# On the Nonlinear Seismic Responses of Shock Absorber-Equipped Porcelain Electrical Components

Authors: Zhubing Zhu; Lingxin Zhang; Yongfeng Cheng; Hulun Guo; Zhicheng Lu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Porcelain electrical equipment is prone to brittle failure due to resonance under seismic effects. To improve its seismic resistance, some researchers have conducted research on shock absorption technology for porcelain electrical equipment. However, extant research fails to provide a detailed and systematic study of the effect of the nonlinear characteristics of these shock absorbers on the performance of equipment under seismic effects. This paper provides a theoretical analysis, verified by shaking table testing, of the performance of a 1000 kV pillar-type porcelain lightning arrester. The Bouc–Wen model is fitted to the force-displacement curve of hysteretic nonlinear metal shock absorbers, and a dynamic model of, and equations for, pillar-type porcelain electrical components are derived, taking into account their nonlinear characteristics. This reveals the influence of the nonlinear characteristics of shock absorbers on the nonlinear seismic response characteristics of these components. Our results indicate that the seismic responses of pillar-type porcelain components can be effectively suppressed by hysteretic nonlinear shock absorbers and that the greater the intensity of the seismic waves, the more obvious the efficiency of shock absorption. However, as the installation radius and yield force of the installed shock absorbers increase, their shock absorption efficiency gradually decreases.


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## Body

## 1. Introduction

As voltage classes have improved and the volume and mass of substation electrical equipment have been optimised, the increasing complexity of its structural systems has been paralleled by increases in its “height, size, flexibility, and weight” characteristics, exacerbating its seismic vulnerability and increasing risks of damage during earthquakes [1]. The natural frequency of pillar-type porcelain electrical components, especially ultra-high-voltage electrical equipment, is relatively low, mostly ranging from 1 to 2 Hz [2]. As this falls within the predominant frequency range of seismic waves, such equipment is vulnerable to large seismic forces. In addition, such equipment is made of brittle porcelain material, with poor energy dissipation capability, making it susceptible to severe damage under seismic loading [3–6]. Previous earthquake events have revealed that pillar-type porcelain electrical equipment is prone to brittle failure due to resonance under seismic loading [7–11]. Predominant failure modes include bushing root fracture and equipment dislocation at flange joints due to flange cement failure. An additional mode of earthquake damage reflects tensile stresses affecting interconnecting cables [12–15]. Figure 1 depicts typical seismic loading-induced equipment damage: the upper component is a porcelain insulator and the lower one is a frame. Engineering applications generally use lattice and cylindrical steel frames, or cylindrical concrete frames, with lattice steel frames generally adopted for equipment of voltage classes in excess of 500 kV. In addition, pillar-type porcelain insulators are sometimes directly mounted on a firewall or foundation, without a supporting frame.Figure 1
Seismic damage to pillar-type porcelain insulators.Research on shock absorption technology to improve the seismic capacity of pillar-type porcelain insulators has been carried out in China and abroad [16]. China Electric Power Research Institute Co., Ltd. developed a shear-type shock absorber for electrical equipment, which is shown in Figure 2. These are designed to be installed between insulator equipment and their frame, via connecting bolt holes on the flange at the bottom of each equipment. Under normal conditions, these devices act as bolts connecting the insulators with their frames. For equipment that does not need frame in actual engineering, shock absorbers are installed between firewall and foundation. Under seismic loading, each shock absorber’s jacket moves up and down relative to its central shaft, and an internal lead alloy element undergoes shear deformation, dissipating seismic energy to achieve shock absorption. Siemens has developed a viscoelastic damper for installation between electrical equipment frames and foundations, as shown in Figure 3. During earthquakes, this damper’s piston rod is displaced within it, displacing hydraulic oil for a damping effect. Toshiba has developed a centripetal shock absorber suitable for pillar components, as shown in Figure 4. This device is installed between the electrical equipment and its frame, dissipating earthquake energy via compressive deformation of internal discs. The inner part of the centripetal shock absorber is vertically arranged and combined by discs, which can only be pressed and cannot be pulled. Therefore, the device needs to be installed in two layers; two centripetal shock absorbers in the upper layer and the lower layer at the same position are connected in series via a bolt bar to ensure the shock absorber can consume earthquake energy in tension and compression state.Figure 2
China Electric Power Research Institute’s shear-type shock absorber and installation details.Figure 3
Siemens’ viscoelastic damper and installation details.Figure 4
Toshiba’s (Japan) centripetal shock absorber and installation details.To date, numerous scholars have undertaken research into the seismic properties of pillar-type porcelain insulators and nonlinear theoretical analysis of related shock absorber-equipped structural systems. Cheng Yongfeng et al. studied the nonlinear responses of shock absorber-equipped pillar-type porcelain insulators under harmonic excitation [17]. Cheng Yongfeng et al. studied the seismic performance of 1000 kV lightning arresters and transformers connected by rigid tubular busbar through shaking table testing [18]. Ye Lieping et al. proposed a nonlinear seismic analysis model for concrete structure, capable of accurately simulating their nonlinear responses to seismic effects [19]. Jean-Bemard Dastous et al. conducted both static and dynamic tests of fixture wiring in order to assess its influence on the seismic performance of electrical equipment. They determined that wiring forms a nonlinear system during earthquakes and that designs retaining a certain degree of relaxation in wiring proved helpful in reducing the nonlinear coupling of interconnected equipment [20–22]. Robyn et al. used nonlinear frictional contact elements to simulate shock absorbers and performed nonlinear time history analysis on structures equipped with these damping devices. Through comparison of multiple schemes, the accuracy of their analysis was verified, providing a basis for the development of nonlinear shock absorber designs [23].Under seismic loading of shock absorber-equipped pillar-type porcelain insulator component systems, the shock absorbers are the components with most obvious nonlinear characteristics. These dissipate seismic energy in a nonlinear fashion to reduce the insulators’ seismic response. However, while scholars have carried out research on the seismic performance of pillar-type porcelain insulators, as yet this fails to provide a detailed, systematic study of the effects of shock absorbers’ nonlinear characteristics on the performance of porcelain insulators under seismic effects. And when pillar-type porcelain electrical components are equipped with shock absorbers with nonlinear characteristics, the structural system as a whole becomes more complex. The application of nonlinear theory to the seismic analysis of these structural systems can thus make a contribution toward raising the level of seismic analysis and of seismic design theory.


