Multistage Asymmetric Rotors Coaxial Measurement Stacking Method Based on Minimization of Exciting Force

Abstract

The unbalanced exciting force of high-speed rotary asymmetric rotor equipment is the main factor causing rotor vibration. In order to effectively suppress the vibration of the asymmetric rotor equipment, the paper establishes a multistage asymmetric rotor coaxial measurement stacking method that minimizes the exciting force. By analyzing the propagation process of the centroid of the multistage asymmetric rotor assembly and analyzing the relationship between the geometric center and the centroid of a single asymmetric rotor, a multistage asymmetric unbalanced rotor propagation model based on geometric center stacking is established. The genetic algorithm is used to optimize the unbalance of the multistage asymmetric rotors. Combined with the vibration principle under the exciting force, the vibration amplitude of the left bearing at different rotation speeds under the minimization of the exciting force and the random assembly phase is analyzed. Finally, the experimental asymmetric rotors are dynamically measured, combined with the asymmetric rotors’ geometric error measurement experiment. The experimental results confirm that the vibration amplitude of the assembly phase with the minimum exciting force is smaller than the vibration amplitude under the random assembly phase at three-speed modes, and the optimization rate reached 73.2% at 9000 rpm, which proves the effectiveness of the assembly method in minimizing the exciting force.

1. Introduction

For a long time, whole machine vibrations have been a problem that is a typical failure that restricts the development of high-speed rotary equipment, and the unbalanced force generated by the unbalanced quantity when the asymmetric rotor system rotates at high speed is one of the main vibration sources of high-speed rotary equipment [1,2]. Under ideal conditions, the rotor is a precise symmetrical structure, but due to processing errors, the formed rotor will become an asymmetrical structure, which will affect the stability of the system. Simultaneously, high-speed rotary equipment asymmetric rotors belong to high-speed rotating machinery, and the maximum speed generally exceeds 10,000 r/min. If a failure occurs during operation, vibration information is currently recognized as the best judgment basis [2,3,4]. Vibration analysis plays an important role in the fault diagnosis of high-speed rotary asymmetric rotor equipment. According to statistics, about 60% of high-speed rotary equipment vibration problems are caused by asymmetric rotor imbalance [3,5,6,7]. The unbalance of the asymmetric rotors means that centroids of the asymmetric rotors deviate from their rotation centerline so that a certain centrifugal force will be generated when the asymmetric rotors rotate at a high speed [4,5,6]. This centrifugal force is transmitted to the bearing parts such as the gearbox through the bearing base, causing the equipment to vibrate [7,8,9]. Therefore, the study of high-speed rotary equipment vibration optimization is of great significance to its development and application.

There are two ways to reduce the eccentricity of the asymmetric rotors: reducing the manufacturing error of each asymmetric rotor segment and rationally assembling the asymmetric rotor segment to minimize the accumulated error. Assembly is the final stage of high-speed rotary equipment manufacturing, and it is also the core link. The quality of assembly directly affects the service performance of the equipment [8,10]. If the assembly accuracy cannot meet the requirements, it will cause vibration or friction between the asymmetric rotors during operation, and even major failures, which will affect the reliability and stability of the equipment [11]. Hussain et al. proposed a three-step method of direct assembly to obtain the smallest eccentricity [12]. Huang et al. used the Multi-Island Genetic Algorithm (MIGA) to optimize the maximum vibration amplitude encountered by the two asymmetric rotor discs when they cross the first critical speed [13]. Ding et al. proposed a multi-level rotation optimization assembly technology and a revolving part assembly deviation propagation model to achieve vibration suppression [14]. Sergio et al. used the receptance harmonic balance method (RHBM) to generate the reverse operator by measuring the vibration of equipment casing and used the least square method to generate the equivalent unbalanced distribution to reduce the unbalanced amount of asymmetric rotors [15]. Chen et al. developed an assembly error propagation model of multistage asymmetric rotors that take into account the alignment process and distribution of the screw holes of adjacent asymmetric rotors [16]. Kang et al. modified the Jacobian-Torsor model to construct multistage casing component deviations, which can simultaneously solve flexible deformation tolerances and geometric manufacturing tolerances [17]. Eissa et al. proposed a positive position feedback (PPF) controller to control the nonlinear vibration of a horizontally supported rotor system and a system model considering the nonlinear restoring force of the rotor and the weight of the rotor ws established [18]. Lusty et al. used a flexible inner stator active magnetic brake to achieve active vibration control of a flexible rotor and proposed an experimental device with a special topology [19]. Guan et al. used the active bump-type foil bearing with the advantage of actively controlling the mechanical preload in the rotor-GFBs system to achieve real-time suppression of rotor vibration [20]. Yang et al. deduced the motion control equation of the dual-rotor system based on the LaGrange equation, introduced a nonlinear energy sink in the LP rotor and realized the vibration suppression of the dual-rotor system [21]. Yun et al. used a nonlinear particle swarm optimization (PSO) method to solve the inverse problem, which reduced the identification complexity and achieved a good vibration suppression effect on the rotor [22].

