CN113155384A - Sensor arrangement method for reducing uncertainty of structural damping ratio identification - Google Patents
Sensor arrangement method for reducing uncertainty of structural damping ratio identification Download PDFInfo
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Abstract
The invention relates to a sensor arrangement method for reducing the identification uncertainty of a structural damping ratio, which is used for civil engineering structure health monitoring and is characterized by comprising the following steps: a posterior probability density function of the damping ratio and a sensor arrangement based on the information entropy; the posterior probability density function of the damping ratio is used for describing the uncertainty of the damping ratio identification; the sensor arrangement based on the information entropy provides an information entropy criterion to quantify the uncertainty of the damping ratio parameter identification, and a sequential algorithm is utilized through the information entropy criterion to guide the arrangement of the sensors.
Description
Technical Field
The invention belongs to the technical field of sensors, and particularly relates to a sensor arrangement method for reducing uncertainty of structural damping ratio identification in civil engineering structure health monitoring.
Background
The structure damping ratio reflects the energy consumption and damping performance of the structure and is reflected in the vibration attenuation process of the structure. When the structure is subjected to performance evaluation, the energy consumption condition of the structure needs to be analyzed according to the damping ratio of each order of the structure, and the situation that the high-order mode participates in vibration is judged. During operation of the structure, when the measurement data is used to identify the damping ratios of the various orders of the structure, measurement noise may cause a deviation between the identified damping ratio and the true damping ratio. Different segmented sampling data are subjected to damping ratio identification for multiple times, and different results, namely uncertainty of structural damping ratio identification, can be obtained. The sensor arrangement is an important link of structural health monitoring, and the quality of the monitoring data acquired by the sensor directly influences the effect of structural damping ratio identification. On the premise of limited number of sensors, more and better measurement data can be obtained by reasonably selecting the positions of the sensors. After the position of the sensor is optimized, when the structural damping ratio is identified by using the measured data, the uncertainty of the identification result can be effectively reduced, and the identification precision is improved.
At present, sensor arrangement methods are more directed at arrangement of acceleration (displacement) sensors, and the methods can be well applied to acquisition of structural modal parameter information. Existing sensor arrangement methods for structure modal identification are mostly based on identification of modal coordinates of a structure: an effective independence method that makes the modal matrix independently distinguishable; a modal kinetic energy method which comprehensively considers a mass array and a modal matrix; and (3) identifying uncertain time domain information entropy method by using the quantized modal coordinates. However, the theoretical framework of these sensor arrangement methods does not consider the influence of error factors such as measurement noise on the identification of the structural damping ratio parameters, and is not suitable for the identification of the structural damping ratio. The sensor arrangement method provided by the invention is suitable for identifying the structural damping ratio parameter under the structural operation condition, can effectively reduce the uncertainty of the structural damping ratio identification, and has high engineering practical value and important research prospect in structural health monitoring.
Disclosure of Invention
In order to better solve the problems existing in the prior art, the invention provides a sensor arrangement method for reducing the uncertainty of structural damping ratio identification.
The invention adopts a sensor arrangement method for reducing the identification uncertainty of the structure damping ratio, which is used for monitoring the health of a civil engineering structure and is characterized by comprising the following steps: a posterior probability density function of the damping ratio and a sensor arrangement based on the information entropy; the posterior probability density function of the damping ratio is used for describing the uncertainty of the damping ratio identification; the sensor arrangement based on the information entropy provides an information entropy criterion to quantify the uncertainty of the damping ratio parameter identification, and a sequential algorithm is utilized through the information entropy criterion to guide the arrangement of the sensors.
Further, a posterior probability density function of the damping ratio adopts discrete measurement response to perform data sampling; due to the influence of measurement noise, the discrete measurement response is represented in the form:
in the formula (1)
Selecting a matrix for the sensor locations; n is a radical ofLThe degree of freedom of the measuring point; n is a radical ofdIs the total degree of freedom of the structure;
is the structural response measured by the sensor; n denotes a discrete sampling instant, i.e. a sample at time t ═ n Δ t, where Δ t denotes the sampling interval;
is the structural damping ratio to be identified, ζ ═ { ζ ═ ζi,i＝1,2,…,Nm}，ζiRepresenting the i-th order damping ratio of the structure; n is a radical ofmThe order of the structure participating in the vibration mode;
is the true response of the structure;
is the error between the measured value and the true value.
