Study on modal parameter identification of engineering structures based on nonlinear characteristics

Wei Guo; Saeed Rubaiee; Anas Ahmed; Asem Majed Othman; Atul Khosla

doi:10.1515/nleng-2022-0011

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Study on modal parameter identification of engineering structures based on nonlinear characteristics

Wei Guo , Saeed Rubaiee , Anas Ahmed , Asem Majed Othman und Atul Khosla

Veröffentlicht/Copyright: 31. März 2022

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Abstract

To study the nonlinear characteristics of the modal recognition of civil engineering parameters, a method of nonlinear recognition of the parameters of characteristics based on LMD is proposed. The LMD method is applied to decompose the acceleration response signals of the disturbing structure of the building, to obtain the PF components, the instantaneous frequency, and the instantaneous amplitude of each PF component, to determine the modal natural frequency and damping coefficient. To determine the modal parameter based on the LMD, the calculation and analysis results are presented as follows: the frequency of the components fluctuates between the fifth and sixth models, which shows that the components contain the reaction of the fifth and sixth design modes. This is because these two modes (3.101 Hz and 3.147 Hz) are very close to each other, which makes it difficult to distinguish between the responses of these two modes by the LMD method. The frequency of the components is always stable (the first 2.5 s), which indicates that during this period the responses of modes 5e and 6e do not dampen, and the ratio between them in the PF1 components does not differ much. The component frequency curve shows an interesting phenomenon. Starting from about 3.8 s, the frequency curve gradually approaches the first mode, and only the frequency of the first mode is about 6 s, which indicates that the response of the first mode still exists and makes up a significant proportion. Modular response, caused by the damping, is only detected in the first half of the 10 s response, after which it is verified from the nonlinear characteristics of the LMD parameter recognition method that half of the third-order modal response on the scale is very low and almost equal to zero, and despite problems with dense frequency separation mode in the LMD method, the frequency responses of its PF components may reflect the mode combination phenomenon and reflect the duration of each mode throughout the response.

Keywords: local mean decomposition method; dynamic detection; finite element model; PF component; nonlinear characteristics

1 Introduction

Civil engineering structure is an important part of national infrastructure, which directly affects people’s life and safety. With the rapid development of China’s economy, the construction and transportation industries have made great progress. It is worth noting that the working conditions of many structures are worrying, with damage, aging, and even collapse happening from time to time [1]. Due to the rapid development of civil engineering structure construction, its construction quality and design technology are often not satisfactory, which increases the hidden danger of accidents to civil buildings in China. The traditional structural analysis theory is mainly through the strength, stability, and other aspects of the study to ensure the reliability of the structural design. The static load test is carried out on civil structures, especially Bridges, to understand their actual working state under the test load and to determine the strength, stiffness, construction quality of the structure, and whether the structure meets the design and use requirements [2]. Currently, the relevant theory has reached a fairly mature level. However, the safety and the reliability of the structure cannot be guaranteed only by the accurate static design and static load test of the structure because the working environment of civil engineering structure determines that it must bear a large number of dynamic loads, such as wind load, ground vibration load, and so on, and the bridge structure also bears the impact load of water flow and traffic load [4,5,6,7,8,9,10,11,12]. Therefore, the static characteristics of the structure cannot fully and accurately reflect the characteristics of the structure. Although the dynamic response signal analysis and processing of civil engineering structures is not the ultimate goal, it is an important step in the process of damage identification [13,14,15,16,17,18,19,20,21], as well as in the process of the difficulty for bridge structures, as environmental excitation power test operation is simple and does not require the use of large measurement equipment. Moreover, there is no damage to the structure [22,23,24,25,26,27]. The basic principle of detecting structural damage in civil engineering based on modal parameter identification is that various structural damages can cause changes in the mass and stiffness of the entire civil engineering system. These changes in mass and stiffness will cause changes in modal parameters. Therefore, through the changes in modal parameters, the overall structural problems of the civil engineering system can be known, and then, the damage can be detected and analyzed. Therefore, great attention has been paid to using dynamic response test data under environmental excitation to identify the modal parameters of the structure [28,29,30]. Although the modal parameter identification theories of large structures are rapidly developing, there are some deficiencies in these theories. For example, for the commonly used feature system implementation method and random subspace method, the determination of system order and the construction of matrix directly affect the accuracy of the identification results. In addition, the identification accuracy of the damping ratio is poor in the application of the multinumerical modal parameter identification methods. At the same time, these methods only provide the best estimate of the modal parameters of the structure, but cannot intuitively obtain the uncertainty of the identified modal parameters from the perspective of mathematics. In the recent years, the Bayesian theory applied in modal parameter identification, the finite element model modification, and the state evaluation of structures has made it possible to calculate the posterior uncertainty of the best estimate of modal parameters from a mathematical perspective [31,32,33]. Therefore, it is of great practical significance to analyze the parameter identification method based on the Bayesian theory that has obvious advantages, and to compare this method with the widely used modal parameter identification method, so as to grasp its key and advantages. Through the method of the modal parameter identification, the structural damage identification of civil engineering is studied. First, it is necessary to set the extraction method of modal parameters, to establish the state equation of the damaged part of civil engineering, and to obtain the extraction model of modal parameters through the discrete time state model. Then, the detection method of civil engineering structural damage is designed, the characteristic parameters of the relative change of the mode shape before and after the structural damage are defined, the damaged part of the building in the civil engineering is determined, and the structural damage severity of the civil engineering building is judged according to the damage index. The inspection report for damage to civil engineering structures is presented.