Based on the error propagation model of multistage asymmetric rotors assembly, the paper proposes an optimization method for the multistage asymmetric rotors’ exciting force. The method analyzes the centroid eccentricity of multistage asymmetric rotors assembly with uniform mass and establishes the unbalanced exciting force propagation model when processing errors are considered. Based on the model, the exciting force optimization function of multistage asymmetric rotors is established, and the objective exciting force is optimized by a genetic algorithm. The optimization solution for the initial imbalance of the final multistage asymmetric rotors’ assembly is obtained, and the balance machine experiments using different assembly strategies are described. By controlling the assembly angle of each asymmetric rotor, the initial unbalance of the final assembly can be minimized, and the optimal rotation angles of different rotors can also be obtained.

2. Model

2.1. Unbalanced Exciting Force Propagation Analysis

During the assembly of multistage asymmetric rotors, the unbalance of each stage rotor is not only related to its own centroid eccentricity but also related to the geometric machining errors of the previously assembled rotors of the various stages. The geometric processing errors of the asymmetric rotors are propagated and amplified through the contact surface during the assembly process, which affects the centroid positions of the final multistage asymmetric rotors after assembly, and then affects the overall imbalance after assembly. In the actual assembly, the overall imbalance of the multistage asymmetric rotors after assembly can be effectively adjusted by rotating the rotors at various stages and changing the rotor installation phase.

If the machining error is not considered and the mass distribution is uniform, the centroid position is the same as the geometric center in the X and Y directions and is located at half the height of the rotor. The centroid propagation of the multistage rotors is shown in Figure 1. C₁, C₂, …, C_n−1, C_n are the geometric centers of the 1, 2, …, n − 1, n stages rotors, respectively; O₁, O₂, …, O_n−1, O_n are the centroids of the 1, 2, …, n − 1, n stages rotors, respectively. By rotating the rotors of various stages, the eccentricity between the centroids of the rotors and the axis of rotation can be effectively reduced, thereby reducing the overall imbalance of the rotors after assembly.

Based on the study of the unbalanced exciting force propagation process of the 3 stage rotors [10], the general law of the unbalanced exciting force propagation of the n stage asymmetric rotors assembly can be obtained. When n ≥ 3, the unbalanced exciting force propagation process is:

$$T_n^Q={\displaystyle \prod _{i=1}^{n-1}\left(T_i^R{T}_i^o\right)}{T}_n^R{T}_{nj}^Q$$

(1)

where ${\mathit{T}}_n^Q$ represents the transformation matrix of the unbalanced exciting force of the n-th stage rotor; ${\mathit{T}}_i^R$ represents the rotation matrix of i-th stage rotor; $T_i^o$ represents the unbalanced exciting force vector transformation matrix from the lower end surface of the i-th rotor to the upper-end assembly joint surface; $T_n^R$ represents the rotation matrix of n-th stage rotor; $T_{nj}^Q$ represents the unbalanced exciting force vector transformation matrix from the center of the lower end face of the n-th stage rotor to the j-th study section. The rotation matrix $T_i^R$ of i-th stage rotor is:

$${\mathit{T}}_i^R=\left[\begin{array}{cc}{\mathit{R}}_i^R& 0\\ 0^T& 1\end{array}\right]=\left[\begin{array}{cccc}\cos {\theta }_{ri}& -\sin {\theta }_{ri}& 0& 0\\ \sin {\theta }_{ri}& \cos {\theta }_{ri}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(2)

