Prediction and Validation of Impact Noise Radiation from Ball Bearings under Elastic Contact

1  Introduction

As global production systems continue evolving toward digitalization and intelligent automation, the significance of mechanical transmission components has become increasingly evident [1]. These components form the fundamental basis for power conversion and transmission, supporting a wide range of mechanical operations. Among them, ball bearings are recognized as essential due to their ability to guide rotational motion while reducing friction between moving parts, thereby enhancing system reliability and extending service life [2,3].

The dynamic behavior of ball bearings involves a complex interplay of contact impact, frictional forces, elastic deformation, and vibration-induced noise. One of the earliest analytical studies on ball-type transmission elements was conducted by Heathcote [4], who sought to identify suitable materials and installation strategies for ball bearings. By formulating mathematical expressions to characterize contact interactions between the steel balls and bearing races, the resulting stress distributions were evaluated in detail. The study concluded that sliding motion at the contact interfaces was a primary cause of localized stress accumulation and bearing failure. Subsequent theoretical advancements were introduced by Jones [5], who incorporated Coulomb friction into bearing models and reported friction coefficients ranging from 0.06 to 0.07 under real-world conditions. Building upon this foundation, dynamic models developed by Walters [6] and Gupta [7,8] integrated six degrees of freedom into rolling element analysis, further emphasizing the critical role of lubrication in bearing performance. To address the system-level behavior of bearings, Wensing [9] introduced a spring-based simplification of contact forces and utilized Component Mode Synthesis (CMS) for vibration analysis under modular structural decomposition. This approach emphasized the importance of localized modal frequencies in shaping the overall dynamic response. The interplay between structural vibration and acoustic radiation has gained attention in recent years, especially in high-speed and fault-prone applications. Xue et al. [10] experimentally correlated the severity of bearing defects with frequency-domain features in both vibration and radiated sound. As rotational speeds increase, corresponding growth in vibration and noise levels highlights the need for integrated vibro-acoustic analysis in predictive modeling and bearing system design.

The dynamic characteristics of ball bearing motion have been extensively investigated using numerical methods. For instance, Ratiu and Van Moerbeke [11] analyzed the Lagrangian behavior of a three-dimensional rigid body rotating about a fixed point under gravitational influence, offering early insights into rigid-body dynamics. Building upon this foundation, Ceanga and Hurmuzlu [12] proposed an impulse–momentum framework to address the complexities of multi-body collision problems. By introducing impulse-related correction ratios, their method refined classical rigid-body theory and was experimentally validated through high-speed camera observations. Meanwhile, in the study of elastic contact phenomena, Love [13] pioneered the investigation of elastic body properties by classifying the vibrations of elastic spheres into those induced by purely normal motion and those involving coupled normal and tangential displacements. To further describe elastic impact behavior, Goldsmith [14] employed Hertzian contact theory to quantify the dynamic response of elastic bodies during collision, emphasizing the transient nature of stress development in such events. Subsequently, Richards et al. [15] contributed to bridging structural dynamics and acoustics by conducting pendulum-based impact experiments. Their results revealed a strong correlation between peak contact acceleration, sound pressure levels, and radiated acoustic energy, thus highlighting the interdependence between mechanical impact and sound generation. This progression of research led to increased interest in structural–acoustic coupling. Notably, Koss [16–18] extended Hertzian contact models by incorporating the Helmholtz equation, deriving a time-domain function for predicting impact-induced sound pressure. His theoretical results, which showed that the maximum radiated pressure occurs along the direction of impact, were supported by experimental validation and laid the groundwork for analytical treatment of ball-on-ball acoustic radiation problems. Subsequent validation by Mehraby et al. [19] and Wang and Tong [20] demonstrated the limitations and applications of Hertz theory under different geometries and impact conditions.

Vibro-acoustic coupling plays a pivotal role in the analysis of noise, vibration, and harshness (NVH) characteristics, and three primary numerical strategies are commonly adopted: FEM (finite element method for structure) + FEM (acoustic), FEM (structure) + BEM (boundary element method for acoustic), and FEM (structure) + IEM (infinite element method for acoustics). Mehraby et al. [19] employed an FEM + FEM approach for acoustic simulation. While effective for interior acoustic problems, this method cannot inherently satisfy far-field radiation conditions and typically requires the addition of artificial boundaries such as perfectly matched layers (PML) or automatically matched layers (AML) to approximate infinite domains. The IEM was developed to address these limitations by extending FEM through the inclusion of infinite elements. However, IEM performance depends heavily on the user-defined expansion order. Although it provides high accuracy at high frequencies, it is prone to numerical instability and divergence at lower frequencies, making it less suitable for full-spectrum analysis. In contrast, the FEM + BEM approach has become the preferred method for acoustic radiation problems, especially in unbounded domains. This is due to the BEM’s use of Green’s functions to reduce problem dimensionality and to automatically satisfy Sommerfeld radiation conditions. Unlike FEM, BEM does not require discretization of the entire acoustic domain, making it highly efficient for far-field computations. Zhang et al. [21] employed a coupled FEM–BEM approach to investigate far-field acoustic radiation from submerged shell structures, demonstrating its effectiveness in modeling acoustic wave propagation in unbounded domains. Further validation was provided by Stütz and Ochmann [22], who conducted both time-domain and frequency-domain BEM simulations and verified their precision against analytical solutions. As a result, FEM + BEM has been widely applied in structural-acoustic coupling problems. For example, Liu et al. [23] analyzed acoustic scattering from thin-shell structures, while Chen et al. [24] simulated impact noise generated by colliding spheres. Cho et al. [25] applied FEM + BEM to investigate tire radiation noise, showing that the dominant vibration modes stemmed from the tire’s natural frequencies. Citarella et al. [26] compared FEM + FEM and FEM + BEM for automotive applications and concluded that although FEM + BEM is more computationally intensive, it delivers superior accuracy in far-field predictions. To address mesh incompatibility in large-scale coupled simulations, Fritze et al. [27] proposed optimized coupling schemes for FEM-BEM interfaces. Building upon these findings, this study adopts the FEM + BEM approach to investigate the structural dynamics and radiated noise of ball bearings under high-speed rotational conditions, aiming to achieve accurate vibro-acoustic modeling across a wide frequency range and spatial domain.

