为研究末端带载荷的单连杆机械臂振动模态问题, 首先利用材料力学方法, 给出了末端带载荷的单连杆机械臂以积分方程表示的弯矩, 然后将该积分方程转化成形式上与 Euler-Bernoulli方程相同, 但边界条件不同的偏微分方程。通过解微分方程组确定了振型函数。给出了末端带载荷的单连杆机械臂振动固有频率满足的方程, 提出了计算固有频率的迭代方法, 同时也给出了计算固有频率的近似公式。与固有频率精确值的对比结果说明, 除第一个外, 其余固有频率的近似值与精确值的误差不超过 10-4, 且随着频率序数的增加, 近似值与精确值的误差趋于零。

In order to deal with the problem of modal analysis of the single-link manipulator with end-effector payload, an integral equation is used to express the bending moment of the single-link manipulator with end-effector payload by using the method of materials mechanics. Then the integral equation is transformed into a partial differential equation that has the same expression and different boundary conditions as Euler-Bernoulli equation. The mode functions can be obtained by solving the partial differential equation. As the parameter of the mode functions, natural frequency of the single-link manipulator with end-effector payload is expressed by an equation. An iterative method to solve the natural frequency from the equation and an approximate expression of the natural frequency are given. The comparison result of real values and approximate values of the natural frequencies shows that the errors do not exceed 10-4 except the first natural frequency, and the errors are close to zero with the increase of the natural frequency ordinal.