Am ir YOUNESPOUR,Hosein GHAFFARZADEH,Bahman Farahmand AZAR

Faculty of Civil Engineering,University of Tabriz,Tabriz 51666,Iran

1 Introduction

In the analysis of linear system s,the physical solution can be obtained,both in time and frequency domains,by using analytical approaches that are able to describe response moments statistics considering input characteristics and system parameters.Moreover,there are many cases where structures exhibit a nonlinear behavior,such as the buildings subject to strong earthquakes and ships in ocean.In those situations analytical approach does not work for many nonlinear system s because the theoretical solutions of nonlinear system s have been found for only some significant special cases[1].The perturbation method is one of the most effective methods in the analysis of deterministic nonlinear system s.It is worthy of pointing out that many of the existing methods only focus on development of approximate methods,since the exact methods require significantly m ore com putational efforts,which,som etimes is not available.Therefore,it is necessary to find approximate solutions for nonlinear system s.An alternative approach is the stochastic equivalent linearization(EL)method,proposed independently but sim ultaneously by the three authors,Caughey[2],Booton[3]and Kazakov[4].This method is popular in the engineering due to its capability for various applications in engineering.The coefficients of linearized term s are obtained m inim izing a stochastic m easure of differences between the tw o solutions.In order to predict the response of this kind of system or to get an approximation solution of nonlinear equation this method is applied to estim ate the accurate equivalent linear param eters.All m ethods of EL can be considered in different fields such asstate space,tim e domain,distribution space,frequency domain and characteristic function space.Usually this technique consists of two main steps.In the first one finding explicit or im p licit analytical form ulas for linearization coefficients based on the linearization criterion w hich is depending on unknown response characteristics such as mean value,variance,and higher-order moments.In the second step,replacing the unknown characteristics of the corresponding ones determ ined for linearized system s.Itis im portant to bear inm ind the accuracyand feasibility of thesesolutionsdepend on the type of nonlinearity and am plitude of external excitation forces.In EL method the coefficients of the equivalent system can be found from a specified optim ization criterion,such as the m ean-square criterion[2],spectral criteria[5],probability density criteria[6]and energy criteria[7]in some probabilistic sense.Foster[8],Iwan and Yang[9],and Atalik and Utku[10]generalized this m ethod to random vibration of m ulti-degree of freedom(MDOF)system s.Later,the m ethod of stochastic EL generalized to a nonlinear system subjected to both parametric and external random white noise excitations by Bruckner and Lin[11].Grigoriu[12,13]and Proppe[14]used equivalent linearization for different classes of input processes(e.g.,Levy w hite noise,Poisson process).Ricciardi[15]used a modified Gram-Charlier series approximation of the probability density function to develop a non-Gaussian stochastic linearization method of nonlinear structural system s under white noise excitation.Som e new ideas about the stochastic linearization method are presented in[16,17]by several authors.For hyperbolic tangent oscillators,Colajanni and Elishakoff[16]proposed a new method of stochastic linearization technique.In 1993, and Faravelli[17]discussed a new philosophy for stochastic equivalent linearization.In[18,19],Anh et al.have proposed a dual criterion of stochastic EL method for nonlinear single and multi-degree-of freedom system s subjected to random excitation.In 2013,based on the dual conception,a consideration at global level to the local mean square error criterion of the stochastic EL method has been studied by Anh et al.[20].

The concept of the orthogonal functions is old and well described in the literature.Depending upon the structure,the orthogonal functions may be broadly classified into three families,including piecewise constant orthogonal functions,orthogonal polynomials and Fourier functions[21].The orthogonal functions play a prominent role in the num erical analysis and approximation theory for improving its accuracy.Since each class of these functions,form s a basis for the series expansion of a square-integrable function,orthogonal functions are comm only referred to as Basis Functions.These functions have been used since the m id-1970s for the identification of dynamic system s[22].More recently,these functions have found application in control system s,identification and sensitivity analysis[23–27].If an orthogonal function is converted to an orthonormal one,it would not only yield a m ore accurate approximation,but also simplify the mathematical operation.Among these functions,Block Pulse(BP)functions are naturally orthonormal.Because of this property,in comparison with other basis functions or polynomials,the block pulse functions can lead m ore easily to recursive computations to solve concrete problem s[28].The BP functions are a set of orthogonal functions with piecewise constant values and are usually app lied as a useful tool in the analysis,identification and other problem s of control and system s science.

