A Vibration Response Prediction Model for Multi-Stage Assembled Rotors Based on Synchronous Excitation of Mass Eccentricity Error and Spigot Eccentricity Error

Abstract

The precise prediction of vibration response is crucial for optimizing the assembly quality of multi-stage rotors. Existing models possess two key limitations: they neglect the geometric displacement excitation from spigot eccentricity error and oversimplify rotor behavior by not accounting for the excitation redistribution caused by significant dynamic deflection at high speeds, particularly near critical speeds. To overcome these shortcomings, this study establishes a novel dynamic model based on the synchronous excitation of both mass and spigot eccentricity errors, which simultaneously incorporates the coupling mechanism of dynamic deflection. System dynamics equations are developed using a finite element approach combined with Timoshenko beam theory and solved via the Newmark-β method. Simulations and experiments on a four-stage rotor demonstrate that the proposed model provides significantly improved accuracy. At sub-critical, first, and second critical speeds, it reduces the maximum prediction error for nodal displacement amplitudes by 6.1%, 9.2%, and 36.4%, respectively, compared to a model neglecting dynamic deflection. Furthermore, analysis confirms that the targeted assembly error excitation exists solely at the fundamental frequency. The developed model, which uniquely integrates dual eccentricity sources with dynamic deflection coupling, is essential for reliable high-speed vibration prediction and assembly optimization, especially for flexible rotors operating near critical speeds.

1. Introduction

Vibration response serves as a critical indicator for evaluating the assembly quality of multi-stage rotors [1]. To suppress vibration, existing research primarily focuses on optimizing assembly angles to minimize errors introduced by manufacturing and assembly. These studies can be broadly categorized into three evolving stages, each with distinct objective functions and physical considerations.

The first stage centered on geometric tolerance analysis. Early research focused on controlling the geometric (spigot) eccentricity error of multi-stage rotors. Hussain, Yang et al. pioneered methods such as “straight-build assembly” and “parallel assembly” [2,3,4,5,6], which involve rotating assembly phases to align the geometric centers of individual stages into a nearly straight line or parallel end faces, thereby minimizing the radial and angular accumulated errors of the assembly. This work, based on statistical tolerance theory (e.g., Monte Carlo method [7,8]), provides guidance for tolerance design. However, its essence lies in evaluating error distribution within given tolerance zones and cannot directly yield a deterministic optimal assembly phase.

The second stage shifted towards direct optimization using overall geometric error as the objective. To further improve assembly precision, research introduced overall geometric metrics such as coaxiality as the objective function. Wang, Sun et al. minimized the coaxiality of multi-stage rotors by establishing measurement models and traversing assembly phases [9,10]. Subsequently, Sun, Ding et al. combined error propagation models with coaxiality objectives, achieving higher accuracy in assembly phase prediction and correction [11,12,13]. Nevertheless, a fundamental limitation persists in these methods: their optimization target (e.g., coaxiality measured relative to the first stage’s datum) is a static geometric quantity, not the dynamic response of the rotor under actual operating conditions. Consequently, they fail to accurately reflect the true vibration excitation caused by geometric eccentricity on the rotor system. Furthermore, they generally neglect the influence of mass eccentricity errors.

The third stage began incorporating dynamic factors, but significant simplifications remain. Recent studies attempted to establish a direct link between assembly phase and dynamic response. Liu, Sun et al. applied error propagation models to the transfer of mass eccentricity, using the resultant unbalance as the optimization target [14,15]. More notably, the work by Sun, Liu et al. [16,17,18] represents a significant advancement by integrating the multi-stage rotor assembly error propagation model with a rotor dynamics model for the first time. They solved for the optimal assembly phase using the vibration displacement at the bearing housing under operating speed as the objective function, marking an important transition from “geometric quantity optimization” to “dynamic quantity optimization”. However, the dynamic models employed in these pioneering studies remain relatively simplified, unable to fully capture the complete physical picture of assembly error excitation and the complex rotor behavior at high speeds. This is manifested in three key aspects:

• Incomplete Excitation Sources: Existing models only consider the excitation from the resultant mass eccentricity (unbalance force) after assembly, completely neglecting the geometric displacement excitation generated by spigot eccentricity [16,17,18]. Yet, both error types change simultaneously during assembly angle variation.
• Neglect of High-Speed Coupling Effects: Existing models fail to account for the significant influence of dynamic deflection at high speeds, particularly near critical speeds. When the rotor enters the flexible motion stage, the overall dynamic deflection causes a redistribution of all initial eccentricity errors, generating new dynamic unbalance and deflection—a coupled feedback process. Ignoring this effect severely limits the model’s predictive accuracy for high-speed vibration response, especially near critical speeds.
• Unclear Excitation Frequency Characteristics: The frequency characteristics of the vibration response excited by assembly errors lack clear analysis. This makes it difficult to effectively isolate the vibration components directly related to assembly optimization from complex experimental signals containing multiple fault frequency components (e.g., harmonics caused by misalignment or oil film instability [19,20,21]).

To fundamentally overcome the aforementioned limitations, this study establishes a novel dynamic model for multi-stage assembled rotors. The core innovations of this work are:

(1)

For the first time, the synchronous excitation from both mass eccentricity and spigot eccentricity is incorporated into a unified equation of motion, more accurately reflecting the nature of assembly error excitation;

(2)

The model systematically includes the excitation redistribution mechanism induced by dynamic deflection, enabling a more precise description of the rotor’s dynamic behavior throughout the transition from rigid to flexible motion, particularly near critical speeds;

(3)

It clarifies that the dual-eccentricity excitation exists solely at the fundamental frequency, providing a clear basis for signal processing in experimental validation.