where ${\mathit{R}}_i^R$ is the rotation vector of the i-th stage rotor; θ_ri is the rotation angle of the i-th stage rotor about the Z-axis. If the rotation error of the geometric eccentricity is small enough, the transformation matrix of the unbalance exciting force vector of the i-th stage rotor is:

$${\mathit{T}}_i^o=\left[\begin{array}{cc}\mathit{d}{\mathit{R}}_i^o& \mathit{d}{\mathit{p}}_i^o\\ 0^T& 1\end{array}\right]=\left[\begin{array}{cccc}1& -\sin {\theta }_{zi}^{}& \sin {\theta }_{yi}^{}& d{x}_i^{}\\ \sin {\theta }_{zi}^{}& 1& -\sin {\theta }_{xi}^{}& d{y}_i^{}\\ -\sin {\theta }_{yi}^{}& \sin {\theta }_{xi}^{}& 1& z_i^{}+d{z}_i^{}\\ 0& 0& 0& 1\end{array}\right]$$

(3)

where ${\mathit{dR}}_i^o$ is the 3 × 3 rotation transformation matrix indicating i-th stage rotor is rotated about the Z-axis; ${\mathit{dp}}_i^o$ is the 3×1 position vector of the i-th stage rotor centroid; θ_xi is the rotation angle of the i-th stage rotor around the X-axis; θ_yi is the rotation angle of the i-th stage rotor around the Y-axis; θ_zi is the rotation angle of the i-th stage rotor around the Z-axis; d_xi is the component of the i-th stage rotor position translation vector along the X-axis; d_yi is the component of the i-th stage rotor position translation vector along the Y-axis; z_i+d_zi is the component of the i-th stage rotor position translation vector along the Z-axis.

Due to machining errors in actual machining processes, the rotor mass distribution is uneven. The unbalance of a single rotor is not only caused by machining errors but also caused by uneven mass distribution. ${\mathit{T}}_{nj}^Q$ is the unbalanced exciting force vector transformation matrix from the center of the lower end face of the n-th stage rotor to the j-th study section and its expression is:

$${\mathit{T}}_{nj}^Q=\left[\begin{array}{cc}\mathit{d}{\mathit{R}}_{nj}^Q& \frac{1-k_n}{k_n}\mathit{d}{\mathit{p}}_{nj}^Q\\ 0^T& 1\end{array}\right]=\left[\begin{array}{cccc}1& -\sin {\theta }_{zn}^Q& \sin {\theta }_{yn}^Q& \frac{1-k_n}{k_n}d{x}_n^Q\\ \sin {\theta }_{zn}^Q& 1& -\sin {\theta }_{xn}^Q& \frac{1-k_n}{k_n}d{y}_n^Q\\ -\sin {\theta }_{yn}^Q& \sin {\theta }_{xn}^Q& 1& \frac{1-k_n}{k_n}d{z}_n^Q\\ 0& 0& 0& 1\end{array}\right]$$

(4)

where ${\mathit{dR}}_{nj}^Q$ is the 3 × 3 rotation transformation matrix indicating i-th stage rotor is rotated about the Z-axis, $\mathit{d}{\mathit{p}}_{nj}^Q$ is the 3 × 1 position vector of the i-th stage rotor centroid; k_i represents the influence factor of the i-th stage rotor centroid, k ∈ [0, 1]. Using Equations (2)–(4) into Equation (1). When the n-th (3 ≤ n) stage rotor is assembled, the unbalanced exciting force propagation matrix of the n-th stage rotor assembly is:

$${\mathit{T}}_n^Q=\left[\begin{array}{cc}{\displaystyle \prod _{i=1}^{n-1}\left(R_i^R{R}_i^o\right)}& \frac{1}{k_n}{\displaystyle \prod _{i=1}^{n-1}\left(R_i^R{R}_i^o\right)}·R_{nj}^Q{p}_{nj}^Q+{\displaystyle \sum _{i=2}^{n-1}\left({\displaystyle \prod _{j=1}^{i-1}\left(R_j^R{R}_j^o\right)}{R}_i^R{p}_i^o\right)}+R_1^R{p}_1^o\\ 0^T& 1\end{array}\right]$$

(5)

