# A Two-Disk Extended Jeffcott Rotor Model Distinguishing a Shaft Crack from Other Rotating Asymmetries

Authors: Xi Wu; Jim Meagher
Journal: International Journal of Rotating Machinery
 (2008-01-23)

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## Abstract

A mathematical model of a cracked rotor and an asymmetric rotor with two disks representing a turbine and a generator is utilized to study the vibrations due to imbalance and side load. Nonlinearities typically related with a “breathing” crack are included using a Mayes steering function. Numerical simulations demonstrate how the variations of rotor parameters affect the vibration response and the effect of coupling between torsional and lateral modes. Bode, spectrum, and orbit plots are used to show the differences between the vibration signatures associated with cracked shafts versus asymmetric shafts. Results show how nonlinear lateral-torsional coupling shifts the resonance peaks in the torsional vibration response for cracked shafts and asymmetric rotors. The resonance peaks shift depending on the ratio of the lateral-to-torsional natural frequencies with the peak responses occurring at noninteger values of the lateral natural frequency. When the general nonlinear models used in this study are constrained to reduce to linear torsional vibration, the peak responses occur at commonly reported integer ratios. Full spectrum analyses of theX and Y  vibrations reveal distinct vibration characteristics of both cracked and asymmetric rotors including reverse vibration components. Critical speeds and vibration orders predicted using the models presented herein include and extend diagnostic indicators commonly reported.


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## Body

## 1. Introduction

The purpose of this investigation is to develop and test models for the vibration response of cracked and asymmetric shafts. Some asymmetries are geometric while others may be due to a shaft crack. In this paper, an asymmetric shaft refers to geometric asymmetry other than that due to a crack. The vibration response of asymmetric and cracked shafts shares characteristics such as2x response which makes them hard to distinguish. A distinct crack diagnostic measure observable with measurable vibration data is a further goal of this study. This topic is widely studied because of possible sudden catastrophic failure of a rotor from fatigue. Stress concentrations and high-rotational speeds exacerbate the problem. This is especially dangerous because the torsional response of the rotor is often unmeasured and lightly damped. A comprehensive literature survey of various crack modeling techniques and system behavior of cracked rotor was given by Wauer [1]. This paper contains the
modeling of the cracked components of the structures and searches for different detection strategies to diagnose fracture damage. A more recent survey paper by Sabnavis et al. [2] divides the current research into three categories: vibration-based methods, modal testing, and nontraditional methods such as wavelets or neural networks. Dimarogonas [3] provided an earlier literature review of the vibration of cracked structures and cites more than 300 papers. His review is categorized according to methods that describe local flexibility due to cracks, nonlinearities introduced into the system, and local stiffness matrix descriptions of the cracked section. The crack leads to a coupled system that can be recognized from additional
harmonics in the frequency spectrum. The subharmonic resonances at approximately half and one third of the bending critical speed of the rotor are reported to be the prominent crack indicators by Gasch [4, 5] and Chan [6]. By
utilizing a single parameter “hinge” crack model, Gasch provided an overview of
the dynamic behavior of a simple rotor with transverse crack. He assumed weight
dominance and employed a perturbation method into his analysis. Cross-coupling
stiffness and dynamic response terms were not included in his
analysis. Mayes
model [7] is more practical for deep cracks than a hinged model. Based on Mayes modified model,
Sawicki et al. [8, 9] studied the transient vibration response of a cracked Jeffcott rotor under constant acceleration ratios and under
constant external torque. The angle between the crack centerline and the rotor
whirl vector is employed to determine the closing and opening of the crack.
This allows one to study the rotor dynamic response with or without the rotor
weight dominance assumption by taking nonsynchronous whirl into account. Sawicki
et al. [10] investigated the nonlinear dynamic response of a cracked one-mass Jeffcott rotor by means of bifurcation plots. When a rotor with the crack depth of 0.4 spins at some speed ranges, both the lateral and torsional vibration responses sustain periodic, quasiperiodic, or chaotic behavior. Some researchers [11, 12] have investigated using additional external excitations, such as active magnetic bearings, to create combination resonances for crack identification. Muszynska et al. [13]
and Bently et al. [14] discuss rotor-coupled lateral and torsional vibrations due to unbalance as well as due to shaft asymmetry under a constant radial preload force. Their experimental results exhibited the existence of significant torsional vibrations due to coupling with the lateral modes. In Bently’s and Muszynska’s experiments, an asymmetric shaft was used to simulate the behavior of a crack.This paper extends the research investigations of both Bently et al.’s [14] and Wu’s work [15, 16]. The unique features in this work are the use
of full spectrum and the incorporation of Mayes and Davies [7] crack steering function into an extended Jeffcott rotor model. This causes the stiffness to change with orientation as opposed to the asymmetric stiffness model which is constant in a rotating coordinate system. Another difference is that the equations of motion herein are expressed and solved in inertial coordinates. While anisotropic shafts share some common characteristics with cracked shafts, the crack opening and closing introduce different behavior. Therefore, in this study, an accurate and realistic crack model is introduced for a two-mass rotor in which the first mass represents a turbine and the second mass represents a generator. Starting from energy equations, an analytical model with four degrees of freedom for a torsional/lateral-coupled rotor due to a crack is developed. A radial constant force is applied to the outboard disk to emphasize the effects of the gravity force which plays a critical role for the “breathing” of a crack. As preload increases, the vibration amplitudes in both lateral and torsional directions increase. The “second-order” nonlinear coupling terms due to a crack introduce supersynchronous peaks at certain rotational speeds, which is unique for a cracked rotor and might be used as an unambiguous crack indicator. Computer
simulations also show that the rotational speeds at which amplitudes of the torsional vibrations reach maximum are governed by the ratio of lateral to torsional natural frequency.


