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Open Physics

formerly Central European Journal of Physics

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Volume 15, Issue 1

Issues

Volume 13 (2015)

Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers

Vasile Marinca
  • University Politehnica Timişoara, Department of Mechanics and Strength of Materials, 300222, Timişoara, Romania
  • Department of Electromechanics and Vibration, Center for Advanced and Fundamental Technical Research, Romania Academy, 300223, Timişoara, Romania
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  • De Gruyter OnlineGoogle Scholar
/ Remus-Daniel Ene / Liviu Bereteu
Published Online: 2017-10-31 | DOI: https://doi.org/10.1515/phys-2017-0072

Abstract

Dynamic response time is an important feature for determining the performance of magnetorheological (MR) dampers in practical civil engineering applications. The objective of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to give approximate analytical solutions of the nonlinear differential equation of a modified Bingham model with non-viscous exponential damping. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. OHAM is very efficient in practice for ensuring very rapid convergence of the solution after only one iteration and with a small number of steps.

Keywords: MR dampers; optimal homotopy asymptotic method; auxiliary functions; optimal parameters

PACS: 02.30.Hq; 02.30.Rz; 02.60.Cb; 02.70.-c; 83.60.Df; 83.80.Gv

1 Introduction

Over the past decade, much attention has been given to magnetorheological (MR) dampers for their characteristics in applications of civil engineering including earthquake hazard mitigation and design for high strength, insensitivity to contamination, and low power requirement. MR fluids have also attracted considerable interest due to their wide range of uses in vibration dampers [1, 2, 3, 4]. Based on experimental data, the mechanical model for an MR damper is often established through an optimization method [5, 6, 7, 8]. Although there are some models which can simulate nonlinear dynamic characteristics, these are established by strong nonlinear equations with a lot of parameters which result in complicated numerical computations. In civil engineering, the Bingham model is often used for emulating the dynamic behavior of MR dampers. The so-called Bingham model can be expressed as [4, 6, 7, 9] τ=τy(H)sgn(γ˙)+ηγ˙,(1)

where sgn( ⋅ ) is the signum function, γ is the shear strain rate, η is the post-yield viscosity independent of the applied magnetic field, τ is the shear stress in the fluid and τy is the yielding shear stress controlled by the applied field H.

Stanway, et al. [5] proposed an idealized mechanical system which consists of a Coulomb friction element placed in parallel with a viscous damper: F(t)=fcsgn(x˙)+c0x˙+f0,(2)

where fc is the coefficient of the frictional force which is related to the fluid yield stress, c0 is the damping coefficient, and f0 denotes an offset in the force. Lee and Wereley[10] and Wang and Gordaninejad [11] used the Herschel-Bulkley model expressed by τ=τy(H)sgn(γ˙)+kγ˙1msgn(γ˙),(3)

where m ≠ 1 is the fluid behavior index of the magnetorheological fluid and k is the consistency parameter. For m = 1, the Herschel-Bulkley model reduces to the Bingham model [12]. Based on the original Bingham plastic and Herschel-Bulkley plastic models, Zubieta, et al. [13] gave plastic models for magnetorheological fluids.

The Bingham body model presented in Figure 2 differs from the Bingham model (Figure 1) by the introduction of a spring k. The Bingham body model contains in parallel three elements, connecting the St. Venant, Newton and Hooke elements. For an applied force up to fc (the static friction force of the St. Venant element) only the spring will deform, similar to an elastic Hooke body.

Bingham model
Figure 1

Bingham model

Bingham body model
Figure 2

Bingham body model

If this force is greater than fc, the Bingham body will elongate. The rate of the deformation will be proportional to the difference of the applied force and the friction force of the St. Venant element [14]. In this case the damping force F can be expressed as F(t)=fcsgn(x˙1)+c0x˙1+f0for|F|>fck(x2x1)+f0for |F|fc,(4)

where k is the stiffness of the elastic body (Hooke model) and the other parameters have the same meaning as in Eq. (2).

An extension of the Bingham model has been formulated by Gamota and Filisko [15]. This viscoelastic-plastic model depends on connection of the Bingham, Kelvin-Voight body and Hooke body models (Figure 3).

