An efficient frequency domain method for random vibration analysis of structures subjected to time-varying coherent non-stationary excitation

Abstract

The present invention provides an efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation, which includes the following steps: Step 1: According to the evolution power spectrum density (EPSD) function of the target excitation, the time-frequency interpolation points are reasonably determined. The time-frequency interpolation points are determined according to the change of EPSD with time and frequency; Step 2: The EPSD matrix of multi-point excitation x(t) at the time-frequency interpolation points is calculated and decomposed, and the decomposed spectrum can be obtained by Cholesky decomposition. The decomposed spectrum is decoupled by using the diagonal proper orthogonal decomposition (DPOD) technology, and the frequency response matrix is ​​quickly calculated by using the fast Fourier transform (FFT). By using the present invention to perform time-varying coherent non-stationary random vibration analysis, it is only necessary to calculate the generalized impulse response, and then the response variance of the structure can be obtained efficiently and accurately by using DPOD and FFT.

Description

Technical Field

[0001] The present invention relates to the field of random vibration analysis of structures, in particular to an efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation. Background Art

[0002] In modern engineering structures, many buildings and infrastructures are subject to complex time-varying coherent non-stationary excitations, which can cause non-stationary random vibration responses in the structures. At present, traditional analysis methods mainly include time domain analysis and frequency domain analysis. Time domain analysis methods such as Monte Carlo simulation require a large number of sample calculations to capture non-stationary characteristics, which is computationally expensive and inefficient. Frequency domain analysis methods can only handle random vibration problems under time-invariant coherent multi-point non-stationary excitations, and their application still has limitations. In addition, especially for multi-degree-of-freedom structures, traditional frequency domain methods still have shortcomings in computational efficiency.

[0003] Therefore, an efficient and accurate analysis method is needed that can accurately calculate the random vibration response of the structure under time-varying coherent multi-point non-stationary excitation and significantly improve the calculation efficiency. Summary of the invention

[0004] In view of the deficiencies in the prior art, the present invention provides an efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation, thereby solving the above-mentioned problems.

[0005] To achieve the above objectives, the present invention is implemented by the following technical scheme: an efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation comprises the following steps:

[0006] Step 1: According to the evolution power spectral density (EPSD) function of the target excitation, reasonably determine the time-frequency interpolation point The time-frequency interpolation point Determined based on the change in EPSD over time and frequency;

[0007] Step 2: Calculate the EPSD matrix of the multi-point excitation x(t) at the time-frequency interpolation point And decompose it, through Cholesky decomposition, we can get

[0008] Step 3: Use the diagonal proper orthogonal decomposition (DPOD) strategy to decompose the decomposed spectrum Decoupling into time factors and frequency factor These time and frequency factors are then interpolated to obtain and

[0009] Step 4: Establish a finite element model (FEM) of the structure; construct the pulse excitation δ(t i ), respectively loaded on each excitation degree of freedom, and time history analysis is performed to obtain the generalized impulse response matrix Θ(t);

[0010] Step 5: Calculate the evolution frequency response matrix I(ω,t) by fast Fourier transform (FFT) or inverse fast Fourier transform (IFFT), and then calculate the EPSD matrix S of the structural response y (ω,t) and the variance vector Get the random vibration response of the structure.

[0011] Preferably, the time-frequency interpolation point Frequency and time interpolation points Can be calculated by the following formula:

[0012]

[0013] Among them, ω1 and ω u denote the first frequency and cutoff frequency respectively, and Respectively represent the number of frequency and time interpolation points, and satisfy and N ω and N t are the number of complete frequency points and time points of the excitation power spectrum, respectively.

[0014] Preferably, assume that x(t) = [x1(t), x2(t), …, x n (t)] T is an n-point time-varying coherent non-stationary excitation with a mean of zero, where t represents time, T represents vector transpose, and the EPSD matrix It is expressed as follows:

[0015]

[0016] Where ω represents the circular frequency; Represents x j (t)'s own score, satisfying Represents x j (t) and x k The cross-spectrum between (t) satisfies and (ω, t) (where * denotes complex conjugate), EPSD matrix of multi-point excitation x(t) at the interpolation point It is calculated by the given EPSD function and coherence function, namely:

[0017]

[0018] By Cholesky decomposition, can be broken down into:

[0019]

[0020] in It is called the decomposed EPSD matrix and can be expressed as:

[0021]

[0022] in is a non-negative real number, Usually plural.

