Research ArticleA Load Identification Application Technology Based onRegularization Method and Finite Element Modified Model

Bingrong Miao , Feng Zhou, Chuanying Jiang, Yaoxiang Luo, and Hui Chen

State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China

Correspondence should be addressed to Bingrong Miao; brmiao@home.swjtu.edu.cn

Received 22 April 2020; Revised 10 August 2020; Accepted 18 August 2020; Published 31 August 2020

Academic Editor: Giuseppe Petrone

Copyright © 2020 BingrongMiao et al.'is is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An improved load identification technology of a beam based on different regularization methods andmodel modifying methods ispresented in an attempt to minimize the estimation error at several periodic loads. A hybrid model is developed to simulate suchill-posed problem interactions under different noise levels. 'e finite element model is modified with the different optimizationalgorithms to obtain the equivalent constraint condition. Experimental verification is also carried out to obtain correct modes andfrequencies by considered vibration response and different boundary conditions. 'e measured results demonstrate the goodagreement with the identification results. 'e results are shown that the improved method not only has more adaptive range andhigher identification accuracy but also has effective identification ability for loads under different noise levels.

1. Introduction

In recent decade years, extensive application of regulari-zation method to engineering structures’ health monitoringand inverse problems has been reported, as it has beenrecognized that using such an approach yields to improvedquasi-static and dynamic load identification ill-posedproblem characteristics [1]. 'e general concept for thesolution of an ill-posed problem is to transform it into a well-posed problem by using additional information about thesolution [2]. 'e methods, which transform the problem tothe form of a well-posed problem, are regularization pro-cesses [3]. For these reasons, the regularization method hadfound its ways into aeronautical, civil, and mechanical en-gineering applications and started to become a standardmethod of solving inverse problem equation [4].

'e literature on structural load identification is quiteextensive and the research activities in this field have focusedon a wide variety of applications. 'e identification problemwas treated as an inverse problem to control certain ob-jective functions. In other words, the structural load iden-tification is reconstructed to achieve the optimization goalbounded by various constraints, among which is as con-sistent as possible with real loads under limited response

data. Many scholars have proposed different identificationmethods for periodic and impact dynamic loads [2]. 'iteand 'ompson [4] discussed the relationship betweenTikhonov regularization and the selection of regularizationparameters by conventional crossvalidation methods andstudied the iterative inversion technique of this methodconsidering the influence of regularization matrix inversion.Liu and Han [5] proposed an inverse method which com-bines the interval analysis with regularization to stablyidentify the bounds of dynamic load acting on the uncertainstructures. Liu et al. [6] found that most time-domain-basedload identification methods based on state space have thedisadvantages of large discrete error and long sampling time.'e implicit Newmark-β algorithm is transformed into adisplay algorithm to solve the load identification equationand uses Tikhonov regularization to solve the ill-conditionedproblem. Sun et al. [7] combined a new improved regula-rization operator with L-curve method to overcome the ill-condition of load reconstruction and obtained the stable andapproximate solutions of inverse problems. Jia et al. [8]proposed a weighted regularization approach based on theproper orthogonal decomposition (POD), in which theregularization parameter was selected by the GCV method.Gao Wei et al. [9] proposed a new method for selecting

HindawiShock and VibrationVolume 2020, Article ID 8875697, 12 pageshttps://doi.org/10.1155/2020/8875697

mailto:brmiao@home.swjtu.edu.cnhttps://orcid.org/0000-0001-9909-3873https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/8875697

regularized parameters. 'e method is based on the entropyfunction method of quadratic programming theory. Nu-merical examples and experiments verify the effectiveness ofthe proposed method by identifying the sinusoidal loadsacting on the cantilever beam. Prawin and Rao [10] usedynamic principal component analysis (DPCA) to proposean online identification load algorithm from the accelerationtime history response measurement of moving windows.MAO et al. [11] proposed a time-domain force identificationapproach for the linear system, which is based on Markovparameter fine integration in the state space, and discussedthe influence of the position of the measuring point andnoise on the load identification result. In addition, Liu et al.[5] use Green pulse function or shape function to discretelyrepresent the dynamic response relationship of the system,thus establishing a positive problem of load identificationand combining different regularization methods to deal withthe ill-posedness in the inverse problem of load identifi-cation to improve identification accuracy [12–15]. Miao et al.[16] also used the Green kernel function to establish the loadidentification equation and compared several regularizationmethods to solve the identification problem of sine andtriangle loads.

