CN109709150B - Laminated rubber vibration isolation support damage identification method based on piezoelectric impedance information - Google Patents

Laminated rubber vibration isolation support damage identification method based on piezoelectric impedance information Download PDF

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CN109709150B
CN109709150B CN201811595124.3A CN201811595124A CN109709150B CN 109709150 B CN109709150 B CN 109709150B CN 201811595124 A CN201811595124 A CN 201811595124A CN 109709150 B CN109709150 B CN 109709150B
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damage
admittance
substructure
resonance frequency
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CN109709150A (en
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朱宏平
张莹
翁顺
雷鹰
袁涌
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Huazhong University of Science and Technology
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Abstract

The invention discloses a laminated rubber vibration isolation support damage identification method based on piezoelectric impedance information, and belongs to the field of civil engineering structure detection. The method comprises the following steps: (1) establishing an original point anti-resonance frequency characteristic equation of a single-damage and non-damage single-coupling periodic structure; (2) simplifying the laminated rubber shock-insulation support into a limited single-coupling period structure, and calculating the dimensionless origin point anti-resonance frequency in a non-damage state; (3) calculating the sensitivity of the dimensionless origin anti-resonance frequency to the change of the shear stiffness of the basic periodic unit, and establishing a sensitivity identification equation set; (4) acquiring admittance signals before and after damage, and extracting the anti-resonance frequency of the original point of the structure; and solving a sensitivity identification equation set based on the change rate of the original point anti-resonance frequency before and after the damage to complete the damage identification. The method can accurately identify the multiple damages of the periodic structure only by measuring the change of the anti-resonance frequency of the original points of a few measuring points before and after the damage of the structure without accurate model parameters of the original structure.

Description

Laminated rubber vibration isolation support damage identification method based on piezoelectric impedance information
Technical Field
The invention belongs to the field of civil engineering structure detection, and particularly relates to a method for identifying damage of a laminated rubber vibration isolation support based on piezoelectric impedance information.
Background
The shock isolation device bears a large amount of seismic energy consumption, the performance of the shock isolation device is continuously deteriorated under the long-term action of multiple factors such as load, environment and the like, and the shock isolation device is a key part which is most easily damaged in the seismic process. Among them, the laminated rubber vibration isolation support is one of the vibration isolation devices widely used at present. The laminated rubber shock-insulation support is generally formed by mutually staggering a layer of rubber and a layer of reinforcing steel plate through a special process by bonding and pressing, and can be regarded as a chain-shaped harmonious periodic structure system formed by connecting a plurality of repeated substructures (or called periodic units) end to end.
The anti-resonance of the structural system refers to the situation that under the harmonic excitation action of certain specific frequencies, harmonic reaction or zero dynamic compliance occurs at certain parts of the system in the elastic system. Compared with the traditional modal parameters, the anti-resonance frequency has the obvious advantages of representing the overall characteristics of the structure and reflecting the local physical parameter change of the structure. At present, anti-resonance is mainly applied to finite element model modification and dynamic modification of non-periodic structures, and is less applied to periodic structure damage identification.
Piezoelectric impedance (EMI) technology based on piezoelectric ceramic sensors/drivers (abbreviated as PZT) has great advantages in the aspect of identifying tiny damage, and is particularly suitable for local online monitoring and accurate damage identification of structures. The method is based on the basic principle that a high-strength adhesive is used for adhering the surface of the PZT structure or implanting the PZT structure into the structure, and the occurrence of damage is judged by monitoring the change of an electric admittance signal of the PZT self-driven sensor. Sensor damage and bond line defects can interfere with identification.
Disclosure of Invention
Aiming at the defects or improvement requirements of the prior art, the invention provides a laminated rubber vibration isolation support damage identification method based on piezoelectric impedance information, and aims to fully utilize the characteristic that the laminated rubber vibration isolation support is periodic along the axial direction, and solve the technical problem of multi-damage identification of the laminated rubber vibration isolation support based on PZT intelligent sensing monitoring data and according to the change of the original point anti-resonance frequency of a few measuring points before and after structural damage.
In order to achieve the above object, according to an aspect of the present invention, there is provided a method for identifying damage to a laminated rubber-vibration-isolated mount based on piezoelectric impedance information, comprising the steps of:
(1) constructing an original point anti-resonance frequency characteristic equation of a single-damage and non-damage single-coupling periodic structure;
(2) simplifying the laminated rubber shock-insulation support into a single-coupling periodic structure, wherein a basic periodic unit of the laminated rubber shock-insulation support is composed of a second-order shear beam and concentrated masses at two ends of the second-order shear beam, and calculating the dimensionless origin point anti-resonance frequency of the single-coupling periodic structure in a non-damage state;
(3) introducing the increase of the shear stiffness into the admittance of the damage unit, and calculating the sensitivity coefficient of the anti-resonance frequency of the dimensionless origin to the change of the shear stiffness of the basic period unit; establishing a sensitivity identification equation set according to approximate linear superposition of the change of the antiresonance frequency under multiple damages into the change caused by single damage;
(4) acquiring admittance signals before and after damage, and extracting the origin point anti-resonance frequency of the single-coupling periodic structure; and solving a sensitivity identification equation set based on the change rate of the original point anti-resonance frequency before and after the damage, and identifying the damage.
