# Damage Detection of Laminated Rubber Bearings Based on the Antiresonance Frequencies of Periodic Structure and Its Sensitivity Analysis

Authors: Ying Zhang; Hongping Zhu; Shun Weng
Journal: Shock and Vibration
 (2022)

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## Abstract

An isolation bearing consumes most of the seismic energy of a structure and is vulnerable to destruction. The performance of isolation bearings is usually evaluated according to the global stiffness and energy dissipation capacity. However, the early minor damage in isolation bearings is difficult to identify. In this study, a damage detection scheme for the isolation bearing is proposed by focusing on the antiresonance of the quasiperiodic structure. Firstly, a laminated rubber bearing was simplified as a monocoupled periodic rubber-steel structure. The characteristic equation of the driving point antiresonance frequency of the periodic system was achieved via the dynamic stiffness method. Secondly, the sensitivity coefficient of the driving point antiresonance, which was obtained from the first-order derivative of the antiresonance frequency, with respect to the damage scaling parameter was derived using the antiresonance frequency characteristic equation. Thirdly, the optimised driving points of the antiresonance frequencies were selected by means of sensitivity analysis. Finally, from the measured changes in the antiresonance frequencies, the damage was identified by solving the sensitivity identification equation via a numerical optimisation method. The application of the proposed method to laminated rubber bearings under various damage cases demonstrates the feasibility of this method. This study has proven that changes in the shear modulus of each rubber layer can be identified accurately.


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## Body

## 1. Introduction

During an earthquake disaster, the seismic isolator consumes a large amount of structural vibration energy. The isolator is the most vulnerable part of the structure that is expected to be destroyed. If the early damage of the seismic isolator cannot be identified and dealt with in time, then it will continue to accumulate and may lead to the sudden structural failure. However, the early minor damage in the isolation bearing cannot be reflected from the test results of mechanical properties. Therefore, the effective monitoring and evaluation of the seismic isolator is of great significance.The laminated rubber bearing is widely used in seismic isolation structures. Most theoretical studies have investigated the behaviour of laminated rubber bearings in terms of Haringx’s theory [1,2]. In the model of Haringx’s equivalent column, the rubber layers and steel shims are treated as homogeneous and isotropic materials. For moderate amounts of shear strain, horizontal stiffness of the elastomeric bearing in Haringx’s theory agrees with the experimental results [3,4]. Haringx’s theory has been applied in the investigation of the stability of rubber bearings [5–8]. On the basis of Haringx’s theory, Koh and Kelly [9] proposed a nonlinear mechanical model to predict the horizontal displacement of the elastomeric bearing under large axial force, and the experimental results of Buckle et al. [10] validated this model. The limitations of the investigations lie in the inability to address the details of the mechanical behaviours of interlaminated rubber layers [4], and the traditional approach cannot handle the influence of geometric and material parameters of each rubber layer. In addition, according to the results of finite element analysis, Kalfas et al. [11, 12] found the damage magnitude of each rubber layer and steel shim is different after the bearing under a certain load. When Haringx’s theory is applied to the damage identification of laminated rubber bearings, the damage of certain layers cannot be located; consequently, the early minor damage in the bearings is difficult to identify.The periodical model can be used to analyse the internal deformation and force of individual cells. Therefore, the laminated rubber bearing model that considers the periodicity can be used to identify the damage of each layer. As a typical one is consisting of several alternating layers of rubber bonded to interleaving steel shims [13]; it can be considered a chain-like periodic structure system composed of an end-to-end repeating structure. Static models [4, 14, 15] and dynamic models [13, 16] of laminated rubber bearing that consider periodicity have been developed. However, the dynamic characteristics will change when damage occurs in a periodic structure.Dynamic characteristics, such as natural frequencies and mode shapes, have been used in the damage detection of periodic structures [17–19]. Zhu [17] used the characteristic receptance approach to obtain the frequency characteristic equation of a periodic spring-mass system with a single damage and identified the damage of it based on the sensitivity of natural frequencies. This method can be applied to various types of finite monocoupled periodic systems with multiple damage [17]. To extend the work [17], Yin [18] used natural frequencies to identify the damage of periodically supported structures. Zhu [19] used the slopes and curvatures of mode shapes to localise damage, which were then quantified using natural frequencies. The abovementioned studies showed that the damage of periodic structures can be identified by using natural frequencies and mode shapes.On the one hand, natural frequencies are less sensitive to local damage compared with mode shapes [19]. On the other hand, the mode shapes have generally lower accuracy than natural frequencies during measurements. In addition, as the total number of natural frequencies is always limited in high-quality measurements [20], the available modal data that can be used in damage detection is usually incomplete. Natural frequencies and mode shapes are clearly inadequate in damage identification; thus, antiresonance frequencies have been applied to the finite element model updating and damage identification of structures [21–27]. Dilena [22] found that the appropriate use of natural frequencies and antiresonance frequencies could avoid the nonuniqueness of the damage location problem, which occurs in symmetrical beams when only frequency data are employed. Wang [23] utilised the first antiresonance to localise the crack of a Timoshenko beam. Hanson [24] used antiresonance frequencies to update the model of a two-degrees-of-freedom (DOF) system. Sinou [25] detected the breathing crack of a pipeline beam based on the antiresonances of higher-order frequency response functions (FRFs). In an experimental study on the crack identification of thick beams, Hou [26] proved that the incorporation of antiresonance frequencies could enhance the modal dataset in finite element model updating. Another study involving model updating found that antiresonance frequencies could be used as an alternative of natural frequencies and mode shapes [27]. Antiresonance frequencies reflect not only the intrinsic characteristics of structures but also their local physical characteristics. Therefore, the antiresonance frequency is further extended to the damage identification of periodic structures in this study.Two methods are generally used to determine antiresonance frequencies. The first method is to extract data from the frequency response or impedance curve [23–26], which are the zeros of the curves. The other method is to calculate the eigenvalues of a structure under a variety of artificial constraints applied to the ground combined with transducer location data [28,29]. A set of sensitivities of antiresonance frequencies [28] was used for damage identification. According to Mottershead [28], the sensitivity of an antiresonance frequency is a linear combination of the sensitivity of mode shapes at the same point and the sensitivity of natural frequencies. However, the calculation of antiresonance frequencies and its sensitivity of periodic structures has not yet been studied. As for the periodic structure, each cell is the same, which implies that the periodicity of the structure can be exploited to simplify the analytic process.At present, the damage detection of laminated rubber bearing based on the antiresonance frequencies of periodic structure is at an early stage. According to the damage detection method for laminated rubber bearing based on the characteristic receptance method developed by Zhang [30], only single damage can be identified. And the influence of driving points on identification results has not been studied. However, as each node of the structure can be configured as the driving point, additional appropriate driving points should be selected to be able to extract the antiresonance frequencies and improve the modal identification. Antiresonance frequencies can be obtained from the point FRFs and transfer FRFs, in which the former is more robust than the latter [21]. Mao [29] studied the selection of optimal antiresonance frequencies but did not distinguish the driving point antiresonance from the transfer point antiresonance. Therefore, the selection of the optimal driving point is investigated in this study.This study proposes a damage detection method of laminated bearings in service based on the sensitivity analysis of the antiresonance frequency of periodic structures. Firstly, the laminated rubber bearing is modelled as a monocoupled periodic rubber-steel structure. The antiresonance frequency of the periodic structure is derived using Bloch’s theorem and dynamic stiffness method. Secondly, the relationship between the antiresonance frequency and damage scaling parameter is derived according to the antiresonance frequency characteristic equation. From the results of the sensitivity analysis, more suitable driving points of the antiresonance frequencies are selected and used in damage identification. Finally, by taking the antiresonance frequencies before and after the damage as the objective function, the sensitivity identification equation is solved via a numerical optimisation method. By detecting the simulated damage of one or more parts of the laminated rubber bearing, the accuracy of the proposed method was verified.


