This article was downloaded by: [Aston University]On: 06 October 2014, At: 06:58Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

ErgonomicsPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/terg20

Resonant frequencies ofstanding humansJ. M. RANDALL , R. T. MATTHEWS & M. A.STILESPublished online: 10 Nov 2010.

To cite this article: J. M. RANDALL , R. T. MATTHEWS & M. A. STILES (1997)Resonant frequencies of standing humans, Ergonomics, 40:9, 879-886, DOI:10.1080/001401397187711

To link to this article: http://dx.doi.org/10.1080/001401397187711

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of allthe information (the “Content”) contained in the publications on ourplatform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy,completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions andviews of the authors, and are not the views of or endorsed by Taylor& Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information.Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly inconnection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access

and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

Resonant frequencies of standing humans

J. M. RANDALL, R. T. MATTHEWS and M. A. STILES

Bio-Engineering Division, Silsoe Research Institute, Wrest Park, Silsoe,Bedford MK45 4HS, UK

Keywords: Human; Resonance; Standing; Vertical; Vibration; Beam.

Many forms of industrial illness are thought to result from the eŒects of vibrationon the human body. Prolonged exposure causes undue stress and discomfort. Atthe human whole-body resonant frequency there is maximum displacementbetween the organs and the skeletal structure and thus this is one frequency ofvibration that should be minimized in the workplace and elsewhere. The verticalwhole-body resonant frequencies of 113 fully-clothed standing humans weremeasured using a vibrating beam method, which imposed a very low accelerationmagnitude at the subjects’ feet. The overall range of resonant frequencies wasfound to be from 9 to 16 Hz and independent of mass, height and mass to heightratio. The mean values ( 6 1 s.e.) were 12 × 2 6 0 × 1 Hz for males and 12 × 8 6 0 × 2 Hzfor females with an overall mean population value of 12 × 3 6 0 × 1 Hz.

1. Introduction

When exposed to vibration, the human body responds in a complex way. It has been

identi® ed that whole-body vibration is the cause of many forms of injury or industrial

illness occurring in workplaces where workers are exposed for long periods of time to

low-frequency vibration (Dupuis and Zerlett 1986). Those working with rotating and

reciprocating machinery are most at risk from these eŒects, often suŒering from back

pain and other spinal disorders. Knowledge of the resonant frequency of the human

body could aid the design of industrial buildings and transport systems so that

exposure to vibration close to the body’s resonant frequency may be minimized. At

the resonant frequency there is maximum displacement between the organ and the

skeletal structure, placing biodynamic strain on the body tissue involved. Measure-

ments of vibration transmissibility from the point of excitation (usually the seat) to

the head (or other organ) reveal frequencies of maximum transmissibility that can be

attributed to resonance of the organ. However, such resonances are in¯ uenced by the

response of other organs and the whole-body resonance. It is almost impossible to

stimulate the natural frequency of one organ alone without exciting the whole-body

resonances (Dupuis and Zerlett 1986).

Many measurements have been made of the whole-body resonance of seated

persons as this is relevant to vibration during driving or travelling and much of this

research has been used to assess and minimize the danger of vibration-related injury

to the drivers of military vehicles. The whole-body resonant frequency of the seated

person is typically 5 Hz (Gri� n 1990), but this is in¯ uenced by many factors. Neither

body posture (e.g. normal or erect) nor body weight has a large eŒect on resonant

frequency. Muscle tension can have a signi® cant eŒect with most subjects showing a

higher frequency when tense (Fairley and Gri� n 1989). The height of a foot rest

changes the impedance at low frequencies, but has little eŒect on seated whole-body

resonant frequency.

ERGONOMICS, 1997, VOL. 40, NO. 9, 879 ± 886

0014± 0139/97 $12 × 00 Ó 1997 Taylor & Francis Ltd

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

There has been relatively little work concerning the whole-body resonant

frequencies of standing humans. Using the transmission response from the feet to the

lumbar spine, the cervical spine and the forehead, Herterich and Schnauber (1992)

found resonances of 4, 8 and 16 Hz, but these can be attributed to spinal and head

resonances rather than whole-body resonances. Again using transmission from the feet

to the head, Paddam (1987) found resonances of vertical vibration at 2 × 5 Hz and a

second smaller peak at 13 Hz for the three postures of legs locked at the knees, legs

unlocked and legs bent. Gri� n (1990) quotes an unpublished study with eight subjects,

which indicated a resonant frequency of about 5 Hz for the standing person with

straight legs (locked at the knees) and about 3 Hz with knees bent. However, using an

indirect technique, Ji et al. (1993) measured the resonant frequency of a standing person

in the range of 8 to 10 Hz. Further work with this technique indicated a range of 10 to

