Nonlinear spring

In Chapter 4, the response of these models to dynamic (i.e., sinusoidal) loads or strains is illustrated. In Chapter 5, the stress-strain response in constant rate experiments is described. Models with nonlinear springs and nonlinear dashpots (i.e., stress not proportional to strain or to strain rate)... 

Figure 13.9 Schematic showing compaction of previously wound layers when the k + 1 layer is wound. The fiber bed behaves as a nonlinear spring...

In the squeezed sponge model [11,13], compaction is not necessarily sequential and the applied pressure is shared by both the matrix and the fiber bed. The fiber bed behaves as a rapidly stiffening nonlinear spring. Pressure in the fiber bed is a function of fiber bed compaction. The rate of change of the fiber volume fraction is related to the pressure drop... 

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... 

When a nonlinear spring law is used in the dumbbell model, the Smoluchowski equation (3-28) is changed to... 

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. 

As a second example, we consider a simple model of an atom bonded to a pair of fixed neighbors by (nonlinear) springs, as depicted in Fig. 3.12. The coefficients k and k2 determine the potential energy function U of the system (a function of the position of the moving body q = x, y), which we assume to lie in a plane). The... 

MD simulations using TersofiF- Brenner potential to simulate covalent bonds while using Lennard-Jones to model interlayer interactions. Finite element analysis employing nonlinear spring element and equivalent beam element to model in-layer non-bonded interactions and covalent bond of two neighbor atoms, respectively... 

The nonlinear spring model described above has been computationally implemented in the computer code ANSRuop (www.ansruop.net) and used for the nonlinear static (pushover) or dynamic analyses in Sect. 15.4. 

Tandon GP, Weng GJ (1984) The effect of aspect ratio of inclusion on elastic properties of unidirectionally aligned composites. Polym Compos 5 327-333 Tanner RI (1960) Full-film lubrication theory for a Maxwell liquid. Int J Mech Sci 1 206-215 Tanner RI (1975) Stresses in dilute solutions of bead-nonlinear-spring macromolecules, ii. Unsteady flows and approximate constitutive relations, constitutive relations. J Rheol 19 37-65... 

Box 1 Figure Fermi, Pasta, and Ulam modeled a long chain of masses connected to one another by nonlinear springs. The two end particles, 1 and n, of the chain are assumed to be fixed. The particles can move back and forth according to Newton s law of motion f = mx (force equals mass times acceleration x) along the line that connects them. 

One of the most commonly used methods for analyzing laterally loaded piles is the p-y method, in which soil reactions to the lateral deflections of a pile are treated as localized nonlinear springs based on the Winkler s assumption. The pile is modeled as an elastic beam that is supported on a deformable subgrade. 

The phenomenological expression of may be given by the elastic energy of a nonlinear spring with the two independent constants ki and k2-... 

Figure 2.10 Model elements for viscoelastic simulations (from left linear spring element, linear dash pot element, nonlinear spring element, nonlinear dash pot element, sliding block (Saint-Venant body) with yield stress 9 (after Hennicke, 1978).

Nonlinear Mechanical Models It is possible to represent nonlinear behavior by introducing nonlinear spring and damper elements into the derivation of differential stress-strain relations. For example, for the four-parameter fluid shown in Fig. 10.5, the spring moduli, damper viscosities and relaxation times are functions of stress, i.e.,... 

Palmer [9] examined the function of ankle joint at different gait speeds in sagital plane and divided moment-angle curve into three sub-phases Controlled Plantarflexion (CP), Controlled Dorsiflexion (CD) and Powered Plantarflexion (PP). He concluded that ankle function can be mimicked by linear spring in CP and by nonlinear spring in CD. In PP a torque actuator should be added to the non-linear spring. 

---