Rayleigh dissipation function

We now need to add the dissipation term. A Rayleigh dissipation function will suffice for this purpose, since the hydrodynamic interactions in the.elastomer should be well screened. Let F be a matrix such that XF has elements of the form... 

This positively determined quadratic expression is referred to as the Rayleigh Onsager dissipative function, d S... 

Wherein the definition of the thermodynamic fluxes and forces of (A3,2,13) and (A3,2,14) have been used. Onsager defined [5] the analogue of the Rayleigh dissipation function by... 

Rayleigh s dissipation function is applied in order to properly handle the viscous component of the given dynamic system. In this case, it is assumed that the viscous nature of the dashpot will be linearly dependent upon the velocity of the mass. More specifically, the viscous damping force is considered to be proportional to the velocity of the mass and is given by the following relation. 

Equation (7.9) is expressed in terms of the generalized coordinate r, where c is defined as the coefficient of proportionality for viscous damping. Rayleigh s dissipation function is defined as... 

Further information concerning the definition and use of Rayleigh s dissipation function can be found in the references. 

The steps needed to determine the equations of motion for this system are essentially the steps for the previous system in Fig. 7.2. In this example, the effects of viscous damping are neglect, thereby omitting Rayleigh s dissipation function within the solution. This is a simple omission where removal of the dissipation function from Lagrange s equation will have an impact on the final solutions. 

Classical Viscous Damping The model of damping based on Rayleigh s dissipation function where the damping force depends on instantaneous velocities. 

A variational formulation of the dynamics of systems of chemical reactions is presented which utilizes a Rayleigh dissipation function to extend the Lagrangian approach to conservative systems to dissipative systems [l]. The formulation is valid arbitrarily far from equilibrium and is based on macroscopic, deterministic chemical kinetics. It thus stands in contrast to variational principles for dissipative systems which are based on studies of fluctuations and are restricted to the near equilibrium regime [z] or, far from equilibrium, to the vicinity of stable stationary states [3]. 

Rayleigh. The Rayleigh dissipation function represents only the effect of viscosity on the entropy production ... 

When kx < k the last term in the numerator and the denominator are real, and the reflectance function is real, so that there is no variation of phase with incident angle. When kx = k all terms except the first term in the numerator and the denominator vanish, and R(kx) = 1. When k square brackets depends only on kx and the properties of the solid. Provided that there is no dissipation in the solid, it vanishes when kx is equal to the Rayleigh wavenumber kR = co/vR, as can be seen by substituting X = (ks/kR)2, Y = (k /ks)2, and comparing with (6.55). 

---