## 2. Antiresonance Frequency Characteristic Equation of the Periodic Rubber-Steel Structure

The laminated rubber bearing is considered a periodic rubber-steel structure, as shown in Figure1(a). Each periodic cell consists of a rubber layer and two steel shims, as depicted in Figure 1(b). Figure 2 illustrates a monocoupled NP-cell periodic structure terminating at the fixed end O and the free end N.Figure 1
(a): The model of a laminated rubber bearing. (b): Rubber–steel cell.
(a)(b)Figure 2
The monocoupled periodic structure ofNP cells with a free end (N) and a fixed end (O).In the horizontal direction, the steel plates do not affect the shear deformations of the rubber layers in the bearing. Since the ratio of the shear modulus of steel to that of rubber is greater than 1 × 104, the deformation of the steel shim is negligible in the analysis. Therefore, the deformation of the isolation bearing is considered to be entirely caused by the rubber layer. The rubber layer is simplified as a shear beam. Here, only the shear deformation is considered, and the bending deformation is neglected. The parameters of the rubber layer are as follows: the thickness of each layer is given by l, the cross-sectional area is given by A, the shear modulus is given by G, and the density is given by ρ. The stiffness and mass matrix of the rubber layer are as follows:(1a)kr=∫0lGAdNwdxTdNwdxdx=GAl1−1−11,(1b)mr=∫0lρANwTNwdx=ρAl62112,where Nwx=1−x/lx/l is the shape function vector.The lumped mass is used for describing the steel shim. The mass of each steel shim in the symmetrical cell is calculated asms/2. With the introduction of the mass ratio of steel to rubber, θ=ms/mr, where mr=ρAl is the mass of each rubber layer. The mass matrix of the composite cell is given by(2)m=mr+ms=ρAl3+ms2ρAl6ρAl6ρAl3+ms2=ρAl13+θ2161613+θ2.When the structure vibrates harmonically with angular frequencyω, the dynamic stiffness matrix of the composite periodic cell can be expressed as(3)d=kr−ω2m=dLLdLRdRLdRR,where(4)dLL=dRR=GAl−ω213+θ2ρAl,dLR=dRL=−GAl−ω2ρAl6.For a monocoupledNP-cell periodic structure with one fixed end and one free end, the global dynamic stiffness matrix can be expressed as(5)Dω=dRR1+dLL2dLR2dRL2dRR2+dLL3dLR3dRL3dRR3+dLL4dLR4⋱⋱⋱dRLNP−2dRRNP−2+dLLNP−1dLRNP−1dRLNP−1dRRNP−1+dLLNPdLRNPdRLNPdRRNPNP×NP.For an undamped structure, the FRFs of a driving point, i.e.,j-th node, can be expressed as(6)Hj=detDjdetD.det· represents the determinant of the matrix D, and subscript j means that the j-th row and the j-th column are deleted. For the driving point FRFs of an undamped structure, the antiresonance frequencies zj are the zeros of the numerator polynomial [27], which satisfies(7)detDj=0.The determinantdetDj can be rewritten as the determinant of a partitioned matrix as(8)detDj=DL00DR=DL⋅DR,where DL and DR are the characteristic determinants of the substructures on both sides of the driving point, and the detailed expressions of DL and DR are provided in Appendix.By substituting equation (7) into (6), the characteristic equation of the driving point antiresonance frequencies can be obtained.(9)DL⋅DR=0.As shown in equation (8), antiresonance occurs provided that the driving frequency reaches a natural frequency of the substructure on the left or right side of the driving point. DL=0 gives the characteristic equation of periodic structure 1 consisting of j cells with two fixed ends. DR=0 gives the characteristic equation of periodic structure 2 consisting of (NP − j) cells and with one free end and one fixed end. In other words, the antiresonance frequencies reflect the natural frequencies of the substructure on either side of the driving point.For a perfectly periodic structure, the dynamic stiffness matrix of each cell is exactly the same. Take theλ-th (λ = 1, 2, …, NP) cell as an example. The displacements and forces at the ends of the cell can be expressed in relation to the dynamic stiffness matrix as follows:(10)dLLdLRdRLdRRXLλXRλ=FLλFRλ.In terms of the vibration of the monocoupled periodic structure, according to Bloch’s theorem, the state vector at the left end of a cell must be equal toeμ times the state vector at the left end of the next cell, where μ is the propagation constant [31, 32]. The relationship between the displacements and forces of the two adjacent cells can be determined in relation to the propagation constant μ.(11)XLλ+1FLλ+1=e−μXLλFLλ,where the superscript λ and λ + 1 refer to cell λ and the next cell λ + 1.The displacements and equilibrium of the forces at the interface of theλ-th cell and the (λ + 1)-th cell are compatible. Thus,(12)XLλ+1FLλ+1=100−1XRλFRλ.By combining equations (10) and (11), the displacements and forces at the right and left ends of the λ-th cell become related to the propagation constant μ.(13)XRλFRλ=e−μ00−e−μXLλFLλ.A periodic structure with symmetric cells hasdLL=dRR and dLR=dRL. Substituting equation (11) into (9) yields the expression of the propagation constant μ.(14)coshμ=−dLLdLR.For the perfectly periodic structure, the determinantDL in equation (A.1a) can be expressed by the dynamic stiffness of any cell. When the dynamic stiffness of the λ-th (λ = 1, 2, …, j) cell, i.e., dLLλ and dLRλ, is reserved, DL can be rewritten using equation (13) as follows:(14a)DL=−dLRj−1D¯L,where D¯L is the determinant of order (j − 1),(14b)D¯L=dLLλ2−dLRλ2dLR2sinhλ−1μsinhj−λμsinh2μ−dLLλdLRsinhj−1μsinhμ+coshj−λμcoshλ−1μ.Furthermore, by settingdLLλ=dLL and dLRλ=dLR in equation (14b), D¯L can be rewritten as(14c)D¯L=sinhjμsinhμ.Similarly, the determinantDR in equation (A.1 b) can be expressed by the dynamic stiffness of the λ-th (λ = j+1, j+2, …, NP) cell,(15a)DR=−dLRNP−jD¯R,where D¯R is the determinant of order (NP − j),(15b)D¯R=dLLλ2−dLRλ2dLR2sinhλ−j−1μsinhNP−λμsinhμ−dLLλdLRcoshNP−j−1μ+coshλ−j−1μsinhNP−λμsinhμ=coshNP−jμ.The detailed calculation process of determinantsD¯L and D¯R are provided in Appendix.By substituting equations (14) and (15) into (8), the characteristic equation of the driving point antiresonance frequencies of the periodic structure containing the information of the location of different cells can be obtained.(16a)coshNP−jμzdLLλ2−dLRλ2dLR2sinhλ−1μzsinhj−λμzsinh2μz−dLLλdLRsinhj−1μzsinhμz+coshλ−1μzcoshj−λμz=0λ=1,2,...,j.(16b)sinhjμzdLLλ2−dLRλ2dLR2sinhNP−λμzsinhλ−j−1μzsinh2μz−dLLλdLRcoshNP−j−1μzsinhμz+sinhNP−λμzcoshλ−j−1μz=0,λ=j+1,j+2,...,NP.The propagation constantμ and arbitrary vibration frequency ω have a one-to-one correspondence. Here, μz denotes the propagation constant corresponding to the antiresonance frequency zj. The characteristic equation of the driving point antiresonance frequencies of the periodic structure expressed as the propagation constant can be obtained using (17)sinhjμzcoshNP−jμz=0.When the vibration frequencyω is equal to the antiresonance frequency zj, combining equations (3) and (13) yields the relationship between the propagation constant μz and antiresonance frequency zj.(18)coshμz=6G−zj22+3θρl26G+zj2ρl2.The propagation constant is usually written in complex form, i.e.,μ=δ+iγ, where ‘i’ is the imaginary unit, namely, i=−1. The real part δ represents the attenuation constant, whereas the imaginary part γ represents the phase constant [33]. When δ≠0, the wave is in a frequency stopband; when δ=0, the wave is in a passband [34]. For an undamped periodic structure with symmetric cells, the natural frequencies lie within the passband [35]. In addition, the driving point antiresonance frequencies interlace the natural frequencies [21]. Therefore, the driving point antiresonance frequencies also lie within the passband (i.e., δ=0). Substituting μ=iγ and equation (18) into (16) yields the antiresonance frequencies of the periodic rubber-steel structure in the healthy status.(19)zju=2sinγu23Gcosγu+2+3θρl2,where γu=γLu=πk1/jk1=1,2,…,j−1orγu=γRu=π/22k2−1/NP−jk2=1,2,…,NP−j, in which the superscript ‘u’ means undamaged.


