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Unformatted text preview: Page 1 of 2 Dimensionless Time & Step Responses of 2ndorder G(s) CHE 361
On pages 82 and 83 of SEMD3 are Figs5.8 & 5.9 = two plots of normalized
y/(KM) step responses of simple secondorder systems with various damping
⎛t⎞
coefficients, where the plots are made with dimensionless time ⎜ ⎟ on the xaxis.
⎝τ ⎠ Here are two quantitative examples. In the first case, different taus are used for the
zetas shown in Fig. 5.9. In the second, τ is always 6 and since the simulation ends
at t = 60, this is equivalent to dimensionless time up to t / τ = 60/6 = 10. The plot
on the right below is thus equivalent to the first half of Fig. 5.9.
Fig. 5.9 BUT Different Taus Fig. 5.9, tau = 6 so t/tau up to 10 not 20 in SEMD3 1.2 1.2 1 1 0.8
Outputs, y' Outputs, y' 0.8 0.6
zeta = 3, tau = 2
zeta = 2, tau = 3
zeta = 1.5, tau = 4
zeta = 1, tau = 6 0.4 zeta = 3, tau = 6
zeta = 2, tau = 6
zeta = 1.5, tau = 6
zeta = 1, tau = 6 0.4 0.2 0 0.6 0.2 0 10 20 30
Time , [min] 40 50 60 0 0 10 20 30
Time , [min] 40 50 These plots should make it clear that a critically damped process is not always
“faster” than an overdamped process – the speed depends on the τ value.
For the same tau, a critically damped system does respond fastest without
overshooting the final value. 60 Page 2 of 2 For constant zeta, the smallest tau is the fastest response. Note the same y value is
reached by the fastest reponse at t = 10, by the intermediate response at t = 20
(twice as long) and by the slowest response at t = 40 (four times as long as the
fastest response). The time ratios are the same as the ratios of the time constants.
Thus if dimensionless time were used for the xaxis, these three curves would be
identical.
Fig. 5.9, All zetas = 1, tau = 3, 6, 12
1.2 1 Outputs, y' 0.8 0.6 zeta = 1, tau = 3
zeta = 1, tau = 6
zeta = 1, tau = 12 0.4 0.2 0 Step Input 0 10 20 30
Time , [min] 60 zeta = 1, tau = 3
zeta = 1, tau = 6
zeta = 1, tau = 12 Zeta1_Tau3 1 50 Fig5_9_Zeta_1.mdl
Same Zeta for Different Taus 1
2+6s+1
9s 36s2+12s+1 40 M ux
PC_graph Zeta1_Tau6 1
2+24s+1
144s
Zeta1_Tau12
M ux ...
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 Winter '08
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