Numerical Computation and Experimental Research for Dynamic Properties of Ultra-High-Speed Rotor System Supported by Helium Hydrostatic Gas Bearings

Abstract

This study delves into the dynamic behavior of ultra-high-speed rotor systems underpinned by helium hydrostatic gas bearings, with a focus on the impact of rotational velocity on system performance. We have formulated an integrative dynamic model that harmonizes the rotor motion equation with the transient Reynolds equation. This model has been meticulously resolved via the Finite Difference Method (FDM) and the Wilson-Θ technique. Our findings unveil intricate nonlinear dynamics, including 2T-periodic and multi-periodic oscillations, and underscore the pivotal role of first-order temporal fluctuations, which account for over 20% of the transient pressure at rotational speeds exceeding 95.0 krpm. Further, we have executed empirical studies to evaluate the system’s performance in practical settings. It is observed that when the ratio of low-frequency to fundamental frequency approaches 0.3 and the amplitude ratio exceeds 3, the vigilant monitoring of system stability and reliability is imperative. Collective insights from both computational simulations and experimental studies have enriched our understanding of the dynamic attributes of ultra-high-speed rotor systems. These revelations are crucial for the advancement of more efficacious and resilient rotor systems designed for high-speed applications.

1. Introduction

The bearing–rotor system is an essential component of cryogenic turbo-expanders, with its reliability and stability being paramount to the efficiency of cryogenic turbines. Aerostatic bearings, characterized by low friction, no oil contamination, high motion precision, and the capability to support high rotational speeds, have become prevalent in various sectors, particularly in cryogenic helium turbine applications [1,2,3,4,5,6]. The low temperature of the incoming medium and the substantial expansion ratio, especially in smaller capacity plants, necessitate the use of small impeller diameters and significant specific enthalpy drops in the expander. To achieve optimal insulation efficiency, high rotor speeds are required, typically ranging from hundreds of thousands to millions of revolutions per minute (rpm) [7]. Despite this, the dynamics of ultra-high-speed bearing–rotor systems are not well-documented in the literature.

Since the early 1990s, gas film bearings have gained favor over traditional rolling contact and liquid film bearings due to their exceptionally low friction and minimal vibration transmission [8]. However, the compressibility and low viscosity of gas films can lead to instability and nonlinear vibrations in high-speed bearing–rotor systems, thus limiting the development of gas bearings for higher speeds [9]. Advances in computational power have enabled the application of nonlinear theories to gas lubrication studies. Researchers have been able to construct more precise bearing–rotor system models using numerical methods [10]. For example, Jiazhong Zhang et al. developed a gas bearing Jeffcott rotor system model and employed the Galerkin method to approximate the transient gas Reynolds equation, solving for the gas film pressure distribution at any given time, to simulate the system’s nonlinear dynamic behavior [11]. C-C Wang et al. introduced a novel hybrid approach combining the finite element method (FEM) with the differential transformation method to solve the compressible Reynolds equation, using the successive over-relaxation (SOR) method to analyze the system’s bifurcation behavior [12,13,14]. Yuta Otsu et al. numerically and experimentally investigated the instability of a rigid rotor supported by aerostatic journal bearings with compound restrictors, revealing a higher threshold speed for instability compared to inherently compensated bearings [15]. Tengfei Yin et al. conducted a nonlinear analysis of the stability and dynamic rotational accuracy of an unbalanced rotor supported by aerostatic journal bearings, utilizing the finite element method and the Runge–Kutta fourth-order method for the simultaneous solution of the transient Reynolds equation and rotor dynamics equations [16]. Jianbo Zhang et al. used bifurcation maps, orbits, Poincaré maps, and frequency spectrum diagrams to examine the effects of rotational speed, imbalance, rotor mass, and supply pressure on the nonlinear vibrations of rotor systems supported by aerostatic bearings, finding that increased supply pressure can mitigate nonlinear vibrations and enhance the stability of the rotor system [17].

The current body of research predominantly addresses the linear and nonlinear dynamics of rotor systems, employing hydrostatic journal bearings with air as the lubricant at relatively low rotational velocities, seldom venturing beyond 100 krpm. This focus has left a significant knowledge gap, particularly concerning the dynamics of hydrostatic gas bearings at ultra-high speeds, where helium serves as the working medium. The imperative exploration of helium-supported rotor systems is driven by the pursuit of enhanced isentropic efficiency in cryogenic helium turbo-expanders, necessitating a further increase in rotational speeds, typically reaching around 250 krpm. This advancement in operating velocity imposes more stringent demands on the dynamic characteristics of the gas bearing–rotor system.

