Nonlinear Hamiltonian modelling of magnetic shape memory alloy based actuators

Abstract

This paper proposes an application of the Lagrangian formalism and its Hamiltonian extension to design, model and control a mechatronic system using Magnetic Shape Memory Alloys. In this aim, an original dynamical modelling of a Magnetic Shape Memory Alloy based actuator is presented. Energy-based techniques are used to obtain a coherent modelling of the magnetical, mechanical and thermodynamic phenomena. The Lagrangian formalism, well suited in such a case, is introduced and used to take into account the dynamical effects. Hamilton equations are deduced and used for the computation of the theoretical behaviour of this actuator. These numerical simulations are compared with some experimental measurements permitting the validation of the proposed modelling. Beyond the work presented here, these results will be used to design an energy shaping nonlinear control well-adapted for a strongly nonlinear active material.

Ni-Mn-Ga single crystal Shape Memory Alloy magneto-thermomechanical modeling

“…Nevertheless in practical applications and for specific conditions, some reductions can be made (isothermal process on 2D motions) and in such a case, the usefulness of the model is shown. For example, in the paper [13], an extension of the quasi-static isothermal model to the dynamical case was made, including the magnetic circuit creating the magnetic field and a dynamical load applied to the MSMA sample. A thermodynamic approach with hamiltonian modeling was used by the authors to design actuators [13] and new control laws [14].…”

“…For example, in the paper [13], an extension of the quasi-static isothermal model to the dynamical case was made, including the magnetic circuit creating the magnetic field and a dynamical load applied to the MSMA sample. A thermodynamic approach with hamiltonian modeling was used by the authors to design actuators [13] and new control laws [14]. Moreover, the thermodynamical approach proposed in this paper to model a specific MSMA can be easily extended to other materials, such as future MSMA monocristals and polycristals.…”

Irreversible Thermodynamics and Smart Materials Systems Modelling. Example of Magnetic Shape Memory actuators

“…4 (a) and presented in Gauthier et al (2008), the different energy functions can be divided into parts as shown in Fig. 4 consider the magnetization energy stored in the iron core, at low magnetic field the behaviour is nearly linear: B = µH or M = χH with B, H, and M , being the magnetic induction field, the magnetic excitation field and the magnetization in the core of the magnetic circuit.…”

“…8, these three internal variables are inspired by the microstructure of MSMA sample (NiMnGa material) and are used to clarify the electromechanical energy conversion inside this smart material. The constitutive equation of the independent internal variable ξ describes the irreversible behaviour of MSMA and is used to calculate the dissipation function D. In Gauthier et al (2008), a simple hysteresis loop was chosen, where threshold and rate were identified experimentally (see Fig. 9).…”

Magnetic Shape Memory Alloys as smart materials for micro-positioning devices

“…All other variables are material parameters and can be experimentally identified (see [31]). In this expression, the magneto-thermo-mechanical coupling is expressed by the choice of the internal variable z.…”