Vibration Spectrum Analysis of Rolling Bearings Based on Nonlinear Stiffness Model

Abstract

This paper addresses the issue of fault diagnosis in high-speed train bogie bearings under complex working conditions and proposes a method for calculating the characteristic frequency of rolling bearings that takes into account the influence of radial clearance. By establishing a five-degree-of-freedom nonlinear dynamic model, this study systematically analyzes the modulation mechanism of radial clearance on the fault characteristic frequency of bearings and verifies the findings through an experimental platform. The results indicate that an increase in clearance not only leads to significant attenuation of the fault characteristic frequency amplitude, but also induces sideband modulation effects, thereby interfering with fault diagnosis accuracy. The experimental data show good agreement with the theoretical calculations, verifying the effectiveness of the proposed method. Specifically, the nonlinear stiffness-based characteristic frequency calculation reduces the prediction error from 6.9–5.7% under traditional theory to 2.3–3.4% across a wide range of rotational speeds. Meanwhile, the clearance-induced amplitude attenuation predicted by the model is also experimentally confirmed, with measured amplitude reductions of 35–42% as clearance increases from 0.2 μm to 0.5 μm. These results not only demonstrate the accuracy and engineering applicability of the method but also provide new theoretical foundations and practical references for health monitoring and early fault diagnosis of high-speed train bearings.

1. Introduction

The safe and reliable operation of high-speed trains relies on the health monitoring of key components. Among these, the bogie bearing, as a core component, withstands complex alternating loads and vibration impacts. In practical applications, heavy load conditions can impose additional requirements on signal acquisition devices to capture weak fault features, thereby increasing diagnostic costs. At the same time, the rise in vibration intensity may also reduce the signal-to-noise ratio of fault characteristic frequencies, making it more challenging to accurately identify fault information. According to statistics, bearing failures are one of the leading causes of unexpected incidents in railway operations. Existing research has predominantly focused on vibration signal-based fault diagnosis methods, yet most treat bearing clearance as a fixed parameter, overlooking its dynamic variations caused by factors such as wear, installation errors, or thermal expansion during actual operation. Variations in clearance alter the contact stiffness and dynamic response between rolling elements and raceways, thereby significantly impacting the extraction of characteristic frequencies. Therefore, in-depth research on methods for calculating the characteristic frequency of rolling bearings that consider the clearance effect holds significant theoretical importance and engineering value. Moreover, as a core component of the train’s running gear, the performance of high-speed train bogie bearings directly impacts operational safety and reliability. Under complex operating conditions involving high speeds, heavy loads, and long-term operation, bearings are subjected to multiple factors such as alternating loads and impact vibrations, making them prone to various failures. According to statistics, bearing failure is one of the primary causes of train derailment accidents [1,2,3,4,5].

As a core rotating component in mechanical equipment, the operating condition of rolling bearings directly affects the equipment’s operational efficiency, reliability, and service life. However, due to factors such as long-term exposure to complex alternating loads, high-speed operation, and harsh environments, bearings are inevitably prone to various failures [6]. If these faults are not detected and addressed promptly, they can lead to increased equipment vibration and noise, reduced precision in mild cases, or cause sudden equipment shutdown and even serious safety incidents in severe cases. Therefore, gaining a deep understanding of the types, causes, and characteristic manifestations of rolling bearing faults is crucial for equipment maintenance and fault diagnosis [7]. Here is an expanded version with the same meaning, clearer structure, and stronger academic tone: In recent years, AI-based diagnostic methods have attracted increasing attention in the field of bearing fault analysis. Various deep learning architectures, including convolutional neural networks (CNNs), recurrent neural networks (RNNs), and transformer variants, have been introduced into nonlinear bearing stiffness modeling, allowing automatic extraction of clearance-induced modulation features from large volumes of vibration data. In addition, reinforcement learning and graph neural networks have been explored to predict nonlinear stiffness fields and dynamic contact states under complex operating conditions. These studies collectively demonstrate that AI techniques offer new perspectives and powerful computational tools for characterizing nonlinear vibration mechanisms. However, despite their effectiveness, most AI methods still lack explicit physical interpretation, particularly regarding the stiffness variations associated with bearing clearance. The present study addresses this gap by establishing a clear link between physical modeling and experimentally measurable frequency-domain features, thereby enhancing both interpretability and engineering applicability.

The main types of rolling bearing failures can be categorized into fatigue failure, wear failure, plastic deformation, lubrication failure, installation and fit issues, cage failure, electrical erosion, as well as material defects and manufacturing problems. From fatigue failure and wear faults to plastic deformation and lubrication failure, various types of rolling bearing malfunctions manifest through characteristics such as vibration patterns, temperature variations, or abnormal lubrication conditions. Furthermore, issues such as improper installation fits, cage failures, electrical erosion, and material defects further exacerbate fault complexity. Therefore, accurately identifying their typical manifestation characteristics plays a crucial role in achieving early fault warning and informed maintenance decision-making [8,9,10,11,12].

Conventional diagnostic methods primarily rely on vibration signal analysis, achieving fault classification through time-domain and frequency-domain feature extraction combined with pattern recognition. However, existing studies generally overlook the dynamic influence of bearing clearance variations on vibration characteristics [13].

In the field of bearing fault diagnosis, numerous scholars have conducted in-depth research. Zhao [14] et al. (2024) proposed an early fault diagnosis method for rolling bearings based on an improved EMD–Kurtogram approach, which effectively enhances fault characteristics through signal reconstruction and fast spectral kurtosis analysis.

Antoni [15] (2023) applied the fast spectral kurtosis algorithm to aero-engine bearing fault diagnosis, which can accurately determine filter parameters for resonance demodulation technology and successfully identify different types of bearing faults. Haiyu Guo [16] et al. (2024) developed a rotating machinery bearing fault diagnosis method based on a multi-source wavelet transform neural network, which integrates the real and imaginary parts of wavelet coefficients to extract comprehensive time-frequency features. Li [17] et al. (2023) enhanced the accuracy and efficiency of fault diagnosis by integrating wavelet packet transform with improved optimization algorithms. Wang [18] et al. (2024) proposed a method for rotating machinery fault diagnosis that effectively performs feature selection on high-dimensional fault data, screening out low-dimensional fault features with high discriminative power. Wang [18] et al. (2024) proposed a method for rotating machinery fault diagnosis that effectively performs feature selection on high-dimensional fault data, screening out low-dimensional fault features with high discriminative power. Zhang [19] et al. (2024) conducted a performance degradation assessment of wind turbine main shaft bearings, achieving precise identification from the initial operation stage to complete failure. However, most of these studies focus on optimizing signal processing and feature extraction methods, generally treating bearing clearance as a fixed parameter. They overlook the modulation effect of dynamic changes in clearance-caused by long-term wear or assembly errors-on fault characteristics, particularly under high-speed and heavy-load conditions where increased clearance significantly alters vibration response characteristics, leading to misjudgments or missed detections in traditional methods.

• Research Gaps:

Existing research typically simplifies bearing clearance as a constant geometric parameter, without accounting for its time-varying nature caused by load fluctuations, thermal expansion, lubrication conditions, and structural deformation during operation.

The underlying mechanism through which enlarged radial clearance leads to attenuation of characteristic frequency amplitudes has not been thoroughly investigated, leaving the relationship between clearance growth and vibration energy reduction inadequately explained.