This paper seeks to bridge this gap by providing a comprehensive analysis of the dynamic behavior of ultra-high-speed rotor systems supported by helium hydrostatic gas bearings. Our research is motivated by the need to understand and optimize the performance of these systems, which are critical in high-speed applications where traditional bearings may not suffice. To this end, we have developed a dynamic model that seamlessly integrates the rotor motion equation with the transient Reynolds equation specific to helium hydrostatic gas bearing–rotor systems. This model is solved using a hybrid numerical approach, combining the Finite Difference Method (FDM) with the Wilson-Θ method. Further bolstering our investigation, we have conducted meticulous experimental tests. The experimental data captures the waterfall plots of the bearing–rotor system throughout the acceleration process. By juxtaposing these empirical observations with theoretical predictions, we have gained a profound understanding of the dynamic performance of helium-gas-lubricated bearing–rotor systems at ultra-high speeds. Our analysis reveals that attention to system stability and safety is warranted when the low-frequency components of the rotor’s vibration spectrum are around 0.3 the frequency of the fundament, and the amplitude exceeds three times that of the fundamental frequency’s amplitude.

This threshold is particularly significant as it may indicate the onset of instability or potential operational risks. The identification of such critical parameters is crucial for the development of advanced diagnostic tools and predictive maintenance strategies, ensuring the reliable operation of high-speed machinery, especially in applications such as helium turbo-expanders where operational efficiency is paramount.

2. Mathematical Modeling

2.1. Transient Reynolds Equation for Compressible Fluid Lubrication

The expansion end of a turbo-expander operates in a cryogenic environment. We employ labyrinth seals to minimize the leakage of cold gas to the ambient side, and adjust the leakage of cooling capacity by controlling the flow rate of labyrinth gas. However, to maintain a constant temperature environment for the bearings on the ambient side, we intentionally allow a small amount of cold gas to leak into the bearing–rotor system of the ambient side. This leakage is intended to balance the frictional heat generated by the rotor at high rotational speeds. For the ideal gas, the transient Reynolds equation for compressible fluid lubrication of an isothermal externally pressurized gas journal bearing is derived as follows:

$${\displaystyle \frac{\partial }{\partial t}}(Ph)+\nabla \cdot (-{\displaystyle \frac{P{h}^3}{12\mu }}\nabla P)+\nabla \cdot ({\displaystyle \frac{Ph}{2}}\overrightarrow{V})+\delta \mu \rho {\displaystyle \frac{P\mathrm{a}}{\rho \mathrm{a}}}\tilde{\nu }=0$$

(1)

The relationship between density and pressure is given by the equation of state for ideal gases, which can be expressed as $P/\rho =\mathrm{constant}$, assuming that the gas film thickness direction (the radial direction) is very small compared with the axial and circumferential geometric dimensions, which means that the change in pressure along the radial direction is ignored. The relative motion of the shaft and the bearing in the axial direction is not considered; the fluid at the gas–solid interface also meets the non-slip boundary condition. Then, Equation (1) in the cylindrical coordinate system can be simplified as follows:

$${\displaystyle \frac{1}{R^2}}{\displaystyle \frac{\partial }{\partial \theta }}({\displaystyle \frac{P{h}^3}{\mu }}{\displaystyle \frac{\partial P}{\partial \theta }})+{\displaystyle \frac{\partial }{\partial z}}({\displaystyle \frac{P{h}^3}{\mu }}{\displaystyle \frac{\partial P}{\partial z}})+12\delta \mu \rho {\displaystyle \frac{P\mathrm{a}}{\rho \mathrm{a}}}\tilde{\nu }=12{\displaystyle \frac{\partial }{\partial t}}(Ph)+6U{\displaystyle \frac{\partial }{\partial \theta }}(Ph)$$

(2)