The structure of this paper is as follows: Section 2.1.1 establishes the dynamic equation for a single node simultaneously subjected to both eccentricity errors and provides the formulas for excitation redistribution after considering dynamic deflection. Section 2.1.2 develops the system dynamic equations for the multi-stage rotor incorporating dynamic deflection. Section 3.2 compares the simulation results from the proposed model with those from a model neglecting dynamic deflection. Section 3.3 analyzes the vibration excitation characteristics of the dual eccentricity errors. Finally, Section 4 comprehensively validates the predictive effectiveness of the proposed model for both low-speed and high-speed vibration responses of a multi-stage assembled rotor through experiment.

2. Methods

2.1. Dynamic Equations of Multi-Stage Rotors Considering Dynamic Deflection

2.1.1. Synchronous Excitation from Mass and Spigot Eccentricity Errors in Multi-Stage Rotors

To clearly illustrate the physical mechanism of synchronous excitation from both mass and spigot eccentricity errors, and to derive the subsequent excitation redistribution formula, a simplified conceptual single-node rotor model is first established for analysis. This model serves as the starting point for theoretical derivation rather than for the dynamic calculation of the actual multi-stage rotor.

Figure 1 illustrates the motion for a single node of a rotor with simultaneous mass and spigot eccentricity. Points O and O’ denote the origins of two fixed spatial coordinate systems, OXY and O’X’Y’, representing the ideal rotational centers at two distinct axial references. M is the center of mass; ε is the mass eccentricity vector, with its angle to the horizontal X-axis being φ_ε; C is the spigot center, and C₀ is the geometric center after the disk deviates from the rotational center C due to initial bending; r_c is the initial eccentricity vector of the spigot center, with its angle to the horizontal X-axis being φ_c; r_d is the motion displacement vector of the spigot center during rotor operation; r is the total displacement vector of the disk center point, i.e., the resultant vector of r_c and r_d, with its angle to the horizontal reference direction being φ_r; p is the total displacement vector of the center of mass, i.e., the resultant vector of r and ε; the nodal mass is m; stiffness is K; and damping is η.

Based on the equilibrium relationship among the inertial force, damping force, and elastic force acting on the disk, the differential equation of motion for the disk center point can be derived as

$$m\ddot{\mathit{p}}+\eta \dot{r}+K{r}_d=0.$$

(1)

From the geometric relationships in Figure 1, we have

$$\left\{\begin{array}{l}p=r+\epsilon \hfill \\ r_d=r-r_c\hfill \end{array}\right..$$

(2)

Substituting p and r_d into Equation (1) yields

$$m\ddot{\mathit{r}}+\eta \dot{\mathit{r}}+Kr=K{r}_c-m\ddot{\epsilon }.$$

(3)

r, r_c, and ε are expressed in complex form as follows:

$$\left\{\begin{array}{l}r=r\exp \left(i\left(\omega t+{\phi }_r\right)\right)=r\cos \left(\omega t+{\phi }_r\right)+jr\sin \left(\omega t+{\phi }_r\right)=x_r+j{y}_r\hfill \\ r_c=r_c\exp \left(i\left(\omega t+{\phi }_c\right)\right)=r_c\cos \left(\omega t+{\phi }_c\right)+j{r}_c\sin \left(\omega t+{\phi }_c\right)=x_c+j{y}_c\hfill \\ \epsilon =\epsilon \exp \left(i\left(\omega t+{\phi }_{\epsilon }\right)\right)=\epsilon \cos \left(\omega t+{\phi }_{\epsilon }\right)+j\epsilon \sin \left(\omega t+{\phi }_{\epsilon }\right)=x_{\epsilon }+j{y}_{\epsilon }\hfill \end{array}\right.,$$

(4)

where ω is the rotational speed of the rotor. Substituting Equation (4) into Equation (3), the differential equation of motion for a single node of the rotor system is obtained:

$$\left\{\begin{array}{l}m{\ddot{x}}_r+c{\dot{x}}_r+K{x}_r=m{x}_{\epsilon }{\omega }^2+K{x}_c\hfill \\ m{\ddot{y}}_r+c{\dot{y}}_r+K{y}_r=m{y}_{\epsilon }{\omega }^2+K{y}_c\hfill \end{array}\right.,$$

(5)

where mx_εω² and my_εω² are the nodal excitation forces caused by rotor mass eccentricity error, and Kx_c and Ky_c are the nodal excitation forces caused by rotor spigot eccentricity error. Equation (5) can be expressed in complex form as

$$m\ddot{r}+\eta \dot{r}+Kr=m\epsilon {\omega }^2+K{r}_c.$$

(6)

Based on the Finite Element Method (FEM), assuming the rotor is discretized into a finite number of nodes and elements, the equation of motion for the n–th node can be expressed as

$$m{\ddot{r}}_n^{\omega }+\eta {\dot{r}}_n^{\omega }+K{r}_n^{\omega }=u_{jk}^n{e}_{jk}^n{\omega }^2+K{c}_j^n,$$

(7)

where $r_n^{\omega }$ is the dynamic deflection of the n–th node at speed ω. $u_{jk}^n$ represents the unbalanced mass present at the n–th node. $u_{jk}^n{e}_{jk}^n{\omega }^2$ is the excitation force due to mass eccentricity error from the j–th rotor stage acting on the n–th node. $K{c}_j^n$ is the excitation force due to spigot eccentricity error from the j–th rotor stage acting on the n–th node.

