Question

PROBLEM 3: (40 points) A rigid massless lever ACB, as shown in Figure 3, is pivoting about point C. A mass mis attached at point A. Assume frictionless pivot point, frictionless pulley, massless pulley, and small angles. The parameters are Ki-30N/m, m=1 kg, K2-40N/m, C-4.35N-s/m, L=0.4 m, a=0.2 m, and b=0.15 m. (i) Draw the Free Body Diagram (FBD) (5 points). (ii) Use Newton's approach to derive the equations of the motion (10 points). (iii) Use Lagrange's method to derive the equations of the motion (10 points). (iv) What are the natural frequency and damping ratio of this system? (5 points) (v) If the system shown below is given an initial angular displacement of /12 and zero initial angular velocity, find the solution for o(t). (10 points) L- Figure 3

Answer 1

Natural frequency fon kec for in Natural frequeney take c=0 above ech mq 2 8 + (k₂ b ² + k, 92) o=0 6 FF.B.D) 6 + 1Kz b²+ kg). 191) Lagrange's Eht proteh t mat K2b²+ k, 9² - moz Applying Dalemberts prisciple we have; man & + kabet koko + cl²o-ol equation of motion. 111) Newton's approach Damping ratio comparing equation ☺ wirto Standard form of equation we have azona C1² ma2 CL? zmo2 on a2 -x mga Summation of all torque acting on the system = Inertia of the system. x Angular acceleration - 1K₂ 60 tk, q²o +(2²ė = (mg2a. or mota + K2 6²0 + kako + cL²0 = 07 or ma²ö + K2620 tk 9% 0 + c2²0 =0 Sto zna2 Kz6²+ lyg