STRUCTURAL AND BEHAVIORAL OPTIMIZATION OF THE NONLINEAR HILL MODEL

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 12, Number 3/2011, pp STRUCTURAL AND BEHAVIORAL OPTIMIZATION OF THE NONLINEAR HILL MODEL Tudor SIRETEANU *, Silviu NĂSTAC**, Mihaela PICU** * Institute of Solid Mehanis, Buharest, Romania ** Dunărea de Jos University, Galati, Romania mihaelapiu@yahoo.om The paper is based on the nonlinear biomehanial model developed by Hill. The entral idea of this study is the behavioural and strutural optimization of this model so that it emphasizes the auto adaptive apaity of the real biomehanial system. The paper analyses and ompares the responses to harmoni and step signals. A version of the Hill model whih ontains nonlinear elasti and dissipative elements was developed based on observed results. It has also been established the onstitutive law of these additional elements so that the initial objetive of the study to be met. The evaluation of harateristi measurements and ompleting the final expression of the onstitutive equation have been established taking into aount a series of experimental results developed on the atual model of the hand-arm system subjeted to dynami vibration ations. The results of this researh provide the neessary foundation and support for future implementations of this version of Hill's model in some virtual tools to analyze the dynamis of the human body subjeted to vibration or shok. Key words: Biodynami, Optimization, Hill model, Nonlinear harateristis, Autoadaptive behaviour. 1. INTRODUCTION One of the well known model in biomehanis was proposed by A.V. Hill in 1938, based on a series of pratial observations dated sine 1922 [5]. Thus, Hill was the first who notied that an ativated musle produes a higher fore in isometri onditions (when both ends are fixed) than in normal onditions (when its length dereases) [6, 9]. This is equivalent to the loss of a quantity of energy to overome an internal resistane fore. It is obvious that this resistane fore an not be simulated by the elasti element of the model [7]. However, Hill noted that the total fore is muh less developed as the musle ontration speed is higher. Considering that ative fore (the fore introdued by the ative element) has a onstant value, Hill stated that a high-speed ontration leads to a high resistive fore. For this reason one of the omponents of the model is a visous damper. It simulates very well the real evolution: the higher the speed applied to the ends of the element, the higher value the developed fore reahes, and the general behaviour of the model is a diret onsequene of the fluid visosity value [8, 11, 12]. 2. HILL MODEL DIAGRAM Hill proposed a model (Fig. 1) with a series struture, onsisting of an elasti element (further noted as a serial element) and a parallel type visoelasti onfiguration, like Voight model (further noted as a parallel element) [3]. The differene from lassi visoelasti model Voight type onsists of an additional element mounted in parallel whih simulates the musle ation. In Fig. 1 this additional element is noted with Ψ and along with the visous element forms the ative omponent of Hill's model [2, 8]. The onstitutive equation of this type of model is obtained by onsidering a linear feature for the two elasti elements [4].

2 214 Tudor Sireteanu, Silviu Năsta, Mihaela Piu 2 F k serial k paralel F For the serial omponent of the model the fore value is: F = k ( x x ), (1) serial 1 10 and for the parallel omponent is: parallel ( ) F = k x x + x +ψ, (2) omponenta ativa x 1 x 2 Fig. 1 Hill model diagram. where x and x 0 are the lengths of eah omponent in free and strained state. The fore developed in the parallel omponent will be: F F F = kparallel x x0 + x +ψ, (3) kserial kserial where: k serial F= kserial + kparallel F kparallel x x0 + x +ψ, (4) kserial ( ) where F, F are the fore developed by the model and respetively its time variation; x, x are the strain and respetively the strain speed, onsidered for the entire model; Ψ is the fore from the ative omponent of the model; k serial, k parallel are the rigidities of the serial an parallel elements and is the damping oeffiient of the ative omponent of the model. It must be mentioned that the F, F, x, x, Ψ parameters are time funtions. Equation (4) is the onstitutive equation of the Hill model presented in Fig. 1. Taking into aount the time variation of the fore applied to the model the harateristi equation (6.35) beomes: F k k parallel serial = kparallel ( x x0 ) + x 1+ F +ψ. (5) kserial The ative fore from the parallel omponent is: k parallel F ψ= F 1+ kparallel ( x x0 ) x. (6) kserial kserial Fenn and Marsh in 1935, then Hill in 1938 disovered that the relation between the fore and the ontration speed is nonlinear in ertain strain onditions [1, 10]. This phenomenon inreases the omplexity of the model without a signifiant inrease of the performane obtained by diret stimulation. Taking all this into aount the nonlinear model will be further studied in the initial onfiguration presented in Fig. 1. Integrating the harateristially equation (6.35) results the apable fore formula F(t): t kserial kserial + kparallel d() xu kserial + kparallel F() t = exp u kparallel ( xu x0) ( u) du F0 e t + +Ψ + 0 du, (7) where Ψ(u) is the ative fore speifi to the musular fibre, x(u) is the ompulsory transit law, and the ineptive fore value is F 0 =F(0). If the integration is made onsidering the independent variable x(t) it results: t 1 k parallel d Fu ( ) x() t = exp u kparallelkserial x0 F( u) ( kserial kparallel ) ( u) kserial du x0 k serial Ψ du + 0 (8) k parallel exp t, where Ψ(u) is the ative fore, F(u) is the apable (ompulsory) fore for the whole model and x 0 = x(0) is the initial length of the musle fibre.