Further, the step (1) further comprises the following sub-steps:
(1.1) for a single-coupling periodic structure of N basic periodic units, setting a left boundary A to be fixed and a right boundary B to be free, and dividing the single-coupling periodic structure into a substructure I and a substructure II by taking the point C as a boundary point when an excitation force P acts on the point C;
(1.2) assuming that the excitation point C is at the node j, the substructure II has a damaged unit k, i.e. j < k; the substructure I is regarded as a healthy single-coupling periodic structure with j units and fixed at two ends; the substructure II is regarded as a single-coupling periodic structure with fixed left end and free right end and (N-j) units, wherein the unit k is damaged; the natural frequency characteristic equations of the substructure I and the substructure II are respectively as follows:
substructure I: 1-e-2jμ＝0
Substructure II: 1+ phi is 0
Figure BDA0001921173550000031
Figure BDA0001921173550000032
C0＝A0+αDDαwr-αEEαwt-αwtαwr
Figure BDA0001921173550000033
E0＝A0+αDDαwt-αEEαwr-αwtαwr
A0＝αDDαEE-αDEαED
Where Φ represents the reflected and transmitted wave displacements at C in substructure II
Figure BDA0001921173550000034
And
Figure BDA0001921173550000035
ratio of (a)DDand alphaEEfor direct admittance of the two ends of the damaged element, alphaEDand alphaDEfor indirect admittance between the ends of the damaged element, alphawtand alphawrcharacteristic wave susceptance, respectively of the transmitted wave and the reflected wave, having alpha for the symmetrical cellwt＝-αwr(ii) a μ is the wave propagation constant;
(1.3) based on the condition of occurrence of anti-resonance, that is, the excitation frequency is equal to a certain natural frequency of the substructure on the left or right of the excitation point, obtaining an origin anti-resonance frequency characteristic equation through the natural frequency characteristic equations of the substructures I and II in the step (1.2) as follows:
(1-e-2jμ)(1+Φ)＝0
(1.4) supposing that the excitation point C is at the node j, the substructure I has a damage unit k, i.e. j is more than or equal to k; the substructure I is regarded as a single-coupling periodic structure with j units and fixed at two ends, wherein the unit k is damaged; the substructure II is a healthy single-coupling periodic structure with fixed left end and free right end and (N-j) units; the natural frequency characteristic equations of the substructure I and the substructure II are respectively:
substructure I: 1+ Ψ ═ 0
Substructure II: 1+ e-2(N-j)μ＝0
Figure BDA0001921173550000036
In the formula, Ψ represents the displacement of the reflected wave and the transmitted wave at C in the substructure I
Figure BDA0001921173550000046
And
Figure BDA0001921173550000047
the ratio of (A) to (B);
(1.5) repeating the step (1.3) to obtain an origin anti-resonance frequency characteristic equation corresponding to the step (1.4) as follows:
[1+e-2(N-j)μ](1+Ψ)＝0
(1.6) under a non-damage state, degenerating an origin anti-resonance frequency characteristic equation of the single damage obtained in the steps (1.3) and (1.5) into:
[1+e-2(N-j)μ](1-e-2jμ)＝0。
further, the step (2) further comprises the following sub-steps:
(2.1) the direct admittance and the indirect admittance at the two ends of the rubber layer are:
Figure BDA0001921173550000041
Figure BDA0001921173550000042
in the formula, gammallAnd gammarrRespectively direct admittance of both ends of the rubber layer, gammalrAnd gammarlRespectively, the indirect admittance between two ends of the rubber layer, G is the shear modulus of the rubber, rho is the density of the rubber, L is the thickness of the rubber layer, A is the cross-sectional area of the rubber layer,
Figure BDA0001921173550000043
is a periodic structure wave number, omega is a circle frequency, and omega is ksL is a dimensionless frequency;
(2.2) admittance of the steel sheet is:
Figure BDA0001921173550000044
wherein β is the admittance of the steel plate, omega is the circular frequency,msthe mass of each layer of steel plate;
(2.3) the direct admittance and the indirect admittance of the composite periodic unit of the laminated rubber vibration isolation bearing and the propagation constants are respectively as follows:
Figure BDA0001921173550000045
Figure BDA0001921173550000051
Figure BDA0001921173550000052
in the formula, αlland alpharris a direct admittance across the composite periodic unit, alphalrand alpharlFor transfer admittance between the two ends of the composite periodic unit,
Figure BDA0001921173550000053
m is the mass ratio of rubber to steel platerRho AL is the rubber mass; μ is the wave propagation constant;
(2.4) direct admittance α of the healthy compound periodic unit of step (2.3)ll、αrrand transfer admittance αlr、αrlAnd (4) substituting the obtained result into the dimensionless origin point antiresonance frequency characteristic equation of the nondestructive single-coupling periodic structure in the step (1.6), and calculating the dimensionless origin point antiresonance frequency of the laminated rubber vibration-isolating support in the nondestructive state.
Further, the step (2.4) of calculating the anti-resonance frequency of the dimensionless origin in the non-damage state further comprises the following sub-steps:
(2.4.1) setting γ ═ μ i, and converting the origin antiresonance frequency characteristic equation in the non-invasive state obtained in step (1.6) into:
cos[(N-j)γ]sinγ＝0
the solution to the equation is:
Figure BDA0001921173550000054
or
Figure BDA0001921173550000055
In the formula:
Figure BDA0001921173550000056
is an imaginary unit, mu is a wave propagation constant;
(2.4.2) converting the wave propagation constant calculation formula of the step (2.3) into:
Figure BDA0001921173550000057
and (4) substituting the solution gamma obtained in the step (2.4.1) into the equation to calculate the dimensionless origin point anti-resonance frequency.