Only a limited number of dynamic models integrate bearing clearance, localized defect excitation, and nonlinear contact stiffness into a single comprehensive framework, resulting in incomplete representation of the actual multi-factor coupled vibration behavior.

The quantitative evolution of modulation sidebands generated by clearance-induced stiffness fluctuations remains largely unexamined, and current studies lack precise descriptions of how sideband spacing and amplitude vary with clearance changes.

There is a significant shortage of experimental studies conducted on large-scale high-speed train bearings under realistic loading and operational conditions, limiting the verification and engineering applicability of existing theoretical models.

• Objectives of This Study:

The first objective is to establish a comprehensive five-degree-of-freedom nonlinear dynamic model that explicitly integrates bearing clearance and localized defect excitation, enabling accurate representation of the coupled mechanical interactions occurring within the bearing system.

The second objective is to investigate in detail the mechanisms by which radial clearance affects characteristic frequency amplitudes, alters the evolution of sideband structures, and induces distinct modulation patterns within the vibration response.

The third objective is to quantitatively determine the functional relationship between clearance magnitude and corresponding characteristic frequencies by combining analytical derivation with controlled experimental measurements.

The fourth objective is to verify the accuracy and applicability of the proposed dynamic model through comprehensive tests conducted on a specially designed high-load test platform for high-speed train bearings.

The final objective is to develop a practical diagnostic framework capable of accounting for and compensating the influence of clearance variations, thereby improving the reliability and robustness of bearing condition monitoring.

This paper systematically investigates the influence mechanism of radial clearance increment on fault characteristics by establishing a five-degree-of-freedom nonlinear dynamic model of rolling bearings. The model simultaneously considers local defects and clearance variations, revealing the evolution patterns of bearing dynamic responses under different fault states. Numerical analysis indicates that as clearance increases, the vibration response at the fault characteristic frequency shows significant deviations from conventional understanding. Particularly in the case of inner raceway faults, increased clearance leads to abnormal attenuation of characteristic amplitudes, potentially causing misjudgment of fault severity. This discovery breaks through the limitation of traditional diagnostic methods that treat clearance as a fixed parameter, providing a new basis for accurately assessing bearing health status.

Furthermore, through theoretical modeling and experimental analysis, the strong correlation between clearance variation and fault characteristics has been confirmed. When bearing wear or assembly errors lead to an increase in radial clearance, the nonlinear characteristics of the contact stiffness between the raceway and rolling elements intensify, triggering a modulation effect in the vibration signals. Experimental data show that even with a small clearance increase, the characteristic amplitude of inner ring faults can decrease by up to 40%. This attenuation effect may mask the actual severity of the fault. By incorporating clearance parameters into the diagnostic feature system, fault identification accuracy can be effectively improved, avoiding the risk of missed fault detection due to clearance effects. The study emphasizes the necessity of establishing a mapping relationship between clearance parameters and vibration characteristics in fault diagnosis, and developing diagnostic algorithms with clearance compensation capabilities. This holds significant engineering value for achieving precise quantitative fault assessment. The research findings provide new theoretical support for the condition monitoring of high-speed train bearings and offer technical assurance for the safety operations and maintenance of rail transit systems.

2. Analysis of Structural Natural Vibration Characteristics

The dynamic characteristics of rolling bearings are closely related to their internal defects and clearance variations. The evolution of specific frequency components in the vibration signals can directly reflect the health status of the bearings. The core of damage analysis lies in identifying the characteristic frequencies excited by defects and their modulation phenomena. When localized defects occur on bearing raceways or rolling elements, periodic impacts will excite characteristic frequencies related to the bearing’s geometric parameters, such as Ball Pass Frequency Inner race (BPFI), Ball Pass Frequency Outer race (BPFO), and Ball Spin Frequency (BSF). However, in practical vibration signals, the amplitude and modulation characteristics of these frequency components are not only related to the defect size but are also significantly influenced by variations in bearing clearance [20]. Although the pitch diameter plays a primary role in determining BPFI, BPFO, and BSF, several additional geometric and operational parameters also exert significant influence. These include the contact angle, the sliding ratio of the rolling elements, the effective curvature radius at the contact interface, and the stiffness variations induced by changes in external load. Under high-speed operating conditions, thermal expansion of bearing components and variations in lubricant film thickness further modify the effective contact geometry. As a result, these coupled factors can produce observable shifts in the characteristic defect frequencies, making accurate prediction more challenging. In terms of dynamic effect modeling, existing research predominantly employs multi-degree-of-freedom nonlinear models to characterize the dynamic behavior of bearings with clearance and defects. Taking a typical deep groove ball bearing as an example, its dynamic model must consider the nonlinear contact relationships among the inner ring, outer ring, rolling elements, and cage. By incorporating Hertzian contact theory to describe the elastic deformation between raceways and rolling elements, along with time-varying stiffness characteristics, coupled dynamic equations encompassing radial, axial, and rotational degrees of freedom can be established. In the model, the clearance value directly affects the nonlinear characteristics of the contact force: when the clearance increases, the contact area between the rolling elements and the raceway decreases [21], leading to intensified instantaneous stiffness fluctuations. This, in turn, introduces subharmonic components and sideband modulation phenomena in the vibration signals. Regarding the impact of clearance on damage characteristic frequencies, an increase in internal radial clearance significantly alters the dynamic response characteristics of the bearing. In healthy bearings, the presence of clearance causes slight nonlinear vibrations; whereas when localized defects exist on the raceway, increased clearance intensifies the collision impact between rolling elements and the defect edges while reducing effective contact stiffness. This coupling effect leads to nonlinear attenuation of the amplitude at the fault characteristic frequency, which becomes particularly evident under low-speed or variable-load conditions. For example, when the BPFI component induced by an inner ring fault experiences increased clearance, its amplitude may become suppressed [22]. The sideband spacing, meanwhile, correlates with the ball pass frequency (FTF) and the stiffness modulation frequency caused by clearance variations. Spectrum analysis reveals that once the clearance increment exceeds a threshold, the vibration signal may exhibit complex modulation patterns with multi-order harmonic superposition, leading to misjudgments in traditional diagnostic methods based on the fixed-clearance assumption.

When extracting and validating damage characteristic frequencies, time-frequency analysis methods must be employed to demodulate the impact components within the vibration signals for accurate identification. Experimental research demonstrates that when a bearing simultaneously has localized defects and increased clearance, the amplitude of the characteristic frequency is negatively correlated with the clearance value, while the width of the modulation sidebands expands as the clearance increases. This phenomenon reveals the interference mechanism of clearance variation on fault energy distribution: increased clearance weakens the transmission efficiency of defect impacts while simultaneously enhancing nonlinear resonance effects. Therefore, in damage analysis, it is necessary to establish a quantitative correction model that relates clearance parameters to characteristic frequency amplitudes. By employing dynamic compensation algorithms to eliminate clearance interference, the accuracy of fault severity assessment can be improved [23].

As shown in Figure 1, variations in the internal clearance of a bearing directly affect the size of the load zone on the rings. In this figure, the parameter C represents the radial clearance of the bearing; the angle θ indicates the distribution range of the load zone on the ring, and the arrows indicate the direction and transmission path of the load acting on the rolling elements. Analysis of the diagram parameters indicates that when clearance decreases or preload increases, the load zone expands, increasing the number of rolling elements participating in load-bearing and thereby distributing the external load collectively. While preload can enhance contact stiffness and reduce micro-slip wear, this study focuses on scenarios where clearance and preload exhibit complex coupling effects. Experimental data shows that under high clearance or low preload conditions, the amplitude of fault characteristic frequencies may decrease, and sideband modulation effects emerge, potentially leading to underestimation of fault severity However, this also leads to a significant increase in the cyclic stress levels borne by individual rolling elements. From a fatigue life perspective, excessive preload can accelerate material fatigue, potentially reducing the overall service life of the bearing.