The Newmark-β step-by-step integration method [22] is employed to solve for $r_n^{\omega }$.

The nodal dynamic deflection at any time can be obtained by iteratively solving with the time step Δt. The nodal displacement at time t + Δt can be expressed as

$$K^{\ast }{\left(r_n^{\omega }\right)}_{t+\mathrm{\Delta }t}=R^{\ast },$$

(8)

where K* is the effective stiffness matrix (see Equation (9)) and R* is the effective load vector (see Equation (10)).

$$K^{\ast }=K+{\displaystyle \frac{m}{\gamma \mathrm{\Delta }{t}^2}}+{\displaystyle \frac{\beta \eta }{\gamma \mathrm{\Delta }t}},$$

(9)

$$\begin{array}{l}{R}^{\ast }=\left(u_{jk}^n{e}_{jk}^n{\omega }^2+k{c}_j^n\right)+m\left[{\displaystyle \frac{1}{\gamma \mathrm{\Delta }{t}^2}}{\left(r_n^{\omega }\right)}_t+{\displaystyle \frac{1}{\gamma \mathrm{\Delta }t}}{\left({\dot{r}}_n^{\omega }\right)}_t+\left({\displaystyle \frac{1}{2\gamma }}-1\right){\left({\ddot{r}}_n^{\omega }\right)}_t\right]\\ +\eta \left[{\displaystyle \frac{\beta }{\gamma \mathrm{\Delta }t}}{\left(r_n^{\omega }\right)}_t+\left({\displaystyle \frac{\beta }{\gamma }}-1\right){\left({\dot{r}}_n^{\omega }\right)}_t+\mathrm{\Delta }t\left({\displaystyle \frac{\beta }{2\gamma }}-1\right){\left({\ddot{r}}_n^{\omega }\right)}_t\right]\end{array},$$

(10)

where β and γ are integration parameters chosen based on the assumed variation in acceleration. The integration is unconditionally stable when γ ≥ 0.5 and β = 0.25 (0.5 + γ) ². This paper uses β = 0.7.

When the rotor transitions from the rigid motion stage to the flexible motion stage, especially near critical speeds, its dynamic deflection increases significantly, and its influence on the initial excitation cannot be neglected. The dynamic deflection of the entire rotor causes a redistribution of the mass and spigot eccentricity excitations, which in turn generates new dynamic unbalance and dynamic deflection (as shown in Figure 2). The equation of motion for a single node considering dynamic deflection $r_n^{\omega }$ can then be expressed as

$$m{\ddot{s}}_n^{\omega }+\eta {\dot{s}}_n^{\omega }+K{s}_n^{\omega }=K\left(c_j^n+r_n^{\omega }\right)+u_{jk}^n{\omega }^2\left(e_{jk}^n+r_n^{\omega }\right),$$

(11)

where $u_{jk}^n{\omega }^2\left(e_{jk}^n+r_n^{\omega }\right)$, $\left(c_j^n+r_n^{\omega }\right)$ and $s_n^{\omega }$ represent the dynamic unbalance, dynamic deflection, and nodal displacement, respectively, formed after the redistribution of mass and spigot eccentricity errors.

Assuming a multi-stage rotor has n shaft elements and n + 1 nodes, where the spigot eccentricity error of the j–th stage acts on node i, and the k–th mass eccentricity error of the j–th stage acts on node f. If the vector direction of the first mass eccentricity error is taken as the measurement direction for the X-direction vibration response, the phase difference between the nodal excitation generated by all other mass and spigot eccentricity errors can be expressed as

$$\left\{\begin{array}{l}{\theta }_{j\to 1}=\arccos \left({\displaystyle \frac{e_{11}\cdot c_j^i}{\left|e_{11}\right|\left|c_j^i\right|}}\right)\hfill \\ {\theta }_{jk\to 1}=\arccos \left({\displaystyle \frac{e_{11}\cdot e_{jk}^f}{\left|e_{11}\right|\left|e_{jk}^f\right|}}\right)\hfill \end{array}\right.\left(1\le i,f\le n+1,n\in {\mathrm{N}}^{\ast }\right).$$

(12)

The mass eccentricity excitation U_f at node f of the multi-stage rotor decomposed into the X and Y directions is

$$\left\{\begin{array}{l}{U}_{fx}=u_{jk}^f{\omega }^2\left|e_{jk}^f+r_j^{\omega }\right|\cos \left({\theta }_{jk\to 1}+\omega t\right)\hfill \\ U_{fy}=u_{jk}^f{\omega }^2\left|e_{jk}^f+r_j^{\omega }\right|\sin \left({\theta }_{jk\to 1}+\omega t\right)\hfill \end{array}\right..$$

(13)

The spigot eccentricity excitation F_i at node i of the multi-stage rotor decomposed into the X and Y directions is

$$\left\{\begin{array}{l}{F}_{ix}=\left|c_j^i+r_j^{\omega }\right|\cos \left({\theta }_{j\to 1}+\omega t\right)\hfill \\ F_{iy}=\left|c_j^i+r_j^{\omega }\right|\sin \left({\theta }_{j\to 1}+\omega t\right)\hfill \end{array}\right..$$

(14)