Further, the step (3) further comprises the following sub-steps:
(3.1) when the rubber is aged, the shear modulus of the corresponding rubber layer is increased, and a damage state characterization parameter is introduced, wherein the direct admittance and the indirect admittance of a damage unit are as follows:
Figure BDA0001921173550000061
Figure BDA0001921173550000062
Figure BDA0001921173550000063
Ω′＝k′L
wherein Δ G is the increase in unit shear modulus;
(3.2) shear modulus change rate xi of the damaged unit k before and after damagekAnd (3) evaluating the damage degree:
Figure BDA0001921173550000064
in the formula, ξkWhen 0, unit k is intact;
(3.3) the rate of change of the antiresonance frequency of the nth order dimensionless origin of the excitation point j before and after the damage is as follows:
Figure BDA0001921173550000065
in the formula:
Figure BDA0001921173550000066
respectively representing dimensionless origin point anti-resonance frequencies before and after damage, wherein the superscript u represents an undamaged state, and the superscript d represents a damaged state;
(3.4) based on the perturbation theory and the sensitivity analysis principle, obtaining the sensitivity of the n-th order dimensionless origin point anti-resonance frequency of the excitation point j to the k-th unit damage
Figure BDA0001921173550000067
Figure BDA0001921173550000068
In the formula:
Figure BDA0001921173550000069
representing the origin antiresonant frequency characteristic equation of a single lesion by steps (1.3) and (1.5)
Figure BDA0001921173550000071
Paxi xikpartial derivatives of (c), then make xi in the expression of resultk＝0；
(3.5) approximately linear superposition of the change of the dimensionless origin antiresonance frequency under multiple damages into the change caused by single damage, and accordingly, the vector of the total dimensionless origin antiresonance frequency change rate at the excitation point caused by multiple damages is established
Figure BDA0001921173550000072
and a damage state identification equation between the shear modulus change rate vector { ξ } of each layer of rubber:
Figure BDA0001921173550000073
Figure BDA0001921173550000074
in the formula, [ S ] is a sensitivity matrix of the dimensionless origin antiresonance frequency, p and q represent different nodes where the excitation point C is located, and p is 1, 2.
Further, the step (4) further comprises the following sub-steps:
(4.1) sticking PZT along the axial direction of the laminated rubber shock-insulation support, and collecting admittance signals Y before and after damage;
(4.2) separating the mechanical impedance Z of the single-coupling periodic structure from the PZT electric admittance signal Y according to the one-dimensional impedance models；
(4.3) velocity admittance H based on Single-coupling periodic StructurevAnd the displacement admittance HdIn relation to (3), the mechanical impedance Z of the single-coupled periodic structuresStructure displacement admittance H converted into single coupling periodd：
Figure BDA0001921173550000075
Extracting a valley value of the displacement admittance curve, namely the original point anti-resonance frequency of the structure;
and (4.4) obtaining the change rate of the original point anti-resonance frequency before and after damage based on the step (4.3), and thus carrying out damage identification on the laminated rubber vibration isolation support.
Further, step (4.2) separates the mechanical impedance Z of the structure from the PZT electrical admittance signal YsFurther comprising the substeps of:
(4.2.1) calculating the mechanical impedance Z of the PZT in the short-circuited statea：
Figure BDA0001921173550000081
Figure BDA00019211735500000812
In the formula (I), the compound is shown in the specification,
Figure BDA0001921173550000082
wave number of PZT, ω circular frequency of excitation frequency, la、ba、haRespectively the length, width and thickness of the PZT,
Figure BDA0001921173550000083
is the composite elastic modulus of PZT when the electric field is constant,
Figure BDA0001921173550000084
is the modulus of elasticity, η is the mechanical loss factor,
Figure BDA0001921173550000085
is a plurality of units;
(4.2.2) the PZT electrical admittance expression is:
Figure BDA0001921173550000086
Figure BDA0001921173550000087
in the formula (I), the compound is shown in the specification,
Figure BDA0001921173550000088
is the complex dielectric constant of PZT when the stress is constant,
Figure BDA0001921173550000089
for a real dielectric constant, δ is a dielectric loss factor, d31Is the piezoelectric strain coefficient of PZT;
further, the step (4.4) is used for identifying the damage of the laminated rubber vibration isolation support based on the measured change rate of the anti-resonance frequency of the original point before and after the damage, and further comprises the following substeps:
(4.4.1) calculating the change rate of the dimensionless origin point antiresonance frequency through the measured before and after damage:
Figure BDA00019211735500000810
in the formula (I), the compound is shown in the specification,
Figure BDA00019211735500000811
respectively representing the measured original point anti-resonance frequency before and after damage, the superscript u representing the undamaged state and the superscript d representing the damaged state;
(4.4.2) converting the equation set solving problem of step (3.5) to a non-negative least squares curve fitting problem based on damage such that the shear stiffness of the rubber increases:
Figure BDA0001921173550000091
and (5) solving [ S ] according to the fitting result of the formula to finish the damage identification.
In order to achieve the above object, the invention also provides a laminated rubber vibration-isolating support damage identification device based on piezoelectric impedance information, which comprises a processor and a damage identification program module; and the damage identification program module executes any one of the above-mentioned laminated rubber vibration-isolating support damage identification methods when being called by the processor.
In general, compared with the prior art, the technical scheme of the invention combines the periodic characteristics of the laminated rubber vibration isolation support and the high sensitivity of the PZT technology to tiny damage, so that the following beneficial effects can be obtained:
1) the periodic characteristic of the laminated rubber shock-insulation support is considered, and the multi-damage identification and accurate positioning of the laminated rubber support is realized. Preferably, relatively many frequency variation data can be obtained using the origin anti-resonance frequency: when periodic structure damage identification is carried out through the structure natural frequency, the order of the available frequency is less and is generally smaller than the structure period number. And damage identification is carried out by utilizing the origin point anti-resonance frequency, one structure can be provided with a plurality of driving points, and each driving point can obtain the multi-order origin point anti-resonance frequency.
