# Nonlinear Dynamic Performance of a Bolt-Disc Rotor with the Position Error of Circumferential Bolt-Holes

Authors: Haibo Zhang; Yi Liu; Xin Huang
Journal: Shock and Vibration
 (2021)

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## Abstract

The bolt-holes in the assembly discs are designed to limit the circumferential displacement of bolts for the bolt-disc rotor. The position error of circumferential bolt-holes is created in a three-dimensional model of bolt-disc rotor. The distribution of nonuniform stress and deformation is acquired according to finite element approach. Static results demonstrate that the position error of bolt-holes leads to obvious concomitant unbalances including constant mass eccentricity and speed-variant bending under the influence of large tightening force. When these unbalance factors are taken into consideration, dynamic performance such as instability areas and nonlinear motions are analyzed by Newton iterative process and a prediction-correction calculation method. Dynamic results show that rotor flexure enables the systematic stability decreased obviously because of this position error. There is a special phenomenon compared to monobloc rotor that the vibration amplitude proceeds to rise when rotating speed exceeds the critical speed. Moreover, the allowable position error of bolt-holes is obviously smaller than that of monobloc rotor and uneven tightening is a feasible way to reduce adverse effects on the dynamic properties when position error appears. This work proposes a static-dynamic approach to investigate the dynamics of imprecise bolt-disc rotor and establishes the relationship between machining error and dynamic features.


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## Body

## 1. Introduction

Bolt-disc rotor is a sort of typical combined rotor, in which many separated parts such as discs are fastened together by one central bolt or several circumferentially distributed bolts. This kind of rotor structure has wide applications in gas turbines, aero engines, and so on. The bolt-holes in the assembly discs are designed to limit the circumferential displacement of bolts. In order to satisfy the high assembly demand, the bolt-holes are manufactured very precisely before the discs are assembled. For example, the position error of bolt-holes in the 17 assembly discs in an F-class gas turbine rotor (Figure1) is controlled to be less than 50 μm when the distributed diameter of bolts is 980 mm. Why the position precision of bolt-holes in the bolt-disc rotor is so high? In this paper, this basic problem will be discussed from the perspective of 3D nonlinear rotor dynamics.Figure 1
Structure of an F-class gas turbine rotor.For traditional rotor-dynamic research studies, rotor models are usually treated as rigid rotor, Jeffcott rotor, 1D finite element rotor (e.g., Timoshenko-beam rotor), and so on [1–5]. These models are very classic and useful to analyze the basic dynamic properties, but detailed features (such as holes and defect) in a rotor cannot be reflected entirely because structure simplification exists in these basic rotor models inevitably. In order to describe the actual influence of machining errors, the bolt-disc rotor model should be established in a 3D way which can contain the entire structure features of the position error of bolt-holes. For the actual needs of the engineering, 1D bolt-disc rotor is presented [6], but 3D dynamic effects of the bolt-disc rotor cannot be considered in the combined model. Afterwards, 3D nonlinear rotor-dynamic analysis approach is proposed for acquiring the characteristics of the monobloc rotor [7].However, the position error of bolt-holes causes the mixed static and dynamic problem because uniformly distributed tightening force of bolts can lead to distinct deformation. There is a report [8] that only considers the roughness of interfaces about this static-dynamic uniting problem, but the method is inadequate to analyze the position error because it has larger shape scale and will cause more obvious effects than surface roughness.In this paper, an actual position error of bolt-holes is considered in a three-dimensional bolt-disc rotor model. The finite element method including contact calculation [9, 10] is adopted to analyze the static influences. The consequent 3D rotor-dynamic properties such as stability and vibration are obtained by Newton iterative process, a prediction-correction calculation method, and Floquet theory [11–14]. The comparison between bolt-disc rotor and monobloc rotor is also widely discussed to deduce relative conclusions.


