Research ArticleStochastic Response Characteristic and EquivalentDamping of Weak Nonlinear Energy Dissipation System underBiaxial Earthquake Action

Yu Xia ZeWu Zhemin Kang and Chuangdi Li

Civil Engineering and Architecture Department Guangxi University of Science and Technology Liuzhou 545006 China

Correspondence should be addressed to Yu Xia summ-rain163com

Received 24 January 2017 Accepted 12 April 2017 Published 10 May 2017

Academic Editor Roman Lewandowski

Copyright copy 2017 Yu Xia et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The random response characteristic of weak nonlinear structure under biaxial earthquake excitation is investigated The structurehas a SDOF (single degree of freedom)with supporting braces and viscoelastic dampers First it adopts integral constitutive relationand establishes a differential and integral equations of motion Then according to the principle of energy balance the equation islinearized Finally based on the stochastic averagingmethod the general analytical solution of the variance of the displacement andvelocity response and the equivalent damping is deduced and derived At the same time the joint probability density function of theamplitude and phase and displacement and velocity of the energy dissipation structure are also given The dynamic characteristicsof a structure with viscoelastic dampers are determined as a solution to the variance of displacement response so the equivalentdamping is taken into consideration as a solution to replace the original nonlinear damping It means it has established a unifiedanalytical solution of stochastic response analysis and equivalent damping of a SDOF nonlinear dissipation structure with the braceunder biaxial earthquake action in this paper

1 Introduction

In addition to the two seismic force components in thehorizontal direction there is still vertical seismic forcecomponents Actual earthquake structure is always subjectto vertical and horizontal earthquake actions Under a largeraction of earthquake the response of structure is furtherincreased At this time the vertical earthquake action can notbe ignored Therefore it is important to research structureresponse in the horizontal and vertical earthquake As landin big cities is limited buildings are located close to eachother To reduce the seismic responses of buildings adjacentbuildings are linked together by connecting dampers such asthe Triple Towers in Downtown Tokyo [1] Researchers haveproposed different types of connecting devices to connectadjacent buildings These devices include passive dampers[2ndash5] semiactive dampers [6ndash8] and active dampers [910] It is now well recognized that seismic responses ofadjacent buildings can be mitigated by connecting dampersBiaxial earthquake action will aggravate the vibration of

the structure the devices of passive control will reduce thestructural vibration The passive control techniques such asviscous and viscoelastic dampers have been widely used [11]Linear viscoelastic damper is a kind of excellent performanceof energy dissipation device and is widely used in seismicengineeringThe integral model is the most general model ofviscoelastic dampers [12] Other models such as the complexmodulus model [13] the fractional derivative model [14ndash16] and the general differential model are all approximatemodel Analytical modeling of a novel type of passive frictiondamper for seismic hazard mitigation of structural systemsis present [17] numerical results show that the proposeddamper is more efficient in dissipating input seismic energythan a passive linear viscous damperwith same force capacityThe equivalent linearization of the motion equations withMaxwell dampers will effectively solve the problem of non-linear equations A system with nonlinear dampers is usuallyreplaced by an equivalent linear system with its propertiesdetermined by using different methods like equating theenergy dissipated [18] equating power consumption [19]

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 7384940 10 pageshttpsdoiorg10115520177384940

2 Mathematical Problems in Engineering

replacing the nonlinear viscous damping by an array offrequency and amplitude-dependent linear viscous models[20] and other random vibration theories [21] Malone andConnor [22] have reported a method setting a new degree offreedom at mass-less point between a dashpot and a stiffnessspring of the Maxwell model and then apply a commonnumerical integration scheme In this method it is necessaryto consider twice the degrees of freedom of the originalsystem It has been mainly applied to the analyses of materialstress-strain relationships Kitagawa et al [23] have reportedthe analysis of reinforced concrete elements by consideringthe effect of strain speed They treated the Maxwell model asa supplementary restoring force on the equation of motionof the system discretized by a central difference methodwhich is categorized into an explicit integration scheme Itcan play a better role of shock absorption by adding brace tothe viscoelastic damper The brace is widely used in damperThe integral model is a typical type this kind of damper [12]can be used to describe the instantaneous elasticity creeprelaxation and strain memory of viscoelastic dampers Parket al [24] and Singh et al [25] describe the use of gradient-based optimization algorithms to obtain the optimal param-eters of dampers and their supporting braces in structuressubjected to seismic motions More recently Chen and Chai[26] also proposed a gradient-based numerical procedure fordetermining the minimum brace stiffness together with aset of optimal damper coefficients to meet a target responsereduction They used Maxwell model-based brace-dampersystems and concluded that brace stiffness equal to the firststorey stiffness would be adequate for the desirable levels ofresponse reduction in typical applications Since the structureis installed with the damper and then turned into the energydissipation structure its design can not be directly appliedto the response spectrum method At the same time thismakes the design of actual engineering very inconvenientThe damping ratio of the dissipation system is the sum ofthe damping ratio of the structure itself and the equivalentdamping ratio of the damper The linear response spectrummethod can be used to calculate the equivalent damping ratioof the damper [27 28] Therefore it is greatly significant toestablish a equivalent structureThen the response spectrummethod can be used directly used to structural analysis andengineering design The relationships between equivalentdamping and ductility for the direct displacement-basedseismic design (DBSD) method are proposed [29] As theconcept of theDBSD is addressed to highlight the importanceof the proper determination of equivalent damping in theDBSD the equivalent stiffness is taken as the secant stiffnessat maximum deformation so the appropriate equivalentdamping should be determined based on such a prescriptionAnd twenty-one SDOF systems are designed according to theDBSD procedure and analyzed to indicate that the proposedequivalent damping relationships are suitable for the DBSDIn addition stochastic averaging method is an effectiveapproximationmethod for predicting the stochastic responseof a structure The basic assumption is small damping andweak broadband excitation Compared with the modal strainenergymethod it is easy to understand andobtain the generalanalytical solution under a close theoretical basis the same

result of decoupling method of the forced vibration modeunder the case of linear small damping can be concludedIn fact in recent years the important theoretical resultsof linear and nonlinear random vibration are obtained byusing the stochastic averaging method It investigates thestochastic response of vibroimpact system with fractionalderivative under Gaussian white noise excitation the nons-mooth transformation and stochastic averaging method areused to obtain the analytical solutions of the equivalentstochastic system [30] The first-passage statistics of Duffing-Rayleigh-Mathieu system under wide-band colored noiseexcitation are studied by using stochastic averaging methodThe motion equation of the original system is transformedinto two time homogeneous diffusion Markovian processesof amplitude and phase after stochastic averaging [31] Theequivalent linearization can solve the problem of nonlinearstructure a nonlinear stochastic optimal control strategy forsingle degree of freedom viscoelastic system with actuatorsaturation is proposed based on the stochastic averagingmethod and stochastic dynamical programing principleAs the viscoelastic system is converted into an equivalentnonlinear nonviscoelastic system by replacing the viscoelas-tic force with amplitude-dependent stiffness and damping[32] in this paper it has used the equivalent linearizationmethod and the stochastic averaging method and it has alsoused the general integral model of viscous and viscoelasticdampers Considering the comprehensive effect of bracestrain history of damper dynamic characteristics of structureand excitation it establishes a complete analytical solutionof stochastic response analysis and equivalent damping of aSDOF nonlinear dissipation structure with the brace underbiaxial earthquake action The new approach can be directlyapplied to damping engineering design with the responsespectrum method

2 Constitutive Equation of Damper with Brace

21 Motion Equation of Maxwell Damper with Braces Themass matrix stiffness matrix and damping matrix of thestructure are 119898 119896 and 119888 respectively A viscoelastic damper(1199011(119905)) of the general integral type is equipped betweenfloors The modified damper with supporting braces (1198961198871) is1199011198661(119905) The complex modulus storage modulus and energydissipationmodulus of1199011(119905) and1199011198661(119905) are1198641198761(1119908)11986411987611(120596)11986411987621(120596) and 1198641198661(1120596) 11986411986611(120596) 11986411986621(120596) respectively Therelaxation modulus equilibrium modulus and relaxationfunction of 1199011(119905) and 1199011198661(119905) are 1198761(119905) 1198961198761 ℎ1198761(119905) and 1198661(119905)1198961198661 ℎ1198661(119905) respectively The displacement vector of thestructure with respect to the ground is 119906 when the horizontaland vertical ground motion are 119892(119905) and V(119905) the relativedisplacement of damper 1199011(119905) and its supporting braces (1198961198871)are 1199061199011 and 1199061198871 respectively two dampers mentioned aboveare shown in Figures 1 and 2

The motion equation can be expressed as follows119898 + 119888 + 119896 119906 + 1198751198661 (119905)

= minus119898 [119877119906 119892 (119905) + 119877V V (119905)] (1)

where 119898 is the mass 119888 is the damping 119896 is the stiffness and119877119906 and 119877V are horizontal and vertical inertial force vector

Mathematical Problems in Engineering 3

m

c

k

p1(t)

u1

kb1

u

ug

(a) The original calculation diagram of structure

m

c

k

u1

u

ug

kG1

P0G1(t)

(b) The modified calculation diagram of structure with brace

Figure 1 Calculation diagram of structure

PG1(t) PG1(t)kb1 P1(t)

u1

(a)

PG1(t)PG1(t) kb1

u1

ub1 up1

kQ1

P01(t)