Extended Bingham model for the Gamota-Filisko model
Figure 3

Extended Bingham model for the Gamota-Filisko model

The damping force in the Gamota-Filisko model is given by F(t)=k1(x2x1)+c1(x˙2x˙1)+f0==c0x˙1+fcsgn(x˙1)+f0==k2(x3x2)+f0 for |F|>fck(x2x1)+c1x˙2+f0==k2(x3x2)+f0 for |F|fc,(5)

where c0, f0 and fc are known from the Bingham model (2) and the parameters c1, k1 and k2 are associated with the fluid’s elastic properties in the pre-yield region. We remark that if |F| ≤ fc then x˙1=0, which means that when the friction force fc related to the new stress in the fluid is greater than the damping force F, the piston remains motionless. Another view of viscoelastic-plastic properties of MR damper behavior is proposed by Li, et al. [16]. In essence the damping force is equal to the visco-plastic force, to which besides the friction force connected with the fluid shear stress fc, the viscotic force and inertial force contribute, which can be written in the form [14] F(t)=fcsgn(x˙)+c0x˙+mx¨,(6)

where c0 is a co-factor of viscotic friction and m is the mass of replaced MR fluid dependent on the amplitude and frequency of a kinematic excitation applied to the piston.

A discrete element model with similar components, referred to as the BingMax model, is reviewed by Makris, et al. [17] and is depicted in Figure 4.

BingMax model
Figure 4

BingMax model

The force F(t) is given by F(t)=c0tμeμ(tτ)x˙(τ)dτ+fcsgn(x˙(t)),(7)

where μ is a parameter representing the non-viscous damping effect (or the relaxation parameter [18]). The model for the damping force expressed as fd(t)=c0tμeμ(tτ)x˙(τ)dτ,(8)

was originally proposed by Biot [19]. Cremer and Heckl [20] concluded that “of the many after-effect functions, that are possible in principle, one—the so-called relaxation function—is physically meaningful”.

Based on the above considerations, in what follows we consider the behavior of a magnetorheological damper on Bingham models which in addition contain a nonlinear term.

The motion equation is established in the form [7, 9, 18, 19] md2x(t¯)dt¯2+c¯0t¯μ¯eμ¯(t¯τ¯)x˙(τ¯)dτ¯+kx(t¯)+αkx3(t¯)+kβsgn(x˙(t¯))+kf0=0,(9)

where m, c, μ, k, αk, βk and kf0 are mass, damping relaxation, linear stiffness, nonlinear stiffness, coefficient of frictional force and offset of the force, respectively.

The initial conditions are: x(0)=A,x˙(0)=v0km.(10)

In recent years, research effort has been devoted to study of some approaches like the optimal homotopy analysis method (OHAM) [21], homotopy analysis method [22, 23, 24, 25], integral mean value method [26], Adomian decomposition method [27] and other methods [28].

The purpose of the present paper is to analytically solve the nonlinear Bingham model given by Eq. (9) using OHAM. A new version of OHAM is applied in this study.

2 Basic ideas of the optimal homotopy asymptotic method

Eq. (9) with initial conditions (10) can be put in a more general form [29]: N[x(t)]=0,(11)

with N an known nonlinear differential operator depending on the unknown function x(t). The initial conditions are: Bx(t),dx(t)dt=0.(12)

Let x0(t) be an initial approximation of x(t). L and B are arbitrary linear operators such that Lx0(t)=0,Bx0(t),dx0(t)dt=0.(13)

The linear operator L is not unique, as in [29].

For p ∈ [0, 1] an embedding parameter and X an analytic function, we propose to construct a homotopy [29, 30, 31, 32, 33, 34, 35, 36]: HLX(t,p),H(t,Ci),NX(t,p),i=1,2,,s,(14)

with properties HLX(t,0),H(t,Ci),NX(t,0)==LX(t,0)=Lx0(t)=0,(15) HLX(t,1),H(t,Ci),NX(t,1)==H(t,Ci)Nx(t)=0,i=1,2,,s,(16)

with H(t, Ci) ≠ 0 an arbitrary auxiliary convergence-control function depending on variable t and on s arbitrary parameters C1, C2, …, Cs, unknown now and to be determined later.

Let us consider the function X in the form X(t,p)=x0(t)+px1(t,Ci).(17)

By substituting Eq. (17) into the equation obtained by means of homotopy (14), HLX(t,p),H(t,Ci),NX(t,p)=0,i=1,,s(18)

and then equating the coefficients of p0 and p1, we obtain HLX(t,p),H(t,Ci),NX(t,p)==Lx0(t)+pLx1(t,Ci)Lx0(t)H(t,Ci)Nx0(t)=0,i=1,2,,s.(19)

From the last equation, we obtain the governing equation of x0(t) given by Eq. (13) and the governing equation of x1(t, Ci), i.e.: Lx1(t,Ci)=H(t,Ci)Nx0(t),Bx1(t,Ci),dx1(t,Ci)dt=0,i=1,,s,(20)

where the expression for the nonlinear operator has the form Nx0(t)=i=1mhi(t)gi(t).(21)

In Eq. (21), the functions hi(t) and gi(t), i = 1, …, m are known and depend on the initial approximation x0(t) and also on the nonlinear operator, m being a known integer number. In this way, taking into account Eq. (16), from Eq. (17) for p = 1, the first-order approximate solution becomes [29]: x¯(t,Ci)=x0(t)+x1(t,Ci),i=1,,s.(22)

It should be noted that x0(t) and x1(t, Ci) are governed by the linear Eqs. (13) and (20) respectively with initial / boundary conditions that come from the original problem.

