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

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 the X 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.


INTRODUCTION
vey paper by Sabnavis et al. [2] divides the current research into three categories: vibration-based methods, modal test-The purpose of this investigation is to develop and test mod-ing, and nontraditional methods such as wavelets or neu els for the vibration response of cracked and asymmetric ral networks.Dimarogonas [3] provided an earlier literashafts.Some asymmetries are geometric while others may be ture review of the vibration of cracked structures and cites due to a shaft crack.In this paper, an asymmetric shaft refers more than 300 papers.His review is categorized according to geometric asymmetry other than that due to a crack.The to methods that describe local flexibility due to cracks, nonvibration response of asymmetric and cracked shafts shares linearities introduced into the system, and local stiffness ma characteristics such as 2x response which makes them hard trix descriptions of the cracked section.The crack leads to a to distinguish.A distinct crack diagnostic measure observable coupled system that can be recognized from additional har with measurable vibration data is a further goal of this study.monics in the frequency spectrum.The subharmonic res-This topic is widely studied because of possible sudden catas-onances at approximately half and one third of the bendtrophic failure of a rotor from fatigue.Stress concentrations ing critical speed of the rotor are reported to be the promi and high-rotational speeds exacerbate the problem.This is nent crack indicators by Gasch [4,5] and Chan [6].By utiespecially dangerous because the torsional response of the ro-lizing a single parameter "hinge" crack model, Gasch protor is often unmeasured and lightly damped.A comprehen-vided an overview of the dynamic behavior of a simple rotor sive literature survey of various crack modeling techniques with transverse crack.He assumed weight dominance and and system behavior of cracked rotor was given by Wauer employed a perturbation method into his analysis.Cross [1].This paper contains the modeling of the cracked com-coupling stiffness and dynamic response terms were not in ponents of the structures and searches for different detection cluded in his analysis.Mayes model [7] is more practical for strategies to diagnose fracture damage.A more recent sur-deep cracks than a hinged model.Based on Mayes modified  model, Sawicki et al. [8,9] studied the transient vibration re sponse of a cracked Jeffcott rotor under constant accelera tion ratios and under constant external torque.The angle be tween the crack centerline and the rotor whirl vector is em ployed 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] in vestigated the nonlinear dynamic response of a cracked onemass Jeffcott rotor by means of bifurcation plots.When a ro tor with the crack depth of 0.4 spins at some speed ranges, both the lateral and torsional vibration responses sustain pe riodic, quasiperiodic, or chaotic behavior.Some researchers [11,12] have investigated using additional external excita tions, such as active magnetic bearings, to create combina tion resonances for crack identification.Muszynska et al. [13] and Bently et al. [14] discuss rotor-coupled lateral and tor sional vibrations due to unbalance as well as due to shaft asymmetry under a constant radial preload force.Their ex perimental results exhibited the existence of significant tor sional 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 fea tures in this work are the use of full spectrum and the in corporation of Mayes and Davies [7] crack steering func tion into an extended Jeffcott rotor model.This causes the stiffness to change with orientation as opposed to the asym metric stiffness model which is constant in a rotating coor dinate system.Another difference is that the equations of motion herein are expressed and solved in inertial coordi nates.While anisotropic shafts share some common charac teristics with cracked shafts, the crack opening and closing introduce different behavior.Therefore, in this study, an ac curate and realistic crack model is introduced for a two-mass rotor in which the first mass represents a turbine and the sec ond mass represents a generator.Starting from energy equa tions, 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 em phasize 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 rota- tional speeds, which is unique for a cracked rotor and might be used as an unambiguous crack indicator.Computer sim ulations also show that the rotational speeds at which am plitudes of the torsional vibrations reach maximum are gov erned by the ratio of lateral to torsional natural frequency.

Physical system
Figure 1 illustrates the system schematic configuration used to model a turbo machine with a cracked rotor or an asym metry at the same location.The rotor is driven through a flexible coupling and is supported by bearings which con strain 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).
The angular position of the outboard disk is expressed as Φ(t) = Ωt + ϕ(t) − ϕ 0 , where Ω is the rotational speed of the rotor, ϕ(t) is the angular position of the outboard disk rela tive to the motor, and ϕ 0 is the initial angular position.Sim ilarly, the angular position of the inboard disk is expressed as Θ(t) = Ωt + θ(t) − θ 0 , where θ(t) is the angular posi tion of the inboard disk relative to the motor.The outboard disk's vibration is represented by the angular coordinate Φ(t) X. Wu and J. Meagher and two lateral displacements in inertial coordinates.The in board disk's vibration is described by the angle Θ(t).The lo cation of the center of mass of the outboard disk can be ex pressed as the following:

Equations of motion
The kinetic energy, potential energy, and dissipation func tion for the rotor system can, respectively, be expressed as the following: Loads applied to the system include a driving torque ap plied to the inboard disk, C c (Ω − θ ˙) + K c (Ωt − θ), and a ver tical 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 stiff ness matrix for a Jeffcott rotor with a cracked shaft in inertial coordinates, K Ic , is given by [5,8,9].Details can be found in [15], where Δk ξ , Δk η 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 coor dinates; and Δk η = Δk ξ /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 where T is the coordinate transformation matrix, T = K ξ 0 cos Φ − sin Φ and K R = is the stiffness matrix in ro sin Φ cos Φ 0 Kη tating coordinates: Note that for the asymmetric shaft, the stiffness param eters 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 asym metric stiffness factors: with these factors

