# Dynamics of a Upright Pole Coupled with Nonlinear Hysteretic Isolators under Harmonic Base Excitation

Authors: Yongfeng Cheng; Hulun Guo; Zhubing Zhu; Xiaochao Su; Yushu Chen
Journal: Shock and Vibration
 (2018)

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## Abstract

Leady isolator shows hysteretic nonlinearity. The isolation efficiency of leady isolator is an important problem in many engineering structures. In this paper, vibrations of a ceramic upright pole coupled with four leady isolators under harmonic base excitation are studied. A hysteretic force-deformation model of leady isolator is derived from experimental results. With this model, the vibration of the pole in rotation is studied. The frequency response is obtained analytically by employing harmonic balance method. The analytical results are agreed well with numerical results. The vibration of the pole is decreased greatly by leady isolator, especially near resonant case. The influences of system parameters on vibration response and resonant peak are discussed in detail.


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## Body

## 1. Introduction

High voltage insulator is widely used in transmission equipment. They are usually designed as slender upright pole; see Figure1. Such engineering structures often suffer various base excitation. The stability and vibration of these structures under base excitation are an important problem.Figure 1
High voltage insulator.Dynamics of a structure subjected base excitation were studied by many authors. Shenton and Jones [1] present the general, two-dimensional formulation for the response of free-standing rigid bodies to base excitation. They gave a complete and consistent formulation for five possible modes of response (rest, slide, rock, slide-rock, and free flight) and impact between the body and foundation. Capecchi et al. [2] used numerical method to study the asymptotic response of a rigid block to harmonic force. Huang et al. [3] employing the incremental harmonic balance (IHB) method studied the nonlinear steady-state response of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. Kocatürk [4] analyzed the steady-state response with respect to base movement of a viscoelastically supported cantilever beam. Ervin and Wickert [5] investigated the forced complex responses of a clamped-clamped beam attached a rigid body, subjected to base excitation. Periodic responses, period-doubling bifurcations, grazing impacts, subharmonic regions, fractional harmonic resonances, and apparently chaotic responses were found in this system. Feng et al. [6] investigated the nonlinear motion of a slender cantilever beam subject to axial narrow-band random excitation.Because of the complex dynamics of cantilever beam due to base excitation, vibration suppression of cantilever beam is of great practical interest in industrial sector. Sometimes, tip mass modify the vibrating behavior of cantilever beam. Esmailzadeh and Nakhaie-Jazar[7] studied the periodic behavior of a cantilever beam with end mass subjected to harmonic base excitation by using Green’s function and employing Schauder’s fixed point theorem. Sayag and Dowell [8] computationally and experimentally studied a cantilever beam with a tip mass under base excitation. Damping and yield stress of the beam were both considered. They observed that the damping was likely nonlinear. Eftekhari et al. [9] investigated the vibration suppression of a symmetrically cantilever composite beam coupled with an oscillator at the tip using internal resonance under chordwise base excitation.However, tip mass or oscillator attached on the top of high voltage insulator is hard to realize. An isolator attached on the bottom of high voltage insulator is more realizable. New structures were applied in many nonlinear isolators. Han et al. [10] presented a nonlinear isolator constituted by three linear spring. They found complex equilibrium bifurcations in this system. Xu et al. [11] designed a nonlinear