Now, we do not solve Eq. (20), but from the theory of differential equations, it is more convenient to consider the unknown function x1(t, Ci), as x1(t,Cj)=i=1nHi(t,hj(t),Cj)gi(t),j=1,,s,Bx1(t,Ci),dx1(t,Ci)dt=0,(23)

where within the expression Hi(t, hj(t), Cj) there appear combinations of some functions hj, some terms of which are obtained from the corresponding homogeneous equation and the unknown parameters Cj, j = 1, …, s, as in [29]. In the sum given by Eq. (23), thenre appear an arbitrary number n of such terms.

We cannot demand that x1(t, Ci) be solutions of Eq. (20), but x(t, Ci) given by Eq. (22) with x1(t, Ci) given by Eq. (23) are the solutions of Eq. (11). This is the underlying idea of our method. The convergence of the approximate solution x(t, Ci) given by Eq. (22) depends upon the auxiliary functions Hi(t, hi, Cj), j = 1, …, s. There exist many possibilities for choosing these functions Hi. We choose the auxiliary functions Hi so that within Eq. (23), the term i=1n Hi(t, hj(t), Cj) gi(t) has the same form as the term i=1m hi(t) gi(t) given by Eq. (21) (see [29]). The first-order approximate solution x(t, Ci) also depends on the parameters Cj, j = 1, …, s. The values of these parameters can be optimally identified via various methods such as the least-squares method, the Galerkin method, the collocation method, the Ritz method, or by minimizing the square residual error: J(C1,C2,,Cs)=DR2(t,C1,C2,,Cs)dt,(24)

where the residual R is given by R(t,C1,C2,,Cs)=N(x¯(t,Ci)).(25)

The unknown parameters C1, C2, …, Cs can be identified from the conditions JC1=JC2==JCs=0.(26)

Thus, the first-order analytic approximate solution given by Eq. (22) is well-known.

3 Application of OHAM to nonlinear Bingham fluid dampers

In what follows, we apply our procedure to analytically solve Eqs. (9) and (10). For this purpose, we introduce the dimensionless variables t=t¯mk,τ=τ¯mk(27)

such that Eq. (9) can be expressed as x¨(t)+2c0tμeμ(tτ)x˙(τ)dτ+x(t)+αx3(t)++βsgn(x˙(t))+f0=0,(28)

where c=c¯2kmμ=mkμ¯ and the overdot represents differentiation with respect to dimensionless time. The initial conditions (10) become x(0)=A,x˙(0)=v0.(29)

Making the transformation x(t)=Aeλtz(t),(30)

where λ is for the moment an unknown real parameter, Eq. (28) can be written z¨2λz˙+(λ2+2μc+1)z2cμ20te(λμ)(tτ)z(τ)dτ+αA2e2λtz3++βAeλtsgn(z˙λz)+f0eλtA2cμe(λμ)t=0.(31)

The initial conditions (29) become z(0)=1,z˙(0)=λ+v0A.(32)

For the nonlinear differential equation (31), we try to choose a linear operator of the form Lz(t)=z¨(t)+ω2z(t),(33)

where ω is an unknown parameter.

The nonlinear operator corresponding to Eqs. (31) and (33) is Nz(t)=2λz˙+(λ2+2μc+1ω2)z2cμ20te(λμ)(tτ)z(τ)dτ+αA2e2λtz3++βAeλtsgn(z˙λz)+f0Aeλt2cμe(λμ)t.(34)

The initial approximation z0(t) can be deduced from Eq. (13) with initial / boundary conditions: z0(0)=1,z˙0(0)=ω2λ.(35)

Eq. (13) with the linear operator (33) and with initial / boundary conditions (35) has the solution z0(t)=cosωtωλsinωt.(36)

According to Eq. (21), the nonlinear operator satisfies Nz0(t)==λ(1+λ2+ω2+2μc)2μ2c(ω2λ2+λμ)λ(ω2+(μλ)2)cosωt++ω(λ2+ω22μc1)(λ2+ω2+λ22μλ)λ(ω2+(μλ)2)++2μ2cω(μ2λ)λ(ω2+(μλ)2)sinωt++2μ2c(ω2λ2)2μcλ(ω2+λ22λμ)λ(ω2+(μλ)2)e(λμ)t+f0Aeλt++αA24e2λt3(ω2+λ2)λ2cosωt+3ω(ω2+λ2)λ3sinωt++λ23ω2λ2cos3ωt+ω(ω23λ2)λ3sin3ωt4βλAeλtcosωt13cos3ωt+15cos5ωt17cos7ωt+19cos9ωt.(37)