For nonlinear system s subjected to random excitations,though the EL method is a simple approach from the theoretical point of view,the implementation by numerical techniques needs lots of computational efforts.Tim e domain EL methods have much higher computational efficiency than the existing methods in other domains such as frequency domain.The methodology used in the present contribution exploits the BP function as an orthogonal function in the linearization procedure through its simple and easy operation.In the proposed method,the mathematical expectations of the stationary system responses can be directly determined in the time domain by using the BP functions and there is no need to transform to other domains.Thus,the proposed approach is more efficient when com pared with the frequency and the mixed time-frequency domain methods,of which a large number of time-history integrals are required at different frequency intervals when random excitations are involved.Besides,the formulation of the proposed method might allow it to be applicable to more general types of excitations,such as the ground motion excitation.In current study,a SDOF nonlinear Van der Pol oscillator system subjected to stationary Gaussian white noise excitation has been considered.The effectiveness of the proposed method is validated by com paring the mean-square responses and FRF of the system s that linearized by different approaches.The comparison revealed that the results of the proposed method based on orthogonal functions are similar to other existing approaches.Also,Monte Carlo(MC)simulation approach has been used as a benchmark to com pare the accuracy of existing methods.Results revealed that com pared to the existing EL methods,the proposed EL method based on BP functions can give more accurate approximations.

The rest part of this paper is organized as follows.We start with a review of orthogonal functions in Section 2.Section 3 presents the equivalent linearization process using orthogonal functions.The case studies have been carried out for comparison,in Section 4.The conclusion in Section 5 closes this paper.

2 Orthogonal function review

The set of functions φi(t)(i=1,2,3,...)is said to be orthogonal over the interval[a,b]if

where

where Kmnis a nonzero positive constant.If Kmnis the Kronecker’s delta,the set of functions φi(t)is said to be orthonormal.The following property,related to the successive integration of the vectorial basis,holds for a set of r orthonormal functions:

where[P]∈Rr,ris a square matrix with constant elements,called operator or operational matrix which is depend on the type of orthogonal function and{φ(t)}={φ0(t),φ1(t),...,φr?1(t)}Tis the vectorial basis of the orthonormal series.This operator has a key role in the methodology procedure.The operators give a proper mathematical frame for the orthogonal functions and are advantageous to the convergence analysis of their series expansions.In other words,the operators produce an image matrix or vector of the function f(t)in the orthogonal function domain.

A set of BP function on a unit time interval[0,1)is defined as[29]

where i=0,1,2,...,m?1 with a positive integer value for m.Also,consider h=1/m,and φiis the i th BP function.

In the BP domain the block pulse operator B determines as follows:

where the vector F is evaluated from

The BP operator has lots of operation rules.Some of them which will be applied in the next section are following[28].

For a real constant k,we have

where ETis a constant vector with all entries ones.

For Addition and subtraction of functions f(t),g(t)∈[0,T),we have

This equation can be derived directly from the linearity of BP operator.

For integration of a function f(t)∈[0,T),we have

where P is the conventional integration operational ma-trix,defined as follow s:

For the convolution integral of functions f(t)and g(t)∈[0,T),we have

where JGand JFare the convolution operational matrices defined in equations(11)and(12).

and for multiple integrals,we have the following rule:

3 The equivalent linearization technique through orthogonal functions

A system which is a SDOF with the nonlinear function only depending on two argum ents of displacem ent and velocity has been considered:

w here ξ and ω are the dam ping ratio and the frequency of the system,respectively;g(x(t),˙x(t))is a nonlinear function,w(t)is a zero mean Gaussian stationary process with the following correlation function:

in which E[·]denotes the expectation value;σ2is variance and δ(τ)is Dirac delta function.

