MODEL OF A PERSON WALKING AS A STRUCTURE BORNE SOUND SOURCE

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 27 MODEL OF A PERSON WALKING AS A STRUCTURE BORNE SOUND SOURCE PACS: 43.5.Pn Matthias Lievens 1 ; Jonas Brunskog 2 1 Institute of Technical Acoustics - RWTH Aachen University, Germany. mli@akustik.rwth-aachen.de 2 Ørsted DTU Acoustic Technology - Technical University of Denmark, Denmark. jbr@oersted.dtu.dk ABSTRACT The behaviour of a person walking as a source of impact sound or walking sound is not yet fully understood. Especially for lightweight structures the coupling between the human body and the floor will determine the power flow into the floor, and therefore the mobility of both source and receiver has to be considered and the contact history must be integrated in the model. This is complicated by the fact that nonlinearities occur at different stages in the system either on the source or receiver side. Not only lightweight structures but also soft floor coverings would benefit from an accurate model of a person walking. In the future such a model would enable us to obtain a correct ranking between the subjective noise caused by a person walking and the objective parameters used to measure impact sound or walking sound. INTRODUCTION The nuisance caused by a person walking consists of three aspects: the well known impact sound heard in the room below, walking sound (footfall noise) radiated directly from the floor covering, and low frequency mechanical vibrations of the floor structure coupled to the surrounding walls and floors. Although the human source behaves as an ideal source on heavyweight structures, the system becomes much more complicated when dealing with the interaction between the person walking and lightweight floors. This aspect has been pointed out in the context of impact sound by many researchers in the field [1, 2, 3, 4]. However the problem has not been completely solved. The correlation between mechanical excitation systems (e.g. a tapping machine, a rubber ball,... ) and the actual human body as a structure borne sound source is still being discussed. To find out more about the details of the coupling between a person walking and a floor, a time-varying mass-spring-damper model designed closely to the physics of the human gait cycle is presented here. Auralisation and tapping machine measurements Auralisation of airborne sound transmission in buildings has been thoroughly described in [5]. It was shown that by using information about the sound reduction index of all separate building elements and the room acoustics in the receiving room it is possible to accurately predict the hearing impression caused by the sound reduction. For the auralisation of impact sound the matter is complicated because the interaction between the structure borne sound source (the person walking) and the floor is a crucial factor for the power transfer between the source and the receiver. The source can not be treated as an ideal source with an infinite internal mobility acting independent of the coupled receiver mobility. Such interaction also exists while measuring impact sound insulation with the ISO standardised tapping machine [6] on floors with a high mobility. By using a detailed description of the coupling between the tapping machine and a certain floor (with its known impact sound level), the coupling with a different source can be accounted for if the mobility of the floor is known from predictions (e.g. [7, 8]) or measurements. In the case of a substitution by a person walking it will enable the prediction of the sound pressure level in the receiving room. For this purpose the interaction between the floor and a person walking has to be investigated in more detail. 1

2 Floor versus body mobility An accurate model of the interaction between the human body and the floor will be presented in the next sections. Before going into detail it is interesting to compare the order of magnitude of average floor mobilities with the mobility of a person standing on one leg. The interaction will only be significant for the power transfer if the mobilities are in the same range. The mobility of nine subjects standing barefoot on a plate in comparison with a simulation of the mobility of a lightweight floor is shown in figure 1(a). The curves reveal that in the frequency range between 6-3 Hz the mobilities are comparable in magnitude, so strong interaction is likely to exist. This means that it is worthwhile to investigate the interaction into more detail to get a better understanding of the behaviour of the human body as structure borne sound source. The setup used to measure the mobility of a person standing on one leg is shown in figure 1(b). It consists of an LDS 51 shaker connected to a plate which is suspended by four springs (k = 1.93kN/m) of adjustable height and 8 steel cables. With a body mass of 8 kg this would result in a resonance frequency of 2.8 Hz if the shaker stiffness is accounted for. The cables prevent the plate from moving horizontally. On top of the plate a second aluminium plate (1x1x15 mm) is attached by three B&K 82 force transducers. The mobility is calculated by use of a B&K 457 accelerometer mounted in the middle of the top plate. The first resonance frequency of the top plate is at about 25Hz. Below this frequency, the force signals can simply be added to get the total force acting on the plate. Y [db re. 1 ms 1 N 1 ] barefoot male 1 (9kg) barefoot male 2 (58kg) barefoot male 3 (71kg) barefoot male 4 (65kg) barefoot male 5 (62kg) barefoot male 6 (7kg) barefoot male 7 (95kg) barefoot male 8 (7kg) barefoot female 1 (52kg) timber joist floor Brunskog(23) (a) mobility of floor and human body (b) mobility measurement setup Figure 1: Comparison of the mobility of a human body with a common lightweight floor construction MODEL OF A PERSON WALKING Ideally a model of a person walking should represent the full gait cycle of one footstep as shown in figure 2. One complete cycle consist of three main phases. During phase 1 both trailing and leading leg are in contact with the floor just after the impact of the heel with the floor. Phase 2 is the phase in which the trailing leg swings past the leading leg and the body goes through the upright position. During phase 3 the body is supported by the toes of the trailing leg and it is pushed forward by the muscle activity in the ankle. The variation of the contact surface of both shoes and the vertical component of the ground force of the left foot is also shown in figure 2. In order to obtain a good model of the human body as a structure borne sound and vibration source the most important aspects of the process should be extracted from the above analysis and represented into a mass-spring-damper model exciting the floor at one location. As a consequence the gait cycle 2 19 th INTE19 th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

