小波基下的结构地震反应及动力可靠性分析
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摘要
针对小波的基本概念、小波变换的基本原理，本文提出了用四种小波：Littlewood—Paley小波，meyer小波，谐波小波，单边指数小波来进行结构地震反应分析、并对二层框架模型进行振动台试验研究、也进行了在小波基下的地震地面运动能量分析和结构地震能量反应、以及结构地震反应在小波基下的动力可靠性分析。
通过对这四种小波的结构地震反应分析研究说明：Littlewood—Paley小波不适合于用来作结构地震反应分析，因为在Littlewood—Paley小波下的结构地震反应太小，不符合实际情况；用meyer小波作结构地震反应分析比较合适，和有限元法的结果比较接近；也可以用谐波小波来作结构地震反应分析，只是在谐波小波下的结果略为偏大；单边指数小波下的结构地震反应分析比有限元法稍大一点，它通过小波变换大大简化了结构地震反应分析。用小波变换方法来进行结构地震反应分析和以往方法不同的是：它不仅可以知道地震波的具体频率段对结构反应的影响，而且同时考虑了地震波的幅值非平稳性以及频率非平稳性；另外与以前方法得到的结果有差异的是，第二振型及以后的高一点的振型的反应没有以前的方法衰减得快。
通过对二层框架模型进行振动台试验研究说明：从小波变换得到的加速度反应和模型上的第一层及第二层的试验测得的加速度比较得出的结论是，Littlewood—Paley小波不适合于用来作结构地震反应分析，因为在此小波下的结构加速度反应太小，和实验情况不符；meyer小波、谐波小波、单边指数小波这三种小波从理论上得到的加速度反应同实验测得的加速度过程比较吻合，因此从试验上证明用meyer小波、谐波小波和单边指数小波来作结构地震反应分析是比较合适的。
通过三种小波基(Littlewood—Paley小波、meyer小波、谐波小波)下的地震地面运动能量分析以及结构地震能量反应分析说明：直接通过加速度记录可以求得地震动能量，由于小波系数中同时含有时域和频域的能量，由此得出的能量比以往的方法更准确；在此基础上求得了结构的能量反应时程，为结构设计及结构中设计耗能体系提供了具体的方法；通过这一章对比分析得出，Littlewood—Paley小波不适合于作地震动能量分析和结构能量反应分析。由于求能量的特殊性，稍微修改了meyer小波，使得改造的meyer小波的频率积分为一常数，由改造的meyer小波求得的能量和谐波小波求得的能量是比较合适的，在改造的meyer小波下求得的地震动能量及结构地震反应能量比谐波小波求得的相应能量要小一些，但都处于同一能量级。
通过结构地震反应在小波基下的动力可靠性分析说明：无论是单自由度还是多自由度从理论上推导在这四种小波下的结构动力可靠性是可行的，而且考虑的结构地震反应都是非平稳的高斯过程，方法实现也比较容易；在Littlewood—Paley小波基下求得的结构的动力可靠度比较大，由改造的meyer小波和谐波小波求得的动力可靠度略有差异；单边指数小波下的结构动力可靠性分析非常简单，不用求功率谱函数来积分，只需求得结构地震反应的位移方差，速度方差，利用最大值分布公式就可得到结构的动力可靠时程；另外也考虑了超越界限对结构动力可靠度的影响，超越界限越小，结构的动力可靠度越小，即破坏概率越大。
最后，本文给出了研究的主要结论，并对今后进一步的研究工作提出了建议。
For the purpose of wavelet's basic concept and wavelet transform fundamental principle, four wavelets: Littlewood-Paley wavelet,meyer wavelet, harmonic wavelet and odd exponent wavelet are used to analyze structural response under earthquake; experimental investigation has been carried out for two-stories frame model; this paper also analyzes earthquake ground motion energy and structural energy response based on wavelet; this paper proposes dynamic reliability analysis for structure seismic response based on wavelet.
It is shown by structural seismic response of four wavelets that Littlewood-Paley wavelet is not suitable for structural seismic response, because structural response is too small, meyer wavelet is a better wavelet for structural seismic response, for it's structural response is agreement with the finite element method, and also harmonic wavelet, structural response under earthquake is a little bigger than finite element method, structural response under odd exponent wavelet is also bigger than finite element method, this method is very simple by wavelet transform, wavelet transform method is different from old methods, one is with which not only knows the effects of earthquake wave detail frequency-band on structural response, but also considers earthquake wave's non-stationary of frequency and time-domain value, another is the second mode shape and higher mode shape response that don't attenuate so fast.