[0023] Preferably, based on the Diagonal Proper Orthogonal Decomposition (DPOD) strategy, each element can be decomposed into several products of time and frequency factors:

[0024]

[0025] in is the rth time factor; is the rth frequency factor; N R Represents the number of POD approximation terms. Usually a very small value can achieve high decoupling accuracy.

[0026] Based on the decoupling results and r=1,2,...,N R , we can use one-dimensional interpolation techniques (such as Lagrange interpolation) to obtain the corresponding results of all time-frequency discrete points (ω, t) and r = 1, 2, ..., N R , so, corresponding to equation (7), L jk (ω,t) can be reconstructed as:

[0027]

[0028] Preferably, according to the definition of the generalized impulse response matrix Θ(t), each column vector Θ k (t)(k=1,2,…,n) represents the structural response under the unit pulse excitation δ(t),

[0029]

[0030] It is assumed that the initial state of the structure is static; Γ k is the kth column vector of Γ; δ(t) is the Dirac-delta function satisfying

[0031]

[0032] The unit pulse excitation can be approximately discretized into a triangular pulse excitation, that is,

[0033]

[0034] Where Δt represents the time step; t i =(i-1)Δt represents the discrete time, substitute into equation (11) into equation (9), and numerically solve equation (9) by stepwise integration method. Since Γ k δ(t i )(k=1,2,…,n) represents a single-point unit pulse excitation, so a total of n time history analyses are required to calculate the matrix Θ(t).

[0035] Preferably, similar to the evolution spectrum method of random vibration analysis under time-invariant coherent non-stationary excitation, the EPSD matrix S of the structural response under time-varying coherent non-stationary excitation is y (ω,t) can be expressed as:

[0036] S y (ω,t)=I(ω,t)I T* (ω,t)

[0037] In the formula, the evolution frequency response matrix I(ω,t) can be directly calculated by the following formula:

[0038]

[0039] Substituting equation (8) into equation (13), and rewriting the time integral into Fourier integral, the FFT technique can be used for efficient calculation, that is,

[0040]

[0041] Among them, "F τ ” refers to the Fourier transform of τ;

[0042] In addition, formula (13) can also be efficiently calculated by using IFFT technology.

[0043]

[0044] in,

[0045]

[0046] in refers to The inverse Fourier transform of t ” refers to the Fourier transform of t,

[0047] The EPSD matrix S of the structural response is obtained by formula (12): y After (ω,t), its variance response can be calculated by the following formula:

[0048]

[0049] The present invention provides an efficient frequency domain method for analyzing random vibration of structures under time-varying coherent non-stationary excitation. It has the following beneficial effects:

[0050] 1. The present invention uses the diagonal proper orthogonal decomposition (DPOD) technology to decouple the decomposed spectrum, and then uses the fast Fourier transform (FFT) to quickly calculate the frequency response matrix. By using the present invention to perform time-varying coherent non-stationary random vibration analysis, it is only necessary to calculate the generalized impulse response, and then use DPOD and FFT to efficiently and accurately obtain the response variance of the structure.

[0051] 2. The present invention constructs the form of FFT through DPOD technology, which greatly improves the calculation efficiency and can perform separate calculations for any degree of freedom and any time. Compared with the traditional Monte Carlo simulation method, the present invention avoids simulating a large number of samples to reflect non-stationarity when calculating the variance of the structural response, and also avoids traversing calculations of a large number of samples in the time domain, thereby greatly improving the calculation efficiency. The efficient frequency domain method proposed in the present invention is suitable for processing random vibration analysis of structures under time-varying coherent non-stationary excitation, especially for structures with multiple degrees of freedom, which can better reflect the calculation efficiency of this method. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 is a flow chart of the present invention;

[0053] Figure 2 This is a schematic diagram of the plane frame structure of the present invention;

[0054] Figure 3 is the standard deviation of the response of the top and middle layers of the structure (FFT method);

[0055] Figure 4 is the standard deviation of the response of the top and middle layers of the structure (IFFT method). DETAILED DESCRIPTION

[0056] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0057] like Figure 1-Figure 4 As shown, the embodiment of the present invention provides an efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation. A case of non-stationary response analysis of a plane frame structure under multi-point time-varying coherent non-stationary excitation is used to further illustrate the present invention:

[0058] (1) According to the evolution power spectral density (EPSD) function of the target excitation, the time-frequency interpolation points are reasonably determined. Assume that the evolution power spectral density function of the non-stationary excitation is:

[0059] S(ω,t)=S0|A(ω,t) 2

[0060]

[0061] Where, S0 = 35000N 2 s / rad; κ = 1 / 0.5413. The time-varying coherence function between each excitation is expressed as:

[0062]

[0063] Among them, Δ jk =3m is the distance between the two excitation points; g(t) is a time modulation function, which is:

[0064]

[0065] Where, t0 = 16s, D t =8s;

[0066] Time-frequency interpolation points It can be determined based on the change of EPSD over time and frequency. and time interpolation points Can be calculated by the following formula:

[0067]

[0068] Among them, ω1 = 0.711Hz and ω u =8Hz; frequency interpolation point Time interpolation points Complete frequency points N ω =256 and time point N t =1024.

[0069] (2) Calculate the EPSD matrix of multi-point excitation at the interpolation point And perform Cholesky decomposition on it. The specific method is as follows:

[0070] Assume x(t)=[x1(t),x2(t),…,x20 (t)] T is a 20-point time-varying coherent non-stationary excitation with a mean of zero, where t represents time and T represents vector transpose. Its EPSD matrix can be expressed as

[0071]

[0072] Through Cholesky decomposition, Decompose into

[0073]

[0074] in It is called the decomposed EPSD matrix and can be expressed as

[0075]

[0076] The decomposed spectrum is transformed into Decoupling into time factors and frequency factor And interpolate it, the specific method is:

[0077] (3) Decompose each by diagonal proper orthogonal decomposition (DPOD) Get the corresponding time factor and frequency factor

[0078]

[0079] Then the obtained small amount (N R =10) time factor and frequency factor Use one-dimensional interpolation techniques (such as Lagrange interpolation) to obtain the corresponding results of all time-frequency discrete points (ω, t) and You can get the L of the complete time-frequency point jk (ω,t);

[0080]

[0081] (4) Establish a finite element model (FEM) of the structure; construct a pulse excitation, load it on each excitation degree of freedom, perform time history analysis, and obtain a generalized pulse response matrix. The specific method is as follows:

[0082] The corresponding finite element model of the plane frame is established. This plane frame structure has 3 spans and 20 floors, such as Figure 1As shown. The cross-sections of its beams and columns are 0.3m×0.4m and 0.4m×0.4m respectively. The elastic modulus of the structure is E=3×10 10 N / m 2 , density is ρ = 2400 kg / m 3 Assuming that the damping of the structure conforms to the Rayleigh damping model, the damping matrix can be calculated as:

[0083] C=αM+βK

[0084] Where α and β are the Rayleigh damping coefficients, which can be calculated by assuming that the first two-order damping ratio of the structure is ξ1=ξ2=0.05.

[0085] The unit pulse excitation is approximately discretized into a triangular pulse excitation, that is,

[0086]

[0087] Where, Δt = 1 / 32s, N t =1024.

[0088] Then the triangular pulse excitation is loaded on each excitation degree of freedom respectively, and the time history analysis is performed to obtain the generalized impulse response matrix Θ(t).

[0089] (5) Calculate the evolution frequency response matrix I(ω,t) by fast Fourier transform (FFT) or inverse fast Fourier transform (IFFT), and then calculate the EPSD matrix S of the structural response y (ω,t) and the variance vector

[0090] EPSD matrix S of structural response y (ω,t) can be expressed as:

[0091] S y (ω,t)=I(ω,t)I T* (ω,t)

[0092] In the formula, the evolution frequency response matrix I(ω,t) can be directly calculated by the following formula:

[0093]

[0094] Substituting the decomposed L(ω,τ) and rewriting the time integral as Fourier integral, the FFT technology can be used for efficient calculation:

[0095]

[0096] Among them, "F τ ” refers to the Fourier transform of τ;

[0097] In addition, efficient calculation can also be achieved by using IFFT technology, that is,

[0098]

[0099] in

[0100]

[0101] in refers to The inverse Fourier transform of t ” refers to the Fourier transform of t.