(b) The original calculation diagram of damper

PG1(t)PG1(t)

u1

P0G1(t)

kG1

(c) The modified calculation diagram of damper with brace

Figure 2 Calculation diagram of damper

1199011198661(119905) is the viscoelastic dampers force Relevant parametersare listed as follows

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

1199011198661 (119905) = 1198961198661119906 + 11990111986610 (119905) 11990111986610 (119905) = int

119905

0ℎ1198661 (119905 minus 120591) (120591) 119889120591

11986411986611 (120596) = 1198961198871 [119864211987611 (120596) + 119864211987621 (120596) + 119896119887111986411987611 (120596)][1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596)

11986411986621 (120596) = 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596)

(2)

3 The Vibration Equation of WeakNonlinear System with Single Degree ofFreedom and Its Linearization

31 The Transfer of the Weak Nonlinear System EquationConsidering the weak nonlinear SDOF system the generalenergy dissipation structural equation can be expressed asfollows (see [33 34])

119898 + 119888 + 119896119906 + 120576119891 (119906 ) + 1198961198661119906+ int1199050ℎ1198661 (119905 minus 120591) (120591) 119889120591 = minus119898(119892 (119905) + V (119905))

(3)

where 119898 is the mass 119888 is the damping 119896 is the stiffness120576119891(119906 ) is the weak nonlinear force including the non-linear damping and the spring forces (1198961198661119906 + int119905

0ℎ1198661(119905 minus120591)(120591)119889120591) is themodified damperwith supporting forces and

4 Mathematical Problems in Engineering

minus119898(119892(119905) + V(119905)) is a biaxial excitation The main aim is toreplace (3) with an equivalent linear one (see [35])

119898 + 119888119890 + 119896119890119906 + 1198961198661119906 + int119905

0ℎ1198661 (119905 minus 120591) (120591) 119889120591

= minus119898(119892 (119905) + V (119905)) + 1198650(4)

According to article (see [35]) 1198650 can be expressed asfollows

1198650 = minus 12120587 [int

2120587

0119891119898 (119860 120593) 119889120593 + int

2120587

0119891119896 (1198600 119860 120593) 119889120593] (5)

where 119888119890 and 119896119890 are the equivalent damping and stiffnessrespectively then the error between solutions of these twosystems is minimized with the mean-square method Thedifference between (3) and (4) is shown in the following1205760 = 119898 + 119888 + 119896119906 + 120576119891 (119906 ) minus 119898 minus 119888119890 minus 119896119890119906 minus 1198650 (6)

To get a relative precise result the error 1205760 should beapproximating to minimum It is better to solve the followinginstead of (6)

1205760 = 119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650 (7)

In order to choose the best equivalent damping 119888119890 andthe equivalent stiffness 119896119890 it is necessary to minimize theerror with statistical procedure which requires (7) to beapproximating to minimum

It means 119864 (12057602) = Minimum (8)

where 119864(12057602) denotes the mathematical expectation

119864 (12057602) = 119864 [(119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650)2] (9)

According to the method of multivariate function thenecessary and sufficient condition (see [36]) for the mini-mum of 119864[12057602] is obtained it requires that

120597119864 (12057602)120597119888119890 = 0

120597119864 (12057602)120597119896119890 = 0

(10)

Equations (10) lead to two linear equations and determinethe optimal values of 119888119890 and 119896119890

119864 [119891 (119906 )] minus 119888119890119864( 1199062) minus 119896119890119864 (119906 ) = 0119864 [119906119891 (119906 )] minus 119888119890119864 (119906 ) minus 119896119890119864 (1199062) = 0

(11)

The required parameters can be obtained simultaneouslyas follows

119888119890 = 119864 (1199062) 119864 [119891 (119906 )] minus 119864 (119906 ) 119864 [119906119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119888

119896119890 = 119864 ( 1199062)119864 [119906119891 (119906 )] minus 119864 (119906 ) 119864 [119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119896

(12)

It is known from the paper (see [37 38]) that 119888119890 and 119896119890determined by the above formula lead to the minimum valueof 119864[12057602] It is important to note that it has to solve the linearrandom vibration system (4) to obtain the optimal values of119888119890 and 1198961198904 Statistical Characteristics of Displacementand Velocity Response of Weak NonlinearEnergy Dissipation System under BiaxialEarthquake Action

41 The Transform of the Time Domain Dynamic EquationThe motion equation of equivalent linear structure withviscoelastic dampers (4) could be written in the followingform

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591)

= [minus (119892 (119905) + V (119905)) + 1198650]119898119890

(13)

where

12059612 = 119896119890 + 1198961198661119898119890

212058511205961 = 119888119890119898119890

1205730 = 1119898119890

119898119890 = 119898

(14)

where the symbols 1205961 1205851 and 1205730 are structure self-vibrationfrequency damping ratio and the reciprocal of structuremass respectively Moreover 119888119890 and 119896119890 are the equivalentdamping and stiffness respectively

According to the seismic code [39] 119878119864119896 should be ascer-tained by the maximum between the following

119878119864119896 = radic1198781199092 + (085119878119910)2

119878119864119896 = radic1198781199102 + (085119878119909)2(15)

So 119906119864119896 can be determined by the following

119906119864119896 = radic1198922 (119905) + (085V (119905))2 (16)

where 119892 and V are the horizontal and vertical accelerationrespectively

Assume that

[minus119898119890 (119892 (119905) + V (119905)) + 1198650]119898119890 = [119898119890119906119864119896 + 1198650]119898119890

= 1198911 (119905) (17)

Mathematical Problems in Engineering 5

So the time domain dynamic equation of the energydissipation structure of a single degree of freedomwith linearviscoelastic damper could be expressed in the following form

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591) = 1198911 (119905) (18)

42 Stochastic Averaging Equation According to the stochas-tic averaging theory the standard Van-der-Pol transform isintroduced

119906 (119905) = 1198601 (119905) cos 1205791 (119905) (119905) = minus1198601 (119905) 1205961 sin 1205791 (119905) 1205791 (119905) = 1205961119905 + Φ1 (119905)

(19)

The stochastic averaging equations that fit the amplitude1198601(119905) are shown in the following

1198891198601 = [minus12058512059611198601 + 1205871198781198911 (1205961)2120596121198601 ]119889119905

+ [1205871198781198911 (1205961)]121205961 119889V1 (119905)(20)

119889Φ1 (119905) = 121205730119867119888 (1205961) 119889119905 +

[1205871198781198911 (1205961)]1211986011205961 119889V2 (119905) (21)

where 119889V1(119905) and 119889V2(119905) are Wiener process of independentunits and 1198781198911(1205961) is the power spectrum function of 1198911 in thevalue of 1205961 the expression of 120585 is shown in (22)

120585 = 1205851 + 119867119888 (1205961)21205961119898119890 (22)

119867119888 (1205961) = intinfin

0ℎ1198661 (119905) cos1205961120591119889120591 = 11986411986621 (1205961)1205961 (23)

where 11986411986611(1205961) = 1198961198661 + 1205961 intinfin0 ℎ1198661(119905) sin1205961119905 119889119905 11986411986621(1205961) =1205961 intinfin0 ℎ1198661(119905) cos1205961119905 11988911990543 The Transient Joint Probability Density Function of EachMode Shape of the Nonlinear Structure with Braces Assumethat the state variables of 1198601(119905) and Φ1(119905) are 1198861 and 1205931respectively Probability density function of 1198601(119905) is 1198751(1198861 119905)The transient joint probability density function of 1198601(119905)and Φ1(119905) is 1198751(1198861 1205931 119905) and the transient joint probabilitydensity function of 119906(119905) and (119905) is 1198751(119906 119905) where 119906(119905) isstructure displacement and (119905) is the velocity Accordingto Ito equation (21) the transient joint probability densityfunction 1198751(1198861 1205931 119905 | 1198860 1205930 1199050) that fits the FPK equation isshown in the following

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] minus

1205971205971205931 [1198981205931198751]

+ 12120597212059711988612 [12059011

21198751] + 12120597212059712059312 [12059022

21198751] (24)

Because (20) does not depend on Φ1(119905) the probabilitydensity function 119875(1198861 119905 | 1198860 1199050) determined by FPK equationis as follows

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] +

1212059721205971198862 [1205901121198751] (25)

The initial conditions of (24) and (25) are respectively asfollows

1198751 (1198861 1205931 1199050 | 1198860 1205930 1199050) = 1205751 (1198861 minus 1198860) 1205751 (1205931 minus 1205930) (26)

1198751 (1198861 1199050 | 1198860 1199050) = 1205751 (1198861 minus 1198860) (27)

Comparing with (24) and (25) we obtain the relationshipof solution under the static initial conditions the following

1198751 (1198861 1205931 119905) = 121205871198751 (1198861 119905)

119875 (1198861 0) = 120575 (1198861) (28)

Meanwhile we obtain the transient joint probabilitydensity function of the original weak nonlinear structurefrom transient displacement 119906(119905) and transient velocity (119905)under the static initial condition

1198751 (119906 119905)= 112059611198861 1198751 (1198861 1205931 119905)

10038161003816100381610038161198861=1198860 12120587120596111988611198751 (1198861 119905) | 1198861

= 1198860(29)

where 1198860 = (1199062 + 212059612)12When the expression of 1198751(1198861 119905) is obtained the original

structure of random response characteristics can be fullydetermined

The solution of (22) and (25) should also fit 1198751(1198861 119905)under the static initial condition 1198751(1198861 119905) could be writtenas follows

