A Model for a Thin Magnetostrictive Actuator in Nonlinear Dynamics

Abstract: In this study, we propose a model for the dynamics of magnetostrictive hysteresis in a thin rod actuator when mechanically the material works in his nonlinear domain. We derive two equations that represent magnetic and mechanical dynamics equilibrium. Our model results from an application of the energy balance principle. The numerical simulations of the model with sinusoidal periodic external fieldgenerate the hysteresis curve and show the equivalent mechanic model. By using the method of multiple scales we analyze the effects of the nonlinear parameter in the system response of magnetostrictive materials. With Routh-Hurwitz theorem, the stability and bifurcation analysis are carried out. Analytical and numerical methods are used to investigate the dynamics of the materials.


INTRODUCTION
Magnetostriction is the phenomenon of strong coupling between magnetic properties and mechanical properties of some ferromagnetic materials: displacements are generated in response to an applied magnetic field, while conversely, mechanical stresses in the materials produce measurable changes in magnetization.This phenomenon can be used for actuation and sensing.Figure 1 shows a schematic of a Terfenol-D actuator manufactured by Etrema Products Inc.The magnetic field generated by the coil current controls the strain in the Terfenol-D rod, which translates into displacement or force (if blocked) output of the actuator.Like other smart materials (e.g., piezoelectrics and shape memory alloys), such materials exhibit complex nonlinear and hysteretic responses.Modeling and control of their behavior is a challenge.We are interested in obtaining low dimensional models for magnetostriction actuators that show a constitutive coupling in their elastic and magnetic behaviors.
Eddy current losses and magnetoelastic dynamics of the magnetostrictive rod were considered to be the origin of the rate-dependent hysteresis in Vankataraman and Manservisi (2006), Venkataraman andKrishnaprasad (1999, 2005) and Venkataraman et al. (1998), where the eddy current losses were modeled by placing a resist or in parallel with a hysteretic inductor and the magneto elastic dynamics was modeled by a second order linear system.Considering a lowdimensional ferromagnetic hysteresis model led to an overall model for magnetostrictive actuators described by a set of switching ordinary differential equations (Venkataraman et al., 1998).Tan andBaras (2002, 2004) the authors suggested using a cascade of a Preisach operator with a linear system to model magnetostrictive actuators.However, these previous authors have not taken into account the nonlinear aspect of the displacement in their modeling process.In practical situation the nonlinear terms exist in the displacement and can affect considerably the real dynamics of the material.Thus, our aim in this study is to derive a nonlinear equation for the displacement and use the methods of dynamical systems (lyapunov exponent, bifurcation diagram, Poincare section, spectral diagram), to study the obtained equation.