## 2. Static Characteristics of an Imprecise Bolt-Disc Rotor with Position Error of Bolt-Holes

### 2.1. Structure of Bolt-Disc Rotor Bearing System

Taking the basic structure of compressor in the F-class gas turbine rotor (Figure1) as reference, an analogous bolt-disc rotor will be studied in Figure 2. This combined rotor has three separated discs (“a,” “b,” and “c”) which are fastened by some circumferential distributed bolts which go through the holes in discs (green lines in Figure 2 and being called bolt-holes). Every bolt has the same size and elongation which is mounted to tighten each rotating disc of this rotor.Figure 2
Typical bolt-disc rotor bearing system.During an assembly procedure, the bolt-disc rotor is combined vertically for avoiding the gravity influence. The axial assembly interfaces (blue lines) between three discs are used to ensure sufficient contact stiffness. The concentricity between discs is guaranteed by some coaxial-locating interfaces (red lines). Bolt surfaces are designed to fit against the corresponding disc-hole side faces.After the above regulated assembly process, the bolt-disc rotor is placed horizontally and supported by 2 cylindrical sliding bearings (“j” and “k”). The rotating speed ranges from 0 to 7500 rpm (working speed). Table1 shows the variables of bolt-disc rotor and bearing.Table 1
Variables of the bolt-disc rotor bearing system.
ItemVariableValueBolt-disc rotorDiameter of shaft,ds0.08 mLength of shaft,Ls1.25 mDiameter of discs,dd0.24 mThickness of disc “a,”la0.075 mThickness of disc “b,”lb0.1 mThickness of disc “c,”lc0.075 mNumber of bolts,n12Tightening elongation,lF0.75 mmDiameter of bolts,dbolt0.02 mDiameter of circumference for bolts,Dbolt0.156 mJournal bearingsSpan of bearing,Lb1.05 mRadius,R0.04 mWidth,Bj0.08 mRadial clearance,c00.0002 mLubricant viscosity,μ0.0182 Pa s


### 2.2. Position Error of Bolt-Holes

The position error of circumferential bolt-holes in this paper is shown in Figure3(a). The center of the circumferentially distributed bolt-holes on the disc “b” shifts from the O (ideal position) to O’ (actual position) because of this positional error. The deviation h = |OO’| is set to be 50 μm which compared to the F-class gas turbine rotor. Meanwhile, other two discs are manufactured perfectly.Figure 3
3D model of bolt-disc rotor with the position error of bolt-holes. (a) Imprecise disc “b” and (b) imprecise bolt-disc rotor.
(a)(b)The 3D imprecise disc and corresponding bolt-disc rotor are discretized bym-node (m = 8) elements by applying the FE method which are shown in Figure 3(b).


### 2.3. Static Analysis

The key point of static analysis is to calculate stress and status of contact surfaces in the bolt-disc rotor that are obtained by the iteration [10]:(1)KR+KΛHTH0uη=F−FΛ−h0,where KR is the stiffness matrix of imprecise bolt-disc rotor; u is displacement vector; η is Lagrangian multiplier; h is contact constraints; F is force vector; Λ is penalty factor; and KΛ = HTΛH and FΛ = HTΛh0 in which H = ∂h/∂u.Since the constraint conditions change constantly with contact state, the incremental iteration is used to solve the above equations.For comparison, static analysis is carried out to investigate the relative characteristics of the perfect and imprecise bolt-disc rotor. In Figure4, at a constant speed (7500 rpm), results indicate that the stress of a bolt (No. 4 bolt, for example) in the perfect bolt-disc rotor is distributed uniformly. However, obviously uneven stress appears on the corresponding bolt in the imprecise bolt-disc rotor.Figure 4
Stress distribution of No.4 bolt in both perfect and imprecise bolt-disc rotor.Due to the existence of the position error of bolt-holes, the bolt-hole surfaces of disc “b” press several bolts when the disc shifts transversely after assembly. As a result, inhomogeneous stress and bending deformation are caused by the pressing force.Consequently, the position error of bolt-holes leads to mass eccentricitye; nonuniform stress triggers axis-center shift sd. Compared to a precise-machining rotor, the position error on disc “b” causes a constant e (equals to 2.98 μm) during the whole procedure and a speed-variant sd in the working procedure which increases gradually up to 6.28 μm at working speed (7500 rpm).On the other hand,sd and e of precise disc are extremely close to zero. It proves that calculating accuracy is absolutely adequate to analyze the static influences of position error of bolt-holes in the rotor.Then disc-center shiftsd enables the combined rotor to have an entire bending rd whose vector is defined as rd = (s1,…, sd,…, sk) in which k is the node number on the axis (Figure 5). The way to obtaining sk is similar with calculating sd. Rotor bending occurs in assembly procedure, and it has been enhanced in working procedure due to centrifugal pull. The bending variation does not absolutely exist in traditional monobloc rotor which does not have circumferential bolts and tightening force.Figure 5
Rotor bending of imprecise bolt-disc rotor.