However, fewer studies have considered the applicationof load identification to different regularization, and themodified model used different optimization method inter-action problems [17–23]. Hence, it is the main objective ofthe current paper to solve load identification problemmodelusing the modified finite element model. 'rough consid-ering the boundary constraint and different optimizationmethods, the main vibration frequency of the measuredmode is corrected.

2. Theoretical Background andMathematical Modelling

2.1. Dynamics Equations Based on Duhamel IntegratedMethod. 'e structural dynamic equation of motion for aproportional damping system with multidegrees of freedomis described as

M€y (t) + C _y(t) + Ky(t) � F(t), (1)

where M, C, and K are the total mass matrix, dampingmatrix, and stiffness matrix, respectively, F(t) is the loadvector, and y(t), _y(t), and €y(t) are the displacement, ve-locity, and acceleration response vectors.

According to the characteristics of the Dirac function,the external excitation f(t) of the system can be expressed asan integral form in which a plurality of pulse functions aresuperimposed. 'e equation can be described as

f(t) � t

0f(τ) δ(t − τ)dτ. (2)

It is assumed that the response of the system under theunit pulse load f(τ) is represented by the Green functionG(t) from the load action point to the responsemeasurementpoint [3]. According to the principle of Dirac function

superposition, the dynamic response caused by the externalload of the system is described as

y(t) � t

0f(τ)G(t − τ)dτ. (3)

Considering the zero initial condition system, the convo-lution integral of equation (3) is transformed, where Δ(t) isrepresented as a discrete sampling time interval and m is thenumber of sampling points. 'e equation can be discretizedinto a set of linear equations whose matrix form is described as

y1

y2

⋮

ym

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

�

G1

G2 G1

⋮ ⋮ ⋱

Gm Gm−1 · · · G1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f0

f1

⋮

fm−1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

Δt, (4)

where yi, Gi, and fi are the response of t� iΔt, the Greenfunction, and the unknown external load. Furthermore,equation (4) can be simply described as

Y � GF. (5)

Similarly, the identification problem of multiple loadeffects can be expressed as a matrix which is described as

Y1

Y2

⋮

Ym

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

�

G11 G12 · · · G1n

G21 G22 · · · G2n

⋮ ⋮ ⋱ ⋮

Gm1 Gm2 · · · Gmn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

⋮

Fn

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

, (6)

where m is the number of applied loads, n is the number ofresponse points, Yi is the response, Fi is the column vector ofthe load, and Gij is the Green Kernel Function between theload Fj and the response point Yi.

Since the actually measured response signal alwayscontains noise, there is a small singular value in the Greenkernel function matrix [17]. Equations (5) and (6) will be-come typical ill-conditioned equations. If not appropriatelyto directly invert the matrixG, it is generally necessary to usean appropriate regularization technique to obtain an ap-proximate solution to approximate the true result.

2.2. Regularization Technique. Regularization technologyimproves the ill-conditioning of load identification problemby modifying the singular value of system matrix andtransforms the ill-conditioned problem into a steady-stateproblem [24–28]. 'erefore, it provides an effective methodfor accurate load identification. 'e Tikhonov and TSVDregularization methods are used to be compared.

2.2.1. Tikhonov Regularization Technique. Tikhonov is de-rived from the variation method and is implemented byintroducing stable functions [19]. For any α> 0, the qua-dratic function defined on the Hilbert space X is described as

Φα(x) � Ax − yδ

�����

����� + α‖x‖2, (7)

2 Shock and Vibration

where α is a regularization parameter and yδ is represents yvalue containing an error. 'e Tikhonov regularizationmethod is to find the minimum point of the function Vα(x)as an approximate solution to the equation.