1205971198751 (1198861 119905)120597119905 = 1205871198781198911 (1205961)2120596121205972119875112059711988612

+ 1205971205971198861 [120585112059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751 (30)

where 119875(1198861 0) = 120575(1198861)Assume that the form of 1198751(1198861 119905) is described as follows

1198751 (1198861 119905) = 11988611198881 (119905) exp[minus1198861221198881 (119905)] (31)

where 1198881(119905) is the undetermined functionEquation (31) is substituted into (28) we transform the

system of (31) into the following form

1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (32)

6 Mathematical Problems in Engineering

u

ug

me

ke

ce

kG1

P0G1(t)

u

ug

me

ke

ce

kG1

cG

u1

Figure 3 Calculation diagram

Then (32) is substituted into (31) we can obtain theanalytical solution of 1198751(1198861 119905)

According to (29) and (32) we can obtain the responsevariance of the structural displacement and velocity respec-tively

119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (33)

119864 [ 1199062 (119905)] = 120596121198881 (119905) = 1205871198781198911 (1205961)212058511205961 [1 minus 119890minus212058511205961119905] (34)

5 Equivalent Damping of WeakNonlinear Structure with the ViscoelasticDamping and the Braces

The actual ground motion is highly random characteristicsBecause of the rationality and practicality of the earthquakethe ground motion model still needs to be further improvedSo the the response spectrum method is adopted in mostcountries Once the structure is installed with the damperand it turns into an energy dissipation structure the responsespectrum method can not be directly applied to thesestructures Therefore it is greatly significant to establishthe equivalent structure which can be used directly withthe response spectrum method The calculation diagram isshown in Figure 3

Where 11987501198661(119905) = int1199050 ℎ1198661(119905 minus 120591)(120591)119889120591 is the equivalent toa damping force of 119888119866 from (4) the motion equation of thestructure can be described as follows

119898119890 + (119888119890 + 119888119866) + (119896119890 + 1198961198661) 119906= minus119898119890 (119892 (119905) + V (119905)) + 1198650

(35)

In this case (35) may be written as the following form

+ 2 (1205851 + 120585119866) 1205961 + 12059612119906 = 1198911 (119905) (36)

where 120585119866 = 11988811986621198981198901205961 1198911(119905) = (minus119898119890(119892(119905) + V(119905)) + 1198650)119898119890According to the stochastic averaging method it is

known that the probability density function of the amplitude

response (1198601(119905)) of the equivalent structure is 1198751(1198861 119905) Theprobability density function fitting the FPK equation is asfollows

1205971198751 (1198861 119905)120597119905= 1205871198781198911 (1205961)212059612

1205972119875112059711988612

+ 1205971205971198861 [(1205851 + 120585119866) 12059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751

(37)

The amplitude probability density function of the originalstructure can be applied to (30) the amplitude probabilitydensity function of the equivalent structure is appropriate for(37)We will know the difference by comparing with (30) and(37) After the following processing the expression can beexpressed as follows

120585119866 = 119867119888 (1205961)21205961119898119890 =11986411986621 (1205961)1205961 sdot 1

21205961119898119890 =11986411986621 (1205961)212059612119898119890

119888119866 = 11986411986621 (1205961)1205961

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

(38)

where 120585119866 is the equivalent damping ratio of damper it isconsistent with the equivalent damping ratio of the Maxwelldamper with the general integral model For arbitraryrandom biaxial earthquake excitations 119892(119905) and V(119905) allstochastic response characteristics calculated with the pro-posed method in equivalent structure are the same as theseof the original structure The equivalent damping ratio of thewhole weak nonlinear dissipation structure is established asfollows

120585119911 = 1205851 + 120585119866 (39)

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

m

c

k

p1(t)

u1

kb1

u

ug

(a) The original calculation diagram of structure

m

c

k

u1

u

ug

kG1

P0G1(t)

(b) The modified calculation diagram of structure with brace

Figure 1 Calculation diagram of structure

PG1(t) PG1(t)kb1 P1(t)

u1

(a)

PG1(t)PG1(t) kb1

u1

ub1 up1

kQ1

P01(t)

(b) The original calculation diagram of damper

PG1(t)PG1(t)

u1

P0G1(t)

kG1

(c) The modified calculation diagram of damper with brace

Figure 2 Calculation diagram of damper

1199011198661(119905) is the viscoelastic dampers force Relevant parametersare listed as follows

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

1199011198661 (119905) = 1198961198661119906 + 11990111986610 (119905) 11990111986610 (119905) = int

119905

0ℎ1198661 (119905 minus 120591) (120591) 119889120591

11986411986611 (120596) = 1198961198871 [119864211987611 (120596) + 119864211987621 (120596) + 119896119887111986411987611 (120596)][1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596)

11986411986621 (120596) = 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596)

(2)

3 The Vibration Equation of WeakNonlinear System with Single Degree ofFreedom and Its Linearization

31 The Transfer of the Weak Nonlinear System EquationConsidering the weak nonlinear SDOF system the generalenergy dissipation structural equation can be expressed asfollows (see [33 34])

119898 + 119888 + 119896119906 + 120576119891 (119906 ) + 1198961198661119906+ int1199050ℎ1198661 (119905 minus 120591) (120591) 119889120591 = minus119898(119892 (119905) + V (119905))

(3)

where 119898 is the mass 119888 is the damping 119896 is the stiffness120576119891(119906 ) is the weak nonlinear force including the non-linear damping and the spring forces (1198961198661119906 + int119905

0ℎ1198661(119905 minus120591)(120591)119889120591) is themodified damperwith supporting forces and

4 Mathematical Problems in Engineering

minus119898(119892(119905) + V(119905)) is a biaxial excitation The main aim is toreplace (3) with an equivalent linear one (see [35])

119898 + 119888119890 + 119896119890119906 + 1198961198661119906 + int119905

0ℎ1198661 (119905 minus 120591) (120591) 119889120591

= minus119898(119892 (119905) + V (119905)) + 1198650(4)

According to article (see [35]) 1198650 can be expressed asfollows

1198650 = minus 12120587 [int

2120587

0119891119898 (119860 120593) 119889120593 + int

2120587

0119891119896 (1198600 119860 120593) 119889120593] (5)

where 119888119890 and 119896119890 are the equivalent damping and stiffnessrespectively then the error between solutions of these twosystems is minimized with the mean-square method Thedifference between (3) and (4) is shown in the following1205760 = 119898 + 119888 + 119896119906 + 120576119891 (119906 ) minus 119898 minus 119888119890 minus 119896119890119906 minus 1198650 (6)

To get a relative precise result the error 1205760 should beapproximating to minimum It is better to solve the followinginstead of (6)

1205760 = 119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650 (7)

In order to choose the best equivalent damping 119888119890 andthe equivalent stiffness 119896119890 it is necessary to minimize theerror with statistical procedure which requires (7) to beapproximating to minimum

It means 119864 (12057602) = Minimum (8)

where 119864(12057602) denotes the mathematical expectation

119864 (12057602) = 119864 [(119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650)2] (9)

According to the method of multivariate function thenecessary and sufficient condition (see [36]) for the mini-mum of 119864[12057602] is obtained it requires that

120597119864 (12057602)120597119888119890 = 0

120597119864 (12057602)120597119896119890 = 0

(10)

Equations (10) lead to two linear equations and determinethe optimal values of 119888119890 and 119896119890

119864 [119891 (119906 )] minus 119888119890119864( 1199062) minus 119896119890119864 (119906 ) = 0119864 [119906119891 (119906 )] minus 119888119890119864 (119906 ) minus 119896119890119864 (1199062) = 0

(11)

The required parameters can be obtained simultaneouslyas follows

119888119890 = 119864 (1199062) 119864 [119891 (119906 )] minus 119864 (119906 ) 119864 [119906119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119888

119896119890 = 119864 ( 1199062)119864 [119906119891 (119906 )] minus 119864 (119906 ) 119864 [119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119896

(12)

It is known from the paper (see [37 38]) that 119888119890 and 119896119890determined by the above formula lead to the minimum valueof 119864[12057602] It is important to note that it has to solve the linearrandom vibration system (4) to obtain the optimal values of119888119890 and 1198961198904 Statistical Characteristics of Displacementand Velocity Response of Weak NonlinearEnergy Dissipation System under BiaxialEarthquake Action

41 The Transform of the Time Domain Dynamic EquationThe motion equation of equivalent linear structure withviscoelastic dampers (4) could be written in the followingform

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591)

= [minus (119892 (119905) + V (119905)) + 1198650]119898119890

(13)

where

12059612 = 119896119890 + 1198961198661119898119890

212058511205961 = 119888119890119898119890

1205730 = 1119898119890

119898119890 = 119898

(14)

where the symbols 1205961 1205851 and 1205730 are structure self-vibrationfrequency damping ratio and the reciprocal of structuremass respectively Moreover 119888119890 and 119896119890 are the equivalentdamping and stiffness respectively

According to the seismic code [39] 119878119864119896 should be ascer-tained by the maximum between the following

119878119864119896 = radic1198781199092 + (085119878119910)2

119878119864119896 = radic1198781199102 + (085119878119909)2(15)

So 119906119864119896 can be determined by the following

119906119864119896 = radic1198922 (119905) + (085V (119905))2 (16)

where 119892 and V are the horizontal and vertical accelerationrespectively

Assume that

[minus119898119890 (119892 (119905) + V (119905)) + 1198650]119898119890 = [119898119890119906119864119896 + 1198650]119898119890

