Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks

Abstract

The purpose of this paper is to present an approach for replacing the effects of each rigid disk mounted on the spin shaft by a lumped mass together with a frequency-dependent equivalent mass moment of inertia so that the whirling motion of a rotating shaft-disk system is similar to the transverse free vibration of a stationary beam and the technique for the free vibration analysis of a stationary beam with multiple concentrated elements can be used to determine the forward and backward whirling speeds, along with mode shapes of a distributed-mass shaft carrying arbitrary rigid disks. Numerical results reveal that the characteristics of whirling motions are significantly dependent on the slopes of the associated natural mode shapes at the positions where the rigid disks are located. Furthermore, the results obtained from the presented analytical method and those obtained from existing literature or the finite element method (FEM) are in good agreement.

An efficient approach for whirling speeds and mode shapes of uniform and nonuniform Timoshenko shafts mounted by arbitrary rigid disks

“…Figure 5 shows the mathematical model of the shaft-disk system considered. The given data are as follows 12 In the above Equations (46a-h), the subscripts "s" and "d" refer to "shaft" and "disk," respectively. In addition, the shear modulus of shaft material is determined by (cf.…”

“…The data given by Equations (46a-h) are the same as those for Table 1 of Reference 12, except that the shaft studied in the last reference is an Euler beam, so that the effects of SD, RI, and GM of the spin shaft are neglected. 12 Based on the Timoshenko shaft theory presented in this article and the data given by Equations (46) and (47), the lowest five natural frequencies ( 1 − 5 ) of the stationary shaft-disk system (with = 0) are listed in Table 2A and the lowest five whirling speeds (̃1 −̃5) with = 1.0 are listed in Table 2B. From Table 2A one sees that the lowest five natural frequencies obtained from the presented (exact) method are very close to those obtained from FEM, furthermore, because of the effects of SD and RI, the values of 1 − 5 are smaller than the corresponding ones obtained from the Euler shaft-disk system 12 as shown in the final row of Table 2A.…”

“…12 Based on the Timoshenko shaft theory presented in this article and the data given by Equations (46) and (47), the lowest five natural frequencies ( 1 − 5 ) of the stationary shaft-disk system (with = 0) are listed in Table 2A and the lowest five whirling speeds (̃1 −̃5) with = 1.0 are listed in Table 2B. From Table 2A one sees that the lowest five natural frequencies obtained from the presented (exact) method are very close to those obtained from FEM, furthermore, because of the effects of SD and RI, the values of 1 − 5 are smaller than the corresponding ones obtained from the Euler shaft-disk system 12 as shown in the final row of Table 2A. For the spin Timoshenko shaft-disk system with = 1.0, from Table 2B one sees that, either forward whiling speeds (̃F 1 −̃F 5 ) or backward ones (̃B 1 −̃B 5 ), the results obtained from the presented method are very close to the corresponding ones obtained from FEM.…”

“…The last result agrees with that obtained from the spin Euler shaft-disk system. 12 Furthermore, by means of the presented method, the influence of spin speeds Ω on the lowest five "forward" whirling speeds̃F r (r = 1 − 5) represented by the solid lines (with symbols •, +, ▲, ◼, and ★) and the "backward" ones̃B r (r = 1 − 5) represented by the dashed lines (with symbols ○, ×, △, □, and ⋆) is shown in Figure 11. It is seen that the influence of spin speeds Ω on the whirling speeds of the loaded F I G U R E 9 The lowest five natural mode shapes of the stationary uniform P-P shaft ( = 0) carrying three identical rigid disks (Figure 8) obtained from presented method (solid lines with symbols •, +, ▲, ◼, and ★) and finite element method (dashed lines with symbols ○, × , △, □, and ⋆)…”

The Effect of the Shaft Diameter and Torsional Stiffness on the Whirling Speed of the Ship Propeller Shafting System

“…Wu et al provided a method for replacing the effects of each rigid disk that was mounted on the rotating shaft by a lumped mass along with a frequency corresponding to the equivalent mass moment of inertia, so that the whirling motion of a rotating shaft disk system, like a transverse free vibration of a fixed beam In which the free vibration analysis technique of a fixed beam with multiple concentrated elements can be used to determine the forward and backward whirling speeds along with mode shape of a distributed-mass shaft carrying arbitrary rigid disk [8].…”