A mass-spring-dashpot system, with mass m, damping constant c, and spring c con- mx + cz...

   

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A mass-spring-dashpot system, with mass m, damping constant c, and spring c con- mx" + cz + kr = 0. Assume that stant k, has position function z(t) satisfying the ODE > 4mk, in other words the system is overdamped. a) If ri and r2 are the roots of the characteristic polynomial of the ODE, write the general solution for r(t). b) Use you general solution in part a) to show that the mass can go through the equilibrium position (r = 0) at most 1 time. How can you tell if the mass does or does not go through the equilibrium position (in terms of the constants in your general solution)? A mass-spring-dashpot system, with mass m, damping constant c, and spring c con- mx" + cz + kr = 0. Assume that stant k, has position function z(t) satisfying the ODE > 4mk, in other words the system is overdamped. a) If ri and r2 are the roots of the characteristic polynomial of the ODE, write the general solution for r(t). b) Use you general solution in part a) to show that the mass can go through the equilibrium position (r = 0) at most 1 time. How can you tell if the mass does or does not go through the equilibrium position (in terms of the constants in your general solution)?

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a The characteristic polynomial of the ODE is given by mr2 cr k 0 The roots of this characteristic p... View the full answer