A computational method for coupled responses of engineering structures under multiple vibration sources

Abstract

The present application relates to a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources, comprising performing a modal analysis on the structure to obtain the required modal characteristics; determining the target power spectra of various vibration sources and constructing a multi-vibration source mutual power spectrum matrix; calculating the power spectrum of the coupled response of the structure under the action of multiple vibration sources and calculating the root mean square; selecting a peak factor and multiplying the calculated root mean square to obtain a standard value of the response; the standard value of the response includes a standard value of the response related to deformation or internal force; the calculated standard value of the response related to deformation is used for verification of the overall deformation and comfort of the structure; the calculated standard value of the response related to internal force is used for verification of internal force combination and component section and reinforcement.

Description

Technical Field

[0001] The present application relates to a method for calculating coupled responses of an engineering structure under the action of multiple vibration sources. Background Art

[0002] In normal use, the main structure of a building is susceptible to the coupling of multiple power sources such as traffic, environment, and equipment, causing noise, vibration, and other problems. Multiple vibration sources are generally reflected in the different types of various power sources and their spatial distribution. The difference in the type of vibration source is due to the different occurrence mechanisms. Different vibration sources will produce vibrations with different frequency characteristics; spatial distribution refers to the fact that the same type of vibration source is not uniformly input at different locations of the structure, such as wind vibration and the traveling wave effect of large-span structures. In order to control structural vibration, vibration isolation or vibration reduction devices are often set in the structure. At this time, the structure may become a non-proportional damping system, and the modal method used for the classical proportional damping system cannot accurately evaluate the actual response. For these structures, when faced with multiple uncertain vibration source excitations, how to quickly and accurately evaluate their responses is the key to measuring vibration impact and control effects. In the prior art, the research on the coupled response of structures under the action of multiple vibration sources mainly relies on experimental measurements or numerical simulations based on time-history analysis. The coupling methods are mostly based on simple superposition combinations or semi-empirical formulas of single vibration source responses. The intrinsic connection between the vibration source excitation and the coupled response components has not been established, and there is also a lack of more general calculation methods.

[0003] Therefore, there is an urgent need in the prior art for a more general calculation method that can efficiently handle the coupled response of structures under the action of multiple vibration sources. Summary of the invention

[0004] The present application provides a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources. Compared with simulation calculations performed by applying multiple load histories to the structure, it provides a solution with higher computational efficiency and can well guide engineering vibration-resistant design.

[0005] The present application relates to a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources, comprising the following steps:

[0006] (1) Perform modal analysis on the structure to obtain the required modal characteristics;

[0007] (2) Determine the target power spectrum of various vibration sources and construct the multi-vibration source cross-power spectrum matrix;

[0008] (3) Calculate the power spectrum of the coupled response of the structure under the action of multiple vibration sources and calculate the root mean square;

[0009] (4) selecting a peak factor and multiplying the calculated root mean square to obtain a standard value of the response; the standard value of the response includes a standard value of the response related to deformation or internal force;

[0010] (5) The calculated standard response value related to deformation is used to verify the overall deformation and comfort of the structure; the calculated standard response value related to internal force is used to verify the internal force combination, component section and reinforcement.

[0011] In step (3), the power spectrum of the coupled response under the action of multiple vibration sources can be calculated in the form of multiple summation, and its variance is calculated according to the following formula:

[0012]

[0013] Where, X m (ω) is the mth component of the structural response, When the subscript is j or q, replace i or p in the above formula with j or q;

[0014] Among them, N M is the number of modes considered in the calculation, N P is the number of vibration sources; the subscripts i and j are the mode numbers, and the formula with subscript j is applicable by referring to the formula with subscript i; p and q are the load numbers, and the formula with subscript q is applicable by referring to the formula with subscript p; the positive i is the imaginary unit, ω i is the natural angular frequency of the i-th mode, H i is the frequency response function of the i-th mode, ξ i is the damping ratio of the ith mode, H i * is the conjugate complex number of the frequency response function of the i-th mode, S pq is the cross power spectrum of the pth and qth vibration sources, A ip is the displacement transfer constant between the i-th mode and the p-th vibration source, B ip is the velocity transfer constant between the i-th mode and the p-th vibration source;