## 3. Sensitivity Analysis

### 3.1. Sensitivity of Driving Point Antiresonance

When the shear modulus of theλ-th (λ = 1, 2, …, NP) cell in the periodic structure changes slightly, the first-order Taylor’s expansion of the antiresonance frequency of driving point j is(20)zjG+ΔGλ=zjG+∂zj∂GλΔGλ,where the first-order partial derivate ∂zj/∂Gλ is the sensitivity coefficient of the antiresonance frequency with respect to the shear modulus of the λ-th (λ = 1, 2, …, NP) cell.According to the characteristic equation of the driving point antiresonance frequencies, i.e., equation (16), the sensitivity coefficient of the antiresonance frequency with respect to the shear modulus of the λ-th cell is as follows:(21)∂zj∂Gλ=−Γzj,Gλ/∂GλΓzj,Gλ/∂zj,where the function Γzj,Gλ is set according to equation (16), i.e.,(22a)Γzj,Gλ=coshNP−jμzdLLλ2−dLRλ2dLR2sinhλ−1μzsinhj−λμzsinh2μz−dLLλdLRsinhj−1μzsinhμz+coshλ−1μzcoshj−λμz,λ=1,2,...,j,(22b)Γzj,Gλ=sinhjμzdLLλ2−dLRλ2dLR2sinhλ−j−1μzsinhNP−λμzsinh2μz−dLLλdLRcoshNP−j−1μzsinhμz+sinhNP−λμzcoshλ−j−1μz,λ=j+1,j+2,...,NP.The first partial derivatives∂Γzj,Gλ/∂Gλ and ∂Γzj,Gλ/∂zj in equation (21) can be obtained from (22), which contains the information about the location of the different cells.(23a)∂Γzj,Gλ∂zj=NP−jsinhNP−jμzsinhjμz+jcoshNP−jμzcoshjμzsinhμz∂μz∂zj,(23b)∂Γzj,Gλ∂Gλ=−coshNP−jμzdLRsinh2μz2coshμz∂dLLλ∂Gλ+∂dLRλ∂Gλsinhj−λμzsinhλ−1μz+∂dLLλ∂Gλsinhμzsinhj−1μzλ=1,2,…,j,(23c)∂Γzj,Gλ∂Gλ=−sinhjμzdLRsinh2μz2coshμz∂dLLλ∂Gλ+∂dLRλ∂Gλsinhλ−j−1μzsinhNP−λμz+∂dLLλ∂GλsinhμzcoshNP−j−1μz.λ=j+1,j+2,...,NP.The detailed expressions of∂dLLλ/∂Gλ, ∂dLRλ/∂Gλ, and ∂μz/∂zj in equation (23) can be obtained from equations (3) and (18):(24a)∂dLLλ∂Gλ=Al,(24b)∂dLRλ∂Gλ=−Al,(24c)∂μz∂zj=2+3θ+coshμzρAlzj3dLRsinhμz,where l is the thickness of each layer, A is the cross-sectional area, and θ is the mass ratio of steel to rubber.Combining equations (19), (21), (23), and (24) yields the sensitivity coefficient ∂zj/∂Gλ.(25a)∂zj∂Gλ=2j3cosγu+2+3φGρl2sinγu2cos22λ−12γu,λ=1,2,…,j,0,λ=j+1,j+2,…,NP.γu=γLu=πk1j,k1=1,2,…,j−1,(25b)∂zj∂Gλ=0,λ=1,2,…,j2NP−j3cosγu+2+3θGρl2sinγu2cos22λ−2j−12γu,λ=j+1,j+2,…,NPγu=γRu=π22k2−1NP−j,k2=1,2,…,NP−j.As shown in equation (25), the sensitivity coefficient ∂zj/∂Gλ is dependent on the total number of periodic cells NP, the driving point j, the location of the cell λ, the propagation constant corresponding to the antiresonance frequency γu, the mass ratio θ , the shear modulus G, the thickness l, and the density ρ of the rubber layer. Each cell λ has a specific sensitivity coefficient ∂zj/∂Gλ that can be used for damage identification.


### 3.2. Selection of Optimal Driving Points of Antiresonance for Damage Identification

Given that a structure comprises more than one driving point, the amount of data of antiresonance frequencies is larger than the amount of data of natural frequencies. Besides, in the real practice of realising damage identification, sufficient structural information can be obtained by selecting a small number of driving points. Therefore, the optimal driving point should be selected.In this section, the optimal driving points are selected by conducting a sensitivity analysis of the antiresonance frequency on the location of driving pointj. Supposing the thickness of the steel shim is neglected, the length of each periodic cell denoted by l is equal to the thickness of the rubber layer. Thus, the location of driving point j can be expressed as(26)xj=jlj=1,2,...,NP.By substituting equation (26) into (19), the antiresonance frequency can be rewritten as(27a)zju=2sinπk1l2xj3Gcosπk1l/xj+2+3θρl2k1=1,2,…,j−1(27b)zju=2sinπ2k2−1l4NPl−xj3Gcosπ2k2−1l/2NPl−xj+2+3θρl2k2=1,2,…,NP−j.As shown in equation (27), the expression of antiresonance frequency contains the information of the location of the driving point. Therefore, the sensitivity of the antiresonance frequency zj with respect to the location of driving point j can be derived as follows:(28a)∂zj∂xj=−3Gρ3πk11+θxj2cosπk1l/xj+2+3θ3/2cosπk1l2xjk1=1,2,…,j−1,(28b)∂zj∂xj=3Gρ3π2k2−11+θ4NPl−xj2cosπ2k2−1l/2NPl−xj+2+3θ3/2cosπ2k2−1l4NPl−xjk2=1,2,…,NP−j.The antiresonance frequency with the larger sensitivity coefficient∂zj/∂xj is expected to be much more sensitive to the location of driving point j. However, the sensitivity coefficient ∂zj/∂xj is closely related to the order of antiresonance frequency zj, so the antiresonance frequencies of different orders cannot be directly fused together for the selection of the optimal driving point and damage identification. Therefore, the relative sensitivity ∂z¯j/∂xj is proposed, defined by(29)∂z¯j∂xj=1zj∂zj∂xj,∂z¯j/∂xj represents the sensitivity of the change rate of antiresonance frequency to the location of driving point, which can be used to eliminate the influence of the different orders.In the evaluation of thej-th node contributions in the sensitivities of the antiresonances, the sensitivity coefficient ∂z¯j/∂xj is normalised.(30)cj,n=∂z¯j,n/∂xj∑j=1NP∂z¯j,n/∂xj.Assuming that only the firstNm order antiresonance frequencies of each driving point are available, the vector containing the sensitivity coefficients cj,n can be formed as Cj=cj,1,cj,2,…,cj,Nm. The overall sensitivity of antiresonance frequency of point j can be expressed as(31)C¯j=∑n=1Nmcj,n.With the indexC¯j, the antiresonance frequency sensitivities involving the much higher value are expected to be optimal for damage identification.


