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书籍/力学书籍/CASEstab_theory_manual/auto/CASEstab_theory_manual.md
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@ -678,7 +678,7 @@ M=\frac{l}{2}\sum_{r=0}^{P}c(r)a_{m,r}\tag{1.49}
$$
where the coefficient function is given by
其中系数函数由下式给出
$$
c\left(r\right)={\frac{\left(-1\right)^{r}+1}{1+r}}={\left\{\begin{array}{l l}{2/(1+r)}&{r\;\;{\mathsf{e v e n}}}\\ {\;\;\;\;0}&{r\;\;\;{\mathsf{o d d}}}\end{array}\right.}\tag{1.50}
$$
@ -687,65 +687,109 @@ The element center of gravity position times the element mass and the nodal deri
单元重心位置乘以单元质量和节点导数可以计算为
$$
\begin{array}{l l}{{\displaystyle M\,{\bf r}_{c g}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p}+a_{m x_{c g},r}\,{\bf r}_{x,p}+a_{m y_{c g},r}\,{\bf r}_{y,p}\right)\right)}}\\ {{\displaystyle M\,{\bf r}_{c g,q_{i}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i}}\right)\right)}}\\ {{\displaystyle M\,{\bf r}_{c g,q_{i},q_{j}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i},q_{j}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i},q_{j}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i},q_{j}}\right)\right)}}\end{array}
$$
{{\displaystyle M\,{\bf r}_{c g}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p}+a_{m x_{c g},r}\,{\bf r}_{x,p}+a_{m y_{c g},r}\,{\bf r}_{y,p}\right)\right)}}\tag{1.51a}
$$
$$
{{\displaystyle M\,{\bf r}_{c g,q_{i}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i}}\right)\right)}}\tag{1.51b}
$$
$$
{{\displaystyle M\,{\bf r}_{c g,q_{i},q_{j}}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}c(p+r)\left(a_{m,r}\,{\bf r}_{o,p,q_{i},q_{j}}+a_{m x_{c g},r}\,{\bf r}_{x,p,q_{i},q_{j}}+a_{m y_{c g},r}\,{\bf r}_{y,p,q_{i},q_{j}}\right)\right)}}\tag{1.51c}
$$
where $c(p+r)$ is given by (1.49). The scalars and matrices of (1.36) and (1.37) have the generic form
其中 $c(p+r)$ 由 (1.49) 给出。(1.36) 和 (1.37) 中的标量和矩阵具有通用形式
$$
\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{u}_{o,q},\mathbf{u}_{x,q},\mathbf{u}_{y,q},\mathbf{w}_{o,p},\mathbf{w}_{x,p},\mathbf{w}_{y,p}\right\}\right)\right)
$$
where the generic operator is defined as
where the generic operator is defined as
其中通用算子定义为
$$
\begin{array}{r}{\mathcal{G}\left\{\mathbf{a},\mathbf{u}_{o},\mathbf{u}_{x},\mathbf{u}_{y},\mathbf{w}_{o},\mathbf{w}_{x},\mathbf{w}_{y}\right\}={a}_{m}\,\mathbf{u}_{o}^{T}\mathbf{w}_{o}+{a}_{m x_{c g}}\,\left(\mathbf{u}_{o}^{T}\mathbf{w}_{x}+\mathbf{u}_{x}^{T}\mathbf{w}_{o}\right)+{a}_{m y_{c g}}\,\left(\mathbf{u}_{o}^{T}\mathbf{w}_{y}+\mathbf{u}_{y}^{T}\mathbf{w}_{o}\right)}\\ {+\;{a}_{i_{x x}}\,\mathbf{u}_{x}^{T}\mathbf{w}_{x}+{a}_{i_{x y}}\,\left(\mathbf{u}_{x}^{T}\mathbf{w}_{y}+\mathbf{u}_{y}^{T}\mathbf{w}_{x}\right)+{a}_{i_{y y}}\,\mathbf{u}_{y}^{T}\mathbf{w}_{y}\,\,\,}\end{array}
