[from D. L. Bartel, A. I. Krauter, ASME Paper 70-WA/DE-5]. A simplified model of two railroad cars engaging (hitting) with each other is shown in Fig. P5.39a. Spring and damper constants kP, MP, and...

[from D. L. Bartel, A. I. Krauter, ASME Paper 70-WA/DE-5]. A simplified model of two railroad cars engaging (hitting) with each other is shown in Fig. P5.39a. Spring and damper constants kP, MP, and Ma are given and it is required to choose the absorber parameters ka and ca so that the initial energy is dissipated as rapidly as possible. Data: Wa = WP = 2000 lb, kP = 24,000 lb/ft, velocity Va = –2mph = –2.933 ft/s, VP = 0, XP (0) = Xa (0) = 0. Optimize for ε = 0.01, ε = 0.02. Hint: Let denote E(t) = fraction of energy remaining in the system at time t; note that E ≤ 1. Our goal is to reduce the value of E to, say 0.05 (5 percent), in minimum time T. Thus, the optimization problem is

Note that E is a monotonically decreasing function of time (Fig. P5.39b). All springs and dampers are linear and there are no external forces on the system. The equations of motion, which can be solved analytically, are: