Multibody System Dynamics

, Volume 26, Issue 1, pp 91–106

# A modeling of sliding joint on one-dimensional flexible medium

• Difeng Hong
• Gexue Ren
Article

## Abstract

The dynamic modeling of a sliding joint on a one-dimensional medium, such as a cable or a beam, is studied in this paper. The sliding joint is implemented by positioning it at a moving node on the one-dimensional medium, which is realized by variable-length elements at either side of the joint. The variable-length element is established with an absolute nodal coordinate formulation (ANCF) in the framework of the Arbitrary Lagrange–Euler (ALE) description. The sliding of the joint is described by the increasing of the length on one side of the one-dimensional medium and a corresponding decreasing of the other side. In order to capture the discontinuity of the slopes at the position of the sliding joint, the moving node has two slopes as generalized coordinates which are equal to each other in the case of a beam but not in the case of a cable, and in order to avoid the addition–deletion constraint, the node adjacent to the moving node is added or deleted if the element is too long or too short. The governing equations for the coupled system are derived in terms of D’Alembert’s principle and the resulting equations of motion are formulated in the standard form of differential algebraic equations of multibody systems. Numerical examples are presented to validate the method proposed by comparing with analytical results which are available or are made possible by simplifying the model.

## Keywords

One-dimensional medium Sliding joint Absolute nodal coordinate formulation Arbitrary-Lagrange–Euler Length varying

## Nomenclature

A

area of cross section of the one-dimensional medium

E

Young’s module of the one-dimensional medium

ρ

density of the one-dimensional medium

J

inertia moment of the cross section of the one-dimensional medium

$$\mathbf{r}_{i}, \mathbf{r}_{i}'$$

position and slope vector at the ith node

r,r

position and slope vector at an arbitrary point

q

generalized coordinate of the element of mass-flowing one-dimensional medium

pi

material coordinate at the ith node which equals the un-deformed length from the head point of the one-dimensional medium to the ith node of the element

p

material coordinate at an arbitrary point which equals the un-deformed length from the head point of the one-dimensional medium to the arbitrary point of the element

N

shape function of the element of the one-dimensional medium

t

time

$$\dot{\square}, \ddot{\square}$$

velocity and acceleration of the parameter

δ

variation of the parameter

ε0

Green’s normal strain at an arbitrary point of the element of the one-dimensional medium

κ

curvature at an arbitrary point of the element of the one-dimensional medium

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