## 2.1. Structure of Bolt-Disc Rotor Bearing System

Taking the basic structure of compressor in the F-class gas turbine rotor (Figure1) as reference, an analogous bolt-disc rotor will be studied in Figure 2. This combined rotor has three separated discs (“a,” “b,” and “c”) which are fastened by some circumferential distributed bolts which go through the holes in discs (green lines in Figure 2 and being called bolt-holes). Every bolt has the same size and elongation which is mounted to tighten each rotating disc of this rotor.Figure 2
Typical bolt-disc rotor bearing system.During an assembly procedure, the bolt-disc rotor is combined vertically for avoiding the gravity influence. The axial assembly interfaces (blue lines) between three discs are used to ensure sufficient contact stiffness. The concentricity between discs is guaranteed by some coaxial-locating interfaces (red lines). Bolt surfaces are designed to fit against the corresponding disc-hole side faces.After the above regulated assembly process, the bolt-disc rotor is placed horizontally and supported by 2 cylindrical sliding bearings (“j” and “k”). The rotating speed ranges from 0 to 7500 rpm (working speed). Table1 shows the variables of bolt-disc rotor and bearing.Table 1
Variables of the bolt-disc rotor bearing system.
ItemVariableValueBolt-disc rotorDiameter of shaft,ds0.08 mLength of shaft,Ls1.25 mDiameter of discs,dd0.24 mThickness of disc “a,”la0.075 mThickness of disc “b,”lb0.1 mThickness of disc “c,”lc0.075 mNumber of bolts,n12Tightening elongation,lF0.75 mmDiameter of bolts,dbolt0.02 mDiameter of circumference for bolts,Dbolt0.156 mJournal bearingsSpan of bearing,Lb1.05 mRadius,R0.04 mWidth,Bj0.08 mRadial clearance,c00.0002 mLubricant viscosity,μ0.0182 Pa s


## 2.2. Position Error of Bolt-Holes

The position error of circumferential bolt-holes in this paper is shown in Figure3(a). The center of the circumferentially distributed bolt-holes on the disc “b” shifts from the O (ideal position) to O’ (actual position) because of this positional error. The deviation h = |OO’| is set to be 50 μm which compared to the F-class gas turbine rotor. Meanwhile, other two discs are manufactured perfectly.Figure 3
3D model of bolt-disc rotor with the position error of bolt-holes. (a) Imprecise disc “b” and (b) imprecise bolt-disc rotor.
(a)(b)The 3D imprecise disc and corresponding bolt-disc rotor are discretized bym-node (m = 8) elements by applying the FE method which are shown in Figure 3(b).


## 2.3. Static Analysis

The key point of static analysis is to calculate stress and status of contact surfaces in the bolt-disc rotor that are obtained by the iteration [10]:(1)KR+KΛHTH0uη=F−FΛ−h0,where KR is the stiffness matrix of imprecise bolt-disc rotor; u is displacement vector; η is Lagrangian multiplier; h is contact constraints; F is force vector; Λ is penalty factor; and KΛ = HTΛH and FΛ = HTΛh0 in which H = ∂h/∂u.Since the constraint conditions change constantly with contact state, the incremental iteration is used to solve the above equations.For comparison, static analysis is carried out to investigate the relative characteristics of the perfect and imprecise bolt-disc rotor. In Figure4, at a constant speed (7500 rpm), results indicate that the stress of a bolt (No. 4 bolt, for example) in the perfect bolt-disc rotor is distributed uniformly. However, obviously uneven stress appears on the corresponding bolt in the imprecise bolt-disc rotor.Figure 4
Stress distribution of No.4 bolt in both perfect and imprecise bolt-disc rotor.Due to the existence of the position error of bolt-holes, the bolt-hole surfaces of disc “b” press several bolts when the disc shifts transversely after assembly. As a result, inhomogeneous stress and bending deformation are caused by the pressing force.Consequently, the position error of bolt-holes leads to mass eccentricitye; nonuniform stress triggers axis-center shift sd. Compared to a precise-machining rotor, the position error on disc “b” causes a constant e (equals to 2.98 μm) during the whole procedure and a speed-variant sd in the working procedure which increases gradually up to 6.28 μm at working speed (7500 rpm).On the other hand,sd and e of precise disc are extremely close to zero. It proves that calculating accuracy is absolutely adequate to analyze the static influences of position error of bolt-holes in the rotor.Then disc-center shiftsd enables the combined rotor to have an entire bending rd whose vector is defined as rd = (s1,…, sd,…, sk) in which k is the node number on the axis (Figure 5). The way to obtaining sk is similar with calculating sd. Rotor bending occurs in assembly procedure, and it has been enhanced in working procedure due to centrifugal pull. The bending variation does not absolutely exist in traditional monobloc rotor which does not have circumferential bolts and tightening force.Figure 5
Rotor bending of imprecise bolt-disc rotor.