[0015] r p is the load influence vector of the pth vibration source, that is, the pth column of the matrix R in the structural motion differential equation, is the matrix α i The mth row of is the matrix β i The mth row of β i , γ i is the matrix 2ψ our-i E i The real and imaginary parts of 2ψ out-i E i =β i +iγ i ; E i The calculation formula is

[0016] ψ iis the state space equation The i-th order vibration mode displacement satisfies the generalized eigenvalue equation:

[0017] (λ i A S +B S )φ i =0

[0018]

[0019] In the formula, λ i and φ are the eigenvalue and eigenvector of the generalized eigenvalue equation, respectively;

[0020] ψ our-i for i The corresponding vibration mode response; a i The calculation formula is φ i T A S φ i ;

[0021] A s , B s is the system state matrix,

[0022] M, C, K, and R are the mass, damping, stiffness, and load distribution matrices of the differential equation of structural motion, that is, Where x is the structural displacement response and p(t) is the multi-vibration source load vector.

[0023] In step (3), the power spectrum of the coupled response under the action of multiple vibration sources can be calculated in the form of matrix multiplication, and its variance is calculated according to the following formula:

[0024] S Xm (ω)=(H (m) ) * RS P (ω)R T (H (m) ) T

[0025] In the formula,

[0026] Where, X m (ω) is the mth component of the structural response, S P (ω) is the multi-vibration source mutual power spectrum matrix; (H (m) ) * H (m) The conjugate matrix of

[0027] Among them, N Mis the number of modes considered in the calculation, the subscript i is the mode number, the positive i is the imaginary unit, ω i is the natural angular frequency of the i-th mode, H i is the frequency response function of the i-th mode, ξ i is the damping ratio of the i-th mode; is the matrix α i The mth row of is the matrix β i The mth row of β i , γ i is the matrix 2ψ out-i E i The real and imaginary parts of 2ψ out-i E i =β i +iγ i ; E i The calculation formula is

[0028] ψ i is the state space equation The i-th order vibration mode displacement satisfies the generalized eigenvalue equation:

[0029] (λ i A S +B S )φ i =0

[0030]

[0031] In the formula, λ i and φ i are the eigenvalues ​​and eigenvectors of the generalized eigenvalue equation, respectively;

[0032] ψ our-i for i The corresponding vibration mode response; a i The calculation formula is φ i T A S φ i ;

[0033] A s , B s are the system state matrix and input matrix,

[0034] M, C, K, and R are the mass, damping, stiffness, and load distribution matrices of the differential equation of structural motion, that is, Where x is the structural displacement response and p(t) is the multi-vibration source load vector.

[0035] Among them, S Xm(ω), the zero-order moment is calculated to get the variance of the response, and the square root of the obtained variance is obtained to get the root mean square σ of the response Xm ,Right now

[0036]

[0037] The present application proposes a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources. This method can simultaneously consider the situation of multiple vibration source input and non-proportional damping, accurately consider the coupling relationship between multiple vibration sources and multiple vibration modes, and has a more rigorous theoretical nature than empirical methods. In addition, the matrix multiplication formula proposed in the present application is more suitable for calculation using a computer program, which significantly improves the calculation efficiency compared to the multiple summation form. The present application uses the calculated standard response values ​​related to deformation to verify the overall deformation and comfort of the structure; and uses the calculated standard response values ​​related to internal forces to verify the combination of internal forces, component sections, and reinforcement. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] The accompanying drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation of the present invention. In the accompanying drawings:

[0039] Figure 1 It is a flow chart of the calculation method of the structural coupling response under the action of multiple vibration sources of the present application.

[0040] Figure 2 It is a schematic diagram of the three-degree-of-freedom multi-vibration source non-proportional damping model of Example 1 of the present application.

[0041] Figure 3 It is a schematic diagram of the three-degree-of-freedom complex vibration type of Example 1 of the present application.

[0042] Figure 4 It is the target power spectrum of the vertical ground acceleration of Example 1 and the power spectrum and time history of the generated random process.

[0043] Figure 5 It is the target power spectrum of the vertical concentrated force of each mass block in Example 1 and the power spectrum and time history of the generated random process.