magnetic low-frequency vibration isolator with the characteristic of quasi-zero stiffness. Zheng et al. [12] employed a negative stiffness magnetic spring to reduce the resonance frequency of linear isolator. Liu et al. [13] designed a negative stiffness corrector which is formed by Euler buckled beams and studied the characteristics of this passive nonlinear isolator. Huang et al. [14] analytically and experimentally studied the vibration isolation characteristics of a nonlinear isolator using Euler buckled beams as negative stiffness corrector. Friswell and Flores [15] investigated the design of nonlinear high-static-low-dynamic-stiffness isolation mounts using beams of tunable geometric nonlinear stiffness. Dutta and Chakraborty [16] studied the effectiveness of a nonlinear isolator with magneto rheological damper (MR). The MR damper was modeled including Bouc-Wen hysteretic element and the spring was modeled by cubic nonlinearity. Jangid and Datta [17] investigated the dissipation of hysteretic energy in the isolator of the base isolated structure under seismic excitation. Zhu et al. [18] used the quasistatic method and digital simulation studied the stationary response of smooth and bilinear hysteretic systems to narrow-band random excitations. Samani et al. [19] investigated the effect of dynamic loading on hysteretic behavior of a special kind of frictional damper (cylindrical frictional damper) by experimental and numerical methods.New materials were also used to constitute nonlinear isolators. Heertjes and van de Wouw [20] studied the nonlinear dynamics of a single-degree-of-freedom pneumatic vibration isolator. Based on a physical model, the stiffness property of the isolator was described by a nonsymmetric stiffness nonlinearity. Chen et al. [21] used nonlinear polynomial functions to present the stiffness and damper of rubber isolators. The coefficients were identified by experimental data. Wang and Zheng [22] studied the coupling vibration of nonlinear isolators and two flexible beams by experimental and analytical methods. Shaska et al. [23] explored the advantages and characteristics of nonlinear butyl rubber isolators in vibratory shear by comparison with linear isolators. Nonlinear characteristics in shear deformation, which was dependent on frequency and temperature, were reflected on stiffness and damping. Carboni et al. [24] found that rheological device provided nonlinear hysteretic forces to a one-degree-of-freedom mass through suitable assemblies of NiTiNOL and steel wire ropes subject to tension-flexure cycles. Then, they studied the frequency-response curves of the isolator system subject to base excitation numerically and experimentally [25]. The dynamic responses of a multistory steel building model were also mitigated well [26]. Jiang and Yuan [27] studied the finite element matrix updating problem in a hysteretic damping model based on measured incomplete vibration modal data.In order to study the vibration problem of a hysteretic system, averaging method was employed by Ding et al. [28]. They integrated the hysteretic function in period by using averaging method. Harmonic balance method can also solve hysteretic problems. Xiong et al. [29] used incremental harmonic balance method to analyze the vibration of a nonlinear system with the bilinear hysteretic oscillator. Based on these methods, we employ the harmonic balance method to investigate the dynamical behavior of a upright pole coupled with leady isolators (see Figure 2) under harmonic base excitation. Firstly, a hysteretic force-deformation model of leady isolator is derived from experimental results. With this model, the dynamic model of upright pole in rotation is established. Then, the frequency response is obtained analytically by employing harmonic balance method. The analytical results are compared with numerical results. Lastly, the influences of system parameters on vibration response and resonant peak are discussed in detail.Figure 2
Leady isolator.


