Biomedical Engineering Reference
In-Depth Information
=
[0]. They may be caused by joints (i.e., a pin joint forces two points on
adjacent segments to have the same trajectory) or by an externally imposed
motion pattern. The constraints enter the equations of motion in the form
of constraint forces [ λ ] t rather than in geometric terms. There exists a con-
straint force associated with each constraint equation and analogous to a
reaction force. The constraint forces [ λ ] t are known as Lagrange multipli-
ers. In the previous example, the particle has two DOF since it is restricted
to planar motion. Hence, the constraint equation is in the form z
The constraints in a system are described by equations of constraint [ ]
0.
The constraint force associated with this constraint equation is the force in
the z axis that keeps the particle in the xy plane of motion. A totally con-
strained system (DOF
=
0) can be solved kinematically. It is impossible to
solve an overconstrained system (DOF < 0) without removing the redundant
constraints.
The independent generalized coordinates are those coordinates that can
be varied independently without violating the constraints, and must equal
the degrees of freedom of the model. The use of the independent gener-
alized coordinates allows the analysis of most models to be made with-
out solving for the forces of constraint. Any additional coordinates in the
model are known as superfluous or dependent coordinates. The relations
between the independent and the superfluous coordinates, are in fact, con-
straint equations [ ]
=
[0]. If the dependent coordinates can be eliminated,
then the system is called holonomic . Nonholonomic systems always require
more coordinates for their description than there are degrees of freedom.
By definition, the first and second derivatives of a generalized coordinate q i
with respect to time are called the generalized velocity ,
=
q i , and the gen-
˙
eralized acceleration ,
q i , respectively. The relation between the position
vector r i of a point i in the system and the generalized coordinates [ q ] t are
called the transformation equations . It is assumed that these equations are in
the form:
¨
x i
=
f xi (q 1 , q 2 , q 3 , ... , q n , t )
y i
=
f yi (q 1 , q 2 , q 3 , ... , q n , t )
(8.3)
z i
=
f zi (q 1 , q 2 , q 3 , ... , q n , t )
where some or all of the generalized coordinates may be present. The velocity
component in the x direction is obtained by taking the time derivative of the
x i equation,
∂x i
∂q j
∂q i
∂t
n
∂x i
∂t
x i
˙
=
+
(8.4)
j
=
1
Similar relations can be written for the
y i and
˙
z i components.
˙
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