Further, the posterior probability density function of the damping ratio performs discrete Fourier transform on discrete measurement response sampling data;
discrete sampling data is obtained by substituting equation (1)Into formula YNAnd (f, N), wherein N is 1, … N, and N is the number of sampling points, and the discrete fourier transform is performed to obtain:
in the formula (2)
Indicating the operation of solving the root number; sigma []Represents a summation symbol; e is a natural constant;
for the resulting discrete Fourier transform vector, the corresponding frequency scale is fk＝k/NΔt，k＝1,…,NqIn which N isqInt (N/2) +1 corresponds to the Nyquist frequency, int [ ·]Expressing the integer sign; i represents a unit imaginary number.
Further, according to the posterior probability density function of the damping ratio, each order mode of the structure is in a frequency band near a certain order damping ratio, and the vibration response of the structure is mainly controlled by the order mode; the number of sampling points in the damping ratio near frequency band is far more than 1, and the result obtained by the formula (2)
As a complex gaussian vector with zero mean, the covariance matrix form is as follows:
in the formula (3) [. above]TRepresenting an operation of transposing a matrix or a vector;
for a certain order mode shape matrix of the structure, it is assumed that the vibration is controlled by the ith order modei，
Is the overall modal shape matrix of the structure; siA spectral density representing an i-th order modal excitation; seA spectral density representative of measurement noise;
is a number Nd×NdThe identity matrix of (1); alpha is alphai＝||Lφ||＝||LΦiThe | | | is a normalization coefficient of the ith-order modal shape vector, and | | · | | represents the modulo operation;
wherein the formula (4) is a power amplification factor part in the formula (3); in the formula (4) < beta >ik＝fi/fkRepresenting the ratio between the damping ratio and the frequency scale corresponding to the Fourier transform; f. ofiIs the ith order natural frequency of the structure; [. the]-1Representing an inversion operation.
Further, a posterior probability density function of the damping ratio is obtained by
The value of (b) satisfies complex multivariate Gaussian distribution, a certain level of damping ratio zetaiA posterior probability density function of (a);
due to the formula (2)
Satisfy complex multiple Gaussian distribution and a certain damping ratio zetaiThe posterior probability density function of (a) is of the form:
oc in the formula (5) represents a proportional symbol;
set of Fourier transform vectors in the near band representing the damping ratio of the ith order；Π[·]Representing a continuous multiplication and product-solving symbol; det [. to]Representing determinant of the matrix; exp [ A ]]An exponential function with a natural constant e as a base and A as an exponential term is represented; superscript []*Representing the conjugate transposed symbol.
Further, the posterior probability density function of the damping ratio is represented by the posterior probability density function p (ζ) of equation (5)i|Zi) The calculation is done in logarithmic form:
p(ζi|Zi)∝exp(-L(ζi)) (6)；
in the formula (6)
Set so that L (ζ)i) Taking the ith order optimal damping ratio parameter estimation quantity of the minimum value, and applying a negative log-likelihood function L (zeta)i) Do it with
The second order taylor expansion of (a) yields:
in the formula (7)
Representing a negative log-likelihood function L (ζ)i) In that
The Hessian matrix, when using single order frequency parameter estimation,
and
degenerates into single variables from vectors and matrices, respectively.
Further, a posterior probability density function of the damping ratio,
by the following formula (7)
The value can be determined by:
scale factor e in formula (8)kCalculated from the following formula:
the calculation of equation (8) is performed while ensuring that the mode shape in equation (3) has undergone the operation of normalization, i.e., | | Φ | | |, 1.