[0044] Figure 6 are the target power spectral density of the additional concentrated force of the top mass block in Example 1 and the power spectral density and time history of the generated random process.

[0045] Figure 7 It is the structural response power spectrum density of mass block m1 obtained by using the method of the present application and the MCS method in Example 1.

[0046] Figure 8It is the structural response power spectrum density of mass block m2 obtained by using the method of the present application and the MCS method in Example 1.

[0047] Fig. 9 It is the structural response power spectrum density of mass block m3 obtained by using the method of the present application and the MCS method in Example 1.

[0048] Fig.10 It is a schematic diagram of the lateral force resistance system of the super high-rise structure of Example 2.

[0049] Fig.11 This is the wind tunnel test model of Example 2 of the present application.

[0050] Fig.12 It is a typical pulsating wind time history and power spectrum diagram obtained from the wind tunnel test of Example 2 of the present application.

[0051] Fig.13 This is a comparison chart of the X-direction floor displacement, velocity, and acceleration root mean square obtained by using the method proposed in this application and the time history analysis method in Example 2 of this application.

[0052] Fig.14 This is a comparison chart of the Y-direction floor displacement, velocity, and acceleration root mean square obtained by using the method proposed in this application and the time-history analysis method in Example 2 of this application.

[0053] Fig.15 This is a comparison chart of the root mean square of the interlayer displacement angles in the X and Y directions obtained by using the method proposed in this application and the time history analysis method in Example 2 of this application.

[0054] Fig.16 This is a comparison chart of the root mean square internal force of the giant column obtained by using the method proposed in this application and the time history analysis method in Example 2 of this application. DETAILED DESCRIPTION

[0055] In order to make the purpose, technical solutions and advantages of this application clearer, the technical solutions in this application will be clearly and completely described below in conjunction with the drawings in this application. Obviously, the described embodiments are part of the embodiments of this application, not all of the embodiments. Based on the embodiments in this application, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of this application.

[0056] This application proposes a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources. In actual engineering, the amplitude or phase of various vibration sources often has uncertainty and can be regarded as a random process. In random vibration theory, the statistical characteristics of input and response are usually more concerned, such as mean, variance or vibration level. The relationship between extreme values ​​and statistical values ​​can be determined based on probability theory, such as peak factor.

[0057] According to the present application, a method for calculating the coupled response of an engineering structure under the action of multiple vibration sources comprises the following steps:

[0058] Step 1: Perform modal analysis on the structure to obtain the required modal characteristics, which may include modal characteristics such as frequency, damping ratio, vibration mode displacement or vibration mode force.

[0059] When performing modal analysis, the modal characteristics can be either real modal characteristics of a proportional damping system or complex modal characteristics of a non-proportional damping system, and the modal characteristics are related to the type of response to be sought. If the type of response to be sought is deformation-related, such as displacement, velocity, or acceleration, the modal characteristics can include frequency, damping ratio, and mode displacement. If the type of response to be sought is force-related, such as internal force, the modal characteristics can include frequency, damping ratio, mode displacement, and mode force.

[0060] Step 2: Determine the target power spectra of various vibration sources and construct the multi-vibration source cross-power spectrum matrix.

[0061] The target power spectra of various vibration sources can be selected independently according to the type of vibration source, such as the filtered white noise spectrum of earthquake action, the Davenport spectrum or white noise spectrum of pulsating wind, the harmonic power spectrum of equipment or pedestrian excitation, etc. The power spectrum can also be obtained by transforming the time domain signal of the vibration source into the frequency domain according to the test or actual measurement using detection equipment.

[0062] Use S P (ω) represents the multi-vibration source cross-power spectrum matrix, each element S pq (ω) represents the cross power spectrum of the pth and qth vibration sources.

[0063] Step three, calculate the power spectrum of the coupled response under the action of multiple vibration sources and calculate the root mean square.

