Page 284 - Biomedical Engineering and Design Handbook Volume 1, Fundamentals

VIBRATION, MECHANICAL SHOCK, AND IMPACT 261
Let us integrate for one period of the waveform, so that T = 2π /ω (any complete number of periods
will give the same result). Then:
A
.
a RMS = = 0 707 A
2
From Eq. (11.1):

⎡ 1 T 4 ⎤ 14 / ⎡ A 4 T 2 ⎤ / 14
A
a = ⎢ ∫ [sin (ω t)] dt or a = ∫ [ [1− cos (2ωt )] dt
RMQ ⎣ T 0 ⎥ ⎦ RMQ ⎢ ⎣ 4 T 0 ⎥ ⎦
⎡ A 4 T ⎡ 3 1 ⎤ ⎤ / 14
a = ∫ − 2 coss(2ωt − cos(4ωt ) dt
)
RMQ ⎢ 0 ⎣ ⎢ ⎥ ⎥
⎣ 4T 2 2 ⎦ ⎦
Again, integrating for one period of the waveform:
/
⎡ 3 A 4 ⎤ 14
.
a RMQ = ⎢ ⎥ = 0 7825 A
⎣ 8 ⎦
Equinoxious Frequency Contours. Human response to vibration, shock, and impact depends on
the frequency content of the stimulus, as well as the magnitude. This may be conveniently introduced
electronically, by filtering the time history of the stimulus signal, and has led to the specification of
vibration magnitudes at different frequencies with an equal probability of causing a given human
response or injury, so defining an equinoxious frequency contour. The concept, while almost
universally employed, is strictly only applicable to linear systems. The biomechanic and biodynamic
responses of the human body to external forces and accelerations commonly depend nonlinearly on
the magnitude of the stimulus, and so any equinoxious frequency contour can be expected to apply
only to a limited range of vibration, shock, or impact magnitudes.
Equinoxious frequency contours may be estimated from epidemiological studies of health effects,
or from the response of human subjects, animals, cadavers, or biodynamic models to the stimuli of
interest. Human subjects cannot be subjected to injurious accelerations and forces for ethical reasons,
and so little direct information is available from this source. Some information has been obtained from
studies of accidents, though in most cases the input acceleration-time histories are poorly known.

Frequency Weighting. The inverse frequency contour (i.e., reciprocal) to an equinoxious contour
should be applied to a stimulus containing many frequencies to produce an overall magnitude that
appropriately combines the contributions from each frequency. The frequency weightings most com-
monly employed for whole-body and hand-transmitted vibration are shown in Fig. 11.1 (ISO 2631-
1, 1997; ISO 5349-1, 2001). The range of frequencies is from 0.5 to 80 Hz for whole-body vibration,
and from 5.6 to 1400 Hz for vibration entering the hand. A frequency weighting for shocks may also
be derived from a biodynamic model (see “Dynamic Response Index (DRI)” in Sec. 11.3.1).

Vibration Exposure. Health disturbances and injuries are related to the magnitude of the stimulus,
its frequency content, and its duration. A generalized expression for exposure may be written

T
(
(
(
Ea , ) m r = ⎡ ⎣ ⎢∫ 0 T [ F a t))] m dt ⎤ ⎦ ⎥ r / 1 (11.3)
w
w
,
where E(a , T) is the exposure occurring during a time T to a stimulus function that has been
w m,r
frequency weighted to equate the hazard at different frequencies, F(a (t)). In general, F(a (t)) may
w
w
be expected to be a nonlinear function of the frequency-weighted acceleration-time history a (t).
w
Within this family of exposure functions, usually only those with even integer values of m are of
interest. A commonly used function is the so-called energy-equivalent vibration exposure for which
F(a (t)) = a (t) and m = r = 2:
w w