Further, the posterior probability density function of the damping ratio adopts a multi-order damping ratio, and the posterior probability density function of the single-order damping ratio is subjected to continuous multiplication operation to obtain the following formula:
in formula (10), Z ═ Zi,i＝1,2,…,NmDenotes a set of discrete fourier transforms of various orders;
the optimal damping ratio vector of all orders at the position of the corresponding sensor is obtained;
for all orders the damping ratio to be identified is
A Hessian matrix of the same multi-order modal parameters
Satisfying block diagonal matrixForms, corresponding to modal parameters of respective orders
The composition is shown as the following formula:
further, after the posterior probability density function of the structural damping ratio is determined, the information entropy is utilized to quantize the uncertainty of frequency identification corresponding to different sensor arrangements, the optimal sensor position is selected according to the quantized value, and the information entropy formula of the damping ratio parameter to be identified is as follows:
ln [. in formula (12)]Representing a logarithmic operation; n is a radical ofθRepresenting the degrees of freedom of the damping ratio of each order considered.
Further, the sensor arrangement based on information entropy comprises the following steps: step one, establishing a finite element model of a structure, taking the freedom degrees of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting 0 number of the sensors; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating h (L) numerical values according to an information entropy formula (12) of the damping ratio parameter to be identified, and selecting the sensor position corresponding to the minimum numerical value; thirdly, deleting the selected positions from the remaining position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the residual positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement and jumping out of the cycle.
The invention has the beneficial effects that: the sensor arrangement method based on the information entropy can reduce the uncertainty of structural damping ratio identification. By the proposed theory, a theoretical link between sensor arrangement and bayesian frequency modal identification is established. And constructing a posterior probability density function of the damping ratio parameter to be identified so as to describe the uncertainty of the damping ratio identification. The information entropy index can well quantify the identification uncertainty of the damping ratio corresponding to different sensor arrangements, and the optimal sensor arrangement corresponds to the minimum information entropy value. By the sensor arrangement method provided by the invention, the adverse effect of measurement noise on the identification result of the damping ratio of each order can be effectively reduced, and the accuracy of the identification of the structural damping ratio is improved.
Drawings
In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings of the embodiments will be briefly described below, and it is obvious that the drawings in the following description only relate to some embodiments of the present invention and are not limiting on the present invention.
FIG. 1 is a schematic diagram of a finite element model of a simply supported beam according to the present invention;
FIG. 2a is a sensor layout of the first order damping ratio of the present invention;
FIG. 2b is a sensor layout of the second order damping ratio of the present invention;
FIG. 2c is a sensor layout of the third order damping ratio of the present invention;
FIG. 2d is a sensor layout of the first and second order damping ratios of the present invention;
FIG. 2e is a sensor layout of the second and third order damping ratios of the present invention;
FIG. 2f is a sensor layout of the first, second and third order damping ratios of the present invention.
Detailed Description
Figures 1, 2a, 2b, 2c, 2d, 2e, 2f, discussed below, and the various embodiments used to describe the principles of the present invention in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the invention. Those skilled in the art will understand that the principles of the present invention may be implemented in any suitable sensor placement method for reducing structural damping ratio identification uncertainty. The terminology used to describe various embodiments is exemplary. It should be understood that these are provided solely to aid in the understanding of this specification and their use and definition do not limit the scope of the invention in any way. The use of the terms first, second, etc. to distinguish between objects having the same set of terms is not intended to represent a temporal order in any way, unless otherwise specifically stated. A group is defined as a non-empty group containing at least one element.
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the drawings of the embodiments of the present invention. It is to be understood that the embodiments described are only a few embodiments of the present invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the described embodiments of the invention without any inventive step, are within the scope of protection of the invention. It should be understood that the exemplary embodiments described herein should be considered in descriptive sense only and not for purposes of limitation. Descriptions of features or aspects in each exemplary embodiment should generally be considered as available for similar features or aspects in other exemplary embodiments.
Example one
The invention adopts the posterior probability density function of the structural damping ratio parameter to describe the uncertainty of the damping ratio identification; an information entropy criterion is provided to quantify the uncertainty of the damping ratio parameter identification; a sequential algorithm is utilized to guide the placement of the sensors based on the magnitude of the information entropy criterion. Considering the damping ratio parameter identification situation in the structural operation state, Gaussian noise is adopted for both environmental excitation and measurement noise. The sensor arrangement method provided by the invention is based on a Bayesian frequency domain modal identification theory. A posterior probability density function of the structural damping ratio parameter is constructed by defining a selection matrix corresponding to the position of a specific sensor, and the theoretical relation between the sensor arrangement and the structural damping ratio identification is established. The information entropy is used as an index of quantization uncertainty, and the smaller the numerical value of the information entropy is, the more concentrated the distribution of the posterior probability density function is, and the smaller the uncertainty of representing the modal identification is. The optimal sensor arrangement corresponds to the minimum information entropy value. A sequential placement algorithm is presented to guide the placement of the sensors.