3 3 Strutural and behavioural optimization of the nonlinear Hill model RESPONSE ANALYSIS OF THE HILL MODEL Hereinafter the response of this model will be analyzed in the following ases: Compulsory harmoni fore Compulsory fore step-type Fig. 2 Fig. 3 Displaement response ompulsory displaement step-type ompulsory unitary displaement Fig. 4 Fig. 5 Fore response For eah ase, the ative fore Ψ is null or has a unit value, the analyse was made by omparing the two onditions of the ative omponent and for three values of the time onstant of the model: τ1 = 0.01; τ2 = 0.10; τ3 = 1.00 whih we onsidered to be representative. a b d e f Fig. 2 Displaement response for the ompulsory harmoni fore and the null ative fore: a) τ1 = 0.01; b) τ2 = 0.10; ) τ3 = 1.00; unit ative fore: d) τ1 = 0.01; e) τ2 = 0.10; f) τ3 = a b d e f Fig. 3 Displaement response for the ompulsory step type fore and for the null ative fore: a) τ1 = 0.01; b) τ2 = 0.10; ) τ3 = 1.00; unit ative fore: d) τ1 = 0.01; e) τ2 = 0.10; f) τ3 = 1.00.

4 216 Tudor Sireteanu, Silviu Năsta, Mihaela Piu 4 a b d e f Fig. 4 Fore response for the ompulsory displaement step type and null ative fore: a) τ 1 = 0.01; b) τ 2 = 0.10; ) τ 3 = 1.00; unit ative fore: d) τ 1 = 0.01; e) τ 2 = 0.10; f) τ 3 = a b d e f Fig. 5 Fore response for the ompulsory unitary displaement and null ative fore: a) τ 1 =0.01; b) τ 2 =0.10; ) τ 3 =1.00; unit ative fore: d) τ 1 =0.01; e) τ 2 =0.10; (f) τ 3 =1.00. Comparing the diagrams from in Figs. 2 5, whih are related to the Hill model behaviour to the ompulsory exitation signals, in the presene or absene of the ative fore, it was determined: In the presene of the ative fore, this is an unitary step type with a period from t 1 = 0,5 s. up to t 2 = 1,7s with the origin being the starting point of the analysis t 0 = 0,0s. these two temporal oordinates were established in relation with the response time of the nervous stimulation on the musle fibre and with the step type exitation period (ases presented in Fig. 3 and Fig. 4). For the other two analysed ases (harmoni displaement response Fig. 2, respetively response to the initial unitary displaement Fig. 5) the same time values were maintained to ease the omparative analysis. Regarding the displaement responses (Fig. 2 and Fig. 3) it is noted that in the absene of the ative omponent (a, b, diagrams) the model follows the assertion of the exitation signal as being exlusively influened by the speifi time onstant value. So we have an amplitude derease of the