It is known from the nature of the quadratic functionthat if the equation is satisfied, it can be obtained byequations (8) and (9):

xδα � A

∗A + αI( −1A∗yδ, (8)

Fδα � GTG + αI

−1GTYδ. (9)

2.2.2. TSVD Regularization Technique. Assuming that thegeneralized inverse of a linear equation exists, the solutioncan be described as

x+

� A+y

� ∞

i�1σ−1i y, vi( ui,

(10)

where σi is the singular value of the compact operator A. 'esingular values are arranged in order from largest tosmallest. ui and vi are the standard orthogonal bases of thesubspace N(A) ⊂ X and R(A) ⊂ Y. If the right end y of thelinear equation has a perturbation error δy, such asyδ � y+ δy, the approximate solution of the linear equation isdescribed as

xδ

� ∞

i�1σ−1i y

δ, vi ui. (11)

And the error between xδ and x+ is described as

δx � xδ

− x+

� ∞

i�1σ−1i δy, vi ui.

(12)

It can be seen from the above equation that if thevalue of σi is smaller, the influence of the disturbanceerror δy on the resulting error δx becomes larger. 'etruncated singular value decomposition regularizationmethod is to cut off the right end of the equation forcalculating the generalized solution. 'e solution onlyretains the first k large singular values and filters thesmaller singular values behind. 'us, the method canavoid the disturbance error being overamplified. 'eTSVD solution of the load identification equation (5) isdescribed as

fδk � ∞

i�1σ−1i y

δ, vi ui, (13)

where fδk is the solution of the equation and k is called thetruncation step of the TSVD method and has the effect ofcorrecting regularization parameters. σi, ui, vi

∞i�1 is a sin-

gular system for the matrix G.How to choose the optimal regularization parameter k

will directly affect the error and identification accuracy

between the identification load and the real load. 'e GCVcriterion is selected as the parameter selection method forTSVD regularization.

2.2.3. Selection of Regularization Parameters. It can beknown from (9) and (13) that it is essential to select theoptimal regularization parameters when the regulariza-tion technique solves the load identification equation.'e Tikhonov regularization method uses the L-curvecriterion for regularization parameter selection, whilethe TSVD regularization method is combined with theGCV criterion for research. 'e size of the regularizationparameter will directly affect the error between theidentification load and the real load. Here is a briefdescription of two regularization parameter selectionstrategies.

(1) L-Curve Criterion. In the optimization problem ofTikhonov regularization, the norm ‖fδα‖2 of the regula-rization and the residual norm ‖yδ − Gfδα‖2 are functionsof the regularization parameter. Taking these two vari-ables as the horizontal and vertical coordinates, and theresulting curve is as follows. In the general coordinatesystem, the curve will appear as a distinct L shape. 'ecorner of L curve is the norm between the regularizationand the residual value to obtain the optimal equilibriumposition. 'e corresponding parameter of this cornerpoint is the optimal regularization parameter. In order toreasonably determine the position of the corner of thecurve, the maximum curvature point of the curve at thescale is generally selected. Assign the parameterρ � log‖yδ − Hfδα‖2, θ � log‖fδα‖2; then, the curvatureequation of the L curve is described as

L(α) �ρ′θ″ − ρ′θ′

ρ′( 2 + θ′( 2 3/2, (14)

where ρ′, θ′, and θ′′ are the first and second derivatives of theregularization parameter, respectively. 'e optimal regula-rization parameters determined by the L-curve criterionshould be satisfy the following condition:

L αop � maxα>0 L(α). (15)

(2) GCV Curve Criterion. 'e generalized crossvalidation(GCV) curve criterion is based on the PRESS criterion instatistical theory. And it is more robust than the PRESScriterion. 'e basic idea of the GCV method is thatreasonable regularization parameters can effectivelypredict any new unknown load.'e equation defining theGCV function is described as

GCV(α) �I − A(α)yδ

�����

�����2

2

Tr(I − A(α))( 2,

(16)

where A(α) � G(GTG + αI)−1GT and Tr represents the traceof the matrix. And the equation satisfies the condition

Shock and Vibration 3

α(GTG + αI)− 1 � I − G(GTG + αI)−1GT; then, equation (16)can be simplified to the following equation:

GCV(α) �GTG + αI

− 1yδ������

������2

Tr GTG + αI −1

. (17)

'e criteria for selecting the optimal regularizationparameters according to the GCV method are described as

GCV αop � minα>0 GCV(α). (18)

In solving this minimization problem, the numeratorterm of the GCV function in equation (17) is actually theresidual norm of the regularization solution. 'is item iseasier to obtain, but when the matrix dimension is large, thefunction of the denominator matrix trace of the function isvery large.