## 2. Extended Jeffcott Rotor Mathematical Model

### 2.1. Physical System

Figure1 illustrates the system schematic configuration used to model a turbo machine with a cracked rotor or an asymmetry at the same location. The rotor is driven through a flexible coupling and is supported by bearings which constrain lateral motion. A crack or asymmetry is located near the outboard disk where a downward constant radial force P is also applied. The coupled torsional-flexural vibrations are modeled using four degrees of freedom; torsional rotation at each disk and lateral motion at the outboard disk. Figure 2 shows the section view of the cracked shaft in both inertial (X,Y), and rotating coordinates (ξ,η).Figure 1
Configuration of the cracked extended Jeffcott rotor with two disks.Figure 2
Section view of cracked shaft.The angular position of the outboard disk is expressed asΦ(t)=Ωt+φ(t)−φ0, where Ω is the rotational speed of the rotor, φ(t) is the angular position of the outboard disk relative to the motor, and φ0 is the initial angular position. Similarly, the angular position of the inboard disk is expressed as Θ(t)=Ωt+θ(t)−θ0, where θ(t) is the angular position of the inboard disk relative to the motor. The outboard disk’s vibration is represented by the angular coordinate Φ(t) and two lateral displacements in inertial coordinates. The inboard disk’s vibration is described by the angle Θ(t). The location of the center of mass of the outboard disk can be expressed as the following:
(1)xcm=X+εcos(Φ+δ),ycm=Y+εsin(Φ+δ).


### 2.2. Equations of Motion

The kinetic energy, potential energy, and dissipation function for the rotor system can, respectively, be expressed as the following:(2)T=12I(Ω+φ˙)2+12I0(Ω+θ˙)2+M2{X˙2+Y˙2−2εX˙(Ω+φ˙)sin(Φ+δ)+2εY˙(Ω+φ˙)cos(Φ+δ)+ε2(Ω+φ˙)2},U=12(k11X2+k22Y2)+k12XY+12Kt(φ−θ)2,D=12CX˙2+12CY˙2+12Ct(φ˙−θ˙)2.Loads applied to the system include a driving torque applied to the inboard disk,Cc(Ω−θ˙)+Kc(Ωt−θ),
and a vertical side load, P, applied to the outboard disk. The damping is modeled as lumped viscous damping at the outboard disk and lumped torsional viscous damping of the shaft. The stiffness matrix for a Jeffcott rotor with a cracked shaft in inertial coordinates, KIc, is given by [5, 8, 9]. Details can be found in [15],
(3)KIc=(k11k12k21k22)=(K00K)−f(Φ)K2(Δk1+Δk2cos2ΦΔk2sin2ΦΔk2sin2ΦΔk1−Δk2cos2Φ),
where
(4)Δk1=Δkξ+ΔkηK,Δk2=Δkξ−ΔkηK,Δkξ,Δkη are, respectively, the reduced stiffness in ξ and η directions in a rotor-fixed coordinate system,
and f(Φ)=(1+cos(Φ))/2 is a steering function which Mayes and Davies [7] proposed to illustrate a smooth transition between the opening and closing of a “breathing” crack in rotating coordinates; and Δkη=Δkξ/6 is assumed to describe the stiffness variation for deep cracks.The stiffness matrix for a rotor with an asymmetric shaft in inertial coordinates is given by(5)KIasym=TKRT−1,
where T is the coordinate transformation matrix, T=(cosΦ−sinΦsinΦcosΦ) and KR=(Kξ00Kη) is the stiffness matrix in rotating coordinates:
(6)KIasym=(K+ΔKcos(2Φ)ΔKsin(2Φ)ΔKsin(2Φ)K−ΔKcos(2Φ)).Note that for the asymmetric shaft, the stiffness parameters differ from the parameters used for a cracked shaft. In the asymmetric shaft model,K is the average stiffness rather than the uncracked shaft stiffness, and ΔK
and q are asymmetric stiffness factors:
(7)K=Kξ+Kη2,ΔK=Kξ−Kη2,q=Kη−Kξ2K,
with these factors
(8)KIasym=(k11k12k21k22)=K[1−qcos2Φ−qsin2Φ−qsin2Φ1+qcos2Φ].


