Mass spring assumption

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

Consider a three-mass generalization of the system shown in fig. 5.3. Compute the vibrational frequencies and associated eigenvectors on the assumption that each mass has only one degree of freedom, namely, motions along the line joining the masses. Assume that the masses are all identical as are the spring constants. 

The basic assumption in MM methods is that a molecule can be considered as a collection of classical Newtonian masses held together by hypothetical springs holding the atoms at an equilibrium position in configuration space [10]. The generic form of an MM force field is given 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. 

There are two theoretical models to determine the relaxation time, -zp, and selfdiffusion time of the PE in the solution. One is the Rouse model, which is based on the assumption that the PEs are comprised of mass centered points connected with springs these points only interact with these springs. The other is the Zimm model which is an extension of the Rouse model and takes hydrodynamic effects into account [24, 30]. The Rouse model was used to establish the scaling theory of PEs in solution, which allows an estimation of the PE structure from the viscosity in the different solution regimes. 

The mass times acceleration term is omitted, an assumption that is common to virtually all kinetic theories for polymers. The four forces in equation (50) are respectively the hydrodynamic, Brownian, spring and external forces in this discussion we neglect the external forces entirely (see ref. 66, chapters 13 to 18, where these forces are included). The expressions for the three remaining forces are... 

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