5 5 Strutural and behavioural optimization of the nonlinear Hill model 217 harmoni signal response (Fig. 2), respetively a more and more pronouned highlight of the integration response of the initial signal (Fig. 3) by inreasing the value of the time onstant of the model. The ative omponent leads to ontrations of the musle fibre length whih is highlighted by the derease of the absolute value reorded by the model under the external exitation ation (d, e, f diagrams). The value of the time onstant assess diretly the way in whih the translation of the transit takes plae, meaning: one the τ 1 value inreases the translation of the transit assumes a nature ontinuous in time, so is highlighted the natural transit way from an ative state to an inative one speifi to the musular struture. Regarding the responses in fore (Fig. 4 and Fig. 5) it is notied that in the inative state (a, b, diagrams) the model regards the exterior ompulsory restrition (ompulsory displaement step-type in the initial moment or at two arbitrary time moments) and the evolution is in greater aordane with the value of the time onstant (the highlight of the visous omponent of the model). The ineption of the ative state of the musular fibre leads to an inrease of the immediate value of the fore during the entire ative state (d, e, f diagrams). It is obvious that the passing from the inative state to the ative one and bakwards happens with a diret influene on the value of the time onstant of the model, highlighting one again the natural transition between the two states speifi to the musular fibre. The diagrams orresponding to the fore responses (Fig. 4 and Fig. 5) also show the seeming pointlessness of initiating an ative state when the model is subjeted to external ompulsory displaement. The pointless omes from the fat that the only visible onsequene is the inrease of the value of the immediate fore. The speiosity is justified by the fat that by setting a fix value (immediate) of displaement at the end of the model, aording to the initial theory suggested and studied by Hill, the apable fore from the musle fibre should inrease to maximum values whih an be seen in the evolution of the analysed model (Fig. 4 and Fig. 5, d, e, f diagrams). 4. THE ACTIVE AUTOADAPTIVE COMPONENT OF THE NONLINEAR MODEL An upgrade to the Hill model is suggested by taking into aount of some nonlinear material harateristis. So the serial element will be still linear, while the parallel element will have a nonlinear harateristi: k serial =t. ; k parallel =f k ( i k p, t, x, x ) with i=1, 2,... ; parallel = f ( i p, t, x, x ) with i=1, 2,... (9) Thus the model built-up (Fig. 6) is ative beause it keeps the ative element Ψ=Ψ(t) in its struture, but aquires an autoadaptive harateristi due to the modifiation of the essential parameters rigidity, damping with the parameters of the disturbing movement strain, respetively the strain flutuation in time. Fig. 6 Ative autoadaptive model of the musular fibre.

6 218 Tudor Sireteanu, Silviu Năsta, Mihaela Piu 6 This omes to support the real behaviour of the human body (or of a body part) under the ation of the dynami external fators namely: when the grasp value of these parameters exeeds a limit threshold an arrest of the body from the vibration soure is tried by reduing the transmissibility of the propagation way. Furthermore, the lab test showed that the body s adaptability is a ontinuous value and it is diretly proportional with the stress level, so that an inrease of the level piked up by the organism asserts a derease in transmissibility performanes of the propagation way of the dynami perturbation. In mathematial formulae, aording to its use in a lurative model all the above an be put into pratie using different funtions. In this paper it is proposed the following funtional dependene: k parallel =f k ( i k p, t, x, x )= lin k parallel β i= 2... n 1 1+ β i ( x) where lin k parallel is the known onstant value of the stiffness of the parallel element, β 1 is a dimensionless nonzero onstant, β i [m -i ], with i =1, 2,... are the summation polynominal oeffiients (an have any real value). It an be seen that when all the polynominal oeffiient are null, the rigidity funtion beomes linear, and the stiffness value is multipliative influened with β 1 : k parallel = f k ( i k p, t, x, x )= lin β1 k parallel 1+ 0 (11) k parallel = f k ( i k p, t, x, x )= lin k parallel β 1. If β 1 = 1, for the linear ase we will obtain: k paralel = lin k paralel. (12) Fig. 7 presents a simulation of the elasti harateristi of an autoadaptive element aording to the relation (10), for: lin k parallel =1 000 N/m, β 1 = 1, β 2 = β 5 = β 12 = 10, and the rest of the polynominal oeffiients were onsidered null. i, (10) Fig. 7 Non-linear elasti harateristi of the autoadaptive model.