## 2. Dynamic Model

The model considered is shown in Figure3. It consists of a ceramic upright pole located on an elastic circular platform, and four leady isolators are uniformly distributed around the platform. The upright pole is considered as a rigid body. Its density isρ, and length isl. The Young’s modulus, diameter, and thickness of the platform areE,r, andh, respectively. The space between leady isolators and central point isR. The base moves asxb inx direction. The angle between the first isolatorξ1 andy direction is α which will be 0 < α <π/2.Figure 3
Geometry of ceramic upright pole: (a) side view; (b) top view.All forces of the upright pole under base excitation are shown in Figure4. Due to base excitation, the upright pole will rotate around pointO, and pointO moves with the base asxb. The rotating angle of the upright pole isθ. Deformation of four isolators are(1)ξ1=ξ3=θRsin⁡αξ2=ξ4=θRcos⁡αFigure 4
Dynamic equilibrium of ceramic upright pole under base excitation.The forces from the platform can be simplified on pointO as supporting forceN, shear forceQ, dampingMc, and momentMO. The damping and moment can be obtained from(2)Mc=cθ˙MO=∫-rrEθy2h2r2-x2dx=πEr44hθThere are four elastic forcesFki when the isolators are stretched or constringent. We consider that there are no deformation of four isolators when the upright pole is static. In this case, the elastic forces of four isolators are zero, and the supporting forceN from platform is equal to the gravity of the upright polemg. When the base is moving, the inertia forces of the upright pole are FIx=mx¨b, FIn=1/2mlθ˙2, and FIτ=1/2mlθ¨, and the inertia moment is MI=1/12ml2θ¨. Then, the governing equation of the upright pole in rotating can be written:(3)112ml2θ¨-mgl2sin⁡θ+mx¨bl2cos⁡θ+12mlθ¨l2+Mc+MO+Fk1Rsin⁡α+Fk3Rsin⁡α+Fk2Rcos⁡α+Fk4Rcos⁡α=0According to experimental results, the elastic force of leady isolator appears hysteresis nonlinearity. The load-deformation curve is shown in Figure5(a). The blue line is obtained from experiment, and the black line is approximately fitted curve. The slopesk1 andk2 are 2.3 kN/mm and 220 kN/mm, respectively. The max deformationξm and middle break point deformationξd are dependent on the max loading.Figure 5
The load-deformation curve of leady isolator.
(a)
Experimental results (b)
