SZENT ISTVÁN UNIVERSITY FACULTY OF MECHANICAL ENGINEERING NUMERICAL METHODS FOR DETERMINING LOCAL MOTIONS OF HUMAN KNEE JOINT

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1 SZENT ISTVÁN UNIVERSITY ACULTY O MECHANICAL ENINEERIN NUMERICAL METHODS OR DETERMININ LOCAL MOTIONS O HUMAN KNEE JOINT Author: usztáv ekete (PhD student) Supervisor: Prof. Dr. En. Béla M. Csizmadia 1. INTRODUCTION The Biomechanical Team of the Szent István University has been carryin on a comprehensive project of the human knee joint. The conception is oriented to the local motions and the phenomenon (and its influence) of wear. By establishin different experimental models, in order to verify and universalize the results, numerical models must be developed and applied. In these special fields, several numerical models have been introduced such as the classical J. Heeaard model [1]. Heeaard mostly investiated the stress and strain relation and the kinematics and dynamics of the patella, in the parameter of flexion. The above-mentioned model was investiated by the inite Element method wherein the non-linear elastic material laws were taken into account (iure 1.1). Besides the examination of the iure 1.1. iure 1.. local motions, another important question is the wear of the contact surfaces. Amon others, ranck Jourdan [] dealt with the phenomenon of wear with his two-dimensional patellofemoral joint model (iure 1..). The third important effect on the connectin surfaces is the actin forces. Koehle and Hull [3] used in their research a commonly applied software proram, called SIMM, which includes a complete musculoskeletal model of the lower limbs, allowin the reaction forces of the knee to be computed (iure 1.3.). The authors had only investiated the internal forces actin on the connectin points of the condyles durin the flexion of the knee. The oriinal SIMM D lower limb model was developed by Delp [4]. The knee model included a tibiofemoral and a patellotibial joint. No patellar liament was modeled, but an articulation was defined between the patella

2 and the tibia by the SIMM proram. Since the knee joint is defined this way, the force enerated by the quadriceps is transmitted to the tibia via a patellotibial joint reaction force and moment, instead of by the patellar liament. Accordin to this effect, the physioloic knee joint reaction forces substantially differ from the results iven by the SIMM. In this way, the authors revealed the weaknesses of the SIMM proram, and suested further modifications.. MAIN OBJECTIVES Since the physioloic knee joint reaction forces are only calculated in certain anles, and there are no allusions about their distribution, one of the main objectives is to introduce the distribution of the contact forces between the tibiofemoral, and the patellofemoral relation durin the flexion of the knee joint. The other objective is to investiate the local motions of the tibiofemoral connection, which are based on slidin and rollin features. iure METHODS AND MODELS 3.1 Problems. Creation of models. A numerical model was built in the MSC.ADAMS proram. This system is able to analyze dynamical problems between flexible or riid bodies. Since the motions of the 3D human knee are very difficult to model, a simpler,.5d model was created (D model with a finite third extent). The eometrical model was mapped from MRI pictures, and used basic CAD methods to create the computational model (iure 3.1.1). iure The properties of the model are the followins:.5d model, Continuum with mass, Riid body. In the case of any mechanical problem, one of the most important questions is the definition of the constraints. Since only ideal constraints are available in the ADAMS, the proper combination of the connectin constraints must be chosen in order to obtain a realistic motion.

3 The applied kinematical constraints of the model are the followins: All derees of freedom of the femur were taken (ixed Joint), The tibia is only allowed to carry out planar motions (Planar Joint), The patella is only allowed to carry out planar motions (Planar Joint), Patello-femoral and tibio-femoral contact relation (Contact). In contrast with the SIMM model, the connectin liaments such as the patellar liament, the quadriceps tendon, and the collateral medial liament were taken into account. The bones were attached to each other by sprins which denote the connectin liaments of the joint. These liaments and the bones were coupled toether accordin to functional anatomy. It is worth notin, that the patellar liament is unable to elonate, thus its sprin constant was set to infinite, while the collateral medial liament and the quadriceps tendon was set to 5 [N/mm]. The applied load: Normal ravity field, T 4 [N] force vector actin on the end of the tibia, Total applied weiht of the tibia. After settin the initial conditions (μ=.11 and μ=.8 [6]) and the solver alorithm (STI) the model was executed (iure ). Althouh the motion seemed to iure be proper, the iven results (s/n; the radient of the friction and normal force quotient) could not be expounded upon (iure ). iure iure Durin the simulation the tibia started chatterin as it was rollin and slidin alon the condyles of the femur. The chatterin caused the oscillation in the phenomena shown in iure This made it impossible to interpret. The problem was attributed to the default settins of the MSC.ADAMS. Since the proram was not oriinally adjusted to handle contact problems, the simulation called forth false results. The solution can be obtained by settin the appropriate parameters, but to achieve that, a basic model must be developed and tested, where the analytical solution can be derived from the classical mechanical laws.

