Chaos in a new bistable rotating electromechanical system

Chaos, Solitons and Fractals 93 (2016) 48–57

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Chaos in a new bistable rotating electromechanical system R. Tsapla Fotsa, P. Woafo∗ Laboratory of Modelling and Simulation in Engineering, Prototypes and TWAS Research Unit, Department of Physics, Cameroon

a r t i c l e

i n f o

Article history: Received 3 July 2016 Revised 24 September 2016 Accepted 29 September 2016

Keywords: Electromechanical systems Rotating arm Bistability Dissipative chaos Melnikov chaos

a b s t r a c t A device consisting of an induction motor activating a rotating rigid arm is designed and comprises a bistable potential due to the presence of three permanent magnets. Its mathematical equations are established and the numerical results both in the absence and in the presence of magnets are compared. The generation of chaotic behavior is achieved using two different external excitations: sinewave and square wave. In the presence of magnets, the system presents periodic and dissipative chaotic dynamics. Approximating the global potential energy to a bistable quartic potential, the Melnikov method is used to derive the conditions for the appearance of Hamiltonian chaos. Such a device can be used for industrial and domestic applications for mixing and sieving activities. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Electromechanical systems (EMS) play an important role in various branches of engineering. In recent years, considerable efforts have been devoted to the study of oscillatory and chaotic states of some vibrational EMS [1–10]. Among these EMS, a particular class is that containing asynchronous motor, which is encountered in various industrial modern processes. Nonlinear dynamics of EMS with translational or pendulum motions has been studied recently [6]. Depending on the form of the external excitation and the set of the chosen parameters, EMS lead to various interesting phenomena such as frequency entrainment, harmonic, subharmonic, super harmonic oscillations, and chaotic behavior [1,3–8]. These studies have been guided by the fact that chaos is useful and beneficial in some applications such.as vibrating sieves [8–10], industrial mixers [11,12], industrial shakers and for monitoring compaction [13]. In most of EMS and particularly the one that consists of asynchronous motor powered by sinusoidal input voltage, chaotic behavior appears for appropriate range of parameters when the device consists of electronic and/or mechanical components which have nonlinear characteristics [9,13,14] such as capacitors with nonlinear chargevoltage characteristics [15,16], inductance with nonlinear term in the flux-voltage characteristics [17], and so on. In practice, electronic components deliver low power and it is difficult to have mechanical components with nonlinear characteristics. Consequently, the chaotification technique which consists to introduce nonlinear

∗

Corresponding author. E-mail address: [email protected] (P. Woafo).

http://dx.doi.org/10.1016/j.chaos.2016.09.025 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

components in the EMS is limited to low power devices (e.g., microelectromechanical systems). In the absence of nonlinear components, one sometimes use anticontrol of chaos methods [13,18–21]. These studies have been extended to hydro-turbine system showing nonlinear dynamics [22–25]. Research is still in progress to find other ways to induce chaos in EMS, particularly in EMS using electric components that can be used to provide high power actuation force. Such a problem is considered in this work. The EMS proposed in this paper is a new model inspired by some previous works such as in [1], where Tcheutchoua Fossi et al. used feedback control to generate complex phenomenon in a simple electromechanical system. By using the power supply feedback control, the authors observed that the system exhibits complex dynamical behaviors such as jump phenomenon, Sommerfeld effect, period-nT and chaotic oscillations. In this work, we provide a new design to chaotify the same device. This is based on the creation of a bistable potential in the electromechanical system. This is done by considering one permanent magnet fixed at the free end of a mechanical rotating arm. In the right and in the left of the equilibrium state of the mechanical rotating arm, there are two other permanent magnets. The goal of this work is thus to present and study the new electromechanical system with rotating arm able to provide complex behavior such as chaos without using chaos anticontrol strategies or feedback control strategies. We expect this device to find applications in various branches of electromechanical engineering, e.g., household electromechanical appliances and industrial food processing systems. The structure of the work is as follows. Section 2 presents the electromechanical system with the bistable potential. The equa-

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49

Fig. 1. Schematic representation of the electromechanical system. Table 1 Parameters of the electromechanical system (some of these parameters are indicated in Fig. 1). Parameters