### 2.3. Cracked Shaft Equations of Motion

The general equations of motion are obtained using Lagrange’s equations. For the cracked shaft, the equations of motion become(9)X¨+CMX˙+ωn2[1−f(Φ)2(Δk1+Δk2cos2Φ)]X−ωn2f(Φ)Δk2sin2Φ2(Ym−PK)=ε(Ω+φ˙)2cos(Φ+δ)+εφ¨sin(Φ+δ),Y¨m+CMY˙m−ωn2f(Φ)Δk2sin2Φ2X+ωn2[1−f(Φ)2(Δk1−Δk2cos2Φ)]Ym=ε(Ω+φ˙)2sin(Φ+δ)−εφ¨cos(Φ+δ)−Pf(Φ)2M(Δk1−Δk2cos2Φ),θ¨+Kt+KcI0θ−KtI0φ=−Ct+CcI0θ˙+CtI0φ˙,φ¨+CtIφ˙−CtIθ˙+KtIφ−KtIθ=Pεf(Φ)2I(Δk1cos(Φ+δ)−Δk2cos(Φ−δ))+P22KI[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Γc,where(10)Γc=−Cερ2M[X˙sin(Φ+δ)−Y˙mcos(Φ+δ)]+εωn2ρ2(1−f(Φ)2Δk1)[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εf(Φ)Δk2ωn22ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]+X2ωn24ρ2∂f(Φ)∂Φ(Δk1+Δk2cos2Φ)−X2ωn22ρ2f(Φ)Δk2sin2Φ+Ym(Ym−2P/K)ωn22ρ2×[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Δk2X(Ym−P/K)ωn22ρ2×[∂f(Φ)∂Φsin2Φ+2f(Φ)cos2Φ].Using nondimensionalized time defined by the following:(11)τ=ωnt,d(⋅)dt=ωnd(⋅)dτ=ωn(⋅)′,d2(⋅)dt2=ωn2d2(⋅)dτ2=ωn2(⋅)′′,
(9) and (10) take the following form:
(12)X″+2ζX′+[1−f(Φ)2(Δk1+Δk2cos2Φ)]X−f(Φ)Δk2sin2Φ2(Ym−PMωn2)=ε(Ωωn+φ′)2cos(Φ+δ)+εφ″sin(Φ+δ),Ym″+2ζYm′−f(Φ)Δk2sin2Φ2X+[1−f(Φ)2(Δk1−Δk2cos2Φ)]Ym=ε(Ωωn+φ′)2sin(Φ+δ)−εφ″cos(Φ+δ)−PMf(Φ)2ωn2(Δk1−Δk2cos2Φ),θ″+RI(1+Kr)(ωtωn)2θ−RI(ωtωn)2φ=−2RIζt(1+Cr)ωtωnθ′+2RIζtωtωnφ′,φ″+2ζtωtωnφ′−2ζtωtωnθ′+(ωtωn)2φ−(ωtωn)2θ=P2Mεf(Φ)ωn2ρ2(Δk1cos(Φ+δ)−Δk2cos(Φ−δ))+P22M21ωn4ρ2[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Γcωn2,(13)Γcωn2=−2ζερ2[X′sin(Φ+δ)−Ym′cos(Φ+δ)]+ερ2(1−f(Φ)2Δk1)×[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εf(Φ)Δk22ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]+X22ρ2[12∂f(Φ)∂Φ(Δk1+Δk2cos2Φ)−f(Φ)Δk2sin2Φ]+Ym2ρ2(Ym−2PMωn2)×[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Δk2X2ρ2(Ym−PMωn2)×[∂f(Φ)∂Φsin2Φ+2f(Φ)cos2Φ],where Kc=KrKt, Cc=CrCt, RI=I/I0.


### 2.4. Asymmetric Shaft Equations of Motion

For an asymmetric shaft, the equations of motion become(14)X¨+CMX˙+ωn2(1−qcos2Φ)X−ωn2q(Ym−PK)sin2Φ=ε(Ω+φ˙)2cos(Φ+δ)+εφ¨sin(Φ+δ),Y¨m+CMY˙m−qωn2Xsin2Φ+ωn2(1+qcos2Φ)Ym=ε(Ω+φ˙)2sin(Φ+δ)−εφ¨cos(Φ+δ)+qPMcos2Φ,θ¨+Kt+KcI0θ−KtI0φ=−Ct+CcI0θ˙+CtI0φ˙,φ¨+CtIφ˙−CtIθ˙+KtIφ−KtIθ=−qPεIcos(Φ−δ)+qP2KIsin2Φ+Γc,
where
(15)Γc=−Cερ2M[X˙sin(Φ+δ)−Y˙mcos(Φ+δ)]+εωn2ρ2[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εqωn2ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]−qPIYmsin2Φ+qωn2ρ2[(Ym(Ym−PK)−X2)sin2Φ+2X(Ym−PK)cos2Φ].Using nondimensionalized time defined by (11), (14), and (15) takes the following form:
(16)X″+2ζX′+(1−qcos2Φ)X−q(Ym−PMωn2)sin2Φ=ε(Ωωn+φ′)2cos(Φ+δ)+εφ″sin(Φ+δ),Ym″+2ζYm′−qXsin2Φ+(1+qcos2Φ)Ym=ε(Ωωn+φ′)2sin(Φ+δ)−εφ″cos(Φ+δ)+PMqωn2cos2Φ,θ″+RI(1+Kr)(ωtωn)2θ−RI(ωtωn)2φ=−2RIζt(1+Cr)ωtωnθ′+2RIζtωtωnφ′,φ″+2ζtωtωnφ′−2ζtωtωnθ′+(ωtωn)2φ−(ωtωn)2θ=(PM)2qωn4ρ2sin2Φ−PMqεωn2ρ2cos(Φ−δ)+Γcωn2,(17)Γcωn2=−2ζερ2[X′sin(Φ+δ)−Ym′cos(Φ+δ)]+ερ2[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+qερ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]−PqYmMρ2ωn2sin2Φ+qρ2[Ym(Ym−PMωn2)−X2]sin2Φ+2qXρ2(Ym−PMωn2)cos2Φ.