From Eqs. (37), (32) and (23), we choose z1(t)=C1t2cosωt+(C2t2+C3t)sinωt++[C4t2+(v0+ω2+λ2λA+3λC7)tcos3ωt++(C5t2+C6t)sin3ωt+C7]e2λtC7eλtcosωt.(38)

The first-order analytic approximate solution of Eqs. (28) and (29) given by Eq. (22) is obtained from Eqs. (36), (38) and (30): x(t)=[(C1t2+A)cosωt++(C2t2+C3tωλA)sinωt]××eλt+{[C4t2+((v0+ω2+λ2λ)A+3λC7)t]cos3ωt++(C5t2+C6t)sin3ωt+C7}e3λtC7cosωt.(39)

4 Numerical results

We highlight the accuracy of our technique for the following values of the parameters: c = 0.1, μ = 20, α = 1, β = 0.1, f0 = 0.1 A = 5, v0 = 0.1. The optimal convergence-control parameters Ci are determined by means of the least-square method in three steps as follows:

For t ∈ [0, 7/2] we obtain: C1=8.3835528344,C2=0.4059363155,C3=9.8101433224,C4=12.7300955924,C5=8.3640772984,C6=14.6926248464,C7=15.2775231617,λ=0.4221369200,ω=1.1700000000.

The first-order analytic approximate solution given by Eq. (39) becomes for this first step: x¯(t)=(8.3835528344t2+5)cosωt++(0.4059363155t2+9.8101433224t13.8580629241)sinωte0.4221369200t++[(12.7300955924t21.0230014969t)cos3ωt++(8.3640772984t214.692624846t)sin3ωt15.2775231617e1.2664107600t++15.2775231617cosωt.(40)

For the second step, when t ∈ [7/2, 7] we obtain: x¯(t)=[(18.7159298670t2++1.3931962652)cosωt+(2.5809325241t212.3307330274t7.2827523763)sinωt]××e0.3726878974t+[(16.1687136688t276.6284415838t)cos3ωt+(1.8324463202t215.4843220143t)sin3ωt68.7655829044]××e1.1180636924t+68.7655829044cosωt,(41)

where ω = 1.9481775386, λ = 0.3726878974.

In the last case, when t ∈ [7, 10], the first-order approximate solution is: x¯(t)=[(0.1521594849t2++2.6425574994)cosωt+(1.4223903751t2++13.8451447750t10.3560770110sinωt××e0.1870115667t+[(7.9954565850t2++12.3657070213tcos3ωt+(1.5073646864t20.3337501284tsin3ωt++1.1037184321××e0.5610347000t1.1037184321cosωt,(42)

where ω = 0.7328908403, λ = 0.1870115667.

In Figure 5 is plotted a comparison between the first-order approximate solution and numerical results. In Table 1 a comparison between the approximate solutions x given by Eqs. (40), (41), (42) with numerical results and corresponding relative errors are presented.

Comparison between the approximate solution (41), (42) and (43) and numerical solution: —–- numerical solution; ……… approximate solution
Figure 5

Comparison between the approximate solution (41), (42) and (43) and numerical solution: —–- numerical solution; ……… approximate solution

Table 1

Some numerical values of the approximate solutions x given by Eqs. (40), (41), (42) and numerical results for c = 0.1, μ = 20, α = 1, β = 0.1, f0 = 0.1 A = 5, v0 = 0.1 and different values of the variable t (relative errors: ϵ = |xnumericalxOHAM|)

From Figure 5 and Table 1 it can be seen that the solution obtained by the proposed procedure is nearly identical with the numerical solution obtained using a fourth-order Runge-Kutta method.

5 Conclusions

The optimal homotopy asymptotic method (OHAM) was used to propose an analytic approximate solution for the nonlinear Bingham model. Our procedure is valid even if the nonlinear differential equation does not contain any small or large parameters. In the construction of the homotopy appear some distinctive concepts: the linear operator, the nonlinear operator, the auxiliary functions Hi(t, Ci) and some optimal convergence-control parameters λ, ω, C1, C2, … which guarantees a fast convergence of the solutions. The example presented in this paper leads to the conclusion that our procedure is very accurate using only one iteration and three steps. OHAM provides us with a simple and rigorous algorithm to control and adjust the convergence of the solution through the auxiliary functions Hi(t, Ci) involving several parameters λ, ω, C1, C2, … which are optimally determined. The fast convergence of the OHAM method proves that our technique is very efficient in practice.

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About the article

Received: 2016-08-05

Accepted: 2017-04-28

Published Online: 2017-10-31


Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.


Citation Information: Open Physics, Volume 15, Issue 1, Pages 620–626, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0072.

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© 2017 V.Marinca et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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