Values

Inductance: L Resistance: R Rotor inertia moment: Jr Viscous damping coefficient: cv Back electromotive force constant: KE Stiffness coefficient: cr Torque constant: KT Length of the mechanical arm  Magnetic moment MA of the magnet Magnets positionh (see Fig. 1) Distance s of the free end of the arm and middle of the distance between the magnets

0.2 2 4.1 × 10 − 2 0.13 × 10 − 3 0.75 × 10 − 2 4 × 10 − 3 0.75 × 10 − 1 0.5 1.2 0.02 0.05

tions of motion are then derived. In Section 3, the dynamical behaviors are determined using both mathematical investigation and numerical simulation. Section 4 concludes the work. 2. Description of the model and its equations The device shown in Fig. 1 is an electromechanical system which consists of a mechanical rotating arm activated by a rotor. Its shaft is fixed mechanically at its two ends by two spiral torsion springs put up in anti-parallel manner [1]. The coupling between the electrical and mechanical parts is realized through the electromagnetic force due to a permanent magnet. It creates a Laplace force in the mechanical part and the Lenz electromotive voltage in the electrical part. The electrical part of the system consists of a resistor R, an inductor L and a voltage source u(t), all connected in series. One permanent magnet is fixed at the free end of the mechanical arm. In the right and in the left of the mechanical arm at equal distances to the equilibrium position, there are two other permanent magnets placed on top of a non-ferromagnetic bearer. The values and dimensions of the parameters of the device are listed in Table 1. These values are selected in order to provide to the device the appropriate dynamical states that can be beneficial for applications. One expects some changes if the device is to be tested experimentally. Using the electrical and mechanical laws, and taking into account the contribution of the Laplace force and that of the Lenz

Dimensions H

 kg.m2 N.m.s/rad V.s/rad N.m/rad N.m/A m A.m2 m m

electromotive voltage, it is found that the system is described by the following set of differential equations:

L

di(t ) dθ + Ri(t ) + KE = u(t ) dt dt

(1a)

Jr

d2 θ dθ + cv + cr θ +  fmag = KT i(t ) dt dt 2

(1b)

Assuming that the potential U is the sum of the elastic potential energy and magnetic potential energy, one obtains:

U =

π − 32 μ0 MA2 cos θ  q1 − p1 sin −θ −ϕ 2π 2  π − 32  + q1 − p1 sin −θ +ϕ cr 2 θ + 2

2

(2)

The first term is the elastic potential energy due to the linear springs and the second term is the mechanical potential energy due to the magnets [26]. These potential energies as well as the total potential energy are respectively represented in Fig. 2(a)–(c). As it can be seen, the association of the elastic potential energy and magnetic potential leads to the bistable nature of the total potential energy. The rotating arm presents two stable points located on both sides of an unstable point.

50

R. Tsapla Fotsa, P. Woafo / Chaos, Solitons and Fractals 93 (2016) 48–57

Fig. 2. Potential energy due to springs (a), potential energy due to magnets (b) and total potential energy (c) as function of the angular displacement with the parameters of Table 1.

3. Dynamical behaviors of the device

y¨ = λ3 x − λ1 y˙ − y

In this section, the external excitation source u(t) is considered as the sinusoidal input voltage of the form u0 sin(2π ft) (u0 , f, t being, respectively the amplitude, frequency, and time). Now use the dimensionless variables:

=

d , y= dτ

θ i , x = , τ = ωe t θ0 i0

3.1.1. Stability analysis Considering external excitation as a constant source (u(t) = E), the equilibria of the new system are the solutions of the following algebraic equations

(3)

where i0 = 0.5 and θ 0 = 0.25 radian are the normalization current and angular displacement. Replacing Eq. (6a) and (b) into (4a) and (b), one obtains the following dimensionless form Replacing relations (3) into equations (1a) and (1b), one obtains. . .

x˙ = −λ2 x − λy˙ + E sin( τ )

(4a)

y¨ = λ3 x − λ1 y˙ − y − β g(yθ0 )

(4b)

where

⎡ 

− 32

q1 + p1 sin(yθ0 − π2 + ϕ )