$U$ is the relative motion speed between shaft and bearing in the circumferential direction, which can be the rotational speed $\omega$$(rad/s)$. Ignoring the change in dynamic viscosity of gas film, and introducing dimensionless physical quantities:

$$\overline{P}={\displaystyle \frac{P}{P_a}}, \overline{h}={\displaystyle \frac{h}{c}}, \overline{z}={\displaystyle \frac{z}{L}}, \tau =\omega t, \Lambda ={\displaystyle \frac{6\mu \omega }{P_a}}{({\displaystyle \frac{R}{c}})}^2, Q={\displaystyle \frac{12\mu R^2}{c^3{P}_a{\rho }_a}}\rho \tilde{\nu }$$

where $P_a$ is the ambient pressure, $c$ is the nominal clearance of the bearing, $L$ is the axial length of the bearing, $R$ is the bearing radius, $\mu$ is the gas dynamic viscosity, ${\rho }_a$ is the density of gas at atmospheric pressure, $\rho$ is the density of gas, $\tilde{\nu }$ is the velocity flowing through the orifice, $\delta$ is the Kronecker function ($\delta =0$ without orifices and $\delta =1$ with orifices), $\Lambda$ is bearing number, $Q$ is the dimensionless mass flow rate; $P$ and $\overline{P}$ are the dimensional and dimensionless pressure; $h$ and $\overline{h}$ are the dimensional and dimensionless film thickness; $t$ and $\tau$ are the dimensional and dimensionless time.

Thus, the dimensionless form of the Reynolds equation for transient state is expressed as follows:

$${\displaystyle \frac{\partial }{\partial \theta }}(\overline{P}{\overline{h}}^3{\displaystyle \frac{\partial \overline{P}}{\partial \theta }})+{({\displaystyle \frac{R}{L}})}^2{\displaystyle \frac{\partial }{\partial \overline{z}}}(\overline{P}{\overline{h}}^3{\displaystyle \frac{\partial \overline{P}}{\partial \overline{z}}})+\delta Q=2\Lambda {\displaystyle \frac{\partial }{\partial \tau }}(\overline{P}\overline{h})+\Lambda {\displaystyle \frac{\partial }{\partial \theta }}(\overline{P}\overline{h})$$

(3)

The gas through the orifice is regarded as a point source and the dimensionless mass flow rate can be given as:

$$Q={\displaystyle \frac{12\mu R^2}{c^3{P}_a{\rho }_a}}{\displaystyle \frac{\dot{m}}{R\Delta \theta \cdot L\Delta \overline{z}}}={\displaystyle \frac{12\mu R}{L{c}^3{P}_a{\rho }_a\Delta \theta \Delta \overline{z}}}\dot{m}$$

(4)

And the mass flow rate $\dot{m}$ can be given as:

$$\dot{m}=C_d{\displaystyle \frac{\pi {d_0}^2}{4}}{P}_s\sqrt{{\displaystyle \frac{2}{R_g{T}_a}}}\psi$$

(5)

where $\psi =\left\{\begin{array}{l}{\left[{\displaystyle \frac{\kappa }{2}}{({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa +1}{\kappa -1}}}\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},{\displaystyle \frac{P}{P_s}}\le {({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa }{\kappa -1}}}\\ {\left[{\displaystyle \frac{\kappa }{\kappa -1}}\left[{({\displaystyle \frac{P}{P_s}})}^{{\displaystyle \frac{2}{\kappa }}}-{({\displaystyle \frac{P}{P_s}})}^{{\displaystyle \frac{\kappa +1}{\kappa }}}\right]\right]}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},{\displaystyle \frac{P}{P_s}}>{({\displaystyle \frac{2}{\kappa +1}})}^{{\displaystyle \frac{\kappa }{\kappa -1}}}\end{array}\right.$ and the discharge coefficient ($C_d$) of the mass flow rate through the orifice can be chosen as 0.8 [18], $d_0$ is the orifice diameter, $P_s$ is the supply pressure, $R_g$ is the helium gas constant, $T_a$ is room temperature, $\kappa$ is the heat capacity ratio. As shown in Figure 1, the dimensionless gas film thickness can be calculated by:

$$\overline{h}=1+\epsilon \sin (\theta -{\phi }_0)$$

(6)

where $\epsilon$ is the eccentricity ratio, ${\phi }_0$ is the attitude angle. And Figure 2 shows the computational domain of the gas bearing.

