Numerical and Experimental Analysis for the Dynamics of Flawed–Machining Rod–Disk Rotor with Inner Misalignment

: The nonlinear dynamic effects of the misalignment between inner disks in a ﬂawed– machining rod–disk rotor is studied. Non–uniform stress distribution due to inner misalignment is obtained based on 3D static solution. The concomitant unbalances, including constant mass eccentricity and speed–variant rotor bending, are taken into account in the nonlinear dynamics. The dynamic results show that rotor bending leads to stability reduction and vibration growth. There is a distinctive feature in that the rotor’s vibration goes up again after critical speed. The maximum allowable inner misalignment is obtained according to its stability boundaries. An uneven tightening method is also presented to reduce adverse effects when the inner misalignment exists. Moreover, an experiment is designed to measure the vibration characteristics for the rod–disk rotor bearing system with inner misalignment. The results show that the theoretical result of vibration amplitude of the ﬂawed rod–rotor bearing system is basically consistent with the experimental value. It is also found that the precise rotor performs the periodic motion, but the ﬂawed rod–disk rotor exhibits the period– doubling orbit. This phenomenon proves that the ﬂawed rod–disk rotor loses stability more easily than the precise rotor due to inner misalignment. However, the amplitude of harmonic frequency components for the precise rod–disk rotor system is obviously larger than the ﬂawed rod–disk rotor system with inner misalignment. The peak value of the vibration amplitude increases when the inner misalignment becomes larger. On the whole, this work presents numerical and experimental analysis to study the dynamics of ﬂawed-machining rod–disk rotor with inner misalignment. It also establishes the relationship between centration precision and dynamic features.


Introduction
The rod-disk rotor has a special structure in which all rotational disks are combined by several rods. This kind of rotor has been used in some gas turbines and aeroplane engines. Due to flawed machining, there is usually centration error after rotor assembly which can cause inner misalignment between different rotational disks.
Misalignment is one of the most common faults in rotor systems. The current studies mainly consider the shaft-to-shaft misalignment as the common sources of vibrations. When two or more shafts are rigidly coupled together and supported by three or more bearings, the bearing reaction forces are sensitive to lateral misalignment [1][2][3]. Kumar et al. [4] proposed a novel trial misalignment approach to estimate the misalignment with a similar concept as the trial unbalance in the rotor balancing. Active magnetic bearing misalignment with the rotor has been investigated with residual misalignment and additional trial misalignment cases. Lu et al. [5] applied the proper orthogonal decomposition method to the dimension reduction of a dual rotor-bearing experiment rig which is similar to an aeroengine rotor. Liu et al. [6] built a rotor system with a misalignment fault experiment table and verified the effectiveness of this new index for detecting the slight misalignment fault of the rotor system. Abdou et al. [7] studied the dynamic response of fluid film bearing of finite length which is subjected to rotor misalignment. Srinivas et al. [8] presented the modelling, analysis, and identification of various system faults in the presence of angular misalignment in coupled rotor-train systems integrated with an auxiliary AMB support. Sawalhi et al. [9] shed light on vibration modelling and diagnosis of misalignment beyond traditional approaches. Li et al. [10] investigated the nonlinear dynamic behaviours of an unsymmetrical generator rotor system supported on the journal bearings. Fu et al. [11] investigated the propagation of bounded uncertainties in the dynamic response of a misaligned rotor using a Legendre collocation based non-intrusive analysis method. Pennacchi et al. [12] aimed to model angular and parallel misalignment and ascertain pertinent diagnostic information. The proposed method is suitable for every type of shaft-line supported by journal bearings. Verma et al. [13] inspected the different types of misalignments using diagnostic media such as stator current signature as well as rotor vibration signal, and found that current signature alone can predict the misalignment effect without the use of vibration signal. Tapia et al. [14] established the criteria for implementing a system for detecting and locating faults using state observers based on the mathematical model of a rotor bearing mass system. Mogal et al. [15] proposed an order analysis technique of vibration analysis for unbalance and misalignment fault diagnosis. The fault type and location are identified from phase and amplitude. Ma et al. [16] systematically investigated oil-film instability laws of an overhung rotor system with parallel and angular misalignments in the runup and run-down processes, aiming at the oil-film instability of the sliding bearings at high speeds.
It is obvious that the misalignment appears between two or more rotors in the present works. However, there is probably misalignment between different inner parts, such as rotational disks in a rod-disk rotor. When these disks do not have the same axis, there is definitely inner misalignment in this combined rotor.
In this paper, the inner misalignment between rotational disks in one rod-disk rotor is considered and investigated using a static-dynamic coupling calculating process. The nonlinear behaviours of the rod-disk rotor bearing system have been obtained by combining the 3D finite element method and rotor-dynamics analysis. The particular features of the flawed rod-disk rotor, such as stability, vibration, and dynamic balance, are studied, and some designing suggestions are proposed based on the analysis results. Moreover, the theoretical data of vibration amplitude of the flawed rod-disk rotor system are basically consistent with the experimental value.

