Negative damping

This expression shows diat if die detuning Acuj is negative (i.e. red detuned from resonance), dieii die cooling force will oppose die motion and be proportional to die atomic velocity. The one-diniensional motion of die atom, subject to an opposing force proportional to its velocity, is described by a damped haniionic oscillator. The Doppler damping or friction coefficient is die proportionality factor. 

When the damping eoeffieient C of a seeond-order system has its eritieal value Q, the system, when disturbed, will reaeh its steady-state value in the minimum time without overshoot. As indieated in Table 3.4, this is when the roots of the Charaeteristie Equation have equal negative real roots. 

Equation (3.79) shows that the third-order transient response eontains both first-order and seeond-order elements whose time eonstants and equivalent time eonstants are 2 seeonds, i.e. a transient period of about 8 seeonds. The seeond-order element has a predominate negative sine term, and a damped natural frequeney of 4.97 rad/s. The time response is shown in Figure 3.23. 

For < 0 (unstable system). If the damping coefficient is negative, the exponential term increases without bound as time becomes large. Thus the system is unstable. 

If all tbe roots lie on the negative real axis, we know the system is overdamped or critically damped (all real roots). 

Thus the location of a complex root can be converted directly to a damping coefficient and a time constant. The damping coefficient is equal to the cosine of the angle between the negative real axis and a radial line from the origin to the root. The time constant is equal to the reciprocal of the radial distance from the origin to the root. 

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. 

A gain of 17 gives a closedloop damping coefficient of 0.316 and a dominant second-order closedloop time constant of 0.85 minutes. The third root is real and lies far out on the negative real axis at —2.3. Thus the largest first-order time constant is 0.43 minutes. 

Notice the very significant result that the damping coefficient is less than one on the negative real axis. This means that in sampled-data systems a real root can give underdamped response. This can never happen in a continuous system the roots must be complex to give underdamped response. 

So positioning the closedloop root on the negative real axis at —0.372 will give a closedloop system with a damping coefficient of 0.3. Solving for the required gain gives... 

Further extensions of the model are required to address the dynamical consequences of these additional regulatory loops and of the indirect nature of the negative feedback on gene expression. Such extended models have been proposed for Drosophila [112, 113] and mammals [113]. The model for the circadian clock mechanism in mammals is schematized in Fig. 3C. The presence of additional mRNA and protein species, as well as of multiple complexes formed between the various clock proteins, complicates the model, which is now governed by a system of 16 or 19 kinetic equations. Sustained or damped oscillations can occur in this model for parameter values corresponding to continuous darkness. As observed in the experiments on the mammalian clock. Email mRNA oscillates in opposite phase with respect to Per and Cry mRNAs [97]. The model displays the property of entrainment by the ED cycle... 

Oscillations of the segmentation clock with a period of 2 h have also been observed in fibroblast cell cultures following serum shock. There also, oscillations in the expression of the gene Hesl related to the Notch pathway have been attributed to negative feedback on transcription [171]. The periodic operation of the segmentation clock was recently demonstrated in cells of the PSM, where intercellular coupling is needed to prevent damping of the oscillations [172]. 

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