A perturbation approach combining FDM is applied to the above transient Reynolds equation to obtain the bearing dynamic coefficients [19,20]. In this approach, the pressure and the film thickness are expanded in Taylor’s series to the first order in terms of the small perturbations of the shaft motion as follows:

$$\overline{P}={\overline{P}}_0+{\overline{P}}_x\Delta X+{\overline{P}}_y\Delta Y+{\overline{P}}_{\dot{x}}\Delta \dot{X}+{\overline{P}}_{\dot{y}}\Delta \dot{Y}$$

(7)

$$\overline{h}={\overline{h}}_0+{\overline{h}}_x\Delta X+{\overline{h}}_y\Delta Y+{\overline{h}}_{\dot{x}}\Delta \dot{X}+{\overline{h}}_{\dot{y}}\Delta \dot{Y}$$

(8)

where the subscript 0 denotes the steady state, the subscript $x$ represents a partial derivative with respect to the $x$ perturbation, the subscript $\dot{x}$ represents a partial derivative with respect to the $\dot{x}$ perturbation, and it is similar for $y$ and $\dot{y}$.

In summary, according to the coordinate system shown in Figure 1, the dimensionless gas film force in the $x$ and $y$ directions are calculated by integrating the pressure field using Simpson’s one-third rule in the following equations:

$$\left\{\begin{array}{l}{\overline{F}}_x={\displaystyle \frac{L}{R}}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{-\frac{1}{2}}^{\frac{1}{2}}(}}\overline{P}-1)\sin \theta d\theta d\overline{z}\\ {\overline{F}}_y=-{\displaystyle \frac{L}{R}}{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_{-\frac{1}{2}}^{\frac{1}{2}}(}}\overline{P}-1)\cos \theta d\theta d\overline{z}\end{array}\right.$$

(9)

2.2. Rotor Dynamics Equations

The model of the gas bearing–rotor system is shown in Figure 1, which consists of a shaft with one disk, two hydrostatic gas bearings distributed on both sides of the disk, and an impeller and braking wheel. The parameters of the hydrostatic gas bearing–rotor system are shown in

Table 1. The operation parameters of the hydrostatic gas bearing–rotor system.

Parameter	Value
Bearing diameter	16 mm
Bearing length	24 mm
Radial clearance	15 μm
Orifice diameter	0.3 mm
Helium density	0.16698 kg/m3
Helium dynamic viscosity	1.9388 × 10−5 Ns/m2
Supply pressure	7.5 bar
Number of column	2
Number of orifices in each column	8

.

The governing equation of the system dynamic motion for the investigated bearing–rotor system in the coordinate $xoy$ is derived as:

$$M\ddot{u}(t)+C\dot{u}(t)+Ku(t)=F_{\mathrm{total}}(t)$$

(10)

where $M$, $C$ and $K$ are the mass, damping and stiffness matrices, respectively. $F_{\mathrm{total}}$ is the external force matrix including the gas film force, the force-induced imbalance mass of rotor, and the gravity force vector in case of horizontal installation. $\ddot{u}$, $\dot{u}$ and $u$ are the acceleration, velocity and displacement matrices, respectively. Due to the pronounced symmetry of the bearing–rotor system, the equations have been simplified for the solution of rotor dynamics within the framework of a single-mass model.

The Wilson-Θ method is used to solve the nonlinear dynamic Equation (10). It assumes a linear change in acceleration within a time interval [$t$, $t+\Theta \Delta t$] with time variable $\varsigma$, and $\Theta \ge 1$, see Figure 3. With the variable Θ, the numerical stability and damping can be influenced; it has been proved that the Wilson method is unconditionally stable for $\Theta \ge 1.38$ [21,22], and the optimal value $\Theta =1.4208$ is adopted in the numerical simulation to obtain the stable prediction [23].

Where $\Delta t$ specifically denotes a discrete time step in the immediate next moment from $t$, while $\varsigma$ is used in a more general sense to indicate a time increment after $t$. Under this assumption, velocity can be derived by integrating acceleration with respect to time, and displacement is obtained by performing a double integration of acceleration over time. A linear variation of acceleration as a function of time yields the subsequent formulation for acceleration at time $t+\varsigma$.

$$\ddot{u}(t+\varsigma )=\ddot{u}(t)+{\displaystyle \frac{\varsigma }{\Theta \Delta t}}[\ddot{u}(t+\Theta \Delta t)-\ddot{u}(t)]$$

(11)

