9.11 EFFECTS OF VIBRATION ON HUMANS

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The human body is a relatively complex vibratory system, because it contains

both linear and nonlinear ‘‘springs’’ and ‘‘dampers.’’ As in the case of

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Copyright © 2003 Marcel Dekker, Inc.

hearing damage studies, it is difficult (and unethical, in extreme cases) to

conduct research on vibratory damage on living human subjects. As a consequence

of this difficulty, much of the research data on vibratory effects on

humans have been obtained from experiments on animals or by simulation.

For the frequency range below about 40 Hz, the human body can be

modeled approximately by a system of masses (the head, upper torso, hips,

legs, and arms), spring elements, and damping elements (Coermann et al.,

1960).

Generally, exposure to vibration at the workplace is more severe than

vibration exposure at home, in terms of both levels of vibration and duration

of vibration exposure. Most of the work-related whole-body vibration

exposure arises from forces transmitted through the person’s feet while

standing, or the buttocks while seated (Von Gierke and Goldman, 1988).

Hand–arm vibration exposure may also occur while holding tools.

There are two important frequency regions as far as vibration of the

whole human body is concerned: (a) from 3 Hz to 6 Hz, where resonance of

the thorax–abdomen system occurs, and (b) from 20 Hz to 30 Hz, where

resonance of the head–neck–shoulder system occurs. The resonance of the

thorax–abdomen system is expecially important, because this resonance

places stringent requirements on the vibration isolation of a sitting or standing

person. For example, at a frequency of 4 Hz, the acceleration of the hip

region of a standing person is approximately 1.8 times the acceleration of

the surface on which the person is standing. For a person seated, the acceleration

of the head–shoulder region is about 3.5 times the acceleration of the

surface on which the person is seated, for a frequency of 30 Hz.

In the frequency region between 60 Hz and 90 Hz, resonance in the

eyeballs occurs. There is a resonant effect in the lower jaw–skull system in

the frequency range between 100 Hz and 200 Hz. Resonance within the skull

occurs in the frequency region between 300 Hz and 400 Hz. Human response

to vibration at frequencies above about 100 Hz is influenced significantly by

the clothing or shoes at the point of application of the vibratory force.

Vibration at frequencies below about 1 Hz affects the inner ear and

produces annoyance, such as cinerosis (motion sickness). For frequencies

greater than about 100 Hz, the perception of vibration is noticed mainly on

the skin, and depends on the specific body region affected and on the clothing,

shoes, etc., that the person is wearing.

Criteria for acceptable vibration exposure have been developed by

national (ANSI, 1979) and international (ISO, 1985) standards organizations.

The rms acceleration levels corresponding to fatigue-induced decrease

in work proficiency are given by the following relationships. If a person is

exposed to rms acceleration levels that exceed the values given by the fol-

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Copyright © 2003 Marcel Dekker, Inc.

lowing relationships, the person will generally experience noticeable fatigue

and decreased job proficiency in most tasks:

for 1Hz _ f < 4Hz

La ј 90_10log10р f =4ЮюCFt (9-168)

for 4Hz _ f _ 8Hz

La ј 90dBюCFt (9-169)

for 8Hz < f _ 80 Hz

La ј 90ю20log10р f =8ЮюCFt (9-170)

The rms acceleration level must not exceed LaрmaxЮ ј 116:8dB, which corresponds

to an acceleration of 0.707g or 6.94 m/s2 (22.75 ft/sec2).

The factor CFt is a correction for the duration of the acceleration

exposure, and may be estimated by the following relationships:

for t _ 8hours

CFt ј 20Ѕ1_рt=8Ю1=2_ (9-171)

for 8 < t _ 16hours

CFt ј 20Ѕр8=tЮ1=2 _1_ (9-172)

The acceleration limits for a condition of ‘‘reduced comfort’’ due to

the vibration may be found by subtracting 10 dB from the values given by

Eqs (9-168), (9-169), or (9-170). The upper bound of allowable acceleration

exposure, which represents a hazard to the person’s health if exceeded, is

found by adding 6dB to the values given by Eqs (9-168), (9-169), or (9-170).

The acceleration level is defined as follows:

La ј 20log10рa=aref Ю (9-173)

The reference acceleration, as given in Table 2-1, is aref ј 10 mm=s2

(0.00039 in/sec2). An acceleration of 1g (g ј 9:806m=s2 ј 32:174 ft=sec2 ј 386.1 in/sec2) corresponds to an acceleration level of the following:

La ј 20 log10р9:806=10 _ 10_6Ю ј 119:8dB _ 120 dB

If the vibrational displacement is sinusoidal, the rms acceleration is

related to the maximum or peak acceleration amax by:

arms ј amax=21=2 ј 0:707amax (9-174)

For a vibrational displacement yрtЮ given by the following sinusoidal relationship,

we may determine the relationship between the acceleration and

displacement:

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yрtЮ ј ymax e j!t (9-175)

aрtЮ ј

dy

dt ј _!2ymax e j!t ј !2ymax e jр!tю_Ю (9-176)

The acceleration of the mass is _ radians or 1808 out of phase with the

displacement. The maximum or peak acceleration is related to the maximum

displacement by the following expression:

amax ј !2ymax ј 4_2f 2ymax (9-177)

If we differentiate the expression for the vibration of a mass subjected

to displacement excitation, given by Eqs (9-124) and (9-130), we obtain the

following relationship for the maximum acceleration of a mass subjected to

displacement excitation:

a2;max ј !2y1;maxTr ј рKS=MЮr2y1;maxTr (9-178)

The quantity r is the frequency ratio, rј !=!n ј f =fn, and Tr is the transmissibility.