It is shown by the two-stories frame model shake-table test that the contrast two response of acceleration based on wavelet and experiment draws on such conclusion: Littlewood-Paley wavelet does not agree with the test, so it is not suitable for structural seismic response, because structural acceleration response is too small. The wavelet transform result of acceleration response based on meyer wavelet, harmonic wavelet and odd exponent wavelet agree with the test, Thus they can be used to analyze structural seismic response.
The analysis of earthquake ground motion energy and structural energy response based on these three wavelets(Littlewood-Paley wavelet,meyer wavelet, harmonic wavelet )shows that we can calculate earthquake ground motion energy by the record of acceleration, because the wavelet coefficient includes time-domain energy and frequency-domain energy. So the result is more accurate than old methods. The result of structural energy response based on wavelet provides a practicable method for structure design and design energy dissipation system in structure, this chapter's analysis and contrast tell us Littlewood-paley wavelet is not suitable for analyzing energy of earthquake ground motion and structural energy response. On the purpose of calculating energy, meyer wavelet must be modified so that its frequency integral is a constant, the result of energy by modified meyer wavelet and harmonic wavelet is practicable, only the later is bigger but in the same energy level.
Dynamic reliability analysis for structure seismic response based on wavelet shows that whether signal of degree or multi of degree's system dynamic reliability analysis are practicable on theory, and also seismic response is to be considered nonstationary Gaussian process, these methods are easy to carry out. dynamic reliability under Littlewood-Paley wavelet is the biggest, dynamic reliability under modified meyer
wavelet and harmonic wavelet are close. Analyzing the dynamic reliability of structure under odd exponent wavelet is very simple, it doesn't calculate Power Spectral Density function integral, it is only to calculate the square of displacement standard deviation and the square of velocity, then the dynamic reliability of structure can be concluded by max distribution. In addition the effects of upper limit is considered, when upper limit is smaller, the dynamic reliability of structure is less, that is, the damage possibility is bigger.
At last, conclusion of the present investigation and suggestion for further study are presented
通过对这四种小波的结构地震反应分析研究说明：Littlewood—Paley小波不适合于用来作结构地震反应分析，因为在Littlewood—Paley小波下的结构地震反应太小，不符合实际情况；用meyer小波作结构地震反应分析比较合适，和有限元法的结果比较接近；也可以用谐波小波来作结构地震反应分析，只是在谐波小波下的结果略为偏大；单边指数小波下的结构地震反应分析比有限元法稍大一点，它通过小波变换大大简化了结构地震反应分析。用小波变换方法来进行结构地震反应分析和以往方法不同的是：它不仅可以知道地震波的具体频率段对结构反应的影响，而且同时考虑了地震波的幅值非平稳性以及频率非平稳性；另外与以前方法得到的结果有差异的是，第二振型及以后的高一点的振型的反应没有以前的方法衰减得快。
通过对二层框架模型进行振动台试验研究说明：从小波变换得到的加速度反应和模型上的第一层及第二层的试验测得的加速度比较得出的结论是，Littlewood—Paley小波不适合于用来作结构地震反应分析，因为在此小波下的结构加速度反应太小，和实验情况不符；meyer小波、谐波小波、单边指数小波这三种小波从理论上得到的加速度反应同实验测得的加速度过程比较吻合，因此从试验上证明用meyer小波、谐波小波和单边指数小波来作结构地震反应分析是比较合适的。
通过三种小波基(Littlewood—Paley小波、meyer小波、谐波小波)下的地震地面运动能量分析以及结构地震能量反应分析说明：直接通过加速度记录可以求得地震动能量，由于小波系数中同时含有时域和频域的能量，由此得出的能量比以往的方法更准确；在此基础上求得了结构的能量反应时程，为结构设计及结构中设计耗能体系提供了具体的方法；通过这一章对比分析得出，Littlewood—Paley小波不适合于作地震动能量分析和结构能量反应分析。由于求能量的特殊性，稍微修改了meyer小波，使得改造的meyer小波的频率积分为一常数，由改造的meyer小波求得的能量和谐波小波求得的能量是比较合适的，在改造的meyer小波下求得的地震动能量及结构地震反应能量比谐波小波求得的相应能量要小一些，但都处于同一能量级。
通过结构地震反应在小波基下的动力可靠性分析说明：无论是单自由度还是多自由度从理论上推导在这四种小波下的结构动力可靠性是可行的，而且考虑的结构地震反应都是非平稳的高斯过程，方法实现也比较容易；在Littlewood—Paley小波基下求得的结构的动力可靠度比较大，由改造的meyer小波和谐波小波求得的动力可靠度略有差异；单边指数小波下的结构动力可靠性分析非常简单，不用求功率谱函数来积分，只需求得结构地震反应的位移方差，速度方差，利用最大值分布公式就可得到结构的动力可靠时程；另外也考虑了超越界限对结构动力可靠度的影响，超越界限越小，结构的动力可靠度越小，即破坏概率越大。
最后，本文给出了研究的主要结论，并对今后进一步的研究工作提出了建议。