The invention adopts a sensor arrangement method for reducing the identification uncertainty of the structure damping ratio, which is used for monitoring the health of a civil engineering structure and is characterized by comprising the following steps: a posterior probability density function of the damping ratio and a sensor arrangement based on the information entropy; the posterior probability density function of the damping ratio is used for describing the uncertainty of the damping ratio identification; the sensor arrangement based on the information entropy provides an information entropy criterion to quantify the uncertainty of the damping ratio parameter identification, and a sequential algorithm is utilized through the information entropy criterion to guide the arrangement of the sensors.
In the embodiment, the posterior probability density function of the damping ratio adopts discrete measurement response to perform data sampling; due to the influence of measurement noise, the discrete measurement response is represented in the form:
in the formula (1)
Selecting a matrix for the sensor locations, the variables in the matrix consisting of 0 and 1, there being only one non-zero entry in each row or column; n is a radical ofLThe degree of freedom of a measuring point, namely the degree of freedom of a sensor position; n is a radical ofdIs the total degree of freedom of the structure;
is the structural response measured by the sensor; n denotes discrete sampling instants, i.e. samples at the instant t ═ n Δ t, where Δ t denotes the sampling interval;
is the structural damping ratio to be identified, ζ ═ { ζ ═ ζi,i＝1,2,…,Nm}，ζiRepresenting the i-th order damping ratio of the structure; n is a radical ofmThe order of the mode of the structure participating in vibration;
is the true response of the structure, here representing the acceleration signal;
is the error between the measured value and the true value, i.e. the measurement noise.
In this embodiment, the posterior probability density function of the damping ratio performs discrete fourier transform on discrete measurement response sampling data;
discrete sampling data is obtained by substituting equation (1) into equation YNAnd (f, N), wherein N is 1, … N, and N is the number of sampling points, and the discrete fourier transform is performed to obtain:
in the formula (2)
Indicating the operation of solving the root number; sigma []Represents a summation symbol; e is a natural constant;
for the resulting discrete Fourier transform vector, the corresponding frequency scale is fk＝k/NΔt，k＝1,…,NqIn which N isqInt (N/2) +1 corresponds to the Nyquist frequency, int [ ·]Expressing the integer sign; i represents a unit imaginary number.
In this embodiment, the posterior probability density function of the damping ratio easily satisfies the following two conditions in actual engineering monitoring: the modes of each order of the structure can be distinguished, namely in a frequency band near a certain order of damping ratio, the vibration response of the structure is mainly controlled by the mode of the order; and satisfies the long sampling condition, i.e. the number of sampling points in the damping ratio near frequency band is far more than 1, and the result obtained by the formula (2)
As a zero mean complex gaussian vectorThe covariance matrix form is as follows:
in the formula (3) [. above]TRepresenting an operation of transposing a matrix or a vector;
for a certain order mode shape matrix of the structure, it is assumed that the vibration is controlled by the ith order modei，
Is the overall modal shape matrix of the structure; siA spectral density representing an i-th order modal excitation; seA spectral density representative of measurement noise;
is a number Nd×NdThe identity matrix of (1); alpha is alphai＝||Lφ||＝||LΦiThe | | | is a normalization coefficient of the ith-order modal shape vector, and | | · | | represents the modulo operation;
wherein the formula (4) is a power amplification factor part in the formula (3); in the formula (4) < beta >ik＝fi/fkRepresenting the ratio between the damping ratio and the frequency scale corresponding to the Fourier transform; f. ofiIs the ith order natural frequency of the structure; [. the]-1Representing an inversion operation.
