Spring, force law

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

A reasonable approximation for the force between two adjacent particles is given by the so-called FENE (finitely extendable non-linear elastic) spring force law (Bird et al. 1987a)... 

Underhill PT and Doyle PS. (2006) Alternative spring force law for bead-spring chain models of the worm-hke chain. Journal of Rheology, 50, pp. 513-529. 

If the spring force law is taken to be Hookean, with = HQ, where H is the spring constant, then it is possible to eliminate (QQ between equations (59) and (62) to obtain the complete... 

It is worth recalling that any of the molecular force laws given by Eqs. (13-16) are derived within the framework of the freely-jointed model which considers the polymer chain as completely limp except for the spring force which resists stretching thus f(r) is purely entropic in nature and comes from the flexibility of the joints which permits the existence of a large number of conformations. With rodlike polymers, the statistical number of conformations is reduced to one and f(r) actually vanishes when the chain is in a fully extended state. 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

In equilibrium, the spring force -kz must be equal to the acceleration force (Newton s second law), i.e.,... 

H, L) Fraenkel (29) dumbbells The beads are joined by a Hookean spring which has a length L if there are no forces on the beads the force law is then F(c) — H(R — L) (R/R). When L=0, one obtains Hookean dumbbells when H approaches oo and R approaches L, rigid dumbbells are obtained. 

The unrealistic behavior of the UCM results from the fact that the Hookean spring allows extensions to go to infinity. A way of improvement is to use a more realistic force law in the model. Warner (1972) replaced the Hookean spring constant Hq by... 

The formulation of Hooke s law rests on the assumption of infinitesimally small deformations. Its apphcation to the simple model of a mass connected with a spring results in a hnear force law and to the well known harmonic oscillation. Investigating even with very modest means the behavior of a real system of this sort shows that the limits of accuracy of this simple description are quite narrow indeed. A more general and accurate description will have to be a nonlinear one. This, in fact mrns out to be tme for all material properties, e.g. dielectric properties and the simple relation (4.2) is valid only for small fields and is an approximation in the same way as Hooke s law (3.51). If we are looking close enough we find that all phenomena aetually are nonlinear, which means that the response of even simple systems to an external influence cannot be precisely described by a direct proportionaUty. 

Notice how this entropic spring force is linearly related to the extension the extended polymer behaves like a classical Hookean spring. The formula can be easily seen as a form of Hooke s law, yielding a spring constant k of... 

The Brownian motion term involves the distribution function 4 (R,r). Whereas the flow field tends to orient the particles, the Brownian forces tend to randomize the orientations. The term in R may be omitted because accelerations are usually quite small. The quantity caimot be specified until the force law for the connector is known. For our case with Hookean springs, P = HR. The notation 9/3R means the vector having components b/dx, 9/3x2, 9/9x3), that is, the gradient operator in R space (the configuration space of the molecule). 

Equation 12.6 resembles the force law of a spring with a spring constant k = 47ta set by the interfacial tension. Therefore, the contact of Figure 12.1, which involves two bubbles, acts like two such springs connected in series. Writing the total overlap 8 = 8q + 8i as the sum of the overlaps in each bubble, the effective force law of the contact can be written as... 

Equation 12.7 resembles the force law of a simple linear spring. The result is only approximate due to the truncated sphere assumption. More accurate calculations that do not presume the deformed shape a priori are available in the literature [16,19]. These show that Equation 12.8 overestimates the magnitude of fceff and misses logarithmic corrections that cause the stiffness to vanish as f 0. Nevertheless, Equation 12.8 establishes useful intuition. 

Consider a spring with a force constant k such that one end of the spring is attached to an immovable object such as a wall and the other is attached to a mass, m (see Figure 1-1). Hamiltonian mechanics will be used hence, the first step is to determine the Hamiltonian for the problem. The mass is confined to the x-axis and will have both kinetic and potential energy. The potential energy is the square of the distance the spring is displaced from its equilibrium position, xo, times one-half of the spring force constant, k (Hooke s Law). 

When the chain is stretched oitiy a httle, this force law agrees with the Gaussian model of Problem 7.A.5. As the spring is stretched more, it becomes stifFer so that the chain can never be extended beyond its contour leugth. 

If the spring follows Hooke s law, the force it exerts on the mass is directly proportional and opposite to the excursion of the particle away from its equilibrium point Xe- The particle of mass m is accelerated by the force F = —kx of the spring. By Newton s second law, F = ma, where a is the acceleration of the mass... 

Three 10,0-g masses are connected by springs to fixed points as harmonic oscillators showui in Fig, 3-12, The Hooke s law force constants of the springs ai e 2k. k, and k as showui, where k = 2.00 N m, What are the pei iods and frequencies of oscillation in hertz and radians per second in each of the three cases a, b, and e ... 

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