Figure 1 presents a think disk suspended through two steel shafts with different

Figure 1 presents a think disk suspended through two steel shafts with different

Figure 1 presents a think disk suspended through two steel shafts with different cross-sectional areas of square and
circle with the dimensions shown. The Young and shear modules of steel are 200(GPa) and 75(GPa), respectively. The
loss factor of steel (η) is η = 0.0004 to be used in the Structural Damping. Note that the shafts are in series. The disk
is also welded to a steel bar at point A; the distance between point A and the disk’s center is rd. The disk is subject to
a Harmonic Excitation of T = T0 sin(ωt) around the axis o − o
0
. The masses of two shafts and bar are negligible in
comparison with the disk’s mass. The geometrical and inertial parameters are shown in Fig. 1.
1- Model the shafts as two sets of equivalent torsional springs (ket) and dampers (cet) which are in series. Also model
the bar as a linear spring (kel). Use the numbers shown in Figure 1. Hint: Use ω = 2ωn
rad
s
; see step (4)
2- Find the equivalent torsional stiffness (ketot) and damping (cetot) of the shaft
3- By having the numerical values of the equivalent torsional stiffness and damping coefficients of both the shafts and
bar, derive the equation of motion (EOM) of the disk (Fig. 2). 1) Use ODE45 or ODE15s of MATLAB and plot α(t)
vs. ω = (1000, 4000, 7000)
rad
s
. Assume T_0=500 N.
4- Find the torsional damping ratio ζt, natural frequency ωn, and possibly the damped natural frequency of the disk.
5- Determine the amplitude of α0 based on the frequency ratio r =
ω
ωn
and torsional damping (ζt) ratio which you can
now calculate from steps (1), (2), and (3) as we have done for a linear case in the class.
6-Use MATLAB and plot α0 vs. r. Find the frequency (ω) in which the resonance happens and show it in the plot.