A conserving formulation of a simple shear- and torsion-free beam for multibody applications

Abstract

Many engineering fields such as aerospace, robotics, and computer graphics, have applications that contain elements amenable to be modeled as slender beams with negligible shear and torsion effects. The literature contains several energy-momentum (EM) formulations for beams based on a nonlinear finite element approach but, to the best of the author’s knowledge, there are not such developments for the finite segment or lumped approach. This work proposes an energy-conserving and symmetry-preserving extension of one of these models recently proposed in the literature; this extension constitutes the main contribution of the paper. The configuration is described by a rotation-free parameterization consisting in inertial Cartesian coordinates of a collection of nodes that defines a chain of articulated truss members. The axial response is derived by from a nonlinear hyperelastic potential and the bending stiffness is represented by another potential defined on overlapped sets composed of two consecutive trusses. The fact that both effects are defined by discrete potentials has an important impact on the simplicity of the EM formulation. The resulting time-integration scheme produces an approximated solution where total mechanical discrete energy and symmetries are exactly preserved, and the numerical stability is enhanced compared to implicit standard methods. Some numerical experiments illustrate the performance of the presented formulation, including some results of well-established beam models from popular commercial software with standard integration.

Appendices

Appendix A: Orthogonality property I

Given the momentum vector ${\mathbf{p}}$ and the $3\times 3M$ matrix $\mathbb{r}$ defined in (15), the following relation holds:

$$\dot{\mathbb{r}}\mathbf{p}=\mathbf{0}.$$

(27)

In order to prove it, let us expand (27) using (14) as

$$\dot{\mathbb{r}}\mathbf{p}=\sum _{i=1}^N{\dot{\mathbf{r}}}_i\times {\mathbf{p}}_i.$$

(28)

On the other hand, velocity $\dot{\mathbf{q}}=(\dot{\mathbf{r}}_{1} \ \dot{\mathbf{r}}_{2} \ ... \dot{\mathbf{r}}_{N})^{ \mathrm{T}}$ and momentum $\dot{\mathbf{p}}$ vectors are related by the mass matrix as $\dot{\mathbf{q}}={\mathbf{M}}{\mathbf{p}}$. The particular expression of the mass matrix (10) allows us to obtain the relations among nodal velocities and momenta:

$$\begin{aligned} \dot{\mathbf{r}}_{1} &= \frac{1}{3}{\mathbf{p}}_{1} + \frac{1}{6}{\mathbf{p}}_{2} , \end{aligned}$$

(29)

$$\begin{aligned} \dot{\mathbf{r}}_{2} &= \frac{1}{6}{\mathbf{p}}_{1} + \frac{2}{3}{\mathbf{p}}_{2} + \frac{1}{6}{\mathbf{p}}_{3} , \end{aligned}$$

(30)

$$\begin{aligned} ... \\ \dot{\mathbf{r}}_{N-1} &= \frac{1}{6}{\mathbf{p}}_{N-2} + \frac{2}{3}{\mathbf{p}}_{N-1} + \frac{1}{6}{\mathbf{p}}_{N} , \end{aligned}$$

(31)

$$\begin{aligned} \dot{\mathbf{r}}_{N} &= \frac{1}{6}{\mathbf{p}}_{N-1} + \frac{1}{3}{\mathbf{p}}_{N}. \end{aligned}$$

(32)

Finally, observe from (28) and (29)–(32) that the terms involving ${\mathbf{p}}_{i} \times {\mathbf{p}}_{j}$ naturally vanish for $i=j$ and cancel within the sum for $i\neq j$, proving Eq. (27). Note that this result stems from the particular structure in diagonal blocks of the mass matrix (10).

Appendix B: Orthogonality property II

Given the approximated momentum vector ${\mathbf{p}}_{n}$ at $t_{n}$ and the $3\times 3M$ matrix $\mathbb{p}$ defined in Sect. 3, the following relation holds:

$${\mathbb{p}}_n{\mathbf{M}}^{-1}{\mathbf{p}}_n=\mathbf{0}.$$

(33)

This result can be readily proved observing that the inverse of a matrix with a diagonal block structure such as the mass matrix (10) is another matrix with identical structure, such as

$$ {\mathbf{M}}^{-1} = \left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} m_{11}{\mathbf{1}} & m_{12}{\mathbf{1}} & \ldots & m_{1N}{\mathbf{1}} \\ m_{21}{\mathbf{1}} & m_{22}{\mathbf{1}} & \ldots & m_{2N}{\mathbf{1}} \\ ... & \ldots & \ldots & ... \\ m_{N1}{\mathbf{1}} & m_{N2}{\mathbf{1}} & \ldots & m_{NN}{\mathbf{1}} \\ \end{array}\displaystyle \right ) $$

(34)

with coefficients $m_{ij}=m_{ji}$. Dropping in (33) the subscript $n$ in order to simplify the notation, the following sum is obtained:

$$\mathbb{p}{\mathbf{M}}^{-1}\mathbf{p}={\mathbf{p}}_1\times (m_{11}{\mathbf{p}}_1+\cdots +m_{1N}{\mathbf{p}}_N)+\cdots +{\mathbf{p}}_N\times (m_{N1}{\mathbf{p}}_1+\cdots +m_{NN}{\mathbf{p}}_N),$$

which is null, since terms involving ${\mathbf{p}}_{i} \times {\mathbf{p}}_{j}$ naturally vanish for $i=j$ and cancel within the sum for $i\neq j$, proving Eq. (33).