### 2.5. Special Cases

Case 1
(Pure Torsional Vibration for a Cracked Rotor).   Assuming no lateral vibration,X=0,Ym=0, and a rigid drive coupling, Θ˙=Ω, leads to the following simplification for the cracked shaft:
(18)φ¨+CtIφ˙+KtIφ=Pε8I(Δk1−Δk2)+{Pε4I(Δk1−Δk2)cosΦ+P28KI(−Δk1+Δk22)sinΦ}+{Pε8I(Δk1−Δk2)cos2Φ+P2Δk24KIsin2Φ}+3P2Δk216KIsin3Φ.
We introduce the following two constants:
(19)E1=Pε2Iωn2=P2Mερ2ωn2,E2=P22KIωn2=P22M21ρ2ωn4.
Using nondimensional time defined by (11) and  (18) takes the following form:
(20)φ″+2ζtωtωnφ′+(ωtωn)2φ=E14(Δk1−Δk2)+{E12(Δk1−Δk2)cosΦ+E24(−Δk1+Δk22)sinΦ}+{E14(Δk1−Δk2)cos2Φ+E2Δk22sin2Φ}+3E2Δk28sin3Φ.Case 2
(Pure Torsional Vibration for an Asymmetric Rotor).  Assuming no lateral vibration,X=0,Ym=0, and a rigid drive coupling, Θ˙=Ω, leads to the following simplification for the asymmetric shaft:
(21)φ¨+CtIφ˙+KtIφ=−qPεIcos(Φ−δ)+qP2KIsin2Φ.
Using nondimensional time defined by (11) and  (21) takes the following form:
(22)φ″+2ζt(ωtωn)φ′+(ωtωn)2φ=−PMqεωn2ρ2cos(Φ−δ)+(PM)2qωn4ρ2sin2Φ.
The equations above were solved using a variable time-step integration algorithm after the following normalization and simplifications, Y=Ym−P/K is used to delineate the static offset from dynamic response.


## 2.1. Physical System

Figure1 illustrates the system schematic configuration used to model a turbo machine with a cracked rotor or an asymmetry at the same location. The rotor is driven through a flexible coupling and is supported by bearings which constrain lateral motion. A crack or asymmetry is located near the outboard disk where a downward constant radial force P is also applied. The coupled torsional-flexural vibrations are modeled using four degrees of freedom; torsional rotation at each disk and lateral motion at the outboard disk. Figure 2 shows the section view of the cracked shaft in both inertial (X,Y), and rotating coordinates (ξ,η).Figure 1
Configuration of the cracked extended Jeffcott rotor with two disks.Figure 2
Section view of cracked shaft.The angular position of the outboard disk is expressed asΦ(t)=Ωt+φ(t)−φ0, where Ω is the rotational speed of the rotor, φ(t) is the angular position of the outboard disk relative to the motor, and φ0 is the initial angular position. Similarly, the angular position of the inboard disk is expressed as Θ(t)=Ωt+θ(t)−θ0, where θ(t) is the angular position of the inboard disk relative to the motor. The outboard disk’s vibration is represented by the angular coordinate Φ(t) and two lateral displacements in inertial coordinates. The inboard disk’s vibration is described by the angle Θ(t). The location of the center of mass of the outboard disk can be expressed as the following:
(1)xcm=X+εcos(Φ+δ),ycm=Y+εsin(Φ+δ).


## 2.2. Equations of Motion

The kinetic energy, potential energy, and dissipation function for the rotor system can, respectively, be expressed as the following:(2)T=12I(Ω+φ˙)2+12I0(Ω+θ˙)2+M2{X˙2+Y˙2−2εX˙(Ω+φ˙)sin(Φ+δ)+2εY˙(Ω+φ˙)cos(Φ+δ)+ε2(Ω+φ˙)2},U=12(k11X2+k22Y2)+k12XY+12Kt(φ−θ)2,D=12CX˙2+12CY˙2+12Ct(φ˙−θ˙)2.Loads applied to the system include a driving torque applied to the inboard disk,Cc(Ω−θ˙)+Kc(Ωt−θ),
and a vertical side load, P, applied to the outboard disk. The damping is modeled as lumped viscous damping at the outboard disk and lumped torsional viscous damping of the shaft. The stiffness matrix for a Jeffcott rotor with a cracked shaft in inertial coordinates, KIc, is given by [5, 8, 9]. Details can be found in [15],
(3)KIc=(k11k12k21k22)=(K00K)−f(Φ)K2(Δk1+Δk2cos2ΦΔk2sin2ΦΔk2sin2ΦΔk1−Δk2cos2Φ),
where
(4)Δk1=Δkξ+ΔkηK,Δk2=Δkξ−ΔkηK,Δkξ,Δkη are, respectively, the reduced stiffness in ξ and η directions in a rotor-fixed coordinate system,
and f(Φ)=(1+cos(Φ))/2 is a steering function which Mayes and Davies [7] proposed to illustrate a smooth transition between the opening and closing of a “breathing” crack in rotating coordinates; and Δkη=Δkξ/6 is assumed to describe the stiffness variation for deep cracks.The stiffness matrix for a rotor with an asymmetric shaft in inertial coordinates is given by(5)KIasym=TKRT−1,
where T is the coordinate transformation matrix, T=(cosΦ−sinΦsinΦcosΦ) and KR=(Kξ00Kη) is the stiffness matrix in rotating coordinates:
(6)KIasym=(K+ΔKcos(2Φ)ΔKsin(2Φ)ΔKsin(2Φ)K−ΔKcos(2Φ)).Note that for the asymmetric shaft, the stiffness parameters differ from the parameters used for a cracked shaft. In the asymmetric shaft model,K is the average stiffness rather than the uncracked shaft stiffness, and ΔK
and q are asymmetric stiffness factors:
(7)K=Kξ+Kη2,ΔK=Kξ−Kη2,q=Kη−Kξ2K,
with these factors
(8)KIasym=(k11k12k21k22)=K[1−qcos2Φ−qsin2Φ−qsin2Φ1+qcos2Φ].