[0064] This application provides two calculation formulas for calculating the power spectrum of coupled response under multiple vibration sources. The first is a multiple summation form, with the mth component X of the structural response m (ω) as an example, its variance can be calculated according to the following formula:

[0065]

[0066] In the formula, When the subscripts are j and q, just replace i and p in the above formula with j and q;

[0067] Among them, N M is the number of modes considered in the calculation, N p is the number of vibration sources; the subscripts i and j are the mode numbers, and the formula with subscript j can refer to the formula with subscript i for application; p and q are the load numbers, and the formula with subscript q can refer to the formula with subscript p for application; the positive i is the imaginary unit, ωi is the natural angular frequency of the i-th mode, H i is the frequency response function of the i-th mode, ξ i is the damping ratio of the i-th mode, H i * is the conjugate complex number of the frequency response function of the i-th mode, S pq is the cross power spectrum of the pth and qth vibration sources, A ip is the displacement transfer constant between the i-th mode and the p-th vibration source, B ip is the velocity transfer constant between the i-th mode and the p-th vibration source. For the proportional damping system, B ip =0;

[0068] r p is the load influence vector of the pth vibration source, that is, the pth column of the matrix R in the structural motion differential equation, is the matrix α i The mth row of is the matrix β i The mth row of β i , γ i is the matrix 2ψ our-i E i The real and imaginary parts of 2ψ out-iE i=β i +iγ i ; E i The calculation formula is

[0069] ψ i is the state space equation The i-th order vibration mode displacement satisfies the generalized eigenvalue equation:

[0070] (λ i A S +B S )φ i =0

[0071]

[0072] In the formula, λ i and φ i are the eigenvalues ​​and eigenvectors of the generalized eigenvalue equation, respectively.

[0073] ψ our-i for i The corresponding modal response can be the modal displacement or modal force output by the system; a i The calculation formula is φ i T A S φ i ;

[0074] A s , B s is the system state matrix,

[0075] M, C, K, and R are the mass, damping, stiffness, and load distribution matrices of the differential equation of structural motion, that is, Where x is the structural displacement response and p(t) is the multi-vibration source load vector.

[0076] The second is the matrix multiplication form, which is significantly more efficient than the multiple summation form. m (ω) as an example, its variance can be calculated according to the following formula:

[0077] S Xm (ω)=(H (m) ) * RS P (ω)R T (H (m) ) T

[0078] In the formula,

[0079] Among them, S P (ω) is the cross-power spectrum matrix of multiple vibration sources; (H (m) ) * H (m) The conjugate matrix of , and the remaining symbols have the same meanings as in the multiple summation form.

[0080] Find S Xm (ω), the variance of the response can be obtained by calculating its zero-order moment, and the root mean square of the response can be obtained by taking the square root, that is,

[0081]

[0082] Step 4: Select an appropriate peak factor and multiply it by the RMS value of the response to obtain the standard value of the response and use it to guide the design.

[0083] In engineering practice, the peak factor can be taken as 3.0. The larger the peak factor, the higher the reliability, but the design cost also increases accordingly. The main contents of engineering structure design include structural deformation verification and component bearing capacity verification. The standard response values ​​related to deformation calculated by the above method of this application, such as displacement angle and acceleration standard values, can be used for overall structural deformation and comfort verification; the standard response values ​​related to internal forces, such as axial force, shear force and bending moment standard values, can be used for internal force combination and component section and reinforcement verification.

[0084] Example 1

[0085] like Figure 2The three-degree-of-freedom mass-spring-damper model shown has a block mass of: m i =200kg (i=1~3); spring stiffness is: k1=200kN / m, k2=80kN / m, k3=100kN / m; damping constant of viscous damper is: c1=50N·s / m, c2=100N·s / m, c3=6000N·s / m. Multi-vibration source excitation includes vertical ground acceleration , the vertical concentrated forces p1(t)~p3(t) of each mass block and the additional concentrated force p of the top mass block h (t). The response of the structure under dynamic load is the relative displacement, relative velocity and absolute acceleration of each mass block.

[0086] First, the complex modal analysis of the three-degree-of-freedom model is performed according to the steps described in this application. The frequency and damping ratio of the complex modes are shown in the table below, and the complex vibration type is shown in Figure 3 .

