CN111859681B - Linear structure damage identification method based on ARFIMA model - Google Patents
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Abstract
The invention discloses a linear structure damage identification method based on an ARFIMA model, which is characterized in that a corresponding ARFIMA model before damage and an ARFIMA model after damage are established for each layer of structure according to acceleration response data of each layer of structure before and after damage of a multi-layer structure; extracting corresponding time sequences from each pre-injury ARFIMA model and each post-injury ARFIMA model respectively, differentiating the time sequences through d-order fractional order differences to obtain a pre-injury time sequence, and establishing an pre-injury fractional difference ARMA model; d, taking the value as a memory parameter of an ARFIMA model; according to the cepstrum coefficient of the ARMA model with the score difference before damage and the cepstrum coefficient of the ARMA model with the score difference after damage, after the cepstrum distance of each layer of structure is calculated, the linear structure damage is identified according to the cepstrum distance: the method can indirectly identify according to the degree of freedom index ARFIMACM or directly identify and locate the damage position according to the linear damage index ARFIMACM, thereby breaking through the limitation that the damage position can not be directly located in the traditional identification method. The invention improves the recognition precision and reliability and has good anti-interference capability.
Description
Technical Field
The invention relates to the technical field of structural damage identification.
Background
Various damages to civil engineering structures often occur during service, so that the study of the health state of the soil-wood structure has been increasingly attracting attention. The dynamic health monitoring of the structure is also gradually becoming a popular research field, especially to the end of the 20 th century, along with the rapid development of the computer Internet and the application of various artificial intelligence algorithms, the structure health detection technology is rapidly developed, the application object of the technology is expanded from the initial aging building structure to various important engineering structures, and the technology gradually forms a more complete emerging comprehensive engineering discipline. Although research in the civil engineering field starts later, and many health detection methods are still in the theoretical research stage, most of the methods are limited to experimental research, and the application in practical engineering is less, although some civil structure engineering sites at home and abroad are applied to the damage real-time detection technology, the research of the technology still has a great development space, so that the identification research on structural damage is necessary.
The damage identification method based on the time domain model is one of the hot spots of current research, and can directly utilize time-course data of structural dynamic response to carry out damage identification. The time series is, as the name implies, a time-ordered sequence of data, with data between time points having a certain relationship. Time series analysis is an important branch of probability statistics and has a wide application range. Common linear time sequences are autoregressive models (AR models), moving average models (MA models), and hybrid autoregressive moving average models (ARMA models), and the accuracy and reliability of linear structure damage identification by adopting the linear models currently need to be improved.
Disclosure of Invention
Aiming at the defects of the technology, the invention provides a linear structure damage identification method based on an ARFIMA model, which solves the technical problem of how to improve the linear structure damage identification precision.
In order to solve the technical problems, the invention provides a linear structure damage identification method based on an ARFIMA model, which comprises the following steps:
establishing a corresponding ARFIMA model for each layer of structure according to acceleration response data of each layer of structure before the damage of the multi-layer structure; extracting corresponding time sequences from each pre-injury ARFIMA modelThen pass->Order fraction order difference versus time sequence +.>Differential to obtain time sequence before injury>According to the time sequence before injuryEstablishing an ARMA model of score difference before injury; wherein i represents an i-th layer structure, i e {1,2,.. M }, m representing the total number of layers of the multilayer structure; />The value is the memory parameter of the ARFIMA model before injury of the ith layer structure,X t 、Z t all represent time slices in the time sequence, t e {1,2,..N }, N representing the total number of time slices;
establishing a corresponding ARFIMA model for each layer of structure according to acceleration response data of each layer of structure after the damage of the multi-layer structure; extracting corresponding time sequences from each post-injury ARFIMA modelThen pass->Order fraction order difference versus time sequence +.>Differentiating to obtain a post-injury time sequence +.>According to the post-injury time series->Establishing an ARMA model of score difference after damage; wherein (1)>Memory parameters of ARFIMA model after injury with value of ith layer structure, ++>
According to the cepstrum coefficient of the ARMA model with the score difference before damage and the cepstrum coefficient of the ARMA model with the score difference after damage, calculating the cepstrum distance according to the following formula:
wherein ,D(L(1) ,L (2) ) i A cepstrum distance of the ARMA model representing the pre-injury score difference and the ARMA model representing the post-injury score difference of the i-th layer structure,cepstrum coefficients of the ARMA model representing the pre-injury score difference of the i-th layer structure,the cepstrum coefficient of the ARMA model representing the score difference after the damage of the ith layer structure, and n represents the cepstrum data interception number;
and after the cepstrum distance of each layer of structure is calculated, the damage of the linear structure is identified according to the cepstrum distance.