⎢  ⎣+ 3 p1 q1 + p1 sin(yθ0 − 2 p1 q1 + p1 sin(yθ0 −

g ( y θ0 ) = ⎢

x˙ = −x − λv + E

(7a)

y˙ = v

(7b)

v˙ = λ3 x − λ1 v − λ2 y

(7c)

λ E This system has a unique equilibrium E1 (E, λ3 , 0 ) with 2

σ 3 + (1 + λ )σ 2 + (λ1 + λλ3 + λ2 )σ + λ2 = 0

− 32 





5 π + ϕ ) − 2 cos(yθ − π + ϕ )+ 0 2 2

5 π − ϕ ) − 2 cos(yθ − π − ϕ ) 0

λ2 = 0.

The characteristic equation of the Jacobian matrix at E1 is:

+ q1 + p1 sin(yθ0 − π2 − ϕ )

2

(6b)

sin yθ0

cos yθ0

(8)

⎤ ⎥ ⎥ ⎦

2

and with the following rescalings:

λ3 =

=

KT i0 Jr ωe2 θ0

; λ1 =

cv

Jr ωe

; λ2 =

ω KE θ0 u0 ;λ = ;E = ωe Li0 Li0 ωe

R Lωe

; ωe2 =

μ0 MA2 cr ;β = ; Jr 2π θ0 Jr ωe2 (5)

3.1. Dynamics in the absence of three magnets In absence of the three magnets, the dimensionless equations of the electromechanical system are given by Eq. (6a) and (b).

x˙ = −λ2 x − λy˙ + E sin( τ )

(6a)

where σ is the eigenvalue of the system at E1 . Using Routh– Hurwitz conditions, this equation has all roots with negative real parts (meaning that E1 is stable) if and only if the following analytic relations are satisfied:

1+λ0

(9a)

(1 + λ )(λ1 + λλ3 + λ2 ) − λ2  0

(9b)

λ2  0

(9c)

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51

Fig. 3. Frequency-responses of the maximal angular displacement (a) and maximal electrical current (b) with the parameters of Table 1, and magnitude of the external excitation E = 0.1 (the results from the analytical expression (15a) and (b)).

Since λ0; λ1 0; λ2 0; and λ3 0, the equilibrium E1 is a stable point for the system (7a)–(c). 3.1.2. Oscillatory states To derive the amplitude of the harmonic oscillatory states (E = 0) delivered by Eq. (6a) and (b), let us express its solution as:

x(τ ) = A0 sin( τ ) + B0 cos( τ )

(10a)

y(τ ) = A1 sin( τ ) + B1 cos( τ )

(10b)

where A0 ; A 1 ; B0 ; B1 are unknown parameters to be determined. ymax = A21 + B21 represents the maximal amplitude of y and



xmax = A20 + B20 represents the maximal amplitude of x. Inserting Eq. (10a) and (b) into Eq. (6a) and (b), and equating the sine (sin(ϖτ )) and cosine (cos(ϖτ )) terms separately, it comes that the unknown parameters satisfy the following expressions:

the mechanical part and anti-resonance for the electrical part. Thus, when the high amplitude is attained in the mechanical part, for a frequency close to 1, the amplitude of the electric signal is at its lowest value. Due to the linear form of Eq. (9a)–(c), one also observes that the maximal dimensionless value of the angular displacement and the electrical current increases linearly with the magnitude of the external excitation (figure not presented here). 3.2. Dynamics in the presence of the magnets The aim of this subsection is to show that the device of Fig. 1 can behave chaotically under certain conditions. Many chaotic indicators are generally used to find chaos in dynamical systems. Amongst them, the Lyapunov exponent is the most precise. The Lyapunov exponent expresses the convergence (when negative) or divergence (when positive) of nearby trajectories.

