Derive the equation (9.3.13) of the book chapter starting from equation (9.3.3).
Figure 4 shows the notation used to represent the motion of the master and slave nodes and the directors at the nodes.
Figure 4. Notation for motion of the continuum-based beam.
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Since the fibers remain straight and do not change length, we have
![{\displaystyle {\text{(5)}}\qquad {\begin{aligned}\mathbf {x} _{i-}(t)&=\mathbf {x} _{i}(t)-{\frac {1}{2}}h_{i}^{0}~\mathbf {p} _{i}(t)\\\mathbf {x} _{i+}(t)&=\mathbf {x} _{i}(t)+{\frac {1}{2}}h_{i}^{0}~\mathbf {p} _{i}(t)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69dce24f13731b18b2bdb364dd959fb411ad4e6d)
where
is the location of master node
,
is the unit director vector at master node
, and
is the initial thickness of the beam.
Taking the material time derivatives of equations (5), we get
![{\displaystyle {\text{(6)}}\qquad {\begin{aligned}\mathbf {v} _{i-}(t)&=\mathbf {v} _{i}(t)-{\frac {1}{2}}{\frac {\partial }{\partial t}}[h_{i}^{0}~\mathbf {p} _{i}(t)]=\mathbf {v} _{i}(t)-{\frac {1}{2}}h_{i}^{0}~{\boldsymbol {\omega }}_{i}(t)\times \mathbf {p} _{i}(t)\\\mathbf {v} _{i+}(t)&=\mathbf {v} _{i}(t)+{\frac {1}{2}}{\frac {\partial }{\partial t}}[h_{i}^{0}~\mathbf {p} _{i}(t)]=\mathbf {v} _{i}(t)+{\frac {1}{2}}h_{i}^{0}~{\boldsymbol {\omega }}_{i}(t)\times \mathbf {p} _{i}(t)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85d850dd4f5d92845d6d85c8dd29d4b63604e66d)
where
is the angular velocity of the director.
From equations (5) we have
![{\displaystyle {\text{(7)}}\qquad {\begin{aligned}\mathbf {p} _{i}(t)&=-{\cfrac {2}{h_{i}^{0}}}[\mathbf {x} _{i-}(t)-\mathbf {x} _{i}(t)]=-{\cfrac {2}{h_{i}^{0}}}\left[(x_{i-}-x_{i})~\mathbf {e} _{x}+(y_{i-}-y_{i})~\mathbf {e} _{y}\right]\\\mathbf {p} _{i}(t)&={\cfrac {2}{h_{i}^{0}}}[\mathbf {x} _{i+}(t)-\mathbf {x} _{i}(t)]={\cfrac {2}{h_{i}^{0}}}\left[(x_{i+}-x_{i})~\mathbf {e} _{x}+(y_{i+}-y_{i})~\mathbf {e} _{y}\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f79cedbf5c68b476d9b9dd583ac89a76b94b812b)
where
is the global basis.
