Experimental Study of a Multipoint Random Dynamic Loading Identification Method Based on Weighted Average Technique

Most of random dynamic loading identification research studies are about the original inverse pseudoexcitation method which does not fundamentally reduce the negative effect of ill-conditioned frequency response function matrix on accuracy of loading identification. ,is paper describes a new improved method based on weighted average technique to reduce peak errors between identified load spectrum and the actual load spectrum near some natural frequencies. Meanwhile, relative error of root mean square value between identified load and the actual load is reduced. ,e introduced selection method of threshold value is innovative which is the key of weighted average technique. ,is improved loading identification method is successfully applied to experiments of cantilever beam and thermal protection composite plate structure. Identification results prove that the proposed method is valid by good agreement between identified power spectrum density and the actual one. Moreover, this method has higher accuracy than inverse pseudoexcitation method in low-frequency band.


Introduction
Dynamic loading identification is the inverse problem in structural dynamics field.Great efforts have been made in dynamic loading identification as well as its engineering applications.
e development and application of dynamic loading identification in frequency domain preceded that in time domain.As pioneers, Bartlett and Flannelly [1] as early as 1979, by experimental tests, verified the accuracy of the dynamic loading identification method in the center of vibration wheel shaft of the dynamic model in American military UH-1H helicopter.Lee and Liu [2] combined the extended Kalman filter with intelligent regression least squares method to identify external dynamic loads acting on nonlinear tower structures.Samagassi et al. [3] reconstructed loads caused by multiple factors in linear elastic structures by the wavelet correlation vector method.
Because dynamic loading identification is an inverse problem in mathematics, many scholars [4][5][6][7][8] have done much work to overcome its ill-posedness in identification procedures.Callahan and Piergentili [9] identified dynamic loads with unknown positions by using frequency response function and singular value decomposition technique.Based on wavelet multiresolution analysis basis function expansion method, Qiao et al. [10] proposed a high precision time domain loading identification method to overcome the defect of ill-posed problem.Jacquelin et al. [11] analyzed a deconvolution loading identification technique in which the ill-posedness is solved by regularization.He et al. [12] combined interval extension with frequency response function-(FRF-) based least squares approach to identify load bounds for uncertain structures.Truncated total least squares (TTLS) method is applied to compute the perturbed part of the load.
Many scholars have made prominent contributions to the development and engineering application of random dynamic loading identification methods.Soize and Batou [13] used an uncertain computational model to identify stochastic loads acting on a nonlinear system for which a few experimental responses are usable.Ren et al. [14] combined maximum entropy regularization with creative conjugate gradient to identify random dynamic loading.Jia et al. [15] proposed a weighted regularization method based on orthogonal decomposition to alleviate the ill-posed problem in random dynamic loading identification.He et al. [16] combined the modified regularization method with matrix perturbation method to identify random loads for stochastic structures.
is method can solve ill-posed problems of traditional inverse pseudoexcitation method and reduce error propagation of the loading identification process.Jia et al. [17] discussed the error source and error range in random dynamic loading identification and then introduced the weighted total least square method to reduce the error expansion and accumulation in the stochastic dynamic loading identification model.Based on inverse pseudoexcitation method, Lin et al. [18] carried out some computer simulation studies on identification of random dynamic loads and then provided suggestions on selection and arrangement of response sensors.Guo and Li [19] drew support from acceleration responses and frequency response function matrix measured by experiments to identify random dynamic loads.
Researchers in References [20,21] have discussed optimal arrangement of response sensors in dynamic loading identification processes through a large number of simulations and experiments.However, in actual engineering conditions, arrangement of response sensors is affected by structural design and external working environment [22,23].Sometimes, the installation positions of response sensors are limited or even fixed.Under these circumstances, it is necessary to optimize traditional methods to improve the accuracy of random dynamic loading identification and make these methods applicable to engineering application.Major contribution of this paper is to reduce peak errors between identified load spectrum and the actual load spectrum near some natural frequencies by introducing condition number weighted average technique to the original inverse pseudoexcitation method.Ultimately, the feasibility and effectiveness of the proposed method are demonstrated by random dynamic loading identification experiments of cantilever beam and thermal protection composite plate structures.e remainder of this paper is organized as follows.In Section 2, the model of improved random dynamic loading identification method based on condition number weighted average technique is introduced; in Sections 3 and 4, the improved method is applied to random dynamic loading identification of cantilever beam and thermal protection composite plate structures; finally, Section 5 gives some conclusions.

