Geometrically nonlinear spring

In this section we describe the variables and unconstrained governing equations for a two-node spring that is constitutively linear and geometrically nonlinear. The spring input properties are is spring constant \(k\) and its unstretched length \(L\). The spring element has reference position defined by its endpoint positions,

\[\begin{split}\begin{aligned} \underline{q}^\mathrm{r} = \begin{bmatrix} \underline{x}_1^\mathrm{r} \\ \underline{x}_2^\mathrm{r} \end{bmatrix} \end{aligned}\end{split}\]

where \(\underline{q}^\mathrm{r} \in \mathbb{R}^6\), and displacement is denoted

\[\begin{split}\begin{aligned} \underline{q} = \begin{bmatrix} \underline{u}_1 \\ \underline{u}_2 \end{bmatrix} \end{aligned}\end{split}\]

The spring internal force contribution to Eq. (2) is

\[\begin{split}\underline{g} = \begin{bmatrix} \underline{f}^\mathrm{sp} \\ -\underline{f}^\mathrm{sp} \end{bmatrix}\end{split}\]

where \(\underline{g} \in \mathbb{R}^6\) and

\[\underline{f}^\mathrm{sp} = -k \frac{\underline{r} }{| \underline{r} |} \left( | \underline{r} | - L \right)\]

with

\[\underline{r} = \underline{x}_2^\mathrm{r} + \underline{u}_2 - \underline{x}_1^\mathrm{r} - \underline{u}_1\]

Variation of the force equation provides the stiffness contribution to the generalized-\(\alpha\) iteration matrix (See Eq. (4)):

\[\begin{split}\underline{\underline{K}} = \begin{bmatrix} \underline{\underline{A}} & -\underline{\underline{A}} \\ - \underline{\underline{A}} & \underline{\underline{A}} \end{bmatrix}\end{split}\]

and

\[\underline{\underline{A}} = k \left( \frac{L}{|\underline{r} |} - 1\right) \underline{\underline{I}} - \frac{k L}{|\underline{r}|^3}\left( \widetilde{r} \widetilde{r} + \underline{\underline{I}} (\underline{r}^T \underline{r} ) \right)\]