Nonlinear spring force

The nonlinearities discussed in Sections 10.1 and 10.1.1 are of a continuous (smooth) nature. That is, the nonlinear addition to spring force, or string tension, is a continuous function of displacement. A more extreme type of nonlinearity—discontinuous nonlinearity—tends to have a more profound effect on the system and spectrum. 

Additional interparticle forces exist in colloidal systems. They can be derived from a potential because they depend only on interparticle distance. Hence they act as springs and as such can cause pronounced elastic effects. However, the spring force depends on interparticle distance, and the springs even rupture when the particles move too far apart. This results in a highly nonlinear material response. During flow, the potential forces will affect the interparticle distances and consequently the frictional forces and the viscosity. We review each of these forces briefly in Sections 10.4.1-10.4.4. More thorough discussion can be found in Russel et al. (1989). 

Fig. 1 Nonlinear force-displacement characteristics of OKION optical mouse spring. The solid line represents the spring force over spring displacement when the switch is compressed. The second dash line is the spring force approximated by 23 order polynomial equation. The last dash line (the lowest) is the spring force over spring displacement when the switch is released...

The accessible motion is generally limited to one third of the inter-electrode initial gap. At this limit, the nonlinear electrostatic force increases more rapidly than a linear restoring force, applied for example by springs attached to the upper electrode. The electrostatic effect then becomes unstable and the mobile electrode drops toward the fixed electrode and sticks to it. 

To prove the influence of T on AU we return to our one dimensional model given by a Lennard-Jones potential. We consider particle B to be fastened between A and C by a nonlinear spring (Fig. 7). The force F is given by... 

For example, suppose a mass m is attached to a nonlinear spring whose restoring force is F(x), where x is the displacement from the origin. Furthermore, suppose that the mass is immersed in a vat of very viscous fluid, like honey or motor oil (Figure 2.6.2), so that it is subject to a damping force bx. Then Newton s law is... 

Contact mechanics is both an old and a modern field. Its classical domains of application are adhesion, friction, and fracture. Clearly, the relevance of the field for technical devices is enormous. Systematic strategies to control friction and adhesion between solid surfaces have been known since the stone age [1]. In modern times, the ground for systematic studies was laid in 1881 by Hertz in his seminal paper on the contact between soHd elastic bodies [2]. Hertz considers a sphere-plate contact. Solving the equations of continuum elasticity, he finds that the vertical force, F , is proportional to where S is the indentation. The sphere-plate contact forms a nonlinear spring with a differential spring constant k = dF/dS oc The nonhnearity occurs because there is a concentration of stress at the point of contact. Such stress concentrations - and the ensuing mechanical nonhnearities - are typical of contact mechanics. 

If the footing s safety factor for vertical loads is high, i.e., N N, there is very little hysteresis in cyclic loading and the cyclic M-0 relation is nonlinear-elastic, returning to about zero displacement at zero moment or force and dissipating very little energy. Then Eqs. (15.7) may be applied also in nonlinear dynamic analysis, with the twin springs taken as nonlinear elastic. 

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