## 2. Dynamic Model and Equations

For this paper, nonlinear dynamic modeling of a 1000 kV pillar-type porcelain lightning arrester with shear-type shock absorbers without frames was conducted, and its dynamic equations were solved, in order to analyze the nonlinear seismic responses of the pillar-type porcelain components. Figure5 provides a schematic diagram of these lightning arresters. They are composed of four pillar-type porcelain elements connected by flanges. These elements are numbered 1–4 from top to bottom. The main structural parameters of the equipment are shown in Table 1, where the elastic modulus of their porcelain material is 1.1 × 1011 N/m2 and the weights of the top equalizing ring and other accessories (joints with electrodes and components) are 290 kg and 1801 kg, respectively. According to the shock-absorption design scheme, 20 shear-type shock absorbers are installed at the bottom of the lightning arrester.Figure 5
Schematic diagram of lightning arrester.Table 1
Lightning arrester structural parameters.
Element no.1234Weight (kg)940121213491571Length (m)2.6622.6622.6622.662Inner diameter (m)0.380.450.550.60Outer diameter (m)0.450.550.650.72


### 2.1. Mechanical and Dynamic Model of the Shock Absorber

The dimensions of the lead alloy in the shock absorber are selected according to the expected seismic intensity, the required damping efficiency, and the number of bolt holes available for installation of these devices. The “restoring force-displacement” test curve of the single shock absorber used in this paper is shown in Figure6. This figure shows obvious hysteretic nonlinearity. Shock absorber’s design parameters are as follows: the yield force is 20 kN, the rigidity before yield is 80 kN/mm, and the equivalent damping ratio is 35%. The commonly used nonlinear models are elastic-plastic mechanical model, viscoelastic mechanical model, and viscous restoring force model. Whether it is an elastic-plastic mechanical model or a viscoelastic mechanical model, there is a sudden change of stiffness at the model’s four corners, which is not consistent with the actual situation. Therefore, a smooth curve should be adopted to fit the restoring force-displacement curve of the damping device. The Bouc–Wen model, one of the viscous restoring force models, which has been widely adopted in engineering, can be well fit to this type of nonlinear hysteretic curve and was therefore used to model the shock absorber.Figure 6
“Restoring force-displacement” curve of shock absorber.A typical Bouc–Wen model can be described by the following equations:(1)F=kLξ+kNZ,Z˙=λ−γsignξ˙Z+βZnξ˙,where F represents the elastic force, ξ represents the displacement, ξ˙ represents the displacement velocity, and γ is a parameter related to the yield force of the model.The parameters of the Bouc–Wen model were determined by comparison of results from the theoretical model and from testing. Figure7 shows a comparison between the Bouc–Wen model estimates and test results for the shock absorber’s force-displacement curve. The Bouc–Wen model uses the following parameters: kL = 1 × 106 N/m, kN = 120 × 106 N/m, λ = 1, γ = 5000, β = 1, and n = 1.Figure 7
Shock absorber force-displacement curve: Bouc–Wen model estimates and test results.In Figure7, the black lines indicate the test results and the colored line represents estimates derived using the calibrated Bouc–Wen model. The two are in good agreement. In addition, the upper half of the Bouc–Wen model’s curve is the result when θ˙>0, while the lower half is the result when θ˙<0l; ξ is a function of θ.We assume thatN shock absorbers are uniformly distributed at the bottom of the porcelain pillar, each separated from the next by an angle of 2π/N, as shown in Figure 8(a). A mechanical model of the shock absorber joints is shown in Figure 8(b).Figure 8
(a) Distribution of shock absorbers at the base of porcelain pillar. (b) Mechanical model of shock absorber joints.
(a)(b)The connection between the shock absorber and the cement part of the flange at the base of the equipment is equivalent to a series spring [17]. The angle of the shock absorbers’ surface of action is φ1, and the spring angle relative to the cement part of the flange at the base of the equipment is φ2. The angle of the first porcelain pillar is thus θ1=φ1+φ2.The elastic force experienced by the shock absorber can be written as follows:(2)Fi=kLξi+kNZi,where i = 1, 2, 3, ..., N. This system is also governed by several other equations as follows:(3)Z˙i=λ−γsignZiξ˙i+βZinξ˙i,ξi=ξ0+φ1Rsinαi,where ξ0 is the static deformation and αi = (i − 1) 2π/N. Denoting the spring stiffness of the flange at the bottom of the first porcelain pillar as k1, this can be expressed as follows:(4)k1φ2=∑i=1NFiRsinαi=Kθ1,where K is the equivalent stiffness. By combining θ1=φ1+φ2 with equation (4), the following equation can be derived:(5)θ1=d1φ1+d2∑i=1NZisinαi,where d1=1+kLR2/k1∑i=1Nsin2αi and d2=kNR/k1.


### 2.2. Calculation of Flange Joint Bending Stiffness

The pillar-type porcelain components are connected by flanges, with one flange at the upper and lower end of each section. Stiffness is generated at both the cement flanges and porcelain bushings, as shown in Figure9.Figure 9
Flange joint.The upper and lower flanges generate total stiffness equivalent to the stiffness of two springs in series. If the stiffness of the flange at the bottom of the upper porcelain pillar is denoted bykid, and the stiffness of the flange at the top of the lower porcelain pillar is denoted by k(i−1)u, the equivalent stiffness, ki, of the two sections of the porcelain pillar can be expressed as follows:(6)ki=kidki−1ukid+ki−1u.The bending stiffness of upper and lower flanges can be calculated according to the equation specified in theTechnical Specification for Seismic Design of Ultra-High-Voltage Porcelain Insulating Electrical Equipment and Installation/Maintenance of Energy Dissipation Devices (Q/GDW 11132-2013) published by the China Electric Power Research Institute [24]:(7)kid=βdddhd2td,kiu=βuduhu2tu,where du and dd are the outer diameters of the cement elements between the porcelain bushing and the upper and lower flanges, respectively; hu and hd are the heights of the cement between the porcelain bushing and the upper and lower flanges, respectively; tu and td are the clearance distances between the porcelain bushings and the upper and lower flanges, respectively; and βu and βd are the bending stiffness coefficients at the connections between the porcelain bushing and the upper and lower flanges, respectively, the value of which is 6.54 × 107 when the outer diameter of the porcelain bushing at the cement element is smaller than 275 mm and is 5.0 × 107 when the outer diameter of the porcelain bushing at the cement element is larger than 275 mm, derived by linear interpolation when the outer diameter of the porcelain bushing at the cement element lies between 275 and 375 mm.