In this embodiment, the posterior probability density function of the damping ratio is determined by
The value of (b) satisfies a complex multiple Gaussian distribution, a certain order damping ratio ζiA posterior probability density function of (a);
due to the formula (2)
Satisfy complex multiple Gaussian distribution and a certain damping ratio zetaiThe posterior probability density function of (a) is of the form:
wherein: oc represents a proportional sign;
representing a set of Fourier transform vectors in an ith order damping ratio near frequency band, and determining a specific frequency band range according to a half-power amplitude value; II []Representing a continuous multiplication and product-solving symbol; det [. to]Representing determinant of the matrix; exp [ A ]]Expressing an exponential function with a natural constant e as a base and A as an exponential term; superscript []*Representing the conjugate transposed symbol.
In this embodiment, the posterior probability density function of the damping ratio is a posterior probability density function p (ζ) of equation (5)i|Zi) The calculation is done in logarithmic form:
p(ζi|Zi)∝exp(-L(ζi)) (6)；
in the formula (6)
Set so that L (ζ)i) Taking the ith order optimal damping ratio parameter estimation quantity of the minimum value, and applying a negative log-likelihood function L (zeta)i) Do it with
The second order taylor expansion of (a) yields:
in the formula (7)
Representing a negative log-likelihood function L (ζ)i) In that
The Hessian matrix is used, when single-order damping ratio parameter estimation is adopted,
and
degenerates into single variables from vectors and matrices, respectively.
In the present embodiment, the posterior probability density function of the damping ratio is represented by the formula (7)
The value can be determined by:
scale factor e in formula (8)kCalculated from the following formula:
the calculation of equation (8) is performed while ensuring that the mode shape in equation (3) has undergone the operation of normalization, i.e., | | Φ | | |, 1.
In this embodiment, the posterior probability density function of the damping ratio needs to consider the multiple-order damping ratio, and considers that each order of mode can be distinguished, and the posterior probability density function of the multiple-order damping ratio is subjected to multiplication to obtain the following formula:
in formula (10), Z ═ Zi,i＝1,2,…,NmDenotes a set of discrete fourier transforms of various orders;
the optimal damping ratio vector corresponding to all orders at the position of the sensor is obtained;
damping ratios to be identified for all orders are
A Hessian matrix of the same multi-order modal parameters
Satisfying the form of block-to-angle array, corresponding to the modal parameters of each order
The composition is shown as the following formula:
as shown in fig. 1, according to the finite element model diagram of the simply supported beam of the exemplary embodiment, in the sensor arrangement method based on the information entropy, after the posterior probability density function of the structural damping ratio is determined, the information entropy is required to quantify the uncertainty of the frequency identification corresponding to different sensor arrangements, and the optimal sensor position is selected according to the quantified value. The information entropy formula of the damping ratio parameter to be identified is as follows:
in the formula ln [. C]Representing a logarithmic operation; n is a radical ofθRepresenting the degrees of freedom of the damping ratio of each order considered.
The figure utilizes a simple beam structure to carry out simulation checking calculation. The length of the beam is 1900mm and the dimensions of the cross section are 50mm x 15.62 mm.
By adopting two-dimensional Euler beam units, the beam of the simple beam structure is divided into 19 units, each node comprises 2 translational degrees of freedom (X and Y-axis directions) and one rotational degree of freedom (in an XY plane), and the total number of the nodes is 20 and 57. The modulus of elasticity of the beam is set to 200GPa, the material density of the beam is 7780 kg/m3, and the damping ratio of each step is 2%. At the resonance peak of the frequency spectrum, the ratio of the spectral density of the response signal to the noise satisfies gammai＝Si/4Seζ230, and the ratio is selected to be 30. The to-be-selected measuring points of the sensor are set to be 18 vertical degrees of freedom, and the sensor arrangement method is verified by adopting the acceleration sensor to serve as the to-be-selected position of the sensor, so that the sensor arrangement reaches the optimal state. In the case of sensor arrangement, the optimal modal parameters for minimizing the log-likelihood function cannot be determined in advance
The values of (1) are obtained by using modal parameters of the finite element model
Namely, the frequency, the damping ratio and the modal shape value of the finite element model are used as the assumed optimal modal parameter value.