2 Literature review

Meng proposed that the essence of the state analysis is a coordinate transformation, whose purpose is to put the response vector described in the original physical coordinate system into the “modal coordinate system” to describe [34]. Astroza et al. proposed that the experimental modal analysis, also known as the experimental process of modal analysis, is an experimental modeling process and belongs to the inverse problem of the structural dynamics. In the inverse process of the theoretical modal analysis, first, the time history of excitation and response is measured experimentally, and the frequency response function (transfer function) or impulse response function is obtained by using digital signal processing technology, and the nonparametric model of the system is obtained. Second, the modal parameters of the system are obtained by the method of parameter identification. Finally, if necessary, further the physical parameters of the system are determined [35]. Deng et al. believe that the theoretical modal analysis is actually a theoretical modeling process, which belongs to the positive problem of structural dynamics. It mainly uses the finite element method to discrete the vibration structure, establishes the mathematical model of the system eigenvalue problem, and uses various numerical methods to solve the system eigenvalue and eigenvector, that is, the modal parameters of the system. By mode superposition, the dynamic response of the structure under the known external load can be analyzed [36]. Based on the current research, this article proposes a nonlinear characteristic parameter identification method based on LMD. By studying the identification of nonlinear characteristic parameters, the establishment of the analysis model and the finite element, and the identification of modal parameters based on LMD, the results of calculation and analysis are as follows: The frequency of the components fluctuates between the frequencies of the fifth and sixth modes, indicating that the components contain the responses of the fifth and sixth modes of the structure. The reason is that the frequencies of the two modes (3.101 and 3.147 Hz) are very close, which makes the responses of the two modes difficult to be separated by the LMD method. The frequency of the component is always stable (the first 2.5 s), indicating that the response of the 5th and 6th modes is not consumed by damping during this period, and the difference in the proportion of the two in the PF1 component is not significant. The frequency curve of the component shows an interesting phenomenon. From about 3.8 s, the frequency curve gradually approaches the first mode, and only the frequency of the first mode exists at about 6 s. This indicates that the response of the first mode always exists and accounts for a large proportion. The response of the third mode only exists in the first half of the entire 10 s response due to the effect of damping, and the response of the third mode in the latter half is very small, almost zero, which verifies the effectiveness of lMD-based structure dynamic detection and analysis method [37].

3 Methods

3.1 Identification of nonlinear characteristic parameters

The transient response of structure displacement is given in formula (1).

(1) x ( t ) = ∑ i = 1 N A i e − n i t sin ( ω d i t + ϕ i ) ,

where n _i = c _i/(2m _i), c _i is the ith mode damping, m _i is the ith order mode mass, and the transient response of acceleration can be obtained after two derivatives, as shown in formula (2).

(2) a ( t ) = ∑ i = 1 N A i ( n ) 2 ( ω d i ) 2 e − n t i sin ( ω d i t + θ i ) ,

where θ _i = φ _i + π. Then, the instantaneous amplitude and frequency of a certain mode of the transient response of acceleration are shown in formulas (3) and (4).

(3) A i ( t ) = A i ( n i ) 2 ( ω d i ) 2 e − n i t ,

(4) f i ( t ) = ω d i 2 π .

Take the logarithm of both sides of formula (3), as shown in formula (5).

(5) I n A i ( t ) = I n [ A i ( n i ) 2 ( ω d i ) 2 ] − n i t ,

where n _i is the slope of the fitting line of the natural logarithm of the amplitude, and the damping is shown in formula (6).

(6) ξ i = n i ω n i .

The relation between undamped natural frequency ω n i and damped natural frequency ω d i is shown in formula (7).