## 2.3. Cracked Shaft Equations of Motion

The general equations of motion are obtained using Lagrange’s equations. For the cracked shaft, the equations of motion become(9)X¨+CMX˙+ωn2[1−f(Φ)2(Δk1+Δk2cos2Φ)]X−ωn2f(Φ)Δk2sin2Φ2(Ym−PK)=ε(Ω+φ˙)2cos(Φ+δ)+εφ¨sin(Φ+δ),Y¨m+CMY˙m−ωn2f(Φ)Δk2sin2Φ2X+ωn2[1−f(Φ)2(Δk1−Δk2cos2Φ)]Ym=ε(Ω+φ˙)2sin(Φ+δ)−εφ¨cos(Φ+δ)−Pf(Φ)2M(Δk1−Δk2cos2Φ),θ¨+Kt+KcI0θ−KtI0φ=−Ct+CcI0θ˙+CtI0φ˙,φ¨+CtIφ˙−CtIθ˙+KtIφ−KtIθ=Pεf(Φ)2I(Δk1cos(Φ+δ)−Δk2cos(Φ−δ))+P22KI[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Γc,where(10)Γc=−Cερ2M[X˙sin(Φ+δ)−Y˙mcos(Φ+δ)]+εωn2ρ2(1−f(Φ)2Δk1)[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εf(Φ)Δk2ωn22ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]+X2ωn24ρ2∂f(Φ)∂Φ(Δk1+Δk2cos2Φ)−X2ωn22ρ2f(Φ)Δk2sin2Φ+Ym(Ym−2P/K)ωn22ρ2×[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Δk2X(Ym−P/K)ωn22ρ2×[∂f(Φ)∂Φsin2Φ+2f(Φ)cos2Φ].Using nondimensionalized time defined by the following:(11)τ=ωnt,d(⋅)dt=ωnd(⋅)dτ=ωn(⋅)′,d2(⋅)dt2=ωn2d2(⋅)dτ2=ωn2(⋅)′′,
(9) and (10) take the following form:
(12)X″+2ζX′+[1−f(Φ)2(Δk1+Δk2cos2Φ)]X−f(Φ)Δk2sin2Φ2(Ym−PMωn2)=ε(Ωωn+φ′)2cos(Φ+δ)+εφ″sin(Φ+δ),Ym″+2ζYm′−f(Φ)Δk2sin2Φ2X+[1−f(Φ)2(Δk1−Δk2cos2Φ)]Ym=ε(Ωωn+φ′)2sin(Φ+δ)−εφ″cos(Φ+δ)−PMf(Φ)2ωn2(Δk1−Δk2cos2Φ),θ″+RI(1+Kr)(ωtωn)2θ−RI(ωtωn)2φ=−2RIζt(1+Cr)ωtωnθ′+2RIζtωtωnφ′,φ″+2ζtωtωnφ′−2ζtωtωnθ′+(ωtωn)2φ−(ωtωn)2θ=P2Mεf(Φ)ωn2ρ2(Δk1cos(Φ+δ)−Δk2cos(Φ−δ))+P22M21ωn4ρ2[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Γcωn2,(13)Γcωn2=−2ζερ2[X′sin(Φ+δ)−Ym′cos(Φ+δ)]+ερ2(1−f(Φ)2Δk1)×[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εf(Φ)Δk22ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]+X22ρ2[12∂f(Φ)∂Φ(Δk1+Δk2cos2Φ)−f(Φ)Δk2sin2Φ]+Ym2ρ2(Ym−2PMωn2)×[12∂f(Φ)∂Φ(Δk1−Δk2cos2Φ)+f(Φ)Δk2sin2Φ]+Δk2X2ρ2(Ym−PMωn2)×[∂f(Φ)∂Φsin2Φ+2f(Φ)cos2Φ],where Kc=KrKt, Cc=CrCt, RI=I/I0.


## 2.4. Asymmetric Shaft Equations of Motion

For an asymmetric shaft, the equations of motion become(14)X¨+CMX˙+ωn2(1−qcos2Φ)X−ωn2q(Ym−PK)sin2Φ=ε(Ω+φ˙)2cos(Φ+δ)+εφ¨sin(Φ+δ),Y¨m+CMY˙m−qωn2Xsin2Φ+ωn2(1+qcos2Φ)Ym=ε(Ω+φ˙)2sin(Φ+δ)−εφ¨cos(Φ+δ)+qPMcos2Φ,θ¨+Kt+KcI0θ−KtI0φ=−Ct+CcI0θ˙+CtI0φ˙,φ¨+CtIφ˙−CtIθ˙+KtIφ−KtIθ=−qPεIcos(Φ−δ)+qP2KIsin2Φ+Γc,
where
(15)Γc=−Cερ2M[X˙sin(Φ+δ)−Y˙mcos(Φ+δ)]+εωn2ρ2[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+εqωn2ρ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]−qPIYmsin2Φ+qωn2ρ2[(Ym(Ym−PK)−X2)sin2Φ+2X(Ym−PK)cos2Φ].Using nondimensionalized time defined by (11), (14), and (15) takes the following form:
(16)X″+2ζX′+(1−qcos2Φ)X−q(Ym−PMωn2)sin2Φ=ε(Ωωn+φ′)2cos(Φ+δ)+εφ″sin(Φ+δ),Ym″+2ζYm′−qXsin2Φ+(1+qcos2Φ)Ym=ε(Ωωn+φ′)2sin(Φ+δ)−εφ″cos(Φ+δ)+PMqωn2cos2Φ,θ″+RI(1+Kr)(ωtωn)2θ−RI(ωtωn)2φ=−2RIζt(1+Cr)ωtωnθ′+2RIζtωtωnφ′,φ″+2ζtωtωnφ′−2ζtωtωnθ′+(ωtωn)2φ−(ωtωn)2θ=(PM)2qωn4ρ2sin2Φ−PMqεωn2ρ2cos(Φ−δ)+Γcωn2,(17)Γcωn2=−2ζερ2[X′sin(Φ+δ)−Ym′cos(Φ+δ)]+ερ2[−Xsin(Φ+δ)+Ymcos(Φ+δ)]+qερ2[−Xsin(Φ−δ)+Ymcos(Φ−δ)]−PqYmMρ2ωn2sin2Φ+qρ2[Ym(Ym−PMωn2)−X2]sin2Φ+2qXρ2(Ym−PMωn2)cos2Φ.