Dynamic evolutionFigure5(b) shows that the deformation varies with different loading. If the max loading is lower thanF0, the deformation varies between (-ξ0,ξ0). The deformation and loading vary as the red route. In this case, the isolator shows as a linear spring, and the elastic coefficient isk2. According to experimental results,ξ0 is 0.206 mm, andF0 is 22.70 kN. If the max loading is higher thanF0, the isolator shows hysteresis nonlinearity. The deformation of isolator changes with the loading in four linear segments which enclose an area. For example, if the max loading of the first and third isolators isFsm, the displacement and loading vary as the black route, and if the max loading of the second and fourth isolators isFcm, the displacement and loading vary as the blue route.When the base excitation is harmonic, the rotating angleθ is periodic. Assuming the amplitude ofθ beingθm, the maxξ1,ξ3, areξsm=θmRsin⁡α, and the maxξ2,ξ4, areξcm=θmRcos⁡α.Ifξsm <ξ0, the first and third isolators are linear.(4)Fk1=Fk3=k2θRsin⁡αIfξsm >ξ0, the first and third isolators are hysteresis.(5)Fk1=Fk3=k1θRsin⁡α+ξsd+Fsd-ξsd<θRsin⁡α<ξsm,θ↑k2θRsin⁡α-ξsm+Fsmξsd<θRsin⁡α<ξsm,θ↓k1θRsin⁡α-ξsd-Fsd-ξsm<θRsin⁡α<ξsd,θ↓k2θRsin⁡α+ξsm-Fsm-ξsm<θRsin⁡α<-ξsd,θ↑Ifξcm <ξ0, the second and fourth isolators are linear.(6)Fk2=Fk4=k2θRcos⁡αIfξcm >ξ0, the second and fourth isolators are hysteresis.(7)Fk2=Fk4=k1θRsin⁡α+ξcd+Fcd-ξcd<θRcos⁡α<ξcm,θ↑k2θRsin⁡α-ξcm+Fcmξcd<θRcos⁡α<ξcm,θ↓k1θRsin⁡α-ξcd-Fcd-ξcm<θRcos⁡α<ξcd,θ↓k2θRsin⁡α+ξcm-Fcm-ξcm<θRcos⁡α<-ξcd,θ↑In order to simplify (5) and (7), the following equations can be obtained from Figure 5(b):(8)ξsd=ξsm-2ξ0,Fsm=k2-k1ξ0+k1ξsm,Fsd=k2+k1ξ0-k1ξsmξcd=ξcm-2ξ0,Fcm=2k1+k2ξ0-k1ξcm,Fcd=k2-k1ξ0+k1ξcmThus, the elastic forces from isolators can be expressed as(9)Fk1=Fk3=GsRsin⁡αFk2=Fk4=GcRcos⁡αwhereGs andGc are dependent onθm and isolator parameters. (10)Gs=k2θθm<θs0Hsθθm>θs0(11)Gc=k2θθm<θc0Hcθθm>θc0whereθs0=ξ0/Rsin⁡α, θc0=ξ0/Rcos⁡α, Δk=k2-k1,(12)Hsθ=k1θ+Δkθs0-θm+2θs0<θ<θm,θ↑k2θ-Δkθm-θs0θm-2θs0<θ<θm,θ↓k1θ-Δkθs0-θm<θ<θm-2θs0,θ↓k2θ+Δkθm-θs0-θm<θ<-θm+2θs0,θ↑(13)Hcθ=k1θ+Δkθc0-θm+2θc0<θ<θm,θ↑k2θ-Δkθm-θc0θm-2θc0<θ<θm,θ↓k1θ-Δkθc0-θm<θ<θm-2θc0,θ↓k2θ+Δkθm-θc0-θm<θ<-θm+2θc0,θ↑The small vibration of the upright pole is studied in this paper. We take the approximation:sin⁡θ≈θ, cos⁡θ≈1. Thus, the rotating equation of the upright pole with four leady isolators can be rewritten as(14)θ¨+2c0ω0θ˙+ω02θ+6R2ml2Gssin2α+Gccos2α=-32lx¨bwhereω02=3πEr4/4hml2-3g/2l and c0=3c/2ω0ml2. If there is no isolator, 3πEr4/4hml2-3g/2l>0 is necessary for the stability of the upright pole.