Further, an indirect method is adopted to identify the damage of the linear structure: at a cepstrum distance of D (L (1) ,L (2) ) i ARFIMACM as a degree of freedom index for structural layers i If the degree of freedom index ARFIMACM of the ith layer structure i Degree of freedom index ARFIMACM with layer i-1 structure i-1 And if the degree of freedom index is larger than that of the rest layer structures, judging that the linear structure damage occurs in the ith layer structure.
Further, the linear structural damage is identified by a direct method:
after the cepstrum distance is calculated, a linear damage identification index is calculated according to the cepstrum distance, and the following formula is adopted:
wherein ARFIMACI i A linear damage identification index representing an i-th layer structure;
if the linear damage identification index ARFIMACI of the ith layer structure i If the linear structure damage is the maximum value, judging that the linear structure damage occurs in the ith layer structure.
Compared with the prior art, the invention has the advantages that:
1. according to the method, the ARFIMA (Autoregressive fractionally integrated moving average) integral autoregressive moving average model is combined with the cepstrum distance for the first time, and is applied to the field of structural damage identification, so that the identification accuracy and reliability are improved. The ARFIMA model contains two classical linear time series processes and one fractional integration process: an autoregressive AR (p) portion, a moving average MA (q) portion, and a fractional integral FI (d) portion. The invention uses the time sequence { X } of ARFIMA model t After fractional differentiation, the original ARFIMA (p, d, q) process is converted into a time series { Z t The ARMA (r, s) procedure of } improves the data accuracy, thereby enabling to base on the time sequence { Z t Building a more accurate ARMA model to obtain more accurate cepstrum coefficients.
2. According to the invention, the cepstrum distance is calculated according to cepstrum coefficients, and two recognition modes of linear structure damage are respectively provided on the basis of the cepstrum distance: the indirect method and the direct method can not directly identify the position of the damage, but the calculated amount is smaller; the direct method needs to further calculate a linear damage index according to the cepstrum distance, and the position of damage can be directly judged according to the linear damage index.
Drawings
FIG. 1 is a flow chart for building an ARFIMA model;
FIG. 2 is a simplified eight degree of freedom architecture diagram of an eight layer shear structure;
FIG. 3 is a graph showing the comparison of damage recognition of the 1-4 layer structure in example 1;
FIG. 4 is a graph showing the comparison of damage detection of the 5-8 layer structure of example 1;
FIG. 5 is a graph showing the comparison of damage detection of 1-8 layer structures in example 2.
Detailed Description
First) establishing ARFIMA model
The ARFIMA model is generally applied in the financial field and is established by adopting financial data, and the process of establishing the ARFIMA model adopts the prior art, but the adopted data is acceleration response data, and is shown by referring to fig. 1.
ARFIMA (p, d, q) time series comprises two classical linear time series processes and a fractional integration process: an autoregressive AR (p) part, a moving average part MA (q) part, and a fractional integral FI (d) part, expressed as follows:
Φ p (B)(1-B) d X t ＝Θ q (B)ε t (1)
wherein the sequence { X } t : t=1, 2, …, N } meets the smoothness requirement; epsilon t Is a residual sequence of the model, is a white noise process, with infinite or finite variance. B is a hysteresis operator, e.g. BX t ＝X t-1 The method comprises the steps of carrying out a first treatment on the surface of the d is the fractional differential order, also called the memory parameter of the ARFIMA model, and the value range of d is-1/2<d<1/2, and d is not equal to 0; (1-B) d Known as a fractional difference operator; phi p (B) Corresponding to the polynomial of the AR (autoregressive) part, Θ q (B) The corresponding polynomial is the MA (mean movement) part, which is expressed as follows:
fractional difference operator (1-B) d Can be regarded as a function f (z) = (1-z) d Its power series expansion:
equation (3) is convergent, where m j (d) Coefficients that are polynomial expansions can be expressed by a function about Γ:
two) converting ARFIMA model into ARMA model
The d-order fractional difference method can be used for converting into an ARFIMA (p, d, q) model which is a d-order fractional difference ARMA time sequence model. The specific method comprises the following steps:
first, the sequence { X } is smoothed by using a fractional difference operator pair t D-order fractional difference is carried out to obtain a new sequence Z t ：
Z t ＝(1-B) d X t (5)
Will Z t Is brought into formula (1) to obtain
Φ p (B)Z t ＝Θ q (B)ε t (6)
Sequence { Z t The meaning of the fractional order difference is to convert complex ARFIMA (p, d, q) sequences into more common time-series ARMA processes, subject to the ARMA process.