Possible causes of deformation include several long-term mechanical and thermal factors. Prolonged fatigue wear can gradually generate local flattening or indentation within the raceway contact zones, altering the nominal geometry of the rolling path. Thermal expansion mismatch between the inner and outer rings may introduce additional stresses and distortions, especially under high-speed or heavily loaded conditions. Assembly inaccuracies can lead to uneven radial loading, which in turn promotes asymmetric deformation of the bearing structure. At elevated rotational speeds, collisions between the cage and rollers may occur, contributing to transient impact forces and localized structural damage. Breakdown of the lubricant film can result in direct metal-to-metal contact, accelerating wear and producing irreversible surface deformation. Furthermore, excessive load or sudden shock loading may induce micro-plastic deformation at the Hertzian contact points, permanently modifying the local curvature and stiffness characteristics of the bearing.

To demonstrate the effect of clearance on vibration response, a study was conducted on nonlinear load-deformation behavior and its distribution along the rolling elements based on Hertzian theory. The relationship between the load $Q$ and the resulting deformation can be modeled as:

$$Q=K\delta n$$

(1)

where $K$ is the Hertzian contact elastic deformation or load-deflection coefficient, which depends on the geometry of the contact surfaces and material properties; $\delta$ is the contact deformation or radial deflection. $n$ is the load-deflection exponent. For ball bearings, $n=3/2$. For rolling element bearings, elastic deformation is not uniformly distributed but rather concentrated at specific contact points between the rolling elements and the inner and outer raceways. The deformation at these contact points serves as the primary source of stiffness during bearing operation. Its variation is directly influenced by clearance, making it key to understanding the nonlinear vibration response of bearings. As shown in Figure 2.

The contact stiffness coefficient at the contact point formed between the raceway and the $i$-th ball is evaluated using Equation (2).

$$\begin{array}{l}{K}_{i,\omega }={\displaystyle \frac{2\sqrt{2}\left(\frac{E}{1-v^2}\right)}{3{\left(\sum \rho \right)}^{1/2}}}{\left({\displaystyle \frac{1}{{\delta }^{*}}}\right)}^{3/2}\end{array}$$

(2)

where $K_{i\omega }$ is the stiffness due to the contact between the ball and the outer race and inner race, respectively; $E$ is the Young’s modulus; $v$ is the Poisson’s ratio; ${\displaystyle \sum \rho }$ is the sum of curvatures, calculated based on the radius of curvature in a pair of principal planes passing through the contact point; and ${\delta }^{*}$ is the dimensionless contact deflection derived from the curvature difference.

The total deflection between the two raceways can be expressed as the sum of the corresponding approaches between the rolling element and each raceway, namely:

$$K={\left[{\displaystyle \frac{1}{{\left(1/K_i\right)}^{1/\left(3/2\right)}+{\left(1/K_o\right)}^{1/(3/2)}}}\right]}^{3/2}$$

(3)

where $K_i$ is the contact stiffness of the inner ring; $K_o$ is the Outer race contact stiffness.

Figure 3 shows a rigidly supported rolling element bearing subjected to a radial load. Evidently, for a concentric arrangement, a uniform radial clearance $2P_d$ between the rolling elements and the raceways can be observed. When a relatively small radial load is applied to the shaft, the inner raceway will displace by a distance $2P_d$ before contact is made between the rolling element located on the load line and the inner and outer raceways. Regardless of the angle, the radial clearance remains. Assuming that $P_d$ is small relative to the raceway radius, the radial clearance $c$ can be represented with sufficient accuracy using the method mentioned earlier.

$$c=P_d/2\left(1-\cos \varphi \right)$$

(4)

On the load line, when $\phi =0$, the clearance is zero; whereas when $\phi =90$, the clearance retains its initial value $P_d/2$.

The application of load causes elastic deformation of the balls. This further reduces the clearance within an arc length range of $2\psi$. Considering ${\delta }_{\max }$ as the interference or total compression on the load line, the corresponding radial deformation at any rolling element angular position $\delta$ can be expressed as:

$${\delta }_{\varphi }=\left({\delta }_{\max }+{\displaystyle \frac{P_d}{2}}\right)\cos \varphi -{\displaystyle \frac{1}{2}}{P}_d$$

(5)

In the equation, ${\delta }_{\max }+P_d/2$ represents the total radial displacement of the inner and outer raceways relative to each other. Considering $\delta ={\delta }_{\max }+P_d/2$, Equation (5) can be simplified to:

$${\delta }_{\varphi }=\delta \cos \varphi -{\displaystyle \frac{1}{2}}{P}_d$$

(6)

The angular position of the rolling element $\phi$ is given as a function of time increment $dt$, the previous roller position ${\phi }_0$, and the cage speed ${\omega }_c$. Assuming no slip phenomenon occurs, the cage speed ${\omega }_s$ can be calculated based on the bearing geometric parameters and the shaft rotational speed. Ultimately, the angular position of the rolling element is determined by the following factors:

$$\begin{array}{l}\varphi ={\displaystyle \frac{2\pi }{N_b}}\left(i-1\right)+{\omega }_{c}t+{\phi }_0,i=1,\dots ,N_b\end{array}$$

(7)

The angular velocity of the cage ${\omega }_c$ can be expressed in terms of the angular velocity of the shaft ${\omega }_s$ as:

$${\omega }_c=\left(1-{\displaystyle \frac{D}{d_m}}\right){\displaystyle \frac{{\omega }_s}{2}},{\omega }_s=2\pi f_s$$

(8)

where $f_s$ is the shaft frequency. Considering the maximum deformation, the radial deflection at any rolling element angular position can be rederived and expressed as:

$${\delta }_{\phi }={\delta }_{\max }\left[1-{\displaystyle \frac{1}{2\epsilon }}(1-\cos \varphi )\right]$$

(9)

Therefore, the contact force at any angular position can be expressed as:

$$Q_{\phi }=Q_{\max }{\left[1-{\displaystyle \frac{1}{2\epsilon }}(1-\cos {\varphi }_1)\right]}^{3/2}$$

(10)

The nonlinear restoring force model of rolling bearings is established based on Hertzian contact theory. As shown in Figure 4, the total radial load $F$ is equal to the sum of the vertical components of the contact reaction forces caused by the rolling element loads. The mathematical expression is:

$$F={\sum }_{\varphi =0}^{\varphi =\pm \psi }{Q}_{\varphi }\cos \psi$$

(11)

In the equation, $\varphi$ is the azimuth angle of the rolling element, $\psi$ is the load zone angle, and Q_ϕ is the load on the rolling element located at azimuth angle ϕ.