(7) ω d i = ω n i 2 − n i 2 .

Because of n i < ω n i , we can think of ω n i ≈ ω d i . From formula (6), formula (8) can be obtained as follows:

(8) ξ i = n i ω d i .

The damping ratio and the natural frequency of the ith-order mode can be obtained from formulas (1)–(8), and the vibration response of the N-order mode is separated by the LMD method.

After the vibration signal is decomposed, the instantaneous amplitude function and the instantaneous frequency function are obtained, as shown in formulas (9) and (10), respectively.

(9) A i ( t ) = A i 1 ( t ) A i 2 ( t ) … A i n ( t ) = ∏ q = 1 n A i q ( t ) ,

(10) f i ( t ) = 1 2 π d { arccos [ sin ( t ) ] } d t .

Take the logarithm of both sides of formula (9), and the result is shown in formula (11).

(11) I n A i ( t ) = I n [ A i 1 ( t ) A i 2 ( t ) … A i n ( t ) ] = I n ∏ q = 1 n A i q .

The natural logarithm curve of instantaneous frequency and instantaneous amplitude can be obtained by formulas (10) and (11), and the modal natural frequency and damping ratio can be identified after fitting.

3.2 Model establishment and finite element analysis

3.2.1 Model establishment

The general finite element software SAP2000 was used to establish the three-dimensional multilayer structure model, and the materials of the model were C45 concrete, longitudinal reinforcement HRB400, and stirrup HPB300. To fit the reality, the structure design and reinforcement are based on the calculation results of PKPM2010, and two damping forms, Rayleigh damping and nonlinear damper, are considered in the SAP2000 model. An arbitrary directional disturbance is applied to the structure, and then, the time–history analysis technique (direct integration method) of SAP2000 is used to analyze the dynamic response of the structure [38]. The dynamic time–history response of the structure is extracted after checking the analysis results to ensure the accuracy of the modeling and results. Here, only one degree of freedom of one node is selected for the analysis. After comparison, it is found that the degree of freedom of other nodes is very similar to this, so there is no too much demonstration here. Figures 1–3 shows the time–history curve of displacement, velocity, and acceleration of structure 4# node in the X-axis direction, in which the sampling time is 10 s, the sampling frequency is 200 Hz, and there are a total of 2001 points.

Figure 1

Offset time history.

Figure 2

Historical ramp time curve.

Figure 3

Historical acceleration time curve.

3.2.2 Time–history response curve and comparative analysis

Figures 1–3 show the displacement, velocity, and acceleration time history curve analysis. In Figure 1, the displacement time–history curve of 0 seconds before curve has a different frequency combination of signs, but the slight change is difficult to use the adaptive algorithm, and in terms of its overall, the displacement time–history curve of the waveform only reflects the basic structure of a modal vibration response. In Figure 2, there is a significant combination of waveforms in the first second of the velocity time–history curve. Theoretically, such waveforms with obvious combination signs can be separated, indicating that the velocity time–history curve reflects the characteristics of the multimodal response combination. The first 2 s of the acceleration time–history curve in Figure 3 shows very significant characteristics of waveform combination, which is very conducive to waveform separation and can obtain more accurate solutions.

From Figures 1–3, it can be seen that the displacement time–history curve only reflects subtle signs of waveform combination and basically only reflects the vibration characteristics of the low-frequency mode of the structure, and it cannot understand the part of the high-frequency mode of the structure. The shape of the acceleration time history curve shows the characteristics of the multimodal combination. This waveform combination not only shows the part of the low-frequency mode of the structure but also fully contains the response information of the high-frequency mode of the structure, which provides effective material for understanding the vibration response characteristics of the structure. The morphology of the velocity time–history curve is between the two, showing the characteristics of the multimode combination, but its combination morphology is not as significant as that of the acceleration time–history curve, resulting in the modal separation effect of the velocity time–history curve is weaker than that of the acceleration time–history curve, especially in the calculation accuracy of high-frequency modes.