## 3. Analytical Study

Since the base is excited by harmonic excitation, the acceleration of the base can be assumed(15)x¨b=fgsin⁡ωtThen, a periodic response ofθ is assumed to take the form(16)θ=θmcos⁡ωt+φSubstituting (16) into (10)-(13) and reserving the first order of Fourier expansion give(17)Gs=a1sin⁡ωt+φ+a2cos⁡ωt+φGc=a3sin⁡ωt+φ+a4cos⁡ωt+φwherea1,a2,a3, anda4 can be obtained by integrating (10) and (11) over ϕ=ωt+φ in one period.(18)a1=1π∫-ππGssin⁡ϕdϕ=0θm<θs01π∫-ππHsθsin⁡ϕdϕθm>θs0(19)a2=1π∫-ππGscos⁡ϕdϕ=k2θmθm<θs01π∫-ππHsθcos⁡ϕdϕθm>θs0(20)a3=1π∫-ππGcsin⁡ϕdϕ=0θm<θs01π∫-ππHcθsin⁡ϕdϕθm>θs0(21)a4=1π∫-ππGccos⁡ϕdϕ=k2θmθm<θs01π∫-ππHcθcos⁡ϕdϕθm>θs0WhenGs andGc are hysteresis, the variateθ in one period is shown in Figure 6. From Figure 5, we know that there are two break points during the deformation of each isolator varying in one period. The break point deformations of the first and third isolators areξsd and -ξsd. The corresponding values of ϕ are ϕs and -π+ϕs, where ϕs=arccos⁡1-2θs0/θm. The variateθ in one period is divided into four segments. They represent four different stiffnesses of a hysteretic isolator in four stages, respectively. Thus, the integrating ofHs can be obtained.(22)1π∫-ππHsθsin⁡ϕdϕ=1π∫-π-π+ϕsk2θmcos⁡ϕ+Δkθm-θs0sin⁡ϕdϕ+1π∫-π+ϕs0k1θmcos⁡ϕ+Δkθs0sin⁡ϕdϕ+1π∫0ϕsk2θmcosϕ-Δkθm-θs0sin⁡ϕdϕ+1π∫ϕsπk1θmcosϕ-Δkθs0sin⁡ϕdϕ=-Δkθm1-cos⁡2ϕs2π(23)1π∫-ππHsθcos⁡ϕdϕ=1π∫-π-π+ϕsk2θmcosϕ+Δkθm-θs0cos⁡ϕdϕ+1π∫-π+ϕs0k1θmcosϕ+Δkθs0cos⁡ϕdϕ+1π∫0ϕsk2θmcosϕ-Δkθm-θs0cos⁡ϕdϕ+1π∫ϕsπk1θmcosϕ-Δkθs0cos⁡ϕdϕ=k1θm+Δkθm2ϕs-sin⁡2ϕs2πFigure 6
Vibration ofθ in one period.The break point deformations of the second and fourth isolators areξcd and -ξcd. The corresponding values of ϕ are ϕc and -π+ϕc, where ϕc=arccos⁡1-2θc0/θm. Thus, the integrating ofHc can also be obtained.(24)1π∫-ππHcθsin⁡ϕdϕ=-Δkθm1-cos⁡2ϕc2π(25)1π∫-ππHcθcos⁡ϕdϕ=k1θm+Δkθm2ϕc-sin⁡2ϕc2πSubstituting (15)-(17) into (14) and collecting the coefficients of cos⁡ϕ and sin⁡ϕ yield(26)
-
ω
2
θ
m
+
ω
0
2
θ
m
+
6
R
2
m
l
2
a
2
s
i
n
2
α
+
a
4
c
o
s
2
α
=
3
2
l
f
g
sin
⁡
φ
-
2
c
0
ω
0
ω
θ
m
+
6
R
2
m
l
2
a
1
s
i
n
2
α
+
a
3
c
o
s
2
α
=
-
3
2
l
f
g
cos
⁡
φEliminatingφ gives(27)ω2θm-ω02θm-b2b42+2c0ω0ωθm-b2b32=b12f2whereb1=3g/2l, b2=6R2/ml2, b3=a1sin2α+a3cos2α, and b4=a2sin2α+a4cos2α. Thus, the relationship between the amplitude of rotating angleθ and the excitation frequency of the base can be obtained analytically from (27).


