WO2018233360A1 - 一种基于kl展开的分布随机动载荷识别方法 - Google Patents
一种基于kl展开的分布随机动载荷识别方法 Download PDFInfo
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- the invention relates to a distributed random dynamic load identification method based on KL expansion, and belongs to the technical field of structural dynamic inverse problems.
- Dynamic load information on the engineering structure is the basis for structural design and safety assessment. In many cases, dynamic loads are difficult to obtain by direct measurement. The dynamic response of the structure is often measured directly, and the dynamic load information on the structure is obtained by means of indirect identification.
- the traditional dynamic load identification method uses the structural dynamic response data of a single actual measurement to identify the excitation information that causes the secondary dynamic response, and is a deterministic dynamic load identification method.
- the existing deterministic dynamic load identification method is used to obtain information such as concentrated dynamic load, moving load and distributed dynamic load on the engineering structure. It is worth noting that the distributed dynamic load identification problem is equivalent to identifying an infinite number of concentrated dynamic loads, which is more difficult. Generally, the distributed dynamic load identification problem needs to be reduced in dimension.
- the dynamic loads acting on the actual engineering structure are not only distributed on the structure, but also random.
- the dynamic response will also appear “randomness”; therefore, the structural dynamic response of a single measured measurement can only be one of the samples of the structural random dynamic response information, and the certainty is utilized based on a certain response sample.
- the dynamic load information obtained by the dynamic load identification method can only partially reflect the random dynamic load excitation; in addition, the dynamic response error contained in a single measurement is also used as part of the “true response” in the deterministic dynamic load identification, causing the load. Identify the deviation of the results.
- the traditional deterministic distributed dynamic load identification method and the centralized random dynamic load identification method are not applicable. It is necessary to develop a new method for distributed random dynamic load identification.
- the object of the present invention is to provide a distributed random dynamic load identification method based on KL expansion, which solves the problem of time-varying statistical characteristics of random dynamic load with spatial distribution in the time domain using the measured structure dynamic response sample identification structure, for serving in random distribution.
- the engineering structure design and safety assessment under dynamic load environment provides a means of indirect acquisition of dynamic loads.
- a distributed random dynamic load identification method based on KL expansion characterized in that the method comprises the following steps:
- the structural random vibration response is developed by using the mode shape to obtain the dynamic response of the structure in the modal space;
- the KL-expanded distributed random dynamic load identification method according to claim 1, wherein the structural random vibration response is expanded by using a modal shape in step S2 to obtain a motion of the structure in the modal space.
- the specific steps are:
- the rth measurement acquisition displacement response sample vector W r is expressed as:
- the KL-expanded distributed random dynamic load identification method according to claim 1, wherein the random dynamic response in the modal space is solved by the KL expansion in step S3 to solve the random dynamic load in the modal space. Specifically, the following steps are included:
- ⁇ i , ⁇ i and m i are the ith order natural frequency, modal damping ratio and modal quality, respectively.
- step S4 The method for identifying a distributed random dynamic load based on KL expansion according to claim 1, wherein the time-varying statistical feature of the spatial distribution of the random dynamic load on the structure in step S4 comprises the following steps:
- the time-varying statistical characteristics of the spatially distributed random loads with spatial distribution including the mean ⁇ f (x, t) and the variance Var f (x, t) are:
- the invention has the following advantages:
- the existing random dynamic load identification technology can only identify the random concentrated dynamic load on the structure by the measured structure dynamic response sample. Most of the distributed random dynamic load identification methods that have appeared at present cannot be applied to the identification of non-stationary random dynamic loads.
- the KL-expanded distributed random dynamic load time domain identification technology provided by the invention can utilize the measured structural dynamic response samples at the finite measurement points to identify the statistical characteristics of the random dynamic load with the spatial distribution, and has certain advancement;
- Figure 1 is a logic flow diagram of the method of the present invention.
- Figure 2 is a schematic diagram of a simply supported beam under distributed random loads.
- Figure 3(a) shows the results of the mean value of the random dynamic load in the beam span.
- Figure 3(b) shows the results of the variance of the random dynamic load in the beam span.