As shown in fig. 2a, which shows a sensor layout of the first order damping ratio of the present invention according to an exemplary embodiment, the horizontal axis is the number of the arranged sensors, the vertical axis is the sensor body position, and the numbering corresponds to the numbered position in fig. 1. On the premise of only considering the first-order damping ratio as a modal parameter to be identified, the method comprises the following steps: firstly, establishing a structure finite element model, taking the frequency, damping ratio and modal shape numerical value of the finite element model as an assumed optimal modal parameter numerical value, taking the self-degree of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating a value h (L) according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and obtaining the optimal arrangement scheme of the sensors under the condition of different numbers of the sensors by jumping out of the cycle.
As shown in fig. 2b, which shows a sensor layout of the second order damping ratio of the present invention according to an exemplary embodiment, the horizontal axis of the diagram is the number of the arranged sensors, the vertical axis is the sensor body position, and the numbering corresponds to the numbering position in fig. 1. On the premise of only considering the first-order damping ratio as a modal parameter to be identified, the method comprises the following steps: firstly, establishing a structure finite element model, taking the frequency, damping ratio and modal shape numerical value of the finite element model as an assumed optimal modal parameter numerical value, taking the self-degree of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating a value h (L) according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and jumping out of the cycle to obtain the optimal sensor arrangement scheme under each number.
As shown in fig. 2c, which shows a sensor layout of the third order damping ratio of the present invention according to an exemplary embodiment, the horizontal axis is the number of the arranged sensors, the vertical axis is the sensor body position, and the numbering corresponds to the numbered position in fig. 1. On the premise of only considering the third-order damping ratio as a modal parameter to be identified, the method comprises the following steps: firstly, establishing a structure finite element model, taking the frequency, damping ratio and modal shape numerical value of the finite element model as an assumed optimal modal parameter numerical value, taking the self-degree of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating a value h (L) according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and jumping out of the cycle to obtain the optimal sensor arrangement scheme under each number.
Fig. 2d shows a sensor layout of the first and second order damping ratios of the present invention according to an exemplary embodiment, wherein the horizontal axis represents the number of sensors arranged and the vertical axis represents the sensor body position, and the numbering corresponds to the numbering in fig. 1. On the premise of simultaneously considering the first-order damping ratio and the second-order damping ratio as modal parameters to be identified, the method comprises the following steps: firstly, establishing a structure finite element model, taking the frequency, the damping ratio and the modal shape numerical value of the finite element model as assumed optimal modal parameter numerical values, taking the freedom degrees of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating values h (L) according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and jumping out of the cycle to obtain the optimal sensor arrangement scheme under each number.
Fig. 2e shows a sensor layout diagram of the second and third order damping ratios of the present invention according to an exemplary embodiment, wherein the horizontal axis represents the number of the arranged sensors, the vertical axis represents the sensor body positions, and the numbering corresponds to the numbering positions in fig. 1. On the premise of simultaneously considering the second-order and third-order damping ratios as modal parameters to be identified, the method comprises the following steps: firstly, establishing a finite element model of a structure, taking the freedom degrees of all unit nodes of the structure as the positions of sensors to be selected, taking the frequency, the damping ratio and the modal shape value of the finite element model as the assumed optimal modal parameter value, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating values h (L) according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and jumping out of the cycle to obtain the optimal sensor arrangement scheme under each number.
Fig. 2f shows a sensor layout of the first, second and third order damping ratios of the present invention according to an exemplary embodiment, wherein the horizontal axis represents the number of sensors arranged and the vertical axis represents the sensor body position, and the numbering corresponds to the numbering in fig. 1. On the premise of simultaneously considering the damping ratios of the first order, the second order and the third order as modal parameters to be identified, the method comprises the following steps: firstly, establishing a finite element model of a structure, taking the frequency, the damping ratio and the modal shape numerical value of the finite element model as assumed optimal modal parameter numerical values, taking the freedom degrees of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, realizing the purpose by rewriting and selecting a matrix L, calculating the values of h and L according to a formula (12), and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement, and jumping out of the cycle to obtain the optimal sensor arrangement scheme under each number.