The linearization of equation(14)leads to following equation:

where the coefficients of linearization ceqand keqare found by EL approach.There are some criteria for determining these coefficients(see[30]).

The replacement of a nonlinear system by a linear system is in some probabilistic sense and it will yield the difference or error.The error defined as

As the excitation is random,an apparently sensible strategy would be to minimize the average difference between the nonlinear system and the linearized system.In fact,this is not sensible as the differences will generally be a mixture of negative and positive and could still average to zero for a wildly inappropriate system.The correct strategy is to minimize the expectation of the squared differences,i.e.,find the ceqand keqwhich minimize

The coefficients ceqand keqare determined by the following equation:

When the excitation to the original nonlinear system is a stationary Gaussian function,assuming the response of the nonlinear system is also Gaussian.Thus,the stationary displacement and velocity responses are uncorrelated,i.e.,E[y˙y]=0.

Therefore,

and

As noted from equations(20)and(21),the linearization coefficients depend on the mean-square response E[y2]of the linearized system.The value of E[y2]is evaluated by

where p(y)is probability density function(PDF).The problem in this procedure is that the exact PDF of the response is available for some special nonlinear systems and currently the PDF of the response is not known for general nonlinear system s.

In the following development of the EL method,a technique proposed in the time domain to evaluate the value of mean square response of the linearized system.If H(ω)is considered as the frequency response function of the linearized system and h(t)is considered as the unity impulse response of the linearized system,the response y(t)of the system(16)can be expressed by the Duhamel integral

The unity impulse response of the linearized system is defined as follow s:

where m and ωnare m ass and natural frequency of the linearized system,respectively.By using equation(24)for evaluating second-order moment of response,we have[31]

Substituting these moments into equations(20)and(21)yields a closed system of unknowns ceqand keq.In general,the obtained system is a nonlinear algebraic one.It may be solved numerically for determining the linearization coefficients.An iterative scheme for finding unknown ceqand keqcan be obtained as follow s(see[8,9]):

a)Assign initial estimations of ceqand keqin order to obtain the mean square response of displacement and velocity(E[y2],E[˙y2]).

b)Substitute the obtained values into equations(20)and(21)to obtain new estimations for ceqand keq.

c)Use these values and return to step a).

d)Repeat steps a),b)and c)until the results from cycle to cycle are similar.

In the proposed approach,the mean square response of the linearized system is calculated by operational rules of the orthogonal functions,i.e.,equations(10)–(13).By using the convolution integral(equation(10))and multiple integrals(equation(13))operators of the BP functions we have

where

and by considering l(t)=˙h(t),we have

4 Illustration

We consider a nonlinear Van der Pol system.The system equation of motion has the form of

where μ and σ are positive real constants.The linearization equation of equation(30)takes the follow ing form:

Based on the criterion equation(19)and using equations(20)and(21)withwe obtain the coefficients ceqand keqof the linearized equation(31)as follow s:

and

Now by using the proposed method and the iteration technique described earlier,the following equation can be solved for computing the mean square response value at each time step.

In fact,one canuseeither the linearization coefficients or responses for the outset of the iterative process.In the frame of this fixed-point iteration,nonlinearities of the system belonging to a stable type are necessary for the convergence of the process.In the case of stochastic EL m ethod,this point was shown and applied successfully in Refs.[9,10]to some nonlinear system s.In this study using the mentioned iterative scheme is successful and the convergence of the process is fast.

The results of the proposed method are com pared with the results calculated via one-step regulation linearization method[32]

and the dual criterion linearization method[20].