7 7 Strutural and behavioural optimization of the nonlinear Hill model 219 The simulation was realised separately for eah ase (i = 2; i = 5; i = 12) and the obtained results were presented omparative. The diagram orrespondent to the linear ase was drawn. Fig. 8 shows orrespondent diagrams of the non-linear elasti fore developed by the autoadaptive model in the same onditions previous presented. Fig. 8 Non-linear elasti fore developed by the autoadaptive model. elasti F parallel = lin β1 k parallel x(t), 1 + β ( ) i i= 2... n ( x t ) where we noted with elasti F parallel the elasti fore of the parallel omponent. It must be mentioned that the simulations were made exlusively for the adaptive omponent of the proposed model, onsidering the total absene of the ative element Ψ = Ψ(t) = 0. It was highlighted the speifi behaviour for the ation of some summation polynominal oeffiients (2, 5, 12). So, we an model a differentiate ative harateristi on adaptability levels, through the fat that every ative element will bring into the model an ation ompulsory by the given summation polynominal oeffiient. The diffiult modelling of this type of element and the neessity for a lot of data explains this stritly theoretial approah, without pratial illustrations. i (13) 5. CONCLUSIONS This paper presented a shortly theoretial approah of the Hill's basi model with a set of an additional improvements regarding behavioural and/or strutural optimization. Additional harges of the Hill's basi model leads to better behavioural approahes, even those harges not involved main strutural modifiations. One of the first harges enjoined to the Hill's basi model supposed the elimination of the damper omponent. This modifiation leads to a strutural simplifiation, but have not the apaity to dignify the natural transitions between the ative and inative states or due to the external stimulus. In the same time, these simplified variants of the Hill's model help to evaluate the natural frequenies of biomehanis systems and also the resonane area taking into aount the ative or inative state of the speial omponent. The last and omplex harges applied to the Hill's model, whih was briefly presented in this paper, not involve main strutural hanges. But through the nonlinear harateristis imposed to elasti and visous

8 220 Tudor Sireteanu, Silviu Năsta, Mihaela Piu 8 elements, the musular fibre responses on external dynami ations respet more preisely the experimental observations. The set of diagrams presented in this paper reveal the major importane of the proper time onstant for eah model. Hereby, the jumping evolutions of the numerial model versus natural transitions of the real system are the two states between the simulations an be evolved. The main advantage of these approahes of the Hill's model is the ative auto-adaptive apaity whih dignifies the natural ability of the musular system to respond on dynami external harge like shok and vibration. This study derived from human biodynami analysis requirements, with the basi appliation framed by the hand-arm dynami behaviour simulation. REFERENCES 1. ABDULLAH, H.A., TARRY, C., DATTA, R., MITTAL, G.S. and ABDERRAHIM, M., Dynami biomehanial model for assessing and monitoring robot-assisted upper-limb therapy, J. of Rehabilitation Researh & Development, 44, 1, pp , ALDIEN, Y., MARCOTTE, P., RAKHEJA, S. and BOILEAU, P.É., Mehanial Impedane and Absorbed Power of Hand- Arm under xh Axis Vibration and Role of Hand Fores and Posture, Industrial Health, 43, pp , BRATOSIN, D., BĂLAN, F.S., CIOFLAN, C.O., Multiple resonane of the site osillating systems, Proeedings of the Romanian Aademy, Series A, 11, 3, pp , DRĂGĂNESCU, M., KAFATOS, M., Categories and Funtors for the Strutural Phenomenologial Modeling, Proeedings of the Romanian Aademy, Series A, 1, 2, pp , HILL, A.V., The heat of shortening and the dynami onstants of musle, Pro. R. So. Lond. B Biol. Si., 126, pp , KNUDSON, D., Fundamentals of Biomehanis, Seond Edition, Springer Siene+Business Media, LLC, KYRIACOS A.A. and ROMAN, M.N., Introdution to Continuum Biomehanis, Synthesis Letures on Biomedial Engineering, 19, Series Editor: John D. Enderle, Morgan & Claypool Publishers, LUNDSTRÖM R. and HOLMLUND P., Absorption of energy during whole-body vibration exposure, Journal of Sound and Vibration, 215, pp , MANSFIELD, N.J., Human response to vibration, CRC Press, PICU, A., NASTAC, S., Advaned Simulations of Human Protetive Devies Against Tehnologial Vibrations, Proeedings of the ACOUSTICS High Tatras th International Aoustial Conferene-EAA Symposium, SIRETEANU, T., GUGLIELMINO, E., STAMMERS, C.W., GHITA, G. and GIUCLEA, M., Semi-ative Suspension Control: Improved Vehile Ride and Road Friendliness, Springer, 06-03, SOUTH, T., Managing Noise and Vibration at Work. A pratial guide to assessment, measurement and ontrol, Elsevier Butterworth-Heinemann, Linare House, Jordan Hill, Oxford, Reeived November 25, 2010