- Figure 4 shows the results of spatial distribution of random dynamic loads on the beam.
- Embodiments For a random dynamic load condition acting on a one-dimensional simply supported beam as shown in FIG. 2, the time-varying statistical characteristics of the random dynamic load with spatial distribution are identified from the measured dynamic random response samples by using the technique of the present invention.
- the damping of the structure is Rayleigh damping
- the trapezoidal distributed random dynamic load distribution function to be identified is:
- the stochastic dynamic load component F(t, ⁇ ) of distributed random dynamic load is divided into two parts: deterministic dynamic load and random dynamic load.
- the first five natural frequencies of the acquisition structure are 3.9 Hz, 15.6 Hz, 35.1 Hz, 62.5 Hz, and 97.6 Hz, respectively, and the mode shapes corresponding to the natural frequencies of the respective orders are obtained;
- the rth measurement acquisition displacement response sample vector W r at the position of the beam structure (x 1 , x 2 , ... x n ) is expressed as:
- w r (x j , t) represents the value of the structural displacement obtained at the rth measurement at time t at x j
- N is the number of measurements, that is, the total number of samples.
- the modal shape function is used to calculate the modal displacement vector corresponding to the rth measurement in the modal space:
- q i,r (t) is the modal displacement of the structural displacement obtained in the rth measurement in the i-th modal space
- the upper right corner + sign indicates the generalized inverse.
- the truncation principle of KL vector in KL expansion is to ensure that the K-L vector component after truncation is close to the original vector under the minimum mean square error criterion, according to the following formula:
- ⁇ i , ⁇ i and m i are the ith order natural frequency, modal damping ratio and modal quality, respectively. with They are the first and second derivatives of z ij (t) versus time t, respectively.
- the modal mass m i of the simply supported beam can be calculated by:
- S4 Solving time-varying statistical characteristics of spatially distributed random loads with spatial distribution, including the following steps:
- the time-varying statistical characteristics of the spatially distributed random loads with the spatial distribution including the mean ⁇ f (x, t) and the variance Var f (x, t), can be obtained from the following formulas:
- Figure 3(a) shows the comparison of the mean value of the random dynamic load in the beam span with time to the true value.
- Figure 3(b) shows the variance of the random dynamic load in the beam span obtained by the identification.
- the comparison between the law of change with time and the true value is shown in Fig. 4.
- the comparison results between the spatial distribution and the true distribution of the random dynamic load on the beam at each time are obtained. It can be seen that the identification method in the present invention can accurately identify the distribution of the random dynamic load with space and the statistical characteristics with time according to the response sample at the limited measurement point, and is suitable for the case of non-stationary random dynamic load; Compared with the Monte Carlo method, when the number of measured response samples is large, there is a significant advantage in computational efficiency.
- the method proposed by the present invention has certain advancement.
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Abstract
一种基于KL展开的分布随机动载荷识别方法。该方法包括步骤：S1.开展模态试验，获取结构的模态参数，包括固有频率和模态振型；S2.将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应；S3.利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷；S4.求解结构上随机动载荷的随空间分布的时变统计特征。解决了在时域内利用实测结构动响应样本识别结构上随机动载荷随空间分布的时变统计特征问题，为服役于分布随机动载荷环境下的工程结构设计与安全评估提供一种动载荷间接获取手段。
Description
本发明涉及一种基于KL展开的分布随机动载荷识别方法，属于结构动力学反问题技术领域。
工程结构上的动载荷信息是结构设计和安全评估的依据。许多情况下，动载荷难以通过直接测量获得，常直接测量结构上的动响应，通过间接识别的手段获取结构上的动载荷信息。
传统的动载荷识别方法是利用单次实测的结构动响应数据识别引起该次动响应的激励信息，是确定性动载荷识别方法。现有的确定性动载荷识别方法被用于获取工程结构上的集中动载荷，移动载荷以及分布式动载荷等信息。值得注意的是，分布式动载荷识别问题相当于识别无穷多个集中动载荷，难度更大，一般需要将分布式动载荷识别问题降维求解。
实际工程结构上作用的动载荷，如建筑物上的风载荷，海洋平台承受的海浪载荷以及飞行器表面的气动载荷等，不仅分布于结构上，而且还具有随机性。随机动载荷施加于结构时，动响应也将随之呈现“随机性”；因此，单次实测的结构动响应只能是结构随机动响应信息的其中一个样本，基于某个响应样本利用确定性动载荷识别方法获得的动载荷信息也只能部分反映该随机动载荷激励；另外，单次测量中包含的动响应误差在确定性动载荷识别中也被作为“真实响应”的一部分，引起载荷识别结果的偏差。针对此类分布式随机动载荷的识别问题，传统的确定性分布动载荷识别方法和集中随机动载荷的识别方法均无法适用，需要发展一种针对分布式随机动载荷识别的新方法。
发明内容
本发明的目的是提供一种基于KL展开的分布随机动载荷识别方法，解决在时域内利用实测结构动响应样本识别结构上随机动载荷随空间分布的时变统计特征问题，为服役于分布随机动载荷环境下的工程结构设计与安全评估提供一种动载荷间接获取手段。
上述的目的通过以下技术方案实现：
一种基于KL展开的分布随机动载荷识别方法，其特征在于，该方法包括如下步骤：
S1.开展模态试验，获取结构的模态参数，包括固有频率和模态振型；
S2.将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应；
S3.利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷；
S4.求解结构上随机动载荷的随空间分布的时变统计特征。
2.根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S2中所述的将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应，具体步骤为：
