The influence on coupling vibration of a rotor system due to a mistuned blade length

doi:10.1016/j.ijmecsci.2006.05.016

International Journal of Mechanical Sciences

Volume 49, Issue 4, April 2007, Pages 522-532

https://doi.org/10.1016/j.ijmecsci.2006.05.016 Get rights and content

Abstract

The influence on coupling vibrations among shaft-torsion, disk-transverse and blade-bending of a rotor system with a mistuned blade length was investigated analytically. The shaft–disk–blades system appeared to have four types of coupling vibrations, shaft–blade (SB), shaft-disk-blade (SDB), disk–blade (DB), and blade–blade (BB) mode. The natural frequencies in a mistuned blade were not only found changed but also the types of coupling modes. First, the DB modes in a tuned system disappeared and instead they are replaced by SDB modes. Second, the disk experienced mode localization due to the mistuned blade. The results also showed that every mode's frequency varied with the mistuned error linearly in one direction, either positive or negative, but not in both directions. At last, the effects of rotation on the changes of the rotor's natural frequencies were illustrated. Frequency loci showed merging phenomenon with an increase in the rotational speed.

Introduction

Rotor systems composed of shaft, disk and blades have been extensively used in the industry. The demands for higher operational speeds require more precise tuning than usual. Addressing the existing relevant studies about shaft, disk and blades, the following were seen. Bauer [1] used the assumed modes method to investigate the vibrational behavior of a beam rotating with a constant spin about its longitudinal axis. Kammer and Schlack [2], [3] utilized the perturbation method to study dynamic characteristics and stability of a rotating Euler beam. Yigit et al. [4] used the Galerkin method for the motion of a rotating beam. Nevzat [5] used the analytical method to explore the shaft–disk system. He found critical speeds of the first and second modes, and verified those with experiments. Shen and Ku [6] used Lagrange's equations and linearized equation of motion to explore the multiple disk system, and found the frequencies of the unbalanced modes were lower than those of disk's one-nodal-diameter modes. Shen [7] further employed the Rayleigh dissipation function and Lagrange's equation to solve for the forced responses of a rotating disk/spindle system. Wu and Flowers [8] adopted the transfer matrix method to solve for the natural frequency and critical rotational speed of multiple disks. Chun and Lee [9] used the assumed modes method to analyze the effects of disk flexibility on the vibrational modes of a flexible disk–blade (DB) rotor system, and obtained more efficiency and correct results, compared to finite element method. Omprakash and Ramamurti [10] used Love–Kichhoff method to study the effects on the natural frequency due to the blade stagger angle and twist angle in a DBs system.

Some studies were focused on the mode localization phenomenon due to mistune, disorder or flaws. Bladh et al. [11] used FEM and employed a component mode synthesis approach to systematically generate a reduced order model (ROM). They concluded that the ROM provided a valuable tool for predicting the statistics of forced response for mistune bladed disks. Kuang and Huang [12] used Hamilton's principle and the Galerkin's method to explore the effects of blade crack on the mode localization of a rotating bladed disk. Huang [13] used the same method to study the effect of number of grouped blades on the mode localization of a mistuned blade system.

Huang and Ho [14] utilized the concept of structure synthesis for a shaft–disk–blade (SDB) system. The system was divided into two subsystems, the shaft–disk and blades. The disk was assumed rigid and transmitted the motion between shaft and blades. The results showed that there existed not only shaft–blade (SB) coupled modes but also the inter-blades coupling modes. Lately, Yang and Huang [15] included the disk flexibility in a rotating SDB system. They studied the free vibration and classified four types of coupling modes, SB, SDB, DB, and blade–blade (BB).

A perfect DB is symmetric and of the characteristics of a periodical structure. Any misorder or mistune, however, will destroy the periodicity and causes some mode localization effects. The mistuned error set in this paper might be exaggerated in contrast to today's precision technology. Nevertheless, the paper is intended to provide a qualitative and quantitative overview of a periodic rotor with a mistune.

Section snippets

Theoretical analysis

The rotor system of interest is shown in Fig. 1. If one of the blades’ length deviates from the others, the cyclic periodicity of blades is destroyed. The effects due to this mistuned blade on the coupling vibration are explored in the research. The system to be analyzed is of a torsional shaft, a flexible disk and flexible blades fixed onto the outer edge of the disk with a stagger angle β. The rotor system can be decomposed into three sub-systems, i.e., a torsional shaft, a transverse disk

Numerical results

To avoid dimensional dependence, all the numerical results are normalized with respect to the cantilevered blade's first natural frequency (ω_b₁), i.e., ω*=ω/ω_b₁ and Ω*=Ω/ω_b₁. Table 1 lists the geometric and material properties of the illustrated examples. Note that the length of blades are deliberately elongated in order to magnify the coupling behaviors. Table 2 gives the natural frequencies of individual components, with the other components temporarily removed or assumed rigid. Table 2

Conclusion

This research explored the coupling modes of a SDB system subjected to a mistuned blade length. The assumed modes method was employed for the analysis. The study began with the evolvement of the modes resulting from a flexible disk. It is arrived at that the SB modes evolve into SDB modes, and the BB modes bifurcate into BB modes and DB modes due to disk flexibility.

A mistuned blade has drawn two important phenomena. First, the original DB modes do not exist any more, and they are instead

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There are more references available in the full text version of this article.

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The influence on coupling vibration of a rotor system due to a mistuned blade length

Abstract

Introduction

Section snippets

Theoretical analysis

Numerical results

Conclusion

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

International Journal of Mechanical Sciences

Effects of nonconstant spin rate on the vibration of a rotating beam

ASME Journal of Applied Mechanics

Dynamic response of a radial beam with nonconstant angular velocity

ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design

On the critical speed of continuous shaft–disk systems

Journal of Vibration, Acoustics, Stress, and Reliability in Design