### 2.3. Dynamic Equation of Shock Absorber and Pillar-Type Porcelain Component System

Taking the 1000 kV pillar-type porcelain lightning arrester as an example, the flange at the bottom of the first pillar-type porcelain component is connected in series with the shock absorber and then consolidated. The elastic potential energy of the shock absorber and the flange joint at the bottom of the first porcelain pillar can now be written as follows:(8)UA=12Kθ12.Differentiating this is equivalent to the following:(9)δUA=Kθ1δθ1.If there areNc porcelain pillar components, there will also be Nc flange joints, and the total elastic potential energy at these flange joints can be calculated as follows:(10)Uk=12k1θ12+∑i=2Nc12kiθi−θi−12.The rotation of the porcelain pillars also causes changes in their gravitational potential energy, and their total gravitational potential energy is as follows:(11)Ug=−12∑i=1Ncmig∑j=1iLjθj2−12Liθi2,where Li and Lj are the lengths of the ith and jth porcelain pillars, respectively.The total potential energy of the system comprising the porcelain pillars and shock absorbers can be calculated as follows:(12)U=Uk+Ug+UA.Figure10 is a schematic diagram of the velocity relationships for the ith porcelain pillar. The dynamic energy of the porcelain pillar includes translational and rotational dynamic energy, described by the following equation:(13)T=12∑i=1Ncmivc,i2+12∑i=1Nc112miLi2θ˙i2,where vc,i is the velocity of the centroid of the ith porcelain pillar.Figure 10
Velocity relationships of theith porcelain pillar.Figure10 depicts the dynamics of a porcelain pillar, again taking a 1000 kV pillar-type porcelain lightning arrester as an example. The velocity of the flange at the bottom of the first porcelain pillar can be expressed as follows:(14)v0=x˙b,where xb is seismic effect-induced ground displacement.The velocity of the centroid of the first porcelain pillar can then be calculated as follows:(15)vc,1=v0+12L1θ˙1.The velocity of the second porcelain pillar’s centroid is then(16)vc,2=v0+L1θ˙1+12L2θ˙2.The velocity of the third porcelain pillar’s centroid can be written as(17)vc,3=v0+L1θ˙1+L2θ˙2+12L3θ˙3.And the velocity of the fourth porcelain pillar’s centroid can be expressed as(18)vc,4=v0+L1θ˙1+L2θ˙2+L3θ˙3+12L4θ˙4.The system’s nonlinear dynamic equation can be derived using Hamilton’s principle. Variation in the system’s kinetic energy can be expressed as follows:(19)δT=−∑i=1Ncmix¨b+∑j=1iLjθ¨j−12Liθ¨iδ∑j=1iLjθj−12Liθi+112miLi2θ¨iδθi.The change in its potential energy can be written as(20)δU=k1θ1δθ1−∑i=1Ncmig∑j=1iLjθjδθj−12Liθiδθi+∑i=2Nckiθi−θi−1δθi−θi−1.The dynamic and potential energy of the system can now be substituted into Hamilton’s variational equation:(21)∫t1t2δT−δUdt=0.This yields the following dynamic equation for the system:(22)13m1+∑i=24miL12θ¨1+12m2+∑i=34miL1L2θ¨2+12m3+m4L1L3θ¨3+12m4L1L4θ¨4+k1θ1−k2θ2−θ1−12m1+∑i=24migL1θ1=−12m1+∑i=24miL1x¨b,12m2+∑i=34miL1L2θ¨1+13m2+∑i=34miL22θ¨2+12m3+m4L2L3θ¨3+12m4L2L4θ¨4+k2θ2−θ1−k3θ3−θ2−12m2+∑i=34migL2θ2=−12m2+∑i=34miL2x¨b,12m3+m4L1L3θ¨1+12m3+m4L2L3θ¨2+13m3+m4L32θ¨3+12m4L3L4θ¨4+k3θ3−θ2−k4θ4−θ3−12m3+m4gL3θ3=−12m3+m4L3x¨b,12m4L1L4θ¨1+L2L4θ¨2+L3L4θ¨3+13m4L42θ¨4+k4θ4−θ3−12m4gL4θ4=−12m4L4x¨b.This equation can be rewritten in the matrix form as(23)Mθ¨+Kθ+Nθ=Fx¨b,where(24)M=M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44,K=K11−k200−k2K22−k300−k3K33−k400−k4K44,Nθ=−k1φ1000,F=−f12m1+∑i=24miL112m2+∑i=34miL212m3+m4L312m4L4.Several other equations complete the system as follows:(25)θ=θ1θ2θ3θ4T,M11=13m1+∑i=24miL12,M12=M21=12m2+∑i=34miL1L2,M13=M31=12m3+m4L1L3,M14=M41=12m4L1L4,M22=13m2+∑i=34miL22,M23=M32=12m3+m4L2L3,M24=M42=12m4L2L4,M33=13m3+m4L32,M34=M43=12m4L3L4,M44=13m4L42,K11=k1+k2−12m1+∑i=24migL1,K22=k2+k3−12m2+∑i=34migL2,K33=k3+k4−12m3+m4gL3,K44=k4−12m4gL4.Taking the damping of the structure into account, its dynamic equation can be written as follows:(26)Mθ¨+Cθ˙+Kθ+Nθ=Fx¨b,where the damping ratio is C=2ζMΦΛΦ−1, Φ is the modal matrix, and Λ is a diagonal matrix consisting of the natural frequencies. For equipment without a shock absorber, Nθ in the dynamic equation can be omitted.