S21：利用多次重复测量方式获取随机振动响应的样本集合；
S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应，具体步骤如下：
在结构表面上(x
1,y
1)，(x
1,y
2)，…，(x
n,y
n)等位置处第r次测量获取位移响应样本向量W
r表示为：
W
r＝{w
r(x
1,y
1,t)w
r(x
1,y
2,t)…w
r(x
n,y
n,t)}
T,r＝1,…,N(1)，
其中w
r(x
i,y
j,t)表示第r次测量获得的结构上空间位置(x
i,y
j)处t时刻的动位移值，N为测量的次数，利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：
其中q
i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，
表示第i阶模态振型函数在(x
j,y
k)处的值，右上角+号表示广义逆。
3.根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S3中所述的利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷，具体包括以下步骤：
S31：利用模态空间内随机动响应的样本集合q
i,r(t)，求解第i阶模态空间内随机位移响应的协方差矩阵Γ
qi；
S32：对协方差矩阵进行特征值分解Γ
qi，计算其特征值λ
ij和特性向量η
ij(t)，进一步获取第i阶模态空间内随机位移响应的第j个KL向量z
ij(t)，可以表示为：
由此，第i阶模态空间内随机位移响应q
i(t,θ)的KL展开可以表示为：
其中ξ
j(θ)为随机变量，θ表示随机维度，当j＝0时ξ
0(θ)＝1；
S33：根据下式，由随机动位移的KL向量z
ij(t)反演随机动载荷对应向量u
ij(t)；
其中ω
i，ζ
i和m
i分别为第i阶固有频率，模态阻尼比和模态质量，
和
分别是z
ij(t)对时间t的一阶和两阶导数；
S34：由随机动载荷对应向量u
ij(t)求解模态空间内随机动载荷f
i(t,θ)，
4.根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S4中所述的求解结构上随机动载荷的随空间分布的时变统计特征，包括以下步骤：
分布随机动载荷f(x,y,t,θ)的表达式为：
结构上随机动载荷的随空间分布的时变统计特征，包括均值μ
f(x,t)和方差Var
f(x,t)分别为：
本发明与现有技术相比，具有以下优点：
1、现有的随机动载荷识别技术一般只能由实测结构动响应样本识别结构上随机集中动载荷，目前已经出现的分布随机动载荷识别方法大多无法适用于非平稳随机动载荷的识别；本发明中提供的基于KL展开的分布随机动载荷时域识别技术能够利用有限测点处的实测结构动响应样本识别随机动载荷随空间分布的统计特征，具有一定的先进性；
2、利用KL展开在模态空间内由随机动响应反演随机动载荷，比基于随机样本的蒙特卡洛法具有更高的计算效率，具有一定的优势。
图1为本发明方法的逻辑流程框图。
图2为分布随机载荷作用下简支梁示意图。
图3(a)为梁跨中处随机动载荷均值识别结果。
图3(b)为梁跨中处随机动载荷方差识别结果。
图4为梁上随机动载荷空间分布识别结果。
下面通过实施例的方式，对本发明技术方案进行详细说明，但实施例仅是本发明的其中一种实施方式，应当指出：对于本技术领域的技术人员来说，在不脱离本发明原理的前提下，还可以以改变结构和载荷形式等方式做出若干改进和等同替换，这些对本发明权利要求进行改进和等同替换后的技术方案，均落入本发明的保护范围。
实施例：对如图2所示一维简支梁上作用分布随机动载荷工况，利用本发明的技术由实测结构随机动响应样本识别随机动载荷随空间分布的时变统计特征。实施例中简支梁长L＝40m，横截面积A＝4.8m
2，截面惯性矩I＝2.5498m
4，结构的阻尼采用瑞利阻尼，各阶模态阻尼比ξ
i＝0.02，材料的弹性模量E＝5×10
10N/m
2，密度ρ＝2.5×10
3kg/m
3。待识别的梯形分布随机动载荷分布函数为：
分布式随机动载荷的随机性动载荷组分F(t,θ)分为确定性动载荷和随机性动载荷两个部分。
F(t,θ)的确定性动载荷部分：
F
d(t)＝20000[1+0.1sin(2πt)]N (2)
F(t,θ)的随机性动载荷部分假定为零均值非平稳高斯随机过程，功率谱函数S(ω,t)为：
S(ω,t)＝C
fP
d(t)Φ(ω) (3)
其中：C
f表示随机水平，取C
f＝0.2；Φ(ω)表示零均值非平稳高斯随机过程的功率谱密度函数，有Φ(ω)＝(1/2π)(2/ω
2+1)。
具体包括以下步骤：
S1：获取结构的前五阶固有频率分别为3.9Hz,15.6Hz,35.1Hz,62.5Hz和97.6Hz，同时获得各阶固有频率所对应的模态振型；
S2：将结构随机振动响应利用模态振型展开，利用多次测量获取位移响应信号，求解结构在模态空间的动响应，包括以下步骤：
S21：在梁结构上均匀布置19个测点，重复测量各测点处的动态位移信号，获取在分布随机激励下随机振动响应的样本集合；
S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应。具体步骤如下：
在梁结构上(x
1,x
2,…x
n)位置处第r次测量获取位移响应样本向量W
r表示为：
W
r＝{w
r(x
1,t)w
r(x
2,t)…w
r(x
n,t)}
T,r＝1,…,N(4)，
其中w
r(x
j,t)表示第r次测量获得的结构位移在x
j处t时刻的值，N为测量的次数，即为样本总数。利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：
