Flexible multibody modeling of reeving systems including transverse vibrations

Abstract

This paper presents a discretization procedure for the flexible multibody modeling of reeving systems. Reeving systems are assumed to include a set of rigid bodies connected by wire ropes using a set of sheaves and reels. The method is capable to model the deformation of the varying-length wire-rope spans. Wire ropes are assumed to deform axially, transversally and in torsion. This paper shows the capability of the presented method to model transverse vibrations. The discretization procedure uses a combination of absolute position coordinates, relative-transverse deformation coordinates and longitudinal material coordinates. Each wire-rope span is modeled using a single two-noded element under an arbitrary Lagrangian–Eulerian approach. The discretization method is validated using analytical and numerical reference solutions found in the literature that describe the dynamics of varying-length strings. In addition, the dynamics of a three-dimensional tower crane is simulated.

Appendix A: Generalized inertia vector based on simplified velocity expression

The generalized momentum is obtained as the partial derivative of the expression of the kinetic energy given in Eq. (16) with respect to the generalized velocities, as follows:

$$ \frac{\partial T}{\partial\dot{\mathbf{q}}} = \left [ \textstyle\begin{array}{c} \mathbf{M}_{a}\dot{\mathbf{q}}_{a} + \mathbf{M}_{am}\dot{\mathbf{q}}_{m} + \mathbf{Q}_{a}\mathbf{q}_{a} \\ \mathbf{M}_{m}\dot{\mathbf{q}}_{m} + \mathbf{M}_{am}^{T}\dot{\mathbf{q}}_{a} + \mathbf{M}_{am}^{T}\mathbf{q}_{a} \\ \mathbf{M}_{as}^{T}\dot{\mathbf{q}}_{a} + \mathbf{Q}_{sa}\dot{\mathbf{q}}_{a} \end{array}\displaystyle \right ]. $$

(56)

The first term in Lagrange equations can be obtained:

$$\begin{aligned} \frac{d}{dt} \biggl( \frac{\partial T}{\partial\dot{\mathbf{q}}} \biggr) =& \left [ \textstyle\begin{array}{c} \mathbf{M}_{a}\ddot{\mathbf{q}}_{a} + \mathbf{M}_{am}\ddot{\mathbf{q}}_{m} \\ \mathbf{M}_{m}\ddot{\mathbf{q}}_{m} + \mathbf{M}_{am}^{T}\ddot{\mathbf{q}}_{a} \\ \mathbf{M}_{as}^{T}\ddot{\mathbf{q}}_{a} \end{array}\displaystyle \right ] + \left [ \textstyle\begin{array}{c} \mathbf{Q}_{a}\dot{\mathbf{q}}_{a} \\ \mathbf{Q}_{am}^{T}\dot{\mathbf{q}}_{a} \\ \mathbf{Q}_{sa}\dot{\mathbf{q}}_{a} \end{array}\displaystyle \right ] \\ &{}+ \left [ \textstyle\begin{array}{c} \frac{\partial ( \mathbf{M}_{a}\dot{\mathbf{q}}_{a} )}{\partial \mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{a}}\dot{\mathbf{q}}_{a} + \mathbf{M}_{as}\ddot{\mathbf{q}}_{s} \\ \frac{\partial ( \mathbf{M}_{m}\dot{\mathbf{q}}_{m} )}{\partial\mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}^{T}\dot{\mathbf{q}}_{a} )}{\partial\mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}^{T}\dot{\mathbf{q}}_{a} )}{\partial\mathbf{q}_{a}}\dot{\mathbf{q}}_{a} + \frac{\partial ( \mathbf{Q}_{am}^{T}\mathbf{q}_{a} )}{\partial\mathbf{q}_{a}}\dot{\mathbf{q}}_{a} + \mathbf{M}_{ms}\ddot{\mathbf{q}}_{s} \\ \frac{\partial ( \mathbf{M}_{as}^{T}\dot{\mathbf{q}}_{a} )}{\partial\mathbf{q}_{a}}\dot{\mathbf{q}}_{a} + \frac{\partial ( \mathbf{Q}_{sa}\mathbf{q}_{a} )}{\partial\mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{Q}_{sa}\mathbf{q}_{a} )}{\partial\mathbf{q}_{a}}\dot{\mathbf{q}}_{a} + \mathbf{M}_{s}\ddot{\mathbf{q}}_{s} \end{array}\displaystyle \right ], \end{aligned}$$