## 2.1. Mechanical and Dynamic Model of the Shock Absorber

The dimensions of the lead alloy in the shock absorber are selected according to the expected seismic intensity, the required damping efficiency, and the number of bolt holes available for installation of these devices. The “restoring force-displacement” test curve of the single shock absorber used in this paper is shown in Figure6. This figure shows obvious hysteretic nonlinearity. Shock absorber’s design parameters are as follows: the yield force is 20 kN, the rigidity before yield is 80 kN/mm, and the equivalent damping ratio is 35%. The commonly used nonlinear models are elastic-plastic mechanical model, viscoelastic mechanical model, and viscous restoring force model. Whether it is an elastic-plastic mechanical model or a viscoelastic mechanical model, there is a sudden change of stiffness at the model’s four corners, which is not consistent with the actual situation. Therefore, a smooth curve should be adopted to fit the restoring force-displacement curve of the damping device. The Bouc–Wen model, one of the viscous restoring force models, which has been widely adopted in engineering, can be well fit to this type of nonlinear hysteretic curve and was therefore used to model the shock absorber.Figure 6
“Restoring force-displacement” curve of shock absorber.A typical Bouc–Wen model can be described by the following equations:(1)F=kLξ+kNZ,Z˙=λ−γsignξ˙Z+βZnξ˙,where F represents the elastic force, ξ represents the displacement, ξ˙ represents the displacement velocity, and γ is a parameter related to the yield force of the model.The parameters of the Bouc–Wen model were determined by comparison of results from the theoretical model and from testing. Figure7 shows a comparison between the Bouc–Wen model estimates and test results for the shock absorber’s force-displacement curve. The Bouc–Wen model uses the following parameters: kL = 1 × 106 N/m, kN = 120 × 106 N/m, λ = 1, γ = 5000, β = 1, and n = 1.Figure 7
Shock absorber force-displacement curve: Bouc–Wen model estimates and test results.In Figure7, the black lines indicate the test results and the colored line represents estimates derived using the calibrated Bouc–Wen model. The two are in good agreement. In addition, the upper half of the Bouc–Wen model’s curve is the result when θ˙>0, while the lower half is the result when θ˙<0l; ξ is a function of θ.We assume thatN shock absorbers are uniformly distributed at the bottom of the porcelain pillar, each separated from the next by an angle of 2π/N, as shown in Figure 8(a). A mechanical model of the shock absorber joints is shown in Figure 8(b).Figure 8
(a) Distribution of shock absorbers at the base of porcelain pillar. (b) Mechanical model of shock absorber joints.
(a)(b)The connection between the shock absorber and the cement part of the flange at the base of the equipment is equivalent to a series spring [17]. The angle of the shock absorbers’ surface of action is φ1, and the spring angle relative to the cement part of the flange at the base of the equipment is φ2. The angle of the first porcelain pillar is thus θ1=φ1+φ2.The elastic force experienced by the shock absorber can be written as follows:(2)Fi=kLξi+kNZi,where i = 1, 2, 3, ..., N. This system is also governed by several other equations as follows:(3)Z˙i=λ−γsignZiξ˙i+βZinξ˙i,ξi=ξ0+φ1Rsinαi,where ξ0 is the static deformation and αi = (i − 1) 2π/N. Denoting the spring stiffness of the flange at the bottom of the first porcelain pillar as k1, this can be expressed as follows:(4)k1φ2=∑i=1NFiRsinαi=Kθ1,where K is the equivalent stiffness. By combining θ1=φ1+φ2 with equation (4), the following equation can be derived:(5)θ1=d1φ1+d2∑i=1NZisinαi,where d1=1+kLR2/k1∑i=1Nsin2αi and d2=kNR/k1.


## 2.2. Calculation of Flange Joint Bending Stiffness

The pillar-type porcelain components are connected by flanges, with one flange at the upper and lower end of each section. Stiffness is generated at both the cement flanges and porcelain bushings, as shown in Figure9.Figure 9
Flange joint.The upper and lower flanges generate total stiffness equivalent to the stiffness of two springs in series. If the stiffness of the flange at the bottom of the upper porcelain pillar is denoted bykid, and the stiffness of the flange at the top of the lower porcelain pillar is denoted by k(i−1)u, the equivalent stiffness, ki, of the two sections of the porcelain pillar can be expressed as follows:(6)ki=kidki−1ukid+ki−1u.The bending stiffness of upper and lower flanges can be calculated according to the equation specified in theTechnical Specification for Seismic Design of Ultra-High-Voltage Porcelain Insulating Electrical Equipment and Installation/Maintenance of Energy Dissipation Devices (Q/GDW 11132-2013) published by the China Electric Power Research Institute [24]:(7)kid=βdddhd2td,kiu=βuduhu2tu,where du and dd are the outer diameters of the cement elements between the porcelain bushing and the upper and lower flanges, respectively; hu and hd are the heights of the cement between the porcelain bushing and the upper and lower flanges, respectively; tu and td are the clearance distances between the porcelain bushings and the upper and lower flanges, respectively; and βu and βd are the bending stiffness coefficients at the connections between the porcelain bushing and the upper and lower flanges, respectively, the value of which is 6.54 × 107 when the outer diameter of the porcelain bushing at the cement element is smaller than 275 mm and is 5.0 × 107 when the outer diameter of the porcelain bushing at the cement element is larger than 275 mm, derived by linear interpolation when the outer diameter of the porcelain bushing at the cement element lies between 275 and 375 mm.


