摩擦作用下直杆在对称斜面间的稳定平衡分析

Analyses on the Stable Equilibrium of a Pole Supported on Symmetrical Plates with Friction

摩擦提供的约束反力在由正应力所确定的范围之内.对称平板支承下的直杆在水平状态时平衡,但重心最高,因而没有摩擦时不能稳定平衡;摩擦作用下直杆平衡的倾角范围与平板坡度相关,可依据摩擦力达到极限值确定;若直杆与平板夹角小于90o,则平衡是稳定的,否则将跳跃到一侧平板,且摩擦角小于板面倾角时下滑到底部.支承板倾角变化时,处于稳定平衡状态的直杆将在一端发生滑移:倾角增大则高端向上、减小则低端向下,以使直杆重心位置较低.若支承板倾角作周期性变化,则直杆将逐渐倾斜而失稳:下滑到底部或跳跃到一侧平板.摩擦使力学系统具有非线性特征,其平衡位置可以在状态参数连续变化时发生跳跃.

Friction provides a resistant force within a range depending on the normal force. A pole supported on symmetrical plates is balanced at horizontal state, but is unstable without friction as the center of gravity is high. The equilibrium range of the inclination angle of the pole with friction is related to the slope of the supporting plates, and is determined from the limit of friction. The equilibrium is stable if the angle between the pole and the plates is less than 90°; otherwise the pole will jump to one plate, and slip to the bottom when friction angle is less than the inclination angle of the plate. When the slope of plates is changed, one end of the pole at the stable equilibrium will move： the upper end goes up when the slope become larger, and vice versa. That results in a low center of gravity of the pole. In addition, under the periodic change of the plate slope, the pole will lean gradually and lose the equilibrium finally： jump to one plate or slip to the bottom. Friction induces nonlinear characteristics to a mechanical system, of which the equilibrium may jump when the state parameters change continuously.