Model of Improved Random Loading Identification Based on Weighted
Average Technique e inverse pseudoexcitation method (IPEM) is applied to solve structural random dynamic loading identification problems.Due to considerable limitations in random dynamic loading identification procedure, some assumptions are given: (1) Problems solved by the proposed method are limited to stationary random dynamic loading identification of linear systems (2) Random responses for loading identification in this method are completely generated by stationary random loads to be identified (3) e loading identification process is discrete including frequency response function matrices and response power spectrum density matrices (4) Position of each random excitation to be identified is fixed, and its position remains unchanged in the whole process In random vibration theory, according to transformation formula of power spectrum density matrices, the relation between response power spectrum density matrix S yy and excitation power spectrum density matrix S FF is represented as follows: where H * is the conjugated matrix of frequency response function matrix H and H T is the transpose of H.
Response power spectrum density matrix S yy is a Hermitian matrix.So, S yy can be expressed as where λ j and ϕ j is the jth characteristic pair, r is the order of S yy , and ϕ H j is the conjugate transpose of ϕ j .Using the characteristics of each order to construct pseudoresponse  y j , the following equation is obtained: where ω is the angular frequency and e iωt is the unit harmonic excitation.en, S yy is transformed as where  y H j is the conjugate transpose of  y j .e original IPEM cannot improve the ill-posed problem of the frequency response function matrix near natural frequencies of the structure.erefore, the identification results of original IPEM have large fluctuations and errors compared with the actual load spectrum near the natural frequencies.In this section, this problem is improved obviously by introducing the weighted average method.
e constructed pseudoresponse  y j is caused by pseudoexcitation  f j , and the mathematic relation between them is shown in the following equation: where H + * is the conjugate inverse of frequency response function matrix H and the dimension of H is m × n, m ≥ n. e frequency response function matrix H in (5) can be written into row vector form: where r is the rank of the response power spectrum matrix S yy , T i (i � e matrix condition number is a rough measurement scale to reflect the influence of coefficient matrix A and constant term b's errors on solution x in equation Ax � b.In this section, weighted average technique is used to improve the accuracy of loading identification.e condition number of a matrix is defined as where ‖ • ‖ p represents matrix norm.Since norms are equivalent, condition numbers defined by different norms are also equivalent.e reciprocal of the condition number of the square matrix H i is t i : In equation ( 9), the condition number of frequency response function matrix is calculated by 2-norm.reshold value t is defined as follows: where t is the 1/k of the arithmetic square root of t i 's quadratic sum.Weight w i is defined as follows: where t i < t; in other words, the condition number of this square matrix H i is big enough, set w i equal to t i /2 for reducing the impact of H i in the inverse process.
Obtain pseudoexcitation  f j by the following formula: hence, excitation power spectrum density matrix S FF can be expressed by pseudoexcitation  f j as where H +T is the transposition inverse of frequency response function matrix H. Flow chart of the above method is shown in Figure 1. e medium-and high-frequency problems need to be solved with the statistical energy method.erefore, this improved method is only used to solve problems of random dynamic loading identification in low frequency.e following experimental studies are all limited in the lowfrequency band (under 200 Hz).

Experimental Validation of Cantilever Beam Structure
In order to verify effectiveness of the proposed improved method described in section 2, a validation experiment was performed for cantilever beam structure.
3.1.Experimental Setup.Diagrammatic sketch of the experimental structure is shown in Figure 2. Setup of the cantilever beam experiment is shown in Figure 3. Four integrated circuit piezoelectric acceleration sensors and two force sensors are employed to measure vibratory signals for random dynamic loading identification proceeding.Experiment conditions are multi-input multioutput.e cantilever beam is divided into ten units, and serial number of each point is 1-10 from fixed end to free end.Sampling frequency of these vibration tests is 2048 Hz.
In the process of cantilever beam experiment, the lowpass filter noise reduction method has been adopted while M + P vibration control and data acquisition system are employed to collect information of frequency response functions and acceleration responses.
Geometry and material parameters of the cantilever beam are listed in Table 1.

