Damping greater than critical

There are three cases. In the first case, the quantity inside the square root in Eq. (8.41a) is positive, so that A.i and X2 are both real. This corresponds to a relatively large value of the friction constant and the case is called greater than critical damping. In this case, the mass at the end of the spring does not oscillate, but returns smoothly to its equilibrium position of z = 0 if disturbed from this position. 

Figure 8.2 shows the position of a greater than critically damped oscillator as a function of time for a particular set of initial conditions. 

From the fact that k, and m are all positive, show that >.1 and A2 are both negative in the case of greater than critical damping, and from this fact, show that... 

Figure 12.2 shows the position of a greater than critically damped oscillator as a function of time for the case that Cl = -C2 so that z = 0 at t = 0. Since the two terms decay at different rates, the oscillator moves away from z = 0 and then returns smoothly toward z = 0. 

For Kc between zero and 4, the two roots are real and lie on the negative real axis. Tbe dosedloop system is critically damped (the dosedloop dampring coeffident is 1) at Xc = I since the roots are equal. For values of gain greater than the roots will be complex. 

The above-mentioned phase shift for the angle of incidence greater than the critical angle implies the existence of a resultant field in the first medium at the interface. In order that the boundary conditions remain satisfied, there must be a resultant disturbance in the second phase. It can readily be shown that this disturbance is of the nature of an exponentially damped wave that penetrates into the second medium. This wave is termed an evanescent wave. 

Here the species Y is taken to be CO and the new quadratic termination step is reaction (cii). The system is reduced to a binary one in [O] and [CO ] by introducing the steady state relations for [H] and [OH] only. Analysis along the lines indicated above then predicts the three types of behaviour (i) no reaction (or very slow reaction controlled by initiation) when 0 < 0 (ii) damped oscillation or a sustained glow when 0 > 0 but less than some critical value and (iii) explosive behaviour when 0 is greater than the critical value. The distinctive difference from the hydrogen oxidation system, where there is a sharp transition from slow reaction to explosion at 0 = 0, is that now there is a more gradual transition within the region 0 < 0 < fep. This is in accord with the experimental observations [511]. 

Gray associates the branching agent X and intermediate Y with the spedes O and CiO respectively. He concludes that this system will show three distinct types of behaviour no reaction if the branching factor (=kb — ktt) is negative, damped oscillations when is greater than 0 but less than a critical value, mid explosion when is greater than a critical value. The scheme cannot exhibit limit-Qrcle behaviour characteristic of successive undamped oscillations. 

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