The invention has the beneficial effects that: the sensor arrangement method based on the information entropy can reduce the uncertainty of structural damping ratio identification. By the proposed theory, a theoretical link between sensor arrangement and bayesian frequency modal identification is established. And constructing a posterior probability density function of the damping ratio parameter to be identified so as to describe the uncertainty of the damping ratio identification. The information entropy index can well quantify the identification uncertainty of the damping ratio corresponding to different sensor arrangements, and the optimal sensor arrangement corresponds to the minimum information entropy value. By the sensor arrangement method provided by the invention, the adverse effect of measurement noise on the identification result of the damping ratio of each order can be effectively reduced, and the accuracy of the identification of the structural damping ratio is improved.
In addition to the above embodiments, the present embodiment is based on the above preferred embodiments of the present invention, and the above description is intended by the workers who work the present invention to make various changes and modifications without departing from the scope of the present invention. The technical scope of the present invention is not limited to the content of the specification, and must be determined according to the scope of the claims.
Claims (10)
1. A sensor placement method for reducing structural damping ratio identification uncertainty, for civil engineering structure health monitoring, comprising: a posterior probability density function of the damping ratio and a sensor arrangement based on the information entropy; the posterior probability density function of the damping ratio is used for describing the uncertainty of the damping ratio identification; the sensor arrangement based on the information entropy provides an information entropy criterion to quantify the uncertainty of the damping ratio parameter identification, and a sequential algorithm is utilized through the information entropy criterion to guide the arrangement of the sensors.
2. The sensor placement method for reducing structural damping ratio identification uncertainty according to claim 1, characterized in that a posterior probability density function of the damping ratio, using discrete measurement responses for data sampling; due to the influence of measurement noise, the discrete measurement response is represented in the form:
in the formula (1)
Selecting a matrix for the sensor locations; n is a radical ofLThe degree of freedom of the measuring point; n is a radical ofdIs the total degree of freedom of the structure;
is the structural response measured by the sensor; n denotes a discrete sampling instant, i.e. a sample at time t ═ n Δ t, where Δ t denotes the sampling interval;
is the structural damping ratio to be identified, ζ ═ { ζ ═ ζi,i＝1,2,…,Nm}，ζiRepresenting the i-th order damping ratio of the structure; n is a radical ofmThe order of the structure participating in the vibration mode;
is the true response of the structure;
is the error between the measured value and the true value.
3. The sensor placement method for reducing structural damping ratio identification uncertainty according to claim 2, characterized in that the posterior probability density function of the damping ratio performs a discrete fourier transform on discrete measurement response sampled data;
discrete sampling data is obtained by substituting equation (1) into equation YNAnd (f, N), wherein N is 1, … N, and N is the number of sampling points, and the discrete fourier transform is performed to obtain:
in the formula (2)
Indicating the operation of solving the root number; sigma []Represents a summation symbol; e is a natural constant;
for the resulting discrete Fourier transform vector, the corresponding frequency scale is fk＝k/NΔt，k＝1,…,NqIn which N isqInt (N/2) +1 corresponds to the Nyquist frequency, int [ ·]Expressing the integer sign; i represents the unit imaginary number.
4. A sensor arrangement method for reducing the identification uncertainty of the damping ratio of a structure according to claim 3, characterized in that said a posteriori probability density function of the damping ratio is such that the modes of the orders of the structure are in the band near the damping ratio of a certain order, and the vibration response of the structure is mainly controlled by the modes of the order; the number of sampling points in the damping ratio near frequency band is far more than 1, and the result obtained by the formula (2)
As a complex gaussian vector with zero mean, the covariance matrix form is as follows:
in the formula (3) [. above]TRepresenting an operation of transposing a matrix or a vector;
for a certain order mode shape matrix of the structure, it is assumed that the vibration is controlled by the ith order modei，
Is an integral modal shape matrix of the structure; siSpectral density representing i-th order modal excitation；SeA spectral density representative of measurement noise;
is a number Nd×NdThe identity matrix of (1); alpha is alphai＝||Lφ||＝||LΦiThe | | | is a normalization coefficient of the ith-order modal shape vector, and | | · | | represents the modulo operation;
wherein the formula (4) is a power amplification factor part in the formula (3); in the formula (4) < beta >ik＝fi/fkRepresenting the ratio between the damping ratio and the frequency scale corresponding to the Fourier transform; f. ofiIs the ith order natural frequency of the structure; [. the]-1Representing an inversion operation.