2) And the original point anti-resonance frequency of the laminated rubber vibration-isolating support is obtained from the measured PZT electric admittance signals, so that the direct measurement of the original point anti-resonance frequency is avoided.
Drawings
FIG. 1 is a schematic flow chart of main steps of a laminated rubber vibration-isolating support damage identification method based on piezoelectric impedance information;
FIG. 2 is a schematic diagram of an experiment for identifying damage to a laminated rubber-vibration-isolating support in accordance with a preferred embodiment of the present invention;
FIG. 3(a) is a schematic diagram of a period system of a laminated rubber vibration-isolating support;
FIG. 3(b) is a schematic diagram of a basic periodic unit of laminated rubber vibration isolation;
FIG. 4(a) is a schematic diagram of single-coupling periodic system wave propagation when the excitation point is on the left side of the damage unit;
FIG. 4(b) is a schematic diagram of single-coupling periodic system wave propagation when the excitation point is at the right side of the damage unit;
FIG. 5(a) is the anti-resonance frequency sensitivity coefficient of node 1;
FIG. 5(b) is the anti-resonance frequency sensitivity coefficient of the node 4;
FIG. 5(c) is the anti-resonance frequency sensitivity coefficient of node 7;
FIG. 5(d) is the anti-resonance frequency sensitivity coefficient of the node 10;
fig. 6 shows the result of lesion recognition.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 1, a method for identifying damage to a laminated rubber-vibration-isolated bearing based on piezoelectric impedance information according to a preferred embodiment of the present invention includes the following steps:
(1) constructing an original point anti-resonance frequency characteristic equation of a single-damage and non-damage single-coupling periodic structure;
(1.1) a single-coupling periodic structure of N basic periodic units, wherein a left boundary A is fixed, a right boundary B is free, an excitation force P acts on a point C, the point C is used as a demarcation point, and the periodic structure is divided into a substructure I and a substructure II;
(1.2) assuming that the excitation point C is at the node j, the substructure II has a damaged unit k, i.e. j < k; the substructure I is a health cycle structure with j units and two fixed ends; the substructure II is regarded as a single-coupling periodic structure with fixed left end and free right end and (N-j) units, wherein the unit k is damaged; the natural frequency characteristic equations of the substructure I and the substructure II are respectively as follows:
substructure I: 1-e-2jμ＝0
Substructure II: 1+ phi is 0
Figure BDA0001921173550000101
Figure BDA0001921173550000111
C0＝A0+αDDαwr-αEEαwt-αwtαwr
Figure BDA0001921173550000112
E0＝A0+αDDαwt-αEEαwr-αwtαwr
A0＝αDDαEE-αDEαED
Where Φ represents the reflected and transmitted wave displacements at C in substructure II
Figure BDA0001921173550000113
And
Figure BDA0001921173550000114
ratio of (a)DDand alphaEEfor direct admittance of the two ends of the damaged element, alphaEDand alphaDEfor indirect admittance between the two ends of the lesion element, subscript D, E distinguishes between the two ends and the admittance direction, αwtand alphawrcharacteristic wave susceptance, respectively of the transmitted wave and the reflected wave, having alpha for the symmetrical cellwt＝-αwr(ii) a μ is the wave propagation constant;
(1.3) based on the condition of occurrence of anti-resonance, that is, the excitation frequency is equal to a certain natural frequency of the substructure on the left or right of the excitation point, obtaining an origin anti-resonance frequency characteristic equation through the natural frequency characteristic equations of the substructures I and II in the step (1.2) as follows:
(1-e-2jμ)(1+Φ)＝0
(1.4) supposing that the excitation point C is at the node j, the substructure I has a damage unit k, i.e. j is more than or equal to k; the substructure I is regarded as a single-coupling periodic structure with j units and fixed at two ends, wherein the unit k is damaged; the substructure II is a healthy single-coupling periodic structure with fixed left end and free right end and (N-j) units; the natural frequency characteristic equations of the substructure I and the substructure II are respectively:
substructure I: 1+ Ψ ═ 0
Substructure II: 1+ e-2(N-j)μ＝0
Figure BDA0001921173550000115
In the formula, Ψ represents the displacement of the reflected wave and the transmitted wave at C in the substructure I
Figure BDA0001921173550000116
And
Figure BDA0001921173550000117
the ratio of (A) to (B);
(1.5) repeating the step (1.3) to obtain an origin anti-resonance frequency characteristic equation corresponding to the step (1.4) as follows:
[1+e-2(N-j)μ](1+Ψ)＝0
(1.6) under a non-damage state, degenerating an origin anti-resonance frequency characteristic equation of the single damage obtained in the steps (1.3) and (1.5) into:
[1+e-2(N-j)μ](1-e-2jμ)＝0。
(2) the laminated rubber shock-insulation support is simplified into a limited single-coupling period structure, a basic period unit of the laminated rubber shock-insulation support is composed of a second-order shear beam and two end concentrated masses, and the dimensionless origin point anti-resonance frequency in a non-damage state is calculated;
(2.1) the direct admittance and the indirect admittance at the two ends of the rubber layer are:
Figure BDA0001921173550000121
Figure BDA0001921173550000122
in the formula, gammallAnd gammarrRespectively direct admittance of both ends of the rubber layer, gammalrAnd gammarlThe subscripts l and r are used for distinguishing the two ends and the admittance direction; g is the shear modulus of the rubber, ρ is the density of the rubber, L is the thickness of the rubber layer, A is the cross-sectional area of the rubber layer,
Figure BDA0001921173550000123
is a periodic structure wave number, omega is a circle frequency, and omega is ksL is a dimensionless frequency;
(2.2) admittance of the steel sheet is:
Figure BDA0001921173550000124
wherein β is the admittance of the steel plate, omega is the circular frequency, msThe mass of each layer of steel plate;
(2.3) the direct admittance and the indirect admittance of the composite periodic unit of the laminated rubber vibration isolation bearing and the propagation constants are respectively as follows:
Figure BDA0001921173550000125
Figure BDA0001921173550000131
Figure BDA0001921173550000132
in the formula, αlland alpharris a direct admittance across the composite periodic unit, alphalrand alpharlSubscripts l and r are used for distinguishing two ends and admittance directions for transfer admittance between two ends of the composite periodic unit;
Figure BDA0001921173550000133
m is the mass ratio of rubber to steel platerRho AL is the rubber mass; μ is the wave propagation constant;
(2.4) direct admittance α of the healthy compound periodic unit of step (2.3)ll、αrrand transfer admittance αlr、αrlAnd (4) substituting the obtained result into the dimensionless origin point antiresonance frequency characteristic equation of the nondestructive single-coupling periodic structure in the step (1.6), and calculating the dimensionless origin point antiresonance frequency of the laminated rubber vibration-isolating support in the nondestructive state.