\begin{array}{r}{\mathcal{G}\left\{\mathbf{a},\mathbf{u}_{o},\mathbf{u}_{x},\mathbf{u}_{y},\mathbf{w}_{o},\mathbf{w}_{x},\mathbf{w}_{y}\right\}={a}_{m}\,\mathbf{u}_{o}^{T}\mathbf{w}_{o}+{a}_{m x_{c g}}\,\left(\mathbf{u}_{o}^{T}\mathbf{w}_{x}+\mathbf{u}_{x}^{T}\mathbf{w}_{o}\right)+{a}_{m y_{c g}}\,\left(\mathbf{u}_{o}^{T}\mathbf{w}_{y}+\mathbf{u}_{y}^{T}\mathbf{w}_{o}\right)}\\ {+\;{a}_{i_{x x}}\,\mathbf{u}_{x}^{T}\mathbf{w}_{x}+{a}_{i_{x y}}\,\left(\mathbf{u}_{x}^{T}\mathbf{w}_{y}+\mathbf{u}_{y}^{T}\mathbf{w}_{x}\right)+{a}_{i_{y y}}\,\mathbf{u}_{y}^{T}\mathbf{w}_{y}\,\,\,}\end{array}\tag{1.52}
$$
where $\mathbf{a}=[a_{m},a_{m x_{c g}},a_{m y_{c g}},a_{i_{x x}},a_{i_{x y}},a_{i_{y y}}]$ is a list with the polynomial coefficients of the material properties (such that ${\bf a}_{r}$ contains the $r$ ’th order coefficients), and the vectors $\mathbf{u}_{o}$ , $\mathbf{u}_{x}$ , $\mathbf{u}_{y}$ , $\pmb{{\mathsf{w}}}_{o}$ , $\mathbf{w}_{x}$ , and $\pmb{w}_{y}$ are 3x1 column
vectors or $\mathsf{1x3}$ row vectors for scalar or matrix evaluations, respectively. The local mass matrix contribution from an element (the entries of the element mass matrix) is
where $\mathbf{a}=[a_{m},a_{m x_{c g}},a_{m y_{c g}},a_{i_{x x}},a_{i_{x y}},a_{i_{y y}}]$ is a list with the polynomial coefficients of the material properties (such that ${\bf a}_{r}$ contains the $r$’th order coefficients), and the vectors $\mathbf{u}_{o}$ , $\mathbf{u}_{x}$ , $\mathbf{u}_{y}$ , $\pmb{{\mathsf{w}}}_{o}$ , $\mathbf{w}_{x}$ , and $\pmb{w}_{y}$ are 3x1 column vectors or $\mathsf{1x3}$ row vectors for scalar or matrix evaluations, respectively. The local mass matrix contribution from an element (the entries of the element mass matrix) is
其中 $\mathbf{a}=[a_{m},a_{m x_{c g}},a_{m y_{c g}},a_{i_{x x}},a_{i_{x y}},a_{i_{y y}}]$ 是一个包含材料属性多项式系数的列表（使得 ${\bf a}_{r}$ 包含 $r$ 阶系数），并且向量 $\mathbf{u}_{o}$ 、 $\mathbf{u}_{x}$ 、 $\mathbf{u}_{y}$ 、 $\pmb{{\mathsf{w}}}_{o}$ 、 $\mathbf{w}_{x}$ 和 $\pmb{w}_{y}$ 分别是用于标量或矩阵评估的 3x1 列向量或 $\mathsf{1x3}$ 行向量。来自一个单元的局部质量矩阵贡献（单元质量矩阵的条目）是
$$
m_{i j}^{11}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\;{\mathcal G}\left\{\mathbf a_{r},\mathbf r_{o,q,q_{i}},\mathbf r_{x,q,q_{i}},\mathbf r_{y,q,q_{i}},\mathbf r_{o,p,q_{j}},\mathbf r_{x,p,q_{j}},\mathbf r_{y,p,q_{j}}\right\}\right)\right)
m_{i j}^{11}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\;{\mathcal G}\left\{\mathbf a_{r},\mathbf r_{o,q,q_{i}},\mathbf r_{x,q,q_{i}},\mathbf r_{y,q,q_{i}},\mathbf r_{o,p,q_{j}},\mathbf r_{x,p,q_{j}},\mathbf r_{y,p,q_{j}}\right\}\right)\right)\tag{1.53}
$$
and its contribution to the local nonlinear gyroscopic matrix (the nonlinear element gyroscopic matrix) is
及其对局部非线性陀螺矩阵（非线性单元陀螺矩阵）的贡献为
$$
h_{i j k}^{111}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,q,q_{i}},\mathbf{r}_{x,q,q_{i}},\mathbf{r}_{y,q,q_{i}},\mathbf{r}_{o,p,q_{j},q_{k}},\mathbf{r}_{x,p,q,q_{j},q_{k}},\mathbf{r}_{y,p,q,q_{k}}\right\}\right)\right)
h_{i j k}^{111}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,q,q_{i}},\mathbf{r}_{x,q,q_{i}},\mathbf{r}_{y,q,q_{i}},\mathbf{r}_{o,p,q_{j},q_{k}},\mathbf{r}_{x,p,q_{j},q_{k}},\mathbf{r}_{y,p,q_{j},q_{k}}\right\}\right)\right)\tag{1.54}
$$
The contribution of an element to the matrix related to the rotational inertia of the substructure about its base is
