A dynamic load frequency domain identification method integrating dynamic response spectrum leakage compensation

Abstract

The present invention proposes a dynamic load frequency-domain identification method integrating dynamic response spectrum leakage compensation. Aiming at the problem that the frequency response function matrix is severely ill-conditioned near the resonance frequency and the inversion is unstable, resulting in large errors in the frequency-domain dynamic load identification, the method adopts the means of integrating the spectrum leakage error compensation into the frequency-domain dynamic load identification process. By constructing a compensation function with clear physical meaning to correct the ill-conditioned characteristics in the resonance frequency band, the effects of reducing the identification error in the resonance frequency band and improving the identification stability are achieved. At the same time, the method maintains the traditional inversion framework, with small computational amount and strong generality, which is convenient for popularization and repeated use in engineering scenarios.

Description

Technical Field

[0001] The present invention relates to the cross - field of structural dynamics and signal processing, and specifically to a dynamic load frequency - domain identification method integrating dynamic response spectrum leakage compensation. Background Technique

[0002] In engineering fields such as civil engineering, aerospace, and machinery, structures are often subjected to unknown time - varying dynamic loads, such as wind vibration, impact, earthquake, or aerodynamic excitation, which pose a severe test to structural safety. Especially in the aerospace field, external dynamic loads often easily trigger the flutter problem of aircraft. Specifically, flutter is a self - excited vibration phenomenon generated by the interaction between aerodynamic loads and the elastic modes of the structure itself. If not controlled, it may quickly lead to structural damage and seriously threaten flight safety. Since such external dynamic loads are difficult to directly measure, it is usually necessary to identify the external excitation by inverting the structural response signal, that is, to carry out dynamic load identification. The dynamic load identification technology not only helps to build a high - precision aerodynamic - structure coupling model and improve the accuracy of flutter prediction and analysis, but also provides input information for the monitoring system and further serves as the feedback signal of the active control system to effectively suppress flutter and ensure the safety of the aircraft structure. And the process of dynamic load identification belongs to a typical second - kind Fredholm inverse problem, with obvious "ill - posed" characteristics, that is, small perturbations will lead to large deviations in the identification results, and it is extremely challenging to solve stably.

[0003] Among the existing dynamic load identification methods, frequency - domain methods are widely used because they can obtain high - resolution spectrum information near the resonance frequency. Such methods generally include: establishing a structural frequency response function model, performing discrete Fourier transform on the response signal, and solving the external excitation in the frequency domain. However, in actual structures, especially in low - damping systems, the frequency response function matrix often shows ill - posedness near the resonance frequency band, with a sharp decay of singular values and a very large condition number, resulting in the inverse solution process being very sensitive to noise and modeling errors and generating relatively serious instability. Summary of the Invention

[0004] Technical Problems to be Solved

[0005] In response to the problems in the background technique, existing research has developed a variety of stabilization methods, including singular value decomposition (SVD) and its variants (such as TSVD, GSVD), Tikhonov regularization, projection methods, sparse optimization methods, and data - driven methods, etc. These methods mostly start from the numerical level, optimize the inversion model or add regularization terms to improve stability and accuracy. However, they generally ignore a deeper source of error in frequency - domain identification: the spectral energy leakage of dynamic response signals.

[0006] In the identification of dynamic loads in the frequency domain, the dynamic response signals are often calculated for their spectra using the finite-length discrete Fourier transform. Due to time-domain truncation, energy diffusion occurs during the frequency-domain transformation of the signal, namely the spectral leakage phenomenon, which leads to distortion of frequency components and energy transfer. This problem is particularly severe in low-damping structures because their dynamic responses exhibit sharp resonance peaks with extremely narrow frequency bands, making it extremely easy for leakage errors to be amplified near the resonance frequencies. Although it is often alleviated by applying window functions (such as the Hanning window, Kaiser window, etc.), spectral leakage cannot be completely eliminated and may still cause significant identification biases under simulation conditions without noise and modeling errors.

[0007] For a long time, spectral leakage has often been regarded as a minor problem in signal processing and has not been incorporated into the core scope of the study on the ill-posedness of dynamic load identification. Most existing methods attribute the identification errors to noise amplification or inaccurate frequency response functions, without realizing that spectral leakage is one of the fundamental mechanisms causing identification instability, featuring systematicity, inevitability, and being easily overlooked. Especially in low-damping structures, even if all other conditions are ideal, spectral leakage is sufficient to cause significant errors in the structural frequency-domain identification results.

