理论力学达郎伯原理15.ppt

1 1 两边对t求导数： Determine the time-derivative: 1 (2) Determine the tensile force of the rope by the angular momentum theorem (differential equation of a fixed-axis rotation). Study the wheel O, form the angular momentum theorem we get (2) 用动量矩定理求绳子拉力 （定轴转动微分方程） 取轮O为研究对象，由动量矩定理得 1 (3) 用质心运动定理求解轴承O处支反力 取轮O为研究对象，根据质心运动定理： (3) Determine the reaction forces on the bearing O by the motion theorem of motion of the center of mass. Investigate the wheel O, from the theorem of motion of the center of mass we get 1 (4) Determine the friction force by the differential equation of plane motion. Investigate cylinder A, from the differential equation of plane motion we gem (4) 用刚体平面运动微分方程求摩擦力 取圆柱体A为研究对象， 根据刚体平面运动微分方程 1 Method 3 determine the angular acceleration of the tub wheel by the theorem of kinetic energy. We can get the reaction forces of the constraints by D’Alembert’s principle (the tensile force of the rope , the reaction forces at the point O on the bearing and and the friction force ). 方法3：用动能定理求鼓轮的角加速度 用达朗伯原理求约束反力（绳子拉力 、轴承O处反 力 和 及摩擦力 ）。 1 P [Example 3] The weight of a homogeneous cylinder is P, its radius is R. It is rolling down from point O along the inclined plane without sliding. At the beginning it does not move. The angle between the plane and the horizon is ?. At OA=S determine the reaction force of the constraint of the board at point O. Neglect the weight of the board. [例3] 均质圆柱体重为P，半径为R，无滑动地沿倾斜平板由静止自O点开始滚动。平板对水平线的倾角为? ，试求OA=S时平板在O点的约束反力。板的重力略去不计。 1 P 解：(1) 用动能定理求速度，加速度 圆柱体作平面运动。在初始位置时，处于静止状态，故T1=0；在末位置时，设角速度为?，则vC = R ?, 动能为： Solution: (1) Determine the velocity and the acceleration by the theorem of kinetic energy: The motion of the cylinder is a plane movement. At the beginning it is at test, at the end the angle velocity is ? and vC = R ? The kinetic energy is. 1 P The work of the positive forces is 主动力的功： From the theorem of kinetic ene