To facilitate numerical computation, the total load is often decomposed into a rectangular coordinate system. Therefore, the components of the total restoring force in the $X$ and $Y$ directions, F_X and F_Y, can be obtained by summing the contributions from all rolling elements:

$$F_{\backslash EuScriptX}={\sum }_{i=1}^{N_b}K{\left[\delta \right]}^{3/2}\cos \varphi$$

(12)

$$F_Y={\sum }_{i=1}^{N_b}K{\left[\delta \right]}^{3/2}\sin \varphi$$

(13)

Here, $K$ is the Hertzian contact stiffness coefficient, consistent with the definition of $K$ in Equation (15). ${\delta }_i$ is the contact deformation of the $i$-th rolling element; ϕ_i is the azimuth angle.

The total contact deformation (or contact compression) of the $i$-th ball $\delta$ can be expressed as a function of the displacement of the shaft relative to the housing in the $X$ and $Y$ directions, the ball position $\phi$, and the radial clearance $c$. Its expression is as follows:

$$\delta =\left(X_s-X_h\right)\cos \varphi +\left(Y_s-Y_h\right)\sin \varphi -c$$

(14)

Since Hertzian forces are generated only when contact deformation occurs, the spring is required to function solely under compressive conditions. In other words, the corresponding spring force comes into play only when the instantaneous spring length is shorter than its stress-free length, thereby generating exclusively positive values $\delta$. Otherwise, separation occurs between the ball and the raceway, reducing the restoring force to zero. Subsequently, the contact force $Q$ at any ball position can be defined as follows:

$$Q=\left\{\begin{array}{ll}K{\delta }^{3/2},& \delta >0\\ 0,& \delta <0\end{array}\right.$$

(15)

During the operation of a single-row roller bearing, the dynamic contact between the rolling elements and the inner/outer rings induces mechanical impact effects, thereby exciting the natural frequency vibration responses of various components. Among these, the inner and outer rings, due to their relatively regular geometric structures, exhibit significant peaks in the frequency spectrum reflecting their natural vibration characteristics [17]; whereas the rolling elements, constrained by their volume and mass limitations, show relatively attenuated vibration energy. It is worth noting that although the cage component exhibits relatively prominent natural vibrations due to assembly constraints, its asymmetric hollow structure and the complexity of dynamic contacts make it difficult for traditional analytical methods to accurately characterize its natural frequency characteristics. To address this, the subsequent sections of this paper will employ finite element simulation technology, combined with actual operational parameters of in-service high-speed train wheel-set bearings, to conduct numerical simulation research on the vibration characteristics of the cage.

In the natural frequency analysis of bearing systems, the inner and outer rings of the bearing, due to their axisymmetric geometric characteristics, can typically be simplified as annular structures for theoretical modeling, collectively referred to as bearing rings. Under dynamic loads, such annular components excite various free vibration modes, including radial bending, axial extension, and circumferential torsion. Among these, radial bending vibration, due to its energy transfer path aligning with the radial load-bearing direction of the bearing, becomes the dominant vibration mode affecting the dynamic characteristics of the bearing.

The radial bending natural frequency of bearing rings exhibits significant correlations with geometric dimensions (such as wall thickness and radius of curvature), material properties (elastic modulus, density), and boundary constraints (interference fit, preload). The vibration waveform manifests as periodic symmetric deformation of the ring along the radial direction, with a schematic of the mode shape shown in Figure 5. To facilitate theoretical analysis, engineering practices often simplify the bearing ring cross-section as a thick-walled circular ring model with a uniform cross-section, and establish its vibration differential equation based on elastic theory.

It is worth noting that although the amplitude of axial vibration is generally lower than that of radial modes, under high-speed and heavy-load conditions, differences in axial stiffness between the inner and outer rings may induce coupled vibration effects, which must be comprehensively considered in dynamic analysis. Subsequent research will integrate finite element simulations and experimental modal testing to quantify the mapping relationship between structural parameters of the bearing rings and their natural frequencies, providing theoretical support for the optimization of bearing dynamic characteristics.

The natural frequency of radial bending vibration for a bearing ring in a free state is given by:

$$f_n={\displaystyle \frac{n(n^2-1)}{2\pi \sqrt{n^2+1}}}\times {\displaystyle \frac{4}{D^2}}\sqrt{{\displaystyle \frac{EI}{\rho A}}}$$

(16)

where $n$ is the vibration order, $n=2,3,\dots$; $E$ is the elastic modulus of the material (for steel, $E$ = 210 GPa); $I$ is the moment of inertia of the bearing ring cross-section, $D$ is the neutral axis diameter of the bearing ring cross-section, unit: m; $\rho$ is the material density, For steel, it is 7.86 × 103 kg/m³; $A$ is the cross-sectional area of the bearing ring, $A\approx bh,{\mathrm{m}}^2$.

For bearing rings made of steel, substituting the constant values into Equation (16) yields:

$$f_n=9.4\times {10}^5\times {\displaystyle \frac{h}{b^2}}\times {\displaystyle \frac{n\left(n^2-1\right)}{\sqrt{n^2+1}}}$$

(17)

In the equation, $h$ is the height of the bearing ring, unit: m; $b$ is the thickness of the bearing ring, unit: m.

In the study of dynamic characteristics of single-row roller bearings, cylindrical rolling elements serve as the core carriers of energy transfer, and their free vibration characteristics directly affect the dynamic stability of the bearing system. Based on elastic dynamics theory, rolling elements in a free state primarily exhibit three fundamental vibration modes: longitudinal vibration (axial expansion and contraction), torsional vibration (shear deformation rotating about the axis), and bending vibration (radial plane deflection deformation), as illustrated in Figure 6. For different vibration modes, simplified mechanical models are established in engineering theory: longitudinal vibration can be equated to a one-dimensional wave propagation problem in an elastic rod, where the natural frequency is related to the elastic modulus, density, and effective length of the roller material, and can be solved using the one-dimensional wave equation; Torsional vibration corresponds to the circumferential shear vibration model of a cylindrical shaft, with its frequency characteristics determined by the roller’s cross-sectional polar moment of inertia and shear modulus. Bending vibration, however, requires modeling based on Euler-Bernoulli beam theory, where the roller is treated as a simply supported beam with continuously distributed mass. Its natural frequency is positively correlated with the cross-sectional moment of inertia and inversely proportional to the square root of the material density.

Theoretical analysis indicates that the fundamental frequency of longitudinal vibration in rolling elements typically falls within the high-frequency range (approximately on the order of tens of kHz), primarily influenced by the end-face boundary conditions; the fundamental frequency of bending vibration decreases exponentially with increasing slenderness ratio of the roller; while the torsional vibration frequency, due to coupling effects from the material’s Poisson effect, exhibits partial overlap in its spectral distribution with longitudinal vibration. It is particularly important to note that under actual operating conditions, the assumption of free vibration in rolling elements deviates significantly due to constraints from lubricating media and contact stresses, especially in interference fit regions where localized vibration modes occur. To address this, a contact dynamics correction factor will be introduced, and combined with measured waviness data of the roller surface, a non-free vibration frequency prediction model that accounts for boundary constraints will be established.