= 1198911 (119905) (17)

Mathematical Problems in Engineering 5

So the time domain dynamic equation of the energydissipation structure of a single degree of freedomwith linearviscoelastic damper could be expressed in the following form

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591) = 1198911 (119905) (18)

42 Stochastic Averaging Equation According to the stochas-tic averaging theory the standard Van-der-Pol transform isintroduced

119906 (119905) = 1198601 (119905) cos 1205791 (119905) (119905) = minus1198601 (119905) 1205961 sin 1205791 (119905) 1205791 (119905) = 1205961119905 + Φ1 (119905)

(19)

The stochastic averaging equations that fit the amplitude1198601(119905) are shown in the following

1198891198601 = [minus12058512059611198601 + 1205871198781198911 (1205961)2120596121198601 ]119889119905

+ [1205871198781198911 (1205961)]121205961 119889V1 (119905)(20)

119889Φ1 (119905) = 121205730119867119888 (1205961) 119889119905 +

[1205871198781198911 (1205961)]1211986011205961 119889V2 (119905) (21)

where 119889V1(119905) and 119889V2(119905) are Wiener process of independentunits and 1198781198911(1205961) is the power spectrum function of 1198911 in thevalue of 1205961 the expression of 120585 is shown in (22)

120585 = 1205851 + 119867119888 (1205961)21205961119898119890 (22)

119867119888 (1205961) = intinfin

0ℎ1198661 (119905) cos1205961120591119889120591 = 11986411986621 (1205961)1205961 (23)

where 11986411986611(1205961) = 1198961198661 + 1205961 intinfin0 ℎ1198661(119905) sin1205961119905 119889119905 11986411986621(1205961) =1205961 intinfin0 ℎ1198661(119905) cos1205961119905 11988911990543 The Transient Joint Probability Density Function of EachMode Shape of the Nonlinear Structure with Braces Assumethat the state variables of 1198601(119905) and Φ1(119905) are 1198861 and 1205931respectively Probability density function of 1198601(119905) is 1198751(1198861 119905)The transient joint probability density function of 1198601(119905)and Φ1(119905) is 1198751(1198861 1205931 119905) and the transient joint probabilitydensity function of 119906(119905) and (119905) is 1198751(119906 119905) where 119906(119905) isstructure displacement and (119905) is the velocity Accordingto Ito equation (21) the transient joint probability densityfunction 1198751(1198861 1205931 119905 | 1198860 1205930 1199050) that fits the FPK equation isshown in the following

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] minus

1205971205971205931 [1198981205931198751]

+ 12120597212059711988612 [12059011

21198751] + 12120597212059712059312 [12059022

21198751] (24)

Because (20) does not depend on Φ1(119905) the probabilitydensity function 119875(1198861 119905 | 1198860 1199050) determined by FPK equationis as follows

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] +

1212059721205971198862 [1205901121198751] (25)

The initial conditions of (24) and (25) are respectively asfollows

1198751 (1198861 1205931 1199050 | 1198860 1205930 1199050) = 1205751 (1198861 minus 1198860) 1205751 (1205931 minus 1205930) (26)

1198751 (1198861 1199050 | 1198860 1199050) = 1205751 (1198861 minus 1198860) (27)

Comparing with (24) and (25) we obtain the relationshipof solution under the static initial conditions the following

1198751 (1198861 1205931 119905) = 121205871198751 (1198861 119905)

119875 (1198861 0) = 120575 (1198861) (28)

Meanwhile we obtain the transient joint probabilitydensity function of the original weak nonlinear structurefrom transient displacement 119906(119905) and transient velocity (119905)under the static initial condition

1198751 (119906 119905)= 112059611198861 1198751 (1198861 1205931 119905)

10038161003816100381610038161198861=1198860 12120587120596111988611198751 (1198861 119905) | 1198861

= 1198860(29)

where 1198860 = (1199062 + 212059612)12When the expression of 1198751(1198861 119905) is obtained the original

structure of random response characteristics can be fullydetermined

The solution of (22) and (25) should also fit 1198751(1198861 119905)under the static initial condition 1198751(1198861 119905) could be writtenas follows

1205971198751 (1198861 119905)120597119905 = 1205871198781198911 (1205961)2120596121205972119875112059711988612

+ 1205971205971198861 [120585112059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751 (30)

where 119875(1198861 0) = 120575(1198861)Assume that the form of 1198751(1198861 119905) is described as follows

1198751 (1198861 119905) = 11988611198881 (119905) exp[minus1198861221198881 (119905)] (31)

where 1198881(119905) is the undetermined functionEquation (31) is substituted into (28) we transform the

system of (31) into the following form

1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (32)

6 Mathematical Problems in Engineering

u

ug

me

ke

ce

kG1

P0G1(t)

u

ug

me

ke

ce

kG1

cG

u1

Figure 3 Calculation diagram

Then (32) is substituted into (31) we can obtain theanalytical solution of 1198751(1198861 119905)

According to (29) and (32) we can obtain the responsevariance of the structural displacement and velocity respec-tively

119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (33)

119864 [ 1199062 (119905)] = 120596121198881 (119905) = 1205871198781198911 (1205961)212058511205961 [1 minus 119890minus212058511205961119905] (34)

5 Equivalent Damping of WeakNonlinear Structure with the ViscoelasticDamping and the Braces

The actual ground motion is highly random characteristicsBecause of the rationality and practicality of the earthquakethe ground motion model still needs to be further improvedSo the the response spectrum method is adopted in mostcountries Once the structure is installed with the damperand it turns into an energy dissipation structure the responsespectrum method can not be directly applied to thesestructures Therefore it is greatly significant to establishthe equivalent structure which can be used directly withthe response spectrum method The calculation diagram isshown in Figure 3

Where 11987501198661(119905) = int1199050 ℎ1198661(119905 minus 120591)(120591)119889120591 is the equivalent toa damping force of 119888119866 from (4) the motion equation of thestructure can be described as follows

119898119890 + (119888119890 + 119888119866) + (119896119890 + 1198961198661) 119906= minus119898119890 (119892 (119905) + V (119905)) + 1198650

(35)

In this case (35) may be written as the following form

+ 2 (1205851 + 120585119866) 1205961 + 12059612119906 = 1198911 (119905) (36)

where 120585119866 = 11988811986621198981198901205961 1198911(119905) = (minus119898119890(119892(119905) + V(119905)) + 1198650)119898119890According to the stochastic averaging method it is

known that the probability density function of the amplitude

response (1198601(119905)) of the equivalent structure is 1198751(1198861 119905) Theprobability density function fitting the FPK equation is asfollows

1205971198751 (1198861 119905)120597119905= 1205871198781198911 (1205961)212059612

1205972119875112059711988612

+ 1205971205971198861 [(1205851 + 120585119866) 12059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751

(37)

The amplitude probability density function of the originalstructure can be applied to (30) the amplitude probabilitydensity function of the equivalent structure is appropriate for(37)We will know the difference by comparing with (30) and(37) After the following processing the expression can beexpressed as follows

120585119866 = 119867119888 (1205961)21205961119898119890 =11986411986621 (1205961)1205961 sdot 1

21205961119898119890 =11986411986621 (1205961)212059612119898119890

119888119866 = 11986411986621 (1205961)1205961

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

(38)

where 120585119866 is the equivalent damping ratio of damper it isconsistent with the equivalent damping ratio of the Maxwelldamper with the general integral model For arbitraryrandom biaxial earthquake excitations 119892(119905) and V(119905) allstochastic response characteristics calculated with the pro-posed method in equivalent structure are the same as theseof the original structure The equivalent damping ratio of thewhole weak nonlinear dissipation structure is established asfollows

120585119911 = 1205851 + 120585119866 (39)

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

minus119898(119892(119905) + V(119905)) is a biaxial excitation The main aim is toreplace (3) with an equivalent linear one (see [35])

119898 + 119888119890 + 119896119890119906 + 1198961198661119906 + int119905

0ℎ1198661 (119905 minus 120591) (120591) 119889120591

= minus119898(119892 (119905) + V (119905)) + 1198650(4)

According to article (see [35]) 1198650 can be expressed asfollows

1198650 = minus 12120587 [int

2120587

0119891119898 (119860 120593) 119889120593 + int

2120587

0119891119896 (1198600 119860 120593) 119889120593] (5)

where 119888119890 and 119896119890 are the equivalent damping and stiffnessrespectively then the error between solutions of these twosystems is minimized with the mean-square method Thedifference between (3) and (4) is shown in the following1205760 = 119898 + 119888 + 119896119906 + 120576119891 (119906 ) minus 119898 minus 119888119890 minus 119896119890119906 minus 1198650 (6)

To get a relative precise result the error 1205760 should beapproximating to minimum It is better to solve the followinginstead of (6)

1205760 = 119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650 (7)

In order to choose the best equivalent damping 119888119890 andthe equivalent stiffness 119896119890 it is necessary to minimize theerror with statistical procedure which requires (7) to beapproximating to minimum

It means 119864 (12057602) = Minimum (8)

where 119864(12057602) denotes the mathematical expectation

119864 (12057602) = 119864 [(119888 + 119896119906 + 120576119891 (119906 ) minus 119888119890 minus 119896119890119906 minus 1198650)2] (9)

According to the method of multivariate function thenecessary and sufficient condition (see [36]) for the mini-mum of 119864[12057602] is obtained it requires that

120597119864 (12057602)120597119888119890 = 0

120597119864 (12057602)120597119896119890 = 0

(10)

