Gibbs–Appell method-based governing equations for one-dimensional finite elements used in flexible multibody systems

Abstract

Lagrange’s equations represent the common approach in finite element analysis of an elastic multibody system. The most important step in this case is to write the governing equations. The work develops an alternative method to obtain these equations, using so-called Gibbs–Appell formalism. The advantage of this method is the decrease in the number of calculations to be made. The acceleration energy will be calculated first for a one-dimensional finite element, and then Gibbs–Appell equations are applied in the classical form. The number of differentiations required, compared to the method of Lagrange’s equations, decreases significantly, with effects on the computational time required to solve such a problem. We can assume that, due to its simplicity, this method will determine the interest of researchers in the case of large industrial applications.

Appendix

The coefficients $r_{ij}$ define the position of the unit vector of the local coordinate system Oxyz. The orthogonality condition leads to:

$$\begin{aligned} r_{ij} r_{kj} =r_{jk} r_{ji} =\bar{{\delta }}_{ij}, \quad i,j,k=1,2,3 \end{aligned}$$

(A.1)

where $\bar{{\delta }}_{ij}$ is the Kronecker delta. If we differentiate this equation, it will result:

$$\begin{aligned} \dot{{r}}_{ij} r_{kj} +r_{ij} \dot{{r}}_{kj} =0,\quad i,j,k=1,2,3. \end{aligned}$$

(A.2)

Denote:

$$\begin{aligned} \omega _{ik} =\dot{{r}}_{ij} r_{kj} . \end{aligned}$$

(A.3)

The relation (20) becomes:

$$\begin{aligned} \omega _{ik} +\omega _{ki} =0. \end{aligned}$$

(A.4)

The skew-symmetric tensor $\omega _{ik}$ is the operator angular velocity (with its components express in the global reference system). To this corresponds the angular velocity vector defined by:

$$\begin{aligned} \omega _{1} =\omega _{32} =-\omega _{23};\quad \omega _{2} =\omega _{13} =-\omega _{31};\quad \omega _{3} =\omega _{21} =-\omega _{12} . \end{aligned}$$

(A.5)

The angular acceleration skew symmetric operator is:

$$\begin{aligned} \varepsilon _{ik} =\dot{{\omega }}_{ik} =\ddot{{r}}_{ij} r_{kj} +\dot{{r}}_{ij} \dot{{r}}_{kj}. \end{aligned}$$

(A.6)

The angular acceleration vector defined by:

$$\begin{aligned} \varepsilon _{1} =\varepsilon _{32} =-\varepsilon _{23};\quad \varepsilon _{2} =\varepsilon _{13} =-\varepsilon _{31};\quad \varepsilon _{3} =\varepsilon _{21} =-\varepsilon _{12}. \end{aligned}$$

(A.7)

We shall have:

$$\begin{aligned} \varepsilon _{ik} =\dot{{\omega }}_{ik} =\ddot{{r}}_{ij} r_{kj} +\dot{{r}}_{ij} \dot{{r}}_{kj} =\ddot{{r}}_{ij} r_{kj} +\dot{{r}}_{ij} r_{jl} r_{ml} \dot{{r}}_{km} =\ddot{{r}}_{ij} r_{kj} -\omega _{il} \omega _{lk}, \end{aligned}$$

(A.8)

from where:

$$\begin{aligned} \ddot{{r}}_{ij} r_{kj} =\varepsilon _{ik} +\omega _{il} \omega _{lk}, \end{aligned}$$

(A.9)

result used in the following calculus.