Modal analysis of bridge under high-speed trains is essential to the design and health monitoring of bridge, but it is difficult to be implemented since the vehicle–bridge interaction (VBI) effect is involved. In this paper, the time–frequency analysis technique is performed on the non-stationary train-induced bridge responses to estimate the frequency variations. To suppress the interference terms in time–frequency analysis but preserve the time-variant characteristics of responses, the enhanced variational mode decomposition (VMD) is proposed, which is used to decompose the train-induced dynamic response into many of envelope-normalized intrinsic mode functions (IMFs). Then the short-time Fourier transform is applied to observe the time–frequency energy distribution of each IMF. The train-induced bridge signals measured from a large-scale high-speed railway bridge are analyzed in this paper. The IMFs associated with the pseudo-frequencies caused by train or the resonant frequencies of bridge are distinguished. And, frequency variations are captured from the time–frequency energy distributions of envelope-normalized IMFs. The results show the proposed method can extract the frequency variations of low-energy IMFs effectively, which are hard to be observed from the time–frequency energy distribution of train-induced bridge response. The instantaneous frequency characteristics extracted from the train-induced bridge response could be the important support for investigating the VBI effect of train–bridge system.

中文翻译：

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通过增强变分模态分解对高速列车通过过程中桥梁的模态分析

高速列车下桥梁的模态分析对于桥梁的设计和健康监测至关重要，但由于涉及车桥相互作用（VBI）效应，因此难以实施。在本文中，对非平稳列车引起的桥梁响应进行时频分析技术来估计频率变化。为了抑制时频分析中的干扰项，但保留响应的时变特性，提出了增强变分模态分解（VMD），用于将列车引起的动态响应分解为许多包络归一化本征模式函数（IMF）。然后应用短时傅里叶变换来观察每个 IMF 的时频能量分布。本文分析了从大型高速铁路桥梁上测得的列车感应桥梁信号。区分了与列车引起的伪频率或桥的共振频率相关的IMF。而且，频率变化是从包络归一化 IMF 的时频能量分布中捕获的。结果表明，该方法可以有效地提取低能量 IMF 的频率变化，这些频率变化很难从列车引起的桥梁响应的时频能量分布中观察到。从车桥响应中提取的瞬时频率特性可以为研究车桥系统的VBI效应提供重要支持。区分了与列车引起的伪频率或桥的共振频率相关的IMF。而且，频率变化是从包络归一化 IMF 的时频能量分布中捕获的。结果表明，该方法可以有效地提取低能量 IMF 的频率变化，这些频率变化很难从列车引起的桥梁响应的时频能量分布中观察到。从车桥响应中提取的瞬时频率特性可以为研究车桥系统的VBI效应提供重要支持。区分了与列车引起的伪频率或桥的共振频率相关的IMF。而且，频率变化是从包络归一化 IMF 的时频能量分布中捕获的。结果表明，该方法可以有效地提取低能量 IMF 的频率变化，这些频率变化很难从列车引起的桥梁响应的时频能量分布中观察到。从车桥响应中提取的瞬时频率特性可以为研究车桥系统的VBI效应提供重要支持。结果表明，该方法可以有效地提取低能量 IMF 的频率变化，这些频率变化很难从列车引起的桥梁响应的时频能量分布中观察到。从车桥响应中提取的瞬时频率特性可以为研究车桥系统的VBI效应提供重要支持。结果表明，该方法可以有效地提取低能量 IMF 的频率变化，这些频率变化很难从列车引起的桥梁响应的时频能量分布中观察到。从车桥响应中提取的瞬时频率特性可以为研究车桥系统的VBI效应提供重要支持。