2.1.2. Dynamic Equations of the Multi-Stage Rotor System

Based on the Timoshenko beam theory [23], the rotor is discretized into a finite element model. The equation of motion for the shaft element adopts the standard form that accounts for shear deformation and rotational inertia:

$$\left[\begin{array}{cc}{M}_{en}& 0\\ 0& M_{en}\end{array}\right]\left\{{\ddot{q}}_{en}\right\}+\left(\omega \left[\begin{array}{cc}0& G_{en}\\ -G_{en}& 0\end{array}\right]+\left[\begin{array}{cc}{C}_{en}& 0\\ 0& C_{en}\end{array}\right]\right)\left\{{\dot{q}}_{en}\right\}+\left[\begin{array}{cc}{K}_{en}& 0\\ 0& K_{en}\end{array}\right]\left\{q_{en}\right\}=\left[\begin{array}{cc}{K}_{en}& 0\\ 0& K_{en}\end{array}\right]\left\{F_i\right\}+\left\{U_f\right\},$$

(15)

where q_en is the element displacement vector {x_n, θ_yn, y_n, −θ_xn, x_(n+1), θ_y(n+1), y_(n+1), −θ_x(n+1)}.

K_en is the element stiffness matrix (see Equation (A1) in Appendix A). M_en is the element mass matrix, M_en = M_Tn + M_Rn, where M_Tn and M_Rn are the translational and rotational inertia matrices, respectively (see Equations (A2) and (A3) in Appendix A). G_en is the element gyroscopic matrix, G_en = 2M_Rn. C_en is the element damping matrix. A stiffness-proportional viscous damping model is adopted, where the damping matrix is proportional to the stiffness matrix, i.e., C_en= μK_en. Here, μ is the stiffness-proportional damping coefficient, set to μ = 0.001, which represents a typical hysteretic loss factor for steel and is commonly used in rotor dynamics simulations of steel shafts to model internal material damping [23].

Assembling the concentrated mass of all shaft elements, support elements, and nodal excitations yields the system equation of motion:

$$\left[M_s\right]\left\{\ddot{q}\right\}+\left[\omega G_s+C_s\right]\left\{\dot{q}\right\}+\left[K_s\right]\left\{q\right\}=\left[K_s\right]\left\{F_s\right\}+\left\{U_s\right\},$$

(16)

where q is the system displacement vector (see Equation (A5) in Appendix A). U_s is the excitation vector from all unbalanced mass eccentricity errors (see Equation (A6) in Appendix A). F_s is the excitation vector from all spigot eccentricity errors (see Equation (A7) in Appendix A). M_s is the 4(n + 1) × 4(n + 1) system mass matrix (see Equation (A8) in Appendix A). K_s is the system stiffness matrix (see Equation (A10) in Appendix A). ω[G_s + C_s] is the system rotational matrix including damping (see Equation (A12) in Appendix A). The system displacement solution equation updated for iterative solution with time step Δt is

$${\left\{q\right\}}_{t+\mathrm{\Delta }t}=\left[R^{\ast }\right]{\left[K^{\ast }\right]}^{-1}.$$

(17)

The iterative solution procedure based on Equation (17) requires consideration of numerical convergence. The chosen parameters for the Newmark-β method (β = 0.7, γ = 0.5) satisfy the condition for unconditional stability, ensuring that the time integration remains stable regardless of the chosen time step Δt. The time step Δt was selected as 1 × 10⁻⁵ s to adequately resolve the system’s dynamic response, being significantly smaller than the period corresponding to the highest frequency of interest (twice the second critical speed, approximately 400 Hz). Convergence to the steady-state periodic response was assumed when the relative difference in the displacement vector norm between two consecutive rotational periods fell below a tolerance of 1 × 10⁻⁶. All simulations performed in this study met this criterion successfully.

3. Simulation

3.1. Finite Element Model of a Four-Stage Assembled Rotor

Figure 3 annotates the key dimensions and node divisions of the finite element model for the four-stage assembled rotor. This rotor model is based on a real aero-engine high-pressure rotor system. It is simplified to a 1:1 scale in dimensions, mass, and inertia, constituting a multi-stage rotor assembled sequentially from four components: the front shaft (Rotor 1), compressor (Rotor 2), high-pressure turbine (Rotor 3), and rear shaft (Rotor 4). The rotor material is defined as steel, with an elastic modulus of 210 GPa and a Poisson’s ratio of 0.3. The selectable assembly angles for each individual stage (where θ_zi denotes the assembly angle for the i–th stage rotor) are consistent with the settings established for the scaled model in Reference [24]. The supporting bearings on both sides are assigned the following stiffness and damping coefficients: K_xx = K_yy = 8 × 10⁷ N/mm, K_xy = K_yx = 0, C_xx = C_yy = 8 × 10⁴ N·s/mm, and C_xy = C_yx = 0. These properties are applied to Nodes 6 and 39, respectively.

Table 1. Mass eccentricity errors for the individual rotor stages.

Rotor Stage No.	Unbalanced Mass Point Number	γjk (mm)	ljk (mm)	ujk (g)	φjk (°)
1	1	70	100	5	0
	2	62	244	5	0
2	1	188	16	5	0
	2	188	306	5	0
3	1	121	39	5	0
	2	261	324	5	0
4	1	62	50	5	0
	2	70	170	5	0

and

Table 2. Spigot eccentricity errors for the individual rotor stages.