3 has to be reduced to a point force injected by one foot. Throughout the rest of the text only the left foot and the vertical force components are considered. contact area contact area contact area force [N] 4 force [N] 4 force [N] phase time [s] phase time [s] phase time [s] Figure 2: Different phases during one complete gait cycle Biomechanical model An analysis of the complete 3D model of the walking process was made by Anderson [9] by using dynamic optimization theory. Although the predicted ground reaction forces correspond well to measurements, such analysis is not able to simulate the impact force caused by the first contact of the leading leg (left foot of phase 1) which is probably the most important phase from the acoustical point of view. To get an insight in the physics behind the walking process, the pendulum model with one or several joints has been used extensively in biomechanics [1, 11, 12, 13]. Again most of the pendulum models concentrate on the support phase and do not model the impact. The simplification of the human gait process by using a pendulum with a limited amount of joints is very appealing because of its physical resemblance with the actual movement of the body but it might be still too complicated to model the important aspects of a structure borne sound source. On the other hand research in running deals with the impact phase in much more detail. A few models that concentrate on the transient ground reaction force during the contact phase use a mass-spring-damper model [14, 15, 16, 17]. Unfortunately these models ignore the rest of the gait cycle. One more disadvantage is the fact that the mass-spring-damper systems are only designed to obtain the right ground reaction force and not to represent the correct source mobility for the interaction with the floor. Ground reaction force The ground reaction force (GRF) of a typical force plate measurement of one footstep is shown in figure 3. By high-pass filtering the force signal below 1 Hz as shown in figure 3(b) it becomes evident that the characteristic shape of the GRF is not relevant for the auditory impression. Only the first contact and the phase during toe impact (at about 4 ms after the first contact depending on the pace of walking) are able to cause impact sound or walking sound. The signal that is left after high-pass filtering follows the original shape very closely and is responsible for low frequency vibration in relation to whole-body vibrations at a receiving room position. It should be pointed out that the force signal in figure 3 was measured barefoot. GRF signals of subjects wearing shoes will also contain high frequency components 4 ms after the first impact when the shoe sole hits the floor at the toes. The amount of high frequency components will depend on the material of the shoes. Mass-spring-damper model The model for the interaction between the human body and the floor is presented in figure 5(a). The human body is represented by its internal mobility m 1...4,k 1...4,c and a blocked force term F BS, and the floor is represented by the impulse response y F of its mobility. The blocked force will be a function of the initial velocity of the foot upon impact. By assuming time-invariance and linearity the coupling with the floor mobility could easily be solved in the frequency domain. However, the human body is characterised by the change of internal impedance over time depending on the contact area of the shoe with the floor and the changing posture of the body itself. Instead of solving the problem 3 19 th INTE19 th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

4 F [db] 5 F [db] (a) ground reaction force low-pass filtered at 1 Hz (b) ground reaction force high-pass filtered at 1 Hz Figure 3: Time signal and spectrum of filtered ground reaction forces in the frequency domain the model has to be dealt with in the time domain and will be calculated numerically. To get an estimate of the mobility of the human body the left foot was measured on the mobility measurement setup in the postures of phase 1 and 3 (figure 2). The measurement curves are shown in figure 4. Phase 2 differs only from phase 1 by the amount of contact area between the shoe sole and the floor. For this reason the difference in mobility could not be measured on the mobility setup. Instead a time-varying contact area as shown in figure 5(b) has to be combined with the measurement of phase 1 and 3 to get a full model of the interaction between the human body and the floor. The multiple mass-spring-damper system of the human body that represents the measurements of phase 1 and 3 is depicted in figure 5(a). Magnitude and phase of the mobility match roughly to the measurement curves as seen from figure 4. The parameters of the system of phase 1 and 3 are given in table 1. The change in mobility was achieved only by varying k 3,c 3 and c 4. Additionally, the stiffness corresponding to the time-varying contact surface can be calculated from k 4 = E A, in which A stands for the contact area (c.f. figure 5(b)) and E denotes the Young s modulus of the material of the shoe sole. m 1 [kg] m 2 m 3 m 4 k 1 [N/m] k 2 k 3 k 4 c 1 [Ns/m] c 2 c 3 c 4 phase e+5 4.6e phase e Table 1: Parameters of the elements of the mass-spring-damper model in figure 5(a) The time-varying contact area in figure 5(b) was taken from an in-shoe dynamic pressure measuring system. Careful estimation of the area during the period of time corresponding to the important phases in figure 3(b) is necessary. Therefore it might be necessary to obtain an even more precise estimate of the change of contact area during heel impact. This can either be done visually by recording the footstep with a high speed camera on a transparent plate or by estimation according to a modified Hertz theory [18]. The time-varying parameters c 4,k 3,c 3 have to be calculated by interpolation of the parameters from the mass-spring-damper models. Additionally the changing centre of mass of the upper part of the human body could be incorporated at a later stage of the modeling th INTE19 th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