[0102] The EPSD matrix S of the structural response is obtained by y After (ω,t), its variance response can be calculated by the following formula:

[0103]

[0104] The above method can be used to analyze the non-stationary response of a 3-span 20-story plane frame structure, and the calculated response variance can be compared with the results calculated by the traditional Monte Carlo simulation method. Figure 2 The verification results of the method of the present invention and the traditional method are given. As can be seen from the figure, the calculation results are consistent with the results of the traditional method, which can illustrate the effectiveness and rationality of the calculation method. Further, the calculation efficiency of the method of the present invention is compared with the traditional method. The results show that the method of the present invention only takes 112 seconds to calculate the 3-span 20-layer plane frame structure through FFT, while the IFFT method takes 210 seconds; while the traditional Monte Carlo simulation method takes 1560 seconds, which shows that the calculation efficiency of this method is greatly improved.

[0105] In the above-mentioned embodiment of the present invention, the method of the present invention is used to perform time-varying coherent non-stationary response analysis, and then diagonal proper orthogonal decomposition (DPOD) is used for decoupling, and then FFT or IFFT is used for calculation, which greatly improves the efficiency of response calculation.

[0106] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation, characterized by: The following steps are involved: Step 1: According to the target excitation evolution power spectrum density EPSD function, reasonably determine the time-frequency interpolation point The time-frequency interpolation point Determined based on the change of EPSD function with time and frequency; Step 2: Calculate the EPSD matrix of the multi-point excitation x(t) at the time-frequency interpolation point And decompose it, through Cholesky decomposition, we can get Step 3: Use the diagonal intrinsic orthogonal decomposition DPOD strategy to decompose the decomposed spectrum Decoupling into time factors and frequency factor These time and frequency factors are then interpolated to obtain and Step 4: Establish the finite element model FEM of the structure; construct the pulse excitation δ(t i ), respectively loaded on each excitation degree of freedom, and time history analysis is performed to obtain the generalized impulse response matrix Θ(t); Step 5: Calculate the evolution frequency response matrix I(ω,t) by fast Fourier transform FFT or inverse fast Fourier transform IFFT, and then calculate the EPSD matrix S of the structural response y (ω,t) and the variance vector Get the random vibration response of the structure; The time-frequency interpolation point Frequency and time interpolation points Calculated by the following formulas: Among them, ω1 and ω u denote the first frequency and cutoff frequency respectively, and Respectively represent the number of frequency and time interpolation points, N ω and N t are the numbers of complete frequency points and time points of the excitation power spectrum, respectively; Assume x(t)=[x1(t),x2(t),…,x n (t)] T is an n-point time-varying coherent non-stationary excitation with a mean of zero, where t represents time, T represents vector transpose, and the EPSD matrix It is expressed as follows: Where ω represents the circular frequency; Represents x j (t)'s own score, Represents x j (t) and x k The cross-spectrum between (t) satisfies and Where * represents complex conjugate, j = 1, 2…, n; k = 1, 2…, n; By Cholesky decomposition, is broken down into: in It is called the decomposed EPSD matrix and is expressed as: in is a non-negative real number, is plural; Based on the diagonal intrinsic orthogonal decomposition DPOD strategy, each element can be decomposed into several products of time and frequency factors: in is the rth time factor; is the rth frequency factor; N R Represents the number of POD approximation terms; Based on the decoupling results and Use one-dimensional interpolation technology to obtain the corresponding results of all time-frequency discrete points (ω, t) and L jk The (ω,t) reconstruction is expressed as: According to the definition of the generalized impulse response matrix Θ(t), each column vector Θ k (t) represents the structural response under the unit pulse excitation δ(t), where k = 1, 2…, n; It is assumed that the initial state of the structure is static; Γ k is the kth column vector of Γ; δ(t) is the Dirac-delta function satisfying and The unit pulse excitation is approximately discretized into a triangular pulse excitation, that is: Where Δt represents the time step; t i =(i-1)Δt represents a discrete moment.

2. The efficient frequency domain method for random vibration analysis of structures under time-varying coherent non-stationary excitation according to claim 1 is characterized by: The EPSD matrix S of the structural response under time-varying coherent non-stationary excitation is y (ω,t) is expressed as: S y (ω,t)=I(ω,t)I T* (ω,t) Where, the evolution frequency response matrix I(ω,t) is directly calculated by the following formula:

Images