Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 3, pp. 839-845, Warsaw 2014

A COMPARISON OF THE HUMAN BODY-SEAT MODEL RESPONSES TO
SEVERAL TYPES OF IMPULSE EXCITATIONS

Marek A. Książek, Łukasz Łacny

Cracow University of Technology, Mechanical Faculty, Kraków, Poland

e-mail: ksiazek@mech.pk.edu.pl; llacny@pk.edu.pl

The purpose of this paper is to present the response of a human body-seat model to the
external disturbance in form of an impulse excitation. A 2-DOFActiveHumanBodyModel
with twodistinctpositions (“back-on”an“back-off”) isused in the simulations.Additionally,
twodifferent seat types are considered: rigid andpassive.Several types of impulse excitations
are used ranging in shape, amplitude and disturbance time. A comparison of the results is
made: the first on the assumption of equal excitation energy, and the secondone for impulses
of equal amplitudes but different duration.

Keywords: impulse excitation, vibroisolation, body-seat model

1. Introduction

Thebehaviour and responseof ahumanbodyto suddenexcitations anddisturbanceshasbecome
an important area of research, judging solely the amount of books andpublications regarding the
topic (Griffin, 1990; Fritz, 1999; Liang and Chiang, 2008; Książek, 2011; Nawayseh and Griffin,
2009). Numerous experiments and tests were performed (Cho and Yoon, 2001; Książek and
Łuczko, 2007; Blood et al., 2010), for the purpose of reducing the influence of such disturbances
to a humanbody. In addition to these experiments andmostly thanks to them, several analytical
human bodymodels were designed (Kimet al., 2005; Hinz et al., 2010; Książek and Ziemiański,
2012) presenting further possibilities for research in this field.

In this paper, one of such models found in the literature is analysed (Książek, 1999). The
primary purpose of this analysis is to determine the effect of several types of impulse excitations
upon the human body – the seat model. These types of excitations differ in terms of shape,
amplitude and length, which allows comparing responses of the models.

2. Description of the Active Human Body Model (AHBM)

The biomechanical model of a sitting human body used for the purpose of this analysis is taken
from (Książek, 1999b). The characteristic feature of this model is the possibility to analyse
two distinct human positions: “back-on”, with the back of the body supported by a chair, and
“back-off” without such a support. It was shown in an earlier research (Książek, 1999a) that
the behaviour of the model subjected to harmonic excitation strongly differs depending on the
position assumed by the passenger.

In Figs. 1a and 1b the 2-DOF model of the human body for two different type of seats is
shown. The seat on the left is a rigid one, represented by a lumpedmass m0. The right seat is
a passive one, consisting of a mass m0, damper αm and spring km. These additional elements
make it possible for the seat to behave as a simple vibroisolation system (VIS).



840 M.A. Książek, Ł. Łacny

Fig. 1. Active human body model with rigid seat (a) and with passive (VIS) seat (b), Książek (1999b)

While the “back-on” and “back-off” bodypositions share the samemodel as shown inFig. 1,
theydiffer in terms of active control forces developed in thebodyaswell as the values of separate
parameters. The control forces for each model can be written using the following equations

Fback−off =−k11(y1−y0)−k12(ẏ1− ẏ0)−k13(y2+y0)−k14(ẏ2+ ẏ0)

Fback−on =−k11(y1+y0)−k12(ẏ1− ẏ0)−k13(y2+y0)−k14(ẏ2− ẏ0)
(2.1)

The parameters on the other hand are calculated based on both these equations and the
values obtained during the experimental phase. The parameters acquired for both positions, for
the human body of mass 70.8kg are provided in Table 1.

