Viscous damping force

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

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

For small ion velocities the Doppler cooling force in Equations 10.11 and 10.12 can be approximated by a viscous damping force that is proportional to velocity, Pooppd =. where Y is a constant that depends on the laser wavelength, inten-... 

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. 

Lagrange s equation can be modified to accommodate the consideration of the viscous damping forces within the dynamic system. Subsequently, Eq. (7.2) can be rewritten as... 

Equation (5.30) is analogous to a Fokkcr-Planck equation with a harmonic-velocity-dependent external field instead of a viscous damping force, and it is therefore not surprising that it can be solved exactly in closed form. Comparing either (5.29) with (4.7b) or (5.21) with (4.7c), we see that (5.2a) and (5.21) resemble the displaced harmonic oscillator of quantum statistical mechanics. 

The simplest linear viscous model is Newton s model. This is shown by a piston-dashpot element [10]. The dashpot is an energy dissipation element, and it represents a viscous damping force. It relates the translational and rotational velocity of a fluid (oil) between two points, and an applied load, by using a damping constant. 

As with the Drude model, the Lorentzian dielectric function is derived by assuming that motion of the electrons is subject to a viscous damping force. However, in this case, the electrons are bound by a harmonic potential to a site in the solid, so that each atom can be considered as an electric dipole. In this manner, the dielectric response function is given by... 

V is the purely repulsive elastic energy (Equation 3.3, here with a = 2) between overlapping disks i and j. The viscous damping force is Pf = -bvi, with a damping coefficient hoslyfrne = 2 chosen to be in the overdamped limit. The force on... 

E Young s modulus of particles i used in the DEM, Pa Ey mean Young s modulus of particles 1 andj. Pa Eyfi real value of Young s modulus of the materials. Pa ic,ij elastic force between particles 1 andj, N ii y viscous damping force between particles 1 andj, N fa,i fluid drag force of particle i, N... 

Figure 5-17. Forced vibration with viscous damping.

ANTI-VISCOUS DAMPING COMPONENT Of OIL WEDGE FORCE... 

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. 

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 Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

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... 

Damping is divided into two categories, where X represents the coefficient of viscous damping between the sliding mass and the drive system and X2 represents the coefficient of viscous damping between the rubbing surfaces. difference between the forces of static and... 

Pressure-driven devices (Figure 4.1-6), such as pressure swirl atomizers, use aerodynamic drag at the gas-liquid interface to amplify natural disturbance to pinch droplets off liquid sheets and columns. Finer mists can be generated by adding a centrifugal force to the process that must overcome viscous damping and surface tension. 

If the kinetic term F(p) belongs to Case A, the potential has a minimum at the point p = 0 and a maximum at p = 1. The second term, -vp, in (4.2) represents a damping force, where v represents the viscosity. Then (4.2) describes the motion of a particle rolling down from the top of the potential at p = 1 to the bottom of the potential well, p = 0, in the presence of a viscous force. If v is small, i.e., the viscosity is small, the particle oscillates near the bottom of the well before it settles down at the minimum p = 0. If u increases, there exists a threshold value at which the oscillations cease. In other words, the particle rolls down monotonically from p = 1 to p = 0 in mechanics this is known as critical damping. If v increases even further, the particle continues to roll down monotonically and has less and less velocity at every point of its trajectory. Consequently, there exists a critical value of V, which we denote by v, such that for v > v, there will be monotonically decreasing solutions to (4.2). The front is said to be propagating into the... 

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