Damping force

As shown in Figure 5-6, viscous damping force is proportional to velocity and is expressed by the following relationship ... 

The system is still comprised of the inertia force due to the mass and the spring force, but a new force is introduced. This force is referred to as the damping force and is proportional to the damping constant, or the coefficient of viscous damping, c. The damping force is also proportional to the velocity of the body and, as it is applied, it opposes the motion at each instant. 

In a damped forced vibration system such as the one shown in Figure 43.14, the motion of the mass M has two parts (1) the damped free vibration at the damped natural frequency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady state harmonic associated with the external force remains as long as the energy force is present. 

With damped forced vibration, the only difference in its equation and the equation for damped free vibration is that it is equal to Fo sin(ft)/) as shown below instead of being equal to zero. 

For damped forced vibrations, three different frequencies have to be distinguished the undamped natural frequency, = y KgJM the damped natural frequency, q = /KgJM — cgJ2M) and the frequency of maximum forced amplitude, sometimes referred to as the resonant frequency. 

The intensity of the energy dissipation can be estimated by the damping force acting on a particle i vibrating in a velocity Vi, which follows a linear relation [21]. 

Now suppose that the harmonic oscillator represented in Fig. 1 is immersed in a viscous medium. Equation (32) will then be modified to include a damping force which is usually assumed to be proportional to the velocity, -hx. Thus,... 

In the Langevin description, one assumes that the degrees of freedom within the system that are not explicitly considered in the simulation, exert, on average, a damping force that is linear in velocity y,-f, along with additional random forces Ti t). This leads to the following equation of motion for particle number i ... 

To simplify, we begin thinking of an ideal metal - that is, a metal withont damping -and later we shall discnss how damping forces can affect the optical properties of this ideal metal. Thns, setting >o = 0 and T = 0 in Equations (4.17) and (4.18), we obtain... 

When a conductor moves in a magnetic field, damping forces are generated by eddy currents induced in the conductor. The magnetic damper, with its reliability and thermal stability, has been utilized in various branches of engineering. A recent analysis of it is given by Nagaya and Kojima (1984). 

The first term is the reversible mechanical force, the second term is the damping. The damping force is that part of the irreversible term in (4.7) that survives in the limit 6 = 0. The other part is represented by the last term in... 

The examples we saw are for L C circuits supplied from a direct current source. What happens when an L C circuit is excited by an alternating current source Once again, oscillatory response will be present. The oscillatory waveform superimposes on the fundamental waveform until the damping forces sufficiently attenuate the oscillations. At this point, the system returns to normal operation. In a power system characterized by low resistance and high values of L and C, the effects would be more damaging than if the system were to have high resistance and low L and C because the natural frequencies are high when the values of L and C are low. The... 

Other methods for performing constant-temperature molecular dynamics calculations have been proposed recently. Evans (72) has introduced an external damping force in addition to the usual intermolecular force in order to keep the temperature constant in the simulation of a dissipative fluid flow. In another method, Haile and Gupta 13) have imposed the constraint of constant kinetic energy on the lagrangian equations of motion to perform calculations al constant temperature. 

The solution to the damped forced oscUlator differential equation proceeds as follows. Begin with (Braun, 1978) ... 

However, full analysis shows that Langevin s assumption that the damping force has no memory implies that < 4(0) A(t)y = 0 for / > 0. Thus if Langevin s equation appears valid within the limits of observation, rj is imobservably short. If, on the other hand, the particle of interest is... 

If we suppose that a suitable damping may be introduced without forcing strong correlation between the and co-ordinates, which can of course be done self-consistently but by a very particular constraint on the damping forces, we obtain... 

The force arising from the potential is F, while R is a gaussian random force. The net effect of the collisions , i.e. dynamical interactions between the particle and solvent molecules, is thus approximately accounted for by the frictional, or damping force, Fj. = —fiC,x, where is a friction constant related to the time correlation of the random force ... 

We shall first summarize the motion of the damped forced oscillator, which includes a force contribution driving the motion and supplying energy to the vibrating system. 

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... 

As before, you should imagine that the particle is heavily damped—its inertia is completely negligible compared to the damping force and the force due to the potential. For example, suppose that the particle has to slog through a thick layer of goo that covers the walls of the potential (Figure 2.7.1). 

Finally, suppose that the motion of the bead is opposed by a viscous damping force bx. 

The simplest model of charge transport in delocalized bands is the Drude model, which assumes the carriers are free to move under the inhuence of an applied electric held, but subject to collisional damping forces. Note that the scattering centers are not the nuclei of the background material, but rather phonons (lattice vibrahons) or impurities. A statistical equahon for estimahng the mean drift velocity of the carriers in the direction of the electric held may be written as... 

It is a matter of observation that the force which exerts the damping action is directed against that of the motion and it also increases as the velocity of the motion increases. The most plausible assumption we can make is that the damping force, at any instant, is directly proportional to the prevailing velocity, and has a negative value. To allow for this, equation (4) must have an additional negative term. We thus get the typical equation of the second order,... 

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