Critical damping coefficient

U3i — rotational angle velocity vector of the ith particle, rad/s Vo — 2y/mKn critical damping coefficient in vibration system of one... 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

At the origin (where z = 0) the damping coefficient is unity. Solving Eq. (19.18) for the value of controller gain that give this critically damped system... 

Thus the closedloop root is located at the origin. This corresponds to a critically damped closedloop system (C = 1). The specified response in the output was for no overshoot, so this damping coefficient is to be expiected. 

The effect of the value of the damping coefficient f on the response is shown in Fig. 7.28. For (< 1 the response is seen to be oscillatory or underdamped when ( >1 it is sluggish or overdamped and when (= 1 it is said to be critically damped, i.e. the final value is approached with the greatest speed without overshooting the Final value. When f = 0 there is no damping and the system output oscillates continuously with constant amplitude. 

Figure 3.6 c and d illustrate amplitude and phase responses of oscillators having different damping coefficients. The step response of a sensor is usually determined by the time constant as well as by the typical rise and response times of the system. Figure 3.6 b shows the response of a critical damped system to a steplike change in the input signal 0 The time constant r (as defined for an exponential response), the 10% to 90% rise time t(o.i/o.9) and the 95% response time t(0 95) are marked. 

For Kc between zero and the two roots are real and lie on the negative real axis. The closedloop system is critically damped (the closedloop damping coefficient is 1) at Kc = since the roots are equal. For values of gain greater than the roots will be complex. 

Since the stiffness and damping coefficients of tilting pad bearing are varying with the variation of exciting frequency, therefore, while the critical speed of instability... 

The value assumed for the equivalent viscous damping ratio is Co = c/Ccrit = 0.02, being Ccrit = 2 /to the viscous damping coefficient for a critically damped system. Even though quite small, this value of Co is reasonable for an undamaged structure within the linear-elastic range. 

A classical example, previously studied by several researchers (Hadi and Arfiadi 1998 Lee et al 2006 Mohebbi et al. 2013), was selected from the literature. The structure is a ten-story shear frame as illustrated in Fig. 1. In the top floor of this stmcture, a TMD is to be installed. The mass of the TMD is chosen a priori to be 3 % of the total mass of the stmcture. The parameters to be determined by the designer are mean values of the stiffness and damping coefficient (c ) of such a TMD. Two variants of the example are solved (a) only the seismic excitation is considered random (b) the seismic excitation is random, but uncertainties are also considered in the mass, stiffness, and damping coefficient of the stmcture and the TMD. These are then modeled as random variables whose characteristics are presented in Table 1. Table 1 also shows the perturbations considered in order to evaluate the robustness of the solutions, following Eq. 9. These perturbations are applied to the random variables of the stmcture. The critical combination of perturbations is given by a reduction of stiffness and damping and an increase of the mass (this explains the signs on the perturbations presented in Table 1). 

The response of the system will depend mainly on the damping coefficient f. When f < 1, the system is underdamped and has an oscillatory response. The smaller the value of f, the greater the overshoot. If f = 1, the system is termed critically damped and has no oscillation. A critically damped system provides the fastest approach to the final value without the overshoot of an underdamped system. Finally, if f > 1, the system is overdamped. An overdamped system is similar to a critically damped system, in that the response never overshoots the final value. However, the approach for an overdamped system is much slower and varies depending upon the value of f. These typical responses are illustrated in Figure 3.27. 

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