Integration over the time control variable $\varsigma$ leads to:

$$\dot{u}(t+\varsigma )=\dot{u}(t)+\varsigma \ddot{u}(t)+{\displaystyle \frac{{\varsigma }^2}{2\Theta \Delta t}}[\ddot{u}(t+\Theta \Delta t)-\ddot{u}(t)]$$

(12)

The displacement can be obtained by integrating the acceleration twice over $\varsigma$:

$$u(t+\varsigma )=u(t)+\varsigma \dot{u}(t)+{\displaystyle \frac{{\varsigma }^2}{2}}\ddot{u}(t)+{\displaystyle \frac{{\varsigma }^3}{6\Theta \Delta t}}[\ddot{u}(t+\Theta \Delta t)-\ddot{u}(t)]$$

(13)

The substitution of the control variable is predicated on the assumption of a linear variation in acceleration over the time interval [$t$, $t$ + $\Theta \Delta t$]. By establishing this assumption, we are able to derive the velocity and displacement at any arbitrary time increment $\varsigma$ within this interval. Once these general expressions for velocity and displacement are obtained as functions of the time increment $\varsigma$, we can then specialize these expressions to the specific time increment $\Theta \Delta t$. This approach allows us to obtain the velocity and displacement at time $t$ + $\Theta \Delta t$, which are crucial for our subsequent analysis. The reason for this variable replacement is to facilitate a more generalized analysis that can be applied to any time increment, thereby enhancing the flexibility and applicability of our control framework. After establishing the generalized expressions, we can easily retrieve the specific case by substituting $\Theta \Delta t$ for $\varsigma$.

So, replace the control variable $\varsigma$ with the expression $\Theta \Delta t$ in Equations (12) and (13):

$$\dot{u}(t+\Theta \Delta t)=\dot{u}(t)+\Theta \Delta t\ddot{u}(t)+{\displaystyle \frac{\Theta \Delta t}{2}}[\ddot{u}(t+\Theta \Delta t)-\ddot{u}(t)]$$

(14)

$$u(t+\Theta \Delta t)=u(t)+\Theta \Delta t\dot{u}(t)+{\displaystyle \frac{{(\Theta \Delta t)}^2}{6}}[\ddot{u}(t+\Theta \Delta t)+2\ddot{u}(t)]$$

(15)

From (14) and (15) we obtain, after rearranging, the following equations for the acceleration and velocity at time $t+\Theta \Delta t$ depending on the displacements:

$$\ddot{u}(t+\Theta \Delta t)={\displaystyle \frac{6}{{\left(\Theta \Delta t\right)}^2}}[u(t+\Theta \Delta t)-u(t)]-{\displaystyle \frac{6}{\Theta \Delta t}}\dot{u}(t)-2\ddot{u}(t)$$

(16)

$$\dot{u}(t+\Theta \Delta t)={\displaystyle \frac{3}{\Theta \Delta t}}[u(t+\Theta \Delta t)-u(t)]-2\dot{u}(t)-{\displaystyle \frac{\Theta \Delta t}{2}}\ddot{u}(t)$$

(17)

In the Wilson-Θ method, equilibrium is enforced at time $t+\Theta \Delta t$, thus

$$M\ddot{u}(t+\Theta \Delta t)+C\dot{u}(t+\Theta \Delta t)+Ku(t+\Theta \Delta t)=F_{\mathrm{total}}(t+\Theta \Delta t)$$

(18)

where, the external load vector satisfies:

$$F_{\mathrm{total}}(t+\Theta \Delta t)=F_{\mathrm{total}}(t)+\Theta \left[F_{\mathrm{total}},(,t,+,\Delta ,t,),-,F_{\mathrm{total}},(t)\right]$$

(19)

Inserting (16), (17) and (19) into (18) yields

$$\begin{array}{l}\left({\displaystyle \frac{6}{{\left(\Theta \Delta t\right)}^2}}M+{\displaystyle \frac{3}{\Theta \Delta t}}C+K\right)u(t+\Theta \Delta t)=F_{\mathrm{total}}(t)+\Theta \left[F_{\mathrm{total}}(t+\Delta t)-F_{\mathrm{total}}(t)\right]+\\ \left[{\displaystyle \frac{6}{{\left(\Theta \Delta t\right)}^2}}u(t)+{\displaystyle \frac{6}{\Theta \Delta t}}\dot{u}(t)+2\ddot{u}(t)\right]M+\left[{\displaystyle \frac{3}{\Theta \Delta t}}u(t)+2\dot{u}(t)+{\displaystyle \frac{\Theta \Delta t}{2}}\ddot{u}(t)\right]C\end{array}$$

(20)

The displacement at time $t+\Theta \Delta t$, namely $u(t+\Theta \Delta t)$, can be obtained by Equation (20).