If we substitute for the transmissibility given by Eq. (9-103), we obtain

the following dimensionless relationships for the maximum acceleration of a

mass subjected to displacement excitation:

рa2;max=gЮ

рKSy1;max=MgЮ ј r2Tr ј r2 1 ю р2_rЮ2

р1 _ r2Ю2 ю р2_rЮ2

" #1=2

(9-179)

If the spring constant KS is the design variable that we are seeking, the

following form is more convenient to use:

рa2;max=gЮ

р4_2f 2y1;max=gЮ ј Tr ј

1 ю р2_rЮ2

р1 _ r2Ю2 ю р2_rЮ2

" #1=2

(9-180)

Example 9-13. A person is seated in a seat that is supported by a spring–

damper system. The mass of the seat and the person is 80 kg (176.4 lbm), and

the damping ratio for the support system is _ ј 0:060. The maximum amplitude

of motion for the foundation to which the support system is attached is

5mm (0.197 in), and the vibration frequency for the foundation is 10 Hz.

The time that the person will be seated is 6 hours per day. Determine the

spring constant for the support such that the person would experience little

fatigue-induced decrease in work proficiency.

The correction for time of vibration exposure may be found from Eq.

(9-171):

CFt ј 20Ѕ1 _ р6=8Ю1=2_ ј 2:68 dB

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Copyright © 2003 Marcel Dekker, Inc.

The rms acceleration level at the fatigue-induced proficiency limit is found

from Eq. (9-170) for a frequency of 10 Hz:

La ј 90 ю 20 log10р10=8Ю ю 2:68 ј 90 ю 1:94 ю 2:68 ј 94:62 dB

arms ј р10Юр10_6Юр1094:62=20Ю ј 0:5383m=s2 р0:1472 in=sec2Ю

For design purposes, let us use an acceleration that is 80% of the limiting

value:

arms ј р0:80Юр0:5383Ю ј 0:4306m=s2 р0:1177 in=sec2Ю

The peak acceleration, assuming sinusoidal excitation, is as follows:

a2;max ј р2Ю1=2р0:4306Ю ј 0:6090m=s2 р0:1665 in=sec2Ю

a2;max=g ј р0:6090Ю=р9:806Ю ј 0:06210

Let us calculate the parameter in Eq. (9-180):

4_2f 2y1;max

g ј р4_2Юр10Ю2р0:0050Ю

р9:806Ю ј 2:0130

The required transmissibility for the support system is found from Eq.

(9-180):

Tr ј

0:06210

2:0130 ј 0:03085 рLTr ј _30:2dBЮ

The parameter _ is as follows:

_ ј 1 ю р2Юр0:06Ю2р1 _ 0:03085Ю2

р0:03085Ю2 ј 1 ю 7:557 ј 8:557

The required frequency ratio is found from Eq. (9-108):

r2 ј 8:557ю р8:557Ю2 ю р1 _ 0:030852Ю

р0:03085Ю2

" #1=2

ј 42:066

r ј р42:066Ю1=2 ј 6:486 ј f =fn

The required undamped natural frequency for the support system is as

follows:

fn ј р10Ю=р6:486Ю ј 1:542 Hz

The required spring constant for the support may now be calculated:

KS ј р4_2Юр1:542Ю2р80Ю ј 7508N=m ј 7:508 kN=m р42:87 lbf=inЮ

Note that the static displacement under the weight of the person and

the seat is as follows:

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Copyright © 2003 Marcel Dekker, Inc.

d ј

Mg

KS ј р80Юр9:806Ю

р7508Ю ј 0:1045 ј 104:5mm р4:11inЮ

The person may experience some problems when sitting in a seat for which

the supports deflect this much, however.

The maximum amplitude of vibration for the seated person during

vibration is as follows:

y2;max=y1;max ј Tr ј 0:03085

y2;max ј р0:03085Юр5:00Ю ј 0:154mm р0:0061 inЮ

The maximum or peak velocity for the foundation is as follows:

v1;max ј 2_fy1;max ј р2_Юр10Юр0:0050Ю ј 0:314m=s

ј 314mm=s р12:4 in=secЮ

According to the data in Table 9-3, this degree of vibration corresponds to a

very rough machine.

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