5. A sensor arrangement method for reducing the identification uncertainty of a damping ratio of a structure according to claim 3, characterized in that the posterior probability density function of the damping ratio is determined by
The value of (b) satisfies complex multivariate Gaussian distribution and a certain damping ratio zetaiA posterior probability density function of (a);
due to the formula (2)
Satisfy complex multiple Gaussian distribution and a certain damping ratio zetaiThe posterior probability density function of (a) is of the form:
oc in the formula (5) represents a proportional symbol;
a set of Fourier transform vectors in a near band representing an ith damping ratio; II []Representing a continuous multiplication and product-solving symbol; det [. to]Representing determinant of the matrix; exp [ A ]]Expressing an exponential function with a natural constant e as a base and A as an exponential term; superscript []*Representing the conjugate transposed symbol.
6. Sensor arrangement method for reducing the identification uncertainty of the damping ratio of a structure according to claim 5, characterized in that the posterior probability density function of the damping ratio is the posterior probability density function p (ζ) of equation (5)i|Zi) The calculation is done in logarithmic form:
p(ζi|Zi)∝exp(-L(ζi)) (6)；
in the formula (6)
Set so that L (ζ)i) Obtaining the ith order optimal damping ratio parameter estimator of the minimum value, and applying a negative log likelihood function L (zeta)i) Do it with
The second order taylor expansion of (a) yields:
in the formula (7)
Representing a negative log-likelihood function L (ζ)i) In that
The Hessian matrix, when using single order frequency parameter estimation,
and
degenerates into single variables from vectors and matrices, respectively.
7. The sensor placement method for reducing structural damping ratio identification uncertainty according to claim 6, characterized in that a posterior probability density function of the damping ratio,
by the following formula (7)
The value can be determined by:
scale factor e in formula (8)kCalculated from the following formula:
the calculation of equation (8) is performed while ensuring that the mode shape in equation (3) has undergone the operation of normalization, i.e., | | Φ | | |, 1.
8. The method as claimed in claim 7, wherein the posterior probability density function of the damping ratio is obtained by multiplying the posterior probability density function of the single-order damping ratio by a multiple-order damping ratio to obtain the following formula:
in formula (10), Z ═ Zi,i＝1,2,…,NmDenotes a set of discrete fourier transforms of various orders;
the optimal damping ratio vector of all orders at the position of the corresponding sensor is obtained;
damping ratios to be identified for all orders are
A Hessian matrix of the same multi-order modal parameters
Satisfy the form of block diagonal matrix, corresponding to the modal parameters of each order
The composition is shown as the following formula:
9. the sensor arrangement method for reducing the structural damping ratio identification uncertainty according to claim 1, characterized in that the sensor arrangement based on the information entropy utilizes the information entropy to quantify the uncertainty of frequency identification corresponding to different sensor arrangements after determining the posterior probability density function of the structural damping ratio, and selects the optimal sensor position according to the quantified value, and the information entropy formula of the damping ratio parameter to be identified is as follows:
ln [. in formula (12)]Representing a logarithmic operation; n is a radical ofθFreedom to express damping ratio of each order consideredAnd (4) degree.
10. A sensor arrangement method for reducing structural damping ratio identification uncertainty according to claim 9, characterized in that said information entropy based sensor arrangement comprises the steps of: firstly, establishing a finite element model of a structure, taking the freedom degrees of all unit nodes of the structure as the positions of sensors to be selected, and initially selecting the number of the sensors to be selected to be 0; secondly, selecting a sensor position from the rest measuring points, adding the sensor position into the existing position, rewriting a selection matrix L, calculating a value h (L) according to an information entropy formula (12) of a damping ratio parameter to be identified, and selecting the sensor position corresponding to the minimum value; thirdly, deleting the selected position from the rest position measuring points; judging the remaining positions, and if the remaining positions do not exist, continuing the next step; if the remaining positions exist, returning to the second step; and fourthly, obtaining the final sensor arrangement and jumping out of the cycle.
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