Cracked shaft equations of motion
The general equations of motion are obtained using La grange's equations.For the cracked shaft, the equations of International Journal of Rotating Machinery where Using nondimensionalized time defined by the following: (9) and (10) take the following form: where

Asymmetric shaft equations of motion
For an asymmetric shaft, the equations of motion become Using nondimensionalized time defined by ( 11), ( 14), and (15) takes the following form:

Special cases
Case 1 (pure torsional vibration for a cracked rotor).Assum ing no lateral vibration, X = 0, Y m = 0, and a rigid drive coupling, Θ ˙= Ω, leads to the following simplification for the cracked shaft: We introduce the following two constants: Using nondimensional time defined by ( 11) and (18) takes the following form: Case 2 (pure torsional vibration for an asymmetric rotor).Assuming no lateral vibration, X = 0, Y m = 0, and a rigid drive coupling, Θ ˙= Ω, leads to the following simplification for the asymmetric shaft: Using nondimensional time defined by ( 11) and (21) takes the following form: The equations above were solved using a variable time-step integration algorithm after the following normalization and simplifications, Y = Y m − P/K is used to delineate the static offset from dynamic response.
International Journal of Rotating Machinery

Pure torsion
Computer simulation results using the parameters in Table 1 for the special cases listed above are shown in Figure 3.The response for a cracked shaft which is cal culated from (18) can be interpreted as a nonlinear os cillator with 1x excitation {(Pε/4I)(Δk 1 − Δk 2 )cosΦ + (P 2 /8KI)(−Δk 1 +Δk 2 /2) sin Φ}, 2x excitation {(Pε/8I)(Δk 1 − Δk 2 )cos2Φ + (P 2 Δk 2 /4KI)sin2Φ}, and a 3x excitation {(3P 2 Δk 2 /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 asym metric shaft, the steady-state responses seen in Figure 3(b) and calculated from (21) can be interpreted as re sponse to 1x{−(qPε/I)cos(Φ − δ)} and 2x excitations {(qP 2 /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].Figure 4 depicts the response for the same system de scribed in Table 1 except that the response is plotted for a range of eccentricities.For the cracked shaft and the asym metric shaft, the critical speed associated with the 1x tor sional 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 con sequence of this is that for well-balanced shafts, the presence of a crack will be more easily detected by monitoring the re sponse at a shaft-rotative speed of ω t /2 or ω t /3.At large val ues 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 shaftrotative speeds, Ω, using the parameters in Table 1 is depicted in Figure 5.Each is dominated by 2x responses with an addi tional 3x-order response for the cracked shaft.

Lateral and torsional coupled vibrations
When the parameters shown in Table 2 are used in the gen eral four degrees of freedom model, the critical frequencies shift as seen by comparing Figure 3 The damaged shaft has a lower natural frequency so that 1.24 ω n corresponds to 0.90 ω t .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, ω t /ω n , 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.
The steady-state response at one third of the lateral natu ral frequency is shown in Figure 8.The trajectory of the whirl for the cracked shaft undergoes three loops per shaft revolu tion, whereas the trajectory for the asymmetric shaft has a double elliptical pattern.
The full spectrum of the lateral vibration response as illustrated in Figure 9 demonstrates the advantage of us ing full-spectrum plots to differentiate between crack asym metries and geometric asymmetries.The full spectrum for the cracked shaft includes reverse order response at −1x in

Figure 1 :
Figure 1: Configuration of the cracked extended Jeffcott rotor with two disks.

Figure 2 :
Figure 2: Section view of cracked shaft.
(a) to Figures 6(a), 6(b) and Figure 3(b) to Figures 7(a), 7(b).Lateral/torsional cou pling causes the lateral natural frequency ω n to appear in the torsional response.Also, critical speeds are no longer at in teger fractional multiples of the torsional natural frequen cies.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 compar ing Figure 6(a) to Figure 6(b) and Figure 7(a) to Figure 7(b).

F 1 1. 5 F
re q u e n c y (r p m ) Ω ( r p m ) re q u e n c y (r p m ) Ω ( r p m ) X. Wu and J. Meagher

Figure 6 :
Figure6: Overall peak-to-peak torsional vibration response, ϕ, general case, for cracked shafts with different lateral stiffness but constant torsional-to-lateral natural frequency ratio.

Figure 7 :
Figure7: Overall peak-to-peak torsional vibration response, ϕ, general case, for asymmetric shafts with different lateral stiffness but constant torsional-to-lateral natural frequency ratio.

Table 1 :
Model physical parameters for pure torsional vibration, special cases.

Table 2 :
Model physical parameters for torsional and lateral vibration, general cases.

Table 3 :
Model physical parameters for torsional and lateral vibration, general case.
general case makes evident the existence of strong coupling between lateral and torsional vibrations where vibration am plitude 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 =2400 rpm, Table3parameters.Δk ξ , Δk η : Reduced stiffness in ξ and η directions, respectively q : S t i ffness asymmetry factor.