Overdamped

Figure Al.6.25. Modulus squared of tire rephasing, (a), and non-rephasing, R., (b), response fiinetions versus final time ifor a near-eritieally overdamped Brownian oseillator model M(i). The time delay between the seeond and third pulse, T, is varied as follows (a) from top to bottom, J= 0, 20, 40, 60, 80, 100,...

Stiff Spring For a stiff spring, satisfying K (fiUjdx, under the overdamped condition assumed in (3) the average force measured by the spring can be expressed as... 

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

The damping strength of the object is such that it may absorb most of its restoritig force before it reaches its original position. There are no oscillations. For overdamped systems, rj > I. 

This result comes from the idea of a variational rate theory for a diffusive dynamics. If the dynamics of the reactive system is overdamped and the effective friction is spatially isotropic, the time required to pass from the reactant to the product state is expected to be proportional to the integral over the path of the inverse Boltzmann probability. 

The diffusion constant should be small enough to damp out inertial motion. In the presence of a force the diffusion is biased in the direction of the force. When the friction constant is very high, the diffusion constant is very small and the force bias is attenuated— the motion of the system is strongly overdamped. The distance that a particle moves in a short time 8t is proportional to... 

There are different conditions of damping critical, overdamping, and under-damping. Critical damping occurs when 11, = Cl). Over-damping occurs when ji, > o). Underdamping occurs when ji, < ai. 

Number and weight average molecular weight transients are siammarized in Figure 7. The more viscous conditions of Run 4 resulted in an overdamped response whereas, the less viscous conditions of Run 5 resulted in overshoot. The simulation was more damped and delayed than the experimental response. 

Figure 3. Phase portrait of the noiseless dynamics (43) corresponding to the linear Langevin equation (15) (a) in the unstable reactive degree of freedom, (b) in a stable oscillating bath mode, and (c) in an overdamped bath mode. (From Ref. 37.)...

Two distinct real poles. The case is named overdamped. Here, we can factor the polynomial in terms of two time constants xi and %2 -... 

This form is unnecessarily complicated. When we have an overdamped response, we typically use the simple exponential form with the exp(-t/xi) and exp(-t/t2) terms. (You ll get to try this in the Review Problems.)... 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

Based onEq. (3-51), the time response y(t) should be strictly overdamped. However, this is not necessarily the case if the zero is positive (or xz < 0). We can show with algebra how various ranges of K and X may lead to different zeros (—l/xz) and time responses. However, we will not do that. (We ll use MATLAB to take a closer look in the Review Problems, though.) The key, once again, is to appreciate the principle of superposition with linear models. Thus we should get a rough idea of the time response simply based on the form in (3-50). 

With respect to the overdamped solution of a second order equation in (3-21), derive the step response y(t) in terms of the more familiar exp(-t/xi) and exp(-t/X2). This is much easier than... 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

Example 6.2 Derive the controller function for a system with a second order overdamped process and system response as dictated by Eq. (6-22). 

The ITAE, with about 14% overshoot, is more conservative. Ciancone and Marlin tuning relations are ultra conservative the system is slow and overdamped. 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

With only open-loop poles, examples (a) to (c) can only represent systems with a proportional controller. In case (a), the system contains a first orders process, and in (b) and (c) are overdamped and critically damped second order processes. 

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