3. The Improved Load Identification Algorithm

'e load identification technology for periodic loadsproposed in this paper is an inverse problem analysismethod based on finite vibration response data, whichmainly considers the problem of large fluctuations in theidentification results due to differences in model accuracyand boundary conditions [29–35]. In order to avoid theuncertainty of the boundary conditions and structuralparameters of the actual structural model from affecting therecognition accuracy, the recognition method proposed inthis paper takes into account the model modification, usingthe results of the calculated and experimental modalitiesand several optimization algorithms to modify the limitedboundary conditions and structural parameters of the metamodel. Based on the abovementioned theoretical formula,load identification is carried out through vibration re-sponse. At the same time, the recognition accuracy ofseveral regularization methods after model modification iscompared in the algorithm.'e specific flow chart is shownin Figure 1.

'e improved algorithm of load identification is listed asfollows:

Step 1: establish numerical and experimental modelsof Euler–Bernoulli beam structure according to ac-tual conditions.Step 2: use experimental modal and computationalmodal analysis methods to obtain the modal and modeshapes of the beam structure under different boundaryconditions.Step 3: determine the type and number of sensors basedon the condition number of the system matrix and thecriteria for obtaining the vibration response accurately.'e selection requirements of sensor location need toimprove the noise immunity, signal-to-noise ratio, andreliability of the data as much as possible.Step 4: according to the analysis results of the calculatedmode and the experimental mode, an optimizationmethod is selected to modify the boundary conditions

and structural parameters of the finite elementmodel toensure that the relative error between the finite elementmode and the experimental mode is minimized.Step 5: after confirming that themodel correction resultsatisfies the conditions, apply the type, size, and po-sition of the periodic load on the numerical model andthe experimental model.Step 6: carry out on-site verification test according tothe measured load to obtain the measured vibrationresponse data.Step 7: use Picard’s condition to determine the well-posedness of the inverse problem.Step 8: perform load identification simulationaccording to the finite element simulation model, andcalculate the vibration response.Step 9: obtain the measured data of the vibration re-sponse according to the experimental model.Step 10: according to Green’s function and differentregularization algorithms, considering different degreesof noise, use vibration response data to calculate thekernel function matrix.Step 11: determine the optimal regularization param-eter (GCV curve and L-curve) according to the regu-larization parameter selection criterion to reduce theinfluence of noise and boundary conditions.Step 12: if it is an ill-posed problem, determine theappropriate regularization technique to solve the loadidentification matrix equation.Step 13: compare and verify the simulation results andexperimental results and evaluate the effectiveness ofthe recognition algorithm.

4. Numerical Example

4.1. Simulation Model for Load Identification. As a typicalforce structure, the cantilever beam structure can be used toidentify the dynamic load, which is of great significance forboth theoretical research and engineering practice. 'ere-fore, the cantilever beam structure is selected as a numericalsimulation model for periodic dynamic load identification, asshown in Figure 2.'e cantilever beam has a structural size of0.8m× 0.06m× 0.004m, and the elastic modulus of the steelmaterial is 2.1× 1011 Pa, the density is 7.8×103 kg/m3, andPoisson’s ratio is 0.3. It is assumed here that the structuraldamping of the cantilever beam is proportional damping.

'e vibration data is obtained to simulate the measuredresponse through adding white Gaussian noise to thesimulation model. 'e response with noise is described as

yδ � yc + lδ · std yc( · randn, (19)

where yc is the response calculated by the simulation, whichcan be displacement, velocity, and acceleration, std(yc) is thestandard deviation of the calculated response, lδ is a per-centage representing the noise level, And randn represents awhite noise random number of the same length as yc, wherethe mean is 0 and the variance is 1.

