9.4.1 Case study: Transient and steady-state responses

For this case study, use the simple free-free torsional mathematical model defined in Chap. 7, having three equal point polar moments of inertia of 1 lb in ² connected by two shafts, each having zero inertia and torsional stiffness of 1 lbf in/rad. The model has the mode shapes shown in Fig. 7.1.

For this case study, show that the transient response torque in shaft 1 settles down under the influence of damping to the same steady-state response solution that would be calculated in a steady-state analysis. The sinusoidal applied torque at node 1 for this study is 10 sin (2 ? 5 t). The applied torques at the other nodes are to be set to zero.

Assume modal damping values ? of 0.5 in each of the three modes. The modal damping has been intentionally set high so that the transient vibration response builds up to the steady-state levels quickly. The initial conditions t = 0 for rotational displacement and velocity at all the nodes are to be set to zero.

Solution to Case Study 9.4.1. Table 9.5 gives the computer program output for the sinusoidal steady-state response for the given conditions. Looking at the shaft torque amplitude line, it is seen that the steady-state torque in shaft 1 equals 3.6035 lbf in.