In rotating systems, if shafts rotate at critical speeds, large vibrations may develop as a result of resonance effects. In Figure 9–28(a), a disk of mass m is mounted on an elastic shaft whose mass...

In rotating systems, if shafts rotate at critical speeds, large vibrations may develop as a result of resonance effects. In Figure 9–28(a), a disk of mass m is mounted on an elastic shaft whose mass is negligible compared with that of the disk, which is placed midway between bearings. Assume that the disk is not perfectly symmetrical and that there is an eccentricity e from the center of the disk.The geometrical center of the disk, the center of mass of the disk, and the center of rotation are denoted by points O, G, and R, respectively. The distance between points R and O is r, and that between points O and G is e. Assume that the equivalent spring constant of the elastic shaft is k, so that the restoring force due to the elastic shaft is kr. What is the critical speed of the system?