4 Shock and Vibration

In order to evaluate the degree of conformity betweenthe identification load and the real load, a relative errorfunction is introduced, which is described as

RE(i) �FReal(i) − Fident(i)

Fident(i)

× 100%, (20)

where i� 1, 2, . . ., n, and n is the number of sampling points.In addition, in order to describe the effectiveness of the

identification method, two indicators of total relative error(RE) and correlation coefficient (CC) are required. 'eequation is described by equations (21) and (22):

RE �fα,δ − f

�����

�����

‖f‖× 100%, (21)

CC �

m

i�1 fα,δi − E f

α,δ fi − E(f)

fα,δ − E fα,δ �����

�����‖f − E(f)‖,

(22)

where E(f ) and E(fα, δ) are the average of the actual load andthe identified load, respectively.

A unit pulse load is applied to the above excitation point(corresponding to the node 16 of the simulation model), andthe Green pulse kernel function response of the excitationpoint to the response point (node 15) is obtained by finiteelement calculation, and a corresponding kernel functionmatrix G is established [36, 37]. At 5% white noise level, theTSVD regularization identification results are comparedwith the traditional Tikhonov regularization method. 'e

identification comparison results of sinusoidal loads at 5%noise level is shown in Figure 3.

'e relative errors of these two regularization methodsin identifying sinusoidal loads in the whole time period areobtained.'e relative errors results are described in Figure 4.

It can be seen that both the TSVD regularization and theTikhonov regularization can identify the added sinusoidalload. Obviously, the load identified by the Tikhonov methodis fluctuated greatly, while the TSVD method is closer to thetrue load value. 'e error of TSVD regularization identifi-cation is lower than the Tikhonov regularization at mosttime points, and the relative error value can be guaranteed tobe around 10% or less. 'e relative error of Tikhonovregularization is basically above 15%, and the overall relativeerror of identification is 21.36%. It shows that TSVD reg-ularization identification can more accurately and stablyidentify the applied sinusoidal load compared to Tikhonov.In addition, the reason for the higher individual peak pointidentification error is due to the smoothing effect of theTSVD method on the identified load during the regulari-zation process.

In order to further analyze the identification effect ofTSVD regularization, Table 1 shows the overall relative errorand correlation coefficient values identified by the TSVDmethod and the Tikhonov method under different noiselevels, and the TSVD regularization is selected at 10%, 20%,and 30% noise levels.

It can be seen from Table 1 that, as the noise level in-creases, the overall relative error of the TSVD and Tikhonovregularization methods increases, and the correlation

Numerical model(known)

Measured model(known)

Measured vibrationresponse

Validate and evaluatethe identification

method

Optimize sensornetwork layout

Calculation andexperimental mode

comparison?

Solving the green functionmatrix

(measured and numerical)

Finite element modelmodification and

update

If met picardcondition?

Numerical vibrationresponse

Direct inversionmethod of loadidentification

Input excitation

Loadidentification

results

Compare the regularization methods(Tikhonov and TSVD)

Selection regularizationparameters (GCV and L

curve)

No

Yes

Noise

If modelcorrection?

No

Figure 1: Load identification improved algorithm based on modified finite element model.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16X

z

0

Sensor

Hammer

FY

Figure 2: Numerical simulation model of the cantilever beam.

Shock and Vibration 5

0 0.02 0.04 0.06 0.08 0.1 0.12–15

–10

–5

0

5

10

15

Time (s)

Forc

e (N

)

Actual loadTikhonov regularizationTSVD regularization

Figure 3: Identification results of sinusoidal loads at 5% noise level.

0 0.02 0.04 0.06 0.08 0.1 0.120

10

20

30

40

50

60

70

80

90

100

110

Time (s)

Rela

tive e

rror

(%)

Tikhonov regularizationTSVD regularization

Figure 4: Relative error of sinusoidal load identification at 5% noise level.

Table 1: Identification error at different noise levels.

Noise levelOverall relative error RE (%) Correlation coefficient CC

TSVD method Tikhonov method TSVD method Tikhonov method5 12.54 21.36 0.9921 0.977710 13.55 31.08 0.9908 0.954320 16.17 47.0 0.9868 0.903330 19.37 63.71 0.9811 0.840240 22.95 80.72 0.9735 0.773250 26.95 97.97 0.9635 0.7077

6 Shock and Vibration

coefficient decreases. However, the relative error identifiedby the Tikhonov method increases by more than 10%, andthe correlation coefficient has dropped to 0.77 when thenoise level is 40%. 'e error identified by the TSVD methodis only increased by about 3% each time, and the correlationcoefficient can be guaranteed to be above 0.96. 'is showsthat the TSVD regularization has a better suppression ofnoise, and the applied load can be more accurately recog-nized even at a higher noise level (20%). 'e results areshown in Figure 5.

A periodic mixed sinusoidal load, a triangular load, anda square wave load are applied to the free end of the beam(node 16), respectively, and 5% of the white noise is addedto the acceleration response on node 15, using the responseand TSVD. Regularization is performed for load identifi-cation, and the results obtained are shown in Figures 6–8.For the triangular and mixed sinusoidal loads, the re-spective identification results at five time points are shownin Table 2.

Tables 2 and 3 show that the Tikhonov identification loaderror is basically higher than TSVD at these five time points,whether it is sinusoidal load or triangular load, and the errorof some points is even more than 20%, and the relative errorof TSVD is 10%. At the same time, the correlation coefficientof the TSVD regularization identification load is above 0.99,while the Tikhonov is below 0.99. 'erefore, from the abovechart, the feasibility of the TSVD regularization method toidentify the periodic load and the superiority with respect toTikhonov are explained in the acceleration response as theload identification input.

'e results in the figures and tables show that the TSVDregularization method is more efficient than the Tikhonovregularization method. Furthermore, the fluctuation andrelative error of the identified loads are smaller. In addition,due to the transient abrupt change of the square wave load,the TSVD method recognizes that when identifying the typeof load, the identification error is large at some points intime, and the overall identification error is higher than theother two types of loads.

4.2. Experiment Validation

4.2.1. Establishment of Experimental Model. From theprevious analysis results, it can be seen that the uncorrectedfinite element model still has a certain error in the loadidentification. As the modal order increases, the error willbe greater. It is necessary to establish a modified finiteelement model to improve the recognition effect.'erefore,this article established a detailed experiment validationplan.'e experimental object is a cantilever steel beam withthe same material and size as the simulation model. 'isarticle mainly analyzes the structural modal test data in theDH5923 software. 'e sampling frequency is 2 kHz. Andthe equivalent constraint boundary conditions can besimplified by linear springs and torsion springs. 'e linearspring stiffness is set to 1× 105 N/m, and the torsion springstiffness is set to 1 × 106 N/m. 'e experimental scene is asshown in Figure 9.

'e modal analysis of the finite element model is carriedout, and the comparison of finite element modal and ex-perimental modal frequencies of the beam structure is listedin Table 4.

From the table, it is found that the relative errors of thefirst and second order modal frequencies obtained by ex-perimental measurements and finite element analysis aresmall, while the relative errors of other orders are greaterthan 5%.'is shows that themodal frequencies calculated bythe finite element method are quite different from themeasured frequencies, and the finite element model needs tobe revised. 'erefore, the first and second order modalfrequency is chosen as the modification target of finite el-ement model updating.

0 0.05 0.1 0.15 0.2–40

–30

–20

–10

0

10

20

30

40

Time (s)

Forc

e (N

)

Actual loadTikhonov regularizationTSVD regularization

Figure 6: Identification results of mixed sinusoidal loads under 5%noise.

0 0.02 0.04 0.06 0.08 0.1 0.12–15

–10

–5

0

5

10

15

20

Time (s)

Forc

e (N

)

Real load10% noise level

20% noise level30% noise level

Figure 5: Identification results of TSVD under 10%, 20%, and 30%noise levels.

Shock and Vibration 7

0 0.01 0.02 0.03 0.04 0.05 0.06–40

–30

–20

–10

0

10

20

30

40

Time (s)

Forc

e (N

)

Actual loadTikhonov regularizationTSVD regularization

Figure 7: Identification results of triangular loads under 5% noise.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08–80

–60

–40

–20

0

20

40

60

80

100

Time (s)

Forc

e (N

)

Actual loadTikhonov regularizationTSVD regularization

Figure 8: Identification results of wave loads under 5% noise.

Table 2: Load identification results at five time points under 5% noise.

Type Time (s) Actual load (N)Tikhonov regularization TSVD regularization

Identified load (N) Error (%) Identified load (N) Error (%)Sine 0.023 13.62 13.59 0.22 13.04 4.26Triangle 0.0025 30 32.29 7.63 29.73 0.90Sine 0.056 –17.63 –20.62 16.96 –17.12 2.89Triangle 0.006 –12 –13.92 16.00 –12.51 4.25Sine 0.0905 –29.91 –31.43 5.08 –30.28 1.24Triangle 0.0095 –6 –6.82 13.67 –6.51 8.50Sine 0.1375 –7.65 –10.65 39.22 –8.66 13.20Triangle 0.013 24 10.13 7.79 23.69 1.29Sine 0.1505 19.94 15.34 23.07 18.84 5.52Triangle 0.0185 –18 –20.80 15.56 –16.35 9.17

8 Shock and Vibration

4.2.2. Model Correction and Sensitivity Analysis. Beforeperforming specific model corrections, it is necessary todetermine the correction parameters from the numerouselements. Choosing correction parameters is the key tosuccessful model correction. 'e interpretability, correct-ness, and reliability of the correction model greatly dependon the composition of the correction parameters. In additionto the empirical screening method, parameters with highersensitivity to vibration measurement parameters can also beselected as correction parameters through sensitivity anal-ysis. Sensitivity analysis method is selected to model cor-rection because it can measure the rate of change ofstructural response caused by the change of parameters ordesign variables. 'rough sensitivity analysis, the uncer-tainty of output variables can be ranked according to thedegree of influence of design variables. In order to achievethe correction of the finite element model, the elasticmodulus, the linear spring stiffness, and the torsional springstiffness of the cantilever beam structure are used to designvariables.

In order to ensure the accuracy of the dynamic loadidentification, it is necessary to carefully modify the finite el-ement model in the load identification process.'is paper usesseveral optimization algorithms to modify the finite elementmodel and compares the relative errors of the natural frequency

changes after different optimization algorithms modify themodel [38].

For the optimization parameters, the equivalent elasticmodulus of the beam structure and the stiffness value ofthe equivalent linear spring are set, and the correspondingupper and lower limits are 1 × 1011 ≤E ≤ 3.0 ×1011MPaand 1 × 104 ≤K1 ≤ 1 × 107 N/m. 'e objective function inthe optimization process is defined as follows:

f � minwc − ws

ws

��������

�������� , (23)

where wc represents the calculated natural frequency and wsis the experimentally measured natural frequency.

'e target of optimization is the first and second orderfinite element modal frequencies, and the tolerance of theobjective function is set to 10−3. Set the optimal con-straints to ensure that the relative error is less than 5% forother finite element modal frequencies. In the optimi-zation, the LSGRG (Large Scale Generalized ReducedGradient), the DS (Downhill Simplex), and the MIGA(Multi-island genetic algorithm) are compared for anal-ysis [39]. 'e results of comparing the natural frequenciesof the 1st to 5th modes of the finite element model with theexperimental modes are shown in Figure 10.

Table 3: Identification results at five time points below 5% noise level.

Type Actual load (N) Tikhonov regularization TSVD regularization

Relative error (%) Sine loads 24.99 7.56Triangle loads 15.63 8.54

Correlation coefficient Sine loads 0.9689 0.9984Triangle loads 0.9881 0.9967

AccelerometerCantilever beam

Exciter

The signal acquisition system

Computer

Figure 9: Field diagram of modal experiments of the cantilever beam.

Table 4: Natural frequency comparison of beam structures and experimental structures.

Modal order Experimental mode (Hz) Finite element modal (Hz) Relative error (%)1 4.91 5.17 4.552 30.64 30.79 −0.493 85.32 76.03 7.034 167.90 132.89 14.285 277.21 226.95 15.49

Shock and Vibration 9

From Figure 10, it can be found that the three opti-mization algorithms LSGRG, DS, and MIGA all have acertain effect on the modified finite element model, espe-cially the correction effect of the MIGA method is the best,and the error rate of the natural frequency is relativelyminimal. It shows that the models modified by these threeoptimization methods can basically meet the requirementsof load identification. Of course, in the load identificationprocess, an accurate finite element model is only the basiccondition, and the vibration response is also an importantindicator to evaluate whether the improved model and theexperimental model are suitable. 'erefore, the influence ofboundary conditions needs to be paid enough attentionwhen carrying out load identification.