Validation Procedure.
Validating the proposed improved method described in Section 2 can be divided into three main steps: (1) e first step is to obtain the frequency response functions between excitation points and response points.Every vibration exciter excites cantilever beam structure respectively and measures frequency response functions simultaneously.Distances from fixed end to each measuring point are listed in Table 2. e first three natural frequencies of the cantilever beam are extracted by a beforehand modal test and listed in Table 3.
(2) e second step is to obtain vibratory responses of the test structure.Two electromagnetic exciters excite the cantilever beam structure at the same time.And condition of the response-measured test is the same as condition in step (1).Moreover, positions of two exciters in these steps remain unchanged.
(3) e last step is to identify the random loads acting on the cantilever beam structure.en compare relative error of the root mean square value between identified load and the actual one.According to random vibration theory, the root mean square value can be obtained by calculating the area under the power spectral curve line.
Shock and Vibration

Results and Discussion
. In comparative gures of identi cation results, "IPEM" represents the identi cation results of original inverse pseudoexcitation method."Actual Load" represents excitation self-power spectrum density measured by force sensors."CWAM" represents identi cation result of improved inverse pseudoexcitation method based on condition number weighted average technique.e relative error is de ned by the following equation: (1) Comparison of the third point's random excitation identi cation results is shown in Figure 4: relative errors of the root mean square value between identi ed load and the actual one in 5 Hz-180 Hz are listed in Table 4.

Shock and Vibration
Relative errors of the root mean square value between identi ed load and the actual one in 5 Hz-180 Hz are listed in Table 4.
With 4.8% decrease of the relative error, compared with IPEM, CWAM signi cantly reduces peak errors between identi ed load spectrum and the actual load spectrum near 8 Hz, 45 Hz, and 115 Hz.
(2) Comparison of the seventh point's random excitation identi cation results is shown in Figure 5: relative errors of the root mean square value between identi ed load and the actual one in 5 Hz-180 Hz are listed in Table 5.
With 9.6% decrease of the relative error, compared with IPEM, CWAM signi cantly reduces peak errors between identi ed load spectrum and the actual load spectrum near 45 Hz, 100 Hz, and 115 Hz.
rough analysis of cantilever beam test results, CWAM signi cantly reduces peak errors between identi ed load spectrum and the actual load spectrum near some natural frequencies.e CWAM's relative error of the root mean square value between identi ed load and the actual one is less than IPEM's relative error in 5 Hz-180 Hz frequency band.

Experimental Validation of Thermal Protection Composite Plate Structure
In this section, a validation experiment was performed for thermal protection composite plate to verify the e ectiveness of the proposed improved method described in Section 2.Moreover, extend the usability of this method for di erent structures.

Experimental Setup.
Setup of the thermal protection composite plate experiment is shown in Figure 6.Diagrammatic sketch of the experimental structure is shown in Figure 7. Tests of thermal protection composite plate adopt multi-input multioutput experimental conditions.Sampling frequency is 1600 Hz.Test structure is thermal protection composite plate for the hypersonic vehicle.ermal protection composite plate and electromagnetic exciters are fastened by bolts through the aluminum plate.In the processes of thermal protection composite plate experiment, the low-pass lter noise reduction method has been adopted while LMS vibration control and test system are employed to collect information of frequency response functions and acceleration responses.
Aluminum plate layer and composite layer are bonded together by glue.Delamination of the composite layer is shown in Figure 8.
Geometry and material parameters of aluminum plate and composite plate are listed in Tables 6-9.
Density of the ber reinforced mullite matrix composite panels (face sheets) is 1580 kg/m 3 .

Validation Procedure.
e validation procedure can also be divided into three main steps:    Shock and Vibration 5 (1) e rst step is to obtain the frequency response functions between excitation points and response points.Every vibration exciter excites thermal protection composite plate structure respectively and measures frequency response functions simultaneously.Sensor arrangement is shown in Figure 9, and their coordinates are listed in Table 10.Natural frequencies of the cantilever beam are extracted by a beforehand modal test and listed in Table 11.
(2) e second step is to obtain vibratory responses.Two electromagnetic exciters excite the thermal protection composite plate structure at the same time.
And condition of response-measured test is the same as condition in step (1).Moreover, positions of two exciters in these steps remain unchanged.(3) e last step is to identify random loads acting on the thermal protection composite plate structure.en, compare relative error of root mean square value between identi ed load and the actual one.According to random vibration theory, the root mean square value can be obtained by calculating the area under the power spectral curve line.

