Biomechanics - Introduction to Biomedical Engineering - page 153

Similarly, summing moments about the -axis,

X M y ¼

y

0

F 1 l þ F 2 l Rx ¼ 0

x ¼ ð F 1 þ F 2 Þ l
R

The coordinates x and y locate the resultant R .

4.2.4 Anthropomorphic Mass Moments of Inertia
A body's mass resists linear motion; its mass moment of inertia resists rotation. The resis-
tance of a body, or a body segment such as a thigh in gait analysis, to rotation is quantified
by the body or body segment's moment of inertia I :

Z

2

I ¼

m r

dm ð 4 : 37 Þ

where m is the body mass and r is the moment arm to the axis of rotation. The elemental

mass . For a body with constant density r, the moment of inertia

can be found by integrating over the body's volume

dm can be written r dV

V :

r Z

2

I ¼

V r

dV ð 4 : 38 Þ

This general expression can be written in terms of rotation about the x , y , and z axes:

I xx ¼ R V ð y

2 2

þ z Þ r dV

I yy ¼ R V ð x

2 2

þ z Þ r dV ð 4 : 39 Þ

I zz ¼ R V ð x

2 2

þ y Þ r dV

is the moment arm between the axis of rotation and a single point
where all of the body's mass is concentrated. Consequently, a body segment may be treated
as a point mass with moment of inertia

The

radius of gyration k

2

I ¼ mk ð 4 : 40 Þ

where m is the body segment mass. The moment of inertia with respect to a parallel axis I is

related to the moment of inertia with respect to the body's center of mass I cm via the parallel

axis theorem :

2

I ¼ I cm þ md ð 4 : 41 Þ

where is the perpendicular distance between the two parallel axes. Anthropomorphic data

for various body segments are listed in Table 4.1.

d