Rotor Stage No.	cj (mm)	θj (°)	pj (mm)	hj (mm)	δj (°)	dj (mm)
1	0.02	0	0.02	315	0	72
2	0.02	90	0.02	410	90	150
3	0.02	180	0.02	338	180	53.5
4	0.02	270	0.02	270	270	70

list the mass eccentricity errors and spigot eccentricity errors, respectively, for the individual rotor stages used in the simulation. Their definitions and measurement methods are consistent with our previous study [24]. C_j and θ_j are, respectively, the eccentric distance and eccentric angle of the centroid of the upper assembly surface, whose coordinate vector can be expressed as C_j = [c_jcos(θ_j), c_jsin(θ_j), h_j]. δ_j is the sampling angle between the center of the calibrated screw hole and highest point. h_j is the vertical distance between the upper and lower assembly surfaces, and p_j is the parallelism error of the upper and lower assembly surfaces. d_j is the radius of the upper assembly surface [24]. φ_jk and γ_jk are the action phase and action radius of the kth unbalanced mass point of the j–th stage rotor, respectively. l_jk is the distance between the kth unbalanced mass point of the j–th stage rotor and the reference support [24]. The validated multi-stage rotor assembly error propagation model from Reference [24] is employed to calculate the resulting excitations at different assembly angles: the geometric eccentricity excitations (applied at Nodes 10, 22, and 34, corresponding to the three assembly spigots) and the mass eccentricity excitations (each rotor stage has two unbalanced masses, applied at Nodes 7, 8, 12, 17, 25, 33, 37, and 38).

3.2. Analysis of Nodal Vibration Response Considering Dynamic Deflection

For vibration suppression in multi-stage rotors, the primary objective is to mitigate vibrations at critical speeds, ensuring smooth traversal and preventing vibration exceedance. Therefore, the critical speeds of the four-stage assembled rotor were first calculated without considering any external excitation or specific assembly angles. The Campbell diagram (Figure 4) reveals that the first and second forward whirl modes occur at 777 rad/s and 1237 rad/s, corresponding to critical speeds of 7421 rpm and 11,810 rpm, respectively.

To independently validate the dynamics model established in this paper using high-fidelity commercial finite element software, the geometric model of the four-stage rotor was accurately reconstructed based on SolidWorks 2022. Rotor dynamics modal analysis was then performed using ANSYS Workbench 2022R1 to obtain the Campbell diagram and critical speeds of the model. Figure 5 presents the Campbell diagram plotted according to the commercial software analysis results, along with the mode shapes corresponding to the first and second critical speeds.

The analysis results indicate that the first and second critical speeds obtained from the commercial software are 7301 rpm and 12,045 rpm, respectively. Compared with the critical speeds calculated using the proposed model in Section 3 (7421 rpm and 11,810 rpm), the relative error is within 2%. This close agreement, from the perspective of an independent commercial software analysis, validates the accuracy of the multi-stage rotor dynamics model developed in this study in computing the system’s natural characteristics.

As shown in Figure 6, the steady-state motion trajectories of the six nodes (Nodes 6, 14, 15, 21, 27, and 39) at a rotational speed of 1000 rpm (v₁) were examined for the four-stage rotor under two assembly angle configurations. Two dynamics models were employed: one considering dynamic deflection $r_n^{\omega }$ (denoted by the symbol √ in the legend) and one neglecting it (denoted by the symbol × in the legend). Figure 7, Figure 8 and Figure 9 share the same underlying principle as Figure 6 but present the results at rotational speeds of 3710.5 rpm (v₂), 7421 rpm (v₃), and 11,810 rpm (v₄), respectively. The vibration displacement amplitudes of the six nodes at the aforementioned speeds, obtained using the two dynamics models for the two assembly angle configurations of the four-stage rotor, are recorded in

Table 3. Displacement amplitudes of the six nodes for the four-stage rotor under at {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°}.

Node	Displacement Amplitude at v1 (mm) × 10−4		Displacement Amplitude at v2 (mm)		Displacement Amplitude at v3 (mm)		Displacement Amplitude at v4 (mm)
	√	×	√	×	√	×	√	×
6	1.8028	1.8078	0.0027	0.0028	0.0091	0.0089	0.0093	0.0112
14	2.2209	2.2283	0.0036	0.0038	0.0177	0.0187	0.0064	0.0039
15	2.3061	2.3139	0.0038	0.0040	0.0197	0.0210	0.0114	0.0081
21	2.4304	2.4386	0.0040	0.0042	0.0220	0.0236	0.0117	0.0140
27	2.5506	2.5588	0.0042	0.0044	0.0238	0.0257	0.0246	0.0209
39	2.2707	2.2773	0.0037	0.0038	0.0214	0.0233	0.0289	0.0270

and

Table 4. Displacement amplitudes of the six nodes for the four-stage rotor under at {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°}.

Node	Displacement Amplitude at v1 (mm) × 10−4		Displacement Amplitude at v2 (mm) × 10−4		Displacement Amplitude at v3 (mm)		Displacement Amplitude at v4 (mm)
	√	×	√	×	√	×	√	×
6	8.2439	8.2581	12.0000	12.0000	0.0057	0.0063	0.0131	0.0132
14	5.4745	5.4880	8.6172	8.9944	0.0052	0.0058	0.0107	0.0102
15	4.3365	4.3501	7.1662	7.5401	0.0046	0.0051	0.0083	0.0077
21	3.6530	3.6667	6.1587	6.5079	0.0037	0.0041	0.0045	0.0038
27	4.8679	4.8792	7.1359	7.3664	0.0024	0.0023	0.0026	0.0032
39	6.9913	7.0023	9.6966	9.8593	0.0022	0.0016	0.0109	0.0121

, respectively.