## 3.1. Sensitivity of Driving Point Antiresonance

When the shear modulus of theλ-th (λ = 1, 2, …, NP) cell in the periodic structure changes slightly, the first-order Taylor’s expansion of the antiresonance frequency of driving point j is(20)zjG+ΔGλ=zjG+∂zj∂GλΔGλ,where the first-order partial derivate ∂zj/∂Gλ is the sensitivity coefficient of the antiresonance frequency with respect to the shear modulus of the λ-th (λ = 1, 2, …, NP) cell.According to the characteristic equation of the driving point antiresonance frequencies, i.e., equation (16), the sensitivity coefficient of the antiresonance frequency with respect to the shear modulus of the λ-th cell is as follows:(21)∂zj∂Gλ=−Γzj,Gλ/∂GλΓzj,Gλ/∂zj,where the function Γzj,Gλ is set according to equation (16), i.e.,(22a)Γzj,Gλ=coshNP−jμzdLLλ2−dLRλ2dLR2sinhλ−1μzsinhj−λμzsinh2μz−dLLλdLRsinhj−1μzsinhμz+coshλ−1μzcoshj−λμz,λ=1,2,...,j,(22b)Γzj,Gλ=sinhjμzdLLλ2−dLRλ2dLR2sinhλ−j−1μzsinhNP−λμzsinh2μz−dLLλdLRcoshNP−j−1μzsinhμz+sinhNP−λμzcoshλ−j−1μz,λ=j+1,j+2,...,NP.The first partial derivatives∂Γzj,Gλ/∂Gλ and ∂Γzj,Gλ/∂zj in equation (21) can be obtained from (22), which contains the information about the location of the different cells.(23a)∂Γzj,Gλ∂zj=NP−jsinhNP−jμzsinhjμz+jcoshNP−jμzcoshjμzsinhμz∂μz∂zj,(23b)∂Γzj,Gλ∂Gλ=−coshNP−jμzdLRsinh2μz2coshμz∂dLLλ∂Gλ+∂dLRλ∂Gλsinhj−λμzsinhλ−1μz+∂dLLλ∂Gλsinhμzsinhj−1μzλ=1,2,…,j,(23c)∂Γzj,Gλ∂Gλ=−sinhjμzdLRsinh2μz2coshμz∂dLLλ∂Gλ+∂dLRλ∂Gλsinhλ−j−1μzsinhNP−λμz+∂dLLλ∂GλsinhμzcoshNP−j−1μz.λ=j+1,j+2,...,NP.The detailed expressions of∂dLLλ/∂Gλ, ∂dLRλ/∂Gλ, and ∂μz/∂zj in equation (23) can be obtained from equations (3) and (18):(24a)∂dLLλ∂Gλ=Al,(24b)∂dLRλ∂Gλ=−Al,(24c)∂μz∂zj=2+3θ+coshμzρAlzj3dLRsinhμz,where l is the thickness of each layer, A is the cross-sectional area, and θ is the mass ratio of steel to rubber.Combining equations (19), (21), (23), and (24) yields the sensitivity coefficient ∂zj/∂Gλ.(25a)∂zj∂Gλ=2j3cosγu+2+3φGρl2sinγu2cos22λ−12γu,λ=1,2,…,j,0,λ=j+1,j+2,…,NP.γu=γLu=πk1j,k1=1,2,…,j−1,(25b)∂zj∂Gλ=0,λ=1,2,…,j2NP−j3cosγu+2+3θGρl2sinγu2cos22λ−2j−12γu,λ=j+1,j+2,…,NPγu=γRu=π22k2−1NP−j,k2=1,2,…,NP−j.As shown in equation (25), the sensitivity coefficient ∂zj/∂Gλ is dependent on the total number of periodic cells NP, the driving point j, the location of the cell λ, the propagation constant corresponding to the antiresonance frequency γu, the mass ratio θ , the shear modulus G, the thickness l, and the density ρ of the rubber layer. Each cell λ has a specific sensitivity coefficient ∂zj/∂Gλ that can be used for damage identification.