## 3. Dynamic Equations and Nonlinear Solution Method

### 3.1. Equations of the 3D Imprecise Rotor and System Reduction

When the bolt-disc rotor has the unbalance factor including mass eccentricity and rotor bending, the dynamic model of imprecise bolt-disc rotor system should take them into account.For the 3D bolt-disc rotor system, the DOFs are divided into nonlinear and linear component. DOFs belong to the bearing nodes (used to add bearing forces) and the disc nodes (used to add unbalance forces) are defined as nonlinear component; all the other DOFs belong to other nodes of the rotor are defined as linear component.For the nonlinear component, eigenmodes can reflect the dynamic characteristics of the linear system. As a result, the eigenvalues of the equations in the linear component are calculated and the first several eigenmodes are reserved.Then, dynamic equations are divided into linear and nonlinear sections:(2)MaaRMaiRMiaRMiiRu¨au¨i+GaaRGaiRGiaRGiiRu˙au˙i+KaaRKaiRKiaRKiiRuaui−rdardi=mω2e0+gagi+fB0,where MR and GR are mass matrix and gyroscope matrix of rotor; g is gravity force vector; subscripts “a” and “i” stand for nonlinear and linear DOFs respectively; and fB is vector of bearing forces which is calculated based on finite difference method [15]. The pad of cylinder bearing is discretized into many nodes in geometry. The total bearing force is regarded as the sum of hydrodynamic pressure which is solved by Reynolds equation.In addition, linear DOFs of dynamic equation (2) are reduced by a transformation matrix T by applying a component mode synthesis method [16]. As a result, the residual equations of imprecise rotor system are listed as(3)M0q¨+G0q˙+K0q=Q0,withq=Tu,Q0=TTKRrd+meω2+g+fBM0=TTMRT,G0=TTGRT,K0=TTKRT.For the reduced bolt-disc rotor system originated from Figure3(b), the reserved DOFs equal 27 which includes 3 disc nodes (9 DOFs), 2 bearing nodes (6 DOFs), and first 12 order eigenmodes (6 rigid eigenmodes and 6 elastic eigenmodes).


### 3.2. Nonlinear Solution Method

In order to obtain displacement and velocity, state parameterY is introduced:(4)dYdt=ddtqq˙=Ft,λ,Y,where Ft,λ,Y=q˙/M0−1Q0−K0q−G0q˙ and λ is the excitation variable such as ω, e, and rd.Sincee and rd meet the periodic feature when rotor rotating, the solutions of equation (4) are determined by(5)dYdt=fY,t,λYt=Yt+T,where T is the rotating period.Whenλ = λ0, equation (4) can be changed into solving target periodic solutions Y(t) according to nonlinear definition(6)Hλ0,Y=Yt+T−Yt=0.Here, the Newton iteration method is applied to approachY(t). H(λ0, Y) is determined by initial value Y0 and Jacobian matrix ∂H/∂Y after Taylor expansion:(7)Hλ0,Y=Hλ0,Y0+∂H∂YY−Y0.After bringing equation (7) into (6), Y is calculated according to the next Newton iterative:(8)Y=Y0+∂H∂Y0−1Hλ0,Y−Hλ0,Y0,where ∂H/∂Y0 is obtained according to equation (8) by integrating T:(9)ddt∂Y∂Y0=F′Xt,λ0,Y⋅∂Y∂Y0.Whenλ alters, equation (6) is replaced by H(λ, Y) = 0. If n substep solution Yn is solved, the n + 1 substep begins with the prediction mode(10)Yn+1=Yn−∂Hλ,Y∂Yn−1⋅∂Hλ,Y∂λ⋅Δλ,λn+1=λn+Δλ,where ∂H(λ, Y)/∂λ is solved by the integration of equation (11):(11)ddt∂Y∂λ=F′Xt,λ,Y⋅∂Y∂λ+Fλ′t,λ,Y,with (∂Y/∂λ)t0 = 0 and (∂Y/∂λ)(t0+T) = ∂H(λ, Y)/∂λ.It ends with the following correction iteration which is based on equation (8) to solve Yn+1:(12)Yn+1=Yn+∂H∂Yn−1Hn+1−Hn.