Then the unbalanced exciting force vector $p_{\mathrm{nj}}^Q$ of the j-th study section of the n-th stage rotor is:

$$p_{nj}^Q=\left[\begin{array}{c}d{x}_{nj}^Q\\ d{y}_{nj}^Q\\ d{z}_{nj}^Q\end{array}\right]=\frac{1}{k_n}{\displaystyle \prod _{i=1}^{n-1}\left(R_i^R{R}_i^o\right)}·R_{nj}^Q{p}_{nj}^Q+{\displaystyle \sum _{i=2}^{n-1}\left({\displaystyle \prod _{j=1}^{i-1}\left(R_j^R{R}_j^o\right)}{R}_i^R{p}_i^o\right)}+R_1^R{p}_1^o$$

(6)

where ${dx}_{nj}^0$, ${dy}_{nj}^0$, ${dz}_{nj}^0$ are respectively the unbalanced exciting force vector components in the X, Y, and Z-axis directions, which are caused by the centroid deviation of the j-th research section of the n-th stage rotor with the center O₁ of the lower end face of the first stage rotor as the reference after the assembly of the n stage rotors.

In the vibration analysis of high-speed rotary equipment rotor system, the connection line between the center O of the lower end surface of the lowest stage asymmetric rotor and the center O_n of the upper-end surface of the highest stage asymmetric rotor is used as the reference axis. Therefore, when considering the vibration problems caused by the unbalanced exciting force due to the centroids deviation of the asymmetric rotors, it is necessary to obtain the centroid coordinates of each stage rotor in the coordinate system OX′Y′Z′. The coordinate system OX′Y′Z′ can be regarded as the coordinate system OXYZ by rotating θ around the rotation axis where the origin O is located. Let the rotation axis direction vector be l(l_x, l_y, l_z)^T and the unit vector w(w_x, w_y, w_z)^T = l/|l|. The rotation axis direction vector l and the rotation angle θ can be obtained by the vector transformation in the coordinate system OXYZ and OX′Y′Z′, respectively.

Suppose the position vector of the centroid of the n-th stage rotor in the coordinate system OX′Y′Z′ is (${d{x}^{\prime }}_{0-n}^c$, ${d{y}^{\prime }}_{0-n}^{ c}$,${d{z}^{\prime }}_{0-n}^{ c}$)^T. According to the coordinate transformation relationship, it can be concluded that:

$$\left[\begin{array}{c}d{x^{\prime }}_{nj}^Q\\ d{y^{\prime }}_{nj}^Q\\ d{z^{\prime }}_{nj}^Q\end{array}\right]=\left[\begin{array}{ccc}\cos \theta +w_x^2\left(1-\cos \theta \right)& w_x{w}_y\left(1-\cos \theta \right)-w_z\sin \theta & w_y\sin \theta +w_x{w}_z\left(1-\cos \theta \right)\\ w_z\sin \theta +w_x{w}_y\left(1-\cos \theta \right)& \cos \theta +w_y^2\left(1-\cos \theta \right)& -w_x\sin \theta +w_y{w}_z\left(1-\cos \theta \right)\\ -w_y\sin \theta +w_x{w}_z\left(1-\cos \theta \right)& w_x\sin \theta +w_y{w}_z\left(1-\cos \theta \right)& \cos \theta +w_z^2\left(1-\cos \theta \right)\end{array}\right]\left[\begin{array}{c}d{x}_{nj}^Q\\ d{y}_{nj}^Q\\ d{z}_{nj}^Q\end{array}\right]$$

(7)

When the rotational angular velocity of the rotor is Ω, the unbalanced exciting force Q_n of each stage rotor due to the centroids deviations in the coordinate system OX′Y′Z′ can be expressed as:

$$Q_{nj}=m_n{\mathsf{\Omega }}^2\left[\begin{array}{l}d{x^{\prime }}_{nj}^Q\\ d{y^{\prime }}_{nj}^Q\\ 0\\ 0\end{array}\right]\cos \mathsf{\Omega }t+m_n{\mathsf{\Omega }}^2\left[\begin{array}{l}-d{y^{\prime }}_{nj}^Q\\ d{x^{\prime }}_{nj}^Q\\ 0\\ 0\end{array}\right]\sin \mathsf{\Omega }t$$

(8)

where m_n represents the mass of the n-th stage rotor. Specify the two correction surfaces A and B on the n-th stage combined rotors, and project the unbalanced exciting force of the two research sections of each stage rotor to the A and B correction surfaces, then the unbalanced exciting force on the correction surfaces A and B is:

$$\{\begin{array}{l}{\mathit{Q}}_A={\displaystyle \sum _{i=1}^n\left({\displaystyle \sum _{j=1}^n\left(\frac{z_B-d{z^{\prime }}_{ij}^Q}{z_B-z_A}{Q}_{ij}\right)}\right)}\\ {\mathit{Q}}_B={\displaystyle \sum _{i=1}^n\left({\displaystyle \sum _{j=1}^n\left(\frac{d{z^{\prime }}_{ij}^Q-z_A}{z_B-z_A}{Q}_{ij}\right)}\right)}\end{array}$$

(9)

where Q_A represents the unbalanced exciting force on the correction surface A of the n-th stage combined rotors; Q_B represents the unbalanced exciting force on the correction plane B of the n-th stage combined rotors; z_A represents the height of the correction plane A position; z_B represents the height of the correction plane B position.

2.2. Genetic Algorithm (GA)

The genetic algorithm (GA) was first proposed by the American professor Holland in 1975 [23]. It is a kind of random search algorithm that draws on the natural selection and natural genetic mechanism of the biological world. The genetic algorithm simulates natural selection, genetic process selection, crossover, and gene mutation phenomena. Each iteration leaves a set of candidate solutions and selects better individuals according to the index. These individuals are combined by genetic operators to generate a new generation of candidate groups, and the process is repeated until the convergence index is met. Its specific operation process is shown in Figure 2.

Aiming at the problem of asymmetric rotor coaxial stacking, the phase of each asymmetric rotor is regarded as a gene, and the different genes of each stage rotor constitute an individual. Different individuals will correspond to the magnitude of the exciting force (each, all, or the maximum value) of the rotor in the assembled state. Assuming that n stage rotors coaxial stacking is optimized, the individual is I= (a₁, a₂, …, a_n), which a_n represents the number of assembly holes of the n-th stage rotor and the number of possible assembly combinations is ${\displaystyle \prod }_1^n{a}_n$.

Constrained nonlinear programming is used to optimize the objective function with equality or inequality constraints. In the paper, the constrained nonlinear programming model is used to optimize the unbalanced exciting force of the combined asymmetric rotors. The multistage asymmetric rotors assembly of high-speed rotary equipment has its particularity, and the overall unbalanced exciting force of the combined asymmetric rotors is an important parameter to evaluate the assembly quality. In addition, based on the requirements of the real high-speed rotary equipment asymmetric rotors assembly process, the assembly angle is limited by the number of bolt holes on each stage rotor flange joint surface. For example, when the number of bolt holes is 36, the minimum adjustment angle of assembly is 10°. The optimization objective of the combined rotors’ unbalanced exciting force is to minimize the rotors’ unbalanced exciting force on the premise of meeting the requirements of the position of the mounting hole. Taking the combined rotors’ unbalanced exciting force as the objective function and the number of bolt holes of the i-th stage rotor as the constraint condition, a nonlinear programming model for the constraint of the rotors’ unbalanced exciting force is established:

$$f({\theta }_{r1}^{},{\theta }_{r2}^{},\cdots ,{\theta }_{rn}^{})=\min \left(\max \left({\mathit{Q}}_A,{\mathit{Q}}_B\right)\right),{\theta }_{ri}^{}=(0^{\circ }{,}^{}\frac{{360}^{\circ }}{t_i}{,}^{}\dots {,}^{}{360}^{\circ }-\frac{{360}^{\circ }}{t_i})$$

(10)

where θ_ri is the installation adjustment angle of the i-th stage rotor; t_i is the number of mounting holes for the i-th rotor.

3. Simulation

The simulation takes a 6-stage asymmetric rotor as an example, the assembly phase of each stage asymmetric rotor is taken as input, the unbalanced exciting force of the asymmetric rotor system on the projection surface is the optimization target. The projection surfaces of the unbalanced exciting force are 100 mm and 500 mm respectively, and the inclination errors of each stage rotor are all 0.005 mm, eccentricity error is 0.02 mm, other parameters are shown in

Table 1. Simulation parameters of 6-stage rotors.

Rotor	1st Stage	2nd Stage	3rd Stage	4th Stage	5th Stage	6th Stage
Radius (mm)	86	285	354	290	248	80
Height (mm)	310	120	130	245	325.5	127
Mass (g)	8360	6990	12,140	19,560	53,110	4960

.

The genetic algorithm parameter mutation probability is set to 0.02, the population size is 2000, and 100 iterations are performed to find the assembly strategy for the minimum unbalanced exciting force of the rotor system. The genetic algorithm runs on an ordinary PC with I7 2.4 GHz 4 GB running memory. The average running time is about 30 s. Figure 3 shows the curve of the unbalance exciting force of the rotor system with iterations. The optimization results are recorded in

Table 2. Comparison of optimized assembly phase and random assembly phase.

Parameter	Random Assembly		Optimized Assembly
Assembly phase	[0,335,280,176,157,161]		[0,301,337,34,83,257]
Projection surface position (mm)	100	500	100	500
Unbalance Exciting Force of Projection Surface (g.mm)	1294.69	2988.72	169.47	195.29
Unbalance Exciting force Phase	−57.06°	−24.58°	−55.40°	34.81°
Vibration amplitude at the left fulcrum (μm)	324.59		43.63

. It can be seen that the assembly with minimal exciting force is optimized by nearly 93.2% compared with the exciting force under random assembly.

In order to meet the dynamic requirements of the high-speed rotary equipment after assembly, analyzing the vibration of high-speed rotary equipment is necessary. Since the left and right supports are assembled by the bearing and the shell, if the vibration of the supports is too intense, the bearing lubricating oil will be lost, which can cause dry friction of the bearing resulting in damage and failure of the bearing. In addition, the vibration will be transmitted to the shell, causing unpredictable vibrations of other parts. At the second-order critical speed, the vibration is the most obvious and the vibration amplitude is the largest. Therefore, we choose the left pivot point as the research object. At the second-order critical speed of 8200 rpm, the obtained vibration value is recorded in Table 2, which shows that the amplitude of this point is optimized by 86.6%.

Generally speaking, the working speed of high-speed rotary equipment is about 7500 rpm, and the vibration amplitude of the supports is smaller than that at the second-order critical speed. At the same time, the high-speed rotary equipment is not always at a constant working speed, and the vibration generated at each speed from start-up to working is different, so it is necessary to check the dynamic optimization effect of vibration at the fulcrum. Choose the two assembly phases in Table 1 and set the starting speed from 0 rmp to 12,000 rmp to simulate the vibration generated at each rotation speed at the fulcrum. Figure 4 shows the vibration diagrams at different rotation speeds at the left fulcrum of the two phases.

Figure 4 shows that the optimized vibration is much smaller than the vibration of random assembly, especially at the second-order critical speed. At this speed, the random assembly vibration amplitude is 43.63 μm, and the optimized assembly vibration amplitude is 324.59 μm, which the optimization is 86.6%. At the first-order critical speed, the random assembly vibration amplitude is 202.9 μm, and the optimized assembly vibration amplitude is 13.36 μm, which the optimization is 93.4%. The vibration at working speed is low, which the optimized assembly vibration amplitude is 18.45 μm and 86.2% better than that of the random assembly method.

4. Experiment

4.1. Geometrical Characteristics and Error Parameter Measurement of Rotor Experimental Parts

Multistage asymmetric rotors centroid propagation needs to be based on the geometric characteristics of each stage rotor, so it is first necessary to measure the geometric error parameters of the single-stage rotor. The rotating measuring instrument is composed of an air-floating turntable, a centering and tilting table, a chuck, an inductive sensor, and a guide rail, as shown in Figure 5.

The specific measurement process is as follows: first, start the air flotation turntable. When the turntable rotates at a stable speed, adjust the lower two inductive sensor probes to make contact with the radial and axial base surfaces of the front axle. Measure radial and axial base planes to determine the geometric axis of the front axle; adjust the centering and tilting table to ensure that the geometric axis of the asymmetric rotor coincides with the axis of rotation; When the air bearing turntable rotates at a stable speed, adjust the upper two inductive sensor probes to contact with the top surface of the front axle in the radial and axial directions. Measure errors such as eccentricity and perpendicularity of the radial and axial end faces of the front axle. The geometric characteristic parameters of each stage rotor components: upper-end face radius, lower end face radius, height, correction surface position, and geometric error parameters: the eccentricity error dx, dy, dz of the upper-end face in the X, Y, and Z-axis directions are recorded in

Table 3. The geometric characteristics and error parameters of the experimental rotor.

Part Name	Front Journal Combined Rotors	3rd Stage Rotor	Turbine	Rear Axle
Upper face radius (mm)	43.32	190.05	99.98	65.99
Radius of lower end surface (mm)	190.02	100.02	65.98	40.04
Height (mm)	561.02	244.93	325.42	124.55
Mass (g)	27558	19578	53112	4961
Eccentricity error dx (mm)	0.0054	−0.0008	0.0088	−0.0008
Eccentricity error dy (mm)	0.0189	0.0034	0.0039	−0.0013
Eccentricity error dz (mm)	0.0030	0.0022	0.0014	0.0025

.

4.2. Dynamic Vibration Measurement

In order to verify the effectiveness of the high-speed rotary equipment rotor vibration response prediction and vibration suppression method, a dynamic vibration measurement test bench was used to measure the vibration response of the experimental rotor. The dynamic vibration measurement test bench includes a basic platform, a speed-increasing drive system, a test system, a lubrication system, a supporting system, and high-speed rotary equipment experimental asymmetric rotor system, etc. The layout diagram is shown in Figure 6.

After the geometric error parameters of the single-stage asymmetric rotor are measured, the combined asymmetric rotor is assembled, and a measurement experiment platform is built in preparation for measuring the vibration response amplitude of the experimental asymmetric rotor. Install the assembled experimental asymmetric rotors on the experimental bench, respectively, to adjust the rotors centering state and install the rolling bearing oil supply and return oil circuits. Arrange the eddy current displacement sensors according to the experimental bench layout plan, connect the signal interface box and the data acquisition industrial control machine. Turn on the oil supply and return the motor to ensure a smooth cooling oil circuit and then turn on the drive motor. Use the inverter to control the motor speed to gradually increase the speed of the experimental rotor and measure the vibration response of the experimental rotor at different speeds. The rotation speed and vibration of the experimental asymmetric rotor will be measured by the eddy current displacement sensor. When the rotation speed reaches 12,000 rpm, the rotation speed is gradually reduced until the machine stops. Turn off the motor power supply and the inverter power supply and turn off the cooling oil circuit power supply.

4.3. Experimental Result

The data collected by the horizontal sensor at the position of the front stop of the turbine of the experimental asymmetric rotor is derived, and the measured vibration response amplitude of the experimental asymmetric rotor assembled with different assembly strategies at different speeds can be obtained. In order to verify the effectiveness of the combined asymmetric rotor vibration suppression method based on the optimization of stiffness and unbalanced exciting force, the actual measured value of the vibration response amplitude of the front end of the combined rotor turbine under the direct assembly and optimal assembly strategies are compared and recorded in Figure 7 and

Table 4. Comparison of the amplitude measurement results of the experimental rotor turbine front stop under different assembly strategies.

Strategies/Rotating Speed	3000 rpm	6000 rpm	9000 rpm
Optimal assembly	5.98 μm	24.59 μm	56.25 μm
Direct assembly	13.42 μm	60.34 μm	210.15 μm
Optimization effect	55.4%	59.2%	73.2%

.

According to Table 4, the vibration amplitude of the experimental rotor under the optimal assembly strategy is much smaller than that of the directly assembled experimental rotor, and it is reduced by more than 50% at most speed conditions. When the speed is 9000 rpm, the optimal assembly strategy reduces the vibration response amplitude by 73.2% compared with the direct assembly of the experimental asymmetric rotor. Therefore, the high-speed rotary equipment multistage asymmetric rotors coaxial stacking method with the minimization of the exciting force can greatly reduce the asymmetric rotor vibration.

5. Conclusions

The paper proposes an assembling method for high-speed rotary multistage asymmetric rotor equipment in which the geometric center replaces centroid. The parameters can be adjusted according to the actual asymmetric rotor shape to obtain the position of the asymmetric rotor centroid, which shows good rationality. This assembly method combines the genetic algorithm to optimize the assembly phase of vibration optimization, which greatly reduces the overall vibration of the asymmetric rotor equipment after assembly, and a lot of assembly time is saved.