其中q
i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，
表示第i阶模态振型函数在x
j处的值，右上角+号表示广义逆。此时，测点数n＝19，模态数m＝5。
S3.利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷，包括以下步骤：
S31：利用模态空间内随机动响应的样本集合q
i,r(t)，求解第i阶模态空间内随机位移响应的协方差矩阵Γ
qi；
S32：对协方差矩阵进行特征值分解Γ
qi，计算其特征值λ
ij和特性向量η
ij(t)。进一步获取第i阶模态空间内随机位移响应的第j个KL向量z
ij(t)，可以表示为：
KL展开中KL向量的截断原则为保证截断后的K-L向量组分在最小均方误差准则下接近原向量，具体根据下式：
其中γ＝0.99。
由此，第i阶模态空间内随机位移响应q
i(t,θ)的KL展开可以表示为：
其中ξ
j(θ)为随机变量，θ表示随机维度，当j＝0时ξ
0(θ)＝1。
S33：根据下式，由随机动位移的KL向量z
ij(t)反演随机动载荷对应向量u
ij(t)；
其中ω
i，ζ
i和m
i分别为第i阶固有频率，模态阻尼比和模态质量，
和
分别是z
ij(t)对时间t的一阶和两阶导数。简支梁的模态质量m
i可由下式计算：
S34：由随机动载荷对应向量u
ij(t)求解模态空间内随机动载荷f
i(t,θ)，
S4：求解结构上随机动载荷的随空间分布的时变统计特征，包括以下步骤：
分布随机动载荷f(x,t,θ)的识别结果可以表示为：
结构上随机动载荷的随空间分布的时变统计特征，包括均值μ
f(x,t)和方差Var
f(x,t)可以分别由下面公式得到：
图3(a)中给出了识别获得的梁跨中处随机动载荷均值随时间变化规律与真实值的对比；图3(b)中给出了识别获得的梁跨中处随机动载荷方差随时间变化规律与真实值的对比；图4中给出了识别获得的各时刻梁上随机动载荷的空间分布与真实分布的对比结果。由此可知，本发明中的识别方法能够利用有限测点处响应样本准确识别随机动载荷随空间的分布以及随时间变化的统计特征，适用于非平稳随机动载荷的情况；同时，跟基于样本的蒙特卡洛法相比，当实测响应样本数量较多时，在计算效率上有明显的优势，例如，在本实施例中，实测样本数等于5000时，在同等识别精度下，基于KL展开的识别方法所用计算时间仅为基于蒙特卡洛法的15％，大幅提高了识别效率。综上所述，本发明提出的方法具有一定的先进性。
Claims (4)
- 一种基于KL展开的分布随机动载荷识别方法，其特征在于，该方法包括如下步骤：S1.开展模态试验，获取结构的模态参数，包括固有频率和模态振型；S2.将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应；S3.利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷；S4.求解结构上随机动载荷的随空间分布的时变统计特征。
- 根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S2中所述的将结构随机振动响应利用模态振型展开，获取结构在模态空间的动响应，具体步骤为：S21：利用多次重复测量方式获取随机振动响应的样本集合；S22：针对单个样本，即单次实测结构振动响应，利用模态振型展开获得结构振动在模态空间内响应，具体步骤如下：在结构上(x 1,y 1)，(x 1,y 2)，…，(x n,y n)位置处第r次测量获取位移响应样本向量W r表示为：W r＝{w r(x 1,y 1,t) w r(x 1,y 2,t) … w r(x n,y n,t)} T,r＝1,…,N(1)，其中w r(x i,y j,t)表示第r次测量获得的结构上空间位置(x i,y j)处t时刻的动位移，N为测量的次数。利用模态振型函数计算模态空间内第r次测量对应的模态位移向量：
其中q i,r(t)为第r次测量获得的结构位移在第i阶模态空间中的模态位移，表示第i阶模态振型函数在(x j,y k)处的值，右上角+号表示广义逆。
- 根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S3中所述的利用KL展开由模态空间内的随机动响应求解模态空间内随机动载荷，具体包括以下步骤：S31：利用模态空间内随机动响应的样本集合q i,r(t)，求解第i阶模态空间内随机位移响应的协方差矩阵Γ qi；S32：对协方差矩阵进行特征值分解Γ qi，计算其特征值λ ij和特性向量η ij(t)，进一步获取第i阶模态空间内随机位移响应的第j个KL向量z ij(t)，可以表示为：
由此，第i阶模态空间内随机位移响应q i(t,θ)的KL展开可以表示为：
其中ξ j(θ)为随机变量，θ表示随机维度，当j＝0时ξ 0(θ)＝1；S33：根据下式，由随机动位移的KL向量z ij(t)反演随机动载荷对应向量u ij(t)；
其中ω i，ζ i和m i分别为第i阶固有频率，模态阻尼比和模态质量，和
分别是z ij(t)对时间t的一阶和两阶导数；
S34：由随机动载荷对应向量u ij(t)求解模态空间内随机动载荷f i(t,θ)，
- 根据权利要求1所述的基于KL展开的分布随机动载荷识别方法，其特征在于，步骤S4中所述的求解结构上随机动载荷的随空间分布的时变统计特征，包括以下步骤：分布随机动载荷f(x,y,t,θ)的表达式为：
结构上随机动载荷的随空间分布的时变统计特征，包括均值μ f(x,t)和方差Var f(x,t)分别为：
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