[0008] Therefore, there is an urgent need to develop a compensation identification strategy for signal spectral leakage. Starting from the mechanism of constructing frequency-domain signals, it systematically compensates for the distortion of the dynamic response spectral leakage caused by the finite-length discrete Fourier transform, thereby improving the stability and accuracy of dynamic load identification. This is precisely the key technical problem to be solved by this invention.

[0009] The technical solution of this invention is as follows:

[0010] A dynamic load frequency-domain identification method integrating compensation for dynamic response spectral leakage, comprising the following steps:

[0011] Step 1: Using the known test dynamic load, obtain the modal response compensation function matrix of the loaded structure through the following steps :

[0012] Step 1.1: Apply the test dynamic load to the loaded structure , and collect the dynamic response time-domain signals ;

[0013] Step 1.2: Model the structural dynamics of the loaded structure to obtain the modal shape matrix and modal dynamic stiffness matrix , where the modal shape matrix includes the modal shape matrix composed of the n-order modal shape vectors of m dynamic response measurement points [ Φ m × n ] and the modal shape matrix composed of the n-order modal shape vectors of l dynamic load application points [ Φ l × n ] , is the frequency;

[0014] Step 1.3: Perform modal decomposition on the loaded structure to obtain the modal dynamic load and the modal dynamic response time-domain signal ;

[0015] Step 1.4: Based on the obtained modal dynamic load and the modal dynamic response time-domain signal , calculate the power spectral density of each order of modal dynamic load and the power spectral density of the modal dynamic response ;

[0016] Step 1.5: Calculate the power spectral density of each order of modal dynamic response of the loaded structure under the test load in the ideal case :

[0017] S y ⌢ i y ⌢ i ( ω ) = 1 K i - M i ω 2 + j C i ω ⋅ S f i f i ( ω ) ⋅ [ 1 K i - M i ω 2 + j C i ω ] H

[0018] where is the modal mass of the i-th order mode of the loaded structure, is the modal stiffness of the i-th order mode of the loaded structure, is the modal damping of the i-th order mode of the loaded structure, is the imaginary unit;

[0019] Step 1.6: According to the power spectral density of the modal dynamic response obtained in Step 1.4 and the power spectral density of each order of modal dynamic response obtained in Step 1.5, calculate the modal response compensation function corresponding to each order of modal dynamic response:

[0020] γ mod al i ( ω ) = S y ⌢ i y ⌢ i ( ω ) S y i y i ( ω ) α mod al i ( ω ) = { γ mod al i ( ω ), ω ∈ [ ω i − ω half , ω i + ω half ] 1 , ω ∉ [ ω i − ω half , ω i + ω half ]

[0021] where, is the ratio function of the spectral leakage of the modal response within the full frequency band, is the natural frequency of the i-th order mode, is the spectral half-bandwidth of the modal response compensation function, used to control the frequency domain range and intensity of the compensation;

[0022] Step 1.7: Construct the modal response compensation function matrix : Based on each order of modal response compensation function , jointly construct the modal response compensation function matrix :

[0023] α M ( ω ) = α mod al ( ω ) ⊗ [ α mod al ( ω ) ] T = [ α mod al 1 ( ω ) ⋅ α mod al 1 ( ω ) α mod al 1 ( ω ) ⋅ α mod al 2 ( ω ) … α mod al 1 ( ω ) ⋅ α mod al n ( ω ) α mod al 2 ( ω ) ⋅ α mod al 1 ( ω ) α mod al 2 ( ω ) ⋅ α mod al 2 ( ω ) … α mod al 2 ( ω ) ⋅ α mod al n ( ω ) ⋮ ⋮ ⋱ ⋮ α mod al n ( ω ) ⋅ α mod al 1 ( ω ) α mod al n ( ω ) ⋅ α mod al 2 ( ω ) … α mod al n ( ω ) ⋅ α mod al n ( ω ) ]

[0024] Among them represents the outer product operation;

[0025] Step 2: When the actual loaded structure is excited by an external dynamic load to be identified, collect the corresponding dynamic response time-domain signal, and calculate the modal dynamic response power spectral density corresponding to the measured dynamic response signal ; Use the modal response compensation function matrix to perform compensation to obtain the power spectral density of the dynamic load to be identified :

[0026] S F ⌢ F ⌢ ( ω ) = ( [ Φ l × n ] T ) + ⋅ Z Φ ( ω ) ⋅ [ α M ( ω ) ∘ S yy ( ω ) ] ⋅ [ Z Φ ( ω )] H ⋅ [ Φ l × n ] +

[0027] Among them ( [ Φ l × n ] T ) + is [ Φ l × n ] T the generalized inverse matrix of, [ Φ l × n ] + is [ Φ l × n ] the generalized inverse matrix of, represents the Hadamard product operation, [ Z Φ ( ω )] H is the conjugate transpose of.