However, despite the extensive literature on bearing dynamics and acoustic modeling, few studies have examined how different contact boundary conditions influence the modal frequency, vibration response, and radiated noise of ball bearings in a fully coupled FEM–BEM framework. Ohta and Kobayashi [28] investigated the vibrational characteristics of ceramic angular contact ball bearings compared to conventional steel ball bearings. Their findings demonstrated that ceramic bearings provide improved thermal resistance and dynamic stability, particularly under high rotational speeds. Expanding on this, Zhang et al. [29] proposed a novel nonlinear dynamic model for rotor-bearing systems that incorporates variable contact angles. By modulating the preload-induced contact geometry, their approach successfully shifted primary resonance zones to lower speed regions, thereby enhancing high-speed operational stability. Further emphasizing the influence of loading conditions, Wang et al. [30] analyzed the dynamic response of ball bearings under unbalanced loads and concluded that axial vibrations significantly affect the behavior of rotor-bearing assemblies.

From an acoustic perspective, Sun et al. [31] conducted mechanical analyses on deep-groove ball bearings and highlighted that surface waviness and reduced machining precision at the raceways amplify axial vibrations of the rolling elements, leading to increased radiated noise. A series of studies by Shi et al. [32] and Bai et al. [33,34] further explored the vibro-acoustic behavior of ceramic angular contact bearings under different operating conditions. Their investigations included model 7009C [32,33] and 7003C [34], focusing on the impact of geometric tolerances, rolling element configurations, and ball counts. The results revealed that dimensional variations—particularly ball diameter tolerances—serve as critical factors influencing sound pressure levels and radiation directivity. These studies collectively underscore the importance of both material selection and geometric precision.

Recent investigations on ceramic angular contact ball bearings have highlighted their acoustic and dynamic advantages under high-speed conditions. Wu et al. [35] demonstrated that bearing cages contribute minimally to radiated noise, with peak sound pressure occurring at 28,000 RPM. Yan et al. [36] further analyzed the directional characteristics of full-ceramic angular contact ball bearings and reported that radial and axial sound pressure components are primarily associated with collision and friction mechanisms, respectively, with radial sound pressure attenuating more rapidly with increasing distance. Beyond ball bearings, Rho and Kim [37] examined hydrodynamic journal bearings and observed harmonics linked to internal components, emphasizing the importance of natural frequencies in bearing system design. Bouaziz et al. [38] showed that increased load and rotational speed elevate noise levels, while proper lubrication effectively reduces acoustic emissions by lowering friction and enhancing damping. These studies collectively indicate that, despite bearing-type differences, fundamental parameters such as stiffness, damping, and excitation mechanisms govern both vibrational and acoustic behavior across transmission systems. More recent work has further emphasized the role of contact boundary conditions, assembly clearances, and structural imperfections. Bai et al. [39] demonstrated that outer ring elastodynamics can introduce additional elastic deformation and alter contact load distribution between rolling elements and raceways, thereby modifying vibration and power loss characteristics, which in turn influence the acoustic radiation behavior of the bearing system. For full ceramic ball bearing systems, temperature-dependent fit clearance between the ceramic outer ring and steel pedestal was shown to significantly affect sound radiation characteristics, including peak angle and directivity [40]. A multi-sound source acoustic radiation model for ceramic angular contact ball bearings was proposed in [41], demonstrating that component-level eigenfrequencies appear clearly in noise spectra and that radiation noise varies nonlinearly with rotational speed. Innovative bearing structures have also been explored: finite element analyses of deep-groove ball bearings with multi-hollow outer rings revealed reduced contact stress and improved fatigue life [42]. Misalignment-related effects have gained increasing attention in recent years. A bearing–rotor dynamic model incorporating inner ring dynamic misalignment and three-dimensional clearance was developed and experimentally validated in [43], showing that misalignment introduces additional excitation forces and significantly alters axial vibration characteristics. Furthermore, combined angular misalignment and cage fracture faults were investigated through numerical and experimental approaches in [44]. The results revealed increased vibration amplitudes, pronounced impact responses, and characteristic low-frequency components associated with cage-related frequencies, providing valuable insight for fault diagnosis and condition monitoring. However, the dynamic response of ball bearings is governed not only by geometric design parameters but also by various external operating conditions, including frictional effects, lubrication behavior, thermal influences, and rotor imbalance excitation. If these factors were simultaneously incorporated into a fully coupled FEM–BEM framework, the superposition of multiple excitations, damping, and stiffness-modifying mechanisms would obscure the fundamental role of structural inertia in shaping contact boundary behavior. In particular, the coupled influence of friction, elastohydrodynamic lubrication (EHL), temperature-dependent material properties, thermal expansion, and imbalance-induced harmonic excitation would make it difficult to clearly identify the intrinsic energy transfer mechanisms associated with contact-induced dynamics. To preserve analytical clarity and establish a well-controlled numerical baseline, the present study adopts several fundamental modeling assumptions: frictional effects are neglected; lubrication and EHL effects are excluded; thermal influences, including temperature-dependent material properties and thermal expansion, are not considered; rotor imbalance excitation is omitted; and the housing structure is assumed to be rigid without structural compliance. Under these controlled conditions, the analysis isolates inertia-driven contact behavior as the dominant excitation mechanism and systematically examines its contribution to structural vibration and subsequent acoustic radiation within an energy-conservation framework.

In this study, a hybrid numerical approach integrating FEM for structural dynamics and BEM for acoustic radiation is employed to investigate ball bearing vibro-acoustic behavior under different contact boundary conditions. The proposed model is validated against experimental sound pressure data and used to analyze the relationship between modal behavior, structural vibration, and radiated noise across varying speeds. The results offer insight into optimal modeling strategies for high-speed bearing noise prediction and contribute to the broader understanding of vibro-acoustic coupling in rotational systems.

2  Theoretical Background

In this study, modal analysis is conducted to evaluate the dynamic characteristics of the bearing components, including natural frequencies and corresponding mode shapes. The goal is to understand how contact boundary conditions influence the structural vibration behavior, which subsequently affects the acoustic radiation patterns in the coupled FEM–BEM model. The calculation process is shown in Fig. 1.

Figure 1: Computational workflow of the proposed modeling framework.

2.1 Theoretical Background of LS-DYNA

LS-DYNA [45] is an explicit finite element solver widely employed for nonlinear transient dynamic simulations, particularly in problems involving high-speed impact, contact, large deformation, and material failure. Its core algorithm performs time integration of the equations of motion without the need for matrix inversion, making it highly efficient for large-scale problems. In the dynamic finite element model, the dynamic response of the structures is obtained by solving the equations of motion, which can be expressed as follows:

Mx¨(t)+Cx˙(t)+Kx(t)=F(t),(1)

where M, C, and K are the mass, damping, and stiffness matrices, respectively; and x(t) and F are the displacement vector and the external force vector. In explicit dynamics, the system state is updated iteratively based on previous time step values of displacement, velocity, and acceleration.

To ensure numerical stability, the critical time step ΔtE must satisfy the propagation time of the elastic wave across the smallest element be larger than the integration time step. ΔtE can be expressed as follows:

ΔtE=LEQE+QE2+cS2,(2)

where LE denotes the characteristic element length, cS is the elastic wave speed, and QE is a function of bulk viscosity coefficients. To ensure numerical stability, the explicit solver evaluates the time step by selecting the minimum allowable value among all elements, expressed as follows:

Δtn+1=asmin{Δt1,Δt2,Δt3,…,ΔtNb},(3)

where Nb is the number of elements. For numerical stability, the scale factor αs is typically set to 0.9 or a smaller value in the LS-DYNA procedure.

Contact interactions among the rolling elements, inner race, and outer race are modeled using a penalty-based contact algorithm. The normal contact force is expressed as:

Fn=kcδd,(4)

where kc is the contact stiffness and δd is the penetration depth. The contact stiffness is automatically computed based on the material properties and element size. In the localized contact region between the ball and the raceway, the deformation behavior is generally governed by Hertzian contact theory. The relationship between the contact force and deformation can be expressed as:

F=kHδ3/2,(5)

where F is the contact force, δ is the contact deformation, and kH is the Hertzian contact stiffness. The equivalent elastic modulus is defined as:

1E∗=1−ν12E1+1−ν22E2,(6)

where E1 and E2 are the Young’s modulus of the two bodies, ν1 and ν2 are the corresponding Poisson’s ratios, and E∗ is the equivalent (effective) elastic modulus. This nonlinear contact behavior governs stress distribution and energy transfer at the contact interface.

Under high contact stress, bearing materials may exhibit elastoplastic behavior. The yielding condition is described by the von Mises criterion:

σeq=12[(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2],(7)

where σeq is the equivalent (von Mises) stress, and σ1, σ2, σ3 are the principal stresses. Plastic deformation occurs when:

σeq≥σy,(8)

where σy is the material yield strength. In LS-DYNA, elastoplastic behavior can be modeled using the bilinear isotropic hardening model. The stress–strain relationship is defined as:

σ={Eε,ε<εyσy+Et(ε−εy),ε≥εy,(9)

where Et is the tangent (hardening) modulus, and εy is the yield strain.

In this study, LS-DYNA is utilized to simulate the transient dynamic behavior of a ball bearing under high-speed operational conditions. As shown in Fig. 2, a three-dimensional explicit model is developed to represent the inner ring, outer ring, rolling elements, and cage in detail. These components are discretized using higher-order solid elements and integrated with nonlinear contact definitions to simulate rotational freedom and relative sliding behavior. The structural analysis is performed using the explicit central difference time integration scheme, and the stable time increment is automatically determined according to the default Courant–Friedrichs–Lewy (CFL) condition, governed by the smallest characteristic element size and the material wave speed. To ensure numerical stability, the time step scale factor (TSSFAC) is set to 0.9, and no mass scaling technique is employed in order to preserve the physical inertia properties of the system. Contact interactions are defined using a penalty-based contact formulation within the explicit framework, and contact enforcement is achieved through the penalty method. The contact stiffness is automatically calculated by LS-DYNA based on the material Young’s modulus and characteristic element size (automatic contact stiffness scaling), without additional manual stiffness adjustment. Stiffness-based hourglass control is activated using default LS-DYNA parameters to suppress zero-energy deformation modes. However, in the present study, all components are assumed to remain within the elastic regime. This assumption allows the isolation of contact-induced dynamic behavior without introducing additional material nonlinearity. To focus specifically on the contact-induced vibration mechanisms, a purely elastic contact formulation is adopted. Accordingly, plastic deformation, frictional effects, and lubrication (including elastohydrodynamic lubrication, EHL) are neglected, thereby avoiding additional nonlinear dissipation and enabling a clearer interpretation of boundary-condition-driven energy transfer under an energy-conservation framework.

Figure 2: Schematic diagram of a three-dimensional ball bearing.

No additional physical damping or numerical regularization treatment is included; numerical damping arises solely from the explicit time integration scheme and the penalty-based contact algorithm. The transient structural responses, including velocity and displacement time histories, are exported for subsequent acoustic analysis using commercial software such as Virtual.Lab Acoustics. The sampling rate is set significantly higher than the highest frequency of interest to prevent aliasing. This mapping procedure preserves temporal consistency and energy integrity of the structural response, enabling detailed investigation of vibration transmission paths, contact-induced excitations, and dominant acoustic radiation modes associated with bearing dynamics.

2.2 Acoustic Theory Background

In this study, transient acoustic responses are analyzed using the time-domain boundary element method (TBEM) implemented in LMS Virtual.Lab [46]. The application of the BEM to time-domain problems was originally developed by Friedman and Shaw [47] and further extended by Cruse and Rizzo [48]. The acoustic field generated by multiple-sphere collisions has also been investigated using TBEM, providing a suitable framework for transient acoustic radiation analysis. The acoustic formulation is established under linear acoustic assumptions, in which pressure, density, and velocity fluctuations are small compared with their ambient mean values. Under the small-perturbation hypothesis, higher-order nonlinear terms are neglected, and the acoustic process is assumed to be adiabatic and inviscid. Consequently, sound radiation in ideal air is governed by the linear time-domain acoustic wave equation:

∇2p=1c2∂2p∂t2,(10)

where c is the sound velocity, p is the acoustic pressure, t is the time, and ∇2 is the Laplacian operator.