( 4 + λ22 + (λ3 λ1 λ + λ21 − 2λ2 ) 2 )E

6 + (λ21 − 2λ3 λ − 2λ2 + 1 ) 4 + (λ22 + 2λ3 λ2 λ + λ21 + 2λ3 λ1 λ + λ23 λ2 − 2λ22 ) 2 + λ22 (− 4 − λ2 λ3 λ − λ2 + (λ3 λ − λ21 + 2λ2 ) 2 )E

B0 =

6 + (λ21 − 2λ3 λ − 2λ2 + 1 ) 4 + (λ22 + 2λ3 λ2 λ + λ21 + 2λ3 λ1 λ + λ23 λ2 − 2λ22 ) 2 + λ22 −((1 + λ1 ) 2 − λ2 )λE A1 = 2 6 4

+ (λ1 − 2λ3 λ − 2λ2 + 1 ) + (λ22 + 2λ3 λ2 λ + λ21 + 2λ3 λ1 λ + λ23 λ2 − 2λ22 ) 2 + λ22 −(− 2 + λ1 + λ3 λ + λ2 )λ3 E

B1 = 2 6

+ (λ1 − 2λ3 λ − 2λ2 + 1 ) 4 + (λ22 + 2λ3 λ2 λ + λ21 + 2λ3 λ1 λ + λ23 λ2 − 2λ22 ) 2 + λ22 A0 =

We analyze the behavior of xmax and ymax when the normalized frequency of the external excitation ϖ is varied and the results are presented in Fig. 3. When ϖ increases from 0 to 2, the response amplitude ymax increases from a minimal amplitude to the higher amplitude equal to 2.3 which corresponds to 33° (θ = y × θ 0 ) and decreases thereafter to the lower amplitude. The response amplitude xmax decreases, then increases from a minimal amplitude 2.03 × 10 − 4 which corresponds to 1.015 × 10 − 4 A (i = x × i0 ) to a higher amplitude 1.871 × 10 − 3 which corresponds to 0.9355 × 10 − 3 A. These behaviors are usually called resonance phenomenon for

(11)

Therefore, a state of the system is said to be chaotic if the Lyapunov exponent is positive (which corresponds, in the bifurcation diagram, to a cloud of points). The state of the system is said to be periodic if the exponent is negative (this corresponds, to a curve lines in the bifurcation diagram). The case λmax = 0 corresponds to the quasi-periodic state of the system. The maximum one-dimensional Lyapunov exponent is defined as:

λmax = lim

t→+∞

 1  t



ln [D(t )] ,

(12)

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Fig. 4. Bifurcation diagram depicting the global maxima of the angular rotating arm displacement (a) and the largest Lyapunov exponent (b) versus the amplitude E with the parameters of Table 1 and for ϖ = 6.0.



with D(t ) = δ12 + δ22 + δ32 , where D(t) is the distance between neighboring trajectories. It is computed from the variational equations obtained by perturbing the solutions of Eq. (8) as follows x → x + δ1 , y → y + δ2 , y˙ → y˙ + δ3 . The magnitude of the external source, and the normalized frequency are used as control parameters. The external excitation used here is the sinusoidal input voltage. In order to find the range of E for which the device exhibits chaos, we present the bifurcation diagram in Fig. 4(a) and the Lyapunov exponent in Fig. 4(b) as function of E. Fig. 4 shows that the behavior of the device shows non chaotic oscillations for E ∈ [1; 3.5] and chaotic oscillations for E ∈ [3.75; 10]. It can be well observed that the corresponding Lyapunov exponent plotted converges well to a positive value only when E ∈ [3.75; 10]. This confirms the successfulness of the chaotification procedure with the addition of the magnets. To complement the results presented in Fig. 4, some temporal traces and corresponding phase portraits are displayed in Fig. 5 indicating chaotic states and periodic oscillations. Fig. 5(a) and (c) show 1T-periodic oscillations while Fig. 5(b) and (d) show chaotic oscillations. Similarly, Fig. 6 presents the bifurcation diagram and the corresponding variation of the Lyapunov exponent when the normalized frequency ϖ is varied. When the normalized frequency of the external excitation ϖ increases from the value ϖ = 1.0, the electromechanical device moves from a periodic state to a chaotic state at ϖ = 5.6 with some small windows of quasiperiodic and chaotic behaviors. This persists until ϖ = 7.9 where only periodic oscillations continue to be displayed. The periodic orbit exists until ϖ = 12.0 Fig. 7 shows some typical temporal trace and phase portraits of the rotating arm motion for different values of ϖ. Fig. 7(a) and (c) show periodic 1T oscillations while Fig. 7(b) and d show chaotic oscillations. By using spiral spring, the mechanical rotating arm has the possibility to oscillate with an angle reaching 360° or more depending