[0028] Furthermore, the test dynamic load uses a swept-frequency dynamic load or an impact load.

[0029] Furthermore, in step 1.2, the modal dynamic stiffness matrix Z Φ ( ω ) = diag [ K i - M i ω 2 + j C i ω ] , is the modal mass of the i-th order mode of the loaded structure, is the modal stiffness of the i-th order mode of the loaded structure, is the modal damping of the i-th order mode of the loaded structure, is the frequency.

[0030] Furthermore, in step 1.3, the modal dynamic load and the modal dynamic response time-domain signal According to the formula

[0031] f test ( t ) = [ Φ l × n ] T F test ( t ) y test ( t ) = [ Φ m × n ] + x test ( t )

[0032] Among them [ Φ m × n ] + is [ Φ m × n ] the generalized inverse matrix of, [ Φ l × n ] T is [ Φ l × n ] the transpose of.

[0033] Furthermore, in step 1.4, let the time-domain signal of the i-th order modal dynamic load be , and the sampling time length of the signal be . Then, the finite-length Fourier transform of the time-domain signal of the i-th order modal dynamic load is:

[0034]

[0035] where is the window function in the time domain, is the imaginary unit, and the power spectral density of the modal dynamic load is:

[0036]

[0037] Let the time-domain signal of the i-th order modal dynamic response be . Perform the same finite-length Fourier transform on :

[0038]

[0039] Then, the power spectral density of the modal dynamic response is:

[0040] .

[0041] In addition, the present invention also proposes an electronic device and a readable storage medium:

[0042] An electronic device includes a processor and a memory. The memory is used to store one or more programs;

[0043] When the one or more programs are executed by the processor, the above method is implemented.

[0044] A readable storage medium stores a computer program, and when the computer program is executed by a processor, the above method is implemented.

[0045] Beneficial effects:

[0046] Aiming at the problem that the frequency response function matrix is severely ill-conditioned near the resonance frequency and the inversion is unstable, resulting in large errors in the identification of frequency-domain dynamic loads, the present invention adopts the means of integrating the spectral leakage error compensation into the frequency-domain dynamic load identification process. By constructing a compensation function with clear physical meaning to correct the ill-conditioned characteristics in the resonance frequency band, the effects of reducing the identification error in the resonance frequency band and improving the identification stability are achieved. At the same time, this method maintains the traditional inversion framework, has a small computational amount and strong generality, and is convenient for promotion and repeated use in engineering scenarios.

[0047] Additional aspects and advantages of the present invention will be given in part in the following description, become apparent in part from the following description, or be learned through the practice of the present invention. Brief Description of the Drawings

[0048] The above and / or additional aspects and advantages of the present invention will become apparent and be readily understood from the description of embodiments in conjunction with the following drawings, in which:

[0049] Figure 1 : 8-degree-of-freedom discrete vibration system;

[0050] Figure 2 : Modal response compensation functions of the 8-degree-of-freedom discrete vibration system;

[0051] Figure 3 : Comparison diagram of dynamic load identification results. Detailed Embodiments

[0052] Embodiments of the present invention will be described in detail below. The embodiments are exemplary and intended to explain the present invention, and should not be construed as a limitation to the present invention.

[0053] There are problems of unstable inversion calculation and insufficient accuracy of identification results in existing frequency-domain dynamic load identification methods. Especially in low-damping structures, due to the concentrated response spectrum and sharp resonance peaks, serious spectral energy leakage is easily caused during the frequency-domain conversion process, which further exacerbates the ill-posedness of the frequency response function inversion. Existing technologies mainly rely on numerical regularization means to deal with matrix ill-conditioning and noise interference, and fail to start from the distortion mechanism of the frequency-domain response signal itself for compensation, resulting in limited identification effects and difficult error control.

