9.3.4 Case study: Quantifying accuracy of forced response calculation

For a simple cylinder that is built in at one end and has a sinusoidal torque applied at the other end, calculate the forced response torsional deflections as a function of position along the length of the cylinder. The modal damping ? is to be assumed to equal zero in each mode. The amplitude of the applied torque is 1000 lbf in, and the frequency is first to be set at 100 Hz and then to 200 Hz.

In this case study, compare the finite element results with the exact analytical solution results determined from the formulas for continuous systems given in App. C. Comment on the shape of the displacement response pattern as a function of axial position for each of the two specified applied torque frequencies.

The cylinder is defined as follows. Outside diameter equals 10 in, and length is 300 in. The rigidity modulus is 11.538 10 ⁶ lbf/in ², and the material density = 0.283 lb/in ³.

For the computer finite element simulation, use 10 identical, three-node finite elements to represent the inertial and stiffness properties of the cylinder.

Solution to Case Study 9.3.4. The first step is calculation of the natural frequencies and mode shapes that are required for interpreting the forced response results needed for this case study. The exact frequencies are given in Table 9.4 using the frequency equation derived in App. C.