And then the expression of the score difference operator is expressed as follows:
when the d value is determined, there is only one variable j in equation (7), this is noted:
then equation (5) may be rewritten as:
by time series { X } t After fractional differentiation, the original ARFIMA (p, d, q) process is converted into sequence { Z t ARMA Process, to distinguish from ARFIMA (p, d, q) models, we label the transformed ARMA model as a d-th order fractional differential ARMA (r, s) model.
Three) cepstrum distance calculation
Since { Z t The sequence is the ARMA (r, s) process after fractional differentiation, and then the system transfer function of the sequence is expressed in the z-domain as:
in the formula ：λt Zero point, eta of model after fractional difference t Representing poles. For a single-input single-output linear time-invariant model, the system function is H (z), the power spectrum is P (z), and the cepstrum of the model can be obtained through inverse Fourier transform after the logarithm of the power spectrum P (z) is taken:
wherein: v is the standard deviation of the model variance. Its cepstrum can be represented by the poles and zeros of the model:
since the poles and zeros are conjugated in the complex plane, the above equation can be further expressed as:
wherein n represents, eta t Representing poles, lambda of ARMA model t The zero point of the ARMA model is represented, v represents the sample variance, r represents the AR order after the fractional difference, and s represents the MA order after the fractional difference.
For the two ARFIMA models L (1), L (2), assuming an autoregressive order r and a moving average partial order s after the d-order fractional difference, the cepstrum distance between the two ARFIMA models can be defined as
in the formula ： and />The cepstral coefficients of the two models are respectively. If a same linear filter L (3) is further introduced, the moving average s term can be filtered out by the transfer function of the filter, and the cepstrum distance between the two ARMA models can be finally reduced to
wherein , and />The t th pole of the model before and after injury, < -> and />The j th conjugate pole of the model before and after injury,> and />Pi in the equation is the multiplicative symbol in the high number.
Fourth) numerical simulation experiment
Numerical simulation was performed on a certain eight-layer shear structure model, see fig. 2, assuming a mass m=100 kg per floor, a stiffness k=1 MN/m, and a damping coefficient of 3%. And the structure is simplified into an eight-degree-of-freedom system. The damage of the structure is realized by reducing the rigidity of a single layer and setting different rigidity reduction coefficients, 8 working conditions and a rigidity reduction coefficient method are adopted to study the damage identification of the structure, and the adopted reduction coefficients alpha are respectively 30%, namely the rigidity after damage is reduced to 1 MN/mX (1-30%). The 8 working conditions are considered, namely, the stiffness of the single layer is respectively reduced from the first layer to the eighth layer in sequence, for example, when the fifth layer is damaged, the stiffness of the fifth layer is only reduced, and the rest floors are unchanged, so that the total of 8 damaged working conditions exist.
A section of the white noise process is used herein as the load excitation of the structure, which is treated with a butterworth low pass filter to make it smoother in order to simulate the structure being subjected to a natural excitation that is relatively smooth in the long term natural environment. The filtered white noise y (t) is taken as the basic excitation f (t) of the eight-layer laminated shear structure, the acceleration response of the structure is calculated by a Wilson-theta method, and in order to ensure that the calculation result is accurate enough and the convergence speed is proper, the calculation step length is set to be 0.005s, and the total calculation length is 9000 steps. In the real environment where the structure is located, a certain amount of noise exists, and according to experience, each calculated acceleration response sequence is added with measurement noise interference of 5% level.
Fifth) different examples of logarithmic simulation experiments
Example 1
In the embodiment, an indirect method is adopted to identify the damage of the linear structure: at a cepstrum distance of D (L (1) ,L (2) ) i ARFIMACM as a degree of freedom index for structural layers i If the degree of freedom index ARFIMACM of the ith layer structure i Degree of freedom index ARFIMACM with layer i-1 structure i-1 And if the degree of freedom index is larger than that of the rest layer structures, judging that the linear structure damage occurs in the ith layer structure.
According to the cepstrum coefficient of the ARMA model with the score difference before damage and the cepstrum coefficient of the ARMA model with the score difference after damage, calculating the cepstrum distance according to the following formula:
wherein ,D(L(1) ,L (2) ) i A cepstrum distance of the ARMA model representing the pre-injury score difference and the ARMA model representing the post-injury score difference of the i-th layer structure,cepstrum coefficients of the ARMA model representing the pre-injury score difference of the i-th layer structure,and (3) representing the cepstrum coefficient of the ARMA model of the post-injury score difference of the ith layer structure, wherein n represents the cepstrum data cut-off number.