4 Results and analysis

4.1 Modal parameter identification based on LMD

4.1.1 Decomposition of acceleration time–history curve

In Figures 1–3, by comparing and analysing the time– history curve of shape and characteristics of selecting the best acceleration time– history curve of calculation and analysis as an original signal, the endpoint effect uses the following method to deal with: consider the left endpoint effect, the original signal of the 1 s data according to the zero point vertical axis image processing and image part don’t show. Considering the end effect on the right of subsequent data processing, the actual sampling time is 3.5 s, and the data in the last 1 s are not shown. The components and allowances of each order PF obtained by decomposition are shown in Figures 4 and 5. Finally, the time–history curve of acceleration is decomposed into the time–history response of two modes, and the proportion of allowance is small and can be ignored. By analysing the working condition of modal SAP2000, one can get this nonlinear structure modal (linear) of each order natural frequency of vibration, which can produce response in 4 # node X axis component of (due to order 7 and above the high frequency of the modal participation is low, the response end quickly, almost did not reflect in the response curve, so temporary not consider) 1, 3, 5, and 6 order modal. The frequencies are 0.901, 1.065, 3.101, and 3.147 Hz. Among them, the first and fifth modes are translational, and the third and sixth modes are rotational. Although the modal information does not fully and accurately show the nonlinear characteristics of the structure, it is accurate.

$Figure 4 The PF 1 {\text{PF}}_{1} components for accelerating time history.$

Figure 4

The PF 1 components for accelerating time history.

$Figure 5 The PF 2 {\text{PF}}_{2} weight and acceleration history allowance.$

Figure 5

The PF 2 weight and acceleration history allowance.

The frequency characteristics of each PF component are analyzed and compared with the linear modal frequency, as shown in Figures 6 and 7.

Figure 6

Frequency and contrast elements without damping.

Figure 7

Component without damping frequency and contrast map.

4.1.2 Analyze the aforementioned two graphs

As shown in Figure 6, the frequency of component PF 1 fluctuates between the frequencies of the fifth and sixth modes, indicating that component PF 1 contains the response of the fifth and sixth modes of the structure. The reason is that the frequencies of the two modes (3.101 and 3.147 Hz) are very close, which makes the response of the two modes difficult to be separated by the LMD method. The frequency of the PF 1 component remained stable all the time (the first 2.5 s), indicating that the response of the 5th and 6th modes was not completely consumed by damping during this period, and the difference in the proportion of the two in the PF1 component was not significant [39]. The frequency curve of PF 2 component in Figure 7 shows an interesting phenomenon. From about 3.8 s, the frequency curve gradually approaches the first mode, and only the frequency of the first mode exists at about 6 s. This indicates that the response of the first mode always exists and accounts for a large proportion. The response of the third mode only exists in the first half of the whole 10 s response due to the effect of damping, and the proportion of the response of the third mode in the latter half is very small, almost zero. The aforementioned phenomenon indicates that although the LMD method has problems in the separation of dense frequency modes, the frequency characteristics of its PF component can reflect the phenomenon of this mode combination and the duration of each mode in the whole response [38,40,41,42,43,44,45,46,47].

5 Conclusion

A nonlinear characteristic parameter identification method based on LMD is proposed. By studying the identification of nonlinear characteristic parameters, the establishment of the analysis model and finite element, and the identification of modal parameters based on LMD, the results of calculation and analysis are as follows: The frequency of component PF 1 fluctuates between the frequencies of the fifth and sixth modes, indicating that component PF 1 contains the response of the fifth and sixth modes of the structure. The reason is that the frequencies of the two modes (3.101 Hz and 3.147 Hz) are very close, which makes the response of the two modes difficult to be separated by the LMD method. The frequency of the PF 1 component remained stable all the time (the first 2.5 s), indicating that the responses of the 5th and 6th modes were not consumed by damping during this period, and the difference in the proportion of the two modes in the PF1 component was not significant. The frequency curve of PF 2 component shows an interesting phenomenon. From about 3.8 s, the frequency curve gradually approaches the first mode, and at about 6 s, only the frequency of the first mode exists. This indicates that the response of the first mode always exists and accounts for a large proportion. The response of the third mode only exists in the first half of the whole 10 s response due to the effect of damping, and the proportion of the response of the third mode in the latter half is very small, almost zero. These phenomena indicate that although there are problems in the separation of dense frequency modes in the LMD method, the frequency characteristics of its PF component can reflect the phenomenon of the combination of modes and reflect the duration of each mode in the whole response. In addition, it is worth noting that the frequency morphology of the aforementioned two PF components all shows some regular fluctuations. The reason for the formation of such morphology is estimated to be related to the included modal frequency, but the specific reason needs further study. In the future, the local wave method will be applied to the engineering problem of multi-degree-of-freedom nonlinear system modal parameter identification, which will further show its superiority in nonlinear and nonstationary signal processing.

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Funding information: The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.

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Received: 2021-12-11

Revised: 2022-02-12

Accepted: 2022-02-22

Published Online: 2022-03-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2022-0011

Schlagwörter für diesen Artikel

local mean decomposition method; dynamic detection; finite element model; PF component; nonlinear characteristics

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BY 4.0