## 3.2. Selection of Optimal Driving Points of Antiresonance for Damage Identification

Given that a structure comprises more than one driving point, the amount of data of antiresonance frequencies is larger than the amount of data of natural frequencies. Besides, in the real practice of realising damage identification, sufficient structural information can be obtained by selecting a small number of driving points. Therefore, the optimal driving point should be selected.In this section, the optimal driving points are selected by conducting a sensitivity analysis of the antiresonance frequency on the location of driving pointj. Supposing the thickness of the steel shim is neglected, the length of each periodic cell denoted by l is equal to the thickness of the rubber layer. Thus, the location of driving point j can be expressed as(26)xj=jlj=1,2,...,NP.By substituting equation (26) into (19), the antiresonance frequency can be rewritten as(27a)zju=2sinπk1l2xj3Gcosπk1l/xj+2+3θρl2k1=1,2,…,j−1(27b)zju=2sinπ2k2−1l4NPl−xj3Gcosπ2k2−1l/2NPl−xj+2+3θρl2k2=1,2,…,NP−j.As shown in equation (27), the expression of antiresonance frequency contains the information of the location of the driving point. Therefore, the sensitivity of the antiresonance frequency zj with respect to the location of driving point j can be derived as follows:(28a)∂zj∂xj=−3Gρ3πk11+θxj2cosπk1l/xj+2+3θ3/2cosπk1l2xjk1=1,2,…,j−1,(28b)∂zj∂xj=3Gρ3π2k2−11+θ4NPl−xj2cosπ2k2−1l/2NPl−xj+2+3θ3/2cosπ2k2−1l4NPl−xjk2=1,2,…,NP−j.The antiresonance frequency with the larger sensitivity coefficient∂zj/∂xj is expected to be much more sensitive to the location of driving point j. However, the sensitivity coefficient ∂zj/∂xj is closely related to the order of antiresonance frequency zj, so the antiresonance frequencies of different orders cannot be directly fused together for the selection of the optimal driving point and damage identification. Therefore, the relative sensitivity ∂z¯j/∂xj is proposed, defined by(29)∂z¯j∂xj=1zj∂zj∂xj,∂z¯j/∂xj represents the sensitivity of the change rate of antiresonance frequency to the location of driving point, which can be used to eliminate the influence of the different orders.In the evaluation of thej-th node contributions in the sensitivities of the antiresonances, the sensitivity coefficient ∂z¯j/∂xj is normalised.(30)cj,n=∂z¯j,n/∂xj∑j=1NP∂z¯j,n/∂xj.Assuming that only the firstNm order antiresonance frequencies of each driving point are available, the vector containing the sensitivity coefficients cj,n can be formed as Cj=cj,1,cj,2,…,cj,Nm. The overall sensitivity of antiresonance frequency of point j can be expressed as(31)C¯j=∑n=1Nmcj,n.With the indexC¯j, the antiresonance frequency sensitivities involving the much higher value are expected to be optimal for damage identification.


## 4. Sensitivity-Based Damage Detection

When the ideal periodicity is disrupted due to the damage, the laminated rubber bearing becomes a quasiperiodic structure. The changes in then-th antiresonance frequency of the j-th node before and after the damage can be expressed as(32)Δzj,n=zj,nd−zj,nu,where zj,nd and zj,nu are the n-th antiresonance frequencies of the j-th node in the damaged and undamaged statuses, respectively.The damage of the shear modulus is considered in this study. Take as an example the case in which the driving point is at thej-th node.(33)∂zj,1∂G1ΔG1+∂zj,1∂G2ΔG2+⋯+∂zj,1∂GNPΔGNP≅Δzj,1∂zj,2∂G1ΔG1+∂zj,2∂G2ΔG2+⋯+∂zj,2∂GNPΔGNP≅Δzj,2⋮∂zj,Nm∂G1ΔG1+∂zj,Nm∂G2ΔG2+⋯+∂zj,Nm∂GNPΔGNP≅Δzj,Nm.Suppose the number of driving points selected isND. Thus, the total number of available changes in the antiresonance frequency is q=ND×Nm. Equation (33) is applied to all driving points, and a matrix form is written as(34)SΔG=Δz,where S∈Rq×NP is the sensitivity matrix of the antiresonance frequencies with respect to the shear modulus. Furthermore, Δz∈Rq×1 is the vector consisting of the change in measured antiresonance frequencies, and ΔG=ΔG1,ΔG2,…,ΔGNPT∈RNP×1 is the parameter vectors to be identified.In practical engineering structures, the damage types of the isolation bearings are varied. As the ageing of rubber material leads to the increase in the shear modulus, the inequality constraintΔG≥0 is introduced. The optimality criterion used in [17, 18] is adopted to solve equation (34).Thus, by solving the nonnegative least-square curve fitting problem [36], the optimal solution of the change in shear modulus of each cell can be obtained.(35)minΔG≥012SΔG−Δz2.It should be pointed out that the damage usually only occurs in a small part of the entire structure, which means the damage vectorΔG is sparse. It guarantees the uniqueness of the solutions to underdetermined linear systems of equations (37).