MATERIALS AND METHODS
According to energy balance principle, the work done by external source (both magnetic and mechanic) is transformed in the change in the free energy of the rod, change in kinetic and losses in the magnetizationprocess and mechanical deformation.This energy balance principe is given by the following equation: The deformation elastic energy is given by (Pérignon, 2004): where, u is the displacement in the rod and F the corresponding force.In the case of large displacement, the corresponding force takes the polynomial form (Chengying and Jin, 2010).When one restricts the development to the cubic order term, the corresponding force becomes: where, 1 k is the elasticity linear coefficient and 3 k the cubic coefficient.The elastic energychange in one cycle of magnetization is given by (Venkataraman et al., 1998): The magnetoelastic energy writes (Venkataraman et al., 1998;Krzysztof and Szczyglowski, 2007;Wang and Zhou, 2010): where, b is the magneto-elastic coupling constant, ν the volume of magnetostrictive rod and M the average magnetic moment of the rod.The magneto-elastic energy change in one cycle of magnetization is given by: . 2 According to Jiles and Atherton postulate (Venkataraman et al., 1998;Xiaojing and Le, 2007;Jiaju et al., 2007), the losses due to hysteresis in one cycle is: where, The losses due to mechanical damping are assumed to be (Venkataraman et al., 1998;Chengying and Jin, 2010;Jiaju et al., 2007) where, c l is the damping co efficient.
The change in the kinetic energy is given by (Venkataraman et al., 1998;Jiaju et al., 2007) Let an external force ' F in the u direction produce a uniform stress u σ in the u direction within the actuator.Thus the mechanic work done by the external force in one cycle of magnetization is given by: .' The work done by battery during one cycle of the magnetization process is (Venkataraman et al., 1998;Venkataraman and Krishnaprasad, 2005): where, µ 0 is the permeability of free space.
Substituting Eq. ( 4), ( 6), ( 7), ( 8), ( 9), ( 10), ( 12) in to Eq. ( 1) we obtain the following equation: We define effective field to be: By integration of the first term of Eq. ( 13) over one cycle of magnetization, we have: The magnetic potential energy for the lossless case is given by (Venkataraman et al., 1998;Calkins et al., 2000): Thus Eq. ( 13) can be rewritten as: . 0 are periodic functions of time.We make the hypothesis that the following equation is valid when we go from one point to another point on this periodic orbit: This equation is assumed to hold only on the periodic orbit.Since du and dH eff are independent variations arising from independent control of external prestress and applied magnetic field respectively, the integrands must be equal to zero: .0 ) 1 )( ( Jiles and Atherton relate the irreversible and the reversible magnetization as follows (Krzysztof and Szczyglowski, 2007;Xiaojing and Le, 2007): where, M δ is defined by: Finally after some algebraic manipulations, the equations of our model can be obtained as follows:

RESULTS AND DISCUSSION
Mechanical part of the model: The material using in this analysis is Terfenol-D rod, for his best magnetostrictive performance.The parameters given by Venkataraman et al. (1998), Venkataraman and Krishnaprasad (1999)  Figure 2 and 3 show that the frequency and amplitude of exciting field have remarkable influence on the magnetic hysteresis loops.Figure 4 on his part shows that, the mechanical nonlinear parameter does not have remarkable influence on the magnetic hysteresis loops.Figure 5, the magnetization take the sinusoidal form with some parameter.
According to Willams et al. (2006) and Kalmar-Nagy and Shekhawat (2009) and Fig. 5 the magnetization takes the sinusoidal form with some amplitude and frequency exciting.In that case: . sin where Ω is the natural frequency; and order the equation by introducing the small parameter ε.
where, Approximate solution: An approximate solution is generally obtained as follows (Chengying and Jin, 2010;Lévine, 2004;Nayfeh and Mook, 1979) = n yet to be determined and ε is arbitrarily small parameter.The derivative perturbations rely on the notion that the real time τ, can be expressed in the form of set of successively independent time scales, T n given by: ,...
In Eq. (30) T 0 is nominally considered as a fast time-scale and T 1 as slower time scale, such that ετ τ = = 1 0 T , T as from Eq. ( 31).It follows that the derivatives with respect to τ become expansions in terms of the partial derivatives with respect to the T n according to: ...
Substituting Eq. ( 30), ( 32), (33) into Eq.( 29) and collecting the coefficients of like order of , n ε and equating them to zero in order to construct the perturbation equations, leads to: • Ordre ε 0 : . sin 2 2 2 0 3 0 Harmonic solution of the zeroth order perturbation is: Substitutingzeroth order perturbation solution into the first order perturbation give: To include near-resonant terms within the secular term, a detuning parameter σ is introduced by: . 2 The secular condition become: Then Eq. ( 41) become: Which give the amplitude of response a as function of the detuning parameter σ.
Different parameters effect to system response: Figure 6 to 8 show that nonlinear parameter, amplitude force and damping parameter affect the amplitude response of the system.
Figure 9 shows that, with some value of detuning parameter σ the amplitude response is simple and for other the amplitude response is multiple.
Stability analysis for vibration system: Let us take Eq. ( 41) in the follows: The solution of Eq. ( 41) is given in the following form: Eq. ( 44) in Eq. ( 43) gives: The transformation of Eq. ( 45) in sin and cos terms with the fact that The left hand side of Eq. ( 46) is equal to zero at the equilibrium point.Eq. ( 46) becomes: . 0 sin 4 8 The solution of Eq. ( 47) is follows as: where, * a δ and * δγ are small perturbation of amplitude and phase respectively.
Eq. ( 48) in Eq. ( 47) give Eq.( 49) in the follows form: where, the stability matrix is given as the follows expression: The characteristic equation of Eq. ( 50) is follows as: According to the stability criterion of Routh-Hurwitz (Lacheisserie and Cyrot, 2000;Lakshmanan and Rajaseekar, 2003), the characteristic equation must have real solutions or with negative real parts.What stipulates that: Bifurcation and chaotic analysis for vibration system: Let us take Eq. ( 28) in form: 53) is solved numerically to define routes to chaos in our model using the standard fourth-order Runge-Kutta algorithm.For the set of parameters used in this work, the time step is always 005 .0 ≤ ∆t and computations are performed using variables and constants parameters in extended mode.For each parameters combination, the system is integrated for sufficiently long time and transient is discarded.Two indicators are used to identify the type of transition leading to chaos (Lévine, 2004;Mabekou, 2008;Zeng et al., 2008).The first indicator is the bifurcation diagram and the second is the largest one dimensional (1D) numerical Lyapunov exponent defined by: We now focus on the effects of biasing on the dynamics of the system modelled by Eq. ( 53).To achieve this goal, 0 is chosen as control parameter and the rest of system parameters are assigned the following values: is considered to monitor the bifurcation control parameter.It is found that the system can exhibit complex dynamic motions including periodic, multiperiodic and chaos states.Indeed, for the values of system parameters defined above, various scemarios/routes to chaos are observed such as period doubling and crisis scemarios to chaos.Sample results are provided in Fig. 13 where we show a bifurcation diagram associated with the corresponding graph of largest 1D numerical lyapunov exponent.This bifurcation diagram is obtained by plotting the displacement in the material in terms of the control parameter 0 Q whereas the lyapunov exponent graph is obtained by simultaneously integrating Eq. ( 53) and Eq.(54) -Eq.( 55).The positive value of λ max is signature of chaotic behaviour.Figure 14 shows the Poincare section.Figure 13 shows a sample result of the scenario

CONCLUSION
In this study we have used the energy principle balance to obtain the equation of model by taking into account the nonlinear effect of material.The mathematical model of the material is presented.Using the multiple time scale method, the approximate solution of our model is obtained.The numerical results present the effect of the nonlinearity of the material, damping and the amplitude of the magnetic force on the stability and chaotic response of the system.In the future work we aim to extend the nonlinearity order five and consider the eddy current, temperature and external force effect.

Fig. 1 :
Fig. 1: Sectional view of a Terfenol-D actuator manufactured by Etrema done in changing the kinetic energy of the system consisting of the magnetoelastic rod, due to the elastic deformation of the rod.

Fig. 4 :
Fig. 4: Mechanical nonlinear effect on the magnetic hysteresis loops for Hz f kV U 160 ; 700 0 = = (40)  in to Eq. (39) and then separating out the real and imaginairy parts of the resulting equation, we obtained:

Fig. 6 :
Fig. 6: Nonlinear effect on the system response for the linear terms in a δ and δγ , Eq. (45) becomes:

Figure 10
Figure 10 and 11, we have the domain of stability in blue and field of instability in red.
following variational equation obtained by perturbing the solutions of Eq. (53) as follows: the torus state of the system.
Figure 12 shows the bifurcation diagram and the corresponding lyapounov function.Therefore the scanning process is performed to investigate the sensitivity of the system to tiny change in 0 Q .The range 100 00 .0 0 ≤ ≤ Q