单元对与子结构绕其基座的转动惯量相关的矩阵的贡献是
$$
\mathbf{l}_{b a s e}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\,\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T},\mathbf{r}_{o,p}^{T},\mathbf{r}_{x,p}^{T},\mathbf{r}_{y,p}^{T}\right\}\right)\right)
\mathbf{I}_{b a s e}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c\left(q+r+p\right)\,\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T},\mathbf{r}_{o,p}^{T},\mathbf{r}_{x,p}^{T},\mathbf{r}_{y,p}^{T}\right\}\right)\right)\tag{1.55}
$$
Finally, the matrices needed for the computation of the remaining inertia forces from an element are
最后，计算一个单元的剩余惯性力所需的矩阵是
$$
\begin{array}{r l}&{\displaystyle\mathsf{A}_{b a s e,i,i}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{i}}^{T},\mathbf{r}_{x,p,q_{i}}^{T},\mathbf{r}_{y,p,q_{i}}^{T},\mathbf{r}_{o,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T}\right\}\right)\right)}\\ &{\displaystyle\mathsf{A}_{b a s e,1,i,j}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{j}}^{T},\mathbf{r}_{x,p,q_{j}}^{T},\mathbf{r}_{y,p,q_{j}}^{T},\mathbf{r}_{o,q_{i}}^{T},\mathbf{r}_{x,q,q_{i}}^{T},\mathbf{r}_{y,q,\bar{q}_{i}}^{T}\right\}\right)\right)}\\ &{\displaystyle\mathsf{A}_{b a s e,2,i,j}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{i},q_{j}}^{T},\mathbf{r}_{x,p,q_{i},q_{j}}^{T},\mathbf{r}_{y,p,q_{i},q_{j}}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T},\mathbf{r}_{y,q}^{T}\right\}\right)\right)}\end{array}
$$
{\displaystyle\mathsf{A}_{b a s e,i}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{i}}^{T},\mathbf{r}_{x,p,q_{i}}^{T},\mathbf{r}_{y,p,q_{i}}^{T},\mathbf{r}_{o,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T}\right\}\right)\right)}\tag{1.56a}
$$
$$
{\displaystyle\mathsf{A}_{b a s e,1,ij}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{j}}^{T},\mathbf{r}_{x,p,q_{j}}^{T},\mathbf{r}_{y,p,q_{j}}^{T},\mathbf{r}_{o,q,q_{i}}^{T},\mathbf{r}_{x,q,q_{i}}^{T},\mathbf{r}_{y,q,{q}_{i}}^{T}\right\}\right)\right)}\tag{1.56b}
$$
$$
{\displaystyle\mathsf{A}_{b a s e,2,ij}=\frac{l}{2}\sum_{p=0}^{P+3}\left(\sum_{r=0}^{P}\left(\sum_{q=0}^{P+3}c(q+r+p)\;\mathcal{G}\left\{\mathbf{a}_{r},\mathbf{r}_{o,p,q_{i},q_{j}}^{T},\mathbf{r}_{x,p,q_{i},q_{j}}^{T},\mathbf{r}_{y,p,q_{i},q_{j}}^{T},\mathbf{r}_{o,q}^{T},\mathbf{r}_{x,q}^{T},\mathbf{r}_{y,q}^{T}\right\}\right)\right)}\tag{1.56c}
$$
The generic form of these contributions to the inertia forces from a single element can be implemented in a single object function that only need the polynomial coefficient sub­vectors (1.46) and their first nodal derivatives
单个单元对惯性力的这些贡献的通用形式可以在一个单一的目标函数中实现，该函数只需要多项式系数子向量（1.46）及其一阶节点导数。
$$
\begin{array}{r l}&{\mathbf{r}_{o,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i}}+\delta_{0p}\,\mathbf{r}_{\mathrm{mid},b,n,q_{i}}+\delta_{1p}\,\frac{l_{n}}{2}\mathbf{e}_{3,b,n,q_{i}}}\\ &{\mathbf{r}_{x,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\mathsf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\mathsf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{,q_{i}}+\delta_{0p}\,\mathbf{e}_{1,b,n,q_{i}}}\\ &{\mathbf{r}_{y,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\mathsf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\mathsf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{,q_{i}}+\delta_{0p}\,\mathbf{e}_{2,b,n,q_{i}}}\end{array}