## 5. Analytical Examples

A rubber-steel periodic structure with one fixed end and one free end is employed to demonstrate the feasibility of the proposed method in the damage identification of a laminated rubber bearing. As shown in Figure1, the rubber-steel quasiperiodic structure consists of NP cells, in which the cells are numbered starting at 1 from the bottom cell to the top one. The structural parameters are as follows: the thickness of each rubber layer is l = 0.01 m, the cross-sectional diameter is φ = 0.8 m, the shear modulus is G = 1 × 106 N/m2, the density is ρ = 1000 kg/m3, the mass ratio of steel to rubber is θ = 2, and the total number of cells is NP = 10. The details are listed in Table 1.Table 1
The geometric and material parameters of periodic rubber–steel structure.
ParametersValuesStatic shear modulus (G)1 × 106 N/m2Cross section diameter of the rubber layer (φ)0.8 mDensity of the rubber material (ρ)1000 kg/m3Thickness of each rubber layer (l)0.01 mMass ratio of steel to rubber (θ)2The total number of cells (NP)10The method is tested for its ability to detect damage by using five cases. The detailed damage configuration is shown in Table2. The cases simulate the increase in shear modulus of the rubber layer due to the ageing of the rubber material. Cases 1 and Case 2 are single damage conditions in which the corresponding shear moduli of cells 1 and 10 are increased by 5%. Case 3 has two damaged cells, which shows the condition in which the shear moduli of cells 1 and 10 are increased by 5% simultaneously. Cases 4 and 5 have three damaged cells. Case 4 shows the condition in which the shear moduli of cells 2, 7, and 10 are increased by 8%, 2%, and 5%, respectively. In Case 5, the damage appears in cells 2, 4, and 6, and the damage magnitudes of the three cells are all increased by 5%.Table 2
Damage cases for the 10-cell periodic rubber–steel structure.
CasesCase 1Case 2Case 3Case 4Case 5Damage cell (k)1101, 102, 7, 102, 4, 6Increase in shear modulus (ΔG/G)/%555, 58, 2, 55, 5, 5The optimal driving points are selected according to the selection criteria in accordance with the more sensitive driving points proposed in Section3.2. The specific process is as follows. Firstly, by substituting the geometric and material parameters into equation (19), the n-th antiresonance frequencies of the j-th node in the healthy status, which is given by zj,nu (j = 1, 2, …, NP; n = 1, 2, ..., Nm), can be calculated. Assume that only the first four antiresonance frequencies of each driving point are used similar to that in a real situation in which only limited measurement information is available, i.e., Nm = 4. The first four antiresonance frequencies of the different driving points are listed in Table 3. Secondly, the sensitivity coefficients of the antiresonance frequencies with respect to the location of driving points are calculated using equation. (28), as depicted in Figure 3(a). Finally, the overall sensitivity of the antiresonance frequencies of driving point j is obtained using equation (31). As shown in Figure 3(c), the three most sensitive points are the 6th, 5th, and 7th nodes, whereas the two least sensitive points are the 9th and 10th node.Table 3
The first four antiresonance frequencies of different driving points (j = 1, 2, …, 10) for undamaged 10-cell periodic rubber–steel structure.
Order (n)z1,nuz2,nuz3,nuz4,nuz5,nuz6,nuz7,nuz8,nuz9,nuz10,nu1318.5167358.2907409.4071477.5191572.7775715.4009817.0391715.4009636.2086572.77752952.18731070.03561220.89732320.48511140.5367952.1873952.18732102.02591265.43681140.536731574.75131765.43901878.67292738.61281697.05841878.67291618.53642738.61281878.67291697.058442175.43602427.57322006.84873123.58742233.74962102.02592381.86693300.16762462.91912233.7496Figure 3
(a) The sensitivity coefficients∂zj,n/∂xjj=1,2,…,NP;n=1,2,…,Nm for Nm = 4 and NP = 10. (b) The normalised sensitivity coefficients cj,nj=1,2,…,NP;n=1,2,…,Nm. (c) The overall sensitivity C¯j of driving point jj=1,2,...,NP.
(a)(b)(c)Figure 4
The procedure of damage detection in laminated rubber bearing.From the calculation of the overall sensitivity of the driving points, five schemes are formulated for driving point selection in damage identification, as shown in Table4. In Schemes 1 and 2, the 6th and 5th nodes are selected as the driving points, respectively. In Scheme 3, the 6th and 5th nodes are simultaneously selected as the driving points. In Scheme 4, the 6th, 5th, and 7th nodes are selected as the driving points to investigate the influence of the size of the “optimal driving point” dataset on the damage identification. Scheme 5, which consists of the “worst two” driving points (9th and 10th nodes), is also investigated.Table 4
Selection of driving points.
Schemes(1)(2)(3)(4)(5)Driving points655, 65, 6, 79, 10Take Case 1 as an example for demonstrating the process of calculation when the 1st node is used as the only driving point, i.e.,j = 6 and ND = 1. And the procedure of damage detection is illustrated in Figure 4. Firstly, by substituting the geometric and material parameters into equation (25), the sensitivity coefficients of antiresonance frequencies with respect to the shear modulus are calculated. As shown in Figure 5, the sensitivity coefficients of the element in the substructure on one side of the driving point are equal to zero. The sensitivity of the driving point antiresonance frequency represents the sensitivity of the natural frequency of the substructure on either side of the driving point. Consequently, the sensitivity of the antiresonance frequency to the element in another substructure equals zero.Figure 5
The sensitivity coefficients∂zj,n/∂Gλj=1,2,…,NP;n=1,2,…,Nm;λ=1,2,…,NP. (a) j = 5. (b) j = 6. (c) j = 7. (d) j = 9. (e) j = 10.
(a)(b)(c)(d)(e)Secondly, by substituting the geometric and material parameters for Case 1 (i.e.,ΔG1=5%G) into equation (8), the antiresonance frequencies in the damaged status zj,nd are calculated. Then, the changes in antiresonance frequencies before and after the damage are computed according to equation (32). Finally, by using the change in antiresonance frequencies and the sensitivity matrix, the damage is identified by solving the optimisation problem of equation (35). The results of the damage identification are expressed using the change rate of the shear modulus, which is defined as ΔG/G. Subsequently, Case 1 is detected using the other four schemes of the driving point selection. As shown in Figure 6, when only the 5th or 6th node is used, the damage in cell 1 is misidentified due to the symmetry. In other words, the sensitivity of antiresonance frequency of the 5th node to cell 1 is equal to that of cell 5, and the sensitivity of antiresonance frequency of the 6th node to cell 1 is equal to that of cell 6, as shown in Figure 5(a) and 5(b). These findings indicate that the antiresonance frequencies of a single driving point cannot be used for damage identification even if it is the optimal driving point. The results of damage identification, which uses the “best two” and “best three” driving points, suggest that the location and magnitude of the damage can be both predicted reasonably. Therefore, more than one driving point should be selected for damage identification.Figure 6
Detection of Case 1 using Schemes 1–4.From the aforementioned analyses, Schemes 3 and 4 are selected for damage identification. Scheme 5, which consists of the “worst two” driving points, is also studied to verify the proposed optimal driving point selection criteria. In addition, the damage detection result of the proposed method that uses antiresonance frequencies is compared with that of the scheme that uses natural frequencies (the first five orders). In this study, only the first five natural frequencies are used to simulate the situation in which only the fewer-order modes can be measured in practical application [17]. The accuracy of the identification results is measured using the root mean square error (RMSE), which is normalised by the initial parameter value, RMSE=1/NP∑λ=1NPΔGexpectedλ−ΔGactualλ/G2λ=1,2,...,NP [29]. Table 5 shows the RMSE of the identification results of each damage identification scheme under different damage conditions. The process is repeated to detect a single damage in the 10th cell. As shown in Figures 7(1) and 7(2), both of the antiresonance frequencies extracted from more than one driving point and natural frequencies offer good predictions of the single damage, with the RMSE approaching zero. The small deterioration in terms of accuracy of the damage magnitude prediction can be attributed to the sensitivity coefficients being deduced around the healthy structure.Table 5
RMSE (%) of the 5 cases.
Case 1Case 2Case 3Case 4Case 5Scheme 30.04250.10600.11990.16740.1126Scheme 40.04310.10790.12280.16640.1082Scheme 50.04820.04950.06380.18452.2470Natural frequencies0.05080.12261.68461.77710.1114Figure 7
Detection of Cases 1–5. Actual damage;  identification result (Scheme 3);  identification result (Scheme 4);  identification result (Scheme 5);  identification result (natural frequencies).
(a)(b)(c)(d)(e)Then, a detection of the two damaged cells is conducted on the basis of the abovementioned two single damage cases. The detection result of Case 3 is shown in Figure7(3). The damage in cells 1 and 10 can be identified using the antiresonance frequencies of more than one driving point. When natural frequencies are used, the damage in cell 10 cannot be identified, and the cell is even incorrectly identified as cell 9. The corresponding RMSE approaches a value as high as 1.7%.Subsequently, the three cells in the structure are utilised for damage detection. In Case 4, the damage types in cells 2, 7, and 10 can be identified using the antiresonance frequencies of more than one driving point. When natural frequencies are used, the damage in cell 10 cannot be identified, and the cell is even incorrectly identified as cell 1, as shown in Figure7(4). In Case 5, the damage types in cells 2, 4, and 6 can be predicted reasonably using either the antiresonance frequencies of more than one optimal driving point or the natural frequencies. When the “worst two” driving points are used, the damaged cells cannot be identified. The RMSE of the damage identification results entailing the “worst two” driving points selected by the proposed criterion is much greater than that involving the “best two” driving points, as shown in Table 5. On the one hand, as shown in Figure 8, the performance of the antiresonance frequencies of the “best two” driving points selected by the proposed criterion is better than that of the natural frequencies. On the other hand, the performance of the antiresonance frequencies of the “best three” driving points selected by the proposed criterion is less superior to that of the “best two” driving points. The analytic results demonstrate that the “best two” driving points can provide sufficient information for identification.Figure 8
Root mean square errors (RMSE) of the detection results for different cases.