As can be seen from Figure 6, during the rigid motion stage at v₁, the steady-state motion trajectories of the six nodes solved by the two dynamics models essentially coincide. The data in Table 3 and Table 4 also indicate that under the assembly angle configurations {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} and {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°}, the maximum displacement amplitude differences for the six nodes are 8.2 × 10⁻⁷ mm (occurring at Nodes 21 and 27) and 1.4 × 10⁻⁷ mm (occurring at Node 6), respectively. When the speed increases to the sub-critical speed of v₂, although the rotor has not yet entered the flexible motion stage, minor deviations in the motion trajectories of the six nodes solved by the two dynamics models are already observed, as shown in Figure 7. Under the assembly angle configurations {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} and {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°}, the maximum displacement amplitude differences for the six nodes are 0.0002 mm (occurring at Nodes 14, 15, 21, and 27) and 3.8 × 10⁻⁵ mm (occurring at Node 14), respectively. These values are higher than the corresponding differences observed at v₁ for the same assembly phase sequences.

When the speed further increases to the first critical speed of v₃, the rotor enters the flexible motion stage. As shown in Figure 8, significant deviations appear in the motion trajectories of the six nodes solved by the two dynamics models. Under the assembly angle configurations {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} and {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°}, the maximum displacement amplitude differences for the six nodes are 0.0019 mm (occurring at Nodes 27 and 39) and 0.0006 mm (occurring at Nodes 6, 14, and 39), respectively. These differences are significantly higher than those observed during the aforementioned rigid motion stage for the corresponding assembly angle sequences. When the speed increases to the second critical speed of v₄, the rotor is also in the flexible motion stage. As shown in Figure 9, significant deviations are again observed in the motion trajectories of the six nodes solved by the two dynamics models. Under the assembly angle configurations {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} and {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°}, the maximum displacement amplitude differences for the six nodes are 0.0037 mm (occurring at Node 27) and 0.0007 mm (occurring at Node 21), respectively.

It is noteworthy that when the rotor is not yet in the flexible motion stage, the vibration displacement amplitudes of the six nodes obtained from the model considering dynamic deflection are consistently greater than those from the model neglecting it, under both assembly phase sequences. At the first critical speed, under the {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} configuration, the model considering dynamic deflection yields larger vibration displacement amplitudes only at Nodes 14, 15, 21, 27, and 39 compared to the model neglecting dynamic deflection. Under the {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°} configuration, this occurs only at Nodes 6, 14, 15, and 21. At the second critical speed, under the {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} configuration, the model considering dynamic deflection yields larger amplitudes only at Node 6. Under the {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°} configuration, this occurs only at Nodes 6, 27, and 39.

The aforementioned results indicate that during the rigid motion stage, the generated dynamic deflection is minimal, causing insignificant changes to the magnitude and phase of the initial rotor eccentricity excitation. Consequently, the vibration responses solved by the two dynamics models exhibit only minor deviations. However, when the rotor enters the flexible motion stage, significant dynamic deflection arises under the influence of various mode shapes. This leads to a redistribution of the initial mass and spigot eccentricity errors within the rotor. The resultant pronounced changes in the input excitation therefore cause substantial deviations in the vibration responses predicted by the two models. Moreover, the nature of these deviations varies with changes in the assembly phase sequence and does not follow a fixed pattern. Hence, to solve the high-speed vibration response of multi-stage rotors—especially the response near critical speeds—the dynamics model that considers dynamic deflection should be adopted.

Based on the complete simulation data, a systematic quantitative assessment of the influence of dynamic deflection was conducted. To fully characterize its effect, the relative prediction error between the models considering and neglecting dynamic deflection, defined as $E_{rel}^i= \mid A_{\mathrm{neglect}}^i-A_{\mathrm{consider}}^i\mid /A_{\mathrm{consider}}^i$, was calculated for all six nodes (6, 14, 15, 21, 27, 39) at the four characteristic rotational speeds and for two assembly sequences: {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°} (Sequence A) and {θ_z2 = 90°, θ_z3 = 90°, θ_z4 = 90°} (Sequence B). Here, $i$ denotes a combination of node and assembly sequence.

The analysis reveals that the influence of dynamic deflection follows a clear and consistent pattern:

• Dominance of rotational speed: Regardless of the assembly sequence, both the mean and maximum values of $E_{rel}^i$ increase monotonically with rotational speed, closely linked to the dynamic characteristics of the rotor system. In the rigid motion stage (1000 rpm), all $E_{rel}^i$ values are below 1%, indicating a negligible influence.
• Significance in the critical speed region: $E_{rel}^i$ increases sharply near or at the critical speeds. For Sequence A at the first critical speed (7421 rpm), $E_{rel}^i$ across nodes ranges from 2.2% to 9.2%, with a mean of 5.8%. At the second critical speed (11,810 rpm), the range expands to 1.7–36.4%, with a mean of 15.1%. A significant increase in $E_{rel}^i$ is also observed for Sequence B within the critical speed regions.
• Determination of the effect threshold: Statistical analysis of all data points indicates that when the operational speed exceeds approximately 60% of the first critical speed (corresponding to the sub-critical speed of 3710.5 rpm in this study), $E_{rel}^i$ generally surpasses 2%. When the speed exceeds about 85% of the first critical speed, E_rel consistently exceeds 5%, marking the point where the influence of dynamic deflection enters a range that must be considered in engineering practice.

