1、附录英文文献及翻译 The Two-Dimensional Dynamic Behavior of Conveyor Belts 3.1.1 NON LINEAR TRUSS ELEMENT If only the longitudinal deformation of the belt is of interest then a truss element can be used to model the elastic response of the belt. A truss element as shown in Figure 2 has two nodal points, p and
2、 q, and four displacement parameters which determine the component vector x: xT = up vpuqvq (1) For the in-plane motion of the truss element there are three independent rigid body motions therefore one deformation parameter remains which describes Figure 2: Definition of the displacements of a truss
3、 element the change of length of the axis of the truss element 7: 1 = D1(x) = o ds - dso d (2) 2dso wheredso is the length of the undeformed element, ds the length of the deformed element and a dimensionless length coordinate along the axis of the element. Figure 3: Static sag of a tensioned belt Al
4、though bending, deformations are not included in the truss element, it is possible to take the static influence of small values of the belt sag into account. The static belt sag ratio is defined by (see Figure 3): K1 = /1 = q1/8T (3) where q is the distributed vertical load exerted on the belt by th
5、e weight of the belt and the bulk material, 1 the idler space and T the belt tension. The effect of the belt sag on the longitudinal deformation is determined by 7: s = 8/3 Ks (4) which yields the total longitudinal deformation of the non linear truss element: 3.1.2 BEAM ELEMENT Figure 4: Definition
6、 of the nodal point displacements and rotations of a beam element. If the transverse displacement of the belt is being of interest then the belt can be modelled by a beam element. Also for the in-plane motion of a beam element, which has six displacement parameters, there are three independent rigid
7、 body motions. Therefore three deformation parameters remain: the longitudinal deformation parameter, 1, and two bending deformation parameters, 2 and 3. Figure 5: The bending deformations of a beam element The bending deformation parameters of the beam element can be defined with the component vect
8、or of the beam element (see Figure 4): xT = up vp p uqvq q (5) and the deformed configuration as shown in Figure 5: 2 = D2(x) = e2p1pq (6) 1o 3 = D3(x) = -eq21pq 1o 3.2 THE MOVEMENT OF THE BELT OVER IDLERS AND PULLEYS The movement of a belt is constrained when it moves over an idler or a pulley. In
9、order to account for these constraints, constraint (boundary) conditions have to be added to the finite element description of the belt. This can be done by using multi-body dynamics. The classic description of the dynamics of multi-body mechanisms is developed for rigid bodies or rigid links which
10、are connected by several constraint conditions. In a finite element description of a (deformable) conveyor belt, where the belt is discretised in a number of finite elements, the links between the elements are deformable. The finite elements are connected by nodal points and therefore share displace
11、ment parameters. To determine the movement of the belt, the rigid body modes are eliminated from the deformation modes. If a belt moves over an idler then the length coordinate , which determines the position of the belt on the idler, see Figure 6, is added to the component vector, e.g. (6), thus re
12、sulting in a vector of seven displacement parameters. Figure 6: Belt supported by an idler. There are two independent rigid body motions for an in-plane supported beam element therefore five deformation parameters remain. Three of them, 1, 2 and 3, determine the deformation of the belt and are already given in 3.1. The remaining two, 4 and 5, determine the interaction between the belt and the idler, see Figure 7. Figure 7: FEM beam element with two constraint conditions. These deformation parameters can be imagined as springs of infinite stiffness. This implies that:
