4 2.7 INCLUDING FRICTIONAL FORCES

“ Sliding is defined as the relative linear velocity between two surfaces at their

point of contact Rolling is defined as the relative angular velocity between two

bodies about an axis lying in the tangent plane of the point of contact Spin is

defined as the relative angular velocity about the common normal of the bodies in contact.” [Johnson, 1985, p.3,4]. The friction effects for these three cases had to be determined and included in the rigid body analysis.

Before limiting friction occurs, the opposing frictional forces are greater than the applied forces. In order to be able to use the equilibrium equations during the period when the femoral component was stationary on the tibial insert, the value of the static friction coefficient and the friction directions were iterated until the equilibrium equations were satisfied. This gave the stationary position a slight advantage over the other positions. However, the acceptable residual limit for the stationary position was set to a lower value compared to the other options of motion to make sure that it did not have an unfair advantage.

The same values for static and dynamic coefficients of friction were set, after observing results from laboratory tests (see next chapter), though there was the option to have different values. For sliding between the present and new

SL = |i.RL (12)

S M = n .R M (1 3 )

= coefficient of friction, taken to be 0.07 under all conditions.

For zero relative sliding between the femoral and tibial surfaces, which could occur if the shear force did not exceed the limiting friction in a time interval with zero flexion change, or if pure rolling occurred:

p . R L > S L > - p . R L (1 4 )

p . R M > S M > - p . R M (1 5 )

The speeds of the cycles of flexion, compression load, anterior-posterior force and torque input to the rigid body analysis were irrelevant, however it was important to take into account the relative surface velocities of the femoral and tibial inserts as these would affect the frictional forces. Consider the following cases of rolling and sliding in a simplified two-dimensional model, points 1,2,3,4 on the femoral component make contact with points a,b,c,d on the tibial insert : (a) A large anterior-posterior force and zero flexion were applied causing one point of the femoral component to slide across the tibial insert (Fig.3a).

(b) Zero anterior-posterior force and a large flexion were applied causing the femoral component to roll on one point of the tibial insert (Fig.3b).

(c) A moderate anterior-posterior force and a small flexion were applied causing the femoral component to slide across the tibial insert while rolling slightly (Fig.3c).

(d) A moderate anterior-posterior force and a large flexion were applied causing the femoral component to roll while sliding across the tibial insert slightly (Fig.3d). These cases show that when motions caused by flexion and anterior-posterior forces oppose each other, the frictional forces produced change direction depending on their surface velocities.

Friction directions were calculated by comparing the relative motion of the femoral and tibial contact points between times t+dt and time t on the timescale of

positive anterior force makes the femoral component slide from a to b, so a frictional force from b to a is exerted by the tibial component on the femoral component to oppose its motion. However, if a positive flexion is also applied, a frictional force in the direction of 1 to 2 is superimposed. This method of determining the friction direction, applied in three-dimensions, could cope with all cases of rolling and sliding. Spin occurs when one condyle sticks and the other condyle moves. For this case, the friction directions were iterated in the plane of contact and the friction coefficient value was also iterated for the condyle about which spin occurred, this being a similar situation to when the component was stationary.

As the tibial insert remained stationary, the direction of the frictional force exerted by the tibial component on the femoral component (vector b to a) was obtained by comparing the contact points of the current force increment (at time t+dt) with the contact points of the previous one (at time t). However, it was not as

simple to find (vector 2 to 1) the direction of the friction force exerted by the

femoral component on the tibial component because the former was mobile. Therefore once the femoral contact points were determined, they were temporarily translated and rotated so that they were in the original (reference) orientation of the femoral component at the beginning of the rigid body analysis. When the old and new contact points were all in the reference orientation, the vectors between them could be calculated and input to the friction direction equation.

If the friction direction was zero because the vector from b to a was equal and opposite to the vector from 2 to 1, pure rolling occurred (Fig.4b):“ ldeally rolling contact should offer no resistance to motion, but in reality energy is dissipated in various ways which give rise to ‘rolling friction’ “[Johnson, 1985, p306]. This was found to be true for knee replacements, where tractive forces produced during rolling proved to be more detrimental compared to friction forces produced during sliding [Wimmer & Andriacchi, 1995]. However, pure rolling did not occur in the model described in this chapter because the incremental nature of

the analysis meant that it was unlikely that the two vectors would ever be exactly equal.

A few analyses were carried out in order to èxamine the effects of including tractive forces. Tolerances of relative motion were set within which pure rolling was allowed to occur, even though the relative displacements of the contact points on the femoral and tibial components were not equal and opposite. Within these tolerances the friction coefficent values were iterated up to the static coefficient of friction and friction directions were iterated in the plane of contact.