两端任意约束的弹性支撑梁在移动荷载下的动力响应

Dynamic response of elastic supported beams with arbitrary constraints at both ends under moving loads

摘要:

为研究任意边界条件连续梁的动力响应问题, 建立了两端任意约束中间弹性支撑梁的力学模型, 给出了求解其振动频率的方法, 并采用振型叠加法推导了其在移动荷载作用下的动力响应的理论解。通过编写相应的MATLAB计算程序进行求解, 结合算例用所提出的理论方法得到的结果与有限元方法进行对比, 验证了方法的正确性与精度以及适用范围。分析了边界条件对结构振动响应的影响, 结果表明:两端任意约束中间弹性支撑连续梁的高阶模态与两端简支或固定支撑的连续梁明显不同, 由于边界条件为弹性, 其高阶模态曲线边界处的幅值反而更大；在移动荷载作用下, 梁体向下动位移的最大值出现在跨中, 而向上动位移的最大值出现在梁端, 同时梁体在各个截面处均交替出现了正弯矩和负弯矩, 跨中位置处的正弯矩和端点处的负弯矩最为显著；当荷载的移动速度增大时, 跨中位移的波动周期也越长；对比不同的边界条件, 两端弹性支撑的跨中挠度变化幅值介于两端固支和两端简支之间, 两端线弹簧刚度主要影响结构的振动幅度, 而转动弹簧刚度则对结构的最大振动位移影响更为明显。

Abstract:

In order to study the dynamic response of continuous beams with arbitrary boundary conditions, a mechanical model of intermediate elastic supported beams with arbitrary constraints at both ends was established, and the method of solving its vibration frequency was given. The theoretical solution to the dynamic response of the beam under moving load was derived by means of mode superposition method. A MATLAB program was written to solve the equation, and the results obtained by the proposed method were compared with the values obtained by the finite element method through calculation example, which verified the correctness, precision, and application scope of the proposed method. The influence of the boundary conditions on the vibration response of the structure was analyzed. Results show that the higher order modes of the continuous beam with intermediate elastic support under arbitrary constraints at both ends were obviously different from those of the continuous beam with simple or fixed supports at both ends. The amplitude at the boundary of the higher order mode curve was larger because of the elastic boundary condition. Under the action of moving load, the maximum downward displacement of the beam appeared in the middle of the span, while the maximum upward displacement appeared in the ends of the beam. Positive bending moment and negative bending moment alternately appeared in each section of the beam body, and the positive bending moment in the middle of the span and the negative bending moment at the end points were the most significant. The fluctuation period of the mid-span displacement became longer when the moving velocity of the load increased. Compared with different boundary conditions, the variation amplitude of mid-span deflection of elastic support at both ends was between fixed support at both ends and simple support at both ends. The stiffness of rotational spring at both ends affected the vibration amplitude of the structure, while the stiffness of rotational spring had a more obvious effect on the maximum vibration displacement of the structure.