4 3. Validation by an elementary model Let there be a cylinder with a radius denoted by R1, and a smaller cylinder denoted by R. The R cylinder starts rollin on the R1 cylinder from a φ position. The initial conditions are the followins: Initial velocity is zero. φ = 3 riction is defined between the two cylinders (μ =.3). As the R cylinder starts rollin upon the R1 cylinder (iure 3..1), the motion can be characterized as pure rollin, ce accordin to Coulomb s theorem the normal force is reater than the iure friction force. At φ position the friction force and the normal turns to be equal, that is the point where the motion reaches the threshold limit, and beyond this position the movin object will have a slidin feature as well. If the threshold position could be calculated with instantaneous values between φ and φ, then compared with a numerical model with same properties, the parameters of the contact module in the MSC.ADAMS could be set to obtain realistic results. As a first step to obtain φ, the principle of impulse for a riid body must be applied; m a I (1) m a K () S m R ( ) (3) e S m an N v( ) m (4) N R1 R Accordin to Coulomb s theorem: S N (5) v( ) m R ( ) m (6) R R If φ = O then the threshold position is reached! Since both ε and v are the function of φ, two more equations must be used to solve the system to φ. rom the principle of anular momentum for a riid body around an axis: P M P (7) J P R R R R (8) J P J P P is the connectin point of the two cylinders, and J P 3/ m R. 1

5 Now, the anular acceleration is obtained, by the use of the principle of work and enery for a riid body, the last demanded parameter, the velocity can be easily derived: E E W (9) 1 v J P R 1 R R1 R (1) R Substitutin (8), (9) into (6), and carryin out a little alebraic manipulation, the (6) equation can be put in more convenient form: (11) Now, the alebraic equation is ready to be solved, and the analytical solutions are: φ1 =.1866 [rad]. φ =.1689 [rad]. Obviously, the demanded solution is the φ1, which is cca As the threshold position is obtained, the radient of the s/n can be calculated. Aain a little manipulation on the (11) equation: S N By substitutin different φ into the (1) equation, the followin diaram (iure 3..) can be created. One can see, that at φ the value is.3, which expresses the equilibrium of the friction and normal forces. Beyond that point, a slidin feature will be added to the motion. As for the next step, a numerical model must be created in the ADAMS (iure 3..3) with the same properties iven in the analytical model. iure 3... The elementary model has the same kinematical constraints as the knee model. The followin parameters could be set as CONTACT settins in the IMPACT type: Penetration depth, orce exponent, Stiffness and Dampin. After several trials, the system responded sensitively to the radient iure of two parameters, which are the penetration depth and the force exponent. By settin the exponent to 1, and the penetration depth to 3, the proram provides the demanded solution (iure 3..4). (1)

6 iure RESULTS AND CONCLUSIONS Now, the parameters are properly set to execute the simulation of the knee model as well. irst, the radient of the normal and friction unit forces have been estimated in the function of the unit lenth of the connectin curvature (iure 4.1.). iure 4.1. iure 4.. Secondly, the radient of the quotient of friction and normal forces have been calculated in the function of the unit lenth of the connectin curvature (iure 4..). Accordin to the solution of the elementary model the followins can be noted from this diaram: Durin the initiative motion of the tibia alon the femur the condyles rolls on the surface of the femur. Then, until cca. 8% of the unit lenth of the curvature is slidin and up to cca. 83% there is rollin. Over 83% slidin is dominant. Obviously, these relations are hihly depended on the connections of the liaments, the shape of the tibia plateau and the condyles. To universalize the iven results, more proper calculations and investiations must be done both numerically and experimentally, but accordin to the introduced methods the required examinations can be carried out. REERENCES [1] J.Heeaard, P.. Leyvraz, A. Curnier, L. Rakotomanana, R. Huiskes: The biomechanics of the human patella durin passive knee flexion. Journal of Biomechanics 8, (1995), pp. [].Jourdan: Numerical wear modellin in dynamics and lare strains. Wear 61, (6), pp. [3] M.J.Koelhe, M.L.Hull: A method calculatin physioloically relevant joint reaction forces durin forward dynamic simulations of movement from an existin knee model. Journal of Biomechanics, 7, (Under press). [4] A.S.Arnold, S.L.Delp: Rotational moment arm of the medial hamstrins and adductors vary with femoral eometry and limbs position: Implications for the treatment of internally rotated ait. Journal of Biomechanics 34, (1), pp. [5] A. Wan: A unified theory of wear for ultra-hih molecular weiht polyethylene in multi-directional slidin, Wear 48, (1), pp.