$$
and second nodal derivatives
{\mathbf{r}_{o,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i}}+\delta_{0p}\,\mathbf{r}_{\mathrm{mid},b,n,q_{i}}+\delta_{1p}\,\frac{l_{n}}{2}\mathbf{e}_{3,b,n,q_{i}}}\tag{1.57a}
$$
\begin{array}{r l}{\mathbf{r}_{o,b,n,p,q_{i},q_{j}}=\mathbf{E}_{b,n,q_{i},q_{j}}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathbf{E}_{b,n,q_{j}}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i}}}&{}\\ {+\,\mathbf{E}_{b,n,q_{i}}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{j}}+\mathbf{E}_{b,n}\,\bar{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i},q_{j}}+\delta_{1p}\,\frac{l_{n}}{2}\mathbf{e}_{3,b,n,q_{i},q_{j}}}&{}\\ {\mathbf{r}_{x,b,n,p,q_{i},q_{j}}=\mathbf{E}_{b,n,q_{i},q_{j}}\,\mathbf{p}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathbf{E}_{b,n,q_{j}}\,\mathbf{p}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i}}}&{}\\ {+\,\mathbf{E}_{b,n,q_{i}}\,\mathbf{p}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{j}}+\mathbf{E}_{b,n}\,\mathbf{p}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i},q_{j}}+\delta_{0p}\,\mathbf{e}_{1,b,n,q_{i},q_{j}}}&{}\\ {\mathbf{r}_{y,b,n,p,q_{i},q_{j}}=\mathbf{E}_{b,n,q_{i},q_{j}}\,\mathbf{p}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathbf{E}_{b,n,q_{j}}\,\mathbf{p}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i}}}&{}\\ {+\,\mathbf{E}_{b,n,q_{i}}\,\mathbf{p}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{j}}+\mathbf{E}_{b,n}\,\mathbf{p}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{q_{i},q_{j}}+\delta_{0p}\,\mathbf{e}_{2,b,n,q_{i},q_
$$
$$
{\mathbf{r}_{x,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\mathsf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\mathsf{P}_{x}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{,q_{i}}+\delta_{0p}\,\mathbf{e}_{1,b,n,q_{i}}}\tag{1.57b}
$$
$$
{\mathbf{r}_{y,b,n,p,q_{i}}=\!\!\mathsf{E}_{b,n,q_{i}}\,\mathsf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}+\mathsf{E}_{b,n}\,\mathsf{P}_{y}\,\tilde{\mathbf{N}}_{n,p}\,\mathbf{g}_{,q_{i}}+\delta_{0p}\,\mathbf{e}_{2,b,n,q_{i}}}\tag{1.57c}
$$
where the element­index $n$ and substructure­index $b$ are included again. Note that the coefficient sub­vectors (1.46) and its first and second derivatives (1.57) and (1.58) must be evaluated at the current nodal DOFs $\mathbf{q}_{b,n}$ of the element. The $\mathsf{3}\mathsf{x}\mathsf{3}$ element coordinate system $\mathsf{E}_{b,n}$ , the $7\times1$ vector function g, and their derivatives are derived as explicit functions of the nodal DOFs in Appendix A. The mid­element position vector is simply a linear function of the $_{2\times3}$ displacements of the end­nodes (A.2).
