf{K} \mathbf{U}_s \right)^{-1} \left( \right) \text{diag} \left[ \frac{1}{m_s (s^2 + 2 \xi_s \omega_s s + \omega^2_s) } \right] \mathbf{U}_s^* \mathbf{f} - \mathbf{K}^{-1} \mathbf{f} \ . \end{aligned}\end{split}\]

8.2. 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…

Modal methods for structural problems

Contents