For the purpose of wavelet's basic concept and wavelet transform fundamental principle, four wavelets: Littlewood-Paley wavelet,meyer wavelet, harmonic wavelet and odd exponent wavelet are used to analyze structural response under earthquake; experimental investigation has been carried out for two-stories frame model; this paper also analyzes earthquake ground motion energy and structural energy response based on wavelet; this paper proposes dynamic reliability analysis for structure seismic response based on wavelet.
It is shown by structural seismic response of four wavelets that Littlewood-Paley wavelet is not suitable for structural seismic response, because structural response is too small, meyer wavelet is a better wavelet for structural seismic response, for it's structural response is agreement with the finite element method, and also harmonic wavelet, structural response under earthquake is a little bigger than finite element method, structural response under odd exponent wavelet is also bigger than finite element method, this method is very simple by wavelet transform, wavelet transform method is different from old methods, one is with which not only knows the effects of earthquake wave detail frequency-band on structural response, but also considers earthquake wave's non-stationary of frequency and time-domain value, another is the second mode shape and higher mode shape response that don't attenuate so fast.
It is shown by the two-stories frame model shake-table test that the contrast two response of acceleration based on wavelet and experiment draws on such conclusion: Littlewood-Paley wavelet does not agree with the test, so it is not suitable for structural seismic response, because structural acceleration response is too small. The wavelet transform result of acceleration response based on meyer wavelet, harmonic wavelet and odd exponent wavelet agree with the test, Thus they can be used to analyze structural seismic response.
The analysis of earthquake ground motion energy and structural energy response based on these three wavelets(Littlewood-Paley wavelet,meyer wavelet, harmonic wavelet )shows that we can calculate earthquake ground motion energy by the record of acceleration, because the wavelet coefficient includes time-domain energy and frequency-domain energy. So the result is more accurate than old methods. The result of structural energy response based on wavelet provides a practicable method for structure design and design energy dissipation system in structure, this chapter's analysis and contrast tell us Littlewood-paley wavelet is not suitable for analyzing energy of earthquake ground motion and structural energy response. On the purpose of calculating energy, meyer wavelet must be modified so that its frequency integral is a constant, the result of energy by modified meyer wavelet and harmonic wavelet is practicable, only the later is bigger but in the same energy level.
Dynamic reliability analysis for structure seismic response based on wavelet shows that whether signal of degree or multi of degree's system dynamic reliability analysis are practicable on theory, and also seismic response is to be considered nonstationary Gaussian process, these methods are easy to carry out. dynamic reliability under Littlewood-Paley wavelet is the biggest, dynamic reliability under modified meyer
wavelet and harmonic wavelet are close. Analyzing the dynamic reliability of structure under odd exponent wavelet is very simple, it doesn't calculate Power Spectral Density function integral, it is only to calculate the square of displacement standard deviation and the square of velocity, then the dynamic reliability of structure can be concluded by max distribution. In addition the effects of upper limit is considered, when upper limit is smaller, the dynamic reliability of structure is less, that is, the damage possibility is bigger.
At last, conclusion of the present investigation and suggestion for further study are presented
引文
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