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

  • Vasile Marinca , Remus-Daniel Ene EMAIL logo und Liviu Bereteu
Veröffentlicht/Copyright: 31. Oktober 2017
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Open Physics
Aus der Zeitschrift Open Physics Band 15 Heft 1

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)=fc⋅sgn(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.

Figure 1 Bingham model
Figure 1

Bingham model

Figure 2 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(x2−x1)+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).

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(x2−x1)+c1(x˙2−x˙1)+f0==c0x˙1+fcsgn(x˙1)+f0==k2(x3−x2)+f0 for |F|>fck(x2−x1)+c1x˙2+f0==k2(x3−x2)+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)=fc⋅sgn(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.

Figure 4 BingMax model
Figure 4

BingMax model

The force F(t) is given by

F(t)=c⋅∫0tμe−μ(t−τ)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)=c⋅∫0tμe−μ(t−τ)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¯e−μ¯(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)=v0⋅km.(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=1nHi(t, hj(t), Cj) gi(t) has the same form as the term ∑i=1mhi(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

∂J∂C1=∂J∂C2=…=∂J∂Cs=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)+2c⋅∫0tμe−μ(t−τ)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)=Ae−λtz(t),(30)

where λ is for the moment an unknown real parameter, Eq. (28) can be written

z¨−2λz˙+(λ2+2μc+1)z−−2cμ2⋅∫0te(λ−μ)(t−τ)z(τ)dτ+αA2e−2λtz3++βAeλtsgn(z˙−λz)+f0eλtA−2cμe(λ−μ)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+1−ω2)z−−2cμ2⋅∫0te(λ−μ)(t−τ)z(τ)dτ+αA2e−2λtz3++βAeλtsgn(z˙−λz)+f0Aeλt−2cμe(λ−μ)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)=cos⁡ωt−ωλsin⁡ωt.(36)

According to Eq. (21), the nonlinear operator satisfies

Nz0(t)==λ(1+λ2+ω2+2μc)−2μ2c(ω2−λ2+λμ)λ(ω2+(μ−λ)2)cos⁡ωt++ω(λ2+ω2−2μc−1)(λ2+ω2+λ2−2μλ)λ(ω2+(μ−λ)2)++2μ2cω(μ−2λ)λ(ω2+(μ−λ)2)sin⁡ωt++2μ2c(ω2−λ2)−2μcλ(ω2+λ2−2λμ)λ(ω2+(μ−λ)2)e(λ−μ)t+f0Aeλt++αA24e−2λt3(ω2+λ2)λ2cos⁡ωt+3ω(ω2+λ2)λ3sin⁡ωt++λ2−3ω2λ2cos⁡3ωt+ω(ω2−3λ2)λ3sin⁡3ωt−4βλAeλtcos⁡ωt−13cos⁡3ωt+15cos⁡5ωt−−17cos⁡7ωt+19cos⁡9ωt−….(37)

From Eqs. (37), (32) and (23), we choose

z1(t)=C1t2cos⁡ωt+(C2t2+C3t)sin⁡ωt++[C4t2+(v0+ω2+λ2λA+3λC7)tcos⁡3ωt++(C5t2+C6t)sin⁡3ωt+C7]e−2λt−C7eλtcos⁡ωt.(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)cos⁡ωt++(C2t2+C3t−ωλA)sin⁡ωt]××e−λt+{[C4t2+((v0+ω2+λ2λ)A+3λC7)t]cos⁡3ωt++(C5t2+C6t)sin⁡3ωt+C7}e−3λt−C7cos⁡ωt.(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)cos⁡ωt++(0.4059363155t2+9.8101433224t−−13.8580629241)sin⁡ωte−0.4221369200t++[(12.7300955924t2−1.0230014969t)cos⁡3ωt++(−8.3640772984t2−14.692624846t)sin⁡3ωt−−15.2775231617e−1.2664107600t++15.2775231617cos⁡ωt.(40)

For the second step, when t ∈ [7/2, 7] we obtain:

x¯(t)=[(−18.7159298670t2++1.3931962652)cos⁡ωt+(2.5809325241t2−−12.3307330274t−7.2827523763)sin⁡ωt]××e−0.3726878974t+[(16.1687136688t2−−76.6284415838t)cos⁡3ωt+(1.8324463202t2−−15.4843220143t)sin⁡3ωt−68.7655829044]××e−1.1180636924t+68.7655829044cos⁡ωt,(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)cos⁡ωt+(−1.4223903751t2++13.8451447750t−10.3560770110sin⁡ωt××e−0.1870115667t+[(−7.9954565850t2++12.3657070213tcos⁡3ωt+(1.5073646864t2−−0.3337501284tsin⁡3ωt++1.1037184321××e−0.5610347000t−1.1037184321cos⁡ωt,(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.

Figure 5 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 = |xnumerical − xOHAM|)

txnumericalxOHAMϵf″(0)
0550
1−1.9027612276−2.14731847620.2445572485
2−1.3080397375−1.50600688540.1979671479
33.09562073083.24511082890.1494900980
4−3.6155447279−3.66039124270.0448465147
53.48173143823.43449110320.0472403350
6−3.2054313558−3.13235489390.0730764619
72.64265749882.64255749940.0000999994
8−1.6161442490−1.74126860790.1251243589
90.15844562790.13091967010.0275259577
101.42318045361.42031386140.0028665921

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.

