# 1. A system is vibrating with very small angle 0, which is shown in...

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1. A system is vibrating with very small angle 0, which is shown in figure 1 with massless lever a =0.8 m and b = 1 m and mass m1 = 1.5 kg and m2 = 2.0 kg. A stiffness k = 100 N/m and a damper c = 0.5 N/m-s are vertically connected to mi and m2, respectively. The force f = 10 N cos(2t) / / m2 k f b c m, a 45° / 7 4 X Figure 1. Problem 1 a) Derive the equation of motion in terms of coordinate y. b) Find the natural frequency and damping ratio. 2. A machine is subjected to harmonic ground excitation. The driving frequency f = 2 Hz. The machine with base isolator can be modeled a SDOF system. The mass is, m = 500kg and the damping ratio is E = 0.05. a) Calculate the damping coefficient. b) Calculate the corresponding relative displacement c) Determine the stiffness of the base isolator so that the acceleration of the machine can be reduced to 1/2 of the level when the stiffness is considered to be infinity. 3. An under-damped 2-DOF system has mass and stiffness matrix M and K. We also know that the system has damping ratio E1 = 0.5 and E2 = 0.4. The damping matrix is written as a) Try to find the coefficients a., ß and Y- b) Find if this system proportionally damped? 4. Write the equation of motion of the system shown in figure 2. a) Calculate the natural frequency and modal shape by assuming c = 0, where the rotating stiffness k1 = k2 = k; the moment of inertia J1 = 3 J2. 01 w 02 . V k1 J1 k2 J2 2/1 c Figure 2 Problem 4 5. A system has mass matrix M = 0 1 4 0 and eigenvector matrix U = "1 U2 0.9673 0.2535 0.9979 -0.0654 And the natural frequencies of the system are: Onl = 1.3582 and On2 = 6.3761. a) Normalize U with respect to M. (Hint: Uc is an eigenvector of M-K. Let =1) 7. A system has = and K = -100 400 -100 100 and its natural frequencies are 50 and 150 respectively. Check which of the following can be eigenvectors of matrix M-1/2 KM-1/2 2 Suppose this system has initial conditions: x(0) 2 and x(0) = {-0.05} 0 = Calculate the responses of displacements. 8. a) Find the dynamic magnification factor (amplitude of the transfer function) for a first order vibration system = = - - (p8) and E are respectively the natural frequency and damping ratio of this system. b) if equation (p8) is the equation of a modal response of one part of a complex conjugate pair, then can you find the transfer function of this particular mode? plot the transfer function by letting E = 0.3 and On = 10.

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