(2.4.1) setting γ ═ μ i, and converting the origin antiresonance frequency characteristic equation in the non-invasive state obtained in step (1.6) into:
cos[(N-j)γ]sinγ＝0
the solution to the equation is:
Figure BDA0001921173550000134
or
Figure BDA0001921173550000135
In the formula:
Figure BDA0001921173550000136
is an imaginary unit, mu is a wave propagation constant;
(2.4.2) converting the wave propagation constant calculation formula of the step (2.3) into:
Figure BDA0001921173550000137
and (4) substituting the solution gamma obtained in the step (2.4.1) into the equation to calculate the dimensionless origin point anti-resonance frequency. The ratio of the dimensionless origin antiresonance frequency value omega to the structural period number N, the excitation point j and the mass of the rubber and the steel plate
Figure BDA0001921173550000138
And (4) relevant, independent of other geometrical and physical parameters.
(3) Introducing the increase of the shear stiffness into the admittance of the damage unit, and calculating the sensitivity coefficient of the anti-resonance frequency of the dimensionless origin to the change of the shear stiffness of the basic period unit; establishing a sensitivity identification equation set according to approximate linear superposition of the change of the dimensionless origin point antiresonance frequency under multiple damages into the change caused by single damage;
(3.1) when the rubber is aged, the shear modulus of the corresponding rubber layer is increased, and a damage state characterization parameter is introduced, wherein the direct admittance and the indirect admittance of a damage unit are as follows:
Figure BDA0001921173550000141
Figure BDA0001921173550000142
Figure BDA0001921173550000143
Ω′＝ks′L
wherein Δ G is the increase in unit shear modulus;
(3.2) shear modulus change rate xi of the damaged unit k before and after damagekAnd (3) evaluating the damage degree:
Figure BDA0001921173550000144
in the formula, ξkWhen 0, unit k is intact;
(3.3) the rate of change of the antiresonance frequency of the nth order dimensionless origin of the excitation point j before and after the damage is as follows:
Figure BDA0001921173550000145
in the formula:
Figure BDA0001921173550000146
respectively representing dimensionless origin point anti-resonance frequencies before and after damage, wherein the superscript u represents an undamaged state, and the superscript d represents a damaged state;
(3.4) based on the perturbation theory and the sensitivity analysis principle, obtaining the sensitivity of the n-th order dimensionless origin point anti-resonance frequency of the excitation point j to the k-th unit damage
Figure BDA0001921173550000147
Figure BDA0001921173550000148
In the formula:
Figure BDA0001921173550000151
representing the origin antiresonant frequency characteristic equation of a single lesion by steps (1.3) and (1.5)
Figure BDA0001921173550000152
Paxi xikpartial derivatives of (c), then make xi in the expression of resultk＝0；
(3.5) approximately linear superposition of the change of the dimensionless origin antiresonance frequency under multiple damages into the change caused by single damage, and accordingly, the vector of the total dimensionless origin antiresonance frequency change rate at the excitation point caused by multiple damages is established
Figure BDA0001921173550000153
and a damage state identification equation between the shear modulus change rate vector { ξ } of each layer of rubber:
Figure BDA0001921173550000154
Figure BDA0001921173550000155
in the formula, [ S ] is a sensitivity matrix of the dimensionless origin antiresonance frequency, p and q represent different nodes where the excitation point C is located, and p is 1, 2.
(4) Acquiring admittance signals before and after damage, and extracting the origin point anti-resonance frequency of the structure; and solving a sensitivity identification equation set based on the change rate of the original point anti-resonance frequency before and after the damage, and identifying the damage.
