Biomimetic Robotics - Engineered Biomimicry - page 88

1. A plane section of the cross-section before
bending remains planar after bending.
2. Every cross-section of the beam is symmetri-
cal with respect to the loading plane.
3. The constitutive equations relating the stresses
and strains are defined by Hooke's law.
4. Deformations are assumed to be dynamic
and slowly varying, so all stresses other
than those used to define the in-plane and
the bending stress resultants do not change.

FIGURE 4.5 The NASA SC(2) 0712 supercritical airfoil
overlaid on top of a symmetric NACA 0012 airfoil in the
background.

( Figure 4.5 ). The net increase of the maximum
lift coefficient could be almost as high as 34%
with a slight reduction in the stall angle, the
airfoil's angle of attack beyond which the airfoil
experiences a marked loss of lift.
To obtain a smoother shape, one could in prin-
ciple use flexible or compliant links that could be
either passively or actively controlled [15-17] . In
the latter case, one could precisely control the
shape of the airfoil as each elastic segment or link
in the manipulator is also actively controlled by
a suitable actuator. The key to achieving shape
control is the availability of typical actuators that
can affect the shape with the expenditure of min-
imum energy. This aspect has been researched
by several investigators [13, 18-21] . Apart from
conventional actuators, one could use PZT-based
piezoceramic or SMA actuators.
In the remainder of this section, let us con-
sider the derivation of the general equations
describing the deformation and morphing
dynamics of a structure [22] . We seek to change
the radius of curvature of a curved beam from
one continuously varying function to another by
the simultaneous application of longitudinal
and transverse forces as well as a distributed
bending moment along the track of the beam.
Let us consider a uniform, curved beam with
variable radius of curvature and its dynamics
under the action of (1) external distributed control
forces acting both tangentially and along the radi-
als as well as (2) an external distributed bending,
control moment acting about the sectional neutral
axes. The fundamental assumptions made to sim-
plify the analysis are as follows:

The geometry of a uniform symmetric curved
beam with a variable radius of curvature R ( α ) is
illustrated in Figure 4.6 . The span of the beam,
maximum rise, height of the middle surface, and
inclination to the x axis are, respectively, denoted
by L , h , y ( x ), and α . The mass per unit length of
the beam is denoted by m , and the section mass
moment of inertia about an axis transverse to the
plane of bending and passing through the section

FIGURE 4.6 (a) Geometry of uniform curved beam, and
(b) in-plane and shear forces and bending moment acting on
an element of the beam.