## 4. Discussions on Dynamic Characteristics

In this section, analytical results and numerical results are compared, and dynamic characteristics are discussed. The influences of natural frequency of the pole and isolator parameters on dynamic response are also discussed. All the results of amplitude in rotating angleθ are presented in dB scale.In this paper, the parameters without special statement are considered asc0 = 0.01,ω0 = 80 Hz,m= 980 kg,l= 2.8 m, α =π/6, R = 0.25 m, andf= 0.5,g= 9.8 m/s2.Analytical results and numerical results are compared in Figure7. The dash line is the analytical result of the pole without isolator. It is a typical damped SDOF vibrator. The vibration amplitude becomes the max value as the excitation frequencyω near the natural frequencyω0. The solid line and hollow dotted line are the analytical and numerical results of the pole with four isolators, respectively. They show that the analytical results are agreed well with the numerical results. More important, due to the leady isolator, the resonant frequency is increased, and resonant peak is decreased greatly.Figure 7
Influence of isolator on dynamic response.In this case,α =π/6, i.e.,θc0 <θs0. Whenθm <θc0,Gs =Gc =k2θ. The four isolators appear as linear spring. The natural frequency of this system is increased due to the linear stiffness of isolators. Whenθc0 <θm <θs0,Gs =k2θ, andGc is hysteresis function. The natural frequency of this system is changed. The second and fourth isolators dissipate some of the vibration energy. The curve of amplitude versus frequency is inclined to lower frequency zone. Whenθm >θs0, bothGs andGc are hysteresis. The vibration amplitude decreases greatly under resonant zone. This phenomenon is totally due to energy dissipation of the four isolators. The max amplitude ofθ is determined by energy dissipation of isolators. It is an important parameter for vibration mitigation of upright pole when the base is excited by wide band frequency. The influence of different parameters on the maxθm is studied in the following.The effects of pole damping on the amplitude ofθ are given in Figure 8. The cases both with isolator and without isolator are considered. The damping coefficientc0 of pole is varied from 0.01 to 0.05. One can see that the vibration amplitude peak of the pole system either with or without isolator is decreased as pole damping increasing. More important, the maxθm decreases 1.068 dB, 0.788 dB, and 0.473 dB due to isolator when the pole damping is 0.01, 0.02, and 0.05, respectively. It means that the smaller the pole damping the more decreasing of amplitude peak is caused by isolators.Figure 8
Effect of pole damping.Different amplitudes ofθ are show in Figure 9 when the natural frequency is chosen as 40 Hz, 60 Hz, 80 Hz, and 100 Hz. The resonant frequency of the system increases as the natural frequency increased. Moreover, the maxθm is slightly changed for different natural frequency.Figure 9
Effect of natural frequency.Figure10 showsθm vary with excitation frequency for different isolator spaceR. We can see that the resonant frequency is increased as the isolator space increased. More important, the vibration amplitude is decreased greatly when the isolator space is increased.Figure 10
Effect of isolator space.Given different excitation amplitudef, Figure 11 shows the amplitude ofθ varies with excitation frequency. The vibration amplitude increases as the excitation amplitude increased. When the excitation amplitude is enough small, the maxθm is larger thanθc0 but no more thanθs0. It means that only the second and fourth isolators dissipate vibration energy, and the first and third isolators are vibrate in linear elasticity. Practically, the maxθm is no more lessθc0. The vibration amplitude will be infinite when the excitation frequency nears the resonant frequency without isolator. Due to energy dissipation of isolator, the vibration amplitude becomes a limit value. Onlyθm >θc0, and the isolator can dissipate vibration energy.Figure 11
Effect of excitation amplitude.The amplitude-frequency curve for different excitation directionα is shown in three-dimensional space (see Figure 12). It can be seen that the resonant peak would be minimum when α is 0,π/4 orπ/2. Which resonant peak is the smallest? It is an important problem for vibration mitigation, especially for the base excited by wide band frequency. Thus, the influence of different parameters on the maxθm is studied in detail in the following.Figure 12
Effect of excitation direction.


