The stability properties of a freely rotating rigid body are governed by the intermediate axis theorem, i.e., rotation around the major and minor principal axes is stable whereas rotation around the intermediate axis is unstable. The stability of the principal axes is of importance for the prediction of rockfall. Current numerical schemes for 3D rockfall simulation, however, are not able to correctly represent these stability properties. In this paper an extended intermediate axis theorem is presented, which not only involves the angular momentum equations but also the orientation of the body, and we prove the theorem using Lyapunov’s direct method. Based on the stability proof, we present a novel scheme which respects the stability properties of a freely rotating body and which can be incorporated in numerical schemes for the simulation of rigid bodies with frictional unilateral constraints. In particular, we show how this scheme is incorporated in an existing 3D rockfall simulation code. Simulations results reveal that the stability properties of rotating rocks play an essential role in the run-out length and lateral spreading of rocks.

自由旋转的刚体的稳定性由中间轴定理控制，即，围绕主轴和次主轴的旋转是稳定的，而围绕中间轴的旋转是不稳定的。主轴的稳定性对于预测落石非常重要。但是，当前用于3D落石模拟的数值方案不能正确表示这些稳定性。本文提出了一个扩展的中间轴定理，该定理不仅涉及角动量方程，而且还涉及物体的方向，并且我们使用Lyapunov的直接方法证明了该定理。根据稳定性证明，我们提出了一种尊重自由旋转体稳定性的新颖方案，可以将其纳入数值方案中，以模拟具有摩擦单边约束的刚体。特别是，我们展示了如何将该方案结合到现有的3D落石模拟代码中。仿真结果表明，旋转岩石的稳定性在岩石的跳动长度和横向扩展中起着至关重要的作用。