Equations (10) lead to two linear equations and determinethe optimal values of 119888119890 and 119896119890

119864 [119891 (119906 )] minus 119888119890119864( 1199062) minus 119896119890119864 (119906 ) = 0119864 [119906119891 (119906 )] minus 119888119890119864 (119906 ) minus 119896119890119864 (1199062) = 0

(11)

The required parameters can be obtained simultaneouslyas follows

119888119890 = 119864 (1199062) 119864 [119891 (119906 )] minus 119864 (119906 ) 119864 [119906119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119888

119896119890 = 119864 ( 1199062)119864 [119906119891 (119906 )] minus 119864 (119906 ) 119864 [119891 (119906 )]119864 (1199062) 119864 ( 1199062) minus [119864 (119906 )]2 + 119896

(12)

It is known from the paper (see [37 38]) that 119888119890 and 119896119890determined by the above formula lead to the minimum valueof 119864[12057602] It is important to note that it has to solve the linearrandom vibration system (4) to obtain the optimal values of119888119890 and 1198961198904 Statistical Characteristics of Displacementand Velocity Response of Weak NonlinearEnergy Dissipation System under BiaxialEarthquake Action

41 The Transform of the Time Domain Dynamic EquationThe motion equation of equivalent linear structure withviscoelastic dampers (4) could be written in the followingform

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591)

= [minus (119892 (119905) + V (119905)) + 1198650]119898119890

(13)

where

12059612 = 119896119890 + 1198961198661119898119890

212058511205961 = 119888119890119898119890

1205730 = 1119898119890

119898119890 = 119898

(14)

where the symbols 1205961 1205851 and 1205730 are structure self-vibrationfrequency damping ratio and the reciprocal of structuremass respectively Moreover 119888119890 and 119896119890 are the equivalentdamping and stiffness respectively

According to the seismic code [39] 119878119864119896 should be ascer-tained by the maximum between the following

119878119864119896 = radic1198781199092 + (085119878119910)2

119878119864119896 = radic1198781199102 + (085119878119909)2(15)

So 119906119864119896 can be determined by the following

119906119864119896 = radic1198922 (119905) + (085V (119905))2 (16)

where 119892 and V are the horizontal and vertical accelerationrespectively

Assume that

[minus119898119890 (119892 (119905) + V (119905)) + 1198650]119898119890 = [119898119890119906119864119896 + 1198650]119898119890

= 1198911 (119905) (17)

Mathematical Problems in Engineering 5

So the time domain dynamic equation of the energydissipation structure of a single degree of freedomwith linearviscoelastic damper could be expressed in the following form

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591) = 1198911 (119905) (18)

42 Stochastic Averaging Equation According to the stochas-tic averaging theory the standard Van-der-Pol transform isintroduced

119906 (119905) = 1198601 (119905) cos 1205791 (119905) (119905) = minus1198601 (119905) 1205961 sin 1205791 (119905) 1205791 (119905) = 1205961119905 + Φ1 (119905)

(19)

The stochastic averaging equations that fit the amplitude1198601(119905) are shown in the following

1198891198601 = [minus12058512059611198601 + 1205871198781198911 (1205961)2120596121198601 ]119889119905

+ [1205871198781198911 (1205961)]121205961 119889V1 (119905)(20)

119889Φ1 (119905) = 121205730119867119888 (1205961) 119889119905 +

[1205871198781198911 (1205961)]1211986011205961 119889V2 (119905) (21)

where 119889V1(119905) and 119889V2(119905) are Wiener process of independentunits and 1198781198911(1205961) is the power spectrum function of 1198911 in thevalue of 1205961 the expression of 120585 is shown in (22)

120585 = 1205851 + 119867119888 (1205961)21205961119898119890 (22)

119867119888 (1205961) = intinfin

0ℎ1198661 (119905) cos1205961120591119889120591 = 11986411986621 (1205961)1205961 (23)

where 11986411986611(1205961) = 1198961198661 + 1205961 intinfin0 ℎ1198661(119905) sin1205961119905 119889119905 11986411986621(1205961) =1205961 intinfin0 ℎ1198661(119905) cos1205961119905 11988911990543 The Transient Joint Probability Density Function of EachMode Shape of the Nonlinear Structure with Braces Assumethat the state variables of 1198601(119905) and Φ1(119905) are 1198861 and 1205931respectively Probability density function of 1198601(119905) is 1198751(1198861 119905)The transient joint probability density function of 1198601(119905)and Φ1(119905) is 1198751(1198861 1205931 119905) and the transient joint probabilitydensity function of 119906(119905) and (119905) is 1198751(119906 119905) where 119906(119905) isstructure displacement and (119905) is the velocity Accordingto Ito equation (21) the transient joint probability densityfunction 1198751(1198861 1205931 119905 | 1198860 1205930 1199050) that fits the FPK equation isshown in the following

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] minus

1205971205971205931 [1198981205931198751]

+ 12120597212059711988612 [12059011

21198751] + 12120597212059712059312 [12059022

21198751] (24)

Because (20) does not depend on Φ1(119905) the probabilitydensity function 119875(1198861 119905 | 1198860 1199050) determined by FPK equationis as follows

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] +

1212059721205971198862 [1205901121198751] (25)

The initial conditions of (24) and (25) are respectively asfollows

1198751 (1198861 1205931 1199050 | 1198860 1205930 1199050) = 1205751 (1198861 minus 1198860) 1205751 (1205931 minus 1205930) (26)

1198751 (1198861 1199050 | 1198860 1199050) = 1205751 (1198861 minus 1198860) (27)

Comparing with (24) and (25) we obtain the relationshipof solution under the static initial conditions the following

1198751 (1198861 1205931 119905) = 121205871198751 (1198861 119905)

119875 (1198861 0) = 120575 (1198861) (28)

Meanwhile we obtain the transient joint probabilitydensity function of the original weak nonlinear structurefrom transient displacement 119906(119905) and transient velocity (119905)under the static initial condition

1198751 (119906 119905)= 112059611198861 1198751 (1198861 1205931 119905)

10038161003816100381610038161198861=1198860 12120587120596111988611198751 (1198861 119905) | 1198861

= 1198860(29)

where 1198860 = (1199062 + 212059612)12When the expression of 1198751(1198861 119905) is obtained the original

structure of random response characteristics can be fullydetermined

The solution of (22) and (25) should also fit 1198751(1198861 119905)under the static initial condition 1198751(1198861 119905) could be writtenas follows

1205971198751 (1198861 119905)120597119905 = 1205871198781198911 (1205961)2120596121205972119875112059711988612

+ 1205971205971198861 [120585112059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751 (30)

where 119875(1198861 0) = 120575(1198861)Assume that the form of 1198751(1198861 119905) is described as follows

1198751 (1198861 119905) = 11988611198881 (119905) exp[minus1198861221198881 (119905)] (31)

where 1198881(119905) is the undetermined functionEquation (31) is substituted into (28) we transform the

system of (31) into the following form

1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (32)

6 Mathematical Problems in Engineering

u

ug

me

ke

ce

kG1

P0G1(t)

u

ug

me

ke

ce

kG1

cG

u1

Figure 3 Calculation diagram

Then (32) is substituted into (31) we can obtain theanalytical solution of 1198751(1198861 119905)

According to (29) and (32) we can obtain the responsevariance of the structural displacement and velocity respec-tively

119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (33)

119864 [ 1199062 (119905)] = 120596121198881 (119905) = 1205871198781198911 (1205961)212058511205961 [1 minus 119890minus212058511205961119905] (34)

5 Equivalent Damping of WeakNonlinear Structure with the ViscoelasticDamping and the Braces

The actual ground motion is highly random characteristicsBecause of the rationality and practicality of the earthquakethe ground motion model still needs to be further improvedSo the the response spectrum method is adopted in mostcountries Once the structure is installed with the damperand it turns into an energy dissipation structure the responsespectrum method can not be directly applied to thesestructures Therefore it is greatly significant to establishthe equivalent structure which can be used directly withthe response spectrum method The calculation diagram isshown in Figure 3

Where 11987501198661(119905) = int1199050 ℎ1198661(119905 minus 120591)(120591)119889120591 is the equivalent toa damping force of 119888119866 from (4) the motion equation of thestructure can be described as follows

119898119890 + (119888119890 + 119888119866) + (119896119890 + 1198961198661) 119906= minus119898119890 (119892 (119905) + V (119905)) + 1198650

(35)

In this case (35) may be written as the following form

+ 2 (1205851 + 120585119866) 1205961 + 12059612119906 = 1198911 (119905) (36)

where 120585119866 = 11988811986621198981198901205961 1198911(119905) = (minus119898119890(119892(119905) + V(119905)) + 1198650)119898119890According to the stochastic averaging method it is

known that the probability density function of the amplitude

response (1198601(119905)) of the equivalent structure is 1198751(1198861 119905) Theprobability density function fitting the FPK equation is asfollows

1205971198751 (1198861 119905)120597119905= 1205871198781198911 (1205961)212059612

1205972119875112059711988612

+ 1205971205971198861 [(1205851 + 120585119866) 12059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751

(37)

The amplitude probability density function of the originalstructure can be applied to (30) the amplitude probabilitydensity function of the equivalent structure is appropriate for(37)We will know the difference by comparing with (30) and(37) After the following processing the expression can beexpressed as follows

120585119866 = 119867119888 (1205961)21205961119898119890 =11986411986621 (1205961)1205961 sdot 1