(4.1) sticking PZT along the axial direction of the laminated rubber shock-insulation support, and collecting admittance signals Y before and after damage;
(4.2) separating the mechanical impedance Z of the single-coupling periodic structure from the PZT electric admittance signal Y according to the one-dimensional impedance models；
(4.2.1) calculating the mechanical impedance Z of the PZT in the short-circuited statea：
Figure BDA0001921173550000156
Figure BDA0001921173550000157
In the formula (I), the compound is shown in the specification,
Figure BDA0001921173550000161
wave number of PZT, ω circular frequency of excitation frequency, la、ba、haRespectively the length, width and thickness of the PZT,
Figure BDA0001921173550000162
is the composite elastic modulus of PZT when the electric field is constant,
Figure BDA0001921173550000163
is the modulus of elasticity, η is the mechanical loss factor,
Figure BDA0001921173550000164
is a plurality of units;
(4.2.2) the PZT electrical admittance expression is:
Figure BDA0001921173550000165
Figure BDA0001921173550000166
in the formula (I), the compound is shown in the specification,
Figure BDA0001921173550000167
is the complex dielectric constant of PZT when the stress is constant,
Figure BDA0001921173550000168
for a real dielectric constant, δ is a dielectric loss factor, d31Is the piezoelectric strain coefficient of PZT;
(4.3) velocity admittance H based on Single-coupling periodic StructurevAnd the displacement admittance HdIn relation to (3), the mechanical impedance Z of the single-coupled periodic structuresStructure displacement admittance H converted into single coupling periodd：
Figure BDA0001921173550000169
Extracting a valley value of the displacement admittance curve, namely the original point anti-resonance frequency of the structure;
and (4.4) obtaining the change rate of the original point anti-resonance frequency before and after damage based on the step (4.3), and thus carrying out damage identification on the laminated rubber vibration isolation support.
(4.4.1) calculating the change rate of the dimensionless origin point antiresonance frequency through the measured before and after damage:
Figure BDA00019211735500001610
in the formula (I), the compound is shown in the specification,
Figure BDA00019211735500001611
respectively representing the measured original point anti-resonance frequency before and after damage, the superscript u representing the undamaged state and the superscript d representing the damaged state;
(4.4.2) converting the equation set solving problem of step (3.5) to a non-negative least squares curve fitting problem based on damage such that the shear stiffness of the rubber increases:
Figure BDA0001921173550000171
and (5) solving [ S ] according to the fitting result of the formula to finish the damage identification.
The damage identification process based on the periodic structure theory is described below by taking the laminated rubber vibration isolation bearing experimental model shown in fig. 2 as an object. FIG. 3(a) is a schematic diagram of a period system of a laminated rubber vibration-isolating support, and the model consists of 10 nodes and 10 units. FIG. 3(b) is a schematic diagram of basic cycle units, with the following parameters: the shear modulus of the rubber was 8X 105N/m2The density of the rubber is 1000kg/m3The thickness of the rubber layer was 3.14mm, and the cross-sectional area of the rubber layer was 0.16m2The steel sheet had a mass of 2.512 kg. Fig. 4(a) is a schematic diagram of single-coupling period system wave propagation when the excitation point is on the left side of the damage unit, and fig. 4(b) is a schematic diagram of single-coupling period system wave propagation when the excitation point is on the right side of the damage unit.
In order to verify the invention, a damage working condition is set for the laminated rubber vibration isolation support: the stiffness of cell 1 increased by 5% and the stiffness of cell 10 increased by 10%. And (3) pasting the PZT on the positions of the nodes 1, 4, 7 and 10 for measurement, converting PZT electric admittance signals into structural mechanical impedance signals, and extracting ninth-order antiresonance frequency. The anti-resonance frequency sensitivity coefficients of the nodes 1, 4, 7, 10 obtained by the above-described method according to the present invention are shown in fig. 5(a) to 5 (d). Based on the antiresonance frequency change rate before and after the damage, the sensitivity identification equation set is solved, and the comparison between the identification result obtained by solving and the actual damage is shown in fig. 6. Therefore, the damage identification result of the method is very close to the actual damage, and the damage position and the damage degree can be accurately identified.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (4)

1. A laminated rubber vibration-isolating support damage identification method based on piezoelectric impedance information is characterized by comprising the following steps:
(1) constructing an original point anti-resonance frequency characteristic equation of a single-damage and non-damage single-coupling periodic structure;
(2) simplifying the laminated rubber shock-insulation support into a single-coupling periodic structure, wherein a basic periodic unit of the laminated rubber shock-insulation support is composed of a second-order shear beam and concentrated masses at two ends of the second-order shear beam, and calculating the dimensionless origin point anti-resonance frequency of the single-coupling periodic structure in a non-damage state;
(3) introducing the increase of the shear stiffness into the admittance of the damage unit, and calculating the sensitivity coefficient of the anti-resonance frequency of the dimensionless origin to the change of the shear stiffness of the basic period unit; establishing a sensitivity identification equation set according to approximate linear superposition of the change of the antiresonance frequency under multiple damages into the change caused by single damage;
(4) acquiring admittance signals before and after damage, and extracting the origin point anti-resonance frequency of the single-coupling periodic structure; solving a sensitivity identification equation set based on the change rate of the original point anti-resonance frequency before and after damage, and identifying the damage;
the step (1) further comprises the following substeps:
(1.1) for a single-coupling periodic structure of N basic periodic units, setting a left boundary A to be fixed and a right boundary B to be free, and dividing the single-coupling periodic structure into a substructure I and a substructure II by taking the point C as a boundary point when an excitation force P acts on the point C;
(1.2) assuming that the excitation point C is at the node j, the substructure II has a damaged unit k, i.e. j < k; the substructure I is regarded as a healthy single-coupling periodic structure with j units and fixed at two ends; the substructure II is regarded as a single-coupling periodic structure with fixed left end and free right end and (N-j) units, wherein the unit k is damaged; the natural frequency characteristic equations of the substructure I and the substructure II are respectively as follows:
substructure I: 1-e-2jμ＝0
Substructure II: 1+ phi is 0
Figure FDA0002430101210000021
Figure FDA0002430101210000022
C0＝A0+αDDαwr-αEEαwt-αwtαwr
Figure FDA0002430101210000028