Results and Discussion
. In comparative gures of the following identi cation results, IPEM, actual load, and CWAM have the same meanings as those described in Section 3.3.e relative error is de ned as Equation ( 14).

Excitation 1's Identi cation Result.
Identi cation results of excitation 1 are shown in Figure 10.
Relative errors of the root mean square value between identi ed load and the actual one in 5 Hz-180 Hz are shown in Table 12.With 6.95% decrease of the relative error, compared with IPEM, CWAM signi cantly reduces peak errors between identi ed load spectrum and the actual load spectrum near 30 Hz and 110 Hz.  13.With 4.9% decrease of the relative error, compared with IPEM, CWAM reduces peak errors between identi ed load spectrum and the actual load spectrum near 110 Hz. rough analysis of thermal protection composite plate test results, CWAM reduces peak errors between identi ed load spectrum and the actual load spectrum near some natural frequencies.e CWAM's relative error of the root mean square value between identi ed load and the actual one is less than IPEM's relative error in 5 Hz-180 Hz regions.

Conclusions
A new improved random dynamic loading identi cation method has been proposed.Some conclusions can be summarized as follows:   E X (GPa)      Shock and Vibration (1) e accuracy of random dynamic loading identification method is improved through combining condition number weighted average technique with original inverse pseudoexcitation method.e influence of the matrix with large condition number is reduced by using weighted average technique.
(2) Loading identification experimental results for cantilever beam structure and thermal protection composite plate structure are all in good agreement with the actual self-power spectrum density curves, which proves the feasibility and effectiveness of the new improved method presented in this paper.(3) e improved method's (CWAM) root mean square value relative error between identified load and the actual one is less than relative error of inverse pseudoexcitation method (IPEM) in the lowfrequency band (under 200 Hz).Meanwhile, peak errors of load spectrum near some natural frequencies are reduced significantly.(4) e proposed method in this paper can be further extended to identify multisource random dynamic loads, and the method can be optimized to improve the accuracy of multipoint random dynamic loading identification in future.

Figure 1 :Figure 2 :
Figure 1: Flow chart of improved random dynamic loading identi cation method.

4. 3 . 2 .
Excitation 2's Identi cation Result.Identi cation results of excitation 2 are shown in Figure 11.Identi cation results of excitation 2 in 75 Hz-150 Hz are shown in Figure 12.Excitation 2's relative errors of the root mean square value between identi ed load and the actual one in 5 Hz-180 Hz are shown in Table

Figure 5 :
Figure 5: Comparison of the seventh point's random excitation identi cation results.

8
rows of frequency response function matrix H can compose square matrix H i (i � 1, 2, . . ., k); meanwhile, the corresponding n pseudoresponse elements are chosen to compose new pseudoresponse vector  y i j (i � 1, 2, . . ., k), and the new pseudoexcitation  f 1, 2, . . ., m) is the row vector of H, and  y 1 ,  y 2 , . . .,  y m are elements of pseudoresponse vector  y j . 2 Shock and Vibration e arbitrary n

Table 2 :
Distances between each measuring point and the xed end.

Table 3 :
First three natural frequencies of the cantilever beam.

Table 5 :
e seventh point's excitation relative errors of root mean square value.Figure 7: Connection of vibration exciter, force sensor, and composite plate.

Table 6 :
Geometry and material parameters of the aluminum plate layer at reference temperature (25 °C).

Table 7 :
Geometry parameters of the composite plate.

Table 8 :
Material parameters of face sheets at reference temperature (25 °C).

Table 9 :
Material parameters of core at reference temperature (25 °C).

Table 10 :
Coordinates of excitation points and response points.

Table 11 :
Natural frequencies of the thermal protection composite plate.

Table 12 :
Excitation 1's relative errors of the root mean square value.

Table 13 :
Excitation 2's relative errors of the root mean square value.