and second nodal derivatives
以及二阶节点导数
$$
\mathbf{r}_{o,b,n,p,q_i,q_j} = \mathbf{E}_{b,n,q_i,q_j} \bar{\mathbf{N}}_{n,p} \mathbf{g} + \mathbf{E}_{b,n,q_j} \bar{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i} \\ + \mathbf{E}_{b,n,q_i} \bar{\mathbf{N}}_{n,p,q_j} \mathbf{g} + \mathbf{E}_{b,n} \bar{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i,q_j} + \delta_{1p} \frac{l_n}{2} \mathbf{e}_{3,b,n,q_i,q_j} \tag{1.58a}
$$
$$
\mathbf{r}_{x,b,n,p,q_i,q_j} = \mathbf{E}_{b,n,q_i,q_j} \mathbf{P}_{x} \tilde{\mathbf{N}}_{n,p} \mathbf{g} + \mathbf{E}_{b,n,q_j} \mathbf{P}_{x} \tilde{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i} \\ + \mathbf{E}_{b,n,q_i} \mathbf{P}_{x} \tilde{\mathbf{N}}_{n,p,q_j} \mathbf{g} + \mathbf{E}_{b,n} \mathbf{P}_{x} \tilde{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i,q_j} + \delta_{0p} \mathbf{e}_{1,b,n,q_i,q_j} \tag{1.58b}
$$
$$
\mathbf{r}_{y,b,n,p,q_i,q_j} = \mathbf{E}_{b,n,q_i,q_j} \mathbf{P}_{y} \tilde{\mathbf{N}}_{n,p} \mathbf{g} + \mathbf{E}_{b,n,q_j} \mathbf{P}_{y} \tilde{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i} \\ + \mathbf{E}_{b,n,q_i} \mathbf{P}_{y} \tilde{\mathbf{N}}_{n,p,q_j} \mathbf{g} + \mathbf{E}_{b,n} \mathbf{P}_{y} \tilde{\mathbf{N}}_{n,p} \mathbf{g}_{,q_i,q_j} + \delta_{0p} \mathbf{e}_{2,b,n,q_i,q_j} \tag{1.58c}
$$
where the element-­index $n$ and substructure-­index $b$ are included again. Note that the coefficient sub­-vectors (1.46) and its first and second derivatives (1.57) and (1.58) must be evaluated at the current nodal DOFs $\mathbf{q}_{b,n}$ of the element. The $\mathsf{3}\mathsf{x}\mathsf{3}$ element coordinate system $\mathsf{E}_{b,n}$ , the $7\times1$ vector function g, and their derivatives are derived as explicit functions of the nodal DOFs in Appendix A. The mid-­element position vector is simply a linear function of the $_{2\times3}$ displacements of the end­-nodes (A.2).
其中再次包含单元索引$n$和子结构索引$b$。请注意，系数子向量(1.46)及其一阶和二阶导数(1.57)和(1.58)必须在单元的当前节点自由度$\mathbf{q}_{b,n}$处进行评估。$\mathsf{3}\mathsf{x}\mathsf{3}$单元坐标系$\mathsf{E}_{b,n}$、$\mathsf{7}\mathsf{x}\mathsf{1}$向量函数g及其导数在附录A中作为节点自由度的显式函数导出。单元中点位置向量是端节点$2\times3$位移(A.2)的简单线性函数。
The substructure object must return the following scalar, vectors, and matrices:
下部结构对象必须返回以下标量、矢量和矩阵：
$M$ Total mass of the substructure which is constant. rcg = nN=e,1b rcg,n The $3\!\times\!1$ position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51a). $\begin{array}{r}{\mathbf{r}_{c g,q_{i}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i}}}\end{array}$ The first nodal derivatives of the 3x1 position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51b). Maximum two elements will contribute to each DOF derivative. $\begin{array}{r}{\mathbf{r}_{c g,q_{i},q_{j}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i},q_{j}}}\end{array}$ The second nodal derivatives of the 3x1 position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51c). Maximum two elements will contribute to each DOF derivative. $\mathbb{M}^{11}$ The local mass matrix of the co­rotational substructure built up by collecting all 12x12 element mass matrices (1.53) on its diagonal. $\mathsf{H}^{111}$ The local nonlinear gyroscopic matrix of the co­rotational substructure built up by collecting all 12x12x12 element nonlinear gyroscopic matrices (1.54) on its diagonal. This local matrix is three­dimensional where the twelve nodal velocities at the ends of each element cause gyroscopic forces on the twelve DOF equations of the same element. $\begin{array}{r}{\mathbf{l}_{b a s e}=\sum_{n=1}^{N_{e,b}}\mathbf{l}_{b a s e,n}}\end{array}$ The $\mathsf{3}\mathsf{x}\mathsf{3}$ rotational inertia matrix where the contributions from each element are given by (1.55).