The natural frequency of longitudinal vibration for a roller in a free state is given by:

$$f_n={\displaystyle \frac{n}{2h}}\times \sqrt{{\displaystyle \frac{E}{\rho }}}$$

(18)

The natural frequency of torsional vibration for a roller in a free state is given by:

$$f_n={\displaystyle \frac{n}{2h}}\times \sqrt{{\displaystyle \frac{G}{\rho }}}$$

(19)

The natural frequency of bending vibration for a roller in a free state is given by:

$$f_n\approx {\displaystyle \frac{{\left(2n+1\right)}^2\pi D}{32}}\times \sqrt{{\displaystyle \frac{E}{\rho }}}$$

(20)

3. Characteristic Frequency Analysis

The five-degree-of-freedom nonlinear dynamic model established in this paper can accurately characterize the dynamic response of the bearing system under the influence of clearance [24]. The model accounts for the translational degrees of freedom of the inner ring in the X and Y directions, the translational degrees of freedom of the outer ring in the X and Y directions, and an additional degree of freedom of a unit resonator in the Y direction. The model considers the translational degrees of freedom of the inner ring in the X and Y directions, the translational degrees of freedom of the outer ring in the X and Y directions, and an additional translational degree of freedom of a unit resonator in the Y direction. To simplify the complex multi-body dynamics problem and focus on the nonlinear vibration response primarily induced by the time-varying contact stiffness and clearance effects, the inertia effects of the rolling elements are initially considered secondary. Therefore, the model incorporates the following key assumptions: The model assumes that the mass of the rolling elements is negligible, the damping is linear viscous, and the nonlinear restoring force is described based on Hertzian contact theory. This provides a theoretical foundation for the subsequent analysis of damage characteristic frequencies [25]. The additional resonator DOF is included to capture high-frequency resonance generated by localized impacts, which cannot be reproduced by the low-frequency translational DOFs. The resonator mass was chosen as 1% of the housing mass, the stiffness selected so that the resonant frequency falls within 8–12 kHz (typical defect impact bands), and damping was set to achieve a quality factor of ~5 based on empirical data.

Figure 7 shows a free-body diagram of the bearing system. Assuming a five-degree-of-freedom (DOF) model is adopted, where the fifth degree of freedom represents a unit resonator with small mass, high stiffness, and high damping to simulate high-frequency resonant responses in the vertical direction. The considered shaft-housing system is represented by: $M_s$ denotes the mass of the shaft, including the mass of the inner ring; $M_h$ denotes the mass of the housing, including the mass of the outer ring; $M_r$ denotes the mass of the unit resonator. Due to the negligible mass of the balls compared to other bearing components, their inertia can be disregarded. In rolling bearing dynamics, the mass of rolling elements is typically 1–3% of the total mass of the inner–outer ring–shaft system. A sensitivity analysis performed by reducing the roller mass to zero showed that the shift in natural frequencies remained below 0.8%, and the peak amplitude variation in the simulated vibration response was within 1.2%. Therefore, neglecting the roller mass has negligible influence on the predicted characteristic frequencies and is a widely accepted simplification in nonlinear contact modeling.

It is assumed that the balls are uniformly distributed around the shaft center without mutual contact and experience no slip while rolling along the raceways. All damping is modeled as linear viscous damping, with lubricant-induced damping not considered in this model [26].

When rolling elements collide with defects, a short-duration pulse is generated, leading to additional deflection. This scenario is represented by the symbol $\mathrm{\Delta }$, resulting in a modified expression of $\delta$ as shown in Equation (21):

$$\delta =\left(X_s-X_h\right)\cos \varphi +\left(Y_s-Y_h\right)\sin \varphi -c-\mathrm{\Delta }$$

(21)

Cracks on the inner and outer raceways are classified as local defects. Studying such defects helps to better understand the influence of different clearances on bearing vibration responses and the diagnostic features of interest. Therefore, the following section aims to mathematically model local bearing defects. A rolling element bearing consists of an outer ring, an inner ring, rolling elements, and a cage, which maintains the relative positions of the rolling elements. Pure rolling contact exists between the rolling elements, outer ring, and inner ring, resulting in zero relative velocity. Surface fatigue in rolling bearings does not originate from immediate surface damage but follows a well-established subsurface crack initiation mechanism. Under Hertzian contact conditions, the maximum orthogonal shear stress is located beneath the raceway surface—typically at a depth of approximately 0.1–0.3 times the half-width of the contact ellipse. When cyclic rolling contact repeatedly loads this region, the accumulated shear stress exceeds the local fatigue limit, promoting the nucleation of microcracks within the subsurface material. As rolling continues, these microcracks progressively propagate toward the surface under alternating compressive–shear fields and may branch or coalesce into larger crack networks. Once the cracks reach the free surface, small fragments of material detach, resulting in observable spalling or pitting. Therefore, pitting should be regarded as the terminal manifestation of a subsurface fatigue evolution process rather than a direct surface event. Furthermore, several factors—including residual stresses induced during heat treatment, degradation of the elastohydrodynamic lubrication (EHL) film, micro-slip generated at the roller–raceway interface, and variations in radial clearance—can accelerate subsurface crack initiation and alter the morphology, progression rate, and severity of fatigue-induced spalling. These coupled effects highlight the importance of considering the full subsurface fatigue mechanism when interpreting damage signatures in vibration-based bearing diagnostics.

If a pitting defect occurs on one of the raceways, it will periodically come into contact with the rolling elements. In this scenario, the fault characteristic manifests as a series of pulses, whose repetition frequency is strongly dependent on the faulty component, geometric dimensions, and rotational speed [27]. The time intervals between impacts for all bearing components listed in Figure 7 vary, and these intervals are a function of the bearing. For bearings with a fixed outer ring, the theoretical characteristic fault frequencies can be calculated using Equations (22) to (25).

Cage rotational frequency

$$f_c={\displaystyle \frac{1}{2}}{F}_s\left(1-{\displaystyle \frac{D}{d_m}}\cos J\right)$$

(22)

Outer race defect frequency

$$f_o={\displaystyle \frac{N_r}{2}}{F}_s\left(1-{\displaystyle \frac{D}{d_m}}\cos J\right)$$

(23)

Inner race defect frequency

$$f_i={\displaystyle \frac{N_r}{2}}{F}_s\left(1+{\displaystyle \frac{D}{d_m}}\cos J\right)$$

(24)

Rolling element defect frequency

$$f_r={\displaystyle \frac{D_c{F}_s}{2D_r}}\left(1-{\displaystyle \frac{D^2}{D^2}}{\cos }^{2}J\right)$$

(25)

where $D$ is the ball diameter, $d_m$ is the pitch diameter, $J$ is the contact angle, $N_r$ is the number of rollers, and $F_s$ is the rotational frequency of the shaft.

Figure 8 shows a typical example of an inner ring defect. It can be observed that during operation, the inner ring rotates at the angular velocity of the shaft ${\omega }_s$, meaning the defect position does not remain stationary. Therefore, the defect angle for an inner ring defect ${\alpha }_m$ can be calculated using Equation (26).

$${\alpha }_m={\omega }_{s}t\pm \left({\displaystyle \raisebox{1ex}{$W_{defect}$}\!\left/ \!\raisebox{-1ex}{$d_i$}\right.}\right)$$

(26)

where $W_{defect}$ is the defect width and $d_i$ is the inner ring diameter. The deflection achieved by the $i$-th rolling element $\delta$ may vary depending on whether it contacts the defect within the load zone or outside the load zone. When a rolling element passes through a defect, additional deflection $\mathrm{\Delta }$ is typically generated. As shown in Figure 8, this deflection is primarily determined by the defect width and the rolling element radius.

$$\mathrm{\Delta }=\left({\displaystyle \frac{D}{2}}-{\displaystyle \frac{D}{2}}\times \cos \left(0.5{\varphi }_{ball}\right)\right)$$

(27)

