Biomedical Engineering Reference
In-Depth Information
Solving this last expression for the one unknown,
F
A
, the vertical force at the elbow:
F
A
¼
69
:
1lb
To find the unknown horizontal force at the elbow,
F
C
, and the unknown force the biceps must
F
B
, the other equation of equilibrium
P
F
generate,
¼
0 is used:
þð
F
B
cos 75
i
þ
F
B
sin 75
j
F
C
i
F
A
j
Þ
10 lb
j
3
:
5lb
j
¼
0
Summing the
x
and
y
components gives
75
Þ¼
F
C
F
B
cos
ð
0
75
Þ
F
A
þ
F
B
sin
ð
10 lb
3
:
5lb
¼
0
Solving these last two equations simultaneously and using
F
A
¼
69
:
1 lb gives the force of the
biceps muscle,
F
B
, and the horizontal elbow force,
F
C
:
F
B
¼
85
:
5lb
F
C
¼
22
:
1lb
4.2.5 Equations of Motion
Vector equations of motion are used to describe the translational and rotational kinetics
of bodies.
Newton's Equations of Motion
Newton's second law relates the net force
F
and the resulting translational motion as
F
¼
m
a
ð
4
:
42
Þ
where
a
is the linear acceleration of the body's center of mass for translation. For rotation
M
¼
I
ð
4
:
43
Þ
a
where
I
is the body's angular momentum. Hence, the rate of change of a body's angular
momentum is equal to the net moment
M
acting on the body. These two vector equations
of motion are typically written as a set of six
a
x
,
y
, and
z
component equations.
Euler's Equations of Motion
Newton's equations of motion describe the motion of the center of mass of a body. More
generally, Euler's equations of motion describe the motion of a rigid body with respect to its
center of mass. For the special case where the
coordinate axes are chosen to coincide
with the body's principal axes, that is, a cartesian coordinate system whose origin is located
at the body's center of mass, Euler's equations are
X
xyz
M
x
¼
I
xx
a
x
þð
I
zz
I
yy
Þ
o
y
o
z
ð
4
:
44
Þ
X
M
y
¼
I
yy
a
y
þð
I
xx
I
zz
Þ
o
z
o
x
ð
4
:
45
Þ
