Answer Happy

Question WHP7-Q1. The system in figure is composed by a single beam of length L = 2 m, mass m = 100 kg and Io = 58.33 kg-m² with respect to point 0. The beam is pinned to the ground in point 0, located at a distance L/4 from its end A. The centre of mass G of the beam is in its centre, i.e. at a distance L/4 from 0. The beam is connected to the ground by: • A damper with coefficient c = 20 N-s/m in A • A spring with stiffness k = 100 N/m in B Both spring and damper slide on the ground so that they remain horizontal throughout motion. A harmonic force F is applied at point P as in figure. The force remains perpendicular to the beam throughout the motion. Defining as the CCW rotation of the beam with 0 = 0 as in figure (vertical beam with G below 0), and knowing that the spring is unstretched for 0 = 0 (as in figure): Answer to all the following sub-questions: a) Show that the non-linear equation of motion in is the following: to k In m, lo Ө G 58.33.0 +5.0 cos² (0) + 100 cos (0) sin(0) + 490.5 sin(0) = F A Oo B P L F L b) Show that the position 0 = 0 is a stable equilibrium position c) Write the linearised equation of motion in the variable 8 (around 0eq = 0) and find the parameters I*, ct and ki d) Find natural frequency w, and damping ratio e) Knowing that the force is F = Fo cos(wt) with Fo= 10 N and w = 2 rad/s find the steady state vibration amplitude e f) Draw the amplitude e of the response as a function of the force frequency w (all other parameters to stay constant as they were in previous points)