## 2.5. Special Cases

Case 1
(Pure Torsional Vibration for a Cracked Rotor).   Assuming no lateral vibration,X=0,Ym=0, and a rigid drive coupling, Θ˙=Ω, leads to the following simplification for the cracked shaft:
(18)φ¨+CtIφ˙+KtIφ=Pε8I(Δk1−Δk2)+{Pε4I(Δk1−Δk2)cosΦ+P28KI(−Δk1+Δk22)sinΦ}+{Pε8I(Δk1−Δk2)cos2Φ+P2Δk24KIsin2Φ}+3P2Δk216KIsin3Φ.
We introduce the following two constants:
(19)E1=Pε2Iωn2=P2Mερ2ωn2,E2=P22KIωn2=P22M21ρ2ωn4.
Using nondimensional time defined by (11) and  (18) takes the following form:
(20)φ″+2ζtωtωnφ′+(ωtωn)2φ=E14(Δk1−Δk2)+{E12(Δk1−Δk2)cosΦ+E24(−Δk1+Δk22)sinΦ}+{E14(Δk1−Δk2)cos2Φ+E2Δk22sin2Φ}+3E2Δk28sin3Φ.Case 2
(Pure Torsional Vibration for an Asymmetric Rotor).  Assuming no lateral vibration,X=0,Ym=0, and a rigid drive coupling, Θ˙=Ω, leads to the following simplification for the asymmetric shaft:
(21)φ¨+CtIφ˙+KtIφ=−qPεIcos(Φ−δ)+qP2KIsin2Φ.
Using nondimensional time defined by (11) and  (21) takes the following form:
(22)φ″+2ζt(ωtωn)φ′+(ωtωn)2φ=−PMqεωn2ρ2cos(Φ−δ)+(PM)2qωn4ρ2sin2Φ.
The equations above were solved using a variable time-step integration algorithm after the following normalization and simplifications, Y=Ym−P/K is used to delineate the static offset from dynamic response.


## 3. Numerical Results and Discussion

### 3.1. Pure Torsion

Computer simulation results using the parameters in Table1 for the special cases listed above are shown in Figure 3. The response for a cracked shaft which is calculated from (18) can be interpreted as a nonlinear oscillator with 1x excitation {(Pε/4I)(Δk1−Δk2)cosΦ+(P2/8KI)(−Δk1+Δk2/2)sinΦ}, 2x excitation {(Pε/8I)(Δk1−Δk2)cos2Φ+(P2Δk2/4KI)sin2Φ}, and a 3x excitation {(3P2Δk2/16KI)sin3Φ} due to the unbalance, the depth of the crack, and the side load. These excitations cause the critical speeds shown in Figure 3(a). For the asymmetric shaft, the steady-state responses seen in Figure 3(b) and calculated from (21) can be interpreted as response to 1x{−(qPε/I)cos(Φ−δ)} and 2x excitations {(qP2/KI)sin2Φ}. Since lateral motion is restrained, only torsional critical frequencies appear. For the cracked shaft, there is a 3x critical speed in addition to 1x and 2x. For the parameters in Table 1, the 2x response is the largest. Further details of the dependence of the magnitude of response to crack depth can be found in [16].Table 1
Model physical parameters for pure torsional vibration, special cases.
Cracked shaftAsymmetric shaftParametersValuesUnitsParametersValuesUnitsωn12000rpmωn12000rpmωt2400rpmωt2400rpmε7.62×10−5mε7.62×10−5mρ0.0241mρ0.0241mP/M1270m/s2P/M1270m/s2ζt0.02ζt0.02Δkξ/K0.30q0.14δ0radδ0radOverall peak-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14Figure4 depicts the response for
the same system described in Table 1 except that the response is plotted for a range of eccentricities. For the cracked shaft and the asymmetric shaft, the critical speed associated with the 1x torsional natural frequency at 2400 rpm has a magnitude that increases with increasing eccentricity. The response at the other critical speeds is independent of eccentricity. The consequence of this is that for well-balanced shafts, the presence of a crack will be more easily detected by monitoring the response at a shaft-rotative speed of ωt/2 or ωt/3. At large values of eccentricity, the response at shaft-rotative speeds equal to the torsional natural frequency dominates. The sensitivity of response to changing eccentricity is much greater for the asymmetric shaft. The frequency response at various shaft-rotative speeds, Ω, using the parameters in Table 1 is depicted in Figure 5. Each is dominated by 2x responses with an additional 3x-order response for the cracked shaft.Overall peak-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14Half spectrum of zero-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14


