^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

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.

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 (Figure

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 [

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 [

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 [

Taking the basic structure of compressor in the F-class gas turbine rotor (Figure

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). Table

Variables of the bolt-disc rotor bearing system.

Item | Variable | Value |
---|---|---|

Bolt-disc rotor | Diameter of shaft, _{s} | 0.08 m |

Length of shaft, _{s} | 1.25 m | |

Diameter of discs, _{d} | 0.24 m | |

Thickness of disc “a,” _{a} | 0.075 m | |

Thickness of disc “b,” _{b} | 0.1 m | |

Thickness of disc “c,” _{c} | 0.075 m | |

Number of bolts, | 12 | |

Tightening elongation, _{F} | 0.75 mm | |

Diameter of bolts, _{bolt} | 0.02 m | |

Diameter of circumference for bolts, _{bolt} | 0.156 m | |

Journal bearings | Span of bearing, _{b} | 1.05 m |

Radius, | 0.04 m | |

Width, _{j} | 0.08 m | |

Radial clearance, _{0} | 0.0002 m | |

Lubricant viscosity, | 0.0182 Pa s |

The position error of circumferential bolt-holes in this paper is shown in Figure

3D model of bolt-disc rotor with the position error of bolt-holes. (a) Imprecise disc “b” and (b) imprecise bolt-disc rotor.

The 3D imprecise disc and corresponding bolt-disc rotor are discretized by

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 [^{R} is the stiffness matrix of imprecise bolt-disc rotor; _{Λ} = ^{T}Λ_{Λ} = ^{T}Λ_{0} in which

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 Figure

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 eccentricity _{d}. Compared to a precise-machining rotor, the position error on disc “b” causes a constant _{d} in the working procedure which increases gradually up to 6.28

On the other hand, _{d} and

Then disc-center shift _{d} enables the combined rotor to have an entire bending _{d} whose vector is defined as _{d} = (_{1},…, _{d},…, _{k}) in which _{k} is similar with calculating _{d}. 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.

Rotor bending of imprecise bolt-disc rotor.

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:^{R} and ^{R} are mass matrix and gyroscope matrix of rotor; _{B} is vector of bearing forces which is calculated based on finite difference method [

In addition, linear DOFs of dynamic equation (

For the reduced bolt-disc rotor system originated from Figure

In order to obtain displacement and velocity, state parameter _{d}.

Since _{d} meet the periodic feature when rotor rotating, the solutions of equation (

When _{0}, equation (

Here, the Newton iteration method is applied to approach _{0}, _{0} and Jacobian matrix ∂

After bringing equation (_{0} is obtained according to equation (

When _{n} is solved, the _{t0} = _{(t0+T)} = ∂

It ends with the following correction iteration which is based on equation (_{n+1}:

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.

The global stability boundaries and areas of imprecise bolt-disc and monobloc rotor systems are demonstrated in Figure

Global stability boundaries and areas.

The global stability boundaries are determined by the eigenvalues (Floquet multipliers) of the Jacobian matrix ∂

B_{1} and B_{2} 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 (_{d}) and line (_{d}) and B_{1} have a crossing point _{1} have a crossing point

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.

The vibration in Figure _{c} ≈ 4800 rpm), but its amplitude _{c} 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.

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.

There is a distinctive feature here for imprecise bolt-disc rotor. Before balance, the amplitude rises again when _{c}; after balance (balance speed equals _{c}), the vibration also grows and becomes larger than critical amplitude _{c} when _{c}.

This performance is also found in the vibration response of gas turbine rotor (Figure _{c}.

The reason is clear when analyzing the vibration of imprecise bolt-disc rotor. Because the balance speed equals _{c} to reduce its excitation _{c} as much as possible, the balance vector _{c} has to meet the following relation:

When rotor has a balance vector _{c} at critical speed, the vibration amplitude equals zero theoretically because the excitation is totally balanced at _{c}. However, for other speeds (_{c}), the excitation vector

As a result, the excitation at other speeds is larger than _{c}. In addition, _{c}. 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.

Figure

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.

Figure

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.

The fast Fourier transform (FFT) is performed to obtain the frequency spectrums of the imprecise rotors. Figure

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.

There is an unchanged mass eccentricity _{d} in the bolt-disc rotor when the position error of bolt-holes exists among discs. Excessive _{d} 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 (Figure _{d}) and line (_{d} 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 (_{d}) and line (_{2} and B'_{2}, respectively.

Allowable position error for both rotors: (a) allowable position error for bolt-disc rotor and (b) allowable position error for monobloc rotor.

The analysis results show that the maximum allowable position error of bolt-holes for bolt-disc rotor and monobloc rotor is 137

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:

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.

The unbalance component including mass eccentricity and rotor bending originating from position error of bolt-holes leads to obvious stability reduction and vibration growth.

The vibration amplitude of imprecise bolt-disc rotor rises again when _{c} before (or after) dynamic balance. This property also exists in an F-class gas turbine rotor and is obviously different from monobloc rotor.

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.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest about this paper.

The work described in this paper was funded by a grant from the National Science Funds of China (no. 51705399) and Aeronautics Power Foundation (no. 6141B090547).