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] (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.  proposed an idealized mechanical system which consists of a Coulomb friction element placed in parallel with a viscous damper: (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 and Wang and Gordaninejad  used the Herschel-Bulkley model expressed by (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 . Based on the original Bingham plastic and Herschel-Bulkley plastic models, Zubieta, et al.  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.
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 . In this case the damping force F can be expressed as (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 . This viscoelastic-plastic model depends on connection of the Bingham, Kelvin-Voight body and Hooke body models (Figure 3).
The damping force in the Gamota-Filisko model is given by (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 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. . 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  (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.
The force F(t) is given by (7)
where μ is a parameter representing the non-viscous damping effect (or the relaxation parameter ). The model for the damping force expressed as (8)
was originally proposed by Biot . Cremer and Heckl  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.
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: (10)
In recent years, research effort has been devoted to study of some approaches like the optimal homotopy analysis method (OHAM) , homotopy analysis method [22, 23, 24, 25], integral mean value method , Adomian decomposition method  and other methods .
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
with N an known nonlinear differential operator depending on the unknown function x(t). The initial conditions are: (12)
Let x0(t) be an initial approximation of x(t). L and B are arbitrary linear operators such that (13)
The linear operator L is not unique, as in .
with properties (15) (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 (17)
and then equating the coefficients of p0 and p1, we obtain (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.: (20)
where the expression for the nonlinear operator has the form (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 : (22)
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 (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 . 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 Hi(t, hj(t), Cj) gi(t) has the same form as the term hi(t) gi(t) given by Eq. (21) (see ). 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: (24)
where the residual R is given by (25)
The unknown parameters C1, C2, …, Cs can be identified from the conditions (26)
Thus, the first-order analytic approximate solution given by Eq. (22) is well-known.
3 Application of OHAM to nonlinear Bingham fluid dampers
such that Eq. (9) can be expressed as (28)
where and the overdot represents differentiation with respect to dimensionless time. The initial conditions (10) become (29)
Making the transformation (30)
where λ is for the moment an unknown real parameter, Eq. (28) can be written (31)
The initial conditions (29) become (32)
For the nonlinear differential equation (31), we try to choose a linear operator of the form (33)
where ω is an unknown parameter.
The initial approximation z0(t) can be deduced from Eq. (13) with initial / boundary conditions: (35)
According to Eq. (21), the nonlinear operator satisfies (37)
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:
The first-order analytic approximate solution given by Eq. (39) becomes for this first step: (40)
For the second step, when t ∈ [7/2, 7] we obtain: (41)
where ω = 1.9481775386, λ = 0.3726878974.
In the last case, when t ∈ [7, 10], the first-order approximate solution is: (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.
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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Published Online: 2017-10-31
Conflict of interest: The authors declare that there is no conflict of interests regarding the publication of this paper.