## 3.1. Equations of the 3D Imprecise Rotor and System Reduction

When the bolt-disc rotor has the unbalance factor including mass eccentricity and rotor bending, the dynamic model of imprecise bolt-disc rotor system should take them into account.For the 3D bolt-disc rotor system, the DOFs are divided into nonlinear and linear component. DOFs belong to the bearing nodes (used to add bearing forces) and the disc nodes (used to add unbalance forces) are defined as nonlinear component; all the other DOFs belong to other nodes of the rotor are defined as linear component.For the nonlinear component, eigenmodes can reflect the dynamic characteristics of the linear system. As a result, the eigenvalues of the equations in the linear component are calculated and the first several eigenmodes are reserved.Then, dynamic equations are divided into linear and nonlinear sections:(2)MaaRMaiRMiaRMiiRu¨au¨i+GaaRGaiRGiaRGiiRu˙au˙i+KaaRKaiRKiaRKiiRuaui−rdardi=mω2e0+gagi+fB0,where MR and GR are mass matrix and gyroscope matrix of rotor; g is gravity force vector; subscripts “a” and “i” stand for nonlinear and linear DOFs respectively; and fB is vector of bearing forces which is calculated based on finite difference method [15]. The pad of cylinder bearing is discretized into many nodes in geometry. The total bearing force is regarded as the sum of hydrodynamic pressure which is solved by Reynolds equation.In addition, linear DOFs of dynamic equation (2) are reduced by a transformation matrix T by applying a component mode synthesis method [16]. As a result, the residual equations of imprecise rotor system are listed as(3)M0q¨+G0q˙+K0q=Q0,withq=Tu,Q0=TTKRrd+meω2+g+fBM0=TTMRT,G0=TTGRT,K0=TTKRT.For the reduced bolt-disc rotor system originated from Figure3(b), the reserved DOFs equal 27 which includes 3 disc nodes (9 DOFs), 2 bearing nodes (6 DOFs), and first 12 order eigenmodes (6 rigid eigenmodes and 6 elastic eigenmodes).


## 3.2. Nonlinear Solution Method

In order to obtain displacement and velocity, state parameterY is introduced:(4)dYdt=ddtqq˙=Ft,λ,Y,where Ft,λ,Y=q˙/M0−1Q0−K0q−G0q˙ and λ is the excitation variable such as ω, e, and rd.Sincee and rd meet the periodic feature when rotor rotating, the solutions of equation (4) are determined by(5)dYdt=fY,t,λYt=Yt+T,where T is the rotating period.Whenλ = λ0, equation (4) can be changed into solving target periodic solutions Y(t) according to nonlinear definition(6)Hλ0,Y=Yt+T−Yt=0.Here, the Newton iteration method is applied to approachY(t). H(λ0, Y) is determined by initial value Y0 and Jacobian matrix ∂H/∂Y after Taylor expansion:(7)Hλ0,Y=Hλ0,Y0+∂H∂YY−Y0.After bringing equation (7) into (6), Y is calculated according to the next Newton iterative:(8)Y=Y0+∂H∂Y0−1Hλ0,Y−Hλ0,Y0,where ∂H/∂Y0 is obtained according to equation (8) by integrating T:(9)ddt∂Y∂Y0=F′Xt,λ0,Y⋅∂Y∂Y0.Whenλ alters, equation (6) is replaced by H(λ, Y) = 0. If n substep solution Yn is solved, the n + 1 substep begins with the prediction mode(10)Yn+1=Yn−∂Hλ,Y∂Yn−1⋅∂Hλ,Y∂λ⋅Δλ,λn+1=λn+Δλ,where ∂H(λ, Y)/∂λ is solved by the integration of equation (11):(11)ddt∂Y∂λ=F′Xt,λ,Y⋅∂Y∂λ+Fλ′t,λ,Y,with (∂Y/∂λ)t0 = 0 and (∂Y/∂λ)(t0+T) = ∂H(λ, Y)/∂λ.It ends with the following correction iteration which is based on equation (8) to solve Yn+1:(12)Yn+1=Yn+∂H∂Yn−1Hn+1−Hn.


## 4. Dynamic Characteristics of Imprecise Bolt-Disc Rotor System

As a comparison, a same-sized monobloc rotor is analyzed simultaneously. The monobloc rotor’s radial-distributed bolts have no elongation, and all the discs are boned together. The monobloc rotor has the same mass eccentricity with the bolt-disc rotor, but it has no bending deformation because there is no assembly procedure for this rotor.


### 4.1. Stability and Bifurcation of the Imprecise Bolt-Disc Rotor

The global stability boundaries and areas of imprecise bolt-disc and monobloc rotor systems are demonstrated in Figure6.Figure 6
Global stability boundaries and areas.The global stability boundaries are determined by the eigenvalues (Floquet multipliers) of the Jacobian matrix ∂H/∂Y in equation (7) in the paper [11–14]. For a stable periodic solution, all the Floquet multipliers should be within the unit circle in the complex plane. The largest magnitude among all eigenvalues of the matrix is called the leading Floquet multiplier. Generally, the common double-periodic will happen when a real leading Floquet multiplier crosses the unit circle on the negative real axis. In addition, the quasiperiodic bifurcation will happen when a pair of complex conjugate leading Floquet multipliers crosses the unit circle.B1 and B2 stand for quasiperiodic (QP) and double-periodic (DP) solution boundaries for the imprecise bolt-disc rotor system. B'1 and B'2 mean QP and DP boundaries for the monobloc rotor system. P represents periodic solution area in which the rotor system has stable periodic orbits. The global stability areas of bolt-disc and monobloc rotor systems are composed of three parts: P, QP, and DP area.The curve (e + rd) and line (e) are extracted from the results of Figure 7. Curve (e + rd) and B1 have a crossing point v; line (e) and B'1 have a crossing point v′. If ω < ωv(ωv′), the bolt-disc (monobloc) rotor will perform P motions; if ω > ωv(ωv′), the bolt-disc (monobloc) rotor behave QP motions.Figure 7
Axis-center shift and mass eccentricity and of disc “b” with the position error of bolt-holes.It can be observed that P area of bolt-disc rotor is obviously smaller than that of monobloc rotor. It means that rotor flexure due to position error of bolt-holes has the lower system stability.


### 4.2. Nonlinear Vibration

The vibration in Figure8(a) indicates that both rotor systems almost have the same critical speed (ωc ≈ 4800 rpm), but its amplitude Ac of bolt-disc rotor is larger than that of monobloc rotor. Thus, speed-variant rotor bending hardly influences critical speed, but it can bring about larger vibration.Figure 8
Vibration for bolt-disc rotor and F-class gas turbine rotor: (a) vibration for bolt-disc rotor and (b) vibration for the F-class gas turbine rotor.
(a)(b)There is a distinctive feature here for imprecise bolt-disc rotor. Before balance, the amplitude rises again whenω > ωc; after balance (balance speed equals ωc), the vibration also grows and becomes larger than critical amplitude Ac when ω > ωc.This performance is also found in the vibration response of gas turbine rotor (Figure1) which is also a typical bolt-disc rotor. The data in Figure 8(b) are measured in the dynamic balancing process which is not influenced by gas and heat. Displacement sensors are used to measure shaft vibration. At a certain speed, the rotor stays for several minutes to detect the vibration value. Similarly, no matter whether the F-class gas turbine rotor has dynamic balance (balance speed equals its second critical speed 2200 rpm), the vibration rises again after ωc.The reason is clear when analyzing the vibration of imprecise bolt-disc rotor. Because the balance speed equalsωc to reduce its excitation Fc as much as possible, the balance vector Bc has to meet the following relation:(13)Bc=−me+KRrdωc2,Fc=Bc+me+KRrdωc2=0.When rotor has a balance vectorBc at critical speed, the vibration amplitude equals zero theoretically because the excitation is totally balanced at ω = ωc. However, for other speeds (ω ≠ ωc), the excitation vector F is rewritten as(14)F=Bc+me+KRrdω2=KRrdωc2ω2ωc2−ω2.As a result, the excitation at other speeds is larger thanFc. In addition, F becomes larger as speed increases when ω > ωc. Therefore, the vibration amplitude rises again after critical speed.This special phenomenon gives the reason why the position error of bolt-holes in the rotor is highly demanded. It also indicates this machining error should be examined carefully when designing this kind of combined rotor.


### 4.3. Nonlinear Vibration Characteristics of Imprecise Rotors

Figure6 shows that the imprecise bolt-disc and monobloc rotor bearing systems lose local stabilities at 8170 rpm and 7850 rpm. In Figures 9(a) and 9(b), the quasiperiodic motions of 13 nodes on the rotor axis are shown and the line connecting the orbits is the transient vibration mode of the deformed rotor.Figure 9
Dynamic motions for the both imprecise rotor: (a) quasiperiodic motion of bolt-disc rotor at 8170 rpm and (b) quasiperiodic motion of monobloc rotor at 7850 rpm.
(a)(b)Figure10(a) compares the whirling orbits of the both imprecise rotors at the node of bearing “k.” Figure 10(b) reveals that the motions of imprecise bolt-disc rotor and imprecise monobloc rotor are typically quasiperiodic because the corresponding Poincaré points form a closed curve.Figure 10
Whirling motions of the both imprecise rotors with position error at the bearing “k”: (a) rotor orbits at bearing “k” and (b) correspondent Poincaré map.
(a)(b)The fast Fourier transform (FFT) is performed to obtain the frequency spectrums of the imprecise rotors. Figure11 shows that both systems have the similar frequency features: harmonic frequency, oil-film instability frequency, and high frequencies.Figure 11
Frequency spectrums of both imprecise rotors with position error at the bearing “k.”Harmonic frequency (136.1 Hz and 130.8 Hz) is calculated according to the rotating speed; oil-film instability frequency (74.87 Hz and 78.48 Hz) approximately equals the half of harmonic frequency which originates from the Hopf periodic solution of the balanced rotor system and the effect of oil-film oscillation occurs to the rotor; the high frequencies stem from the quasiperiodic motion.