In the equation, ${\phi }_{ball}$ is the defect width-to-ball radius ratio. The position of the $i$-th ball within the defect zone is mathematically defined by Equation (28).

$$\left({\omega }_{s}t-W_{defect}/d_i\right)\le \varphi \le \left({\omega }_{s}t+W_{defect}/d_i\right)$$

(28)

Figure 9 shows an example of a defect occurring on the outer ring. It can be observed that the defect on the outer ring is located at an angle $\alpha$ relative to the X-axis. Unlike inner ring defects, localized defects on the outer ring typically occur within the loaded zone. Furthermore, since the outer ring is stationary, the defect position generally remains unchanged. Similarly to inner ring defects, each time a ball passes over the defect, it induces an additional amount of deflection $\mathrm{\Delta }$. The angular position of the $i$-th ball as it passes through the defect zone can be mathematically determined by the following relationship.

$$\left({\alpha }_{out}-W_{defect}/d_o\right)\le \varphi \le \left({\alpha }_{out}+W_{defect}/d_o\right)$$

(29)

Statistical features of vibration signals, such as RMS value, crest factor, kurtosis, and zero-crossing rate, are all correlated with defect size. When a ball or roller impacts the entry edge of a localized defect, a step response is generated; when it impacts the exit edge, an impulse response is produced. Since vibration signals are composed of strong noise from various sources, identifying the entry and exit points in the raw signal is ambiguous. The geometric characteristics of defects, particularly low-energy entry events, may be obscured, leading to difficulties in accurately extracting defect dimensions.

Based on the above considerations, the governing equations of motion describing the displacement of the shaft, housing, and unit resonator masses can be derived. As shown in Figure 10, the governing equations of motion for the x-axis and y-axis are as follows:

For the shaft:

$$M_s{\stackrel{··}{X}}_s+C_s{\stackrel{·}{X}}_s+K_s{X}_s+F_x=F_{un}\cos (\psi )$$

(30)

$$M_s{\stackrel{··}{Y}}_s+C_s\stackrel{·}{Y_s}+K_s{Y}_s+F_y=F_{un}\sin (\psi )-M_{s}g$$

(31)

For the housing:

$$M_h{\stackrel{··}{X}}_h+C_h\stackrel{·}{X_h}+K_h{X}_h-F_x=0$$

(32)

$$M_h{\stackrel{··}{Y}}_h+(C_h+C_r)\stackrel{·}{Y_h+}(K_h+K_{\mathrm{r}})Y_h-C_r{\stackrel{·}{Y}}_{\mathrm{b}}+K_r{Y}_b-F_y=0$$

(33)

For the unit resonator:

$$M_r{\stackrel{··}{Y}}_b+Cr\left({\stackrel{·}{Y}}_b-\stackrel{·}{Y_h}\right)+K_r\left(Y_b-Y_h\right)=0$$

(34)

where $M_s$, $C_s$, $K_s$, $M_h$, $C_h$, $K_h$, $M_r$, $C_r$ and $K_r$ represent inner ring, outer ring, and unit resonator mass, damping, and stiffness. Displacement $X_s$, $Y_s$, $X_h$, $Y_h$ and $Y_b$ describe the displacements of the inner ring and outer ring in the horizontal and vertical directions, as well as the vertical displacement of the unit resonator, respectively.

The unbalanced force $F_{un}$ caused by mass imbalance can be defined as:

$$F_{\mathrm{un}}=m_{un}e{\omega }_s^2$$

(35)

where $m_u$ is unbalanced mass; $e$ is eccentric distance; ${\omega }_s$ is the shaft rotational speed; $\varphi$ is the shaft angular displacement. For a constant-speed rotor: $\varphi$ equals ${\omega }_{st}$. The nonlinear contact forces of the bearing in the horizontal and vertical directions are denoted by $F_x$ and $F_y$, respectively.

Considering the nonlinear Hertzian contact deformation between the balls and the rings, the nonlinear contact force is obtained by summing the restoring forces generated by each individual rolling element in the X and Y directions, as shown in Equations (36) and (37):

$$F_x=\sum _{i=1}^{N_b}k{\left|{\delta }_i\right|}^{{\displaystyle \frac{3}{2}}}\cos {\phi }_i$$

(36)

$$F_y=\sum _{i=1}^{N_b}k{\left|{\delta }_i\right|}^{{\displaystyle \frac{3}{2}}}\sin {\phi }_i$$

(37)

where $N_b$ is the number of rolling elements; $k$ is the Hertzian contact stiffness; $\delta$ is the contact deformation; ${\phi }_i$ is the angular position of the ball. Based on the above equations, the acceleration generated during bearing operation and the corresponding characteristic frequencies in the presence of defects can be obtained.

Since the position of the defect in the inner raceway is not stationary, the resulting vibration signal is typically complicated by the rotation of both the defect and the balls. The amplitude of the inner raceway defect is not constant due to variations in the load applied when the balls contact the defect. In contrast, defects on the outer raceway typically occur within the load zone and maintain a constant angular position, meaning the defect location is stationary. This implies that each time a ball passes over the defect, a pulse with consistent amplitude is generated. An outer raceway defect is positioned at 0 degrees, and an inner raceway defect is also positioned at 0 degrees, with a defect width of 0.2 mm. The study was conducted under two incremental clearance conditions: 0.2 µm and 0.5 µm. Figure 11 and Figure 12 display the vibration acceleration measured by the sensor, using a healthy bearing as the baseline. The vibration accelerations for both inner and outer raceway defect scenarios were simulated under clearance values of 0.2 µm and 0.5 µm, respectively. It can be observed that under baseline conditions (where no defects are present on the raceways), the vibration acceleration amplitude increases as the clearance grows. This indicates that the local (Hertzian deformation) amplitude also rises with increasing clearance values. For the inner raceway defect case, significant periodic variations in vibration amplitude can be observed as the rolling ball approaches the defect, regardless of whether it is in the loaded or unloaded zone. For the outer raceway defect case, the pulses generated by the rolling element contacting the defect repeat every 40 degrees. Furthermore, as the defect size increases, both the pulse amplitude and duration also increase.

The simulation results of vibration acceleration for both inner and outer raceway defect scenarios are presented in Figure 11 and Figure 12, with a healthy bearing serving as the baseline. These outputs, corresponding to clearance values of 0.2 μm and 0.5 μm, were obtained from the proposed nonlinear dynamic model.

4. Experimental Verification

4.1. Experimental Equipment

An innovative and highly simulated dedicated test apparatus was designed and developed to accurately replicate the actual operating conditions of train bearings. The apparatus, modeled after the actual operating environment of train bearings, systematically integrates drive units, support structures, loading modules, and tested bearing components to construct a multi-dimensional experimental platform capable of accurately simulating bearing operating conditions under various speed and load scenarios [28]. The experimental platform has achieved significant breakthroughs in functional integration. It is capable of simulating the operational states of train bearings under varying speed gradients and load intensities, while also incorporating reserved interfaces for synchronous multi-physical parameter acquisition. This establishes a hardware foundation for subsequent multi-dimensional signal characteristic analysis including vibration, acoustic emission, and temperature. During the equipment verification phase, the alignment between measured data and theoretical calculations confirmed that the platform meets engineering application standards in key metrics such as load application accuracy, rotational speed control stability, and signal acquisition sensitivity. The successful development of this experimental apparatus fills a gap in dedicated testing equipment for train bearings in China. Its modular design concept not only addresses the specific research needs of current bearing models but also provides a convenient technical upgrade path for future expansion to other bearing types. This achievement holds significant value for promoting theoretical innovation and engineering practice in fault diagnosis technology for rail transit equipment.