And taking the expression

$$\ddot{u}(t+\Theta \Delta t)=\ddot{u}(t)+\Theta \left[\ddot{u}(t+\Delta t)-\ddot{u}(t)\right]$$

(21)

Rearranging it, such that

$$\ddot{u}(t+\Delta t)={\displaystyle \frac{1}{\Theta }}\ddot{u}(t+\Theta \Delta t)-\left({\displaystyle \frac{1}{\Theta }}-1\right)\ddot{u}(t)$$

(22)

Inserting (16), finally, the following equation for the acceleration at time $t+\Delta t$ is shown

$$\ddot{u}(t+\Delta t)={\displaystyle \frac{6}{{\Theta }^3\Delta t^2}}[u(t+\Theta \Delta t)-u(t)]-{\displaystyle \frac{6}{{\Theta }^2\Delta t}}\dot{u}(t)+\left(1-{\displaystyle \frac{3}{\Theta }}\right)\ddot{u}(t)$$

(23)

Then, setting $\varsigma =\Delta t$, $\Theta =1$ in (12) and (13), we obtain the final expression for the velocity and displacement at time $t+\Delta t$

$$\dot{u}(t+\Delta t)=\dot{u}(t)+{\displaystyle \frac{\Delta t}{2}}\left[\ddot{u}(t+\Delta t)+\ddot{u}(t)\right]$$

(24)

$$u(t+\Delta t)=u(t)+\Delta t\dot{u}(t)+{\displaystyle \frac{\Delta t^2}{6}}\left[\ddot{u}(t+\Delta t)+2\ddot{u}(t)\right]$$

(25)

Finally, Equations (23)–(25) constitute the Wilson-Θ method for the dynamic response analysis of the bearing–rotor system. In the following study, the transient behaviors of a vertical unbalanced rotor with helium hydrostatic gas bearings are investigated as the rotor speed varies.

3. Predicted Dynamic Properties

The computational process for determining the dynamic properties of the bearing–rotor system, as depicted in Figure 4, is an iterative one. It involves the Finite Difference Method (FDM) combined with a perturbation approach for the transient Reynolds equation, and the Wilson-Θ method for the rotor dynamic equation. Initially, the steady-state Reynolds equation, which disregards time (as in Equation (3)), is solved to determine the initial gas film force and the static equilibrium position, thereby establishing the initial displacement. Subsequently, the transient Reynolds equation is solved using the static gas film pressure and thickness distributions to calculate the bearing’s stiffness and damping coefficients.

With these results serving as input conditions, and assuming an initial velocity of zero, the shaft’s displacement, velocity, and acceleration are recalculated by solving Equation (10) using the Wilson-Θ method. This updated displacement is then fed back into the transient Reynolds equation, and the transient gas film force is recalculated, leading to a new set of rotor dynamics equations. This iterative process continues step by step until the predefined time period is reached, ensuring that the rotor system reaches a steady state. The entire numerical computation is implemented using MATLAB R2012a program code.

3.1. First-Order Time Fluctuation of Pressure and the Film Thickness

To highlight the critical role of first-order temporal variations in pressure and film thickness, as defined by Equations (7) and (8), this study meticulously computes the ratio of fluctuation components to the total transient pressure. The total transient pressure refers to the pressure value on the left side of Equation (7), which is expressed as the sum of the steady-state pressure and the pressure fluctuations. The pressure fluctuations encompass both spatial variations and temporal variations. The proportion to total pressure represents the ratio of the pressure fluctuations over the total transient pressure. Specifically, for the x-direction, this is denoted as ${\overline{P}}_{\dot{x}}\Delta \dot{X}/\overline{P}$, and for the y-direction, it is ${\overline{P}}_{\dot{y}}\Delta \dot{Y}/\overline{P}$. When considering both directions, it is the sum of the pressure fluctuations in both directions divided by the total pressure, represented as $({\overline{P}}_{\dot{x}}\Delta \dot{X}+{\overline{P}}_{\dot{y}}\Delta \dot{Y})/\overline{P}$. Figure 5 compellingly demonstrates that an increase in rotational velocity correlates with an escalation in the amplitude of the fluctuation terms. The rotational velocities corresponding to the fluctuation magnitudes of 10% to 30% are 33.3 krpm, 95.0 krpm, and 181.9 krpm, respectively. Specifically, once the rotational velocity exceeds 95.0 krpm (where krpm denotes 1000 revolutions per minute), the temporal pressure fluctuations constitute more than 20 percent of the overall transient pressure. Ignoring these fluctuations can result in significant discrepancies in the estimation of transient pressures, potentially undermining the accuracy of dynamic characteristic forecasts for ultra-high-speed bearing–rotor systems.

3.2. Effect of Rotational Speed on Rotor Response

To elucidate the impact of rotational velocity on the dynamic response of the bearing–rotor system during the acceleration phase, bifurcation diagrams (Figure 6) were constructed for a range of rotor speeds from 10 to 300 krpm. This selection encompasses a spectrum of velocities designed to scrutinize the system’s orbital behavior, spectral characteristics, and Poincaré mappings.

The bifurcation diagrams delineate a critical transition in the system’s dynamical behavior with escalating rotational speeds. At velocities below 150 krpm, the system sustains a stable 1 T-periodic motion. However, at the threshold of 90 krpm, there is an abrupt amplitude escalation, and the orbit morphs into an inclined ellipse, signifying the traversal of the primary critical speed and an attendant alteration in the rotor’s modal attributes. Beyond this juncture, the rotor initiates multi-periodic oscillations accompanied by low-frequency vibration signals in the vicinity of 1500 Hz. Upon reaching 150 krpm, the rotor transitions into a 2T-periodic motion state, which intriguingly reverts to a 1T-periodic motion with an initial surge followed by a diminution in vibration amplitude. Progressively, as the rotational velocity surpasses this point, the amplitude escalates in tandem with the rotational speed, while the system’s multi-periodic motion traits endure.

This analysis underscores the nuanced interplay between rotational speed and the bearing–rotor system’s stability, offering insights into the complex dynamics that emerge as the system navigates through various operational regimes.

Further analysis was performed by integrating the orbit, frequency spectrum, and Poincaré map data as shown in Figure 7. In the 1T-periodic motion state, the rotor’s orbit traces a closed circular path. With an increase in rotational speed, the orbit’s complexity escalates, becoming intricately intertwined. A higher low-frequency amplitude corresponds to increased orbit complexity. The Poincaré map indicates a single point for the 1T-periodic motion state, which evolves to include two or more points as the rotational speed escalates. At 270 krpm, the points on the map exhibit chaotic behavior, losing their regular pattern, and signifying the onset of chaotic motion. At this speed, the low-frequency to 1× frequency ratio is 0.35, and the amplitude ratio is 4.427.

The low-frequency amplitude’s relative increase compared to the 1× frequency amplitude elevates the risk of rotor instability. As the low-frequency vibration’s amplitude surpasses that of the 1× frequency, the low-frequency to 1× frequency ratio further decreases to 0.325 at 300 krpm, while the amplitude ratio soars to 5.231. This suggests that amplified low-frequency amplitude more readily leads to a chaotic rotor motion.

The frequency ratio observed is notably lower than that of traditional oil bearings experiencing half-speed whirl instability, where the whirl frequency to rotor 1× frequency ratio is 0.5 at the onset of instability. This discrepancy is primarily attributed to the lower viscosity of the gas medium compared to oil, which influences the system’s dynamic response.

4. Experimental Test

Additionally, experimental investigations are carried out to study the dynamic properties of the rotor system in a practical setup. The experimental setup mainly includes the turbo-expander, expanding gas source, bearing gas source, eddy current sensors, and data acquisition system as shown in Figure 8. The eddy current sensor is orthogonally arranged and has a measurement range of 0–400 μm. The sensor model is MICRO-EPSILON EU05, produced by MICRO-EPSILON MESSTECHNIK from Ortenburg, Germany. The maximum frequency of the displacement sensor is 20,000 Hz, the linearity is ±1 μm, and the resolution is 0.05 μm. The calibration of the distance coefficient between the measured rotor and the sensor probe is shown in Figure 9. The relationship between the voltage signal and the distance value is characterized by a proportionality coefficient of 54.3 μm per unit volt.