4.2.3. Results Analysis and Discussion. In this paper, themulti-island genetic optimization algorithm is mainly usedto modify the finite element model in the example, and theaccuracy of the TSVD (GCV) regularization method isverified. In order to compare the results of the modifiedfinite element model load identification, an experimentalverification analysis is performed. In the test process, thesampling frequency is 2000Hz and the sampling time is 2 s.In order to improve the calculation efficiency, load identi-fication is performed on 200 data points among them. 'en,the unit pulse loads are applied to the modified model, andthe kernel function response is calculated to construct aGreen function matrix G for load identification. 'e loadsare identified by the TSVD (GCV) and the Tikhonov (L-curve) regularization method, respectively. 'e comparisonresult is shown in Figure 11.

As can be seen in Figure 11, the experimental resultshave shown that TSVD has better identification thanTikhonov regularization. According to the comparison of

simulation and test results, both methods can effectivelyidentify the dynamic load, and the TSVD regularizationmethod has better recognition effect than the Tikhonovmethod. 'ere is only a slight deviation in the loadidentification at the peak, which is mainly due to theexcessive regularization of the model equation.

In addition, results comparison of the finite elementmodel correction based on the Tikhonovmethod is shown inFigure 12.

In Figure 12, in order to verify the before and afterrecognition results of the improved method, the Tikhonovregularization method with better identification effect is

1 2 3 4 50

2

4

6

8

10

12

14

16

Nat

ural

freq

uenc

y ch

ange

rate

(%)

Model order

Initial finite elementmodelModified modelwith LSGRG

Modified modelwith DSModified modelwith MIGA

Figure 10: Comparison of natural frequency difference before andafter model modification.

0.00 0.02 0.04 0.06 0.08 0.10–0.4

–0.2

0.0

0.2

0.4

Forc

e (N

)

Time (s)

Actual loadTSVD regularizationTikhonov regularization

Figure 11: 'e identify results comparison based on differentregularization methods.

Measured dataInitial modelModified model

0.00 0.02 0.04 0.06 0.08 0.10–0.4

–0.2

0.0

0.2

0.4

Forc

e (N

)

Time (s)

Figure 12: Comparison of results after modified finite elementmodel.

10 Shock and Vibration

selected for comparison. 'e study found that using theload identification method of the modified finite elementmodel, the identification result is closer to the measuredload, which is better than the identification effect withoutthe modified model. 'rough the comparison before andafter the recognition results, it is necessary to consider theboundary constraints and the influence of noise. By op-timizing the boundary conditions and structural pa-rameters of the simulation model, the recognition effectand accuracy can be further improved.

5. Conclusion

In order to improve the accuracy of periodic dynamic loadidentification in practical engineering, the paper proposes astructural load identification method based on two regu-larization methods and a finite element correction modelthrough simulation and experimental verification methods.'is method compares the results of TSVD and Tikhonovregularization methods and verifies the effectiveness of themethod through experiments.'e specific conclusions are asfollows:

(1) When performing periodic load recognition, al-though both recognition methods have certainrecognition effects, the TSVD method has a betterrecognition effect on periodic loads and can effec-tively deal with the ill-conditioned problem of ki-nematics equations in the load recognition process.

(2) During the test verification process, the improvedmethod can make use of the optimization algorithmto determine the appropriate design variables toensure that the modified model has a better recog-nition effect.

(3) When carrying out load identification, the influ-ence of boundary conditions needs to be fullyconsidered. And it is necessary to use optimizationalgorithms to select design variables to ensurebetter equivalence of model boundary conditions.At the same time, when performing the finite el-ement model parameter correction based on theactual structural modal measurement results, it isrecommended to select the modal with bettermeasured results for correction.

Data Availability

'e data used to support the findings of this study are in-cluded within the article. 'e data are published on figsharewebsite. 'e files’ link can be found in https://figshare.com/s/79df76d5d71a5606eff8.

Conflicts of Interest

'e authors declare that they have no conflicts of interest.

Acknowledgments

'e authors wish to thank the teachers and students whoparticipated in the research team on Load Identification and

Damage Identification for their contributions. 'e work wasfinanced by the National Natural Science Foundation ofChina (51775456) and Self-Developed Research Project ofthe State Key Laboratory of Traction Power (2019TPL T03).

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