Frequency domain analysis for maximum displacement response of bridges excited by single moving load
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摘要: 为了更加有效地建立列车运营速度与桥梁最大位移响应之间的关系, 针对单个移动荷载激励下桥梁最大位移响应提出了一种频域分析方法; 采用傅里叶变换推导出单个移动荷载匀速通过梁桥时的移动荷载谱和桥梁振动位移响应谱, 通过分析移动荷载幅值谱获得了导致桥梁自由振动位移出现极值响应的移动荷载速度, 并提出了该移动荷载速度的计算公式; 以一简支梁为例, 通过与相关文献结果的对比, 验证了本文数值计算程序的正确性, 进一步基于该程序, 通过数值分析验证了频域分析方法理论推导的正确性和移动荷载速度计算公式的准确性。研究结果表明: 在频域内得到的移动荷载幅值谱与时域内得到的桥梁自由振动幅值响应规律一致, 因此, 移动荷载幅值谱能有效反映桥梁自由振动位移响应; 导致桥梁发生自由振动最大位移响应的移动荷载速度与移动荷载幅值谱最大值对应的速度相等, 且移动荷载幅值谱的其他极值点与桥梁自由振动位移响应极值点对应的移动荷载速度一致; 在自由振动阶段, 桥梁位移响应极值点对应的单个移动荷载速度仅与桥梁自振频率和跨度有关; 单个移动荷载以共振速度通过桥梁时, 桥梁发生的受迫振动与自由振动位移响应均不是最大响应, 因此, 对于高速铁路桥梁的列车运营速度, 除关注列车共振速度外, 需更加重视使桥梁产生最大位移响应的速度。Abstract: To establish the relationship between the operation speed of train and the maximum displacement response of bridge more effectively, a frequency domain analysis method was proposed for the maximum displacement response of bridge excited by a single moving load. The Fourier transform was used to derive the moving load spectrum and the vibration displacement response spectrum of bridge when a single moving load passing through the beam bridge at a constant speed. The moving load speed causing the extreme free vibration displacement response of bridge was obtained by analyzing the moving load amplitude spectrum, and a calculation formula for the moving load speed was proposed. Taking a simply-supported beam as an example, the correctness of numerical calculation procedure in this paper was verified by comparing with the results of related literatures. Based on this procedure, the correctness of theoretical derivation of frequency domain analysis method and the accuracy of moving load speed formula were verified through the numerical analysis. Research result shows that the moving load amplitude spectrum obtained in the frequency domain is consistent with the amplitude response law of free vibration of bridge obtained in the time domain, so the moving load amplitude spectrum can effectively reflect the free vibration displacement response of bridge. The moving load speed corresponding to the maximum free vibration displacement response of bridge is equal to that corresponding to the maximum amplitude spectrum of moving load, and the moving load speeds corresponding to other extreme points of moving load amplitude spectrum are also consistent with those of the extreme points in the free vibration displacement response of bridge. In the free vibration phase, the speed of single moving load correeponding to the extrem point of displacement response of bridge is only related to the natural frequency and span of bridge. When a single moving load passes through a bridge at the resonant speed, the responses of forced and free vibration displacements are not the maximum response. Therefore, for the train operation speed of high-speed railway bridges, in addition to the resonance speed of train, more attention should be paid to the speed causing the maximum displacement response of bridge.
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图 3 不同移动荷载速度下桥梁跨中竖向位移
Figure 3. Vertical displacements of bridge mid-span under different moving load speeds
图 5 桥梁跨中竖向位移时程曲线
Figure 5. Time history curves of vertical displacement of bridge mid-span
图 6 阻尼与模态阶数对桥梁跨中竖向位移的影响
Figure 6. Effects of damping and modal order on vertical displacement of bridge mid-span
图 7 不同移动荷载速度下桥梁振动最大位移响应
Figure 7. Maximum vibration displacement responses of bridge under different moving load speeds
图 8 移动集中荷载下桥梁竖向位移响应幅值谱
Figure 8. Response amplitude spectrums of vertical displacement of bridge under moving concentrated load
图 9 不同移动荷载速度下移动集中荷载的最大幅值
Figure 9. Maximum amplitudes of moving concentrated loads under different moving load speeds
图 10 不同移动荷载速度下桥梁自由振动最大位移响应
Figure 10. Maximum free vibration displacement responses of bridge under different moving load speeds
表 1 桥梁和移动荷载参数
Table 1. Parameters of bridge and moving load
参数 桥跨长度/m 桥梁截面弯曲刚度/(N·m2) 桥梁单位长度质量/(kg·m-1) 移动集中力/kN 阻尼比/% 桥梁第1阶自振频率/(rad·s-1) 参数值 20 109 3 000 6 2.5 14.25 
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表 2 脉冲荷载幅值谱极值点
Table 2. Extreme points of pulse load amplitude spectrum
脉冲荷载幅值谱极值点 极小值点 极大值点 1 3 5 2 4 6 λ 9.42 15.7 4.34 12.15 18.66
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表 3 移动荷载幅值谱极值点对应的速度
Table 3. Speeds corresponding to extreme points of moving load amplitude spectrum
移动荷载幅值谱极值点 极小值点 极大值点 1 3 5 2 4 6 λ 9.42 15.7 4.34 12.15 18.66 v/(m·s-1) 30.25 18.15 65.65 23.44 15.27
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