### 4.4. Allowable Position Error

There is an unchanged mass eccentricitye and speed-variant rotor bending rd in the bolt-disc rotor when the position error of bolt-holes exists among discs. Excessive e and rd will bring bad effects on the dynamic characteristics. For the sake of controlling these factors effectively, the maximum allowable error should be ensured.For the stability diagram (Figure12) of imprecise bolt-disc rotor and monobloc rotor, curve (e + rd) and line (e) gradually approach their stability boundaries when e and rd increase from 0 to working speed (7500 rpm). In this speed range, both rotors should work in the stable area (P area) all the time. So, the maximum allowable position error can be found out when curve (e + rd) and line (e) are tangent to boundary B2 and B'2, respectively.Figure 12
Allowable position error for both rotors: (a) allowable position error for bolt-disc rotor and (b) allowable position error for monobloc rotor.
(a)(b)The analysis results show that the maximum allowable position error of bolt-holes for bolt-disc rotor and monobloc rotor is 137μm and 417 μm. This tolerance also indicates a reference of bolt position precision when designing this combined rotor.


## 4.1. Stability and Bifurcation of the Imprecise Bolt-Disc Rotor

The global stability boundaries and areas of imprecise bolt-disc and monobloc rotor systems are demonstrated in Figure6.Figure 6
Global stability boundaries and areas.The global stability boundaries are determined by the eigenvalues (Floquet multipliers) of the Jacobian matrix ∂H/∂Y in equation (7) in the paper [11–14]. For a stable periodic solution, all the Floquet multipliers should be within the unit circle in the complex plane. The largest magnitude among all eigenvalues of the matrix is called the leading Floquet multiplier. Generally, the common double-periodic will happen when a real leading Floquet multiplier crosses the unit circle on the negative real axis. In addition, the quasiperiodic bifurcation will happen when a pair of complex conjugate leading Floquet multipliers crosses the unit circle.B1 and B2 stand for quasiperiodic (QP) and double-periodic (DP) solution boundaries for the imprecise bolt-disc rotor system. B'1 and B'2 mean QP and DP boundaries for the monobloc rotor system. P represents periodic solution area in which the rotor system has stable periodic orbits. The global stability areas of bolt-disc and monobloc rotor systems are composed of three parts: P, QP, and DP area.The curve (e + rd) and line (e) are extracted from the results of Figure 7. Curve (e + rd) and B1 have a crossing point v; line (e) and B'1 have a crossing point v′. If ω < ωv(ωv′), the bolt-disc (monobloc) rotor will perform P motions; if ω > ωv(ωv′), the bolt-disc (monobloc) rotor behave QP motions.Figure 7
Axis-center shift and mass eccentricity and of disc “b” with the position error of bolt-holes.It can be observed that P area of bolt-disc rotor is obviously smaller than that of monobloc rotor. It means that rotor flexure due to position error of bolt-holes has the lower system stability.


## 4.2. Nonlinear Vibration

The vibration in Figure8(a) indicates that both rotor systems almost have the same critical speed (ωc ≈ 4800 rpm), but its amplitude Ac of bolt-disc rotor is larger than that of monobloc rotor. Thus, speed-variant rotor bending hardly influences critical speed, but it can bring about larger vibration.Figure 8
Vibration for bolt-disc rotor and F-class gas turbine rotor: (a) vibration for bolt-disc rotor and (b) vibration for the F-class gas turbine rotor.
(a)(b)There is a distinctive feature here for imprecise bolt-disc rotor. Before balance, the amplitude rises again whenω > ωc; after balance (balance speed equals ωc), the vibration also grows and becomes larger than critical amplitude Ac when ω > ωc.This performance is also found in the vibration response of gas turbine rotor (Figure1) which is also a typical bolt-disc rotor. The data in Figure 8(b) are measured in the dynamic balancing process which is not influenced by gas and heat. Displacement sensors are used to measure shaft vibration. At a certain speed, the rotor stays for several minutes to detect the vibration value. Similarly, no matter whether the F-class gas turbine rotor has dynamic balance (balance speed equals its second critical speed 2200 rpm), the vibration rises again after ωc.The reason is clear when analyzing the vibration of imprecise bolt-disc rotor. Because the balance speed equalsωc to reduce its excitation Fc as much as possible, the balance vector Bc has to meet the following relation:(13)Bc=−me+KRrdωc2,Fc=Bc+me+KRrdωc2=0.When rotor has a balance vectorBc at critical speed, the vibration amplitude equals zero theoretically because the excitation is totally balanced at ω = ωc. However, for other speeds (ω ≠ ωc), the excitation vector F is rewritten as(14)F=Bc+me+KRrdω2=KRrdωc2ω2ωc2−ω2.As a result, the excitation at other speeds is larger thanFc. In addition, F becomes larger as speed increases when ω > ωc. Therefore, the vibration amplitude rises again after critical speed.This special phenomenon gives the reason why the position error of bolt-holes in the rotor is highly demanded. It also indicates this machining error should be examined carefully when designing this kind of combined rotor.