on the value of the spring constant and the input voltage. In the absence of the permanent magnets, the device exhibit periodic oscillations when the external source is the sinusoidal input voltage. We observed that in the presence of permanent magnets, chaotic behavior appears for the magnitude of the external excitation E ≥ 3.5. These results are very interesting since depending on the application, one can use the chaotic behavior with small or high value of θ . In the situation where chaos is efficient, but with the small values of the magnitude of the external source, one can use the external square signal source with the mathematical expression given by Eq. (13).

 u(t ) = u0 sign(sin(ωt )) where sign(x ) =

1; 0; −1;

if x  0 if x = 0 . if x ≺ 0

(13)

Replacing the sinusoidal input voltage by a square one, and considering the magnitude of the external excitation, and the normalized frequency as control parameters, one obtains the results presented in Fig. 8. Thus, with the square signal, one finds non-chaotic oscillations for E ∈ [1; 2.78] and chaotic oscillations for E ∈ [2.78; 10]. Fig. 9 presents the bifurcation diagram and the corresponding variation of the Lyapunov exponent when the normalized frequency ϖ is varied. The following transitions are observed. When the normalized frequency ϖ of the external excitation increases from the value ϖ = 1.0, the electromechanical device moves from a periodic state to a chaotic state at ϖ = 2.35. This persists until ϖ = 2.65 where only periodic oscillations continue to be displayed. From ϖ = 5.45, there is a transition from periodic orbit to a chaotic behavior with some small windows of period-n orbit. This continues to take place before leading to only periodic orbit at ϖ = 8.4. One notes that the device presents the same dynamics as in the case where sinusoidal voltage source is used as input source. But

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53

Fig. 5. Temporal traces (a), (b) and phase portraits (c), (d) obtained with the parameters of Fig. 4 and E = 2; (a), (c) ; E = 8; (b), (d).

Fig. 6. Bifurcation diagram depicting global maxima of the angular rotating arm displacement (a) and the largest Lyapunov exponent (b) versus the parameter ϖ with the parameters of Table 1 and for E = 17.

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R. Tsapla Fotsa, P. Woafo / Chaos, Solitons and Fractals 93 (2016) 48–57

Fig. 7. Temporal trace and phase portraits in the (y, y˙ ) plane: period-1T oscillations (Fig. 7(a) and (c)) with the parameters of Fig. 6 and ϖ = 2.0 and chaotic oscillations (Fig. 7(b) and (d)) with the parameters of Fig. 6 and ϖ = 6.0.

with square wave signal, chaos behavior appears for smaller values of the signal amplitude and frequency. 4. The Hamiltonian chaos The chaotic dynamics obtained in Section 3 corresponds to dissipative chaos. Because of the bistable nature of the potential energy, one might expect the occurrence of Hamiltonian or Melnikov chaos. In order to find a mathematical condition for the Melnikov chaos, one needs to have an approximate form of the bistable potential from which homoclinic orbits will be calculated. This is a quite complicated issue here because of the complexity of the mathematical expression of the potential energy. However, some crude approximations can be made following the curve in Fig. 2. We thus appromixate the total potential energy by the following quartic expression

U (θ ) = −

A 2 B 4 θ + θ 2 4

(14)

where A and B are positive coefficients. Considering the potential wells of Fig. 2, and those derived from  Eq. (14), one can write the values of the minima as θ =



A = 0.0238144 and B = 0.1 which satisfied

A B

A B

= ±0.488. Taking

= ±0.488. Although,

there is some quantitative difference between the curve generated by Eq. (14) and the potential in Fig. 2(c), expression (14) will be used to determine a condition for the appearance of the Hamiltonian chaos.

