The closed-loop vehicle response in stopping a train

Chapter 8, Problem 53

(choose chapter or problem)

The closed-loop vehicle response in stopping a train depends on the train's dynamics and the driver, who is an integral part of the feedback loop. In Figure P8.3, let the input be R(s) = \(v_r\) the reference velocity, and the output C(s) = v, the actual vehicle velocity. (Yamazaki, 2008) shows that such dynamics can be modeled by G(s) = \(G_d(s) G_t(s)\) where

\(G_d(s)\) = \(h\left(1+\frac{K}{s}\right) \frac{s-\frac{L}{2}}{s+\frac{L}{2}}\)

represents the driver dynamics with h, K, and L parameters particular to each individual driver. We assume here that h = 0.003 and L = 1. The train dynamics are given by

\(G_t(s)\) = \(\frac{k_b f K_p}{M\left(1+k_e\right) s(\tau s+1)}\)

where M = 8000 kg, the vehicle mass; \(k_e\) = 0.1 the inertial coefficient; \(k_b\) = 142.5, the brake gain; \(K_p\) = 47.5, the pressure gain; \(\tau\) = 1.2 sec, a time constant; and f = 0.24, the normal friction coefficient.

a. Make a root locus plot of the system as a function of the driver parameter K.

b. Discuss why this model may not be an accurate description of a real driver-train situation.