212 Question WHP7-01. The system in figure is composed by a single beam of length L = 2 m, mass m = 100 kg and lo = 58.33 kg-m 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, L.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 80 as in figure (vertical beam with G below 0), and knowing that the spring is unstretched for 0 = 0 (as in figure): A Oo: 8 GO k B F P m, lo Answer to all the following sub-questions: a) Show that the non-linear equation of motion in 0 is the following: 58.33 +5 cos2(0)+100 cos(0) sin(0) + 490.5 sin(0) = 1.5 - F b) Show that the position = 0 is a stable equilibrium position c) Write the linearised equation of motion in the variable (around eq = 0) and find the parameters I*, c and k d) Find natural frequency w,, and damping ratio < e) Knowing that the force is F = F, cos(wt) with F = 10 N and w = 2 rad/s find the steady state vibration amplitude e f) Draw the amplitude of the response as a function of the force frequency w (all other parameters to stay constant as they were in previous points) Answers a) To show using steps 1-4 of procedure seen in class b) To show studying eq. and stability (steps 5 and 6 of procedure) c) = 58.33 kg-m, c c = 5 N-s-m/rad, and 16 3.182 rad/s and