21205961119898119890 =11986411986621 (1205961)212059612119898119890

119888119866 = 11986411986621 (1205961)1205961

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

(38)

where 120585119866 is the equivalent damping ratio of damper it isconsistent with the equivalent damping ratio of the Maxwelldamper with the general integral model For arbitraryrandom biaxial earthquake excitations 119892(119905) and V(119905) allstochastic response characteristics calculated with the pro-posed method in equivalent structure are the same as theseof the original structure The equivalent damping ratio of thewhole weak nonlinear dissipation structure is established asfollows

120585119911 = 1205851 + 120585119866 (39)

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

So the time domain dynamic equation of the energydissipation structure of a single degree of freedomwith linearviscoelastic damper could be expressed in the following form

+ 212058511205961 + 12059612119906 + 1205730 int119905

0ℎ1198661 (119905 minus 120591) (120591) = 1198911 (119905) (18)

42 Stochastic Averaging Equation According to the stochas-tic averaging theory the standard Van-der-Pol transform isintroduced

119906 (119905) = 1198601 (119905) cos 1205791 (119905) (119905) = minus1198601 (119905) 1205961 sin 1205791 (119905) 1205791 (119905) = 1205961119905 + Φ1 (119905)

(19)

The stochastic averaging equations that fit the amplitude1198601(119905) are shown in the following

1198891198601 = [minus12058512059611198601 + 1205871198781198911 (1205961)2120596121198601 ]119889119905

+ [1205871198781198911 (1205961)]121205961 119889V1 (119905)(20)

119889Φ1 (119905) = 121205730119867119888 (1205961) 119889119905 +

[1205871198781198911 (1205961)]1211986011205961 119889V2 (119905) (21)

where 119889V1(119905) and 119889V2(119905) are Wiener process of independentunits and 1198781198911(1205961) is the power spectrum function of 1198911 in thevalue of 1205961 the expression of 120585 is shown in (22)

120585 = 1205851 + 119867119888 (1205961)21205961119898119890 (22)

119867119888 (1205961) = intinfin

0ℎ1198661 (119905) cos1205961120591119889120591 = 11986411986621 (1205961)1205961 (23)

where 11986411986611(1205961) = 1198961198661 + 1205961 intinfin0 ℎ1198661(119905) sin1205961119905 119889119905 11986411986621(1205961) =1205961 intinfin0 ℎ1198661(119905) cos1205961119905 11988911990543 The Transient Joint Probability Density Function of EachMode Shape of the Nonlinear Structure with Braces Assumethat the state variables of 1198601(119905) and Φ1(119905) are 1198861 and 1205931respectively Probability density function of 1198601(119905) is 1198751(1198861 119905)The transient joint probability density function of 1198601(119905)and Φ1(119905) is 1198751(1198861 1205931 119905) and the transient joint probabilitydensity function of 119906(119905) and (119905) is 1198751(119906 119905) where 119906(119905) isstructure displacement and (119905) is the velocity Accordingto Ito equation (21) the transient joint probability densityfunction 1198751(1198861 1205931 119905 | 1198860 1205930 1199050) that fits the FPK equation isshown in the following

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] minus

1205971205971205931 [1198981205931198751]

+ 12120597212059711988612 [12059011

21198751] + 12120597212059712059312 [12059022

21198751] (24)

Because (20) does not depend on Φ1(119905) the probabilitydensity function 119875(1198861 119905 | 1198860 1199050) determined by FPK equationis as follows

1205971198751120597119905 = minus 1205971205971198861 [1198981198861198751] +

1212059721205971198862 [1205901121198751] (25)

The initial conditions of (24) and (25) are respectively asfollows

1198751 (1198861 1205931 1199050 | 1198860 1205930 1199050) = 1205751 (1198861 minus 1198860) 1205751 (1205931 minus 1205930) (26)

1198751 (1198861 1199050 | 1198860 1199050) = 1205751 (1198861 minus 1198860) (27)

Comparing with (24) and (25) we obtain the relationshipof solution under the static initial conditions the following

1198751 (1198861 1205931 119905) = 121205871198751 (1198861 119905)

119875 (1198861 0) = 120575 (1198861) (28)

Meanwhile we obtain the transient joint probabilitydensity function of the original weak nonlinear structurefrom transient displacement 119906(119905) and transient velocity (119905)under the static initial condition

1198751 (119906 119905)= 112059611198861 1198751 (1198861 1205931 119905)

10038161003816100381610038161198861=1198860 12120587120596111988611198751 (1198861 119905) | 1198861

= 1198860(29)

where 1198860 = (1199062 + 212059612)12When the expression of 1198751(1198861 119905) is obtained the original

structure of random response characteristics can be fullydetermined

The solution of (22) and (25) should also fit 1198751(1198861 119905)under the static initial condition 1198751(1198861 119905) could be writtenas follows

1205971198751 (1198861 119905)120597119905 = 1205871198781198911 (1205961)2120596121205972119875112059711988612

+ 1205971205971198861 [120585112059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751 (30)

where 119875(1198861 0) = 120575(1198861)Assume that the form of 1198751(1198861 119905) is described as follows

1198751 (1198861 119905) = 11988611198881 (119905) exp[minus1198861221198881 (119905)] (31)

where 1198881(119905) is the undetermined functionEquation (31) is substituted into (28) we transform the

system of (31) into the following form

1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (32)

6 Mathematical Problems in Engineering

u

ug

me

ke

ce

kG1

P0G1(t)

u

ug

me

ke

ce

kG1

cG

u1

Figure 3 Calculation diagram

Then (32) is substituted into (31) we can obtain theanalytical solution of 1198751(1198861 119905)

According to (29) and (32) we can obtain the responsevariance of the structural displacement and velocity respec-tively

119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (33)

119864 [ 1199062 (119905)] = 120596121198881 (119905) = 1205871198781198911 (1205961)212058511205961 [1 minus 119890minus212058511205961119905] (34)

5 Equivalent Damping of WeakNonlinear Structure with the ViscoelasticDamping and the Braces

The actual ground motion is highly random characteristicsBecause of the rationality and practicality of the earthquakethe ground motion model still needs to be further improvedSo the the response spectrum method is adopted in mostcountries Once the structure is installed with the damperand it turns into an energy dissipation structure the responsespectrum method can not be directly applied to thesestructures Therefore it is greatly significant to establishthe equivalent structure which can be used directly withthe response spectrum method The calculation diagram isshown in Figure 3

Where 11987501198661(119905) = int1199050 ℎ1198661(119905 minus 120591)(120591)119889120591 is the equivalent toa damping force of 119888119866 from (4) the motion equation of thestructure can be described as follows

119898119890 + (119888119890 + 119888119866) + (119896119890 + 1198961198661) 119906= minus119898119890 (119892 (119905) + V (119905)) + 1198650

(35)

In this case (35) may be written as the following form

+ 2 (1205851 + 120585119866) 1205961 + 12059612119906 = 1198911 (119905) (36)

where 120585119866 = 11988811986621198981198901205961 1198911(119905) = (minus119898119890(119892(119905) + V(119905)) + 1198650)119898119890According to the stochastic averaging method it is

known that the probability density function of the amplitude

response (1198601(119905)) of the equivalent structure is 1198751(1198861 119905) Theprobability density function fitting the FPK equation is asfollows

1205971198751 (1198861 119905)120597119905= 1205871198781198911 (1205961)212059612

1205972119875112059711988612

+ 1205971205971198861 [(1205851 + 120585119866) 12059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751

(37)

The amplitude probability density function of the originalstructure can be applied to (30) the amplitude probabilitydensity function of the equivalent structure is appropriate for(37)We will know the difference by comparing with (30) and(37) After the following processing the expression can beexpressed as follows

120585119866 = 119867119888 (1205961)21205961119898119890 =11986411986621 (1205961)1205961 sdot 1

21205961119898119890 =11986411986621 (1205961)212059612119898119890

119888119866 = 11986411986621 (1205961)1205961

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

(38)

where 120585119866 is the equivalent damping ratio of damper it isconsistent with the equivalent damping ratio of the Maxwelldamper with the general integral model For arbitraryrandom biaxial earthquake excitations 119892(119905) and V(119905) allstochastic response characteristics calculated with the pro-posed method in equivalent structure are the same as theseof the original structure The equivalent damping ratio of thewhole weak nonlinear dissipation structure is established asfollows

120585119911 = 1205851 + 120585119866 (39)

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

u

ug

me

ke

ce

kG1

P0G1(t)

u

ug

me

ke

ce

kG1

cG

u1

Figure 3 Calculation diagram

Then (32) is substituted into (31) we can obtain theanalytical solution of 1198751(1198861 119905)

According to (29) and (32) we can obtain the responsevariance of the structural displacement and velocity respec-tively

119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)2120585112059613 [1 minus 119890minus212058511205961119905] (33)

119864 [ 1199062 (119905)] = 120596121198881 (119905) = 1205871198781198911 (1205961)212058511205961 [1 minus 119890minus212058511205961119905] (34)

5 Equivalent Damping of WeakNonlinear Structure with the ViscoelasticDamping and the Braces

The actual ground motion is highly random characteristicsBecause of the rationality and practicality of the earthquakethe ground motion model still needs to be further improvedSo the the response spectrum method is adopted in mostcountries Once the structure is installed with the damperand it turns into an energy dissipation structure the responsespectrum method can not be directly applied to thesestructures Therefore it is greatly significant to establishthe equivalent structure which can be used directly withthe response spectrum method The calculation diagram isshown in Figure 3