The significance of dynamic deflection’s influence on the vibration response prediction of multi-stage rotors is primarily governed by the rotational speed and the system’s critical characteristics, rather than by a specific assembly sequence. The quantitative data provided in this study solidly supports the following engineering guideline: For multi-stage rotor systems that need to operate stably within or traverse through speed ranges exceeding 60% of their first critical speed, particularly near the critical speeds, it is essential to employ a dynamic model that accounts for the coupling effect between dynamic deflection and excitation redistribution, as established in this work. This ensures the reliability of vibration prediction and assembly optimization results. This conclusion reinforces the necessity of moving beyond traditional rigid or quasi-static assumptions in the dynamic analysis of high-speed flexible rotors.

3.3. Analysis of Vibration Excitation Characteristics from Mass Eccentricity Errors and Spigot Eccentricity Errors

The dynamics model considering dynamic deflection, as described above, continues to be employed. Three distinct excitation forms are configured separately: Form 1—geometric excitation from spigot eccentricity error only; Form 2—mass excitation from mass eccentricity error only; Form 3—simultaneous excitation from both aforementioned sources. The rotational speed is set to v₃, and the assembly angle sequence is {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°}. The vibration responses at Nodes 6 and 39 are selected for time-frequency analysis. Figure 10 and Figure 11 present the X-direction vibration responses at Node 6 and Node 39, respectively. The frequency-domain curves indicate that the nodal vibration response induced by the synchronous excitation of mass and spigot eccentricity errors exists only at the fundamental frequency.

Two primary conclusions can be drawn from the aforementioned results. First, the geometric excitation caused by spigot eccentricity errors generates a non-negligible vibration response in the rotor. Therefore, it should not be omitted when constructing the dynamics model for a multi-stage rotor. Second, the vibration excitation from both mass and spigot eccentricity errors exists solely at the fundamental frequency. Consequently, when conducting vibration tests on a multi-stage rotor with different assembly angle sequences, only the fundamental frequency component of the system’s vibration response should be extracted, while vibration components in other frequency bands must be filtered out. Failure to do so would prevent an accurate evaluation of the vibration suppression effectiveness achieved by the assembly angle optimization process.

4. Experiment

4.1. Experimental Setup

A four-stage assembled rotor, identical to the one used in the simulations in Section 3, was manufactured from steel for experimental validation. The actual spigot eccentricity errors and mass eccentricity errors of the individual rotor stages were obtained following the measurement method described in Reference [24] (see

Table 5. Mass eccentricity errors for the individual rotor stages.

Rotor Stage No.	Unbalanced Mass Point Number	γjk (mm)	ljk (mm)	ujk (g)	φjk (°)
1	1	70	100	3.7	19
	2	62	244	5.1	67
2	1	188	16	21.5	204
	2	188	306	6.4	131
3	1	121	39	8.9	193
	2	261	324	12.2	297
4	1	62	50	2.6	144
	2	70	170	3.2	31

and

Table 6. Spigot eccentricity errors for the individual rotor stages.

Rotor Stage No.	cj (mm)	θj (°)	pj (mm)	hj (mm)	δj (°)	dj (mm)
1	0.0245	206	0.0263	315.1401	144	72.0240
2	0.0384	27	0.0408	410.1939	231	149.9801
3	0.0477	123	0.0357	337.9233	219	53.5226
4	0.0189	181	0.0230	270.5056	36	70.0195

). Subsequently, the geometric eccentricity excitations and mass eccentricity excitations at different assembly angles were calculated using the validated multi-stage rotor assembly error propagation model presented in the same reference. These measured parameters were then incorporated into the two dynamics models: the multi-stage rotor dynamics model considering dynamic deflection developed in Section 2 and the existing model neglecting dynamic deflection. The vibration displacement amplitudes of the six nodes for the four-stage rotor at the four rotational speeds (v₁, v₂, v₃, v₄) were solved. The effectiveness of the proposed model in this paper was validated by comparing these computational results with the experimental data.

As shown in Figure 12, a high-speed balancing machine served as the driving carrier for the experiment. The four-stage rotor was connected to the drive end via a flexible coupling and mounted on double-sided pendulum supports. All vibration test sensors were arranged according to the axial positions of the six nodes defined in the simulation in Section 3, and their signals were fed into a 16-channel data acquisition unit. Four orthogonally oriented vibration velocity sensors were installed near both bearing ends to measure the vibration response at Nodes 6 and 39, with their displacement output channels selected. Additionally, eight eddy current displacement sensors were orthogonally arranged at Nodes 14, 15, 21, and 27.

4.2. Experimental Results

The four-stage rotor was tested at constant speed under the four aforementioned fixed rotational speeds, with the default assembly angle sequence set to {θ_z2 = 0°, θ_z3 = 0°, θ_z4 = 0°}. Figure 13 presents the experimentally measured time-domain curves for the six nodes of the four-stage rotor at these four speeds. Based on the time-frequency characteristic analysis of the mass and spigot eccentricity excitations conducted in Section 3, which confirmed that their excitation of the rotor vibration response exists solely at the fundamental frequency, a band-pass filter was applied to extract only the fundamental frequency component from the measurement results of each sensor. As shown in Figure 13, the filtered displacement amplitudes are orders of magnitude lower than unfiltered displacement amplitudes. This significant discrepancy arises from the substantial low-frequency vibration of the sensor mounting fixtures during high-speed operation, which is not related to the rotor’s dynamic response. The eddy current sensors were mounted on fixtures inside the protective enclosure, which shared the same massive base as the bearing supports. Significant ground vibration transmitted through this base during high-speed rotation, coupled with airflow induced by the rotor, caused a low-frequency sway of the fixtures themselves. The amplitude of this fixture vibration was orders of magnitude larger than the actual rotor nodal displacement relative to the bearings. However, as established in Section 3.3, the targeted vibration excitation from assembly errors exists solely at the fundamental rotational frequency (1X). Therefore, a band-pass filter was applied to the raw signal to isolate this 1X component, effectively removing the large-amplitude, low-frequency fixture vibration and revealing the true rotor displacement response for comparison with the model predictions.

