Dynamic modeling of drivetrains is a highly required and indispensable task in the efficient and reliable design, development, and optimization of many engineering applications and modern vehicles. More recently, it provides a comprehensive framework in the context of NVH analysis for understanding, predicting, and mitigating vibrations and noise, ensuring optimal performance, passenger comfort, and regulatory compliance. As a determinative factor, damping is well known for its role in the design of structures to minimize noise, structural instability, and fatigue failure of components. Damping mechanisms dissipate energy, reducing the amplitude of vibrations and mitigating the risk of resonance. The internal damping of the elements including bearings and supports, shafts and gears represents different levels of energy dissipation under specific running conditions. Incorporating damping into the analysis, particularly in the gear meshes, shafts, and bearings, is essential for accurately predicting the dynamic response. The presence of damping in these components is crucial for enhancing the stability and longevity of the gearbox under varying torque conditions. Damping in gear meshes primarily arises from three sources: internal hysteresis of gear teeth, lubricant film (oil squeeze), and surrounding structural contributions (Ref. 1). Gear mesh damping plays a critical role in attenuating dynamic responses, especially near resonance conditions. While damping has a minor effect on natural frequencies, it significantly reduces resonance amplitudes and mitigates nonlinear behaviors such as backlash and parametric excitation. Early modeling approaches employed constant viscous damping ratios to capture energy dissipation due to internal and lubricant-induced friction. However, time-varying and nonlinear damping models have been shown to yield more accurate dynamic predictions. For example, Amabili and Rivola (Ref. 2) introduced a single-DOF gear model with meshing damping proportional to time-varying stiffness, while others (Refs. 3, 4) incorporated backlash and impact effects in nonlinear damping frameworks. Li and Kahraman (Ref. 5) developed a viscous damping model derived from elastohydrodynamic lubrication (EHL) theory, linking damping behavior to shear stress and transient contact conditions. Their findings indicate that damping increases with torque and decreases with speed and temperature due to film thickness changes. Further studies (Refs. 6–8) quantified nonlinear damping under varying torque, speed, and lubrication regimes, showing damping ratios typically ranging between 5.3 percent and 8 percent. Yousfi et al. (Refs. 9,10) advanced this by formulating damping in the time-frequency domain using wavelet-based methods, incorporating operating condition variations and demonstrating that damping is inherently dependent on load, speed, and temperature. This study investigates the influence of internal damping within drivetrain components under varying loading conditions, with a particular focus on its impact on NVH characteristics. A representative two-stage gearbox model is employed as the basis of analysis. Using KISSsoft, a forced response analysis is performed to evaluate the dynamic behavior of the gearbox when subjected to multiple excitation sources. This approach offers a comprehensive view of the system’s vibrational response, identifying regions susceptible to elevated noise or vibration. The core of the study lies in isolating and examining the damping contributions of bearings, shafts, and gear meshes individually, thereby clarifying their specific roles in mitigating meshing contact forces and bearing reaction forces. Damping in the NVH Studies The evolution of damping strategies in drivetrain dynamics has significantly advanced the field of noise, vibration, and harshness (NVH) performance evaluation. From structural damping in EV components to torsional vibration mitigation in hybrid systems, damping strategies are integral to optimizing drivetrain design. Recent studies on this topic have mainly focused on developing advanced damping materials and techniques tailored to specific drivetrain configurations to further enhance NVH performance. From foundational studies to advanced methodologies utilized in simulations, research has continuously improved the understanding and application of damping mechanisms. The reviewed studies underscore the significant role of damping in enhancing NVH performance in vehicle drivetrains. NVH performance is primarily governed by the interaction between meshing dynamics and the structural response of the drivetrain. Viscous mesh damping is therefore essential in dynamic models, particularly for polymer and lightweight gear applications. In the absence of adequate damping, dynamic excitations from gear meshing lead to increased noise emissions, elevated dynamic loads, and accelerated wear of gears and bearings. A detailed modeling approach was presented in (Ref. 11), incorporating steady-state dynamics and radiated noise predictions. The influence of harmonic excitations, load torque, and rotational speed on noise and vibration was highlighted. Bozca (Ref. 12) employed a meshing stiffness-proportional damping model to suppress rattle noise in automotive transmissions through transmission-error-based design optimization. Similarly, Zubik et al. (Ref. 13) quantified the effect of design variables—including torque, imbalance, backlash, and torsional stiffness—on gearbox noise, providing a useful diagnostic and design tool.

Material damping has also been studied extensively. O’Rourke and Grander (Ref. 14) analyzed how material selection impacts gear noise, while subsequent work (Ref. 15) compared the damping behavior of unreinforced, glass, and carbon fiber reinforced nylon 6/6 gears. Sharma et al. (Ref. 16) confirmed that glass fiber-reinforced plastic gears offer superior NVH performance under light-load conditions, making them viable alternatives to metal gears in automotive applications. Additional studies, such as those by Inoue et al. (Ref. 17) and Zhou et al. (Ref. 18) have explored advanced vibro-acoustic models that integrate time-varying damping with structural and acoustic coupling, showing improved prediction accuracy of gear noise radiation. Recent efforts also emphasize the role of variable damping in accounting for operational effects such as load-dependent contact damping and temperature-sensitive behavior in polymer gears (Ref. 19). More recently, the study in Ref. 20 presented some guidelines to properly adjust the gear mesh damping for achieving better NVH characteristics of a powertrain system. This goal was achieved by choosing the gear body material with higher damping properties within the required torque ratio specifications. Model Setup The electric axle analyzed in the following is a single-speed, two-stage gearbox powering the front wheels of an electric vehicle, as shown in Figure 1. The analysis is carried out for the nominal input torque of 320 Nm applied to the input shaft, which is considered here as the reference boundary with a nominal speed of 2,742 rpm. Table 1 specifies the gear mesh data of the gearbox. The microgeometry modifications (Table 2) are designed to optimize the tooth profile for reducing the excitations: helix angle modification and crowning are adopted to reduce the face load factor KHb under various load conditions, while tip relief and profile crowning are adopted to eliminate contact shock and reduce the peak-to-peak transmission error. For brevity, the depicted numbering shown in this figure for the bearings and gears is used from now on within the manuscript.

Furthermore, Table 3 shows the bearings specifications for the electric axle. In this Table, following parameters are specified: Di: inner diameter, DO: outer diameter, B: width, C: dynamic load rating, CO: static load rating, PdO: nominal diametral clearance, an PaO: nominal axial clearance.

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