一种模拟冲击荷载作用下非均匀Timoshenko梁弯曲波研究的一种数值方法

为研究冲击荷载作用下非均匀Timoshenko梁弯曲波问题，对考虑剪切变形和转动惯性的Timoshenko梁进行空间离散构造出控制体，并在时间域上交替运用控制体平动方程和剪力-剪应变关系式、交替运用控制体转动方程和弯矩-曲率关系式，通过转角关系式建立平动和转动问题之间的联系，给出研究非均匀Timoshenko梁弯曲波传播的数值方法。通过两端无约束均匀圆形截面梁弯曲波的传播问题为例，将本文方法与有限差分法进行对比验证了本文方法数值计算的准确性和有效性；冲击荷载作用下，针对几何不均匀的阶梯形悬臂梁弯曲波传播问题进行研究，结果表明：梁中控制体的转动惯性不可忽略，整体变形中的剪切变形部分不能忽略；危险截面不发生在悬臂梁的固定端处，也不在截面的变化处。本文方法适合于模拟冲击荷载作用下几何和材料非均匀Timoshenko梁中弯曲波的传播问题，是一种从波动理论角度研究冲击荷载作用下Timoshenko梁动力响应问题的有力工具。

A numerical method for flexural wave propagation in non-uniform Timoshenko under impact loading

A numerical method is presented for simulating flexural wave propagation in elastic non-uniform Timoshenko beam under impact loading. The controlled volumes are constructed among the discrete elements in the beam. By applying the deflection equations of motion of control volumes and the transverse shear force-strain relationship alternately, the rotation equations of motion of control volumes and the bending moment-curvature relationship alternately, and considering the relationship of deflection and rotation, the numerical algorithm is established. The accuracy and the effectiveness of the numerical method are tested against the existing results. Numerical results demonstrate that the rotational inertia and the shear deformation in the total deformation should not be omitted when the beam is subjected to an impact loading. The dangerous cross-section occurs neither at the end nor at the discontinue cross-section of the beam under the action of impacting loading. The method provide a new practicable tool for analyzing dynamic response of elastic Timoshenko beam subjected to impact loading.