## 6. Conclusions

This study presented an effective method of damage detection for rubber bearings based on the antiresonance frequencies of a periodic structure and its sensitivity analysis. The laminated rubber bearing was simplified as a monocoupled rubber-steel periodic structure. The antiresonance frequency of the monocoupled periodic structure was derived using the dynamic stiffness method combined with the periodicity of the structure. The optimal driving points were selected on the basis of the sensitivity analysis of the antiresonance frequency on the location of the driving point. The sensitivity of the antiresonance frequency to the damage scaling parameter was derived from the antiresonance frequency characteristic equation. The expressions of antiresonance frequency as well as its sensitivities are explicit, which have application to engineers and designers. Then, the changes in antiresonance frequencies were measured. The damage was identified by solving the sensitivity identification equation by means of the numerical optimisation method.The numerical case studies were performed under different damage scenarios. The identification results showed that the antiresonance frequencies can be used to identify the shear modulus damage of different rubber layers, and the identification result entailing antiresonance frequencies is better than that involving natural frequencies. The driving point antiresonance frequencies selected by the optimal driving points selection criterion can always produce better identification results. Moreover, this study demonstrated that the “best two” driving points can provide sufficient information for damage identification. It should be emphasized that compared with the most of laminated rubber bearings used in the practical engineering structures, the periodic system model considered in this paper is relatively simple. For example, the bending deformation of the rubber layer is neglected. Therefore, the performance of the approach for multicoupled periodic systems with complex boundary conditions needs to be investigated. Besides, there exists another matter of how to consider dampness and nonlinearity of rubber materials.



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