- $M$ Total mass of the substructure which is constant.
- $\begin{array}{r}{\mathbf{r}_{c g}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,}}\end{array}$ The $3\!\times\!1$ position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51a).
- $\begin{array}{r}{\mathbf{r}_{c g,q_{i}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i}}}\end{array}$ The first nodal derivatives of the 3x1 position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51b). Maximum two elements will contribute to each DOF derivative.
- $\begin{array}{r}{\mathbf{r}_{c g,q_{i},q_{j}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i},q_{j}}}\end{array}$ The second nodal derivatives of the 3x1 position vector for the center of gravity of the substructure where the contributions from each element are given by (1.51c). Maximum two elements will contribute to each DOF derivative.
- $\mathsf{M}^{11}$ The local mass matrix of the co­rotational substructure built up by collecting all 12x12 element mass matrices (1.53) on its diagonal.
- $\mathsf{H}^{111}$ The local nonlinear gyroscopic matrix of the co­rotational substructure built up by collecting all 12x12x12 element nonlinear gyroscopic matrices (1.54) on its diagonal. This local matrix is three­dimensional where the twelve nodal velocities at the ends of each element cause gyroscopic forces on the twelve DOF equations of the same element.
- $\begin{array}{r}{\mathbf{I}_{b a s e}=\sum_{n=1}^{N_{e,b}}\mathbf{I}_{b a s e,n}}\end{array}$ The $\mathsf{3}\mathsf{x}\mathsf{3}$ rotational inertia matrix where the contributions from each element are given by (1.55).
- $M$ 子结构的恒定总质量。
- $\begin{array}{r}{\mathbf{r}_{c g}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,}}\end{array}$ 子结构重心点的 $3\!\times\!1$ 位置向量，其中每个元素的贡献由 (1.51a) 给出。
- $\begin{array}{r}{\mathbf{r}_{c g,q_{i}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i}}}\end{array}$ 子结构重心点 $3\times1$ 位置向量的一阶节点导数，其中每个元素的贡献由 (1.51b) 给出。每个 DOF 导数最多有两个元素贡献。
- $\begin{array}{r}{\mathbf{r}_{c g,q_{i},q_{j}}=\sum_{n=1}^{N_{e,b}}\mathbf{r}_{c g,n,q_{i},q_{j}}}\end{array}$ 子结构重心点 $3\times1$ 位置向量的二阶节点导数，其中每个元素的贡献由 (1.51c) 给出。每个 DOF 导数最多有两个元素贡献。
- $\mathsf{M}^{11}$ 通过收集所有 12x12 单元质量矩阵 (1.53) 在其对角线上构建的共旋转子结构的局部质量矩阵。
- $\mathsf{H}^{111}$ 通过收集所有 12x12x12 单元非线性陀螺矩阵 (1.54) 在其对角线上构建的共旋转子结构的局部非线性陀螺矩阵。该局部矩阵是三维的，其中每个单元两端的十二个节点速度会在同一单元的十二个 DOF 方程上产生陀螺力。
- $\begin{array}{r}{\mathbf{I}_{b a s e}=\sum_{n=1}^{N_{e,b}}\mathbf{I}_{b a s e,n}}\end{array}$ $\mathsf{3}\mathsf{x}\mathsf{3}$ 旋转惯性矩阵，其中每个元素的贡献由 (1.55) 给出。
# Gravity forces
# Elastic stiffness forces