The data acquisition system is configured with a sample size of 50,000 and operates at a sampling frequency of 200,000 Hz. The experimental acceleration waterfall plot, which captures the dynamic response, is presented in Figure 10. During the initial acceleration phase, the rotor’s speed increments in a steady manner. The 1× vibration amplitude initially increases, and then diminishes, before resuming growth. Beyond a rotational speed of 250 krpm, a pronounced low-frequency vibration signal emerges. This low-frequency amplitude continues to escalate as the speed increases, until the shaft makes contact with the bearings. Below a rotational speed of 130 krpm, a vibration signal near 1000 Hz is detected, attributable to noise from the bearing gas supply. Notably, both the first critical speed of the bearing–rotor system and the noise signal from the bearing gas coincide around 1000 Hz. To assess the noise signal’s impact on the data, orbit plots were compared before and after noise filtering, as illustrated in Figure 11. The presence of the vibration signal results in a more complex and overlapping orbit, indicative of multi-periodic or quasi-periodic motion. Post signal processing to filter out the noise, the orbit is rendered more distinct and straightforward. By integrating experimental observations and noise filtration, the rotor’s authentic vibration signal is discernible. During the experiment, instability occurs when the speed hits 273.9 krpm, leading to a collision and friction between the rotor and bearings. At this unstable state, the low-frequency to 1× frequency ratio is 0.27, and the amplitude ratio is 3.112. Future research should continue with the acceleration experiment, with a focus on mitigating the noise from the bearing gas supply.

The appearance of low-frequency signals during the acceleration process is acknowledged as a principal inducement of destabilization within ultra-high-speed bearing–rotor systems. A comprehensive analysis, integrating theoretical computations with experimental data, reveals that at an inflection point where the low-frequency components’ frequency equates to roughly 0.3 of the operational frequency, and their amplitudes exceed threefold that of the operational frequency, the system deviates from straightforward periodic movements, advancing into a complex regimen of multi-periodic or possibly chaotic behaviors. It is imperative that the stability of such systems is subjected to rigorous and continuous oversight to prevent potential system disruptions.

5. Conclusions

In conclusion, the synergistic approach of numerical computation and experimental research has yielded significant insights into the dynamic properties of an ultra-high-speed rotor system supported by helium hydrostatic gas bearings. The study commenced with the development of a numerical model designed to simulate the rotor system’s dynamic behaviors. The numerical simulations elucidated the rotor’s bifurcation diagram and provided detailed analyses of the orbit, frequency spectrum, and Poincaré maps at various rotational speeds. With increasing speed, the rotor’s motion evolves from a simple 1T-periodic state to more intricate 2T-periodic, multi-periodic, and eventually chaotic states, all characterized by the emergence of low-frequency components. Notably, as rotational speed escalates, the ratio of low-frequency to 1× frequency diminishes, while the ratio of low-frequency amplitude to 1× frequency amplitude expands, indicating heightened system instability. At a rotational speed of 300 krpm, the frequency ratio is observed to be 0.325, and the amplitude ratio reaches 5.231. The critical speeds at which these transitions occur are identified as 90 krpm for the first and 165 krpm for the second, marking pivotal thresholds for system stability.

The experimental component of the research furnished empirical data on the rotor system’s dynamic characteristics. Vibration amplitudes and frequencies measured at different rotational speeds were instrumental in evaluating how design and operational parameters affect system performance. A pronounced low-frequency oscillation is detected when the speed surpasses 250 krpm. At the maximum tested speed of 273.9 krpm, the low-frequency to 1× frequency ratio is 0.27, and the amplitude ratio is 3.112. It is imperative to monitor the system’s stability and reliability closely when the low-frequency to 1× frequency ratio approaches 0.3 and the amplitude ratio exceeds 3.

This research has established a robust methodology for analyzing the dynamic properties of ultra-high-speed rotor systems. However, further refinements in the theoretical model and experimental techniques are necessary to more precisely predict and investigate the nonlinear dynamics of such systems. Future work should focus on enhancing the model’s predictive accuracy and developing more sophisticated experimental protocols to ensure the reliability and efficiency of high-speed rotor systems in practical applications.