E0＝A0+αDDαwt-αEEαwr-αwtαwr
A0＝αDDαEE-αDEαED
Where Φ represents the reflected and transmitted wave displacements at C in substructure II
Figure FDA0002430101210000023
And
Figure FDA0002430101210000024
ratio of (a)DDand alphaEEfor direct admittance of the two ends of the damaged element, alphaEDand alphaDEfor indirect admittance between the ends of the damaged element, alphawtand alphawrCharacteristic wave susceptance, pair of transmitted wave and reflected wave respectivelyin a symmetrical unit having awt＝-αwr(ii) a μ is the wave propagation constant;
(1.3) based on the condition of occurrence of anti-resonance, that is, the excitation frequency is equal to a certain natural frequency of the substructure on the left or right of the excitation point, obtaining an origin anti-resonance frequency characteristic equation through the natural frequency characteristic equations of the substructures I and II in the step (1.2) as follows:
(1-e-2jμ)(1+Φ)＝0
(1.4) supposing that the excitation point C is at the node j, the substructure I has a damage unit k, i.e. j is more than or equal to k; the substructure I is regarded as a single-coupling periodic structure with j units and fixed at two ends, wherein the unit k is damaged; the substructure II is a healthy single-coupling periodic structure with fixed left end and free right end and (N-j) units; the natural frequency characteristic equations of the substructure I and the substructure II are respectively:
substructure I: 1+ Ψ ═ 0
Substructure II: 1+ e-2(N-j)μ＝0
Figure FDA0002430101210000025
In the formula, Ψ represents the displacement of the reflected wave and the transmitted wave at C in the substructure I
Figure FDA0002430101210000026
And
Figure FDA0002430101210000027
the ratio of (A) to (B);
(1.5) repeating the step (1.3) to obtain an origin anti-resonance frequency characteristic equation corresponding to the step (1.4) as follows:
[1+e-2(N-j)μ](1+Ψ)＝0
(1.6) under a non-damage state, degenerating an origin anti-resonance frequency characteristic equation of the single damage obtained in the steps (1.3) and (1.5) into:
[1+e-2(N-j)μ](1-e-2jμ)＝0；
the step (2) further comprises the following substeps:
(2.1) the direct admittance and the indirect admittance at the two ends of the rubber layer are:
Figure FDA0002430101210000031
Figure FDA0002430101210000032
in the formula, gammallAnd gammarrRespectively direct admittance of both ends of the rubber layer, gammalrAnd gammarlRespectively, the indirect admittance between two ends of the rubber layer, G is the shear modulus of the rubber, rho is the density of the rubber, L is the thickness of the rubber layer, A is the cross-sectional area of the rubber layer,
Figure FDA0002430101210000033
is a periodic structure wave number, omega is a circle frequency, and omega is ksL is a dimensionless frequency;
(2.2) admittance of the steel sheet is:
Figure FDA0002430101210000034
wherein β is the admittance of the steel plate, omega is the circular frequency, msThe mass of each layer of steel plate;
(2.3) the direct admittance and the indirect admittance of the composite periodic unit of the laminated rubber vibration isolation bearing and the propagation constants are respectively as follows:
Figure FDA0002430101210000035
Figure FDA0002430101210000036
Figure FDA0002430101210000041
in the formula, αlland alpharrIs a composite perioddirect admittance, alpha, at both ends of the celllrand alpharlFor transfer admittance between the two ends of the composite periodic unit,
Figure FDA0002430101210000042
m is the mass ratio of rubber to steel platerRho AL is the rubber mass; μ is the wave propagation constant;
(2.4) direct admittance α of the healthy compound periodic unit of step (2.3)ll、αrrand transfer admittance αlr、αrlSubstituting the obtained result into the dimensionless origin point anti-resonance frequency characteristic equation of the nondestructive single-coupling periodic structure in the step (1.6), and calculating the dimensionless origin point anti-resonance frequency of the laminated rubber vibration isolation support in the nondestructive state; the step (3) further comprises the following substeps:
(3.1) when the rubber is aged, the shear modulus of the corresponding rubber layer is increased, and a damage state characterization parameter is introduced, wherein the direct admittance and the indirect admittance of a damage unit are as follows:
Figure FDA0002430101210000043
Figure FDA0002430101210000044
Figure FDA0002430101210000045
Ω′＝ks′L
wherein Δ G is the increase in unit shear modulus;
(3.2) shear modulus change rate xi of the damaged unit k before and after damagekAnd (3) evaluating the damage degree:
Figure FDA0002430101210000046
in the formula, ξkWhen 0, unit k is intact;
(3.3) the rate of change of the antiresonance frequency of the nth order dimensionless origin of the excitation point j before and after the damage is as follows:
Figure FDA0002430101210000047
in the formula:
Figure FDA0002430101210000051
respectively representing dimensionless origin point anti-resonance frequencies before and after damage, wherein the superscript u represents an undamaged state, and the superscript d represents a damaged state;
(3.4) based on the perturbation theory and the sensitivity analysis principle, obtaining the sensitivity of the n-th order dimensionless origin point anti-resonance frequency of the excitation point j to the k-th unit damage
Figure FDA0002430101210000052
Figure FDA0002430101210000053
In the formula:
Figure FDA0002430101210000054
representing the origin antiresonant frequency characteristic equation of a single lesion by steps (1.3) and (1.5)
Figure FDA0002430101210000055
Paxi xikpartial derivatives of (c), then make xi in the expression of resultk＝0；
(3.5) approximately linear superposition of the change of the dimensionless origin antiresonance frequency under multiple damages into the change caused by single damage, and accordingly, the vector of the total dimensionless origin antiresonance frequency change rate at the excitation point caused by multiple damages is established
Figure FDA0002430101210000056
and a damage state identification equation between the shear modulus change rate vector { ξ } of each layer of rubber:
Figure FDA0002430101210000057
Figure FDA0002430101210000058
in the formula, [ S ] is a sensitivity matrix of a dimensionless origin antiresonance frequency, p and q represent different nodes where an excitation point C is located, and p is 1,2,. and N, q is 1,2,. and N;
the step (4) further comprises the following substeps:
(4.1) sticking PZT along the axial direction of the laminated rubber shock-insulation support, and collecting admittance signals Y before and after damage;
(4.2) separating the mechanical impedance Z of the single-coupling periodic structure from the PZT electric admittance signal Y according to the one-dimensional impedance models；
(4.3) velocity admittance H based on Single-coupling periodic StructurevAnd the displacement admittance HdIn relation to (3), the mechanical impedance Z of the single-coupled periodic structuresStructure displacement admittance H converted into single coupling periodd：
Figure FDA0002430101210000061
Extracting a valley value of the displacement admittance curve, namely the original point anti-resonance frequency of the structure;
and (4.4) obtaining the change rate of the original point anti-resonance frequency before and after damage based on the step (4.3), and thus carrying out damage identification on the laminated rubber vibration isolation support.