For better understanding it is interesting to com pare the results of the approximate mean square response values of the Van der Pol oscillator equation(30)obtained by the proposed method equation(34),one-step regulation equation(35)and the dual criterion equation(36)linearization methods.In Figs.1 and 2 the E[x2]values of the approximate solutions of existing methods for ω =1.0 and ξ =0.05,as well as for various values of nonlinearity parameter and intensity of excitation are com pared.The value of number basis elements is m=100.

Fig.1 Mean square value for different nonlinearity parameter(σ2=2.0).

Fig.2 Mean square value for different intensity of excitation(μ=20.0).

Furthermore,the comparison results have been presented in Tables 1 and 2 for different values of nonlinearity μ and excitation intensity σ2.

Table 1 The mean-square responses E[x2]of the Van der Pol oscillator versus the parameter μ (σ2=1).

Table 2 The mean-square responses E[x2]of the Van der Pol oscillator versus the parameter σ2(μ =1).

Fig.3 for μ=0.1,1,10 and 100 provide a comparison among existing methods by com paring the FRFs of linearized system via present method,regulation method and dual criterion linearization method.

Fig.3 Frequency response function of linearized system.(a)μ=0.1.(b)μ=1.(c)μ=10.(d)μ=100.

The procedure of the proposed EL method based on BP function has been app lied in the time domain.The comparison results of mean square values depicted the results of the proposed method match closely with the regulation and dual criterion linearization approaches.Besides,by com paring the FRF of linearized system,it can be seen the proposed method are in agreement with other methods.

5 Conclusions

In this contribution,we studied the equivalent linearization of nonlinear Van der Pol system in the time domain via orthogonal functions.The m ean square values of the response to a wide range of nonlinearity and different stationary inputs are presented.Results confirm ed the efficiency of the proposed method using BP function as an orthogonal function.with this in view,it may be concluded that the EL method can be applied in the time domain with high accuracy for stationary random excitation.The results of existing methods have been com pared with the results of Monte Carlo simulation m ethod.The comparison results revealed that the proposed EL method based on BP functions can give more accurate approximations.Besides,by com paring the FRF of linearized system demonstrated the proposed method is in agreement with other approaches.It is worth noting that the proposed EL method has been applied in the time domain,thus,it is understandable that there exist some negligible difference com pared with the results of the other existing methods.

[1]A.Nayfeh,D.Mook.Nonlinear Oscillations.New York:John Wiley&Sons,1995.

[2]T.K.Caughey.Response of Van der Pol’s oscillator to random excitations.ASME Journal of Applied Mechanics,1956,26:345–348.

[3]R.C.Booton.The analysis of nonlinear control system s with random inputs.IRE Transactions on Circuit Theory,1954,1(1):9–18.

[4]I.E.Kazakov.An approxim ate method for the statistical investigation of nonlinear system s.Trudy VVIA im Prof.N.E.Zhukovskogo,1954,394:1–52.

[5]M.Apetaur,F.Opicka.Linearization of stochastically excited dynam ic system s.Journal of Sound and Vibration,1983,86(4):563–585.

[6]L.Socha.Probability density equivalent linearization technique for nonlinear oscillator with stochastic excitations.Journal of Applied Mathematics and Mechanics,1998,78(S3):1087–1088.

[7]R. C. Zhang. Work/energy-based stochastic equivalent linearization with optimized power.Journal of Sound and Vibration,2000,230(2):468–475.

[8]E.Foster.Sem i-linear random vibrations in discrete system s.ASME Journal of Applied Mechanics,1986,35(3):560–564.

[9]W.D.Iwan,I.Yang.Application of statistical linearization techniques to nonlinear m ulti-degree of freedom system s.ASME Journal of Applied Mechanics,1972,39(2):545–550.

[10]T.Atalik,S.Utku.Stochastic linearization of multi-degree of freedom nonlinear system.Earthquake Engineering&Structural Dynamics,1976,4(4):411–420.