Static Features of Flawed Rod-Disk Rotor with Inner Misalignment
Referring to the compressor structure of a gas turbine rotor, a similar rod-disk rotor is shown in Figure 1. Three contiguous disks are connected by several radial-distributed rods in this combined rotor. Each rod has identical elongation. Two axial assembly surfaces between the disks (red lines) are designed to be connected, and all rods are mounted to be against disk-holes (blue lines). The above assembly steps can ensure that the rotor has sufficient centration precision. This rod-disk rotor is supported by two cylindrical sliding bearings. The working speed for the rotor bearing system is 7500 r/m, and the other parameters are listed in Table 1.

Inner Misalignment and Static Solution
The inner misalignment is described in Figure 2. The middle disk includes two symmetrical grooves. It is assumed that radial assembly surfaces deviate from designed centre O to actual centre O 1 because flawed machining exists. Meanwhile, two other disks are also precisely machined. As a result, the inner misalignment appears in the rod-disk rotor because there is a centration error between middle disk and two other disks. The deviation distance h between O and O 1 is defined as the misalignment value. Moreover, three other sorts of interfaces exist in this rotor: (1) interfaces (orange) between disks; (2) interfaces (green) between rod-heads and disks; and (3) interfaces (pink) between holes and rods. When misalignment value equals 10µm, the 3D flawed rod-disk rotor can be meshed by 26,592 8-node elements. In order to make it recognizable, the inner misalignment value has been magnified 800 times in Figure 3. For a pair of contact surfaces in 3D static model, the total potential energy E can be calculated according to reference [17].
When ∂E/∂u = 0, the static analysis method can be carried out by solving the following equations for the flawed rod-disk rotor.
As the contact state varied slowly, the above iteration is applied to calculate contact equations and the static stress can be obtained.

Static Analysis of the Flawed Rod-Disk Rotor
The static results of precise rotor are introduced to be a comparative reference. Figure 4 shows that uneven stress and rotor bending appear on the rod of flawed rotor; meanwhile, the corresponding rod in precise rotor has uniform stress distribution at its tensile region. This is because radial disk-holes have downward displacements and their surfaces press rods when the radial assembly surfaces have deviation.   Rod deformation triggers disk-centre shift (z d ) for flawed disks ( Figure 6). In the assembly procedure, z d equals 0.27 µm and inner misalignment causes a mass eccentricity (e d = 9.22 µm). When rotating speed goes up, e d keeps constant but z d increases gradually up to 3.39 µm. Meanwhile, the precise disk hardly has disk-centre shift. Therefore, the accuracy is sufficient to calculate the static stress and contact state for the flawed rod-disk rotor with inner misalignment. Every node of the shaft and disks has a disk-centre shift z d . As a result, the rod-disk rotor has an overall bending y d = (z 1 , . . . , z d , . . . , z k ) which is shown in Figure 7. Initial bending is formed in the assemble process. Moreover, the bending value becomes larger during the working period under the influence of centrifugal force.