12 Hz using four subjects and depending on the modal mass of the person (Ji 1995).

Measurements of the whole-body resonant frequency can be made by one of two

methods. The ® rst, the apparent mass method (Fairley and Gri� n 1989) involves

measurement of force and acceleration over a range of frequencies of a subject on an

oscillating platform. The apparent mass for a given frequency is provided by the

ratio of force to acceleration at that frequency. The resonant frequency appears as a

peak in the frequency against apparent mass curve. This method is unsuitable when

working with animals as it requires a degree of co-operation on the part of the

subject, or certain sensitive humans (e.g. pregnant women) as the magnitude of the

r.m.s. acceleration might cause discomfort or distress.

A second method (Ji et al. 1993, Randall and Peng 1995) makes use of a simple

rectangular beam supported at the ends and made to vibrate in its fundamental

mode by being gently struck near its centre. The process is repeated with a subject

standing at the centre of the beam. In both cases the resonant frequency of the beam

is recorded, from which the resonant frequency of the subject may be deduced. In

this situation, the magnitude of the vibration is negligible and it is therefore safe

when used with sensitive subjects. This method allows a quick and easy ¯ ow of

subjects thus minimizing the time taken for measurements.

2. Materials and methods

2.1. Theory of the beam

The subject-beam system may be adequately modelled by an undamped , 2 degrees-

of-freedom model (® gure 1). The equations of motion of the system are given by

msX..

s 1 ksXs 2 ksXb 5 0 (1)

and

2 ksXs 1 mbX..

b 1 (ks 1 kb)Xb 5 F (2)

where m is the mass, X is the vertical displacement, k is the dynamic stiŒness and F is

the external force on the beam. The subscripts, s and b refer to the subject (human)

and the beam, respectively. Equations (1) and (2) give two natural frequencies for the

system (Randall and Peng 1995), x 1 and x 2 with x 1 < x 2 and related by the equations

x21 1 x

22 5 x

2b 1 (1 1 a ) x

2s (3)

and

x 1 x 2 5 x s x b (4)

880 J.M.Randall et al.

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

where a = ms/mb. In this case, mb refers to the modal mass of the beam, which for the

fundamental mode is equal to one-half the actual mass (Mb) of the beam, i.e.

mb 5 Mb /2 (5)

The use of the modal mass is necessary because the portions of the beam around its

ends contribute very little to the motion of the beam. The resonant frequency of the

subject may now be calculated from the equation

x2s 5

x 2( x 2b 2 x 2)

x 2b 2 (1 1 a ) x 2

(6)

where x is either x 1 or x 2. It can now be seen that by measuring the frequency of the

beam ( x b) and of the system ( x ) with the masses of the beam and subject known

(giving a ), the resonant frequency of the subject ( x s) can be calculated.

2.2. Design and use of the beam

Randall and Peng (1995) conclude that for optimum accuracy in the measurement

of natural frequency (taking into account the separation of the two resonance

peaks, the resolution of the measuring equipment and varying masses to be used

with the same beam) the beam should be designed so that its mass (Mb) is twice that

of the average subject and its frequency is twice that of the expected subject

frequency. With this being the case, the accuracy was calculated to be

approximately 6 0 × 3 Hz.

Figure 1. Human-beam system represented by a two degrees-of-freedom model.

881Resonant frequencies of standing humans

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

The beam used was constructed of two steel channels spot-welded back to back,

and of length 3 × 7 m (® gure 2). Each channel measured 76 by 152 mm and the beam

was supported 50 mm from each end on triangular supports forming a totally rigid

base. The mass outside the boundaries of the supports was deducted from the total

mass of the beam. A 0 × 3 m square steel plate 7 mm thick was bolted to the centre of

the beam adding 5 × 50 kg to the modal mass of the beam. The mass of the beam was

calculated (British Constructional Steelwork Association Ltd 1991) and determined

by weighing as 132 × 1 kg. The modal mass used for calculating results was therefore

69 × 8 kg.