[0054] Therefore, this embodiment proposes a dynamic load frequency-domain identification method that integrates dynamic response spectrum leakage compensation. By constructing a frequency-related compensation function, the spectral leakage error generated by the finite-length discrete Fourier transform is explicitly corrected, improving the accuracy and stability of load identification from the source. This method is applicable to single-degree-of-freedom and multi-degree-of-freedom structural systems, and can combine modal decomposition and regularization strategies, showing good robustness and adaptability under different damping conditions and excitation types. The proposed method is applicable to structural vibration analysis and load inversion tasks in fields such as aerospace structures, civil engineering, and mechanical systems, and has clear engineering practical value and promotion potential.

[0055] The dynamic load frequency-domain identification method that integrates dynamic response spectrum leakage compensation proposed in this embodiment introduces an energy compensation mechanism for dynamic response spectrum leakage on the basis of the traditional dynamic load frequency-domain identification method. The core idea is to construct a compensation model by comparing the difference between the spectrum of the measured dynamic response signal and the ideal frequency-domain response, so as to correct the error caused by spectral energy leakage, specifically including the following steps:

[0056] Step 1: Using the known test dynamic load, obtain the modal response compensation function matrix of the loaded structure through the following steps :

[0057] Step 1.1: Apply the test dynamic load to the loaded structure , and collect the time-domain dynamic response signals ;

[0058] In this embodiment, the test dynamic load applied to the loaded structure by the load application device adopts a swept-frequency dynamic load. Of course, an impact load or the like can also be adopted, and the corresponding time-domain dynamic response signals are collected through sensors ; The load application device and the sensors are all conventional facilities in the art

[0059] Step 1.2: Model the structural dynamics model of the loaded structure;

[0060] For the actual loaded structure, based on the prior accurate simulation model or using the modal parameter identification method, establish an accurate modal shape matrix [ Φ m × n ] and [ Φ l × n ] , as well as the modal dynamic stiffness matrix for characterizing the dynamic equilibrium relationship of the loaded structure in the modal space ;

[0061] where is the number of dynamic response measurement points, is the order number, is the number of dynamic load application points, and there is ;

[0062] [ Φ m × n ] is the modal shape matrix composed of the n-order modal shape vectors of m dynamic response measurement points;

[0063] [ Φ l × n ] is the modal shape matrix composed of the n-order modal shape vectors of l dynamic load application points;

[0064] The modal dynamic stiffness matrix Z Φ ( ω ) = diag [ K i - M i ω 2 + j C i ω ] , is the modal mass of the i-th order mode of the loaded structure, is the modal stiffness of the i-th order mode of the loaded structure, is the modal damping of the i-th order mode of the loaded structure, is the frequency;

[0065] Step 1.3: Decompose the modes of the loaded structure:

[0066] Calculate the test dynamic load and the corresponding time-domain dynamic response signal the corresponding modal dynamic load and the modal dynamic response time-domain signal :

[0067] f test ( t ) = [ Φ l × n ] T F test ( t ) y test ( t ) = [ Φ m × n ] + x test ( t )

[0068] where [ Φ m × n ] + is [ Φ m × n ] the generalized inverse matrix of [ Φ l × n ] T is [ Φ l × n ] the transpose of

[0069] Step 1.4: Based on the obtained modal dynamic load and the modal dynamic response time-domain signal , calculate the power spectral density of each order modal dynamic load and the power spectral density of the modal dynamic response respectively;

[0070] Specifically, let the time-domain signal of the i-th order modal dynamic load be , the sampling time length of the signal be , then the finite-length Fourier transform of the time-domain signal of the i-th order modal dynamic load is:

[0071]

[0072] where is the window function in the time domain, such as Hanning window, Hamming window, etc.; is the imaginary unit, then the power spectral density of the modal dynamic load is:

[0073]

[0074] And let the time-domain signal of the i-th order modal dynamic response be , perform the same finite-length Fourier transform on :

[0075]

[0076] Then the power spectral density of the modal dynamic response is:

[0077]

[0078] Step 1.5: Calculate the power spectral density of each order modal dynamic response of the loaded structure under the test load in the ideal case (i.e., without spectral energy leakage introduced by the finite-length discrete Fourier transform) :

[0079] S y ⌢ i y ⌢ i ( ω ) = 1 K i - M i ω 2 + j C i ω ⋅ S f i f i ( ω ) ⋅ [ 1 K i - M i ω 2 + j C i ω ] H