5 3 Y [db re. 1 ms 1 N 1 ] measurement phase 1 measurement phase 3 model phase 1 model phase 3 2 Y [rad] Figure 4: Mobility measurements of phase 1 and 3 m 1 c 1 k 1 m 2 c 2 k Area Force 7 6 m 3 5 c 3 c 4 m 4 k 3 k 4 F BS Area [cm 2 ] F y F (a) mass-spring-damper model (b) varying contact area of shoe sole Figure 5: Model of a person walking 5 19 th INTE19 th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID

6 Numerical simulation A simulation of the model and the parameters presented above was numerically solved using a Runge-Kutta method. The blocked force of the mass-spring-damper system is shown in figure 6. The general shape of the simulated force matches with a typical ground force of a person as in figure 5(b) Figure 6: Simulated blocked ground force of the mass-spring-damper model in figure 5(a). References [1] W. Scholl. Impact sound insulation: The standard tapping machine shall learn to walk! Building Acoustics, 8: , 21. [2] B. G. Watters. Impact-noise characteristics of female hard-heeled foot traffic. The Journal of the Acoustical Society of America, 37(4):619 63, [3] Jin Yong Jeon, Jong Kwan Ryu, and Jeong Ho Jeong. Review of the impact ball in evaluating floor impact sound. Acta Acustica united with Acustica, 92: , 26. [4] A. C. C. Warnock. Floor impact noise and foot simulators. In Proceedings - International Conference on Noise Control Engineering, volume 2, pages Inst of Acoustics, Edinburgh, Sc, [5] M. Vorländer and R. Thaden. Auralisation of airborne sound insulation in buildings. Acta Acustica united with Acustica, 86:76 89, 2. [6] J. Brunskog and P. Hammer. The interaction between the iso tapping machine and lightweight floors. Acta Acustica united with Acustica, 89:296 38, 23. [7] Jonas Brunskog and Per Hammer. Prediction models of impact sound insulation on timber floor structures; a literature survey. Building Acoustics, 7:89 112, 2. [8] P. Hammer and J. Brunskog. Vibration isolation on lightweight floor structures. Building Acoustics, 9: , 22. [9] Frank C. Anderson and Marcus G. Pandy. Dynamic optimization of human walking. Journal of Biomechanical Engineering, 123(5):381 39, 21. [1] Marcus G. Pandy and Necip Berme. Synthesis of human walking: A planar model for single support. Journal of Biomechanics, 21(12):153 16, [11] Hartmut Geyer, Andre Seyfarth, and Reinhard Blickhan. Compliant leg behaviour explains basic dynamics of walking and running. Proc Biol Sci, 273(163): , 26. [12] S. Onyshko and D. A. Winter. A mathematical model for the dynamics of human locomotion. Journal of Biomechanics, 13(4): , 198. [13] Karin G. M. Gerritsen, Anton J. van den Bogert, and Benno M. Nigg. Direct dynamics simulation of the impact phase in heel-toe running. Journal of Biomechanics, 28(6): , [14] Kai-Jung Chi and Daniel Schmitt. Mechanical energy and effective foot mass during impact loading of walking and running. Journal of Biomechanics, 38(7): , 25. [15] Benno M. Nigg and Wen Liu. The effect of muscle stiffness and damping on simulated impact force peaks during running. Journal of Biomechanics, 32(8): , [16] Wen Liu and Benno M. Nigg. A mechanical model to determine the influence of masses and mass distribution on the impact force during running. Journal of Biomechanics, 33(2): , 2. [17] Wangdo Kim, Arkady S. Voloshin, and Stanley H. Johnson. Modeling of heel strike transients during running. Human Movement Science, 13(2): , [18] G. Gilardi and I. Sharf. Literature survey of contact dynamics modelling. Mechanism And Machine Theory, 37(1): , th INTE19 th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA27MADRID