2. The method for identifying damage to a laminated rubber-vibration-isolated support based on piezoelectric impedance information as claimed in claim 1, wherein the step (2.4) of calculating the dimensionless origin antiresonance frequency in a non-damaged state further comprises the substeps of:
(2.4.1) setting γ ═ μ i, and converting the origin antiresonance frequency characteristic equation in the non-invasive state obtained in step (1.6) into:
cos[(N-j)γ]sinγ＝0
the solution to the equation is:
Figure FDA0002430101210000062
or
Figure FDA0002430101210000063
In the formula:
Figure FDA0002430101210000064
is an imaginary unit, mu is a wave propagation constant;
(2.4.2) converting the wave propagation constant calculation formula of the step (2.3) into:
Figure FDA0002430101210000065
and (4) substituting the solution gamma obtained in the step (2.4.1) into the equation to calculate the dimensionless origin point anti-resonance frequency.
3. The method for identifying damage to laminated rubber-vibration-isolated support based on piezoelectric impedance information as claimed in claim 1, wherein the step (4.2) separates the mechanical impedance Z of the structure from the PZT electrical admittance signal YsFurther comprising the substeps of:
(4.2.1) calculating the mechanical impedance Z of the PZT in the short-circuited statea：
Figure FDA0002430101210000071
Figure FDA0002430101210000072
In the formula (I), the compound is shown in the specification,
Figure FDA0002430101210000073
is the wave number of PZT, and ω is the excitation frequencyCircular frequency of la、ba、haRespectively the length, width and thickness of the PZT,
Figure FDA0002430101210000074
is the composite elastic modulus of PZT when the electric field is constant,
Figure FDA0002430101210000075
is the modulus of elasticity, η is the mechanical loss factor,
Figure FDA0002430101210000076
is an imaginary unit;
(4.2.2) the PZT electrical admittance expression is:
Figure FDA0002430101210000077
Figure FDA0002430101210000078
in the formula (I), the compound is shown in the specification,
Figure FDA0002430101210000079
is the complex dielectric constant of PZT when the stress is constant,
Figure FDA00024301012100000710
for a real dielectric constant, δ is a dielectric loss factor, d31Is the piezoelectric strain coefficient of PZT.
4. The laminated rubber-vibration-isolation bearing damage identification method based on piezoelectric impedance information as claimed in claim 3, wherein the step (4.4) is to identify the damage of the laminated rubber-vibration-isolation bearing based on the measured change rate of the before-damage and after-damage origin point anti-resonance frequency, and further comprises the following substeps:
(4.4.1) calculating the change rate of the dimensionless origin point antiresonance frequency through the measured before and after damage:
Figure FDA00024301012100000711
in the formula (I), the compound is shown in the specification,
Figure FDA00024301012100000712
respectively representing the measured original point anti-resonance frequency before and after damage, the superscript u representing the undamaged state and the superscript d representing the damaged state;
(4.4.2) converting the equation set solving problem of step (3.5) to a non-negative least squares curve fitting problem based on damage such that the shear stiffness of the rubber increases:
Figure FDA0002430101210000081
and (5) solving [ S ] according to the fitting result of the formula to finish the damage identification.
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Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2913530A (en) * 1953-07-03 1959-11-17 Int Standard Electric Corp Line identification in telecommunication systems
CN103528778A (en) * 2013-10-11 2014-01-22 南京航空航天大学 Rapid damage detection method for three-layer-of-base shock isolation system on basis of incomplete measurement
CN104750978A (en) * 2015-03-12 2015-07-01 华中科技大学 Beam member damage recognition method based on antiresonant frequency and particle swarm optimization

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20110260875A1 (en) * 2010-04-21 2011-10-27 Koninklijke Philips Electronics, N.V. Alert device and method

Patent Citations (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US2913530A (en) * 1953-07-03 1959-11-17 Int Standard Electric Corp Line identification in telecommunication systems
CN103528778A (en) * 2013-10-11 2014-01-22 南京航空航天大学 Rapid damage detection method for three-layer-of-base shock isolation system on basis of incomplete measurement
CN104750978A (en) * 2015-03-12 2015-07-01 华中科技大学 Beam member damage recognition method based on antiresonant frequency and particle swarm optimization

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
叠层橡胶隔震支座的振动传递特性;朱宏平等;《工程力学》;19951130;第12卷(第4期);第109-114页 *

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