Nonlinear Equations and Solving Algorithm
For the flawed rod-disk rotor with inner misalignment, the dynamic equations should consider an additional rotor bending on the base of traditional factors. Furthermore, the dynamic equations are divided into linear and nonlinear components for the sake of reducing degrees of freedom and adding forces on the nonlinear DOFs.
in which nonlinear sections have m DOFs and linear sections have n DOFs. Then, the residual flawed rotor bearing system can be written as Equation (4)[18] after carrying out the DOFs reduction process based on transformation matrix P. After reducing the equations, the total DOFs of residual flawed rotor bearing system number 27, composed of 15 nonlinear DOFs (from three disk nodes and two bearing nodes) and 12 linear eigenmodes.
When unbalance force me d ω 2 and rotor bending y d change under the rotating period T, the solutions of dynamic equations can be calculated based on the variable X which contains displacement and speed simultaneously Nonlinear description is applied to define the periodic solutions X(t) at the whirling state whose excitation parameter λ equals to me d ω 2 and y d .
Therefore, Newton's iterative method can be used to approach X(t) when λ = λ 0 based on Taylor's expansion.
In different working conditions, the n + 1 iterative substep is calculated based on n substep solution X n .

Dynamic Features of Flawed Rod-Disk Rotor System
A common integral rotor only has mass eccentricity in the manufacturing process but it does not have bending deformation due to assembly error. There will definitely be different dynamic features for the integral and rod-disk rotor bearing system. In this part, two kinds of rotor with identical size are analysed to find out the essential discrepancy. Figure 8 shows the global stability features of both rotor bearing system. Periodic, quasi-periodic, and period-doubling solutions are defined as P, QP, and DP areas, respectively. B 1 -B 2 (B' 1 -B' 2 ) lines represent the boundaries of quasi-periodic and period-doubling solutions for both rod-disk and integral rotor systems. The curves of (e d + y d ) and e d , which come from the results of disk-centre shift, pass through the lines of B 1 and B' 1 at the crossing points v and v'. It is found that the flawed rod-disk (integral) rotor performs periodic motions when ω < ω v (ω v' ) and exhibits quasi-periodic motions when ω ≥ ω v (ω v' ).

Global Stability Features
It can be seen that the area of stable solutions (P area) is much smaller than that of the integral rotor. It is proven that the stability of rod-disk rotor system decreases significantly due to inner misalignment between disks.

Nonlinear Vibration
The characteristics of nonlinear vibration for both rod-disk rotor and integral rotor are shown in Figure 9a. It is shown that two kinds of rotors have almost identical critical speed ω c which equals roughly 4800 r/m. However, the vibration amplitude of the integral rotor is obviously smaller than that of flawed rod-disk rotor. This indicates that the speedvariant rotor bending has little influence on the critical speed, but it can lead to larger vibration amplitude due to larger unbalance pulls. Moreover, a special feature can be seen that the amplitude of flawed rod-disk rotor increases again after the critical speed ω c no matter whether the dynamic balance is carried out. This phenomenon is also found for a gas turbine (a typical rod-disk rotor) in the dynamic balancing process, whose experimental measuring data are shown in Figure 9b without heat and gas. In the same way, the vibration of the gas turbine increases again regardless of whether dynamic balance happens after the critical speed 2200 r/m, and the vibration amplitude becomes much smaller than at other velocities after the balance speed.
As to the reason the vibration of the flawed rod-disk rotor will rise after dynamic balance, the balance vector f c can be written as the following formula in order to reduce its unbalance component as much as possible at the critical speed ω c .
Obviously, the vibration amplitude theoretically equals zero after adding balance vector f c when ω = ω c . However, the unbalance excitation vector F is much larger than F c when ω > ω c . As the result, the vibration amplitude rises again after critical speed.
Therefore, it is significant that the centration precision of rod-disk rotor should be designed to a high quality in order to reduce the inner misalignment as much as possible.

Dynamic Motions
The dynamic motions of flawed rod-disk and integral rotors are described in Figure 10. Both rotor systems perform quasi-periodic motions when the rotating speed goes through ω v (8280 r/m) and ω v' (7940 r/m). The orbits of each selected node on axis form the dynamic motions of both rotors. Meanwhile, the comparison of dynamic motions and corresponding Poincaré maps are shown in Figure 11. Quasi-periodic orbits ensure Poincaré points to form a closed loop.
As both orbits are identified as quasi-periodic motions, they have similar frequency components ( Figure 12): working frequencies f ω (f ω = ω/60); Hopf semi-frequency f H (f H ≈ f ω /2), which is known as f ω /2 whirling; and another frequency, f QP , which comes from traditional quasi-periodic orbits. In addition, f ω = 138 Hz or 132.5 Hz for rod-disk rotor or integral rotor, and f H = 76 Hz.

