User:EML4500.f08.delta 6.ramirez/10808

Note: 18-2 continued.

Goal: Find $$ \tilde{T}_{4x4}^{(e)} $$ that transforms the set of local element d.o.f.'s $$ d^{(e)} $$ to another set of local element d.o.f.'s $$ \tilde{d}_{4x1}^{(e)} $$ such that $$ \tilde{T}^{(e)} $$ is invertible.



$$ \tilde{d}_{4x1}^{(e)} = \tilde{T}_{4x4}^{(e)} d_{4x1}^{(e)}$$

$$ (1) \hat{d}_{1}^{(e)} = \begin{bmatrix} l^{(e)}\ m^{(e)}\

\end{bmatrix}

\begin{Bmatrix}d_1^{(e)} \\ d_{2}^{(e)} \end{Bmatrix} $$

Where,

$$ \hat{d}_{1}^{(e)} = q_1^{(e)} $$

$$\hat{d}_{2}^{(e)} = \vec{d}_1^{(e)} \cdot \tilde{j}$$

Note: $$\hat{d}_{2}^{(e)}$$ is the component of $$\vec{d}_1^{(e)}$$ along $$\tilde{j}$$, i.e., $$\tilde{y}$$ axis.

Thus,

$$\tilde{d}_2^{(e)} = -\sin \theta^{(e)}d_1^{(e)} + \cos \theta^{(e)} d_2^{(e)}$$

Equation (2) is as follows:

$$(2) \hat{d}_{2}^{(e)} = \begin{bmatrix} -m^{(e)}\ l^{(e)}\

\end{bmatrix}

\begin{Bmatrix}d_1^{(e)} \\ d_{2}^{(e)} \end{Bmatrix}$$

Next, put (1) & (2) into matrix form,

$$\begin{Bmatrix} \tilde{d}_1^{(e)} \\ \tilde{d}_2^{(e)} \end{Bmatrix} = \begin{bmatrix} l^{(e)} & m^{(e)} \\ -m^{(e)} & -l^{(e)} \end{bmatrix} \begin{Bmatrix} d_1^{(e)} \\ d_2^{(e)} \end{Bmatrix}$$

Where,

$$R^{(e)} = \begin{bmatrix} l^{(e)} & m^{(e)} \\ -m^{(e)} & -l^{(e)} \end{bmatrix}$$

Thus,

$$\begin{Bmatrix} \tilde{d}_1^{(e)}\\ \tilde{d}_2^{(e)}\\ \tilde{d}_3^{(e)}\\ \tilde{d}_4^{(e)} \end{Bmatrix} = \begin{bmatrix} R_{2x2}^{(e)} & 0_{2x2}\\ 0_{2x2} & R_{2x2}^{(e)} \end{bmatrix} \begin{Bmatrix} d_1^{(e)}\\ d_2^{(e)}\\ d_3^{(e)}\\ d_4^{(e)} \end{Bmatrix}$$

which represents,

$$\tilde{d}_{4x1}^{(e)} = \tilde{T}_{4x4}^{(e)} d_{4x1}^{(e)}$$

Next, rotate the element achieving the following



$$\tilde{f}^{(e)} = \begin{bmatrix} 1 & 0 & -1 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \hat{d}^{(e)}$$

Note: all zero values are zero because transverse force does not stretch the spring.

Next, we can conclude:

$$\tilde{f}_{4x1}^{(e)} = \tilde{k}_{4x4}^{(e)} \tilde{d}_{4x1}^{(e)}$$

Note: this is the stopping point of transforming the FD relation back to original (x,y) coordinate system.