[0087] Mode shape Natural frequency (Hz) Damping ratio (%) 1 1.77 3.87％ 2 5.27 88.04％ 3 6.09 2.69％

[0088] After completing the above complex modal analysis, the load target power spectrum is selected according to step 2 described in this application. In Example 1 of this application, the ground acceleration Using the filtered white noise model, its power spectrum expression is:

[0089]

[0090] Where ω g ,ω h and g To reflect the parameters of the site characteristics, the values ​​in this embodiment are ω g =37.70rad / s,ω h =75.40rad / s,ξ g =0.15. The vertical concentrated forces p1(t), p2(t), and p3(t) are correlated stationary white noise models. p2(t) and p3(t) are delayed by 0.5s and 1s respectively relative to p1(t), i.e., p2(t) = p1(t-0.5) and p3(t) = p1(t-1.0). The top mass block m3 has an additional excitation p h (t) is the harmonic load containing two frequency components of 5 Hz and 10 Hz.

[0091] After obtaining the load power spectrum information required for step 2 of this application, the power spectrum and root mean square of the response are calculated according to the formula of step 3 of this application. In order to verify the accuracy of the method proposed in this application, Monte Carlo simulation (MCS) is used for verification: the power spectrum of the above-mentioned various excitations is used as the target power spectrum, and the trigonometric series method is used to generate 200 groups of time-course signals of random processes and the direct integration method is used to calculate the response time course. The mean of these 200 groups of responses is then calculated and compared with the method proposed in this application (MICCQC). The power spectrum density and time course of the generated random process are shown in Figure 4-6 For the sake of clarity, only part of the signal duration is shown. By comparing with the target power spectrum, it can be seen that each generated signal meets the preset statistical characteristics. The root mean square results of the response of each mass block are shown in the table below, and the relative error is in brackets. The response power spectrum density results of each mass block are shown in Figure 7 , Figure 8 and Fig. 9 .

[0092]

[0093] The calculation results show that the calculation method of the structural coupling response under the action of multiple vibration sources proposed in this application is consistent with statistics and has good accuracy. After obtaining the root mean square of the response, the standard value of the response is calculated according to step 4 described in this application. In this embodiment, the peak factor is taken as 3.0, and the standard value of the response is obtained as shown in the following table.

[0094]

[0095] Example 2

[0096] Example 2 of the present application is a giant frame-cantilever-core tube super high-rise structure with a building height of 489m, a structural height of 464m, a total of 95 floors above ground, and a structural system as shown in Fig.10 Rayleigh damping is used in the analysis, with boundary periods of 0.4s and 8.2s and a damping ratio of 3%. Since Rayleigh damping is proportional damping, real modal analysis can be performed, and a total of 100 modes are calculated.

[0097] The vibration source excitation of the structure is the fluctuating wind load, and the fluctuating wind time history data comes from Fig.11 The wind tunnel test shown in the figure. X-axis horizontal force, Y-axis horizontal force and torque around the Z axis are applied to each layer, so this example is a special case of spatially distributed multiple vibration sources. At 90 floors or less, every 5 floors are a segment and a set of horizontal force and torque time history is specified, so the vibration source coherence coefficient within the segment is 1. Above 90 floors, a set of time history is specified for each layer, and the vibration source coherence is weak.

[0098] A total of 63 different fluctuating wind time histories are input in this example. Typical fluctuating wind time histories and power spectra are shown in Fig.12The sampling frequency in the wind tunnel test is 2Hz. The power spectrum is calculated by transforming the time history signal into the frequency domain as described in step 2 of this application. After obtaining the load power spectrum information required for step 2 of this application, the power spectrum and root mean square of the response are calculated according to the formula of step 3 of this application. In order to verify the accuracy of the method proposed in this application, the time history analysis calculation is supplemented. The calculation results of the root mean square of the response are compared. Figure 13-16 . The calculation results show that the method (MICQC) proposed in this application is consistent with the time history analysis (THA) and has good accuracy. For the embodiment of the present application, the multiple summation form proposed in step three requires 208 hours of calculation time, while the matrix multiplication proposed in step three only takes 14 seconds, and the calculation efficiency is significantly improved. After obtaining the response root mean square, the standard value of the response can be calculated according to step four described in this application for guiding the design.

[0099] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present application, rather than to limit it. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present application.