Table 1.Parameters for “back-off” and “back-on” position

“back-off” “back-on”

m1 [kg] 9.1 66

k1 [N/m] 11972.5557 51189.32

α1 [Ns/m] 3251.9783 1704.17

m2 [kg] 61.7 4.8

k2 [N/m] 22456.7485 63335.50

α2 [Ns/m] 519.044 1262.59

k11 [N/m] 97323.2354 123251.32

k12 [Ns/m] -2226.0653 -1781.04

k13 [N/m] -1960.5176 -104227.69

k14 [Ns/m] 1164.3525 759.69

By using the models shown in Fig. 1 and described by Eqs. (2.1), it is possible to obtain
the transfer functions between both masses and the ground (H1g,H2g) for every presented
combination of the position and seat type. These, in turn, after substitution of the parameters
(Table 1) are used to obtain the response of the masses to an impulse excitation.

3. Response of the AHBM to impulse excitation

3.1. Choice of impulse types

The aim of the paper, as stated before, is to analyse the behaviour of the human body-seat
model under an impulse excitation. In order to do so, the definition of such an excitation needs



A comparison of the human body-seat model responses to... 841

to be provided first. For this purpose, several types of impulse functions, differing in shape have
been chosen. The list of functions is provided in Table 2with the shapes presented in Fig. 2 and
impulses defined by Eqs. (3.1)-(3.4), respectively.

Table 2. List of impulse excitations

Impulse type Amplitude [–]

square 10

triangle 17.3205

half-sine 14.1421

spline 17.5158

Fig. 2. Shapes of impulse excitations

The value of amplitude associated with each function has been calculated on the assumption
of equal “signal energy” Ex and equal time ∆t of the disturbances. This assumption has been
made to ensure that the results obtained from the analysis for different impulse excitations are
comparable. The energy Ex of each signal x(t) can be calculated using Eq. (3.5). It should be
noted that the energy obtained this waymay have units other than [J], and depends on the unit
of the signal used for the calculation.
While inTable 2nophysical unit is assigned to the amplitude, in this paper it is assumed that

the excitation is in formof a suddendisplacement (bump), and theunits are given in centimeters.
Using the previously found transfer functions between bothmasses of themodel and the ground
(H1g,H2g) it is possible to directly obtain the displacement. From these displacements, it is
possible to calculate the acceleration and, in turn, the forces to which these masses might be
subjected

xsq(t)=Asq t∈ [0,∆t] (3.1)

xtr(t)=















Atr

∆t
t for t∈

[

0,
∆t

2

]

−Atr

∆t
t+2Atr for t∈

[∆t

2
,∆t
]

(3.2)

xhs(t)=Ahs sin
( π

∆t
t
)

t∈ [0,∆t] (3.3)

xsp(t)=



































12Asp
∆t2
t2 for t∈

[

0,
∆t

3

]

12Asp
∆t2

[3

4

(∆t

3

)2

−

(

t−
∆t

2

)2]

for t∈
[∆t

3
,
2∆t

3

]

12Asp
∆t2
(t−∆t)2 for t∈

[2∆t

3
,∆t
]

(3.4)



842 M.A. Książek, Ł. Łacny

Ex =

∆t
∫

0

x
2(t) dt (3.5)

3.2. Response to impulse excitation (equal energy)

The first series of tests have been performed for selected impulse excitations on the assump-
tion of equal energy and equal time, as stated in the previous Subsection.While the shapes and
amplitudes of the impulses are different by this assumption, the results proved to be almost
identical with no regard to the type on the excitation. Therefore, the results for both masses
and both types seats, are presented only for one type of the impulse.

Fig. 3. Response of the human body-seat model to the impulse excitation for both “back-on” and
“back-off” positions: (a) upper mass, rigid seat; (b) lowermass, rigid seat; (c) upper mass, passive seat

(α=260Ns/m); (d) lowermass, passive seat (α=260Ns/m); (e) upper mass, passive seat
(α=2600Ns/m); (f) lowermass, passive seat (α=2600Ns/m)

In Fig. 3, the response of the model to the considered impulse disturbance is presented.
In the case of the model with the rigid seat (Figs. 3a,b), the displacement for both masses is
quite high, especially for the “lower mass” m2 in the “back-on” position, for which it reaches
the value of 25cm. In addition to that, the frequency of oscillations caused by the impulse is