## 2.3. Dynamic Equation of Shock Absorber and Pillar-Type Porcelain Component System

Taking the 1000 kV pillar-type porcelain lightning arrester as an example, the flange at the bottom of the first pillar-type porcelain component is connected in series with the shock absorber and then consolidated. The elastic potential energy of the shock absorber and the flange joint at the bottom of the first porcelain pillar can now be written as follows:(8)UA=12Kθ12.Differentiating this is equivalent to the following:(9)δUA=Kθ1δθ1.If there areNc porcelain pillar components, there will also be Nc flange joints, and the total elastic potential energy at these flange joints can be calculated as follows:(10)Uk=12k1θ12+∑i=2Nc12kiθi−θi−12.The rotation of the porcelain pillars also causes changes in their gravitational potential energy, and their total gravitational potential energy is as follows:(11)Ug=−12∑i=1Ncmig∑j=1iLjθj2−12Liθi2,where Li and Lj are the lengths of the ith and jth porcelain pillars, respectively.The total potential energy of the system comprising the porcelain pillars and shock absorbers can be calculated as follows:(12)U=Uk+Ug+UA.Figure10 is a schematic diagram of the velocity relationships for the ith porcelain pillar. The dynamic energy of the porcelain pillar includes translational and rotational dynamic energy, described by the following equation:(13)T=12∑i=1Ncmivc,i2+12∑i=1Nc112miLi2θ˙i2,where vc,i is the velocity of the centroid of the ith porcelain pillar.Figure 10
Velocity relationships of theith porcelain pillar.Figure10 depicts the dynamics of a porcelain pillar, again taking a 1000 kV pillar-type porcelain lightning arrester as an example. The velocity of the flange at the bottom of the first porcelain pillar can be expressed as follows:(14)v0=x˙b,where xb is seismic effect-induced ground displacement.The velocity of the centroid of the first porcelain pillar can then be calculated as follows:(15)vc,1=v0+12L1θ˙1.The velocity of the second porcelain pillar’s centroid is then(16)vc,2=v0+L1θ˙1+12L2θ˙2.The velocity of the third porcelain pillar’s centroid can be written as(17)vc,3=v0+L1θ˙1+L2θ˙2+12L3θ˙3.And the velocity of the fourth porcelain pillar’s centroid can be expressed as(18)vc,4=v0+L1θ˙1+L2θ˙2+L3θ˙3+12L4θ˙4.The system’s nonlinear dynamic equation can be derived using Hamilton’s principle. Variation in the system’s kinetic energy can be expressed as follows:(19)δT=−∑i=1Ncmix¨b+∑j=1iLjθ¨j−12Liθ¨iδ∑j=1iLjθj−12Liθi+112miLi2θ¨iδθi.The change in its potential energy can be written as(20)δU=k1θ1δθ1−∑i=1Ncmig∑j=1iLjθjδθj−12Liθiδθi+∑i=2Nckiθi−θi−1δθi−θi−1.The dynamic and potential energy of the system can now be substituted into Hamilton’s variational equation:(21)∫t1t2δT−δUdt=0.This yields the following dynamic equation for the system:(22)13m1+∑i=24miL12θ¨1+12m2+∑i=34miL1L2θ¨2+12m3+m4L1L3θ¨3+12m4L1L4θ¨4+k1θ1−k2θ2−θ1−12m1+∑i=24migL1θ1=−12m1+∑i=24miL1x¨b,12m2+∑i=34miL1L2θ¨1+13m2+∑i=34miL22θ¨2+12m3+m4L2L3θ¨3+12m4L2L4θ¨4+k2θ2−θ1−k3θ3−θ2−12m2+∑i=34migL2θ2=−12m2+∑i=34miL2x¨b,12m3+m4L1L3θ¨1+12m3+m4L2L3θ¨2+13m3+m4L32θ¨3+12m4L3L4θ¨4+k3θ3−θ2−k4θ4−θ3−12m3+m4gL3θ3=−12m3+m4L3x¨b,12m4L1L4θ¨1+L2L4θ¨2+L3L4θ¨3+13m4L42θ¨4+k4θ4−θ3−12m4gL4θ4=−12m4L4x¨b.This equation can be rewritten in the matrix form as(23)Mθ¨+Kθ+Nθ=Fx¨b,where(24)M=M11M12M13M14M21M22M23M24M31M32M33M34M41M42M43M44,K=K11−k200−k2K22−k300−k3K33−k400−k4K44,Nθ=−k1φ1000,F=−f12m1+∑i=24miL112m2+∑i=34miL212m3+m4L312m4L4.Several other equations complete the system as follows:(25)θ=θ1θ2θ3θ4T,M11=13m1+∑i=24miL12,M12=M21=12m2+∑i=34miL1L2,M13=M31=12m3+m4L1L3,M14=M41=12m4L1L4,M22=13m2+∑i=34miL22,M23=M32=12m3+m4L2L3,M24=M42=12m4L2L4,M33=13m3+m4L32,M34=M43=12m4L3L4,M44=13m4L42,K11=k1+k2−12m1+∑i=24migL1,K22=k2+k3−12m2+∑i=34migL2,K33=k3+k4−12m3+m4gL3,K44=k4−12m4gL4.Taking the damping of the structure into account, its dynamic equation can be written as follows:(26)Mθ¨+Cθ˙+Kθ+Nθ=Fx¨b,where the damping ratio is C=2ζMΦΛΦ−1, Φ is the modal matrix, and Λ is a diagonal matrix consisting of the natural frequencies. For equipment without a shock absorber, Nθ in the dynamic equation can be omitted.


## 3. Seismic Response Analysis

### 3.1. Ground Motion Input

The seismic wave data adopted were those for artificial wave proposed by the China Electric Power Research Institute. The artificial wave was fitted based on a seismic acceleration response spectrum with a characteristic period of 0.9 s, capable of enveloping the characteristic periods of Category I0–III soil sites, making them suitable for seismic design and testing and for seismic performance evaluation of HV and UHV electrical equipment [25]. The artificial wave time history and the curve fitted to the artificial wave response spectrum and target spectrum at peak seismic acceleration of 1 g are illustrated in Figure 11.Figure 11
(a) Time history curve of artificial seismic wave. (b) Response-spectrum fitted curve.
(a)(b)


### 3.2. Experimental Verification of the Theoretical Model

To ensure that the theoretical model is correct and rational, its results under typical working conditions were compared with results from shaking table testing under the same conditions. During the test, strain gauges were arranged at the root of the porcelain bushing of each lightning arrester element, and the strain converted into a stress response using the elastic modulus. Accelerometers were arranged at the upper flange of each bushing element to provide data-permitting analysis of the dynamics of the equipment. The main operating condition applied during the test was white noise. The frequency of the equipment was tested under this condition and then under 0.3 g seismic test condition and 0.5 g shock absorption test conditions. At the end of the test, the relevant data and signal processing methods in mathematics are used to analyze the test data [26–28].The natural frequencies of the equipment’s seismic model and the shock absorption model were estimated, via solution for the eigenvalues of their dynamic equations, at 2.39 Hz and 2.27 Hz, respectively. These values were similar to the experimental estimates of 2.17 Hz and 2.08 Hz, representing errors of 10.14% and 9.13%, respectively.Figure12 provides a comparison of estimates for stress at the bottom of the porcelain pillars derived via theoretical analysis and testing under 0.3 g seismic effects. The results of the theoretical calculation are basically consistent with the experimental results. The maximum stress on the equipment derived from experimental and theoretical analyses was 22.02 MPa and 20.30 MPa, respectively, representing a difference in maximum stress of only 9.04%, confirming the validity of the seismic model without shock absorbers.Figure 12
Stress at the bottom of the porcelain pillar under 0.3 g seismic effect.Figure13 provides a comparison of time history for stress at the bottom of the porcelain pillar with shock absorption under a 0.5 g seismic effect. Once again, the results of the theoretical calculation are basically consistent with the experimental results. Estimates of the maximum stress on the equipment under a 0.5 g seismic effect were 11.5 MPa and 10.3 MPa, derived from experimental and theoretical analysis, respectively, representing a difference of only 10.4%, confirming the validity of the model with shock absorption.Figure 13
Stress at the bottom of the porcelain pillar under 0.5 g seismic effect with shock absorption.


