vault backup: 2025-08-11 09:57:55

书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md

@@ -476,13 +476,29 @@ using (E.7) and the volume integral notations introduced above.
 Inserting (1.12) into (1.9), expanding and sorting the terms into the $_{2\times2}$ block matrix form (1.19) yield the centrifugal stiffness matrix elements  
 
 $$
-\begin{array}{l}{{k_{c,i j}^{00}=\displaystyle\int_{\mathcal{V}}\rho\left(\mathbf{r}_{0,q_{i}q_{j}}^{T}\tilde{\mathbf{r}}_{0}+\mathbf{r}_{0,q_{i}}^{T}\tilde{\mathbf{r}}_{0,q_{j}}+\mathbf{r}_{0,q_{i},q_{j}}^{T}\tilde{\mathbf{R}}_{0}\mathbf{r}_{1}+\breve{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{i},q_{j}}\mathbf{r}_{1}+\breve{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0,q_{i}}\mathbf{r}_{1}+\mathbf{r}_{0,q_{i}}^{T}\breve{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}+\right.}}\\ {{\left.\qquad\quad+\mathbf{\sigma}_{1}^{T}\mathbf{R}_{0,q_{i}}^{T}\tilde{\mathbf{R}}_{0,q_{j}}\mathbf{r}_{1}+\mathbf{r}_{1}^{T}\mathbf{R}_{0,q_{i},q_{j}}^{T}\tilde{\mathbf{R}}_{0}\mathbf{r}_{1}\right)d\mathcal{V}}}\\ {{k_{c,i j}^{01}=\displaystyle\int_{\mathcal{V}}\rho\left(\breve{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{j}}^{T}\mathbf{r}_{1,q_{i}}+\breve{\mathbf{r}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{0,q_{i}}^{T}\breve{\mathbf{R}}_{0}\mathbf{r}_{1,q_{j}}+\breve{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{i}}\mathbf{r}_{1,q_{j}}\right.}}\\ {{\left.\qquad\quad+\mathbf{\sigma}_{1}^{T}\breve{\mathbf{R}}_{0}^{T}\mathbf{R}_{0,q_{j}}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{1}^{T}\breve{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\mathbf{r}_{1,q_{i}}+\mathbf{r}_{1}^{T}\breve{\mathbf{R}}_{0}^{T}\mathbf{R}_{0,q_{i}}\mathbf{r}_{1,q_{j 
+k_{c,ij}^{00} = \int_{\mathcal{V}} \rho \left( \ddot{\mathbf{r}}_{0,q_i}^T \mathbf{r}_{0,q_j} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{r}}_{0,q_j} + \ddot{\mathbf{r}}_{0,q_i,q_j}^T \mathbf{R}_{0} \mathbf{r}_{1} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1} + \mathbf{r}_{0,q_j}^T \ddot{\mathbf{R}}_{0,q_i} \mathbf{r}_{1} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0} \mathbf{r}_{1} + \mathbf{r}_{1}^T \ddot{\mathbf{R}}_{0}^T \mathbf{R}_{0,q_i} \mathbf{r}_{1} \right) d \mathcal{V} \tag{1.34a}
+$$
+
+$$
+k_{c,ij}^{01} = \int_{\mathcal{V}} \rho \left( \ddot{\mathbf{r}}_{0,q_i}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_i} + \ddot{\mathbf{r}}_{0,q_j}^T \mathbf{R}_{0,q_i} \mathbf{r}_{1,q_i} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1,q_j} + \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_i} + \mathbf{r}_{1}^T \ddot{\mathbf{R}}_{0,q_i}^T \mathbf{R}_{0,q_j} \mathbf{r}_{1,q_i} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \mathbf{r}_{1,q_i} + \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0,q_i} \mathbf{r}_{1,q_j} + \mathbf{r}_{1,q_j}^T \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0} \mathbf{r}_{1,q_j} \right) d \mathcal{V} \tag{1.34b}
+$$
+
+$$
+k_{c,ij}^{11} = \int_{\mathcal{V}} \rho \left( \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0} \mathbf{r}_{1,q_i,q_j} + \mathbf{r}_{1,q_i}^T \mathbf{R}_{0}^T \mathbf{R}_{0} \mathbf{r}_{1,q_j} + \mathbf{r}_{1}^T \mathbf{R}_{0,q_i}^T \mathbf{R}_{0} \mathbf{r}_{1,q_j} \right) d \mathcal{V} \tag{1.34c}
 $$  
 