[0080] Step 1.6: Calculate the modal response compensation function corresponding to each order of modal dynamic response according to the power spectral density of the modal dynamic response obtained in Step 1.4 and the power spectral density of each order of modal dynamic response under ideal conditions obtained in Step 1.5 :

[0081] γ mod al i ( ω ) = S y ⌢ i y ⌢ i ( ω ) S y i y i ( ω ) α mod al i ( ω ) = { γ mod al i ( ω ), ω ∈ [ ω i − ω half , ω i + ω half ] 1 , ω ∉ [ ω i − ω half , ω i + ω half ]

[0082] Wherein, is the proportion function of modal response spectrum leakage within the full frequency band, is the natural frequency of the i-th order mode, is the spectral half-bandwidth of the modal response compensation function, used to control the frequency domain range and intensity of compensation; belongs to empirical parameters, which can be adjusted or determined according to specific usage scenarios.

[0083] Step 1.7: Construct the modal response compensation function matrix : Based on each order of modal response compensation function , jointly construct the modal response compensation function matrix :

[0084] α M ( ω ) = α mod al ( ω ) ⊗ [ α mod al ( ω ) ] T = [ α mod al 1 ( ω ) ⋅ α mod al 1 ( ω ) α mod al 1 ( ω ) ⋅ α mod al 2 ( ω ) … α mod al 1 ( ω ) ⋅ α mod al n ( ω ) α mod al 2 ( ω ) ⋅ α mod al 1 ( ω ) α mod al 2 ( ω ) ⋅ α mod al 2 ( ω ) … α mod al 2 ( ω ) ⋅ α mod al n ( ω ) ⋮ ⋮ ⋱ ⋮ α mod al n ( ω ) ⋅ α mod al 1 ( ω ) α mod al n ( ω ) ⋅ α mod al 2 ( ω ) … α mod al n ( ω ) ⋅ α mod al n ( ω ) ]

[0085] Where represents the outer product operation.

[0086] Step 2: When the actual loaded structure is excited by the external dynamic load to be identified, collect the corresponding dynamic response time-domain signal, and calculate the power spectral density of the modal dynamic response corresponding to the measured dynamic response signal by using the power spectral density calculation method of the modal dynamic response described in Step 1.4 ; Use the modal response compensation function matrix to for compensation to obtain the power spectral density of the dynamic load to be identified :

[0087] S F ⌢ F ⌢ ( ω ) = ( [ Φ l × n ] T ) + ⋅ Z Φ ( ω ) ⋅ [ α M ( ω ) ∘ S yy ( ω ) ] ⋅ [ Z Φ ( ω )] H ⋅ [ Φ l × n ] +

[0088] Where ( [ Φ l × n ] T ) + is [ Φ l × n ] T 's generalized inverse matrix, [ Φ l × n ] + is [ Φ l × n ] generalized inverse matrix denotes the Hadamard product operation [ Z Φ ( ω )] H is conjugate transpose of

[0089] Based on the above process, in this embodiment, the 8-degree-of-freedom discrete vibration system shown in Figure 1 is used to verify the effectiveness of the method proposed by the present invention

[0090] The mass coefficients of the discrete mass blocks of the 8-degree-of-freedom discrete vibration system are all 3.5 kg, and the stiffness coefficients are all 24000 N; here, we set the modal damping ratios of each order to 0.5% respectively, and the modal parameters of the system are shown in Table 1

[0091] Table 1: Modal parameters of the 8-degree-of-freedom discrete vibration system

[0092]

[0093] First, we use a sinusoidal swept-frequency dynamic load with a frequency band width of 0 - 30 Hz and a duration of 64 seconds as the test dynamic load to load the 8-degree-of-freedom discrete vibration system, and collect the displacement response of each dynamic response measurement point of the system to construct the modal response compensation function and the modal response compensation function matrix , where the corresponding curves of each order modal response compensation function are as shown in Figure 2 . Then, a stationary random load sample with an amplitude of 10 N, a frequency band width of 0 - 30 Hz, and a duration of 64 seconds is used to load the 8-degree-of-freedom discrete vibration system, and the displacement response data of each degree of freedom of the system is collected for dynamic load identification verification. Here, we first compare the dynamic load identification results obtained by using the conventional direct inversion method, and compare the dynamic load identification results obtained by the method of the present invention combined with the modal response compensation function matrix with the true dynamic load. The identification results obtained are as shown in Figure 3 . It can be seen that the method of the present invention combined with the modal response compensation function matrix can very effectively reduce the dynamic load identification error brought by the conventional direct inversion method