### 3.3. Component Response under Seismic Excitation

To study the influence of ground motion intensity on the shock absorption of a 1000 kV pillar-type porcelain lightning arrester, a comparison of this equipment’s seismic responses without and with shock absorption was conducted under conditions of 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g ground motion excitation. Figure14 shows the stress response time history of the equipment for different magnitudes of ground motion excitation. The curve in Figure 15 describes the relationship between the magnitude of seismic wave excitation and shock absorption efficiency. Figure 16 shows the force-displacement relationships of a typical shock absorber under seismic effects of different magnitudes.Figure 14
Stress at the bottom of the porcelain pillar under seismic excitation of different magnitudes: (a) 0.1 g, (b) 0.2 g, (c) 0.3 g, (d) 0.4 g, and (e) 0.5 g.
(a)(b)(c)(d)(e)Figure 15
Relationship between seismic wave excitation and shock absorption efficiency.Figure 16
Force-displacement relationship of typical shock absorbers under seismic effects of different magnitudes: (a) 0.1 g, (b) 0.2 g, (c) 0.3 g, (d) 0.4 g, and (e) 0.5 g.
(a)(b)(c)(d)(e)The results of this analysis show that as the intensity of seismic excitation increases, the loop in the hysteretic curve formed by the shock absorber increases in size, indicating that it dissipates more energy and has a larger effect, under increased seismic excitation. With seismic excitation of 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g, corresponding levels of shock absorption efficiency were 27.56%, 47.21%, 62.23%, 71.59%, and 77.49%, respectively. Absorption efficiency can be written as follows:(27)η=σ1−σ2σ1×100%,where η is the absorption efficiency and σ1 and σ2are the stress on equipment under seismic conditions before and after installation of shock absorbers.


### 3.4. Influence of Shock Absorber Parameters on Equipment Seismic Response

Due to requirements concerning the stiffness characteristics of shock absorbers to be connected with electrical components during engineering application, in practice, the stiffness of different types of shock absorbers differs only slightly. Of all the shock absorber parameters, yield force is the main factor affecting their energy dissipation properties. The shock absorber installation radius—the radius of the circle around which the bolt holes for shock absorber installation are arranged—is also a key factor affecting their displacement energy dissipation properties. This study therefore focuses on the influence of shock absorbers’ installation radius and yield force on their shock absorption efficiency.To analyze the influence of the shock absorber installation radius on shock absorption efficiency, this parameter was varied over a 0.4–0.8 m range, and its influence on the shock absorption efficiency of the porcelain pillar system was examined for levels of seismic wave excitation intensity between 0.1 g and 0.5 g. Figure17 depicts analytically derived relationships between shock absorber installation radius and the shock absorption efficiency of the pillar-type porcelain component. Table 2 shows how shock absorption efficiency varies with installation radius under different magnitudes of seismic excitation.Figure 17
Influence of installation radius on porcelain pillar shock absorption efficiency.Table 2
Shock absorption efficiency with different installation radii and magnitudes of seismic excitation.
Seismic excitation (g)Shock absorption efficiency (%)R = 0.4R = 0.6R = 0.80.153.1129.623.940.262.242.1420.850.370.6754.8638.740.477.0564.8552.670.581.5171.8062.20Figure17 shows that for a given level of seismic wave excitation, shock absorption efficiency decreases gradually as shock absorber installation radius increases. Under seismic excitation of 0.5 g, shock absorption efficiency decreases from 81.51% to 62.20% as shock absorber installation radius increases from 0.4 m to 0.8 m. For lower magnitudes of seismic excitation, the porcelain pillar’s shock absorption efficiency decreases more rapidly with increases in shock absorber installation radius: with seismic excitation of 0.1 g and a shock absorber installation radius of 0.8 m, shock absorption efficiency falls to the very low level of 3.94%. This represents a roughly 92% decrease in shock absorption efficiency compared with a shock absorber installation radius of 0.4 m. As seismic excitation gradually increases, the influence of shock absorber installation radius on the shock absorption efficiency of the porcelain pillar assembly gradually decreases. With 0.3 g seismic excitation and a 0.8 m installation radius, shock absorption efficiency is 38.74%, 45% lower than when the installation radius is 0.4 m. With 0.5 g seismic excitation and a 0.8 m installation radius, the shock absorption efficiency is 62.2%, only 24% lower than for 0.4 m installation radius. The degree to which shock absorption efficiency declines with installation radius gradually decreases as seismic excitation increases.The influence of the shock absorber’s yield force on shock absorption efficiency was analyzed by estimating the effect of parameterγ on shock absorption efficiency under different intensities of seismic excitation. Figure 18 shows how shock absorption efficiency changes with γ under different intensities of seismic excitation. Table 3 provides shock absorption efficiency values with different values of γ and seismic excitation. In the Bouc–Wen model, γ is closely related to the shock absorber’s yield force, and the yield force values corresponding to the relevant values of γ are shown in Table 4.Figure 18
Relationship between shock absorption efficiency andγ.Table 3
Shock absorption efficiency with different values ofγ and magnitudes of seismic excitation.
Seismic excitation (g)Shock absorption efficiency (%)γ=3000γ=4000γ=5000γ=6000γ=70000.119.1820.8122.8025.0727.560.226.2931.2936.0042.1247.210.335.9743.8051.5057.2562.240.445.6054.7162.0067.4271.600.553.8663.0269.4774.0776.90Table 4
Yield force of shock absorber for different values ofγ.
γ Value30004000500060007000Yield force (kN)4030252018As can be seen from Figure18 and Table 3, for a given level of seismic excitation, shock absorption efficiency gradually increases as γ increases. With 0.1 g seismic excitation, shock absorption efficiency is 19.18% for γ = 3,000, rising to 27.56% for γ = 7,000, a 44% increase. With 0.3 g seismic excitation, shock absorption efficiency is 35.97% for γ = 3,000, rising to 62.24% for γ = 7,000, a 73% increase. With 0.5 g seismic excitation, shock absorption efficiency is 53.86% for γ = 3,000, rising to 76.9% for γ = 7,000, a 43% increase.For a given value ofγ, shock absorption efficiency gradually increases with seismic excitation. For γ = 3,000, shock absorption efficiency is 53.86% for 0.5 g seismic excitation, representing a significant increase of 180% compared to efficiency with 0.1 g seismic excitation. For γ = 5,000, shock absorption efficiency under 0.5 g seismic excitation is 204% higher than that under 0.1 g seismic excitation. For γ = 7,000, shock absorption efficiency under 0.5 g seismic excitation is 179% higher than that under 0.1 g seismic excitation.These estimates show that for a given level of seismic excitation, the smaller the shock absorber’s yield force, the higher the shock absorption efficiency; with the decrease of yield force, the increasing amplitude of shock absorption efficiency increases first and then decreases.