With the approximate form of the potential energy, the differential Eq. (1a) and (b) becomes

L

di(t ) dθ + Ri(t ) + KE = u(t ) dt dt

(15a)

Jr

d2 θ dθ + cv − Aθ + Bθ 3 = KT i(t ) dt dt 2

(15b)

In order to find the Melnikov chaos in our system, let us assume that the inductance L is negligible. Then Eq. (15a) and (b) reduces to the single Eq. (16)

d2 θ + dt 2

c

v

Jr

+

KE KT Jr R

 dθ dt

−

A B u K θ + θ 3 = 0 T cos(ωt ) Jr Jr Jr R

(16)

In the non-dimensional form, Eq. (16) takes the form

y¨ + γ1 y˙ − y + y3 = γ4 cos(τ ) where

θ ; τ = ω0 t; ω0 = θ0  ω Jr = ; θ = ω0 ω0 0 B y=



(17)

cv + KERKT A − cr u0 KT , γ1 = ; γ4 = ; Jr Jr ω0 Jr θ0 Rω02 (18)

Eq. (17) describes the bistable Duffing oscillator. The fixed points ( − 1; 0) and (1; 0) are the stable equilibrium points while the fixed point (0; 0) is an unstable equilibrium point. Eq. (11) can be viewed as that of a Hamiltonian system plus a perturbation consisting of−γ1 y˙ + γ4 cos(τ ). One can thus use

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55

Fig. 8. Bifurcation diagram depicting global maxima of the angular rotating arm displacement (a) and the largest Lyapunov exponent (b) versus the parameter E obtained with the parameters of Fig. 4.

Fig. 9. Bifurcation diagram depicting global maxima ϖ of the angular rotating arm displacement (a) and the largest Lyapunov exponent (b) versus the parameter ϖ with the parameters of Fig. 6.

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R. Tsapla Fotsa, P. Woafo / Chaos, Solitons and Fractals 93 (2016) 48–57

Fig. 10. Variation of the fractal structure of the basin of attraction for  = 2.5 and (a) γ 4 = 10, (b) γ 4 = 5, (c) γ 4 = 1, (d) γ 4 = 0.02. Other parameters are taken from Table 1.

the Melnikov’s formalism to find the condition for the appearance of horseshoe chaos [27]. This condition is given



5

γ4 ≥

22

3 π 2

γ1 cosh

π 2



(19)

The details for the derivation of Eq. (19) can be found in Ref. [27]. A system whose parameters satisfy condition (19) will have a chaotic dynamics in a sense that it will present fractality in the basin of attraction. To validate the analytical predictions, we have simulated numerically Eq. (17) to look for the effects of the control parameters γ 4 on the appearance of the fractality in the basin of attraction. The results are presented in Fig. 10 in which the white and dark regions represent respectively the set of initial conditions that result in motions around the equilibrium points and motion covering both equilibrium points. One notices that the basin of attraction becomes fractal following the condition defined by Eq. (19). For instance, for  = 2.5, the fractality appears at γ 4 = 10 and γ 4 = 5(see Fig. 10(a) and b, We find that the black space increases and the fractality disappear when γ 4 decreases (see Fig. 10(c)). The white space even completely disappears for γ 4 = 0.02 (see Fig. 10(d)). 5. Conclusion In this paper, we have presented a new electromechanical system with bistable potential energy generated by the combined action of a spiral spring and magnets appropriately placed in the device. We have studied the dynamical behavior of the device both in the absence and in the presence of the magnets. The device has been seen to generate complex behaviors including chaos and angular oscillations with amplitude greater than one turn. In the absence of magnets, the device presents a single stable point and

exhibit periodic behavior. In presence of magnets, we numerically show that the association of elastic potential energy and mechanical potential energy due to the magnets leads to the bistable nature of the total potential energy. This bistablity is responsible for the generation of chaotic dynamics in the electromechanical system. By using square excitation, the device presents chaotic behavior for smaller values (compared to the case with sinusoidal source) of the excitation amplitude and frequency. It has also been shown that the electromechanical system can show Hamiltonian chaos characterized by the fractality of the basin of attraction. These complex behaviors can find applications in engineering, e.g., mixers and sieves in which chaos motion could enhance the dispersion of particles and avoid stagnation and formation of aggregates. The device can also give some hints on how to optimize the operation of home appliances such as those used for mixing various food components (flour, eggs, spices, vegetables, fruits, etc.).

Appendix A. Derivation of the magnetic potential energy Let us note Emag as the potential magnetic energy. To derive this magnetic potential energy, let us consider Fig. A1. The magnets A, B, C are identical. The potential energy of the magnet A in the field generated by the magnets B and C is [26]:

Emag

A = BA + B CA )  M = − (B

μ0 MA2 cos θ 2π



1 1 + 3 3 rBA rCA

 (A1)

BA and B CA are respectively the magnetic field creates on where B magnet A by magnet B and magnet C. The distance between B and A is given by (A2) and the distance between C and A is given by (A3):

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57

magnetic force as:

− → μ M2 Fmag = 0A fmag = 20π A

⎡

1

⎢ [q1 + p1 sin(θ − π +ϕ )] 32 2 ⎢  ⎣ 3 p1 cos(θ − π2 +ϕ ) +2

=

Fig. A.1. Schematic structure for the derivation of the magnetic potential energy.



→ − → − rBA = −OB + OA = − h2 + (s +  )2 sin





h2 + ( s +  )

2

+

cos



3 ⇒ rBA = p1 − q1 sin

π

−θ +ϕ

2



→ − → − rCA = −OC + OA = − h2 + (s +  )2 sin





h2 + ( s +  )

2

+



cos

3 ⇒ rCA = q1 − p1 sin

2

π

 32

π

−θ −ϕ

2



2

(A2)



− θ − ϕ er



− θ − ϕ eθ + er

2

π



− θ + ϕ er

− θ + ϕ eθ + er

2

π

π

 32

(A3)

where:



 q1 = +h +(s+ ) ; p1 =2 2

2

2

h2 +

(s+ ) ; ϕ = arctan 2

h s+

 (A4)

Replacing (A2)–(A4) into (A1), one obtains:

Emag

 μ0 MA2 cos θ 1 =

3 2π q1 − p1 sin( π2 − θ − ϕ ) 2  +

1

(A5)

32

q1 − p1 sin( π2 − θ + ϕ )

The interaction force between the magnets can be obtained by taking the gradient of (A5)

−→ 1 d fmag = −−  (Emag ) = −  dθ (Emag )eθ ⎡grad =

μ0 MA2 ⎢ ⎢ 2π  ⎣

1 π [q 1 + p1 sin (θ − 2 +ϕ )]

+ 32

3 2

+

p1 sin(θ − π2 +ϕ ) 5

[q1 + p1 sin(θ − π2 +ϕ )] 2

 1

[q1 + p1 sin(θ − π2 −ϕ )] +

3 2

⎤ sin θ



p1 sin(θ − π2 −ϕ ) 5

[q1 + p1 sin(θ − π2 −ϕ )] 2

cos θ

⎥ ⎥eθ ⎦ (A6)

By taking into account the fact that the magnets are at the end (distance ) of the rotating arm, the displacement between the magnetic force and the rotational axis () give the moment of the

μ

2π

5

[q1 + p1 sin(θ − π2 +ϕ )] 2

⎡ 2 0 MA



+

1 3

[q1 + p1 sin(θ − π2 −ϕ )] 2 +

p1 cos(θ − π2 −ϕ )

⎤ sin θ



5

[q1 + p1 sin(θ − π2 −ϕ )] 2

− 32

q1 + p1 sin(θ − π2 + ϕ )

cos θ

⎥ ⎥er eθ ⎦ ⎤

⎢ ⎥

− 3  ⎢ ⎥ ⎢+ q1 + p1 sin(θ − π2 − ϕ ) 2 sin θ ⎥ ⎢  ⎥k

− 52 ⎢ 3 ⎥ π π cos(θ − 2 + ϕ ) ⎥ ⎢+ 2 p1 q1 + p1 sin(θ − 2 + ϕ ) ⎣ ⎦ 

− 5 + p1 q1 + p1 sin(θ − π2 − ϕ ) 2 cos(θ − π2 − ϕ ) cos θ

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