Contact point jumps

We now point out some problems, especially when contact point jumps can occur. We demonstrate this type of problems by first considering a circle with radius d moving along a sinusoidal curve, see Fig. 5.16 ... 

If d > there exist points where the curvature of both curves coincide, see Fig. 5.17. In these points does npe not satisfy the conditions of the Implicit Function Theorem and Eq. (5.5.2d) cannot be solved for Pr = tji. If the circle would move beyond such a point both bodies would locally penetrate. There exists no physical solution beyond this point. The solution can be continued if one admits contact point jumps. 

The contact point jump which corresponds to this critical region cannot be determined from this picture and as pointed out before not from standard DAE solvers. 

If the integrator encounters an ill-conditioned iteration matrix a critical region with a potential contact point jump is assumed. The minima of the distance function between the two profiles are computed together with the corresponding distance vectors. Then a backward integration is performed with monitoring these distance vectors until one of those becomes zero. There, a contact point jump has to be performed. 

One uses a sort of preprocessing of the contact planes and establishes where contact point jumps may occur. 

The first variant has the disadvantage that the physically correct contact point jump may not be found. The second variant has the disadvantage that such a point may not exist and if it exists its computation can be a problem. The third variant can be very costly because the whole surface has to be analyzed, though the solution is only a curve on it. 

For a more detailed analysis of contact point jumps and their numerical treatment in the context of wheel/rail dynamics we refer to [AN96]. 

The high value of 234.4 N in the coursewise direction in the jersey structure was closely analysed and it was conclnded that it had a valid result. There was a cut-through even before the contact point at 234.4N, but the blade did not make contact with the force plate as the fabric was pulled by the blade and the accumulation of the loops made the blade to jump. 

Fig. 6.16 (A) Potential energy of an atom probe approaching a surface, as a function of separation, and the resulting normal force, the differential of the energy. Point A is the position of equilibrium contact. (B) Enlarged view of the force versus separation curve near the equilibrium contact point. Smaller separations give a repulsive force, and the probe is said to be in contact with the surface. The straight lines correspond to the stiffness of the cantilever, and greater slopes make the probe position unstable. (C) Cantilever deflection as a function of the distance of the cantilever support above the surface. The vertical lines show how the probe will be unstable and jump into and out of contact with the surface. A shffer cantilever would produce the smaller deflection shown as a dotted line, and suppress the instability.

It would appear that small local forces can be ensured by adjusting the feedback set point to correspond to zero net force or to some point in the attractive region with net negative force. However, operation in the attractive region is usually too unstable a situation. If there is a sudden dip in the surface, the cantilever deflection increases to the point where the tip jumps out of contact before the feedback loop can correct it, and that is the end of that scan. As was pointed out above, zero net force can be a combination of strong repulsive forces at the contact point and attractive forces from the surrounding region. 

The surfaces jump out of contact when the applied load is equal to the pull-off force, i.e. P — —P. The contact radius at the point of separation, a, is given by... 

FIG. 9. The tunnel resistance as a function of tip movement is shown in (a). Positive z displacement corresponds to a decreasing tunnel junction width, and the jump in conductance at is associated with point contact. The inset figure shows the degree of reproducibility associated with the experiment which employed an Ir tip and substrate. The gradient associated with the attractive force between the two electrodes was measured simultaneously as shown in (b). (From Ref. 66.)... 

The one exception in which phase contrast is not due to the dissipation arises when the tip jumps between attraction phases (>90°) and repulsion phases (<90°). Since sine is a symmetric function about 90°, the phase changes symmetric even if there are no losses in the tip-sample interaction. The relative contribution of the repulsive and attractive forces can be estimated experimentally from the frequency-sweep curves in Fig. lib by measuring the effective quality factor as Qe=co0/Ao)1/2, where Ago1/2 is the half-width of the amplitude curve. The relative contribution of the attractive forces was shown to increase with increasing the set-point ratio rsp=As/Af. Eventually, this may lead to the inversion of the phase contrast when the overall force becomes attractive [110,112]. The effect of the attractive forces becomes especially prominent for dull tips due to the larger contact area [147]. 

Figure 5.10d illustrates a case of contact mode. The contact position (z) is on the repulsive side of the short-range force curve. The arrow pointed to the left on Figure 5.10d represents the phenomenon of tip jump-to-contact with the sample caused by cantilever elastic bending when... 

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