9.3.1 Case study: Mode responsiveness

For the simple free-free (unconstrained) torsional mathematical model used at the beginning of Chap. 7, having three equal point polar moments of inertia of 1.0 lb in ² connected by two torsional springs of stiffness 1.0 lbf in/rad and having the mode shapes shown in Fig. 7.1, and assuming modal damping values of 0.01 ( ?) in each of the three modes:

1. What are the undamped torsional natural frequencies?

2. For a stimulus of 1.0 lbf in applied at node 2 at the natural frequency of mode 2, which modes will respond most? To help explain your answer, calculate the modal applied torque vector [see Eq. (8.3)].

3. Why isn t the maximum response in mode 2, as the stimulus frequency equals the natural frequency in this mode?

4. If torques of 1.0 lbf in are applied at nodes 1 and 3, 180 out of phase at the natural frequency of mode 3, why is the response in mode 3 zero?

5. If a torque of 1.0 lbf in is applied at node 2 at the mode 3 natural frequency, which mode will respond most, and why? What are the torques in the two springs, and are they in phase? What happens to the response torques if the modal damping values are increased by a factor of 2?

6. For a torque of 1.0 lbf in applied at node 1 at a frequency of 75 percent of the mode 2 torsional natural frequency,...