True Motion in Paradoxical Situations

In the next section, we use the limiting process approach [51] to determine the true motion of the system in the regions of the paradoxes. 

In Sect. A3 A A we studied an example where an approximate solution was obtained in the region of paradoxes by adding compliance to the two bodies in contact. In Sect. 8.6.1 below, we will present numerical results of a similar approach apphed to the lead screw and nut (i.e. using the 2-DOF model of Sect 5.5). But first, we will take a closer look at the behavior of the rigid body system under the cmiditimis of the paradoxes. The approach adopted here is based on the limiting process described in [51] where the law of motion of the rigid body system is taken as that of the system with compliant ccmtact when the contact stiffness tends to infinity. 

The equations of motion of lead screw and nut with complaint threads are given by (5.16) and (5.17). Instead of (5.24), the contact force is assumed here as (neglecting contact damping)  

Now consider the case where conditions given by (8.21) are satisfied. The differential equation given by (8.10) can be written as  

The fixed point of the first equation in (8.11) is a saddle at x = F equation has a center at x = written as 

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