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基于POD方法的理论基础与数学矩阵分析,通过引入随机场均值的相关矩阵及对均值相关矩阵的POD分解,阐述均值对随机场完整信号和脉动信号POD分解的影响机理。以双曲冷却塔模型和闭口箱梁模型的断面风压场POD分解实例,分析风压均值的影响,并验证上述理论解释的合理性。尽管随机场均值并不影响POD方法对随机场的分解和重建过程,但两种信号分解得到的各阶参数如本征值、本征模态及主坐标都只是所采用信号特性的反映而不能相互混淆。以完整信号分解得到的各阶参数都将受到随机场均值的影响,尤其是低阶参数。由于实际使用中对随机场的信号均值和脉动信号多分别对待,因此在POD分解中应使用脉动信号构建相关矩阵以完全反映脉动信号的特性,从而避免信号均值的影响。
Based on the theoretical basis and mathematical matrix analysis of POD method, the influence mechanism of mean on POD decomposition of complete signal and pulsation signal in random field is illustrated by introducing correlation matrix of random field mean and POD decomposition of mean correlation matrix. The POD decomposition example of the cross-section wind pressure field of the hyperbolic cooling tower model and the closed box girder model is used to analyze the influence of the mean wind pressure and to verify the rationality of the above theoretical explanation. Although the random field mean does not affect the decomposition and reconstruction of the random field by the POD method, the parameters of the two kinds of signal decomposition such as eigenvalue, eigenmode and main coordinates are only the reflection of the signal characteristics used Confused with each other. Each order parameter obtained by complete signal decomposition will be affected by random field mean value, especially low order parameter. Since the average signal and the ripple signal of the random field are treated in actual use, the correlation matrix should be constructed by using the ripple signal in the POD decomposition to fully reflect the characteristics of the ripple signal so as to avoid the influence of the signal mean value.