- 4 Conservative Mechanical System 140 Points The Illustration In Fig 2 Depicts A Con Servative System Which Is Com 1 (184.88 KiB) Viewed 45 times
4. Conservative Mechanical System (140 points). The illustration in Fig. 2 depicts a con- servative system, which is composed of two bodies (with masses mi and m2) and two springs (with constants ki and k2). As customary, the variables 11 and 12 indicate the positions of the two masses with respect to the points at which the springs are relaxed. Note that, in this case, friction is considered to be negligible and, therefore, the system is conservative because energy is not dissipated. (a) Write the mathematical expression for the total kinetic energy of the system, T. 4 (b) Write the mathematical expression for the total potential energy of the system, V. Hint: In this case, the potential function of the system, V, has a gravitational component and an elastic component. (c) Recalling that for a conservative system T +V = constant, find a expression of the form d (T+V) = 0. dt (d) Using your answer for (c), find the two differential equations that describe the dynamics of the system. (e) Using Newton's second law, find the two differential equations that describe the dynamics of the system. (f) Are your answers for (d) and (e) consistent with each other? Explain thoroughly. (g) Using your answer for (d), find a four-state state-space representation of the system, where the output is y = 22 and u is the input. Hint: Note that, in this case, any state-space representation should have the form Ż = Az + Bu, y = Cz + Du. Datum No Friction mi k2 12 WA m2 Figure 2: Conservative Mechanical System.