Where 11987501198661(119905) = int1199050 ℎ1198661(119905 minus 120591)(120591)119889120591 is the equivalent toa damping force of 119888119866 from (4) the motion equation of thestructure can be described as follows

119898119890 + (119888119890 + 119888119866) + (119896119890 + 1198961198661) 119906= minus119898119890 (119892 (119905) + V (119905)) + 1198650

(35)

In this case (35) may be written as the following form

+ 2 (1205851 + 120585119866) 1205961 + 12059612119906 = 1198911 (119905) (36)

where 120585119866 = 11988811986621198981198901205961 1198911(119905) = (minus119898119890(119892(119905) + V(119905)) + 1198650)119898119890According to the stochastic averaging method it is

known that the probability density function of the amplitude

response (1198601(119905)) of the equivalent structure is 1198751(1198861 119905) Theprobability density function fitting the FPK equation is asfollows

1205971198751 (1198861 119905)120597119905= 1205871198781198911 (1205961)212059612

1205972119875112059711988612

+ 1205971205971198861 [(1205851 + 120585119866) 12059611198861 minus

1205871198781198911 (1205961)2119886112059612 ]1198751

(37)

The amplitude probability density function of the originalstructure can be applied to (30) the amplitude probabilitydensity function of the equivalent structure is appropriate for(37)We will know the difference by comparing with (30) and(37) After the following processing the expression can beexpressed as follows

120585119866 = 119867119888 (1205961)21205961119898119890 =11986411986621 (1205961)1205961 sdot 1

21205961119898119890 =11986411986621 (1205961)212059612119898119890

119888119866 = 11986411986621 (1205961)1205961

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761

(38)

where 120585119866 is the equivalent damping ratio of damper it isconsistent with the equivalent damping ratio of the Maxwelldamper with the general integral model For arbitraryrandom biaxial earthquake excitations 119892(119905) and V(119905) allstochastic response characteristics calculated with the pro-posed method in equivalent structure are the same as theseof the original structure The equivalent damping ratio of thewhole weak nonlinear dissipation structure is established asfollows

120585119911 = 1205851 + 120585119866 (39)

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

u

ug

me

ke

ce

P0G1(t)

kb1kQ1

k0

c0 u

ug

me

ke

ce

cG

u1

kG1

Figure 4 Calculation diagram

That is the equivalent structure can be used as a totalequivalent ratio of 120585119911 instead of the original structure damp-ing ratio 1205851 then we can use response spectrum method forstructural analysis and engineering design

6 Numerical Example

It shows a SDOF nonlinear generalized Maxwell damperenergy dissipation structure and the equivalent structure inFigure 4 the earthquake intensity is 8 degrees (02 g) itsmass stiffness damping and damping ratio are respectively119898119890 = 2 kg 119896119890 = 100Nm 119888119890 = 2Nsdotsm and 1205851 = 005The nonlinear structure is subjected to transient forces underbiaxial earthquake 1198781198911 = 1198911(0) = 500 times 10minus6 (m2s3)119879 = 02 s The performance parameters of Maxwell damperin parallel are listed as follows the brace 1198961198871 = 200Nmequilibrium modulus 1198961198761 = 200Nm ℎ1198761 = 200 sminus2element damping coefficient 1198880 = 30Nsdotsm and the stiffness1198960 = 50 kNm The excellent frequency and damping ratioof the site are 1205961198921 = 967 sminus1 and 1205851198921 = 09 respectivelySpectral intensity factor 1198780 = 001387m2s3 According tothe equivalent damping ratio formula when 119905 = 02 s theattached equivalent damping ratio 120585119866 of damper and theresponse variance of equivalent structural displacement arecalculated the response variance of original structure is alsoobtained by the frequency domain method

11986411987611 = 11987000 + 119870012058802120596121 + 1205880212059612

11986411987621 (1205961) = 119888012059611 + 1205880212059612 =30 times 10

1 + 036 times 100 =30037

= 81Nm1198701198761 = 11986411987611 (0) = 119870001205880 = 11988801198960 =

301198960 = 06

11986411987611 (1205961) = 1198961198761 + 1205961 intinfin

0ℎ1198761 (119905) sin1205961119905 119889119905

1198701198761 = 11986411987611 (0) = 11987000 = 1198961198761 + 10 times 0= 200Nm

1198701 = 11986411987611 (1) = 1198961198761 + 1 sdot 200 sdot (minus cos 02)= 200 minus 200 times 098 = 4Nm

11986411987611 = 200 + 1198701 times 036 times 1001 + 036 times 100 = 200 + 14437

= 20389Nm(40)

According to (2) (14) and (35) we can obtain the valueof the following parameters

1198961198661 = 119896119887111989611987611198961198871 + 1198961198761 =200 times 200200 + 200 = 100Nm

12059612 = 119896119890119898119890 +11989611988711198961198761

119898119890 (1198961198871 + 1198961198761)= 100

2 + 200 times 2002 times (200 + 200) = 100 119904minus2

1205961 = 10 sminus1

119888119866 = 11986411986621 (1205961)1205961= 1198961198871211986411987621 (120596)[1198961198871 + 11986411987611 (120596)]2 + 119864211987621 (120596) sdot

11205961

(41)

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Hence

119888119866 = 11989611988712 sdot 1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905(1198961198871 + 1198961198761 + 1205961 intinfin0 ℎ1198761 (119905) sin1205961119905 119889119905)2 + (1205961 intinfin0 ℎ1198761 (119905) cos1205961119905 119889119905)2

119888119866 = 11989611988712 (1198880 (1 + 1205880212059612))[1198961198871 + 11986411987611 (1205961)]2 + 119864211987621 (1205961) =

2002 times (30 (1 + 036 times 100))(200 + 20389)2 + 812 = 324324324

163127132 + 6561 =32432432416319274 = 0199

(42)

The total coefficient of the parallel spring group is equalto the sum of the coefficients of each spring

119896119911 = 1198961198661 + 1198960 + 119896119890 = 100 + 50 + 100 = 250Nm (43)

According to (36) 120585119911 can be calculated as follows

120585119911 = 1205851 + 120585119866 = 11988811989021198981198901205961 +11988811986621198981198901205961 = 005 + 0005

= 0055(44)

From (32) and (34) we can conduct the following calcu-lations

1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961times02]

= 314 times 5002 times 0055 times 103 (1 minus 119890minus2times0055times10times02) times 10minus6

= 1570110 times (1 minus 119890minus022) times 10minus6

= 1427 times 02 times 10minus6 = 2854 times 10minus6m2

(45)

Hence we can obtain the following parameters values

1205901199062 = 119864 [1199062 (119905)] = 1198881 (119905) = 1205871198781198911 (1205961)212058511991112059613 [1 minus 119890minus21205851199111205961119905]

= 2854 times 10minus6m21205902 = 119864 [ 1199062 (119905)] = 120596121198881 (119905)

= 1205871198781198911 (1205961)21205851199111205961 [1 minus 119890minus21205851199111205961119905]

= 314 times 5002 times 0055 times 10 (1 minus 119890minus2times0055times10times02) times 10minus6

= 157011 times (1 minus 119890minus022) times 10minus6

= 142727 times (1 minus 08) times 10minus6

= 0285 times 10minus3m2 sdot sminus2 sdot 1205902119906max = 1205871198781198911 (1205961)212058511991112059613= 314 times 5002 times 0055 times 103 = 1427 times 10minus6m2 times 10minus6

(46)According to the frequency domain method frequency

response function and the variance of displacement areobtained respectively

119867119906 (1205961) = 1198610 + 1198611 (1198941205961)1198600 + 1198601 (1198941205961) + 1198602 (1198941205961)2 + 1198603 (1198941205961)3

1205901199062 = intinfin

minusinfin

1003816100381610038161003816119867119906 (1205961)10038161003816100381610038162 11987811989111198891205961

= 1205871198781198911 (119860011986112 + 119860211986102)1198600 (11986011198602 minus 11986001198603)= 314 times 500 times (100 times 062 + 16 times 12)

100 times (76 times 1 minus 100 times 06)times 10minus6 = 958 times 10minus6m2

(47)

where1198610 = 11198611 = 11988801198960 =

3050 = 06

1198600 = 119896119890 + 1198961198661119898119890 = 2002 = 100

1198603 = 11988801198960 = 06

1198601 = 119888119890119898119890 +(119896119890 + 1198961198661) 11988801198960119898119890 + 1198880119898119890 =

22 +

200 sdot 3050 sdot 2 + 30

2= 76

1198602 = 1 + 11988811989011988801198981198901198960 = 1 +2 sdot 302 sdot 50 = 16

(48)

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 1 Results comparison of the frequency domain method and the proposed method in this paper

Damping coefficient1198880NsdotsmApproximate calculation formula of

frequency domain methodMethod proposed in this paper

Displacement standard deviationm (10minus3) Displacement standard deviationm (10minus3) Relative error10 27 3897 44315 2950 3867 31120 2950 3838 30125 3032 3809 25630 3095 3778 22135 3145 3751 19340 3186 3724 16945 3223 3695 146

The relative error can be calculated

Error =10038161003816100381610038161003816radic14270 minus radic95810038161003816100381610038161003816radic958 = 0221 (49)

It is known that we have calculated the maximum dis-placement standard deviation by frequency domain methodand equivalent structure The results of maximum displace-ment standard deviation are given in Table 1 Results of thetwo methods are gradually approaching with the increase ofthe damping coefficientThemaximum displacement relativeerror is gradually reduced with the increase of the dampingcoefficient When 1198620 increases to a certain value the resultshave a higher precision accuracy

7 Conclusions

In this paper a weak nonlinear structural system with onedegree of freedom is researched and a systematically researchon the random response characteristic of structure wasconducted which is under biaxial earthquake action Firstintegral constitutive relation is adopted it then establishesa differential and integral equations of motion of SDOFweak nonlinear structure containing the general integralmodel viscoelastic dampers and the braces And then themotion equation is linearized according to the principle ofenergy balance Finally based on the stochastic averagingmethod the general analytical solution of the variance of thedisplacement velocity response and equivalent damping isdeduced and derived The joint probability density functionof the amplitude and phase and displacement and velocityof the energy dissipation structure are also given at thesame time Numerical example shows the availability andaccuracy of the proposed method It means it has establisheda complete analytical solution of stochastic response analysisand equivalent damping of a SDOF nonlinear dissipationstructure with the brace under biaxial earthquake actionin this paper The proposed method provides a beneficialreference for the engineering design of this kind of structure

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (51569005 51468005 and 51469005)Guangxi Natural Science Foundation of China (2015GXNS-FAA139279 and 2014GXNSFAA118315) Innovation Projectof Guangxi Graduate Education in China (GKYC201628GKYC201711 and YCSZ2015207) and Innovation Team ofGuangxi University of Science and Technology 2015

References

[1] M Asano Y Yamano K Yoshie Y Koike K Nakagawa andT Murata ldquoDevelopment of active-damping bridges and itsapplication to triple high-rise buildingsrdquo JSME InternationalJournal Series C Mechanical Systems Machine Elements andManufacturing vol 46 no 3 pp 854ndash860 2003

[2] A V Bhaskararao and R S Jangid ldquoSeismic analysis of struc-tures connected with friction dampersrdquo Engineering Structuresvol 28 no 5 pp 690ndash703 2006

[3] Y L Xu Q He and J M Ko ldquoDynamic response of damper-connected adjacent buildings under earthquake excitationrdquoEngineering Structures vol 21 no 2 pp 135ndash148 1999

[4] Y L Xu S Zhan J M Ko et al ldquoExperimental investigationof adjacent buildings connected by fluid damperrdquo EarthquakeEngineering amp Structural Dynamics vol 28 no 6 pp 609ndash6311999

[5] W S Zhang and Y L Xu ldquoVibration analysis of two buildingslinked by maxwell model-defined fluid dampersrdquo Journal ofSound amp Vibration vol 233 no 5 pp 775ndash796 2000

[6] S D Bharti S M Dumne and M K Shrimali ldquoSeismicresponse analysis of adjacent buildings connected with MRdampersrdquo Engineering Structures vol 32 no 8 pp 2122ndash21332010

[7] R E Christenson B F Spencer and E A Johnson ldquoSemi-active connected control method for adjacent multidegree-of-freedom buildingsrdquo Journal of Engineering Mechanics vol 133no 3 pp 290ndash298 2007

[8] Y L Xu and C L Ng ldquoSeismic protection of a building complexusing variable friction damper experimental investigationrdquoJournal of Engineering Mechanics vol 134 no 8 pp 637ndash6492008

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[9] R E Christenson B F Spencer Jr N Hori and K Seto ldquoCou-pled building control using acceleration feedbackrdquo Computer-Aided Civil and Infrastructure Engineering vol 18 no 1 pp 4ndash18 2003

[10] Y Zhang and W D Iwan ldquoStatistical performance analysisof seismic-excited structures with active interaction controlrdquoEarthquake Engineering amp Structural Dynamics vol 32 no 7pp 1039ndash1054 2003

[11] T T Soong and G F Dargush Passive Energy DissipationSystems in Structural Engineering JohnWiley and Ltd England1997

[12] S W Park ldquoAnalytical modeling of viscoelastic dampers forstructural and vibration controlrdquo International Journal of Solidsand Structures vol 38 no 44-45 pp 8065ndash8092 2001

[13] K-C Chang and Y-Y Lin ldquoSeismic response of full-scalestructurewith added viscoelastic dampersrdquo Journal of StructuralEngineering vol 130 no 4 pp 600ndash608 2004

[14] J S Hwang and J C Wang ldquoSeismic response predictionof HDR bearings using fractional derivative Maxwell modelrdquoEngineering Structures vol 20 no 9 pp 849ndash856 1998

[15] A Aprile J A Inaudi and J M Kelly ldquoEvolutionary modelof viscoelastic dampers for structural applicationsrdquo Journal ofEngineering Mechanics vol 123 no 6 pp 551ndash560 1997

[16] R Lewandowski and B Chorązyczewski ldquoIdentification of theparameters of the Kelvin-Voigt and the Maxwell fractionalmodels used to modeling of viscoelastic dampersrdquo Computersand Structures vol 88 no 1-2 pp 1ndash17 2010

[17] M Amjadian and A K Agrawal ldquoAnalytical modeling of asimple passive electromagnetic eddy current friction damperrdquoin Active and Passive Smart Structures and Integrated Systems2016 Proceedings of SPIE 9799 March 2016

[18] J A Fabunmi ldquoExtended damping models for vibration dataanalysisrdquo Journal of Sound amp Vibration vol 101 no 2 pp 181ndash192 1985

[19] G Pekcan B J Mander and S S Chen ldquoFundamentalconsiderations for the design of non-linear viscous dampersrdquoEarthquake Engineering amp Structural Dynamics vol 28 no 11pp 1405ndash1425 1999

[20] S Rakheja and S Sankar ldquoLocal equivalent constant rep-resentation of nonlinear damping mechanismsrdquo EngineeringComputations vol 3 no 1 pp 11ndash17 1986

[21] J B Roberts ldquoLiterature review response of nonlinearmechanical systems to random excitation part 2 equivalentlinearization and other methodsrdquo Shock ampVibration Digest vol13 no 5 pp 13ndash29 1981

[22] D W Malone and J J Connor ldquoTransient dynamic response oflinearly viscoelastic structures and continuardquo in Proceedings ofthe Structural Dynamics Aeroelasticity Specialisted Conferencepp 349ndash356 AIAA New Orleans La USA 1969

[23] Y Kitagawa Y Nagataki and T Kashima ldquoDynamic responseanalyses with effects of strain rate and stress relaxationrdquo Trans-actions of the Architectural Institute of Japan pp 32ndash41 1984

[24] H J Park J Kim and K W Min ldquoOptimal design of addedviscoelastic dampers and supporting bracesrdquo Earthquake Engi-neeringampStructural Dynamics vol 33 no 4 pp 465ndash484 2004

[25] M P Singh N P Verma and L M Moreschi ldquoSeismic analysisand design with Maxwell dampersrdquo Journal of EngineeringMechanics vol 129 no 3 pp 273ndash282 2003

[26] Y Chen and Y H Chai ldquoEffects of brace stiffness on perfor-mance of structures with supplemental Maxwell model-basedbracendashdamper systemsrdquo Earthquake Engineering amp StructuralDynamics vol 40 no 1 pp 75ndash92 2010

[27] S L Xun ldquoStudy on the calculation formula of equivalentdamping ratio of viscous dampersrdquo Engineering EarthquakeResistance and Reinforcement and Reconstruction vol 36 no 5pp 52ndash56 2014

[28] HWenfu C Chengyuan and L Yang ldquoA comparative study onthe calculation method of equivalent damping ratio of viscousdampersrdquo Shanghai Structural Engineer vol 32 no 1 pp 10ndash162016

[29] Y-H Li and B Wu ldquoDetermination of equivalent dampingrelationships for direct displacement-based seismic designmethodrdquo Advances in Structural Engineering vol 9 no 2 pp279ndash291 2006

[30] Y Yang W Xu Y Sun et al ldquoStochastic response of nonlin-ear vibroimpact system with fractional derivative excited byGaussian white noiserdquo Communications in Nonlinear Science ampNumerical Simulation 2016

[31] Y Wu and W Fang ldquoStochastic averaging method for esti-mating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu systemrdquo Acta Mechanica SinicaLixue Xue-bao vol 24 no 5 pp 575ndash582 2008

[32] H Xiong andW Q Zhu ldquoA stochastic optimal control strategyfor viscoelastic systems with actuator saturationrdquo Probabiliste-dic Engineering Mechanics vol 45 pp 44ndash51 2016

[33] L Chuang di G X Guang and L Yunjun ldquoRandom responseof structures with viscous damping and viscoelastic damperrdquoJournal of Applied Mechanics vol 28 no 3 pp 219ndash225 2011

[34] L Chuang di G X Guang and L Yun jun ldquoEffective dampingof damping structure of viscous and viscoelastic dampersrdquoJournal of Applied Mechanics vol 28 no 4 pp 328ndash333 2011

[35] B-C Wen Y-N Li and Q-K Han Analytical Methods AndEngineering Application of The Theory of Nonlinear VibrationNortheastern University Press Shenyang China 2001

[36] Fang-TongVibration of Engineering NationalDefence IndustryPress Beijing China 1995

[37] Y K Lin ldquoSome observations on the stochastic averagingmethodrdquo Probabilistic Engineering Mechanics vol 1 no 1 pp23ndash27 1986

[38] W-Q Zhu Random Vibration Science Press Beijing China1998

[39] GBT 50011-2010 Code for Seismic Design of Buildings Chinabuilding industry press Beijing China 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of