Within the TBEM framework, the wave equation is transformed into a time-domain boundary integral equation through the free-space retarded Green’s function G(x,y,t). Accordingly, the acoustic pressure at a field point can be expressed in terms of boundary pressure and normal particle velocity histories as

C(x)p(x,t)=∫s[∂G(x,y,t)∂nyp(y,t)−G(x,y,t)∂p(y,t)∂ny]dSy,(11)

where C(x) is the geometric coefficient, x denotes the field (collocation) point and y represents the boundary source point.

For structural–acoustic coupling, the normal structural velocity obtained from the LS-DYNA transient finite element analysis is prescribed as the acoustic source term. According to the linearized momentum relation, the normal component of structural surface velocity drives acoustic radiation and serves as the Neumann boundary condition in the TBEM formulation. The FEM–BEM coupling is performed via nodal interpolation and surface projection to ensure spatial consistency and temporal synchronization at the structural–acoustic interface. A one-way coupling strategy is adopted, assuming linear acoustic propagation and neglecting acoustic feedback to the structure, which is justified under weak-coupling conditions where structure-borne radiation dominates. Following the TBEM solution, the acoustic pressure is obtained in the time domain as p(t). The sound pressure level (SPL) is computed based on the root-mean-square (RMS) pressure within a specified time window:

prms=1T∫0Tp2(t)dt,(12)

and

SPL=20log10⁡(prmsp0),(13)

where the reference pressure in air is p0=20 µPa.

In this study, for A-weighted sound pressure levels (dB(A)), the time-domain pressure signal is first transformed into the frequency domain via Fourier transformation, and the A-weighting filter is applied prior to SPL computation. This procedure follows the default acoustic post-processing implementation in LMS Virtual.Lab. Instantaneous acoustic pressure is reported in Pa, and A-weighted sound pressure level is expressed in dB(A).

3  Boundary Condition Verification

This section aims to validate the accuracy of the proposed computational methods in both structural dynamic and acoustic simulations. A verification model consisting of three steel balls is employed to assess the structural dynamics and corresponding acoustic response. The structural simulation is conducted using the explicit time integration scheme in ANSYS LS-DYNA to model the transient impact behavior of the colliding steel balls. Subsequently, the resulting surface velocity data is utilized as input for the acoustic simulation using the BEM implemented in LMS Virtual.Lab. To evaluate the reliability and precision of the acoustic predictions, the simulated sound pressure results are compared with the experimental measurements reported in the literature [24]. This comparison serves to verify the feasibility and accuracy of the adopted structural–acoustic computational framework.

3.1 Modal Analysis of Steel Balls

To investigate the inherent dynamic characteristics of the steel spheres, a modal analysis was performed prior to the transient collision simulation. The analysis aimed to identify the natural frequencies and corresponding mode shapes that may influence the vibration response and acoustic radiation during impact events. After confirming the material properties and geometric specifications of the steel sphere, the finite element mesh was constructed as shown in Table 1 and Fig. 3. The steel ball had a diameter of 22.2 mm, with a material density of 7900 kg/m3, Young’s modulus of 195 GPa, and Poisson’s ratio of 0.3.

Figure 3: Single steel ball: (a) geometric model, and (b) finite element mesh discretization.

The modal analysis was conducted under free–free boundary conditions, with the number of elements increased from 64 to 5832 to examine convergence behavior. As illustrated in Fig. 4a, the frequency of the first elastic mode (Mode No. 7) converged toward 111.36 kHz, with convergence effectively achieved once the element count exceeded 2744, beyond which variations became negligible. Similarly, Fig. 4b shows the convergence of the second elastic mode (Mode No. 8), which stabilized at 117.71 kHz under the same threshold. Based on these results, a mesh size of 2744 elements was adopted for all subsequent structural modal and dynamic validations of the steel ball. Following mesh convergence verification, the first 20 modal frequencies of a single steel ball were extracted and are summarized in Table 2. Modes No. 1–6 correspond to rigid-body modes, while Mode No. 7 and higher represent elastic deformation modes. The first elastic natural frequency appears at 111.36 kHz, confirming that the steel ball’s vibrational response primarily occurs in the high-frequency range.

Figure 4: Mesh convergence plots of the first and second elastic mode frequencies of the steel ball: (a) Mode No. 7, and (b) Mode No. 8.

3.2 Modal Analysis of Contacted Steel Balls under Different Contact Conditions

The objective of this subsection is to investigate the natural frequencies of steel balls under different contact definitions without initial gaps. Modal analysis was conducted in ANSYS using two commonly adopted contact types: bonded and no separation. The bonded condition enforces a fully constrained interface, prohibiting both sliding and separation, while the no separation condition permits frictionless sliding but prevents normal separation, thereby simulating stick–slip behavior without torque transfer.

As shown in Fig. 5, the finite element model consists of three steel balls in contact, each discretized into 2744 solid elements, resulting in 8232 elements in total. Under both contact definitions, the first 20 natural frequencies were extracted, as listed in Table 3. Distinct differences are observed in the 4th to 6th rigid-body rotational modes: with no separation, the frequencies approach zero, indicating nearly unconstrained rigid-body motion, whereas with bonded contact, non-zero frequencies appear in the X, Y, and Z directions.

Figure 5: Finite element mesh discretization of the three steel balls used in the modal analysis.

Mode shape comparisons (Table 4) further clarify this behavior. Under a bonded contact, the three spheres rotate collectively about a common axis, with the constraint introducing rotational stiffness and producing finite natural frequencies. In contrast, the no separation contact allows independent rotation of each sphere, yielding decoupled rigid-body modes with near-zero frequencies.

3.3 Dynamic Validation of Steel Balls

The objective of this subsection is to validate the structural dynamic response through both dynamic theory and modal frequency correlation. A three-ball collision was simulated using the explicit time integration scheme in ANSYS LS-DYNA, with material properties consistent with Table 1.

As shown in Fig. 6a, three steel balls (O1, O2, and O3) are initially in contact without gaps. The coordinate system is defined with the centroid of O2 as the origin, O3 located along the +X-axis, and the +Y-axis pointing upward, while the Z-axis follows the right-hand rule. Ball O1 is assigned an initial velocity of 0.5 m/s toward O3. The Y and Z displacements of all balls are constrained, and steel is modeled as linear elastic to account for deformation during impact. Observation points are illustrated in Fig. 6b: each ball includes three surface points—Point 1 (−X), Point 2 (+Y), Point 3 (+Z)—as well as a centroid, enabling localized analysis of wave propagation and acoustic source formation.