## 5. Resonant Peak Analysis

In order to study the effect of isolator on resonant peak, the damping of the pole is omitted, i.e.,c0 = 0. Actually, whenθm is the resonant peak, we have(28)dθmdω=0Equations (27) and (28) imply(29)f2b12-b22b32=0Thus, the resonant peakθm,p can be obtained.(30)fmgl4R2=a1sin2α+a3cos2αIn order to solve (30), we consider two cases: (1) 0 < α <π/4; (2)π/4 < α <π/2.Case (1) 0 < α <π/4, i.e.,θc0 <θs0.Ifθm,p<θc0<θs0, we havea1 =a3 = 0. There is no solution of resonant peak. Actually, according to the discussions in Section 4, we know that the resonant peakθm,p would be no more less the smaller one betweenθs0 andθc0. Thus, this case will not happen.Ifθc0<θm,p<θs0, the resonant peak is(31)θm,p=16Δkξ0216Δkξ0Rcos⁡α-πfmglConsidering the amplitude ofθ is positive, 16Δkξ0Rcos⁡α-πfmgl>0 should be satisfied.Ifθc0<θs0<θm,p, the resonant peak is(32)θm,p=32Δkξ0216Δkξ0Rsin⁡α+cos⁡α-πfmglwhere 16Δkξ0Rsin⁡α+cos⁡α-πfmgl>0 should be satisfied.C a s e   ( 2 ) π/4 < α <π/2, i.e.,θs0 <θc0.Ifθm,p<θs0<θc0, there is no solution of resonant peak.Ifθs0<θm,p<θc0, the resonant peak is(33)θm,p=16Δkξ0216Δkξ0Rsin⁡α-πfmglwhere16Δkξ0Rsin⁡α-πfmgl>0 should be satisfied.Ifθs0<θc0<θm,p, the resonant peak is(34)θm,p=32Δkξ0216Δkξ0Rsin⁡α+cos⁡α-πfmglwhere16Δkξ0Rsin⁡α+cos⁡α-πfmgl>0 should be satisfied.Figure13 shows the influence of excitation direction on resonant peak. We can conclude that when α is 0 orπ/2, the resonant peak is the smallest. It means that two isolators have the largest deformation, and the most vibration energy is dissipated. But the other two isolators have no deformation.Figure 13
Influence of excitation direction on resonant peak.Figures14–17 show the influences ofR, Δk,ξ0, andf on the resonant peak. It can be seen that whenR < 0.087 m, Δk < 37.469 kN/mm,ξ0 < 0.072 mm, orf > 1.436, the resonant peak will be infinite. Otherwise, the resonant peak will be decreased as increasingR or Δk or decreasingf. It should be noticed that the resonant peak will decrease greatly and then increase rarely whenξ0 increases from 0.072 mm. More important, the resonant peak will decrease greatly during special parameter intervals. We could conclude that the resonant peak is sensitive to these parameters within these parameter intervals. Outside these parameter intervals, the resonant peak is changed rarely as the parameter changing. Furthermore, we assume that dθm,p/dp>0.01 (p is one of the parametersR, Δk,ξ0, orf), when the parameter is within the sensitive parameter interval. According to (31)-(34), we can obtain four sensitive parameter intervals: 0.087 m <R < 0.292 m, 37.469 kN/mm < Δk < 61.475 kN/mm, 0.072 mm <ξ0 < 0.177 mm, and 0.616 <f < 1.436. The best parameters should be chosen outsider these sensitive parameter intervals.Figure 14
Influence ofR on resonant peak.Figure 15
Influence of Δk on resonant peak.Figure 16
Influence ofξ0 on resonant peak.Figure 17
Influence off on resonant peak.From (31)-(34), we also found that the massm and lengthl of the pole have the same effect as the excitation amplitudef. It means that decreasingm orl could also decrease the resonant peak of the pole. One should note that the gravity centre is in the middle of the pole where isl/2. Thus, decreasing the gravity centre also decreases the resonant peak. Figures 18 and 19 reveal the effects of the massm and lengthl on the resonant peak. The sensitive parameter intervals ofm andl are 1771 kg <m < 2814 kg and 4.40 m <l < 8.04 m, respectively.Figure 18
Influence ofmon resonant peak.Figure 19
Influence ofl on resonant peak.


## 6. Conclusions

The dynamical behavior of an upright pole coupled with four leady isolators under harmonic base excitation is addressed. Experimental results show that elastic force of leady isolator is hysteresis with deformation. Analytical solution of periodic response of the upright pole in rotating is obtained by employing harmonic balance method. The result is agreed well with numerical result. Both the analytical and numerical results show that the hysteresis in stiffness could dissipate vibration energy. The amplitude of the pole in rotating is decreased greatly, especially in resonant case. The smaller the pole damping is the more decreasing of amplitude peak is caused by isolators. Moreover, the natural frequency of the pole is almost independent of the resonant peak. In order to decrease the resonant peak of the pole, increasing the isolator spaceR or stiffness D-value of isolator Δk or decreasing the pole massm, lengthl or excitation amplitudef would be viable. When isolators are installed in the excitation direction, energy dissipation would be maximum. More important, there are sensitive parameter intervals for all these parameters. Within the parameter interval, the resonant peak of upright pole could be decreased greatly. Outside the parameter interval, the resonant peak would be not sensitive to the parameter. The best parameters should be chosen outsider these sensitive parameter intervals.



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*Source: 2497035-2018-06-13.xml*