A comparison of the human body-seat model responses to... 843

greater than that for the “back-off” position. By analysing both these factors (amplitude and
frequency of oscillation), a conclusion can be drawn that in the case of the “back-on” position
the accelerations caused by the impulse excitation and by extension forces, to which the “lower
mass” is exposed, are much higher than in the other case (“back-off” position).
InFigs. 3c,d, the response to the same excitation butwith the seat type changed to “passive”

is shown. It can be seen that the simple vibroisolation system works as intended, and the
amplitude of oscillations is much lower than for the model with the rigid seat. However, as
before, it should be noted that the frequency of vibration for the “back-on” position is much
higher than for the “back-off” position. Figures 3e,f used for the comparison show that the
damper parameter αm is ten times larger in this case. The frequency of oscillations is similar,
but the amplitude in each case is much higher. This however is to be expected as the seat
becomesmore rigidwith thismodification.An additional confirmation of these observations can
be obtained by analysing Bode plots of the models under consideration, see Fig. 4.

Fig. 4. Bode plot for both “back-on” and “back-off” positions in the range 0.1-50Hz: (a) upper mass,
rigid seat; (b) lowermass, rigid seat; (c) upper mass, passive seat (α=260Ns/m); (d) lowermass,

passive seat (α=260Ns/m)

3.3. Response to impulse excitation (velocity condition)

The second part of the analysis has been carried out on the assumption that the amplitude
of the impulse excitation is constant, while the time in which this excitation occurrs vary. In
the analysis, this could be interpreted as a model moving with a given velocity over a bump of
a certain height and width. The faster the movement of the model, the shorter the time of the
disturbance.
The shape of the impulse chosen for the purpose of this test is the “spline”. The width of

the bump considered in the analysis is w = 0.5m, while the height h which corresponded to
the amplitude of the impulse is equal to 0.1m. The values of velocity considered in the test are
5, 10 and 20m/s with the impulse duration of 0.1, 0.05 and 0.025s, respectively. The results
obtained for the passive seat with default parameters can be seen in Fig. 5.



844 M.A. Książek, Ł. Łacny

Fig. 5. Response of the human body-seat model with the passive seat to the impulse excitation of
amplitude h equal to 10cm but different time lengths ∆t: (a) upper mass, ∆t=0.1s; (b) lowermass,
∆t=0.1s; (c) upper mass, ∆t=0.05s; (d) lowermass ∆t=0.05s; (e) upper mass, ∆t=0.025s;

(f) lowermass, ∆t=0.025s

It canbenoted inFig. 5 that the responseof themodel is strongly dependenton the duration
of the impulse. This is especially true in the case for the “back-on” position – the oscillation at
velocity 20m/s (0.025s) is noticeably larger than that at 5m/s (0.1s).Worth noting is also the
fact that the amplitude of the response is larger for longer impulse duration. This is related to
the energy of excitation – with an increase in the impulse duration the energy is increased as
well.

4. Conclusions

Several simulations and calculations have been performed in this paper in order to determine
the response of the human body-seatmodel to an external excitation. The conclusions gathered
from the results of these simulations can be written as follows:

• the “back-on” human body position is much more susceptible to an impulse excitation
than the “back-off” one



A comparison of the human body-seat model responses to... 845

• the response of the model is only slightly affected by the shape of the impulse; this is of
course valid on the assumption of the equal signal energy and a short impulse duration

• depending on the velocity at which the obstacle (bump) is encountered, the response of
the system changes; the shorter the duration (and therefore higher velocity), the stronger
are high-frequency oscillations in the model; on the other hand the shorter time leads to
a drop in the energy of the signal, and thus the amplitude of the response is lower.

Additionalmodifications are considered for the presentedmodel and for further testing. The
most important ones are the modification of AHBM equations to include non-linearities of the
model and the addition of an active controller which alongwith the presented passive seatmight
improve the damping properties of the system, thus reducing the vibration to which the human
body is subjected.

References

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Manuscript received December 5, 2013; accepted for print May 7, 2014