There are three main important control techniques based on three different physical phenomena: current control, resistance control, and Peltier effect control. There exist recent advances in the control of SMA actuators, especially of wire and rod actuators for linear movements, which will be discussed. Of partic-ular emphasis is the use of some physical properties to achieve the desired behavior of the considered sys-tem. The scope is a brief mention of the control problems of SMA actuators, referring the reader to the specific bibliography for further detailed explanation. The first type of control approach for SMA actua-tors considered is current control. SMA fibers are electrical conducactua-tors having particular values of elec-trical resistance. Because of this resistance and because of the Joule effect, a current going through the wire is able to generate heat, and consequently the current can be used to control the temperature of the SMA wire.
Many authors reported studies of this type of SMA control: Bhattacharyya et al. (2000) considered a SMA wire subjected to an electric current, under no mechanical load, and under proper assumptions of the material’s properties. Under their assumptions, the energetic balance can be formulated as in Equation 5.54 referring to Figure 5.35:
冤K(ξ) 冥ρE
(
ξ)
J(
t)
2 (TTamb)CV(ξ) H (5.54)where ξ is martensite fraction (Bhattacharyya proposes a cooling and a heating law, to be associated to the thermodynamic energetic equation);K,ρ
E andCV are, respectively, thermal conductivity, electrical resistivity and heat capacity;his the convection coefficient (Bhattacharyya proposes a linear definition of these properties);His the transformation latent heat;R is the cross-sectional radius of the wire;Tand Tambare, respectively, the temperature of the wire and of the environment;tis the time variable; andJis the current density, which is used to generate a forced heating of the wire. The boundary and the initial conditions of the Cauchy problem can be expressed by Equation 5.55, if there is a wire with a 2L length.
冦T(x, t)冨T(x, t)冨xL,∀x∈冨L,L冨,t0∀t∈ℜTTambamb, ξ(T(x, t))|∀x∈冨L,L冨,t0ξ
0 (5.55)
The second control scheme taken into consideration is based on the change of wire resistance during the phase transformation (Airoldi et al., 1995). When the wire is subjected to a generic thermal load and a constant mechanical load, a linear relationship is observed between the strain and the relative variation of the resistance of the wire (Wu et al., 2000). The slope of the relationship is dependent upon the specific
∂ξ
transformation, and the relationship between the stress and the relative variation of resistance can be quite complex. This functional aspect can be used to control the movement of the microactuator with a simple control scheme. A quantitative work in this direction is reported in Sittner et al. (2000). The authors studied the behavior of a Ni50Ti45Cu5alloy and, utilizing the principles of Equation 5.56 during martensitic transformation and some other theoretical and experimental observations, they developed a new behavioral model of this alloy:
(12v)ε (5.56)
where R0andρ
0are, respectively, the resistance and resistivity in the austenitic phase at the reference tem-perature TR;∆Rand∆ρare, respectively, the absolute change of resistance and resistivity during the trans-formation at the generic T temperature; ε is the resulting strain; and v and CR are two material parameters. The authors, along with other researchers (De Araujo et al., 1999), observed a linear relation between relative resistivity and strain in martensitic and austenitic transformation:
Kε (5.57)
where Krepresents the constant slopes associated with the shape memory transformations.
The third, and most promising, control scheme considered is based on the Peltier effect (Lagoudas and Kinra, 1993), which focuses on the ability to increment the working frequency of SMA microactuators.
This special phenomenon is produced using a thermoelectric element, obtained by sandwiching a SMA layer between two semiconductor layers [a positively doped (P) and a negatively doped (N)]. Alternating the current direction, the SMA layer becomes the hot or the cold junction of a thermoelectric couple, because the Peltier effect generated by electric current causes a temperature differential at a junction of the dissimilar metals. The general thermal transfer model for this control scheme is proposed in (Bhattacharyya, Faulkner and Almaraj, 2000):
Ki (x, t)ρ
iJ2(t)H 冢Ti(x, t)T0冣Civ (x, t), x∈Ii,t0,i∈{P, S, N}
IP冥L , 冤,IS冥 , 冤,IN冥 ,L 冤 (5.58)
where P, S, and Nare, respectively, a positively doped semiconductor, a SMA layer, and a negatively doped semiconductor (Figure 5.36);πis the electrical resistivity;Jis the current density;Cviis the heat capacity per unit volume;Tis the temperature with tas its time variable;PCandACare the perimeter and the area of the cross-section; and His the heat convection coefficient.
d
The interface conditions consist in the temperature continuity and the equal exchange of net heat flux
Pare the Seeback coefficients. The end boundary conditions are defined in Equation 5.60 and the initial conditions consist in the thermal equilibrium of the system at the T0 environmental temperature.
TP冢L ,t冣T0, TN冢L ,t冣T0 (5.60)
The solution of the differential problem provides a model to understand the behavior of the SMA system under the presence of the current density J.
Acknowledgments
Part of this research is financed by the Italian Ministry of Research, University and Instruction (MIUR), under the fund COFIN 2003 Project MiniPar, and part is financed by a grant of the University of Brescia on microactuators. The author would like to thank Riccardo Adamini and Enzo Locatelli for their fruit-ful collaboration; Mohamed Gad-el-Hak, Andrew Oliver and Mauro Ferrari for the discussion on the chapter scheme; Rodolfo Faglia for the discussion on the definition of microactuator; Vittorio Ferrari for the discussion on piezoelectric actuators; Giovanni Legnani for the discussion on electromagnetic actua-tors; and Roberto Roberti for the discussion on shape memory alloys.
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