### 3.2. Lateral and Torsional Coupled Vibrations

When the parameters shown in Table2 are used in the general four degrees of freedom model, the critical frequencies shift as seen by comparing Figure 3(a) to Figures 6(a), 6(b) and Figure 3(b) to Figures 7(a), 7(b). Lateral/torsional coupling causes the lateral natural frequency ωn to appear in the torsional response. Also, critical speeds are no longer at integer fractional multiples of the torsional natural frequencies. Instead of a given ratio of torsional to lateral natural frequency, critical speeds occur at fixed noninteger multiples of the lateral natural frequency. This is shown by comparing Figure 6(a) to Figure6(b) and Figure 7(a) to Figure7(b). The damaged shaft has a lower natural frequency so that 1.24 ωn corresponds to 0.90 ωt. Although the absolute frequencies have shifted, the relative critical speeds appear as 1/4, 1/3, and 1/2 of this value. Further details about the variations of the critical speeds due to different stiffness ratios, ωt/ωn, can be found in [16]. By comparing Figure 6 to Figure 7, it is shown that the coupling causes some frequencies to appear in both the asymmetric and cracked shafts, while others appear only in the cracked shaft. The P/M values used in the plots are set intentionally large for the parametric study in order to easily illustrate the coupling and to reproduce values from a paper upon which this study is based [14]. Smaller values would lead to lower peaks and sometimes change the peak response to a different critical speed.Table 2
Model physical parameters for torsional and lateral vibration, general cases.
Cracked shaftAsymmetric  shaftParametersValuesUnitsParametersValuesUnitsε5.08×10−5mε5.08×10−5mρ0.0229mρ0.0229mP/M101.6m/s2P/M101.6m/s2ζt0.02ζt0.02ζ0.1ζ0.1Kr5Kr5Cr1Cr1Δkξ/K0.38q0.18δ0radδ0radTable 3
Model physical parameters for torsional and lateral vibration, general case.
Cracked shaftAsymmetric shaftParametersValuesUnitsParametersValuesUnitsωn2400rpmωn2400rpmωt3312rpmωt3312rpmε5.08×10−5mε5.08×10−5mρ0.0229mρ0.0229mP/M101.6m/s2P/M101.6m/s2ζt0.02ζt0.02ζ0.1ζ0.1Kr5Kr5Cr1Cr1RI1RI1Δkξ/K0.30q0.18δ0radδ0radOverall peak-to-peak torsional vibration response,φ, general case, for cracked shafts with different lateral stiffness but constant torsional-to-lateral natural
frequency ratio.
(a)
ωt/ωn=1.38;ωn=1800 rpm(b)
ωt/ωn=1.38;ωn=2600
rpmOverall peak-to-peak torsional vibration response,φ, general case, for asymmetric shafts with different lateral stiffness but constant torsional-to-lateral natural frequency ratio.
(a)
ωt/ωn=1.38;ωn=1800 rpm(b)
ωt/ωn=1.38;ωn=2600
rpmThe steady-state response at one third of the lateral
natural frequency is shown in Figure8. The trajectory of the whirl for the cracked shaft undergoes three loops per shaft revolution, whereas the trajectory for the asymmetric shaft has a double elliptical pattern.Shaft whirling atΩ=ωn/3; ωt/ωn=1.38; ωn =2400 rpm, Table 3 parameters.
(a)
Cracked shaft(b)
Asymmetric shaftThe full spectrum of the lateral vibration response as illustrated in Figure9 demonstrates the advantage of using full-spectrum plots to differentiate between crack asymmetries and geometric asymmetries. The full spectrum for the cracked shaft includes reverse order response at −1x in addition to 1x and the supersynchronous responses at 2x and 3x. The asymmetric lateral shaft response contains only 1x and 2x without any reverse vibration components.Full spectrum lateral vibration:ωt/ωn = 1.38; ωn =2400
rpm, Table 3 parameters.
(a)
Cracked shaft(b)
Asymmetric shaft


## 3.1. Pure Torsion

Computer simulation results using the parameters in Table1 for the special cases listed above are shown in Figure 3. The response for a cracked shaft which is calculated from (18) can be interpreted as a nonlinear oscillator with 1x excitation {(Pε/4I)(Δk1−Δk2)cosΦ+(P2/8KI)(−Δk1+Δk2/2)sinΦ}, 2x excitation {(Pε/8I)(Δk1−Δk2)cos2Φ+(P2Δk2/4KI)sin2Φ}, and a 3x excitation {(3P2Δk2/16KI)sin3Φ} due to the unbalance, the depth of the crack, and the side load. These excitations cause the critical speeds shown in Figure 3(a). For the asymmetric shaft, the steady-state responses seen in Figure 3(b) and calculated from (21) can be interpreted as response to 1x{−(qPε/I)cos(Φ−δ)} and 2x excitations {(qP2/KI)sin2Φ}. Since lateral motion is restrained, only torsional critical frequencies appear. For the cracked shaft, there is a 3x critical speed in addition to 1x and 2x. For the parameters in Table 1, the 2x response is the largest. Further details of the dependence of the magnitude of response to crack depth can be found in [16].Table 1
Model physical parameters for pure torsional vibration, special cases.
Cracked shaftAsymmetric shaftParametersValuesUnitsParametersValuesUnitsωn12000rpmωn12000rpmωt2400rpmωt2400rpmε7.62×10−5mε7.62×10−5mρ0.0241mρ0.0241mP/M1270m/s2P/M1270m/s2ζt0.02ζt0.02Δkξ/K0.30q0.14δ0radδ0radOverall peak-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14Figure4 depicts the response for
the same system described in Table 1 except that the response is plotted for a range of eccentricities. For the cracked shaft and the asymmetric shaft, the critical speed associated with the 1x torsional natural frequency at 2400 rpm has a magnitude that increases with increasing eccentricity. The response at the other critical speeds is independent of eccentricity. The consequence of this is that for well-balanced shafts, the presence of a crack will be more easily detected by monitoring the response at a shaft-rotative speed of ωt/2 or ωt/3. At large values of eccentricity, the response at shaft-rotative speeds equal to the torsional natural frequency dominates. The sensitivity of response to changing eccentricity is much greater for the asymmetric shaft. The frequency response at various shaft-rotative speeds, Ω, using the parameters in Table 1 is depicted in Figure 5. Each is dominated by 2x responses with an additional 3x-order response for the cracked shaft.Overall peak-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14Half spectrum of zero-to-peak torsional vibration response,φ, special cases, for cracked shaft (a) and asymmetric shaft (b).
(a)
Δkξ/K=0.30(b)
q=0.14