## 3.1. Ground Motion Input

The seismic wave data adopted were those for artificial wave proposed by the China Electric Power Research Institute. The artificial wave was fitted based on a seismic acceleration response spectrum with a characteristic period of 0.9 s, capable of enveloping the characteristic periods of Category I0–III soil sites, making them suitable for seismic design and testing and for seismic performance evaluation of HV and UHV electrical equipment [25]. The artificial wave time history and the curve fitted to the artificial wave response spectrum and target spectrum at peak seismic acceleration of 1 g are illustrated in Figure 11.Figure 11
(a) Time history curve of artificial seismic wave. (b) Response-spectrum fitted curve.
(a)(b)


## 3.2. Experimental Verification of the Theoretical Model

To ensure that the theoretical model is correct and rational, its results under typical working conditions were compared with results from shaking table testing under the same conditions. During the test, strain gauges were arranged at the root of the porcelain bushing of each lightning arrester element, and the strain converted into a stress response using the elastic modulus. Accelerometers were arranged at the upper flange of each bushing element to provide data-permitting analysis of the dynamics of the equipment. The main operating condition applied during the test was white noise. The frequency of the equipment was tested under this condition and then under 0.3 g seismic test condition and 0.5 g shock absorption test conditions. At the end of the test, the relevant data and signal processing methods in mathematics are used to analyze the test data [26–28].The natural frequencies of the equipment’s seismic model and the shock absorption model were estimated, via solution for the eigenvalues of their dynamic equations, at 2.39 Hz and 2.27 Hz, respectively. These values were similar to the experimental estimates of 2.17 Hz and 2.08 Hz, representing errors of 10.14% and 9.13%, respectively.Figure12 provides a comparison of estimates for stress at the bottom of the porcelain pillars derived via theoretical analysis and testing under 0.3 g seismic effects. The results of the theoretical calculation are basically consistent with the experimental results. The maximum stress on the equipment derived from experimental and theoretical analyses was 22.02 MPa and 20.30 MPa, respectively, representing a difference in maximum stress of only 9.04%, confirming the validity of the seismic model without shock absorbers.Figure 12
Stress at the bottom of the porcelain pillar under 0.3 g seismic effect.Figure13 provides a comparison of time history for stress at the bottom of the porcelain pillar with shock absorption under a 0.5 g seismic effect. Once again, the results of the theoretical calculation are basically consistent with the experimental results. Estimates of the maximum stress on the equipment under a 0.5 g seismic effect were 11.5 MPa and 10.3 MPa, derived from experimental and theoretical analysis, respectively, representing a difference of only 10.4%, confirming the validity of the model with shock absorption.Figure 13
Stress at the bottom of the porcelain pillar under 0.5 g seismic effect with shock absorption.


## 3.3. Component Response under Seismic Excitation

To study the influence of ground motion intensity on the shock absorption of a 1000 kV pillar-type porcelain lightning arrester, a comparison of this equipment’s seismic responses without and with shock absorption was conducted under conditions of 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g ground motion excitation. Figure14 shows the stress response time history of the equipment for different magnitudes of ground motion excitation. The curve in Figure 15 describes the relationship between the magnitude of seismic wave excitation and shock absorption efficiency. Figure 16 shows the force-displacement relationships of a typical shock absorber under seismic effects of different magnitudes.Figure 14
Stress at the bottom of the porcelain pillar under seismic excitation of different magnitudes: (a) 0.1 g, (b) 0.2 g, (c) 0.3 g, (d) 0.4 g, and (e) 0.5 g.
(a)(b)(c)(d)(e)Figure 15
Relationship between seismic wave excitation and shock absorption efficiency.Figure 16
Force-displacement relationship of typical shock absorbers under seismic effects of different magnitudes: (a) 0.1 g, (b) 0.2 g, (c) 0.3 g, (d) 0.4 g, and (e) 0.5 g.
(a)(b)(c)(d)(e)The results of this analysis show that as the intensity of seismic excitation increases, the loop in the hysteretic curve formed by the shock absorber increases in size, indicating that it dissipates more energy and has a larger effect, under increased seismic excitation. With seismic excitation of 0.1 g, 0.2 g, 0.3 g, 0.4 g, and 0.5 g, corresponding levels of shock absorption efficiency were 27.56%, 47.21%, 62.23%, 71.59%, and 77.49%, respectively. Absorption efficiency can be written as follows:(27)η=σ1−σ2σ1×100%,where η is the absorption efficiency and σ1 and σ2are the stress on equipment under seismic conditions before and after installation of shock absorbers.


