Damping term

This is the same equation of motion that is satisfied by the original coordinate qa(t), except that the stochastic driving term is absent. The relative dynamics is therefore deterministic. We have chosen the notation accordingly and left out the index a in the definition (41) of Aq (although, of course, we cannot expect the relative dynamics to remain noiseless in the full nonlinear system). Although noiseless, the relative dynamics is still dissipative because Eq. (43) retains the damping term. 

Here, two distinct situations arise depending on the relative magnitudes of the two terms in parentheses, if x/m > h2/4m2, Eq. (42) is of the same form as Eq. (29), whose solutions can be written as C cos a) t + Dsin co t, with oj = x/m - h2j4m2. Note that the presence of the damping term h/m modifies the natural (angular) frequency of the system. Then,... 

Tan landa, a damping term, is a measure of the ratio of energy dissipated as heat to the maximum energy stored in the material during one cycle of oscillation. For small to medium amounts of damping. G is the same as the shear modulus measured by other methods at comparable time scales. The loss modulus G" is directly proportional to the heat H dissipated per... 

Damping is often expressed in terms of quantities conveniently obtained with the type of instrument used. Since there are so many kinds of instruments, there are many damping terms in common use, such as the logarithmic decrement A, the half-width of a resonance peak, the half-power width of a resonance peak, the Q factor, specific damping capacity i <, the resilience R, and decibels of damping dB. 

The strain-stress relationship ay = Cykfiki is valid when strain is instantaneously provoked by stress, i.e., no dephasing process occurs. In fact, it was observed that the magnitude of acoustic waves decreased as a function of the travelling distance. Introducing a damping term proportional to the time course of strain variations can solve this discrepancy ... 

Once again, it should be noted that the polarizabihty is a complex nnmber because of the damping term. 

The damping term in a metal is due to the scattering suffered by the free electrons with atoms and electrons in the solid, which produces the electrical resistivity. 

It must be pointed out that the definition of the damping term depends on the method used. It is, therefore, necessary to express damping in terms of a complex modulus in order to be able to make comparisons. Thus ... 

The imaginary part of the modulus, also called the loss modulus, is a damping term which determines the dissipation of energy as heat upon deformation. G"/G is called the dissipation factor and is proportional to the ratio of energy dissipated per cycle to the maximum potential energy stored during a cycle. 

We may now express the damping term of the torsion pendulum experiment in terms of the dissipation factor by the simple equation (an approximation which holds for most cases) ... 

Because the central-differencing on the continuity equation is only neutrally stable, it is beneficial to introduce an artificial damping term to assist numerical stability [199]. Following common practice in solving Euler equations, a first-order damping term may be... 

In section 2 it was mentioned that a damping term in the Schrodinger equation violates the conservation of probability. It is possible, however, to compensate this loss by adding artificially a suitable fluctuating term. The result will be an equation for if/ resembling the classical Langevin equation. 

The terms on the left are the electric force and the restoring force that is linear in the displacement. A damping term with a proportionally constant r is added to the right hand side to account for dissipation of energy during the polarization response. The solution to this equation leads to an expression for the displacement, r, as... 

We recognize as we could have guessed that this is a wave equation with a source term and a damping term. With the help of Eqns (7.159) and then (7.158), it is possible to get rid of the gravitational perturbations in the right handside of the above equations and obtain a closed equation in term of dtp7. After heavy manipulations, one gets10... 

---