This study aims to correlate the vibration data with quantitative indicators of structural health by comparing and validating the feasibility of identifying unknown excitation forces using output vibration responses. First, numerical analysis was performed to investigate the accuracy, convergence, and robustness of the load identification results for different noise levels, sensors numbers, and initial estimates of structural parameters. Then, the laboratory beam structure experiments were conducted. The results show that using the two identification methods Tikhonov (L-curve) and TSVD (GCV-curve) can successfully and accurately identify the different excitation forces of the external hammer. The TSVD based on GCV method has more advantages than the Tikhonov based on L-curve method. The proposed two kinds of load identification procedure based on vibration response can be applied to the safety performance evaluation of the railway track structure in future inverse problems research.

In the past two decades, a large number of researchers have carried out many effective research studies on inverse problems such as structural load identification and damage identification [

Thite and Thompson considered the influence of regularizing the matrix inversion, which discussed Tikhonov regularization with the ordinary cross-validation method for selecting the regularization parameter, and they also studied an iterative inversion technique of the method [_{1}-norm of the coefficient vector of basic functions [_{2}-norm by _{1}-norm [

In this paper, the TSVD of regularization parameter selection method based on generalized cross-validation (GCV) method and Tikhonov regularization based on L-curve method are compared and analyzed. The identification accuracy and applicability of the two methods have been summarized through the simulation of the single-input single-output model (SISO) and multiple-input multiple-output model (MIMO), and the identification characteristics of sine wave load and triangle wave load have been demonstrated. Finally, the results of simulation analysis are verified by experiments.

This paper establishes the forward problem of load identification based on the green kernel function. According to the characteristics of the pulse function, the integral of the product of the pulse function with any other function is equal to the function value of the pulse moment. Therefore, any dynamic load of a structure can be expressed as a superposition of the unit pulse signal in the time domain:

It is assumed that the Green function from the loading point to the responding measurement point is

Then, it is easily to written as

Equation (

The Tikhonov regularization method can be expressed as an optimization problem as follows [

The more exact representation of the objective function in Equation (

According to the concept that the gradient of the objective function is equal to zero, the least squares solution of Equation (

Deriving from Equation (

The method for dealing with ill-conditioned matrices is to derive new problems by using the rank deficient coefficient matrix of rank deficit. The rank deficient matrix _{k}, which is extended by the decomposition of the truncated singular value of the rank _{k} can be written as

The solution to Equation (

The solution of

Unfortunately, since optimal regularization parameter value search range is too large, the computational cost will be high or the L-curve graph is not satisfied.

The basic idea of generalized cross-validation (GCV) means if the arbitrary element

Equation (

As shown in Figure ^{11} Pa, density is 7.8 × 10^{3} Kg/m^{3}, and Poisson’s ratio is 0.3. The size of the beam is 1 × 0.05 × 0.005 m. The beam is divided into ten elements, marked elements 1 to 10. Then, the number of finite element model nodes is expressed from the mark number of one to eleven. Numerical simulation is used to verify the efficiency of two proposed force identification method as well as compared to the identified results. Furthermore, the identified results of different loads applied on the same structure are analyzed.

The finite element model of cantilever beam.

For the purpose of studying the effect of noise on identified results, the noise is added to the response data.

A sinusoidal load

Comparison of results using (a) Tikhonov regularization based on L-curve and (b) TSVD regularization based on GCV.

Results of load identification of SISO system.

Noise level (%) | TSVD + GCV (RE/%) | Tikhonov + L-curve (RE/%) |
---|---|---|

0 | 7.14 | 7.36 |

1 | 8.20 | 7.35 |

2 | 11.35 | 8.86 |

5 | 24.44 | 16.78 |

As shown in Figure

Comparison of experimental and simulated natural frequencies of test specimens.

Modal orders | Natural frequency value of test (Hz) | Natural frequency value of simulation (Hz) | Error (%) |
---|---|---|---|

1 | 20.42 | 20.271 | 0.7 |

2 | 55.66 | 56.933 | 2.2 |

3 | 184.40 | 186.49 | 1.12 |

4 | 268.93 | 280.64 | 4.17 |

To study the differences between the two methods, a sinusoidal load in the

The sinusoidal load identification results with no noise and different sampling time.

The triangle load identification results with no noise and different sampling time.

The sinusoidal load identification results with 1 percent.

The triangular load identification results with 1 percent.

According to Figures

The cantilever beam structure applied for force identification is set up as shown in Figure ^{11} N/m^{2}, density is 7.8 × 10^{3} Kg/m^{3}, and Poisson ratio is 0.3. Four points of the beam are attached by four accelerometers, which are IEPE voltage output piezoelectric acceleration sensor. A shaker is used to generate the sinusoidal force. The measured signals containing the force and acceleration are synchronously recorded by DH9522 data acquisition system. For the purpose of modal frequency, the finite element model of the structure is established and calculated according to the actual information of the geometry, material, and boundary conditions. A modal impact hammer is used to impose impact forces acting on the cantilever beam. The natural frequencies of simulation and experiment are compared as shown in Table

Experimental setup: (a) site map; (b) sketch map.

From the natural frequency of Table

The dynamic response of the structure is measured by experiment, and the Green kernel function between the loading point and the measuring point is calculated by the finite element method. Therefore, the load identification equation based on the Green kernel function has been established. As shown in Figure

The identified results in the experimental.

This study proposes a comparative research method for the inverse problem of load identification including the TSVD (GCV) method and Tikhonov based on the L-curve method. At the same time, numerical methods and experiments are used to verify the recognition effectiveness and robustness of these two approaches based on vibration response to identify the external excitation force. For these two kinds of regularization methods under different noise levels, the equations based on the established input-output relations for unknown external excitation forces are iteratively estimated. Some detailed conclusions are briefly listed below:

In the absence of noise, both the two mentioned methods of the above identification methods can accurately identify the load imposed on the system.

By virtue of the proposed method, the noise should be considered to the simulation and experiment. The results of numerical simulations and experimental are indicated that the identification accuracy of the TSVD based on the GCV method is much higher than that of the Tikhonov based on the L-curve method. In other words, the former method is less sensitive to noise.

The difference between the two methods is mainly due to the fact that the L-curve generated by the experiment is unobservable, and the optimal point on the L-curve is difficult to localize with accuracy.

The results of different loads in different sampling time can get the conclusion that the identified effect of triangle wave load changes obviously with the change of sampling time. Therefore, the sampling time of linear load such as triangle wave is short; in other words, the sampling frequency could be large.

The data used to support the findings of this study are included within the article. The data are published on figshare website. (

The authors declare that they have no conflicts of interest.

The authors thank the National Natural Science Foundation of China (51375405 and 51775456) and the Self-Developed Research Project of the State Key Laboratory of Traction Power (2016TPL T10).