Page 513 - Handbook of Biomechatronics

Bioinspired and Biomimetic Micro-Robotics for Therapeutic Applications 507

piezo-laminates embedded in while simultaneously being subject to two-
way coupling with the surrounding viscous liquid, for sake of completeness.
However, in other cases, the elastic tail in question may be a just a passive and
slender extremity (Khalil et al., 2016, 2017b). The preferred governing
equation for structural deformation should take internal shear if the undu-
lation amplitude is large, for example, comparable to the wavelength. Thus,
Rayleigh-Timoshenko equations (Kaneko, 1975) are selected here:

∂ φ ∂ φ Jm ∂ φ mEI ∂ φ Jm ∂ F ?
4
2
4
4
2
EI + m + ¼ F ? + J + +
2
∂s 4 ∂t 2 k T AG ∂t 4 k T AG ∂t ∂x 2 k T AG ∂t 2
2
mEI ∂ F ?

k T AG ∂x 2
(29)
where φ (m) is the local planar wave deformation for an active elastic tail, E
2
4
(Pa) is the Young’s modulus, I (m ) is the second moment of area, J (kgm )is
the mass moment of inertia, m (kg) denotes the mass of infinitesimal slice
2
around the mesh node, A (m ) is the total cross-sectional area of the flagel-
lum that is perpendicular to its long axis, G (Pa) is the shear modulus, k T is
the Timoshenko correction coefficients for internal shear, and with F ?
(N/m) being the unequally distributed load per unit length due to external
viscous drag and internal piezoelectric effect combined as (Cook-Chennault
et al., 2008; Jeon et al., 2005)
∂φ E PZT Vd 15 A PZT

F ? ¼ c n + u == + (30)
∂t t PZT L PZT
with u // (m/s) being the velocity component parallel to the direction of
deformation on that particular location with respect to the surrounding fluid

owing to the rigid-body motion of the micro-swimmer. Furthermore, E PZT
2
(Pa) is the Young’s modulus of the PZT-laminate, A PZT (m ) is the cross-
sectional area of the PZT-laminate that is perpendicular to its long axis, L PZT
(m) is the length of the PZT-laminate, and d 15 (C/N) is the piezoelectric
coupling coefficient used to predict the mechanical strain while driven in
the shear mode along the direction of wave propagation. Indeed, piezoelec-
tric laminates can be designed to deform in different directions thus different
coupling coefficients and loading conditions will be used instead (Preumont,
2006; Piefort, 2001) and it will be evident by the references given here that
the stress term due to the piezoelectric effect presented by the second part of
Eq. (30) is a very simplistic approximation for a far more complicated