[11]A.Bruckner,Y.K.Lin.Generalization of the equivalent linearization method for the nonlinear random vibration problems.International Journal of Nonlinear Mechanics,1987,22(3):227–235.

[12]M.Grigoriu.Equivalent linearization for Poisson White noise input.Probabilistic Engineering Mechanics,1995,10(1):45–51.

[13]M.Grigoriu.Equivalent linearization for system s driven by Levy white noise.Probabilistic Engineering Mechanics,2000,15(2):185–190.

[14]C.Proppe.Stochastic linearization of dynamical system s under parametric Poisson white noise excitation.International Journal of Nonlinear Mechanics,2003,38(4):543–555.

[15]G.Ricciardi.A non-Gaussian stochastic linearization method.Probabilistic Engineering Mechanics,2007,22(1):1–11.

[16]P.Colajanni,I.Elishakoff.A new look at the stochastic linearization technique for hyperbolic tangent oscillator.Chaos,Solitons&Fractals,1998,9(9):1611–1623.

[17]F.Casciati,L.Faravelli.Anew philosophy for stochastic equivalent linearization.Probabilistic Engineering Mechanics,1993,8(3/4):179–185.

[18]N.D.Anh,N.N.Hieu,N.N.Linh.A dual criterion of equivalent linearization method for nonlinear system s subjected to random excitation.Acta Mechanica,2012,223(3):645–654.

[19]N.D.Anh,V.L.Zakovorotny,N.N.Hieu,et al.A dual criterion of stochastic linearization m ethod for m ulti-degree of freedom system s subjected to random excitation.Acta Mechanica,2012,223(12):2667–2684.

[20]N.D.Anh,L.X.Hung,L.D.Viet.Dual approach to local mean square error criterion for stochastic equivalent linearization.Acta Mechanica,2013,224(2):241–253.

[21]K.B.Datta,M.Mohan.Orthogonal Functions in System s and Control.Singapore:World Scientific Publishing Co.Pte.Ltd,1995.

[22]C.F.Chen,C.H.Hsiao,Tim e-domain synthesis via Walsh functions.Proceedings of the Institution of Electrical Engineers,1975,122(5):565–570.

[23]P.N.Paraskevopoulos,D.A.Sklavounos,D.A.Karkas.A new orthogonal series approach to sensitivity analysis.Journal of the Franklin Institute,1990,327(3):419–433.

[24]R.P.Pachecoa,V.S.Jr.On the identification of non-linear mechanical systems using orthogonal functions.International Journal of Nonlinear Mechanics,2004,39(7):1147–1159.

[25]H.Ghaffarzadeh,A.Younespour.Active tendons control of structures using block pulse functions.Structural Control and Health Monitoring,2014,21(12):1453–1464.

[26]S.Ghosh,A.Deb,S.R.Choudhury,et al.Modeling and analysis of singular systems via orthogonal triangular functions.Journal of Control Theory and App lications.2013,11(2):141–148.

[27]A.Younespour,H.Ghaffarzadeh.Structural active vibration control using active m ass dam per by block pulse functions.Journal of Vibration and Control,2015,21(14):2787–2795.

[28]Z.H.Jiang,W.Schaufelberger.Block Pulse Functions and their Applications in Control System s.Berlin:Springer,1992.

[29]E.Babolian,Z.Masouri.Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions.Journal of Computational and Applied Mathematics,2008,220(1/2):51–57.

[30]J.B.Roberts,P.D.Spanos.Random Vibration and Statistical Linearization.New York:John W iley&Sons,1990.

[31]L.D.Lutes,S.Sarkani.Random Vibrations Analysis of Structural and Mechanical System s.Oxford:Elsevier,2004.

[32]I.Elishakoff,L.Andrimasy,M.Dolley.Application and extension of the stochastic linearization by Anh and Di Paola.Acta Mechanica,2009,204(1/2):89–98.