Biomedical Engineering Reference
In-Depth Information
center of mass is denoted by I m . The displacements
in the direction of the positive radial and the tan-
gent to the curve are assumed to be w and v ,
respectively. The rotation of the cross-section is
assumed to be ψ . The in-plane, axial stress result-
ant, and bending moment stress resultant are
denoted by N and M , respectively. The external
distributed torque acting is denoted as T . The
shear force acting at any section is given by Q , and
the external longitudinal and transverse forces are
denoted by F and P , respectively.
The displacements in the direction of the pos-
itive radial and the tangent to the curve w and
v , and the rotation of the cross-section ψ, are
related to each other by the relation
The bending moment stress resultant is related
to the transverse displacement by the relation
M =− EI A
R 2 ( W ′′ + W ) .
(4.4)
The in-plane stress resultant is
N = EA
R ( W + V ) − M
R
(4.5)
= EA
R
W + D
M
D α ( W R ψ)
R .
In Eqs. (4.4) and (4.5) , E is the Young's modu-
lus of the material of the beam that is assumed
to be homogeneous, A is the area of cross-section
of the beam, and I a is the section area moment
of inertia about an axis transverse to the plane
of bending and passing through the section's
geometric center. Eliminating M and N from
Eqs. (4.3a)-(4.3c) , we obtain three partial differ-
ential equations that must be solved to obtain w ,
ψ , and Q . Alternately, they may also be expressed
in terms of w , v , and Q .
Given the equations of motion and the con-
trol inputs, one could design an appropriate
control for morphing the structure. In flight, one
would need to include the unsteady loads.
However, the downside of morphing structures
is their susceptibility to divergence and flutter
instabilities, which must be avoided.
DW / D α ≡ W = V + R ψ ,
(4.1)
where the prime denotes differentiation with
respect to α , and R = R ( α ) is the local radius cur-
vature. On considering a typical element of the
beam as shown in Figure 4.7 , the conditions of
force and moment equilibrium are as follows:
DF + DN + QD α = MR D α D 2
DT 2 ( W R ψ),
(4.2a)
DP + DQ ND α = MR D α D 2
DT 2 W ,
(4.2b)
DT DM QR D α = I M RD α D 2
DT 2 ψ.
(4.2c)
4.3 E NGINEERING APPLICATI ONS
Dividing both sides of Eqs. (4.2a)-(4.2c) by
R , we obtain
In this section several key applications are con-
sidered in some detail.
R = M D 2
1
R
DF
D α + 1
DN
D α + Q
4.3.1 Modeling and Control of Robotic
Manipulators
4.3.1.1 Introduction
Motion control of a robot manipulator is a fun-
damental problem that must be addressed at the
design stage. Two categories of motion-control
problems may be identified during the use of
DT 2 ( W R ψ),
R
(4.3a)
R = M D 2
1
R
DP
D α + 1
DQ
D α N
DT 2 W ,
(4.3b)
R
D α + Q = I M D 2
1
R
DT
D α 1
DM
DT 2 ψ.
(4.3c)
R
 
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