Centration Tolerance
As excessive mass eccentricity e d and rotor bending y d will result in terrible effects on the dynamics of rod-disk rotor system, the inner misalignment should be controlled to an allowable extent. Figure 13 indicates the stability figure of both flawed rod-disk rotor and integral rotor. The curves of (e d + y d ) and e d will get close to the stability boundaries of B 2 and B' 2 , respectively, when e d and y d increase gradually. The allowable inner misalignment is defined to make the flawed rotor always perform stable response in P area in which the rotor has periodic motions. When the inner misalignment becomes an excessive value, the flawed rotor exhibits an unstable response which includes quasi-periodic and period-doubling motions. Therefore, the maximum allowable inner misalignment will appear when the (e d + y d ) curve (or e d curve) and the boundary B 2 (or B' 2 ) each have a tangency point in the speed range from 0 to 7500 rpm.
As a result, the maximum allowable inner misalignment for the flawed rod-disk rotor and integral rotor are 21 µm and 28 µm, respectively. This value can be used as a tolerance reference to design a rod-disk rotor.

Uneven Tightening
As tightening elongation is the essential reason rotor bending appears and increases, to cut its amount is a feasible way to reduce adverse effects on dynamic properties.
As Figure 2 shows, middle disk has a downward displacement and presses several rods. If the tightening elongation of No.4 rod reduces to some extent (2.5%-10%), there may be some differences.
After static-dynamic coupling analysis, it is seen that when elongation reduction ∆F equals 7.5%, rotor bending changes slightly as ω rises (Figure 14a). This condition will make dynamic balance much easier. As a result, rod-disk rotor has the lowest vibration amplitude (Figure 14b) at working speed after dynamic balance when ∆F = 7.5%. Therefore, rotor system can also have a relatively satisfying dynamic property by uneven tightening, even if inner misalignment exists. Figure 15 shows the structure of the rod-disk rotor which contains three disks which are fastened by six radial-distributed rods. Every rod has the same size and elongation and rods fit against disk-holes. A 20 µm deviation between the left disk and middle disk is artificially set during the assembly process which has been magnified 1000 times in Figure 15c. Two cylindrical sliding bearings are used to support the combined rotor ( Figure 16). The parameters of the shaft, rods, and bearings are shown in Table 2.

Test Rig System and Its Sensor Arrangement
The test rig system is composed of six parts ( Figure 17): drive motor, coupling, bearings, rod-disk rotor, test rig base, and lubrication structure. Moreover, a series of sensors are installed and debugged to monitor the rotor state and collect test data: two pairs of electric eddy current displacement sensors (C1 and C3) are arranged at the rotating shaft near the sliding bearing to measure the displacement data and establish the whirling orbits of shaft; one pair of displacement sensors (C2) are set to measure the displacement of disk; one displacement sensor (C4) is installed facing the end of the right disk to measure the axial displacement of rotor; two acceleration sensors (A1 and A2) are installed on the bearing seats at both ends to test the vibration state of test rig; one photoelectric sensor (B) is set to measure the rotor speed; and one temperature sensor is designed to measure the temperature of bearing.  Figure 18 shows that the theoretical result of vibration amplitude (at the wheel disk node) of the flawed rod-rotor bearing system is basically consistent with the experimental value when the speed increases. The result shows that the critical speed of this rod-disk rotor bearing system with inner misalignment is about 2620 rpm. The disagreement between the simulation and experimental data may stem from the calculation factor: the theoretical damping of oil film is larger than the real bearing which leads to the fact that the peak value of simulation is smaller than the experimental results; the theoretical stiffness of oil film is smaller than the real bearing, which results in the fact that the calculating critical speed is smaller than the experimental data.  Figure 19 demonstrates the whirling orbits of the precise and flawed rotor before (2250 rpm) and after (2850 rpm) the first critical speed. It is found that the precise rotor performs the periodic motion but the flawed rod-disk rotor exhibits the period-doubling orbit. This phenomenon proves that the flawed rod-disk rotor loses stability more easily than the precise rotor due to inner misalignment. Moreover, the range of period-doubling orbit is obviously larger than the periodic orbit. It shows that the flawed factor leads to large vibration for the rod-disk rotor bearing system. The vibration amplitude in time domain and its frequency spectrum are shown in Figure 20 at 2250 rpm. Besides the main frequency (37.5 Hz), the other harmonic frequency components (such as 75Hz, 112.5hz, 150Hz, and 187.5hz) also exist for both precise and flawed rod-disk rotor bearing system. However, the amplitude of harmonic frequency components for the precise rod-disk rotor system is obviously larger than the flawed rod-disk rotor system with inner misalignment. It also proves that the error of inner misalignment can bring larger and more complicated vibration to the rod-disk rotor system.  Figure 21 shows the overall trend of the vibration amplitude within the speed range of 800~3000rpm for rod-disk rotor systems within different inner misalignments. It can be seen that the peak value of the vibration amplitude increases when the inner misalignment becomes larger. After the critical speed, the vibration amplitude decreases rapidly.