In order to highlight the damage detection effect of the ARFIMACM index, the structural damage detection method directly based on the ARMA model cepstrum distance (Cepstral Metric between ARMA model-CM-ARMA) is also compared with the structural damage detection method. The calculated ARFIMACM index value and CM-ARMA index value result are plotted as a histogram, and as shown in fig. 3 to 4, the titles "1-store, 2-store, …" represent the floor where the Damage source is located, the ordinate represents the degree of freedom number (DOF number) of the eight-layer structure, and the abscissa corresponds to the value of the degree of freedom index ARFIMACM, and the value is denoted as "Damage".
The damage detection results of the working conditions State-1 to State-8, i.e., the stiffness reduction coefficient α=30% are shown in fig. 3. The damage identification based on the cepstrum distance index is judged by two degrees of freedom, so that when a certain layer is damaged, the index value corresponding to the two degrees of freedom is theoretically obviously higher than that of other degrees of freedom. For example, in the working condition "2-store", the degrees of freedom index ARFIMACM of degrees of freedom numbers 1 and 2 has a significantly higher value than the other degrees of freedom, the damage is localized to the second layer, and the source of the damage in State 2 is consistent with the degree of freedom numbers 1 and 2 (the second layer). From the overall results shown in FIG. 3, the ARFIMACM index showed a greater lesion localization than CM-ARMA. In the "1-store", the ARFIMACM index value of the number 1 degree of freedom is obviously higher than the CM-ARMA index value, and the damage values of the CM-ARMA index on the number 1 and 2 degrees of freedom are basically at the same level, which can cause misjudgment of damage positioning; likewise, for the working condition of 2-store, the CM-ARMA index is difficult to judge the floor where the damage source is located, and the degree of freedom index ARFIMACM can be used; in the working condition "3-store", although the ARFIMACM index has larger damage values in the degrees of freedom 4, 5 and 6, the damage source can be positioned to the third layer by using the ARFIMACM. For other conditions, although the ARFIMACM index is not much higher than the CM-ARMA index, the accuracy of lesion localization is increased to some extent.
In conclusion, the degree of freedom index ARFIMACM based on the ARFIMA model cepstrum distance can accurately locate the occurrence position of the structural damage source. In judging the damage degree, compared with the CM-ARMA index, the ARFIMACM index obviously improves the damage identification capability.
Example 2
And (3) identifying the damage of the linear structure by adopting a direct method:
after the cepstrum distance is calculated, a linear damage index is calculated according to the cepstrum distance, and the following formula is adopted:
wherein ARFIMACI i A linear damage index representing the i-th layer structure;
if the linear damage identification index ARFIMACI of the ith layer structure i If the linear structure damage is the maximum value, judging that the linear structure damage occurs in the ith layer structure.
Considering 8 working conditions, the damage degree is 30%, namely 30% damage occurs from the first layer to the eighth layer in sequence, and the identification result is shown in fig. 5. The linear injury index ARFIMACI based on ARFIMA model and the cepstrum distance index (CM-ARMA) based directly on ARMA model are used for injury position identification, and the calculation result is shown in figure 5. From the recognition result shown in fig. 5, the linear injury index ARFIMACI has a higher injury localization capability than CM-ARMA. Under the damage working condition of the 1 st layer, the linear damage index ARFIMACI value of the 1 st stiffness layer is obviously higher than the CM-ARMA index value, and the damage values of the CM-ARMA index values on the 1 st stiffness layer and the 2 nd stiffness layer are basically at the same level, so that misjudgment of damage positioning can be caused; likewise, under the layer 2 damage working condition, the CM-ARMA index is difficult to judge the rigidity layer where the damage source is located, and the linear damage index ARFIMACI can better judge the damage position; in a word, for these 8 damage conditions, the recognition effect of the linear damage index ARFIMACI is significantly better than that of the CM-ARMA index.
The ARFIMA model-based distance function conversion index provided by the invention can accurately identify the position of the damage, and the identification effect is obviously better than that of the traditional CM-ARMA index based on the ARMA model, so that the ARFIMA model-based distance function conversion index obviously improves the accuracy of damage identification.
