Page 283 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

260 BIOMECHANICS OF THE HUMAN BODY

11.1.1 Definitions and Characterization of Vibration, Mechanical Shock,
and Impact
Vibration and Mechanical Shock. Vibration is a time-varying disturbance of a mechanical, or
biological, system from an equilibrium condition for which the long-term average of the motion will
tend to zero, and on which may be superimposed either translations or rotations, or both. The mechani-
cal forces may be distributed, or concentrated over a small area of the body, and may be applied at an
angle to the surface (e.g., tangential or normal). Vibration may contain random or deterministic
components, or both; they may also vary with time (i.e., be nonstationary). Deterministic vibration
may contain harmonically related components, or pure tones (with sinusoidal time dependence), and
may form “shocks.” A mechanical shock is a nonperiodic disturbance characterized by suddenness
and severity with, for the human body, the maximum forces being reached within a few tenths of a
second, and a total duration of up to about a second. An impact occurs when the body, or body part,
collides with an object. When considering injury potential, the shape of the object in contact with, or
impacting, the body is important, as is the posture. In addition, for hand tools, both the compressive
(grip) and thrust (feed) forces employed to perform the manual task need to be considered.
Although vibration, shock, and impact may be expressed by the displacement of a reference point
from its equilibrium position (after subtracting translational and rotational motion), they are more
commonly described by the velocity or acceleration, which are the first and second time derivatives
of the displacement.

Vibration Magnitude. The magnitude of vibration is characterized by second, and higher even-
order mean values, as the net motion expressed by a simple, long-term time average will be zero. For
an acceleration that varies with time t, as a(t), the higher-order mean values are calculated from:
1/ r
⎡ 1 T m ⎤
at)]
a RM = ⎢ ∫ 0 [( dt ⎥ ⎦ (11.1)
⎣
T
where the integration is performed for a time T, and m and r are constants describing the moment
and root of the function. By far the most common metric used to express the magnitude of whole-
body or hand-transmitted vibration is the root mean square (RMS) acceleration a RMS , which is
obtained from Eq. (11.1) with m = r = 2; i.e.,
12 /
⎡ 1 T 2 ⎤
a RMS = ⎢ ∫ 0 [( dt ⎥ ⎦ (11.2)
at)]
⎣
T
Other metrics used to express the magnitude of vibration and shock include the root mean quad
(RMQ) acceleration a RMQ , with m = r = 4 (and higher even orders, such as the root mean sex (RMX)
acceleration a RMX , with m = r = 6).
The RMS value of a continuous random vibration with a gaussian amplitude distribution corre-
sponds to the magnitude of the 68th percentile of the amplitudes in the waveform. The higher-order
means correspond more closely to the peak value of the waveform, with the RMQ corresponding to
the 81st percentile and the RMX to the 88th percentile of this amplitude distribution. The relation-
ships between these metrics depend on the amplitude distribution of the waveform, wherein they find
application to characterize the magnitude of shocks entering, and objects impacting, the body. This
can be inferred from the following example, where the RMS value corresponds to 0.707 of the ampli-
tude of a sinusoidal waveform, while the RMQ value corresponds to 0.7825 of the amplitude.
EXAMPLE 11.1 Calculate the RMS and RMQ accelerations of a pure-tone (single-frequency) vibration
of amplitude A and angular frequency w.
Answer: The time history (i.e., waveform) of a pure-tone vibration of amplitude A can be
expressed as a(t) = A sin wt, so that, from Eq. (11.2):

12 / 2 / 12
⎡ 1 T 2 ⎤ ⎡ A T ⎤
A
)
a RMS = ⎢ ∫ 0 [sin (ω t)] dt ⎥ ⎦ or a RMS = ⎢ ⎣ 2T ∫ 0 [1− cos(2ω tdt ⎥ ⎦
]
⎣
T
T