## 3.2. Lateral and Torsional Coupled Vibrations

When the parameters shown in Table2 are used in the general four degrees of freedom model, the critical frequencies shift as seen by comparing Figure 3(a) to Figures 6(a), 6(b) and Figure 3(b) to Figures 7(a), 7(b). Lateral/torsional coupling causes the lateral natural frequency ωn to appear in the torsional response. Also, critical speeds are no longer at integer fractional multiples of the torsional natural frequencies. Instead of a given ratio of torsional to lateral natural frequency, critical speeds occur at fixed noninteger multiples of the lateral natural frequency. This is shown by comparing Figure 6(a) to Figure6(b) and Figure 7(a) to Figure7(b). The damaged shaft has a lower natural frequency so that 1.24 ωn corresponds to 0.90 ωt. Although the absolute frequencies have shifted, the relative critical speeds appear as 1/4, 1/3, and 1/2 of this value. Further details about the variations of the critical speeds due to different stiffness ratios, ωt/ωn, can be found in [16]. By comparing Figure 6 to Figure 7, it is shown that the coupling causes some frequencies to appear in both the asymmetric and cracked shafts, while others appear only in the cracked shaft. The P/M values used in the plots are set intentionally large for the parametric study in order to easily illustrate the coupling and to reproduce values from a paper upon which this study is based [14]. Smaller values would lead to lower peaks and sometimes change the peak response to a different critical speed.Table 2
Model physical parameters for torsional and lateral vibration, general cases.
Cracked shaftAsymmetric  shaftParametersValuesUnitsParametersValuesUnitsε5.08×10−5mε5.08×10−5mρ0.0229mρ0.0229mP/M101.6m/s2P/M101.6m/s2ζt0.02ζt0.02ζ0.1ζ0.1Kr5Kr5Cr1Cr1Δkξ/K0.38q0.18δ0radδ0radTable 3
Model physical parameters for torsional and lateral vibration, general case.
Cracked shaftAsymmetric shaftParametersValuesUnitsParametersValuesUnitsωn2400rpmωn2400rpmωt3312rpmωt3312rpmε5.08×10−5mε5.08×10−5mρ0.0229mρ0.0229mP/M101.6m/s2P/M101.6m/s2ζt0.02ζt0.02ζ0.1ζ0.1Kr5Kr5Cr1Cr1RI1RI1Δkξ/K0.30q0.18δ0radδ0radOverall peak-to-peak torsional vibration response,φ, general case, for cracked shafts with different lateral stiffness but constant torsional-to-lateral natural
frequency ratio.
(a)
ωt/ωn=1.38;ωn=1800 rpm(b)
ωt/ωn=1.38;ωn=2600
rpmOverall peak-to-peak torsional vibration response,φ, general case, for asymmetric shafts with different lateral stiffness but constant torsional-to-lateral natural frequency ratio.
(a)
ωt/ωn=1.38;ωn=1800 rpm(b)
ωt/ωn=1.38;ωn=2600
rpmThe steady-state response at one third of the lateral
natural frequency is shown in Figure8. The trajectory of the whirl for the cracked shaft undergoes three loops per shaft revolution, whereas the trajectory for the asymmetric shaft has a double elliptical pattern.Shaft whirling atΩ=ωn/3; ωt/ωn=1.38; ωn =2400 rpm, Table 3 parameters.
(a)
Cracked shaft(b)
Asymmetric shaftThe full spectrum of the lateral vibration response as illustrated in Figure9 demonstrates the advantage of using full-spectrum plots to differentiate between crack asymmetries and geometric asymmetries. The full spectrum for the cracked shaft includes reverse order response at −1x in addition to 1x and the supersynchronous responses at 2x and 3x. The asymmetric lateral shaft response contains only 1x and 2x without any reverse vibration components.Full spectrum lateral vibration:ωt/ωn = 1.38; ωn =2400
rpm, Table 3 parameters.
(a)
Cracked shaft(b)
Asymmetric shaft


## 4. Conclusions

This paper documents the effect of a shaft crack versus other geometric asymmetries on lateral and torsional vibrations of a two-mass rotor system. Nondimensional analytical models of extended Jeffcott rotors are derived from  Lagrange’s equations taking into consideration the lateral/torsional vibration coupling mechanism induced by a “breathing” crack or a geometric asymmetry. Four degrees of freedom describe the models; two lateral displacements, one torsional
angular displacement of an outboard disk, and the torsional angular displacement of an inboard disk. The nonlinearities associated with a breathing crack or geometric asymmetry couple the four equations of motion. Two cases are considered in this work: a torsionally rigid rotor without lateral vibration and a general unconstrained solution to the four degrees of freedom model presented. The first case is characterized by torsional vibrations which occur atΩ=ωt/3 and Ω=ωt/2. For a cracked shaft, a 3x torsional vibration also occurs. The general case makes evident the existence of strong coupling between lateral and torsional
vibrations where vibration amplitude increases with crack depth, stiffness asymmetry, and radial load. Nonlinear lateral-torsional coupling from a crack shifts the resonance peaks in the torsional vibration response. The resonance peak frequencies shift depending on the ratio of the lateral to torsional natural frequencies with the peak responses occurring at fixed noninteger values of the lateral natural frequency. Orbit plots can also be used to identify a crack with patterns distinguishable from other geometric asymmetries. The distinct vibration signatures predicted by this model can be used for shaft crack diagnostic purposes.



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*Source: 846365-2008-03-13.xml*