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Kinematics and dynamics of rigid bodies 71
2.13 Equations of motions
If we consider the rigid body, Body 2, shown in Figure 2.36 we can formulate
six equations of motion corresponding with the six degrees of freedom
resulting from unconstrained motion.
From our earlier consideration of linear momentum given in (2.161) we
can write
d
d
Σ{ F } { L } m { V G }
dt
d t
2 1 2 1 2 2 1
Expressing this in the familiar form of Newton’s second law we get
(2.224)
Σ{ F} m { A G }
2 1 2 2 1
(2.225)
The vector equation given in (2.225) will thus yield the three equations
associated with the translational motion of the body. It may be noted that
for these equations the vectors in (2.225) may be conveniently referred to
the fixed ground reference frame O 1 .
In the same way that a resultant force acting on the body produces a change
in linear momentum, a resultant moment will produce a change in angular
momentum. If we consider the expression for angular momentum given in
(2.178) we can obtain the equations of motion associated with rotational
motion. For the rotational equations it is convenient to refer the vectors to
the reference frame O 2 fixed in and rotating with Body 2:
d
d
Σ{ M } { H } I
dt
d t
[ ]{ } 2
G2 1/ 2 2 1/ 2 2 1/
2
It can also be shown that
d
d { } d
H2 { H2}
{ 2} { H2}
t dt
11 / 12 / 12 / 12 /
(2.226)
(2.227)
{ M } / [ I2] / { 2} / { 2} / { H2}
/
G2 1 2 2 2 1 2 1 2 1 2
(2.228)
Body 2
Z 2
{V G2 } 1
Z 1
X 2
O 2 G 2 Y 2
GRF
{ω 2 } 1/2
X 1
O 1
Y 1
Fig. 2.36
Rigid body motion