 which can be rewritten as  
 
 $$
-\begin{array}{r l}&{k_{c,i j}^{00}=M\left(\pmb{\Gamma}_{0,q_{i},q_{j}}^{T}\ddot{\mathbf{p}}_{0}+\pmb{\Gamma}_{0,q_{i}}^{T}\ddot{\mathbf{p}}_{0,q_{j}}\right)+\left(\pmb{\Gamma}_{0,q_{i},q_{j}}^{T}\ddot{\mathbf{R}}_{0}+\ddot{\mathbf{r}}_{0}^{T}\pmb{\mathsf{R}}_{0,q_{i},q_{j}}+\ddot{\mathbf{r}}_{0,q_{j}}^{T}\pmb{\mathsf{R}}_{0,q_{i}}+\pmb{\Gamma}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0,q_{j}}\right)M\mathbf{r}_{c g}}\\ &{\quad\quad\quad+\left(\pmb{\mathsf{R}}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0,q_{j}}+\pmb{\mathsf{R}}_{0,q_{i},q_{j}}^{T}\ddot{\mathbf{R}}_{0}\right):\mathsf{I}_{b a s e}}\\ &{k_{c,i j}^{01}=\left(\ddot{\pmb{\Gamma}}_{0,q_{j}}^{T}\mathbf{R}_{0}+\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{j}}\right)M\mathbf{r}_{c g,q_{i}}+\left(\pmb{\Gamma}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}+\ddot{\mathbf{r}}_{0}^{T}\mathbf{R}_{0,q_{i}}\right)M\mathbf{r}_{c g,q_{j}}}\\ &{\quad\quad\quad+\left(\ddot{\pmb{\mathsf{R}}}_{0}^{T}\mathbf{R}_{0,q_{j}}+\ddot{\mathbf{R}}_{0,q_{j}}^{T}\mathbf{R}_{0}\right):\mathsf{A}_{b a s e,i}+\left(\ddot{\pmb{\mathsf{R}}}_{0}^{T}\mathbf{R}_{0,q_{i}}+\pmb{\mathsf{R}}_{0,q_{i}}^{T}\ddot{\mathbf{R}}_{0}\right):\mathsf{A}_{b a s e,j}}\\ &{k_{c,i j}^{11}=\ddot{\pmb{\Gamma}}_{0}^{T}\mathbf{R}_{0}M\mathbf{r}_{c g,q_{i},q_{j}}+\left(\pmb{\mathsf{R}}_{0}^{T}\
+k_{c,ij}^{00} = M \left( \ddot{\mathbf{r}}_{0,q_i,q_j}^T \mathbf{r}_{0} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{r}}_{0,q_j} \right) + \left( \mathbf{r}_{0,q_i,q_j}^T \ddot{\mathbf{R}}_{0} + \ddot{\mathbf{r}}_{0,q_i}^T \mathbf{R}_{0,q_j} + \ddot{\mathbf{r}}_{0,q_j}^T \mathbf{R}_{0,q_i} + \mathbf{r}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} \right) M \mathbf{r}_{cg} \\ + \left( \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0,q_j} + \mathbf{R}_{0,q_i,q_j}^T \ddot{\mathbf{R}}_{0} \right) : \mathbf{I}_{base} \tag{1.35a}
+$$
+
+$$
+k_{c,ij}^{01} = \left( \ddot{\mathbf{r}}_{0,q_j}^T \mathbf{R}_{0} + \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0,q_j} \right) M \mathbf{r}_{cg,q_i} + \left( \ddot{\mathbf{r}}_{0,q_i}^T \mathbf{R}_{0} + \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0,q_i} \right) M \mathbf{r}_{cg,q_j} \\ + \left( \ddot{\mathbf{R}}_{0}^T \mathbf{R}_{0,q_j} + \mathbf{R}_{0,q_j}^T \ddot{\mathbf{R}}_{0} \right) : \mathbf{A}_{base,i} + \left( \ddot{\mathbf{R}}_{0}^T \mathbf{R}_{0,q_i} + \mathbf{R}_{0,q_i}^T \ddot{\mathbf{R}}_{0} \right) : \mathbf{A}_{base,j} \tag{1.35b}
+$$
+
+$$
+k_{c,ij}^{11} = \ddot{\mathbf{r}}_{0}^T \mathbf{R}_{0} M \mathbf{r}_{cg,q_i,q_j} + \left( \mathbf{R}_{0}^T \ddot{\mathbf{R}}_{0} \right) : \mathbf{A}_{base,1,ij} + \left( \ddot{\mathbf{R}}_{0}^T \mathbf{R}_{0} \right) : \mathbf{A}_{base,2,ij} \tag{1.35c}
 $$  
 
 using (E.7) and the volume integral notations introduced above.