18.1 Intro by example: spring elements
\[\begin{align} u_1 \; &= 0 \rightarrow F_2 = k \cdot u_2 \\ & \Longrightarrow F_1 = -F_2 = -k \cdot u_2 \\ u_2 \; &= 0 \rightarrow F_1 = k \cdot u_1 \\ & \Longrightarrow F_2 = -F_1 = -k \cdot u_1 \\ \end{align}\]
If we apply both displacements: \[\begin{align} F_1 \; &= k \cdot u_1 - k \cdot u_2 \\ F_2 \; &= k \cdot u_2 - k \cdot u_1 \end{align}\]
\[\begin{align} \left\{ \begin{array}{c} F_1 \\ F_2 \end{array} \right\} = \left[ \begin{array}{cc} k & -k \\ -k & k \end{array} \right] \cdot \left\{ \begin{array}{c} u_1 \\ u_2 \end{array} \right\} \\ \end{align}\]
or: \[\{F\} = [K] \{u\}\]
18.1.1 Consider adding a second spring to the system
\[\begin{align} \left\{ \begin{array}{c} F_1 \\ F_2 \end{array} \right\} = \left[ \begin{array}{cc} k_1 & -k_1 \\ -k_1 & k_1 \end{array} \right] \cdot \left\{ \begin{array}{c} u_1 \\ u_2 \end{array} \right\} \end{align}\]
\[\begin{align} \left\{ \begin{array}{c} F_2 \\ F_3 \end{array} \right\} = \left[ \begin{array}{cc} k_2 & -k_2 \\ -k_2 & k_2 \end{array} \right] \cdot \left\{ \begin{array}{c} u_2 \\ u_3 \end{array} \right\} \end{align}\]
\[\begin{align} \left\{ \begin{array}{c} F_1 \\ F_2 \\ F_3 \\ \end{array} \right\} = \left[ \begin{array}{ccc} k_1 & -k_1 & 0 \\ -k_1 & k_1+k_2 & -k_2 \\ 0 & -k_2 & k_2 \\ \end{array} \right] \cdot \left\{ \begin{array}{c} u_1 \\ u_2 \\ u_3 \\ \end{array} \right\} \end{align}\]
Once the element equation is known – assemblies can be done.
18.1.2 Note: this process can also be accomplished through other methods such a through variational principles
\[U = \frac{1}{2} k_1 \left(q_1-q_2\right)^2 +\frac{1}{2} k_2 \left(q_2-q_3\right)^2\] \[W = F_1 u_1 + F_2 u_2 + F_3 u_3\]
\(\Pi = U - W\)
\(\Pi = \frac{1}{2} k_{12} (q_1-q_2)^2 + \frac{1}{2} k_{23} (q_2-q_3)^2 -F_1 q_1-F_2 q_2-F_3 q_3\)
Maxima
Recall \[\frac{\partial \Pi}{\partial q_1} \delta q_1 + \frac{\partial \Pi}{\partial q_2} \delta q_2 + \frac{\partial \Pi}{\partial q_3} \delta q_3 = 0\]
When you consider that \(\delta q_1\), \(\delta q_2\), and \(\delta q_3\) are arbitrary, this yields the exact same set of three equations.