Figure 6: Observation point locations on the steel ball: (a) geometric model, and (b) finite element mesh discretization.

Dynamic validation was conducted using centroid and surface data. The measured quantities include X-direction velocity, X-direction acceleration, and acceleration frequency response functions (FRFs). Velocity histories (Fig. 7a) show that O1 impacts O2, transferring momentum before O2 collides with O3. O2 reaches a maximum velocity of 0.46 m/s, while O3 attains 0.5 m/s, confirming conservation of momentum. Centroid accelerations (Fig. 7b) range from −4 × 104 to 5 × 104 m/s2. O2 exhibits a complete waveform, while O1 and O3 show half-wave responses, consistent with sequential momentum exchange. The acceleration FRFs (Fig. 7c) up to 500 kHz reveal first peaks at 2 kHz for O1 and O3, and 7 kHz for O2. Minimal response occurs near 25 kHz, beyond which multiple high-frequency peaks (173–497 kHz) appear, corresponding to elastic deformation modes. These higher-order frequencies are subsequently compared with surface responses and modal frequencies from structural analysis.

Figure 7: The center of mass of three steel balls: (a) velocity diagram, (b) acceleration diagram in the X direction, and (c) acceleration frequency response diagram.

The physical quantities at the surface observation points were further examined. The velocity time histories at the surface points during collision are shown in Fig. 8. As illustrated in Fig. 8a, the acceleration frequency response within 500 kHz exhibits the maximum peak at approximately 393 kHz at Observation Point 1. In Fig. 8b,c, the excitation frequencies are identical, with the first peak appearing below 25 kHz. The remaining peaks correspond well to the dynamic response frequencies observed at the centroids. The centroid acceleration responses were compared with the modal frequencies, as shown in Fig. 9. The blue solid line, black dashed line, and red dashed line represent the centroid acceleration responses of O1, O2, and O3, respectively. The blue circles on the zero axis denote the modal frequencies of a single ball, the black crosses correspond to the modal frequencies under bonded contact of three balls, and the red triangles represent the modal frequencies under no-separation contact of three balls. The results indicate that, for both boundary conditions considered, the peak response frequencies closely match the modal frequencies.

Figure 8: Acceleration frequency response of the three steel balls measured at surface observation points: (a) Point 1, (b) Point 2, and (c) Point 3.

Figure 9: Center of mass acceleration response vs. modal frequency.

3.4 Acoustic Simulation Verification

In this section, the BEM is employed to calculate the acoustic field generated by steel ball collisions. The accuracy of the simulations is validated against experimental data obtained from the Newton pendulum experiment reported in [24]. The ball surface serves as the acoustic boundary, discretized into 7,056 elements as shown in Fig. 10, with the coordinate system centered on O2. As shown in Fig. 11, the impact direction is aligned with the X-axis, while the Z-axis is oriented upward. Following Reference [24], quantitative validation was performed using a single observation point located 285 mm along the positive Z-axis from O2. Structural displacement results are mapped onto the acoustic model as boundary conditions, with the sound velocity set to 340 m/s and the air density to 1.225 kg/m3.

Figure 10: Steel sphere acoustic mesh discretization.

Figure 11: Time distribution of sound pressure of steel ball impact: (a) 6.4 × 10−6 s; (b) 2.5 × 10−5 s; (c) 4.9 × 10−5 s; (d) 9.9 × 10−5 s; (e) 1.51 × 10−4 s, and (f) 2.0 × 10−4 s.

The sound pressure distributions at different times are shown in Fig. 11. At early stages (6.4 × 10−6–2.5 × 10−5 s), sound pressure is generated during O1–O2 impact. At 4.9 × 10−5 and 9.9 × 10−5 s, the distributions correspond to energy transfers between O1–O2 and O2–O3, respectively, consistent with the acceleration histories in Fig. 7b. At later times (1.51 × 10−4–2.0 × 10−4 s), the collision concludes and only residual natural vibrations remain, producing weak acoustic responses. To validate the numerical method, the sound pressure at the single-point observation (285 mm along +Z) is compared with the experimental and simulated values from [24]. In [24], the FEM–BEM coupling assumed rigid-body behavior, whereas the present study incorporates elastic deformation. The comparison is shown in Fig. 12. Experimental data from [24] are represented by red markers (peak 0.524 Pa), while the numerical predictions of [24] are indicated by the black dashed curve (peak 0.352 Pa).

Figure 12: Comparison of the calculated results with experimental data and results reported in [24].

The present elastic-body simulation is shown by the blue solid curve (peak 0.543 Pa). The experimental and present results exhibit strong agreement, with a maximum difference of only 0.019 Pa. By contrast, the rigid-body predictions of [24] deviate substantially under positive pressures. During the transient stage, the impact contact process is highly nonlinear, resulting in short-duration, high-amplitude excitation. The corresponding time-domain pressure response is highly sensitive to contact stiffness, impact duration, and initial boundary conditions, which may account for discrepancies in waveform trends compared with experimental results. Nevertheless, the time-domain fundamental solution implemented in the LMS TBEM framework effectively captures the peak transient acoustic pressure and accurately reflects the dominant energy transfer mechanism. These findings confirm the necessity of elastic-body modeling and demonstrate the robustness and accuracy of the proposed FEM–BEM coupling framework.

4  Structural Vibration and Radiated Noise Analysis

Ball bearings, as indispensable transmission elements, serve critical roles in guiding shaft rotation, reducing frictional losses, prolonging service life, and sustaining both radial and axial loads. Under high-speed operating conditions, the relative motion of the rolling elements within the raceways gives rise to pronounced vibrations and radiated noise resulting from frictional interactions and impact phenomena. Accordingly, the dynamic and acoustic responses of ball bearings have become central topics of investigation in structural–acoustic research. The acoustic–solid coupling analysis of ball bearings encompasses three primary objectives: structural static analysis, structural dynamic analysis, and acoustic analysis. This study employs a combined FEM-BEM approach. FEM is utilized to model the static and dynamic behavior of the bearing, employing explicit integration for short-duration collision simulations. The results are compared with modal frequencies to identify sources of vibration. Subsequently, BEM calculates the radiated sound pressure for acoustic analysis, which is validated against experimental data. This validation facilitates a deeper correlation between modal and acoustic responses, enabling the identification of noise origins.

