A unitorin rod is anachcd Lo a mass (in1) which is slidine without friction as showw Use x to denote the horiiontal position of the mass relative to the uiìdetrined configuration, and O lo denote the angle that the pendulum makes with the vertical. When x0 the linear spring is unstretched. ihe rod has length I. and mass m:. The acceleration due to gravity is g and is acting downward. The applied external trce is harmonic with tixed frequency and magnitude Fc and i% applied in the x—direction.
(a) Write expressions l’or (lie kinetic energy T and potential energy V in ternis of the generalized coordinates and velocities and the parameters ni1, m. 1., k. etc. (lO pis)
(h) Develop the equations of motion using Lagranges equation. (10 pIs.)
(e) Vrite the equations ot motion ill phase—space form using 1 laniiliomfs equatiOns. (10 pis)
(d) Ifihe rod has length L.50 ni . mt =2 Kg and m = 1 Kg and k3.000 N/in determine the equilibrium
configurations (critical points) of the system. Examine the stability of the system about these
equilibrium points and state if tite system is stable or iton—stable relative to the equilibrium points.
State the classitication of the node or focus associated with the equilibrium point (is it a stable miode.
stable t’ocus. unstable tbcus. center point. etc.) (lo pis.)
(e) Assume friction occurs between the sliding mass nil and the horizontal surfiice. How would YOU
expect this to aftixt the stability of the equilibrium points in (d). That is. how would the
classitications change (stable node, stable focus, etc.). (10 pis.)