Claims

1. A method for calculating the coupled response of an engineering structure under the action of multiple vibration sources, characterized in that: The following steps are involved: (1) Perform modal analysis on the structure to obtain the required modal characteristics; (2) Determine the target power spectrum of various vibration sources and construct the multi-vibration source cross-power spectrum matrix; (3) Calculate the power spectrum of the coupled response of the structure under the action of multiple vibration sources and calculate the root mean square; (4) selecting a peak factor and multiplying the calculated root mean square to obtain a standard value of the response; the standard value of the response includes a standard value of the response related to deformation or internal force; (5) The calculated response standard values ​​related to deformation are used for verification of overall deformation and comfort of the structure; the calculated response standard values ​​related to internal forces are used for verification of internal force combination, component section and reinforcement; In step (3), the power spectrum of the coupled response under the action of multiple vibration sources is calculated using a multiple summation form, and its variance is calculated using the following formula: Where, X m (ω) is the mth component of the structural response, When the subscript is j or q, replace i or p in the above formula with j or q; Among them, N M is the number of modes considered in the calculation, N P is the number of vibration sources; the subscripts i and j are the mode numbers, and the formula with subscript j is applicable by referring to the formula with subscript i; p and q are the load numbers, and the formula with subscript q is applicable by referring to the formula with subscript p; the positive i is the imaginary unit, ω i is the natural angular frequency of the i-th mode, H i is the frequency response function of the i-th mode, ξ i is the damping ratio of the ith mode, H i * is the conjugate complex number of the frequency response function of the i-th mode, S pq is the cross power spectrum of the pth and qth vibration sources, A ip is the displacement transfer constant between the i-th mode and the p-th vibration source, B ip is the velocity transfer constant between the i-th mode and the p-th vibration source; r p is the load influence vector of the pth vibration source, that is, the pth column of the matrix R in the structural motion differential equation, is the matrix α i The mth row of is the matrix β i The mth row of β i , γ i is the matrix 2ψ out-i E i The real and imaginary parts of 2ψ out-i E i =β i +iγ i ; E i The calculation formula is ψ i is the state space equation The i-th order vibration mode displacement satisfies the generalized eigenvalue equation: (l i A S +B S )φ i =0 In the formula, λ i and φ i are the eigenvalues ​​and eigenvectors of the generalized eigenvalue equation, respectively; ψ out-i for i The corresponding vibration mode response; a i The calculation formula is φ i T A S φ i ; A s , B s is the system state matrix, M, C, K, and R are the mass, damping, stiffness, and load distribution matrices of the differential equation of structural motion, that is, Where x is the structural displacement response and p(t) is the multi-vibration source load vector.

2. The calculation method according to claim 1, characterized in that: In step (3), the power spectrum of the coupled response under the action of multiple vibration sources is calculated using matrix multiplication, and its variance is calculated using the following formula: S Xm (ω)=(H (m) ) * RS P (ω)R T (H (m) ) T In the formula, Where, X m (ω) is the mth component of the structural response, S P (ω) is the cross-power spectrum matrix of multiple vibration sources; (H (m) ) * H (m) The conjugate matrix of Among them, N M is the number of modes considered in the calculation, the subscript i is the mode number, the positive i is the imaginary unit, ω i is the natural angular frequency of the i-th mode, H i is the frequency response function of the i-th mode, ξ i is the damping ratio of the i-th mode; is the matrix α i The mth row of is the matrix β i The mth row of β i , γ i is the matrix 2ψ out-i E i The real and imaginary parts of 2ψ out-i E i =β i +iγ i ; E i The calculation formula is ψ i is the state space equation The i-th order vibration mode displacement satisfies the generalized eigenvalue equation: (l i A S +B S )φ i =0 In the formula, λ i and φ i are the eigenvalues ​​and eigenvectors of the generalized eigenvalue equation, respectively; ψ out-i for i The corresponding vibration mode response; a i The calculation formula is φ i T A S φ i ; A s , B s are the system state matrix and input matrix, M, C, K, and R are the mass, damping, stiffness, and load distribution matrices of the differential equation of structural motion, that is, Where x is the structural displacement response and p(t) is the multi-vibration source load vector.

3. The calculation method according to claim 1 or 2, characterized in that: Get S Xm (ω), the zero-order moment is calculated to get the variance of the response, and the square root of the obtained variance is obtained to get the root mean square σ of the response Xm ,Right now

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