4.1 Ball Bearing Model Establishment

The bearing transmission model, illustrated in Fig. 2, is based on the SKF 7003C ceramic angular contact ball bearing. To represent the cage effect, linear springs are introduced between adjacent balls, while the assembly consists of an inner ring, an outer ring, twelve balls, and inter-ball springs. A summary of the principal geometrical parameters is provided in Tables 5 and 6. Rings are modeled with 52100 chrome steel, and the balls with silicon nitride ceramic. The calculated bearing mass (0.0323 kg) shows a minor deviation from the actual mass (0.038 kg), attributable to the excluded cage mass.

4.2 Bearing Spring Stiffness Verification

To examine the influence of the cage on system behavior, explicit finite element integration is conducted using ANSYS LS-DYNA in conjunction with LMS Virtual.Lab for structural dynamic and acoustic–solid coupling analyses. To preserve a constant contact angle among the rolling elements during high-speed operation, linear springs are introduced between adjacent balls to emulate the boundary constraints imposed by the cage. A dynamic simulation under fixed rotational speed is first carried out to evaluate the effect of spring stiffness on ball dynamics, with the stiffness coefficient calibrated against radiated noise levels reported in the literature [35].

4.2.1 Dynamic Simulation of Bearing System

The cage’s dynamic influence is analyzed using explicit time integration within a finite-element framework, examining how varying inter-ball stiffness affects high-speed rotation. The structural mesh (Fig. 13) contains 13,164 elements. The geometric center of the bearing is set as the origin; the X-axis points to the right, the Z-axis upward, and the Y-axis follows the right-hand rule. The red-highlighted ball above the bearing is labeled Ball 1. In the simulation, the inner ring rotates clockwise at 10,000 RPM about the Y-axis, while the outer circumference of the outer ring is fixed. The X-direction acceleration–time response is computed over 5 × 10−3 s. Linear springs are placed between adjacent balls to limit relative displacement and preserve the contact angle. According to the research in reference [49], it is reasonable and reliable to use a linear spring model to simulate the elastic connection of the cage, and it can accurately reflect the influence of the overall flexibility of the cage on the dynamic performance of the bearing. Three stiffness values are examined: 1 × 105, 5 × 106 and 2.5 × 107 N/m. As shown in Fig. 14, the overall dynamic trends are similar across stiffness levels—particularly in the 50–110 kHz band–while high-frequency content (≈400–450 kHz) is essentially unaffected. The response in this upper band is dominated by elastic (body-mode) dynamics, indicating that cage-equivalent stiffness is not a primary driver of the high-speed response. This outcome is consistent with the literature [35], which reports a minimal contribution of the cage to radiated noise at elevated speeds.

Figure 13: Ball bearing mesh discretization plot.

Figure 14: Comparison of k values for sphere-centroid dynamic response.

4.2.2 Structural–Acoustic Coupling Simulation

In this section, LMS Virtual.Lab is used to perform acoustic–solid coupling. The acoustic field is solved with the BEM, and experimental sound-pressure measurements for the 7003C bearing reported in [35] are used for validation. The bearing surface is taken as the acoustic boundary and discretized with a triangular mesh of 21,496 elements (Fig. 15). The coordinate system matches the structural model (origin at the bearing geometric center; +X to the right, +Z upward, +Y by the right-hand rule). The surrounding fluid is air with ρ = 1.225 kg/m3 and c = 340 m/s. Transient structural responses computed in ANSYS LS-DYNA are exported to Virtual.Lab and enforced as external loads via an indirect BEM formulation. The total simulation time is 5 × 10−3 s, and sound pressure is monitored at a single field point 100 mm above the origin along +Z, consistent with the measurement position in [35]. Simulations are conducted at 10,000 rpm for inter-ball spring stiffnesses of 1 × 105, 5 × 106, and 2.5 × 107 N/m. The resulting sound-pressure time histories (Fig. 16) yield peak levels of 72.74, 72.35, and 72.88 dB(A), respectively. Table 7 compares these with the experimental level of 73.96 dB(A) at 10,000 rpm [35]. The absolute errors are 1.22, 1.61, and 1.08 dB(A) (relative errors 1.65%, 2.18%, and 1.46%). These results indicate that the cage exerts only a minor influence on radiated noise, consistent with [35]. Among the cases, the 2.5 × 107 N/m stiffness yields the smallest error and is therefore adopted for subsequent analyses.

Figure 15: Ball bearing acoustic model.

Figure 16: Ball bearing 10,000 RPM sound pressure time diagram: (a) k = 1 × 105 N/m; (b) k = 5 × 106 N/m, and (c) k = 2.5 × 107 N/m.

4.3 Modal Analysis and Verification of Dynamic Response

To investigate the dynamic behavior of ball bearings during operation, the X-direction acceleration response is transformed into the frequency domain and compared with modal analysis results to identify vibration sources. Previous studies [9] have shown that both the global modal frequencies of the bearing assembly and those of its sub-components significantly influence the dynamic response. Extending this concept, the present study examines modal frequencies under different contact boundary conditions, clarifying the correspondence between response peaks and specific contact interactions, while modal shape analysis further elucidates the underlying vibration mechanisms.

4.3.1 Modal Analysis of Ball Bearings under Different Contact Conditions

ANSYS modal analysis was employed to investigate the influence of contact boundary conditions on ball bearing motion, with emphasis on their effects on dynamic and acoustic responses. Contact conditions were classified into two categories: constrained contacts and non-separation contacts. In the present study, frictional effects are not explicitly considered. The two boundary conditions, namely bonded and no separation, are therefore used to represent the rolling state and the combined rolling–sliding state, respectively. Based on this framework, four representative configurations (C1–C4) were defined, as summarized in Table 8: C1 (Fully constrained): The ball is rigidly connected to both the inner and outer rings, with no allowance for separation or sliding. C2 (Semi-constrained I): The ball is fully constrained to the inner ring and in non-separation contact with the outer ring. C3 (Semi-constrained II): The ball is fully constrained to the outer ring and in non-separation contact with the inner ring. C4 (Fully non-separation): The ball maintains non-separation contact with both the inner and outer rings. This classification enables a systematic evaluation of contact interactions and their influence on the vibrational characteristics of the bearing system.

