Page 114 - The Mechatronics Handbook

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Electric Phenomena
In the previous section the laws governing the motion of rigid and deformable bodies were reviewed.
The forces entering these equations are often of electromagnetic origin; thus one has to know the
distribution of electric and magnetic fields. The electromagnetic field is governed by a set of four coupled
equations known as Maxwell’s equations. Similarly, to the momentum equations, these can also be
postulated in integral form. Here we only give the local form

B + ∇ × E = 0
˙
⋅
∇ D = q f
(8.17)
∇ × HD = i
˙
–
∇ B = 0
⋅
where E is the electric field, D is the electric displacement, B is the magnetic induction, H is the magnetic
f
field strength, i is the electric current density, and q is the free charge volume density. Equations (8.17)
require constitutive laws specifying the current density, electric displacement, and magnetic field in terms
of electric field and magnetic induction vectors. A linear form of these laws is given by
E
i = ----, D = e 0 E + P, B = m 0 H + m 0 M = m 0 m r H (8.18)
r e
where ρ e is the electrical resistance. The coupling between electrical and mechanical fields can be linear
or nonlinear. For example, piezoelectricity is a linear phenomenon describing the generation of electric
field as a result of the application of mechanical stress. Electrostriction on the other hand is a second
order effect, resulting in the generation of mechanical strain proportional to the square of the electric
field. Other effects include piezoresistivity, i.e., a change of the electrical resistance due to mechanical
stress. In addition to these material properties, electromechanical coupling can be achieved through direct
use of electromagnetic forces (Lorentz force) as is commonly done in conventional electrical machines.
Lorentz force per unit volume is given by
f = q E + v × B) (8.19)
(
f
L
f
where q is the volume charge density. Equation (8.19) accounts for the forces acting on free charge only.
If the fields have strong gradients, the above expression should be modified to include the polarization
and magnetization terms [Maugin 1988].
EM f  ∂P 
⋅
⋅
f = q E +  i + ------ × B + P ∇E + ∇BM (8.20)
∂t 
Equation (8.19) or (8.20) can be used in the momentum equation (8.10) in place of the body force f.
As mentioned earlier, piezoelectricity and piezoresistivity are the other commonly used effects in
electromechanical systems. The piezoelectric effect occurs only in materials with certain crystal structure.
Common examples include BaTiO 3 and lead zirconia titanate (PZT). In the quasi-electrostatic approxi-
mation (when the magnetic effects are neglected) there are four variables describing the electromechanical
state of the body—electric field E and displacement D, mechanical stress T and strain εε εε. The constitutive
laws of piezoelectricity are given as a set of two matrix equations between the four field variables, relating
one mechanical and one electrical variable to the other two in the set
e ij = s ijkl T kl + d ijk E k , D i = d ikl T kl + e 0 Ξ ij E j (8.21)

where s ijkl is the elastic compliance tensor, d ijk is the piezoelastic tensor, Ξ ij is the electric permittivity tensor.
If the electric field and the polarization vectors are co-linear, the stress and strain tensors are symmetric,
and the number of independent coefficients in s ijkl is reduced from 81 to 21 and for the piezoelastic ten-
sor d ijk from 27 to 18. If further, the piezoelectric is poled in one direction only (for example index 3),

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