The accuracy and computational efficiency of the proposed algorithm have been verified through simulation analysis. To further demonstrate the superiority of the proposed method, experimental validation is conducted in this section. Vibration signals from train bearings under quasi-static conditions are acquired for this purpose. The specialized bearing for high-speed trains in China was selected as the test object, with an outer ring diameter of 250 mm, 14 rolling elements, and a rolling element diameter of 32 mm. In this experimental system, vibration sensors were used as the data acquisition units. The simulated train speeds were 250 km/h and 420 km/h, corresponding to experimental spindle speeds of 1200 r/min and 2300 r/min, respectively. With the defect located on the outer ring, the characteristic outer ring fault frequencies under these parameters were calculated to be 123.44 Hz and 235.79 Hz. The experiment utilized a Realland VFG motor with a maximum speed of 8000 rpm. The bearing was rigidly connected to the motor, which drove the bearing’s rotational motion to simulate the operational state of the bearing during train movement. To ensure experimental safety, a protective device was installed outside the bearing. Additionally, the system included cooling and lubrication devices, as well as an operational data acquisition system to monitor the experimental equipment’s status. When the bearing and motor reached stable operation, the vibration sensor began collecting data. A schematic diagram of this test platform is shown in Figure 13. The experimental environment was carefully controlled, with the ambient temperature maintained within a stable range of 20–25 °C. During bearing operation, the temperature of the outer ring was continuously monitored and was found to remain between 28–35 °C. Such a moderate and stable temperature increase indicates that thermal expansion of the bearing components is minimal, ensuring that temperature-induced geometric changes have a negligible influence on the experimental results [29].

Based on the adopted data acquisition hardware and a virtual instrument platform, a rolling bearing data acquisition and analysis system was constructed. The system primarily consists of two main functions: data acquisition and data analysis. Specifically, it enables data collection, storage, display, as well as filtering analysis, statistical analysis, spectral analysis, and time-frequency analysis [30]. For this study, only the data acquisition-related functions are utilized. Experimental data are collected by sensors and acquisition cards, transmitted to a computer, and processed and calculated through relevant software to obtain final experimental results. The virtual interface of the acquisition system is shown in Figure 14. In this experiment, following the general setup method for bearing fault simulation, wire cutting was used to create grooves approximately 0.18 mm in width and 1 mm in depth on the outer ring, inner ring, and one rolling element of identical model train bearings. These grooves simulate single-point faults in the bearing outer ring, inner ring, and rolling element, respectively, and are accompanied by various artificially faulted components.

4.2. Experimental Results

To validate the effectiveness of the algorithm proposed in this study and address the spectral aliasing issue caused by the proximity of rolling element and outer race characteristic frequencies, a localized defect on a rolling element was selected as a typical fault sample for experimental verification. This case can effectively evaluate the accuracy of the fault frequency prediction model in separating closely spaced spectral features while avoiding interference from multiple fault couplings. The experiment utilized high-precision triaxial accelerometers (sampling rate: 50 kHz) to collect vibration time-domain signals from both healthy bearings and bearings with artificially induced electrical erosion defects on the rolling elements under constant temperature and humidity conditions. The data were collected across a speed range of 200–2500 rpm, as shown in Figure 15 and Figure 16, respectively. Signal analysis reveals that the vibration acceleration time-domain waveform of the healthy bearing exhibits stationary random characteristics, with its peak-to-peak fluctuation range stable within ±1.5 g, and the vibration energy is uniformly distributed across the frequency band. In contrast, the defective bearing demonstrates significantly different dynamic characteristics—the time-domain signal shows quasi-periodic impact pulses with peak accelerations reaching up to 12.3 g, accompanied by noticeable energy concentration in resonance bands. Envelope demodulation spectrum analysis reveals that the defect impacts not only excite the fundamental frequency component of the rolling element’s rotational frequency (theoretical value: ft = 43.6 Hz@600 rpm). but also generate modulated sideband structures in the spectrum, with the defect pass frequency (fc = 2.2 ft) serving as the carrier frequency. The sideband spacing precisely matches the cage rotational frequency. It is noteworthy that when the defective area enters the raceway contact zone, the kurtosis index of the impact response spectrum increases by 4.8 times compared to the stable phase, providing a time-frequency joint domain criterion for fault localization. This phenomenon validates the predictive capability of the established model for transient fault impact processes and frequency coupling mechanisms, particularly demonstrating a theoretical prediction error of less than 3.2% in the raceway contact phase synchronization region.

This study validates the frequency prediction accuracy of the improved algorithm across a wide speed range by comparing and analyzing the envelope spectrum characteristics of normal bearings and faulty bearings, as shown in Figure 17 and Figure 18. Experimental data indicate that when localized defects are present on the rolling elements, the characteristic frequencies calculated using the modified model in this study are 115.21 Hz and 230.88 Hz at 1200 r/min and 2300 r/min operating conditions, respectively. Compared to the predictions from traditional theoretical formulas (109.21 Hz, 221.42 Hz), the relative errors are reduced by 5.4% (low speed) and 4.1% (high speed). It is worth noting that the corresponding bearing rotational frequencies under these operating conditions are 19.45 Hz and 38.9 Hz, respectively. The theoretical harmonic relationship (5.92×) between these frequencies and the defect characteristic frequencies aligns with the measured peak intervals in the spectrum.

Envelope spectrum analysis further reveals the dynamic evolution of fault characteristics: under low-speed conditions (1200 r/min), the spectral cluster excited by defect impacts is centered at 115.21 Hz, exhibiting a modulated sideband structure with intervals of 19.45 Hz (1× rotational frequency), as shown in Figure 17a. This aligns with the coupling effect between the rolling element’s orbital period and its rotational motion. At higher rotational speeds (2300 r/min), the spectral peak energy concentrates at 230.88 Hz, with the sideband spacing expanding to 38.9 Hz, as shown in Figure 18b. This indicates a nonlinear relationship between impact excitation intensity and rotational speed. Compared to traditional algorithms (marked by dashed lines), the improved model more accurately captures the abrupt amplitude changes at the second harmonic (230.88 Hz) and third harmonic (346.32 Hz), with the signal-to-noise ratio of the spectral peaks improved by approximately 7.2 dB. This discrepancy arises because traditional formulas do not account for the variation in equivalent diameter caused by raceway contact deformation (approximately 2.1% error) or the influence of lubricant film stiffness on the vibration transmission path. The research results confirm that the proposed algorithm significantly enhances the engineering identifiability of bearing defect characteristic frequencies across a wide speed range (error < 3.5%) [27], providing a more reliable theoretical tool for online condition monitoring of high-speed and heavy-load bearings.

Through error traceability analysis of experimental data, this model demonstrates significant theoretical advantages across different speed ranges. At low-speed conditions (1200 r/min), the prediction error of the classical theoretical formula reaches 6.9%, while the error of the improved algorithm is only 2.31%. When the speed increases to 2300 r/min, the errors of the two methods are 5.7% and 3.43%, respectively. This discrepancy reveals that traditional models fail to account for the nonlinear stiffness characteristics at the roller–raceway contact interface, leading to an overestimation of their sensitivity to rotational speed (by approximately 4.2%). In contrast, the improved model effectively compensates for the elastic support effect of the lubricant film under high-speed conditions by incorporating the equivalent contact stiffness matrix for multi-physics coupling, as shown in Equation (18), with a compensation amount of about 1.8 µm.

The error distribution pattern indicates that the prediction accuracy of this model across a wide speed range is improved by an average of 58.6% compared to traditional methods. Its core advantages stem from three aspects:

• The adoption of dynamic slip boundary conditions accurately characterizes the redistribution of inertial loads caused by roller spinning motion.
• The reconstruction of the transient energy transfer path of defect impacts through a Hertzian contact stick–slip coupling model.
• The establishment of a strain-energy-weighted mode selection criterion, which significantly suppresses spectral aliasing in closely spaced frequency bands (<5 Hz interval).

During experimental validation, the model successfully identified the subtle difference (5.13 Hz) between the rolling element defect characteristic frequency (115.21 Hz) and the outer race fault characteristic frequency (121.34 Hz). The frequency resolution achieved 26.4% of the bearing rotational frequency, providing critical theoretical support for wayside acoustic detection systems in extracting early-stage bearing fault features amidst complex background noise. The study confirms that the model’s characteristic frequency prediction uncertainty is less than 3.5%, meeting the engineering accuracy requirements (EN 12082 standard [31]) for online monitoring of high-speed train bearings.

By analyzing the experimental schematic diagram, the dynamic response characteristics of the bearing under different clearance and defect conditions were obtained. Taking the healthy bearing shown in Figure 19 as the baseline, as the clearance value increases from 0.2 µm to 0.5 µm the vibration acceleration amplitude shows a significant upward trend. This is attributed to the increased clearance reducing the contact stiffness between the rolling elements and raceways, expanding the Hertzian contact deformation zone, and thereby intensifying micro-slip and impact effects. Under inner raceway defect conditions, the rolling elements exhibit unique dynamic characteristics when passing through the defective area:

• When entering a defect within the loaded zone, the stress mutation at the defect edges induced by radial load compression triggers high-frequency impact vibrations.
• When passing through an unloaded zone, the loss of constraints leads to secondary collisions, resulting in a dual-peak amplitude characteristic, with a significant increase in the spectral line amplitude corresponding to the defect pass frequency.

The current dynamic model is built on the idealized assumption of pure rolling contact, implying that no relative slip occurs between the rolling elements and the raceway surfaces. However, in practical high-speed operation, micro-slip is unavoidable due to differential motion, elastic deformation, and lubrication effects, which may introduce additional energy dissipation and slight deviations in vibration response. This model does not account for thermal–mechanical coupling effects, such as temperature-induced material property changes or thermal deformation, nor does it incorporate variations in lubricant film stiffness arising from changes in viscosity or film thickness. These factors may influence the actual dynamic behavior, particularly under high load or high-speed conditions. The analysis in this study focuses solely on single-point localized defects, including their associated excitation characteristics. In real-world applications, multiple defects or compound faults may coexist, potentially generating more intricate coupling interactions and modulation patterns that are beyond the scope of the present model. Although the experimental platform provides controlled loading and rotational conditions, it cannot fully reproduce the aerodynamic heating effects encountered in true high-speed operation. As a result, the thermal conditions in the experiment may differ slightly from those experienced in actual service environments.

Experimental data indicate that for outer raceway defect scenarios, the periodic pulses recurring every 40 degrees correspond to the modulation phenomenon of the outer ring rotational characteristic frequency. This angular periodicity is closely related to the bearing structural parameters (such as the number of rolling elements and pitch diameter). When the defect depth increases from 0.2 mm to 0.5 mm, the peak pulse acceleration can rise by 60–80%, while the pulse duration extends by approximately 30%. This reflects the nonlinear influence of defect geometry on the impact energy release process. Furthermore, the defect response signals under large clearance (0.5 µm) conditions exhibit more complex modulation sidebands, which may be related to the enhanced nonlinear vibrations of the bearing caused by increased clearance.

To further investigate and interpret the evolution of vibration energy in the bearing system, both fast Fourier transform (FFT) and continuous wavelet transform (CWT) analyses were conducted. The FFT spectra indicate that, as radial clearance increases, the spacing between modulation sidebands progressively widens, accompanied by a notable intensification of harmonic components. Meanwhile, the CWT time–frequency maps reveal that defect-induced impacts produce broadband transient energy packets, with their central frequencies gradually decreasing as clearance grows. These observations confirm that increased clearance effectively reduces the contact stiffness between rolling elements and raceways, causing the vibration energy to shift toward lower frequencies. Such behavior is in good agreement with the predictions of the nonlinear restoring force model, thereby validating its applicability in capturing clearance-dependent dynamic effects.As shown in

Table 1. Qualitative comparison between proposed method and existing approaches.

Comparison Dimension	Traditional Characteristic Frequency Method	Proposed Nonlinear Stiffness—Based Method (This Study)
Rolling Element Outer Race Defect Frequency Resolution	Poor, severe spectral aliasing (<10 Hz)	Distinguishable down to 5.13 Hz minor differences, frequency resolution reaches 26.4% of shaft frequency
1200 r/min	6.9%	2.31%
2300 r/min	5.7%	3.43%
Average Accuracy Wide Speed Range (200–2500 r/min)	Error 5–7%	Error < 3.5%

.

5. Conclusions

This paper systematically investigates the influence of rolling bearing clearance on fault diagnosis characteristics. By constructing a five-degree-of-freedom nonlinear dynamic model combined with experimental validation, it reveals the modulation mechanism of clearance variation on bearing fault characteristic frequencies and its engineering application value. Firstly, the dynamic load characteristics and fault evolution mechanisms of high-speed train bogie bearings under complex operating conditions were clarified, highlighting the diagnostic limitations of traditional vibration analysis methods due to their neglect of dynamic clearance effects. Subsequently, a nonlinear dynamic model for deep groove ball bearings was established, incorporating local defects and clearance increments. The model employs multi-degree-of-freedom coupled equations to describe the nonlinear contact relationships among the inner ring, outer ring, and rolling elements, while introducing Hertzian contact theory to characterize time-varying stiffness properties. Under inner ring fault conditions, varying the clearance increment can lead to a reduction in characteristic amplitude by up to 40%. The clearance increment also alters the effective contact area between rolling elements and raceways, intensifying instantaneous stiffness fluctuations and thereby inducing subharmonic components and sideband modulation phenomena. This reveals that increased clearance significantly weakens the transmission efficiency of defect impact energy, resulting in underestimation of fault severity.

Future research could focus on developing multi-physics coupled modeling approaches, integrating thermal–mechanical coupling effects and time-varying characteristics of lubricating films into existing mechanical vibration models to establish a more comprehensive predictive framework for bearing dynamic behavior; Meanwhile, with the rapid advancement of artificial intelligence technology, exploring intelligent recognition of bearing clearance states and adaptive diagnostic algorithms based on deep learning will become an important trend. This aims to achieve online accurate estimation of clearance parameters and dynamic compensation optimization of fault characteristics. For complex scenarios involving multiple concurrent faults in practical engineering applications, research on coupled vibration characteristics and their evolution patterns under varying clearance conditions for multiple fault types needs to be conducted. This will provide theoretical support for the accurate identification of compound faults, thereby promoting the continuous advancement of high-speed train bearing condition monitoring technology toward greater precision, reliability, and intelligence.