Table 7. Displacement amplitudes of the six nodes calculated by the two dynamics models and the measured data.

Node	Displacement Amplitude at v1 (mm) × 10−4			Displacement Amplitude at v2 (mm) × 10−4			Displacement Amplitude at v3 (mm) × 10−4			Displacement Amplitude at v4 (mm) × 10−4
	√	×	Experiment	√	×	Experiment	√	×	Experiment	√	×	Experiment
6	6	6	6	86	92	97	451	501	547	283	272	269
14	8	8	8	115	123	129	651	742	781	536	409	388
15	8	8	8	117	125	133	680	764	817	611	489	463
21	8	8	8	119	131	137	709	815	859	659	545	512
27	8	8	8	116	122	130	698	781	837	500	386	343
39	6	6	6	87	93	100	542	602	640	393	278	264

and Figure 14 document the displacement amplitudes of the six nodes calculated by the two dynamics models alongside the measured data. The results indicate that:

• At the rigid motion speed v₁, the vibration responses of all six nodes in the four-stage rotor are minimal. When the calculated results from both models and the experimental results are rounded to the same number of significant digits, no deviations are observed among the three sets of data.
• At the sub-critical speed v₂, the maximum prediction error for the six-node displacement amplitudes between the calculated results of the model considering dynamic deflection and the experimental results is 7.0%. In contrast, the maximum prediction error for the model neglecting dynamic deflection rises to 13.1%.
• At the first critical speed v₃, the maximum prediction error for the model considering dynamic deflection is 8.4%, whereas that for the model neglecting dynamic deflection increases to 17.6%.
• At the second critical speed v₄, the maximum prediction error for the model considering dynamic deflection is 12.5%, while the corresponding error for the model neglecting dynamic deflection rises significantly to 48.9%.

5. Discussion

This study addresses Problem 1 summarized in the Introduction, namely, the omission of the geometric vibration excitation on the rotor caused by spigot eccentricity error in existing models. Section 2.1.1 establishes the nodal dynamic equation that simultaneously incorporates both mass and spigot eccentricity errors.

To tackle Problem 2, i.e., the neglect of the influence of dynamic deflection on the high-speed vibration response of the rotor, Section 2.1.2 develops the system dynamic equations for the multi-stage rotor considering dynamic deflection. A comparison of the nodal vibration responses calculated by the two dynamics models (with and without considering dynamic deflection) is presented in Section 3. Both simulation and experimental results demonstrate that while the two models yield only minor deviations in the calculated nodal vibration responses during the rigid motion stage of the rotor—indicating their validity for predicting low-speed vibration responses—significant deviations emerge as the rotational speed increases into the flexible motion stage. The results from the model considering dynamic deflection align more closely with the experimental data. Therefore, the rotor dynamics model that accounts for dynamic deflection should be employed when solving for the high-speed vibration response of multi-stage rotors, particularly the response at critical speeds.

Regarding Problem 3, which concerns the lack of consideration for geometric excitation and the subsequent investigation into the vibration excitation characteristics of mass and spigot eccentricity in existing models, the simulation results from Section 3.3 reveal two key findings. First, the geometric excitation from spigot eccentricity generates a non-negligible vibration response and should therefore be included in the dynamics model of a multi-stage rotor. This finding validates the necessity of the work conducted in Section 2.1.1. Second, the vibration excitation from both mass and spigot eccentricity errors exists solely at the fundamental frequency. This indicates that the vibration response affected by the assembly optimization process is also confined to the fundamental frequency. For existing studies that consider only mass eccentricity, even if the fundamental frequency component is isolated from vibration test results, the effectiveness of altering assembly angles in optimizing the vibration response cannot be accurately assessed due to the presence of Problem 1. This is because the fundamental frequency component inherently includes the contribution from the geometric excitation.

6. Conclusions

This study establishes and validates a novel dynamic model for multi-stage assembled rotors that fundamentally advances the state-of-the-art in vibration prediction for assembly optimization. The core novelty and contributions are explicitly summarized as follows:

• Dual-eccentricity excitation model: for the first time, a nodal dynamic equation simultaneously incorporating both mass eccentricity and spigot (geometric) eccentricity as synchronous excitation inputs was established. This correctly captures the complete physical nature of assembly error excitation, addressing a critical omission in existing models which consider only resultant mass unbalance.
• Dynamic deflection coupling mechanism: the model systematically integrates the excitation redistribution mechanism induced by dynamic shaft deflection. This innovation enables a physically accurate description of the rotor’s transition from rigid to flexible motion, particularly near critical speeds, where this coupling effect becomes significant and was previously overlooked.
• Quantitative accuracy improvement for critical speeds: The proposed model provides substantially improved predictive accuracy for high-speed responses, which is vital for flexible rotors that must traverse critical speeds. For the four-stage test rotor, the model achieved maximum prediction errors of 7.0%, 8.4%, and 12.5% at sub-critical, first, and second critical speeds, respectively. These represent reductions in the maximum error by 6.1%, 9.2%, and 36.4% compared to a model neglecting dynamic deflection.

Future work will focus on defining a generalized dimensionless ratio between the dynamic deflection amplitude and initial assembly errors to establish a universal criterion for the significance of the excitation redistribution effect.