The expected natural frequency of the beam was calculated (Ryder 1971) using

the equation:

f 5p

2l 2

EI

A qHz (7)

where l is the suspended length (3 × 6 m), E is the Young’ s Modulus (2 × 1 ´ 1011

Pa), I is the

moment of inertia (1 × 703 ´ 10Ð 11

m4), A is the cross-sectional area (4 × 555 ´ 10

Ð 3m

2),

and q is the density (7840 kg mÐ 3

). The beam was calculated as having a natural

frequency (not including the mass of the plate and neglecting temperature and other

atmospheric eŒects) of 38 × 36 Hz. An accelerometer (Setra Systems, Acton, MA model

141A, range 6 40 msÐ 1

) was rigidly mounted on the beam using a strong magnet and its

output signal passed through an ampli® er to a dynamic signal analyser (DSA, Hewlett-

Packard 35660A, Santa Clara, CA) with a resolution of 0 × 125 Hz.

First, the natural frequency of the beam was measured by lightly tapping it with a

rubberized hammer near its centre (the exact position was unimportant as long as the

beam was set into resonance in its fundamental mode). The frequency distribution of

the signal was calculated and displayed by the DSA and the frequency of the peak

was noted. The procedure was repeated with the person standing on the plate at the

centre of the beam facing along its length. Three values were taken for each subject

and each individua l value was used to calculate the whole-body resonance from

which a mean was taken. Finally, the beam frequency was recorded once more,

averaged with the ® rst value and used for the calculation of the human frequency.

To check if small variations in the position of the person would aŒect the

measurement technique, the beam was ® rst subjected to a series of static mass

experiments using ® xed weights at various distances from the centre and recording

the resonant frequency of the weighted beam.

2.3. Subjects

The 113 subjects (78 male, 35 female) were broadly representative of the general

population being drawn from the delegates to the Human Response to Vibration

Figure 2. Design of the beam.

882 J.M.Randall et al.

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

Conference 1995 and the staŒof Silsoe Research Institute. However, they were not

selected to be statistically representative of the population . Ages ranged from about 15

to 65 years, weights from 50 to 155 kg and heights from 1 × 55 to 1 × 97 m. The subject’ s

height and weight were measured to 6 0 × 005 m and 6 0 × 1 kg, respectively. They wore

normal clothing and shoes. When standing on the plate, each subject was instructed to

stand in an upright relaxed posture with similar pressure on the heel and ball of the foot.

3. Results

3.1 Preliminary experiments

The results of the static mass experiments using ® xed weights at various distances

from the centre (® gure 3) show that the variation in frequency with displacement is

very small over the small range 6 0 × 4 m. It is therefore not essential for the subject to

stand exactly at the centre of the plate.

3.2. Human resonant frequencies

The resonant frequencies of all 113 males and females are shown agains t their mass

(® gure 4), agains t their height (® gure 5) and their mass to height ratio (® gure 6). The

means of the population s are 12 × 2 6 0 × 1 Hz for males, 12 × 8 6 0 × 2 Hz for females and

12 × 3 6 0 × 1 Hz for all data. The standard deviation of the whole data set was 1 × 4 Hz

with a range of 9 × 1 to 15 × 7 Hz.

Although the linear regressions in ® gures 4 and 5 show a general downward trend

of frequency with mass and height, there is no evidence (R2

= 0 × 05 for mass,

R2

= 0 × 02 for height) to suggest that either of these factors is signi® cant.

4. Discussion

The preliminary experiment to establish the extent to which the position of the

person on the beam is likely to aŒect the measured frequency (® gure 3) shows that

Figure 3. Response of the beam to oŒset static masses.

883Resonant frequencies of standing humans

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

the centre of mass may be up to 6 0 × 4 m from the centre of the beam without

signi® cantly aŒecting the results. In practice, the feet could always be placed within

6 0 × 1 m of the centre and if all the person’s weight were concentrated at the heel or

the toe (contrary to instructions) , the centre of mass could not be more than 6 0 × 2 m

from the centre of the beam.

