14_StepResponses_zetas

14_StepResponses_zetas - Page 1 of 2 Dimensionless Time...

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Unformatted text preview: Page 1 of 2 Dimensionless Time & Step Responses of 2nd-order 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 second-order systems with various damping ⎛t⎞ coefficients, where the plots are made with dimensionless time ⎜ ⎟ on the x-axis. ⎝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 x-axis, 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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