Name ___________________________ EM406 Examination II October 26, 2006 Problem Score 1 /25 2 /35 3 /40 Total /100 Show all work for credit AND Stay in your seat until the end of class AND Turn in your signed help sheet NOTE: Do not get bogged down on short answer problems!Name 25 pts EM406 Examination III Problem 1 October 26, 2006 Problem 1.1 A second order system is forced with a periodic input as shown below (only the portion of the input displacement for t>0 is shown – the dotted line is the beginning of the next cycle). Determine a) What is the fundamental frequency of the input? (4 pts) b) Is the function odd, even or neither? What is the implication of this when you look at your Maple worksheet results? (4 pts) c) What is a0 for this function? (4 pts) Problem 1.2 What is the mass moment of inertia of the system shown below about its center of gravity? Assume each of the masses is a point mass (4 pts) 1.0 0.5 Time (s) y (cm) 1 1.5 m m 2m a aProblem 1.3 (4 pts) Two cycles of a general periodic forcing function is shown below. Write the Maple command to input this particular function into the Fourier series worksheet provided to you earlier this quarter. 0246810120 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Time (s)Force (N) Problem 1.5 (5 pts) Given experimental time responses for a system how can you determine the magnitude and phase of the frequency response function in Matlab?Name 35 pts EM406 Examination III Problem 2 October 26, 2006 A 2-DOF linear dynamic system has the mass and stiffness matrices given below. 1.04 00 1.04M =   and 10 3.23.2 25.36K =  . a) Write down the characteristic polynomial of the system. Do not find the roots of this polynomial, and you do not need to simplify the polynomial in any way. (7 pts) b) The natural frequencies of the dynamic system are ω1 = 3.0 rad/sec and ω2 = 5.0 rad/sec. The mode shape associated with the first natural frequency is 10.2  − . Find the mode shape associated with the frequency ω2 = 5.0. (8 pts) c) Specify an initial displacement pattern, 12XX   , which would produce a free vibration having the single frequency ω1 = 3.0 rad/sec. You may assume zero initial velocity. (5 pts)d) Suppose the initial displacement pattern for a free vibration was 10   . Assume zero initial velocity and determine the time responses for x1 and x2. (15 pts)Name 40 pts EM406 Examination III Problem 3 October 26, 2006 The 2-DOF system shown below is forced harmonically with the force Fcosωt. a) Using Lagrange’s equations determine the equations of motion for the two masses shown. Show all work for credit. Write your final equations in 2nd order matrix form by filling in the matrices shown below. (continued on next page) c1 k2 m2 k1 k3 m1 x2 x1 θ Fcosωt =++ 212121xxxxxx&&&&&&b) Assuming the equations of motion are found to be: determine the steady-state response of mass 2 (neglect the homogeneous solution). c) Determine the values of k3 and m1 so that mass 1 acts like vibration absorber for mass 2 and mass 1 has a displacement less than 0.05. ()( )=++−+−++t4cos020555100 0