The subjects used in the study were chosen deliberately and were generally

representative of the population as a whole, rather than a sub-set such as a young

military population . Consequently, the whole-body resonant frequency of standing

humans subjected to vertical (z-axis) vibration has a fairly wide range. The total

range of 9 × 1 to 15 × 7 Hz encompasses the values of 8 to 10 Hz and 10 to 12 Hz found

by Ji et al. (1993) and Ji (1995), respectively. However, they are considerably higher

Figure 4. Frequency against mass for males ( j ) and for females ( s ); the line shows a linearregression.

Figure 5. Frequency against height for males ( j ) and for females ( s ); the line shows alinear regression.

884 J.M.Randall et al.

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

than the 3 and 5 Hz quoted by Gri� n (1990) and the lower frequency of 2 × 5 Hz

measured by Paddan (1987) for the head. Herterich and Schnauber (1992) attribute

an 8 Hz resonance to the lumbar spine and a 16 to 20 Hz resonance to the head,

Paddan (1987) also measured a higher frequency of 13 Hz for the head, suggesting

that both of these could be contributing to the whole-body resonance of 9 to 16 Hz.

The mean measured resonant frequency ( 6 1 S.E.) of 12 × 3 6 0 × 1 Hz for all

subjects conceals the wide range of values as measured for individuals . The standard

deviation of the data for all subjects of 1 × 4 Hz implies that 68% of the measured

population had frequencies of between 10 × 9 and 13 × 7 Hz. There is no signi® cant

diŒerence between the mean measured resonant frequencies of 12 × 2 6 0 × 1 Hz for

males and 12 × 8 6 0 × 2 for females.

A measure of a person’ s over or under weight from the mean may be drawn from

their body mass per unit height. This quantity is plotted with resonant frequency in

® gure 6. The plot shows a band of values corresponding to a set of ideal’ height and

weight values between two vertical lines. The upper end of this band corresponds

more to the male body, while the lower end holds more relevance for the female

population . It can be seen that the spread of the data in this region is similar to the

spread outside it Ð although the range is narrower owing to the small number of

data points. This suggests that there is no direct correlation between how far away

from their ideal weight’ a person is, and the position of that person’s resonant

frequency within the overall population .

Over the range of subject weights and heights used in this study, there is no

evidence of a relationship between the measured resonant frequency and subject

mass, height or mass-to-height ratio. This is consistent with the response of seated

subjects for which no signi® cant relationship with mass has been found (Fairley and

Gri� n 1989).

ReferencesBRITISH CONSTRUCTIONAL STEELWORK ASSOCIATION LTD 1991, Properties and safe load tables,

Structural Steelwork Handbook (Eleventh impression) (British Constructional SteelworkAssociation Ltd, London).

Figure 6. Frequency against mass/height for males ( j ) and for females ( s ).

885Resonant frequencies of standing humans

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014

DUPUIS H. and ZERLETT G. 1986, The EŒects of Whole-body Vibration (Springer-Verlag, Berlin/Heidelberg).

FAIRLEY, T. E. and GRIFFIN, M. J. 1989, The apparent mass of the seated human body: verticalvibration, Journal of Biomechanics, 22, 81 ± 94.

GRIFFIN, M. J. 1990, Handbook of Human Vibration (Academic Press, London).HERTERICH, J. and SCHNAUBER, H. 1992, The eŒects of whole-body mechanical vibration on

standing man, Journal of Low Frequency Noise and Vibration, 11, 52 ± 61.JI, T. 1995, A continuous model for the vertical vibration of the human body in a standing

position. Proceedings of the UK Group on Human Response to Vibration, Silsoe, 18 ± 20September 1995.

JI, T., ELLIS, B. R. and BEAK, M. 1993, Indirect measurement of human whole bodyfrequencies. Paper presented at the United Kingdom Informal Group on HumanResponse to Vibration at the Army Personnel Research Establishment, Farnborough,20 ± 22 September 1993.

PADDAN, G. S. 1987, Transmission of vertical vibration from the ¯ oor to the head in standingsubjects. Proceedings of the UK Group on Human Response to Vibration, Shrivenham,21 ± 22 September 1987.

RANDALL, J. M. and PENG, C. 1995, Vibrating beam method for measuring animal naturalfrequencies, Journal of Low Frequency Noise and Vibration, 14, 119 ± 133.

RYDER, G. H. 1971, Strength of Materials, SI edition (Macmillan, London).

886 J.M.Randall et al.

Dow

nloa

ded

by [

Ast

on U

nive

rsity

] at

06:

58 0

6 O

ctob

er 2

014