[0094] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention without departing from the principles and purposes of the present invention

Claims

1. A dynamic load frequency domain identification method integrating dynamic response spectrum leakage compensation, characterized by: The following steps are involved: Step 1: Using the known test dynamic load, obtain the modal response compensation function matrix of the loaded structure through the following steps: : Step 1.1: Apply test dynamic load to the loaded structure , and collect dynamic response time domain signals ; Step 1.2: Model the structural dynamics model of the loaded structure to obtain the modal vibration matrix and modal dynamic stiffness matrix , where the modal vibration matrix includes the modal vibration matrix composed of the n-order modal vibration vectors of m dynamic response measurement points and the modal mode matrix composed of the n-order modal mode vectors of l dynamic load application points , is the frequency; Step 1.3: Perform modal decomposition on the loaded structure to obtain modal dynamic loads And the modal dynamic response time domain signal ; Step 1.4: Based on the obtained modal dynamic loads and modal dynamic response time domain signal , calculate the power spectrum density of each modal dynamic load separately and the power spectral density of the modal dynamic response ; Step 1.5: Calculate the ideal power spectrum density of each modal dynamic response of the loaded structure under the test load : in is the modal mass of the i-th mode of the loaded structure, is the modal stiffness of the i-th mode of the loaded structure, is the modal damping of the i-th mode of the loaded structure, is an imaginary unit; Step 1.6: The power spectrum density of the modal dynamic response obtained in step 1.4 And the ideal power spectrum density of each modal dynamic response obtained in step 1.5 , calculate the modal response compensation function corresponding to each order modal dynamic response : in, is the proportional function of the modal response spectrum leakage in the full frequency band, is the natural frequency of the i-th mode, is the spectrum half-bandwidth of the modal response compensation function, which is used to control the frequency domain range and intensity of the compensation; Step 1.7: Construct the modal response compensation function matrix :According to the compensation function of each modal response , together construct the modal response compensation function matrix : in represents the outer product operation; Step 2: When the actual loaded structure is excited by the external dynamic load to be identified, the corresponding dynamic response time domain signal is collected, and the modal dynamic response power spectrum density corresponding to the measured dynamic response signal is calculated. ; Using the modal response compensation function matrix right Compensate and obtain the power spectrum density of the dynamic load to be identified : in for The generalized inverse matrix of for The generalized inverse matrix of represents the Hadamard product operation, for The conjugate transpose of .

2. The method for dynamic load frequency domain identification integrating dynamic response spectrum leakage compensation according to claim 1, characterized in that: The test dynamic load adopts a sweep frequency dynamic load or an impact load.

3. The method for dynamic load frequency domain identification integrated with dynamic response spectrum leakage compensation according to claim 1, characterized in that: In step 1.2, the modal dynamic stiffness matrix , is the modal mass of the i-th mode of the loaded structure, is the modal stiffness of the i-th mode of the loaded structure, is the modal damping of the i-th mode of the loaded structure, is the frequency.

4. The method for dynamic load frequency domain identification integrated with dynamic response spectrum leakage compensation according to claim 1, characterized in that: In step 1.3, the modal dynamic load And the modal dynamic response time domain signal According to the formula in for The generalized inverse matrix of for The transpose of .

5. The method for dynamic load frequency domain identification integrated with dynamic response spectrum leakage compensation according to claim 1, characterized in that: In step 1.4, let the time domain signal of the i-th order modal dynamic load be , the sampling time length of the signal is , then the time domain signal of the i-th order modal dynamic load is The finite length Fourier transform of is: in is the window function in the time domain, is an imaginary unit, then the power spectrum density of the modal dynamic load is: Assume that the time domain signal of the i-th order modal dynamic response is ,right Perform the same finite-length Fourier transform: Then the power spectrum density of the modal dynamic response is: 。 6. An electronic device comprising a processor and a memory, wherein the memory is used to store one or more programs; characterized in that: When the one or more programs are executed by the processor, the method according to any one of claims 1 to 5 is implemented.

7. A readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the method according to any one of claims 1 to 5 is implemented.

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