Discussion
In order to achieve a real rotor structure, rotor models become more and more complicated for the increasing demand of engineering application when investigating the nonlinear dynamic features of rotor system.
At first, the nonlinear models of rotor are often treated as a symmetrical rigid rotor [1,2,4,7,10] whose shaft is regarded as an absolutely rigid part and disk is regarded as a mass point. Then, the Jeffcott rotor model [8,14,19,20] is applied to analyse nonlinear dynamic problems in which the shaft's deformation is considered and the disk is treated as a rigid part. After that, a flexible rotor model is established by introducing a 1D element such as the famous Timoshenko and Euler element [3,5,9,11,21]. Obviously, the above three kinds of simplified rotor models cannot represent the original rotor system fully and some local structures such as holes and grooves cannot be described in detail.
So it will encounter difficulties when analysing the nonlinear dynamic characteristics of highly complicated rotors such as gas turbine rotor and aeroplane engine rotor. Therefore, 3D-element rotor models are built without much structural simplification [6,20,22]. However, the present 3D rotor models are integral rotors which do not have many parts to assemble together.
In comparison, the 3D rotor model presented in this paper is a complicated combined rotor which is closer to the applications in practice. On the other hand, the static-dynamic analysis process will definitely take much time to calculate, and its calculating efficiency is much lower. The comparison between other models and the proposed model in this paper is concluded in Table 3.

Conclusions
This paper investigates a static-dynamic solving process and an experimental test to analyse the effects of inner misalignment in a rod-disk rotor system. The conclusions include: (1) The deviation of radial assembly surfaces causes mass eccentricity for imprecise disks; the associated inner misalignment among disks brings about rotor bending, which leads to obvious stability reduction and vibration growth; the amplitude of flawed rod-disk rotor increases again after the critical speed ω c regardless of if the dynamic balance is carried out.
(2) The maximum allowable inner misalignment of a rod-disk rotor is obviously smaller than that of an integral rotor. This shows that the centration precision should be controlled precisely to reduce inner misalignment as much as possible; moreover, uneven tightening is a feasible way to reduce adverse effects on the dynamic properties when inner misalignment appears.
(3) The theoretical result of vibration amplitude of the flawed rod-rotor bearing system is basically consistent with the experimental value. It is also found that the precise rotor performs the periodic motion but the flawed rod-disk rotor exhibits the period-doubling orbit. This phenomenon proves that the flawed rod-disk rotor loses stability more easily than the precise rotor due to inner misalignment.
The proposed static-dynamic method in this paper has wide application usage in any assembly combined rotor such as gas turbine rotor, aeroplane engine rotor, electric generator rotor, etc. In order to overcome its disadvantage of low calculating efficiency, more attention should be to rapid solving algorithms in the future.