## 3.4. Influence of Shock Absorber Parameters on Equipment Seismic Response

Due to requirements concerning the stiffness characteristics of shock absorbers to be connected with electrical components during engineering application, in practice, the stiffness of different types of shock absorbers differs only slightly. Of all the shock absorber parameters, yield force is the main factor affecting their energy dissipation properties. The shock absorber installation radius—the radius of the circle around which the bolt holes for shock absorber installation are arranged—is also a key factor affecting their displacement energy dissipation properties. This study therefore focuses on the influence of shock absorbers’ installation radius and yield force on their shock absorption efficiency.To analyze the influence of the shock absorber installation radius on shock absorption efficiency, this parameter was varied over a 0.4–0.8 m range, and its influence on the shock absorption efficiency of the porcelain pillar system was examined for levels of seismic wave excitation intensity between 0.1 g and 0.5 g. Figure17 depicts analytically derived relationships between shock absorber installation radius and the shock absorption efficiency of the pillar-type porcelain component. Table 2 shows how shock absorption efficiency varies with installation radius under different magnitudes of seismic excitation.Figure 17
Influence of installation radius on porcelain pillar shock absorption efficiency.Table 2
Shock absorption efficiency with different installation radii and magnitudes of seismic excitation.
Seismic excitation (g)Shock absorption efficiency (%)R = 0.4R = 0.6R = 0.80.153.1129.623.940.262.242.1420.850.370.6754.8638.740.477.0564.8552.670.581.5171.8062.20Figure17 shows that for a given level of seismic wave excitation, shock absorption efficiency decreases gradually as shock absorber installation radius increases. Under seismic excitation of 0.5 g, shock absorption efficiency decreases from 81.51% to 62.20% as shock absorber installation radius increases from 0.4 m to 0.8 m. For lower magnitudes of seismic excitation, the porcelain pillar’s shock absorption efficiency decreases more rapidly with increases in shock absorber installation radius: with seismic excitation of 0.1 g and a shock absorber installation radius of 0.8 m, shock absorption efficiency falls to the very low level of 3.94%. This represents a roughly 92% decrease in shock absorption efficiency compared with a shock absorber installation radius of 0.4 m. As seismic excitation gradually increases, the influence of shock absorber installation radius on the shock absorption efficiency of the porcelain pillar assembly gradually decreases. With 0.3 g seismic excitation and a 0.8 m installation radius, shock absorption efficiency is 38.74%, 45% lower than when the installation radius is 0.4 m. With 0.5 g seismic excitation and a 0.8 m installation radius, the shock absorption efficiency is 62.2%, only 24% lower than for 0.4 m installation radius. The degree to which shock absorption efficiency declines with installation radius gradually decreases as seismic excitation increases.The influence of the shock absorber’s yield force on shock absorption efficiency was analyzed by estimating the effect of parameterγ on shock absorption efficiency under different intensities of seismic excitation. Figure 18 shows how shock absorption efficiency changes with γ under different intensities of seismic excitation. Table 3 provides shock absorption efficiency values with different values of γ and seismic excitation. In the Bouc–Wen model, γ is closely related to the shock absorber’s yield force, and the yield force values corresponding to the relevant values of γ are shown in Table 4.Figure 18
Relationship between shock absorption efficiency andγ.Table 3
Shock absorption efficiency with different values ofγ and magnitudes of seismic excitation.
Seismic excitation (g)Shock absorption efficiency (%)γ=3000γ=4000γ=5000γ=6000γ=70000.119.1820.8122.8025.0727.560.226.2931.2936.0042.1247.210.335.9743.8051.5057.2562.240.445.6054.7162.0067.4271.600.553.8663.0269.4774.0776.90Table 4
Yield force of shock absorber for different values ofγ.
γ Value30004000500060007000Yield force (kN)4030252018As can be seen from Figure18 and Table 3, for a given level of seismic excitation, shock absorption efficiency gradually increases as γ increases. With 0.1 g seismic excitation, shock absorption efficiency is 19.18% for γ = 3,000, rising to 27.56% for γ = 7,000, a 44% increase. With 0.3 g seismic excitation, shock absorption efficiency is 35.97% for γ = 3,000, rising to 62.24% for γ = 7,000, a 73% increase. With 0.5 g seismic excitation, shock absorption efficiency is 53.86% for γ = 3,000, rising to 76.9% for γ = 7,000, a 43% increase.For a given value ofγ, shock absorption efficiency gradually increases with seismic excitation. For γ = 3,000, shock absorption efficiency is 53.86% for 0.5 g seismic excitation, representing a significant increase of 180% compared to efficiency with 0.1 g seismic excitation. For γ = 5,000, shock absorption efficiency under 0.5 g seismic excitation is 204% higher than that under 0.1 g seismic excitation. For γ = 7,000, shock absorption efficiency under 0.5 g seismic excitation is 179% higher than that under 0.1 g seismic excitation.These estimates show that for a given level of seismic excitation, the smaller the shock absorber’s yield force, the higher the shock absorption efficiency; with the decrease of yield force, the increasing amplitude of shock absorption efficiency increases first and then decreases.


## 4. Conclusions

This paper applies the Bouc–Wen model to fit restoring force-displacement curves for hysteretic nonlinear metal shock absorbers, avoiding the sharp inflection points introduced by bilinear models. Dynamic models and equations for pillar-type porcelain electrical components were developed, taking their nonlinear characteristics into consideration, allowing examination of the effects of shock absorbers on these components’ nonlinear seismic responses. This yielded the following conclusions:(1)
Shaking table test results indicated small differences, not exceeding 11%, between the theoretically and experimentally derived results, illustrating the accuracy of the theoretical model and results derived from this theoretical analysis. This validates the correctness and reasonableness of results from the theoretical model.(2)
Under seismic wave excitation, porcelain pillars’ vibration responses can be effectively suppressed using hysteretic nonlinear shock absorbers. The greater the seismic wave intensity, shock absorption devices enter the yield state better and absorb more seismic energy, and thus, the more obvious the efficiency of this shock absorption.(3)
As the installation radius and yield force of the shock absorber increase, its shock absorption efficiency gradually decreases. Thus, during engineering design, reasonable values for the yield force and installation radius of shock absorbers should be selected based on structural characteristics and the required degrees of seismic fortification and the required shock absorption efficiency of the equipment.(4)
For the shock absorption design of electrical equipment in high seismic intensity area, shock absorption efficiency should be more than 50%. During the shock absorption design of 1000 kV arrester equipment, to ensure perfect absorption efficiency under 0.4 g earthquake, the yield force should be between 18 kN∼30 kN, and the installation radius of shock absorbers should be about 0.6 m.



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*Source: 9026804-2020-05-15.xml*