Summarizing, the indirect method and the direct method provided by the invention can improve the identification precision, improve the reliability of damage positioning and have good anti-interference capability. In particular, the linear damage index ARFIMACI provided by the invention can directly locate the damage position, and the limitation that the damage position cannot be directly located by the traditional identification method is broken through.
Claims (5)
1. The linear structure damage identification method based on the ARFIMA model is characterized by comprising the following steps of:
establishing a corresponding ARFIMA model for each layer of structure according to acceleration response data of each layer of structure before the damage of the multi-layer structure; extracting corresponding time sequences from each pre-injury ARFIMA modelThen pass->Order fraction order difference versus time sequence +.>Differential to obtain time sequence before injury>According to the time sequence before injuryEstablishing an ARMA model of score difference before injury; wherein i represents an i-th layer structure, i e {1,2,.. M }, m representing the total number of layers of the multilayer structure; />The value is the memory parameter of the ARFIMA model before injury of the ith layer structure,and->X t 、Z t All represent time slices in the time sequence, t e {1,2,..N }, N representing the total number of time slices;
establishing a corresponding ARFIMA model for each layer of structure according to acceleration response data of each layer of structure after the damage of the multi-layer structure; extracting corresponding time sequences from each post-injury ARFIMA modelThen pass->Order fraction order difference versus time sequence +.>Differentiating to obtain a post-injury time sequence +.>According to the time sequence after injury/>Establishing an ARMA model of score difference after damage; wherein (1)>Memory parameters of ARFIMA model after injury with value of ith layer structure, ++>And->
According to the cepstrum coefficient of the ARMA model with the score difference before damage and the cepstrum coefficient of the ARMA model with the score difference after damage, calculating the cepstrum distance according to the following formula:
wherein ,D(L(1) ,L (2 )) i A cepstrum distance of the ARMA model representing the pre-injury score difference and the ARMA model representing the post-injury score difference of the i-th layer structure,cepstrum coefficients of ARMA model representing pre-injury score difference of ith layer structure, ++>The cepstrum coefficient of the ARMA model representing the score difference after the damage of the ith layer structure, and n represents the cepstrum data interception number; wherein, the calculation formula of the cepstrum coefficient is as follows:
wherein ,ηt Representing poles, lambda of ARMA model t Representing the zero point of the ARMA model, v representing the sample variance, r representing the AR order after the fractional difference, and s representing the MA order after the fractional difference;
after the cepstrum distance of each layer of structure is calculated, the damage of the linear structure is identified according to the cepstrum distance;
wherein, adopt the indirect method to discern linear structure damage: at a cepstrum distance of D (L (1) ,L (2) ) i ARFIMACM as a degree of freedom index for structural layers i If the degree of freedom index ARFIMACM of the ith layer structure i Degree of freedom index ARFIMACM with layer i-1 structure i-1 If the degree of freedom index is larger than that of the other layer structures, judging that the linear structure damage occurs in the ith layer structure;
alternatively, the direct method is used to identify the damage to the linear structure:
after the cepstrum distance is calculated, a linear damage index is calculated according to the cepstrum distance, and the following formula is adopted:
wherein ARFIMACI i A linear damage index representing the i-th layer structure;
if the linear damage identification index ARFIMACI of the ith layer structure i If the linear structure damage is the maximum value, judging that the linear structure damage occurs in the ith layer structure.
2. The method for identifying linear structural damage based on ARFIMA model according to claim 1, wherein the degree of freedom index of the multi-layered structure is shown in the form of a histogram: the abscissa represents the degree of freedom index, and the ordinate represents the layer structure number.
3. The method for identifying linear structural damage based on ARFIMA model according to claim 1, wherein the linear damage index of the multi-layer structure is shown in the form of a histogram: the abscissa represents the layer structure number and the ordinate represents the linear damage index.
4. The method for identifying the damage to the linear structure based on the ARFIMA model according to claim 1, wherein a linear filter is introduced to filter out a moving average term of the ARMA model with the score difference before damage and the ARMA model with the score difference after damage, and a cepstrum distance calculation formula is simplified, and the general formula is as follows:
wherein , and />The t th pole of the model before and after injury, < -> and />Represents the j-th conjugate pole of the model before and after injury.
5. The method for identifying linear structural damage based on ARFIMA model as recited in claim 1, wherein X is t And Z is t The conversion formula of (c) is as follows:
Z t ＝(1-B) d X t ；
wherein B represents a hysteresis operator; d represents the fractional differential order, d takes the value of the memory parameter of the ARFIMA model, d is more than 1/2 and less than 1/2, and d is not equal to 0.
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