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

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-deformationmodel 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.


Introduction
High voltage insulator is widely used in transmission equipment.They are usually designed as slender upright pole; see Figure 1.Such engineering structures often suffer various base excitation.The stability and vibration of these structures under base excitation are an important problem.
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 narrowband 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.

Dynamic Model
The model considered is shown in Figure 3.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 is , and length is l.The Young's modulus, diameter, and thickness of the platform are E, r, and h, respectively.The space between leady isolators and central point is R.The base moves as x b in x direction.The angle between the first isolator  1 and y direction is  which will be 0 <  < /2.
All forces of the upright pole under base excitation are shown in Figure 4. Due to base excitation, the upright pole will rotate around point O, and point O moves with the base as x b .The rotating angle of the upright pole is .Deformation of four isolators are The forces from the platform can be simplified on point O as supporting force N, shear force Q, damping M c , and moment M O .The damping and moment can be obtained from There are four elastic forces F ki 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 force N from platform is equal to the gravity of the upright pole mg.When the base is moving, the inertia forces of the upright pole are and , and the inertia moment is   = (1/12) 2 θ .Then, the governing equation of the upright pole in rotating can be written: According to experimental results, the elastic force of leady isolator appears hysteresis nonlinearity.The loaddeformation curve is shown in Figure 5(a).The blue line is obtained from experiment, and the black line is approximately fitted curve.The slopes k 1 and k 2 are 2.3 kN/mm and 220 kN/mm, respectively.The max deformation  m and middle break point deformation  d are dependent on the max loading.
Figure 5(b) shows that the deformation varies with different loading.If the max loading is lower than F 0 , the deformation varies between (- 0 ,  0 ).The deformation and loading vary as the red route.In this case, the isolator shows as a linear spring, and the elastic coefficient is k 2 .According to experimental results,  0 is 0.206 mm, and F 0 is 22.70 kN.If the max loading is higher than F 0 , 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 is F sm , the displacement and loading vary as the black route, and if the max loading of the second and fourth isolators is F cm , the displacement and loading vary as the blue route.
When the base excitation is harmonic, the rotating angle  is periodic.Assuming the amplitude of  being  m , the max  1 ,  3 , are  sm = m Rsin , and the max  2 ,  4 , are  cm = m Rcos .
If  sm <  0 , the first and third isolators are linear.
If  sm >  0 , the first and third isolators are hysteresis.If  cm <  0 , the second and fourth isolators are linear.
If  cm >  0 , the second and fourth isolators are hysteresis.
In order to simplify ( 5) and ( 7), the following equations can be obtained from Figure 5(b): Thus, the elastic forces from isolators can be expressed as where G s and G c are dependent on  m and isolator parameters. where The small vibration of the upright pole is studied in this paper.We take the approximation: sin  ≈ , cos  ≈ 1.Thus, the rotating equation of the upright pole with four leady isolators can be rewritten as where  2 0 = 3 4 /4ℎ 2 −3/2 and  0 = 3/2 0  2 .If there is no isolator, 3 4 /4ℎ 2 − 3/2 > 0 is necessary for the stability of the upright pole.

Analytical Study
Since the base is excited by harmonic excitation, the acceleration of the base can be assumed Then, a periodic response of  is assumed to take the form Substituting ( 16) into ( 10)-( 13) and reserving the first order of Fourier expansion give where a 1 , a 2 , a 3 , and a 4 can be obtained by integrating (10) and ( 11) over  =  +  in one period.
When G s and G c are hysteresis, the variate  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 are  sd and - sd .The corresponding values of  are   and −+  , where   = arccos(1−2 s0 /  ).The variate  in one period is divided into four segments.They represent four different stiffnesses of a hysteretic isolator in four stages, respectively.Thus, the integrating of H s can be obtained.
The break point deformations of the second and fourth isolators are  cd and - cd .The corresponding values of  are   and − +   , where   = arccos(1 − 2 0 /  ).Thus, the integrating of H c can also be obtained.

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 angle  are presented in dB scale.
Analytical results and numerical results are compared in Figure 7.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 frequency  near the natural frequency  0 .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.
In this case,  = /6, i.e.,  c0 <  s0 .When  m <  c0 , G s = G c = k 2 .The four isolators appear as linear spring.The natural frequency of this system is increased due to the linear stiffness of isolators.When  c0 <  m <  s0 , G s = k 2 , and G c 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.When  m >  s0 , both G s and G c 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 of  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 max  m is studied in the following.
The effects of pole damping on the amplitude of  are given in Figure 8.The cases both with isolator and without isolator are considered.The damping coefficient c 0 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 max  m 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.
Different amplitudes of  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 max  m is slightly changed for different natural frequency.
Figure 10 shows  m vary with excitation frequency for different isolator space R. 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.
Given different excitation amplitude f, Figure 11 shows the amplitude of  varies with excitation frequency.The vibration amplitude increases as the excitation amplitude increased.When the excitation amplitude is enough small, the max  m is larger than  c0 but no more than  s0 .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 max  m is no more less  c0 .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.Only  m >  c0 , and the isolator can dissipate vibration energy.
The amplitude-frequency curve for different excitation direction  is shown in three-dimensional space (see Figure 12).It can be seen that the resonant peak would be minimum when  is 0, /4 or /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 max  m is studied in detail in the following.

Resonant Peak Analysis
In order to study the effect of isolator on resonant peak, the damping of the pole is omitted, i.e., c 0 = 0. Actually, when  m is the resonant peak, we have Equations ( 27) and (28) imply Thus, the resonant peak  m,p can be obtained.
If  , <  0 <  0 , we have a 1 = a 3 = 0.There is no solution of resonant peak.Actually, according to the discussions in Section 4, we know that the resonant peak  m,p would be no more less the smaller one between  s0 and  c0 .Thus, this case will not happen.
Figure 13 shows the influence of excitation direction on resonant peak.We can conclude that when  is 0 or /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.
Figures 14-17 show the influences of R, Δk,  0 , and f on the resonant peak.It can be seen that when R < 0.087 m, Δk < 37.469 kN/mm,  0 < 0.072 mm, or f > 1.436, the resonant peak will be infinite.Otherwise, the resonant peak will be decreased as increasing R or Δk or decreasing f.It should be noticed that the resonant peak will decrease greatly and then increase rarely when  0 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 |d , /d| > 0.01 (p is one of the parameters R, Δk,  0 , or f ), when the parameter is within the sensitive parameter interval.According to ( 31  <  0 < 0.177 mm, and 0.616 < f < 1.436.The best parameters should be chosen outsider these sensitive parameter intervals. From (31)-(34), we also found that the mass m and length l of the pole have the same effect as the excitation amplitude f.It means that decreasing m or l could also decrease the resonant peak of the pole.One should note that the gravity centre is in the middle of the pole where is l/2.Thus, decreasing the gravity centre also decreases the resonant peak.

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 space R or stiffness D-value of isolator Δk or decreasing the pole mass m, length l or excitation amplitude f 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.

Figure 4 :
Figure 4: Dynamic equilibrium of ceramic upright pole under base excitation.

Figure 6 :
Figure 6: Vibration of  in one period.
Figures 18 and 19 reveal the effects of the mass m and length l on the resonant peak.The sensitive parameter intervals of m and l are 1771 kg < m < 2814 kg and 4.40 m < l < 8.04 m, respectively.

Figure 18 :Figure 19 :
Figure 18: Influence of m on resonant peak.