4.3.2 Verification of Ball Dynamic Response

This study investigates the origin of vibration peaks in the dynamic response of a ball bearing by employing a moving coordinate system referenced to the ball’s center of mass. The response, modeled with a spring stiffness of 2.5 × 107 N/m, is compared against the modal frequencies of individual bearing components under various contact conditions, as illustrated in Fig. 17. The black solid line in the figure represents the acceleration amplitude at the ball’s center of mass. Blue asterisks (‘*’) indicate the frequencies at which peaks occur, and the red dotted line marks the highest peak. Component-specific modal indicators are also provided: black triangles (△) correspond to outer ring modes, while blue inverted triangles (▽) represent inner ring modes. The 54.0 kHz peak is found to coincide with an outer ring mode, whereas the 422.9 kHz peak aligns with both inner and outer ring modes.

Figure 17: Dynamic response vs. individual component modal frequencies.

Fig. 18 presents the dynamic response under different contact conditions. The black solid line shows the acceleration amplitude, blue asterisks (‘*’) mark peak positions, and the red dotted line identifies the dominant peak frequency. Contact configurations are labeled below: ‘×’ for C1 (blue), ‘×’ for C2 (black), ‘○’ for C3 (black), and ‘△’ for C4 (red). Each contact scenario exhibits a distinct set of excitation frequencies in the range of 54.0–425.9 kHz, indicating that no single contact condition can explain all observed peaks. The correspondence between the dynamic response peaks and the modal frequencies under different contact conditions is summarized in Table 9. All peak frequencies can be matched to specific modal frequencies, confirming that the observed peaks in the dynamic response arise from the excitation of natural frequencies under varying boundary conditions.

Figure 18: Dynamic response comparison of different contact element modal frequencies.

4.4 Validation of Acoustic Response under Varying Rotational Speeds

In the above section, the acoustic–solid coupling results were validated at a rotational speed of 10,000 RPM. To further evaluate the accuracy of the dynamic characteristics and acoustic–solid coupling of the transmission components under external excitation at different speeds, the present analysis is divided into two parts: comparison with sound pressure levels reported in the literature and assessment of the simulated acoustic response.

4.4.1 Verification of Sound Pressure Level

To validate the accuracy of the acoustic–solid coupling analysis of ball bearing motion, simulations were carried out at inner ring speeds of 10,000, 15,000, and 20,000 RPM. The structural dynamic response data obtained from FEM were exported to LMS Virtual.Lab and applied as boundary conditions for the acoustic analysis.

The sound pressure histories are shown in Fig. 19. At 10,000 RPM (Fig. 19a), the black solid line illustrates the instantaneous sound pressure variation, while the blue dotted line represents the average peak pressure of 67 dB(A) and the red dotted line the maximum pressure of 72.88 dB(A). At 15,000 RPM (Fig. 19b), the maximum pressure of 75.16 dB(A). At 20,000 RPM (Fig. 19c), the sound field exhibits an average peak of 76 dB(A) and a maximum value of 80.28 dB(A). In all cases, the sound pressure stabilizes into a periodic steady-state pattern after approximately 1 × 10−4 s, owing to the assumption of constant rotational speed rather than acceleration from rest. Once the rolling elements complete one full cycle, the acoustic field becomes periodic. As expected, both average and maximum sound pressure levels increase with rotational speed, with the highest recorded value reaching 80.28 dB(A) at 20,000 RPM. This observation agrees with fundamental vibration theory, whereby stronger excitations induce greater responses, and is consistent with previously reported findings [34,35,38]. Quantitative comparison with experimental data [35] further confirms the accuracy of the model. As summarized in Table 10, experimental sound pressures at 10,000, 15,000, and 20,000 RPM were 73.96, 75.29, and 80.27 dB(A), respectively, while the corresponding simulated values were 72.88, 75.16, and 80.28 dB(A). The resulting absolute errors are 1.08, 0.13, and 0.01 dB(A), corresponding to relative errors of 1.46%, 0.17%, and 0.01%. Fig. 20 illustrates this comparison, where experimental values are marked by black ‘○’, simulations by blue ‘X’, and errors by red dotted ‘●’. The close alignment in both trend and magnitude demonstrates that the proposed FEM–BEM coupling accurately reproduces the acoustic–solid interaction of high-speed ball bearings. Notably, the maximum absolute error remains within 1.08 dB(A), and the prediction accuracy improves with increasing speed. These results establish both the qualitative reliability and quantitative precision of the proposed methodology.

Figure 19: Time histories of sound pressure at different rotational speeds: (a) 10,000 RPM, (b) 15,000 RPM, (c) 20,000 RPM.

Figure 20: Comparison of sound pressure levels and simulation errors at different rotational speeds.

4.4.2 Verification of Acoustic Response and Boundary Conditions

Previous studies [33,37] have investigated acoustic responses and identified noise sources by analysing time-domain sound pressure at various rotational speeds, demonstrating that the modal characteristics of bearing components strongly influence radiated noise. However, the effect of different contact boundary conditions on this relationship has not been fully explored. To address this gap, the present study examines the acoustic response by comparing the modal frequencies of individual bearing components and the complete bearing assembly under varying contact boundary conditions, while the associated modal vibration patterns are further analysed to clarify the mechanisms governing noise radiation.

As shown in Fig. 21, the acoustic–solid coupling response was validated by analysing acoustic signals corresponding to modal frequencies at three different rotational speeds. The results indicate that both pressure amplitude and average sound pressure increase with rotational speed. Dominant acoustic amplitudes are mainly concentrated in the low-frequency range of 10–20 kHz, where multiple pronounced peaks are observed, while the amplitudes decrease significantly beyond 30 kHz. To determine the origins of these peaks, the acoustic responses were compared with the modal frequencies of individual bearing components and excitation frequencies under various contact boundary conditions.

Figure 21: Comparison of acoustic response at different speeds.