In terms of the global basis, the angular velocity is given by
![{\displaystyle {\boldsymbol {\omega }}_{i}=\omega _{i}~\mathbf {e} _{z}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a58e1f71adc4d5774897dabe5919f26249d28c)
Therefore,
![{\displaystyle {\text{(8)}}\qquad {\begin{aligned}{\boldsymbol {\omega }}_{i}\times \mathbf {p} _{i}&=-{\cfrac {2}{h_{i}^{0}}}{\begin{vmatrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\0&0&\omega _{i}\\x_{i-}-x_{i}&y_{i-}-y_{i}&0\end{vmatrix}}=-{\cfrac {2}{h_{i}^{0}}}\left[-\omega _{i}(y_{i-}-y_{i})~\mathbf {e} _{x}+\omega _{i}(x_{i-}-x_{i})~\mathbf {e} _{y}\right]\\{\boldsymbol {\omega }}_{i}\times \mathbf {p} _{i}&={\cfrac {2}{h_{i}^{0}}}{\begin{vmatrix}\mathbf {e} _{x}&\mathbf {e} _{y}&\mathbf {e} _{z}\\0&0&\omega _{i}\\x_{i+}-x_{i}&y_{i+}-y_{i}&0\end{vmatrix}}={\cfrac {2}{h_{i}^{0}}}\left[-\omega _{i}(y_{i+}-y_{i})~\mathbf {e} _{x}+\omega _{i}(x_{i+}-x_{i})~\mathbf {e} _{y}\right]~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750e146fb4ff61d3066b129cbc2e02f895f742ae)
Substituting equation (8) into equations (6), we get
![{\displaystyle {\text{(9)}}\qquad {\begin{aligned}\mathbf {v} _{i-}&=\mathbf {v} _{i}-\omega _{i}(y_{i-}-y_{i})~\mathbf {e} _{x}+\omega _{i}(x_{i-}-x_{i})~\mathbf {e} _{y}\\\mathbf {v} _{i+}&=\mathbf {v} _{i}-\omega _{i}(y_{i+}-y_{i})~\mathbf {e} _{x}+\omega _{i}(x_{i+}-x_{i})~\mathbf {e} _{y}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2851df38b09f38254e8be3b122f4c11fb773361f)
Let the velocity vectors be expressed in terms of the global basis as
![{\displaystyle {\begin{aligned}\mathbf {v} _{i}&=v_{i}^{x}~\mathbf {e} _{x}+v_{i}^{y}~\mathbf {e} _{y}\\\mathbf {v} _{i-}&=v_{i-}^{x}~\mathbf {e} _{x}+v_{i-}^{y}~\mathbf {e} _{y}\\\mathbf {v} _{i+}&=v_{i+}^{x}~\mathbf {e} _{x}+v_{i+}^{y}~\mathbf {e} _{y}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697fb5768eee0f7dcecc48254e3543ba3d35a148)
Then equations (9) can be written as
![{\displaystyle {\text{(10)}}\qquad {\begin{aligned}v_{i-}^{x}~\mathbf {e} _{x}+v_{i-}^{y}~\mathbf {e} _{y}&=[v_{i}^{x}-\omega _{i}(y_{i-}-y_{i})]~\mathbf {e} _{x}+[v_{i}^{y}+\omega _{i}(x_{i-}-x_{i})]~\mathbf {e} _{y}\\v_{i+}^{x}~\mathbf {e} _{x}+v_{i+}^{y}~\mathbf {e} _{y}&=[v_{i}^{x}-\omega _{i}(y_{i+}-y_{i})]~\mathbf {e} _{x}+[v_{i}^{y}+\omega _{i}(x_{i+}-x_{i})]~\mathbf {e} _{y}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73093a59c625664648c39745c461bdf81e8f879d)
Therefore, the components of the velocity vectors are
![{\displaystyle {\text{(11)}}\qquad {\begin{aligned}v_{i-}^{x}&=v_{i}^{x}-\omega _{i}(y_{i-}-y_{i})\\v_{i-}^{y}&=v_{i}^{y}+\omega _{i}(x_{i-}-x_{i})\\v_{i+}^{x}&=v_{i}^{x}-\omega _{i}(y_{i+}-y_{i})\\v_{i+}^{y}&=v_{i}^{y}+\omega _{i}(x_{i+}-x_{i})~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76313b1af233f3b7b663ca11b49c566a66c265a9)
Then, the matrix form of equations (11) is
![{\displaystyle {\begin{bmatrix}v_{i-}^{x}\\v_{i-}^{y}\\v_{i+}^{x}\\v_{i+}^{y}\end{bmatrix}}={\begin{bmatrix}1&0&-(y_{i-}-y_{i})\\0&1&(x_{i-}-x_{i})\\1&0&-(y_{i+}-y_{i})\\0&1&(x_{i+}-x_{i})\end{bmatrix}}{\begin{bmatrix}v_{i}^{x}\\v_{i}^{y}\\\omega _{i}\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48fb05ee11a7eb65e55673964f210c223beaaf65)
or
![{\displaystyle {\begin{bmatrix}\mathbf {v} _{i-}\\\mathbf {v} _{i+}\end{bmatrix}}^{\text{slave}}=\mathbf {T} _{i}~{\dot {\mathbf {d} }}_{i}^{\text{master}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7336bd347101fc36223697b9cbb7f39b9122eea3)