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

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Received: 2016-8-5
Accepted: 2017-4-28
Published Online: 2017-10-31

© 2017 V.Marinca et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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https://doi.org/10.1515/phys-2017-0072
Schlagwörter für diesen Artikel
MR dampers; optimal homotopy asymptotic method; auxiliary functions; optimal parameters
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Artikel in diesem Heft

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  117. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  118. Physical-chemical properties studying of molecular structures via topological index calculating
  119. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  121. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  122. City traffic flow breakdown prediction based on fuzzy rough set
  123. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  124. Conservation laws for a strongly damped wave equation
  125. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  126. Blending type approximation by Stancu-Kantorovich operators based on Pólya-Eggenberger distribution
  127. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  128. Computing the Ediz eccentric connectivity index of discrete dynamic structures
  129. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  130. A discrete epidemic model for bovine Babesiosis disease and tick populations
  131. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
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  133. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  134. Relationship between solitary pulmonary nodule lung cancer and CT image features based on gradual clustering
  135. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  136. A novel fast target tracking method for UAV aerial image
  137. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  138. Fuzzy comprehensive evaluation model of interuniversity collaborative learning based on network
  139. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  140. Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation
  141. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  142. After notes on self-similarity exponent for fractal structures
  143. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  144. Excitation probability and effective temperature in the stationary regime of conductivity for Coulomb Glasses
  145. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  146. Comparisons of feature extraction algorithm based on unmanned aerial vehicle image
  147. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  148. Research on identification method of heavy vehicle rollover based on hidden Markov model
  149. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  150. Classifying BCI signals from novice users with extreme learning machine
  151. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  152. Topics on data transmission problem in software definition network
  153. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  154. Statistical inferences with jointly type-II censored samples from two Pareto distributions
  155. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  156. Estimation for coefficient of variation of an extension of the exponential distribution under type-II censoring scheme
  157. Special issue on Nonlinear Dynamics in General and Dynamical Systems in particular
  158. Analysis on trust influencing factors and trust model from multiple perspectives of online Auction
  159. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  161. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  169. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  171. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  173. Special Issue on Advances on Modelling of Flowing and Transport in Porous Media
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  189. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  190. Special Issue: Ever New "Loopholes" in Bell’s Argument and Experimental Tests
  191. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  193. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  194. Erratum to: The ultimate loophole in Bell’s theorem: The inequality is identically satisfied by data sets composed of ±1′s assuming merely that they exist
  195. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  197. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  198. What If Quantum Theory Violates All Mathematics?
  199. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  200. Relativity, anomalies and objectivity loophole in recent tests of local realism
  201. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  203. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  204. Bohr against Bell: complementarity versus nonlocality
  205. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  206. Is Einsteinian no-signalling violated in Bell tests?
  207. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
  208. Bell’s “Theorem”: loopholes vs. conceptual flaws
  209. Special Issue on Ever-New "Loopholes" in Bell’s Argument and Experimental Tests
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  211. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  212. Three-dimensional computer models of electrospinning systems
  213. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  214. Electric field computation and measurements in the electroporation of inhomogeneous samples
  215. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  216. Modelling of magnetostriction of transformer magnetic core for vibration analysis
  217. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  218. Comparison of the fractional power motor with cores made of various magnetic materials
  219. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  220. Dynamics of the line-start reluctance motor with rotor made of SMC material
  221. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  222. Inhomogeneous dielectrics: conformal mapping and finite-element models
  223. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  224. Topology optimization of induction heating model using sequential linear programming based on move limit with adaptive relaxation
  225. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
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  227. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  228. Current superimposition variable flux reluctance motor with 8 salient poles
  229. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  230. Modelling axial vibration in windings of power transformers
  231. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  232. Field analysis & eddy current losses calculation in five-phase tubular actuator
  233. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  234. Hybrid excited claw pole generator with skewed and non-skewed permanent magnets
  235. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  236. Electromagnetic phenomena analysis in brushless DC motor with speed control using PWM method
  237. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  238. Field-circuit analysis and measurements of a single-phase self-excited induction generator
  239. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  240. A comparative analysis between classical and modified approach of description of the electrical machine windings by means of T0 method
  241. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  242. Field-based optimal-design of an electric motor: a new sensitivity formulation
  243. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  244. Application of the parametric proper generalized decomposition to the frequency-dependent calculation of the impedance of an AC line with rectangular conductors
  245. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  246. Virtual reality as a new trend in mechanical and electrical engineering education
  247. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  248. Holonomicity analysis of electromechanical systems
  249. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  250. An accurate reactive power control study in virtual flux droop control
  251. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  252. Localized probability of improvement for kriging based multi-objective optimization
  253. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  254. Research of influence of open-winding faults on properties of brushless permanent magnets motor
  255. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  256. Optimal design of the rotor geometry of line-start permanent magnet synchronous motor using the bat algorithm
  257. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  258. Model of depositing layer on cylindrical surface produced by induction-assisted laser cladding process
  259. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  260. Detection of inter-turn faults in transformer winding using the capacitor discharge method
  261. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  262. A novel hybrid genetic algorithm for optimal design of IPM machines for electric vehicle
  263. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  264. Lamination effects on a 3D model of the magnetic core of power transformers
  265. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  266. Detection of vertical disparity in three-dimensional visualizations
  267. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  268. Calculations of magnetic field in dynamo sheets taking into account their texture
  269. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  270. 3-dimensional computer model of electrospinning multicapillary unit used for electrostatic field analysis
  271. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  272. Optimization of wearable microwave antenna with simplified electromagnetic model of the human body
  273. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  274. Induction heating process of ferromagnetic filled carbon nanotubes based on 3-D model
  275. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering ISEF 2017
  276. Speed control of an induction motor by 6-switched 3-level inverter
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Heruntergeladen am 9.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/phys-2017-0072/html
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