As summarised in Tables 11–13, excitation frequencies at 9.9, 22.5, and 46.0 kHz coincide with the modal frequencies of specific components, and all excitation peaks correspond to modal frequencies under contact conditions C1–C4. Moreover, the frequencies at 9.9 and 67.6 kHz are consistent with peak responses observed under multiple contact scenarios. These results confirm that noise generation is closely governed by the natural frequencies of bearing components under specific contact configurations. Minor discrepancies between modal and excitation frequencies are attributed to time-dependent variations in ball position, which cause slight shifts in excitation frequency. Overall, the modal characteristics of bearing components exhibit strong correspondence with the observed acoustic response peaks. Further comparison of Tables 11–13 reveals consistent peak frequencies at 10,000 and 15,000 RPM. At 10,000 RPM, dominant peaks occur at 9.7, 20.0, 33.2, and 46.0 kHz, while a peak at 29.4 kHz is observed at 15,000 RPM. Among the three rotational speeds, the C1 contact condition yields the lowest excitation frequency due to the conflicting boundary constraints between the rotating inner ring and the fully fixed outer ring, which restrict acoustic response and suppress pressure peaks. Most excitation peaks are concentrated in the 9.7–20.0 kHz range. At 20,000 RPM, an excitation frequency of 12.6 kHz closely matches the 11.8 kHz peak at 15,000 RPM, and both exhibit similar modal vibration patterns, indicating strong ball–outer ring interaction. Modal shape analysis confirms that the outer ring is the dominant vibration source, consistent with previous dynamic response studies. Overall, these results demonstrate that acoustic response peaks are strongly correlated with the modal frequencies of bearing components and are significantly influenced by contact boundary conditions, validating the effectiveness of the proposed modal-based acoustic analysis methodology.

4.5 Spatial Distribution of Acoustic Radiation at Different Rotational Speeds

To further investigate the spatial characteristics of acoustic radiation from the ball bearing, a three-dimensional directivity analysis was conducted in three orthogonal planes. With the geometric center of the bearing defined as the origin, field points were distributed along circular arcs of 100 mm radius in the XY, YZ, and XZ planes as shown in Fig. 22, with an angular sampling interval of 1°, yielding 360 observation points per plane. Simulations were performed at rotational speeds of 10,000, 15,000, and 20,000 RPM, and the maximum, minimum, and peak-to-peak SPL differences across the three planes are summarized in Table 14.

Figure 22: Maximum SPL at different plane directivity: (a) XY plane, (b) YZ plane, (c) XZ plane.

As observed across all three planes, the overall SPL increases with rotational speed, consistent with the single-point validation results reported in Section 4.4.1 and Table 10. The XZ plane exhibits the smallest angular variation, with peak-to-peak differences of 5.68, 5.37, and 4.52 dB(A) and a maximum SPL of 82.50 dB(A) at 20,000 RPM. The XY plane, perpendicular to the rotation axis, shows differences of 8.71, 6.42, and 7.44 dB(A) and a maximum of 85.45 dB(A), reflecting the non-uniform circumferential distribution of ball–raceway impact forces in the radial direction. The YZ plane, containing the rotation axis, exhibits the largest peak-to-peak differences of 9.39, 8.28, and 7.91 dB(A) and the highest maximum SPL of 86.42 dB(A) at 20,000 RPM, attributable to the simultaneous capture of both radial and axial radiation components, which is consistent with the outer ring mode dominance identified in Section 4.3, and further supported by the acoustic response analysis in Section 4.4.2. Notably, the peak-to-peak differences in both the YZ and XZ planes decrease monotonically with increasing speed, indicating a trend toward more spatially uniform radiation at higher rotational speeds.

The multi-lobe directivity patterns observed in all three planes reflect the superposition of excitation contributions from multiple rolling elements at different angular positions, and the progressive outward expansion of the contours with increasing speed is consistent with the escalating contact-induced excitation energy identified in Section 4.3.2. These findings demonstrate that the proposed FEM–BEM framework is capable of reproducing not only the overall sound pressure magnitude but also the three-dimensional spatial distribution of the acoustic field, providing a comprehensive validation of the model’s directivity prediction capability.

5  Conclusion

This study developed and validated a structural–acoustic coupling methodology that integrates FEM with BEM for analyzing the vibration and noise of high-speed ball bearings. First, the explicit FEM integration scheme was verified through a three-ball collision model, thereby demonstrating accurate momentum transfer and stable dynamic responses. Subsequently, modal analysis under different contact conditions confirmed that vibration frequencies originated from elastic modes activated by contact interactions. Moreover, acoustic validation against experimental data showed excellent agreement, with maximum discrepancies below 0.02 Pa. In addition, a systematic investigation of a 7003C ceramic angular contact ball bearing indicated that cage stiffness had negligible influence on dynamic and acoustic responses, which is consistent with prior studies. Furthermore, simulations at 10,000–20,000 RPM revealed increasing acceleration and radiated noise with speed, while deviations from experimental results remained within 1.08 dB(A). By correlating response peaks with modal frequencies, it was identified that contact-induced modes—particularly outer ring excitations—served as the dominant sources of acoustic peaks. Therefore, the proposed framework not only confirms the accuracy of the FEM–BEM coupling approach but also provides practical guidance for NVH optimization in ball bearing design. Although the present study successfully captures frequency response peaks governed by primary structural modes, the detailed influences of contact angle variations, material property differences, lubrication conditions, bearings with defects, and preload effects warrant further investigation. Future work will systematically incorporate these critical parameters to more comprehensively represent dynamic and acoustic behaviors under realistic operating conditions.

Acknowledgement: Not applicable.

Funding Statement: Financial support from the National Science and Technology Council (NSTC), Taiwan, under Grant No. 114-2221-E-019-049, is gratefully acknowledged.

Author Contributions: Chiao-Yang Kuan: Conceptualisation, Investigation, Data curation, Writing—original draft. Yung-Wei Chen: Conceptualisation, Investigation, Supervision, Writing—original draft, review & editing. Jian-Hung Shen: Supervision, Writing—original draft, review & editing. Yen-Shen Chang: Supervision, Writing—original draft, review & editing. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: All data generated or analyzed during this study are included in this published article.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

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