(57)

where the terms on the r.h.s. which are proportional to the generalized accelerations are already accounted for in the generalized inertia force vector $\mathbf{M}\ddot{\mathbf{q}}$. The second term in Lagrange equations can be obtained:

$$ \frac{\partial T}{\partial\mathbf{q}} = \left [ \textstyle\begin{array}{c} \dot{\mathbf{q}}_{a}^{T}\frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{a}} + \mathbf{q}_{a}^{T}\frac{\partial ( \mathbf{Q}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{a}} \\ \mathbf{0} \\ \dot{\mathbf{q}}_{a}^{T} ( \frac{1}{2}\frac{\partial ( \mathbf{M}_{a}\dot{\mathbf{q}}_{a} )}{\partial \mathbf{q}_{s}} + \frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{s}} + \frac{1}{2}\frac{\partial ( \mathbf{P}_{a}\mathbf{q}_{a} )}{\partial\mathbf{q}_{s}} ) + \frac{1}{2}\dot{\mathbf{q}}_{m}^{T}\frac{\partial ( \mathbf{M}_{m}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{s}} \end{array}\displaystyle \right ]. $$

(58)

The generalized quadratic-velocity inertia vector $\mathbf{Q} _{v}$ collects all inertia terms which are not proportional to the generalized accelerations, as follows:

$$ \mathbf{Q}_{v} = \frac{\partial T}{\partial\mathbf{q}} - \biggl( \frac{d}{dt} \biggl( \frac{\partial T}{\partial\dot{\mathbf{q}}} \biggr) - \mathbf{M}\ddot{\mathbf{q}} \biggr). $$

(59)

Substituting Eqs. (58), (57) and (18) into (59) and rearranging yields

$$\begin{aligned} \mathbf{Q}_{v} =& \left [ \textstyle\begin{array}{c} ( \frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{a}} )^{T}\dot{\mathbf{q}}_{a} + ( \frac{\partial ( \mathbf{Q}_{am}\dot{\mathbf{q}}_{m} )}{\partial\mathbf{q}_{a}} )^{T}\mathbf{q}_{a} \\ \mathbf{0} \\ ( \frac{\partial ( \frac{1}{2}\mathbf{M}_{a}\dot{\mathbf{q}}_{a} + \mathbf{M}_{am}\dot{\mathbf{q}}_{m} + \frac{1}{2}\mathbf{P}_{a}\mathbf{q}_{a} )}{\partial \mathbf{q}_{s}} )^{T}\dot{\mathbf{q}}_{a} + \frac{1}{2} ( \frac{\partial\mathbf{M}_{m}\dot{\mathbf{q}}_{m}}{\partial \mathbf{q}_{s}} )^{T}\dot{\mathbf{q}}_{m} \end{array}\displaystyle \right ] \\ &{}+ \left [ \textstyle\begin{array}{c} - ( \mathbf{Q}_{a}\dot{\mathbf{q}}_{a} + \frac{\partial ( \mathbf{M}_{a}\dot{\mathbf{q}}_{a} + \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}\dot{\mathbf{q}}_{m} )}{\partial \mathbf{q}_{a}}\dot{\mathbf{q}}_{a} ) \\ - ( \mathbf{Q}_{am}^{T}\dot{\mathbf{q}}_{a} + \frac{\partial ( \mathbf{M}_{m}\dot{\mathbf{q}}_{m} + \mathbf{M}_{am}^{T}\dot{\mathbf{q}}_{a} )}{\partial \mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{am}^{T}\dot{\mathbf{q}}_{a} + \mathbf{Q}_{am}^{T}\mathbf{q}_{a} )}{\partial \mathbf{q}_{a}}\dot{\mathbf{q}}_{a} ) \\ - ( \mathbf{Q}_{sa}\dot{\mathbf{q}}_{a} + \frac{\partial ( \mathbf{Q}_{sa}\mathbf{q}_{a} )}{\partial \mathbf{q}_{s}}\dot{\mathbf{q}}_{s} + \frac{\partial ( \mathbf{M}_{as}^{T}\dot{\mathbf{q}}_{a} + \mathbf{Q}_{sa}\mathbf{q}_{a} )}{\partial \mathbf{q}_{a}}\dot{\mathbf{q}}_{a} ) \end{array}\displaystyle \right ]. \end{aligned}$$

(60)