Page 293 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

270 BIOMECHANICS OF THE HUMAN BODY

Mechanical Impedance of the Hand-Arm System. The idealized mechanical input impedance of
the hand-arm system when the hand is gripping a cylindrical or oval handle has been derived for the
three directions of the basicentric coordinate system shown in Fig. 11.2 (ISO 10068, 1998). The
transmissibility of vibration through the hand-arm system has not been measured with noncontact or
lightweight transducers satisfying the conditions described in Sec. 11.2.1. However, it has been
demonstrated that the transmissibility from the palm to the wrist when a hand grips a vibrating
handle is unity at frequencies up to 150 Hz (Boileau et al., 1992).

11.3 MODELS AND HUMAN SURROGATES

Knowledge of tolerable limits for human exposure to vibration, shock, and impact is essential for
maintaining health and performance in the many environments in which man is subjected to dynamic
forces and accelerations. As already noted, humans cannot be subjected to injurious stimuli for
ethical reasons, and so little direct information is available from this source. In these circumstances,
the simulation of human response to potentially life-threatening dynamic forces and accelerations is
desirable, and is commonly undertaken using biodynamic models, and anthropometric or anthropo-
morphic manikins. They are also used in the development of vehicle seats and, in the case of hand-arm
models, of powered hand held tools.

11.3.1 Biodynamic Models
Simple Lumped Models. At frequencies up to several hundred hertz, the biodynamic response of
the human body can be represented theoretically by point masses, springs, and dampers, which
constitute the elements of lumped biodynamic models. The simplest one-dimensional model consists
of a mass supported by a spring and damper, as sketched in Fig. 11.6, where the system is excited at
its base. The equation of motion of a mass m when a spring with stiffness k and damper with resis-
tance proportional to velocity, c, are base driven with a displacement x (t) is:
0

ma t) + c x t) − 0 ( k x t) − x t)) = 0 (11.7)
x t)) +
(
(
((
((
1
0
1
1
where the displacement of the mass is x (t), its acceleration is a (t), and differentiation with respect
1 1
to time is shown by dots.
For this simple mechanical system, the apparent mass may be expressed as a function of frequency
by (Griffin, 1990):
mk i c)ω
( +
M()ω = (11.8)
k ω 2 m i c
−
ω
+
1/2
where ω is the angular frequency (= 2πf ), i = (−1) , and the transmissibility from the base to the
mass is
ω
+
ki c
H() = 2 (11.9)
ω
+
k ω m i c
ω
−
The modulus of the transmissibility may then be written
/ 12
⎡ 1 + 2 ) r 2 ⎤
( ξ
|( )| = ⎢ 22 2 ⎥ (11.10)
H ω
( ξ
1
⎣ ( − r ) + 2 ) r ⎦