## 4.3. Nonlinear Vibration Characteristics of Imprecise Rotors

Figure6 shows that the imprecise bolt-disc and monobloc rotor bearing systems lose local stabilities at 8170 rpm and 7850 rpm. In Figures 9(a) and 9(b), the quasiperiodic motions of 13 nodes on the rotor axis are shown and the line connecting the orbits is the transient vibration mode of the deformed rotor.Figure 9
Dynamic motions for the both imprecise rotor: (a) quasiperiodic motion of bolt-disc rotor at 8170 rpm and (b) quasiperiodic motion of monobloc rotor at 7850 rpm.
(a)(b)Figure10(a) compares the whirling orbits of the both imprecise rotors at the node of bearing “k.” Figure 10(b) reveals that the motions of imprecise bolt-disc rotor and imprecise monobloc rotor are typically quasiperiodic because the corresponding Poincaré points form a closed curve.Figure 10
Whirling motions of the both imprecise rotors with position error at the bearing “k”: (a) rotor orbits at bearing “k” and (b) correspondent Poincaré map.
(a)(b)The fast Fourier transform (FFT) is performed to obtain the frequency spectrums of the imprecise rotors. Figure11 shows that both systems have the similar frequency features: harmonic frequency, oil-film instability frequency, and high frequencies.Figure 11
Frequency spectrums of both imprecise rotors with position error at the bearing “k.”Harmonic frequency (136.1 Hz and 130.8 Hz) is calculated according to the rotating speed; oil-film instability frequency (74.87 Hz and 78.48 Hz) approximately equals the half of harmonic frequency which originates from the Hopf periodic solution of the balanced rotor system and the effect of oil-film oscillation occurs to the rotor; the high frequencies stem from the quasiperiodic motion.


## 4.4. Allowable Position Error

There is an unchanged mass eccentricitye and speed-variant rotor bending rd in the bolt-disc rotor when the position error of bolt-holes exists among discs. Excessive e and rd will bring bad effects on the dynamic characteristics. For the sake of controlling these factors effectively, the maximum allowable error should be ensured.For the stability diagram (Figure12) of imprecise bolt-disc rotor and monobloc rotor, curve (e + rd) and line (e) gradually approach their stability boundaries when e and rd increase from 0 to working speed (7500 rpm). In this speed range, both rotors should work in the stable area (P area) all the time. So, the maximum allowable position error can be found out when curve (e + rd) and line (e) are tangent to boundary B2 and B'2, respectively.Figure 12
Allowable position error for both rotors: (a) allowable position error for bolt-disc rotor and (b) allowable position error for monobloc rotor.
(a)(b)The analysis results show that the maximum allowable position error of bolt-holes for bolt-disc rotor and monobloc rotor is 137μm and 417 μm. This tolerance also indicates a reference of bolt position precision when designing this combined rotor.


## 5. Conclusions

This work presents a static-dynamic uniting analysis to study the influences of position error of bolt-holes in a typical bolt-disc rotor system. The conclusions include the following:(1)
The deviation of position error of bolt-holes causes mass eccentricity and rotor bending due to tightening elongation and working speed for the imprecise disc.(2)
The unbalance component including mass eccentricity and rotor bending originating from position error of bolt-holes leads to obvious stability reduction and vibration growth.(3)
The vibration amplitude of imprecise bolt-disc rotor rises again whenω > ωc before (or after) dynamic balance. This property also exists in an F-class gas turbine rotor and is obviously different from monobloc rotor.(4)
The allowable position error of bolt-holes is obviously smaller than that of monobloc rotor. It shows that this assembly precision should be restricted to avoid position error as much as possible.



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*Source: 6639375-2021-05-07.xml*