Surface Waviness of EV Gears and NVH Effects—A Comprehensive Review

Abstract

Electric vehicle (EV) drivetrains operate at high rotational speeds, which makes the noise, vibration, and harshness (NVH) performance of gear transmissions a critical design factor. Without the masking effect of an internal combustion engine, gear whine can become a prominent source of passenger discomfort. This paper provides the first comprehensive review focused specifically on gear tooth surface waviness, a subtle manufacturing-induced deviation that can excite tonal noise. Periodic, micron-scale undulations caused by finishing processes such as grinding may generate non-meshing frequency “ghost orders,” leading to tonal complaints even in high-quality gears. The article compares finishing technologies including honing and superfinishing, showing their influence on waviness and acoustic behavior. It also summarizes modern waviness detection techniques, from single-flank rolling tests to optical scanning systems, and highlights data-driven predictive approaches using machine learning. Industrial case studies illustrate the practical challenges of managing waviness, while recent proposals such as controlled surface texturing are also discussed. The review identifies gaps in current research: (i) the lack of standardized waviness metrics for consistent comparison across studies; (ii) the limited validation of digital twin approaches against measured data; and (iii) the insufficient integration of machine learning with physics-based models. Addressing these gaps will be essential for linking surface finish specifications with NVH performance, reducing development costs, and improving passenger comfort in EV transmissions.

1. Introduction

The electrification of automotive powertrains has heightened the importance of quiet gear transmissions. In conventional vehicles, engine noise often masked gear whine, but electric vehicles (EVs) lack this masking effect [1]. Even minor gear acoustic disturbances that were previously inaudible can now become prominent to passengers. As a result, gear noise ranks among the top noise, vibration, and harshness (NVH) concerns in EV design. Gear whine is primarily excited by gear transmission error (TE)—deviations in the smooth rotation ratio due to imperfections in tooth geometry and meshing [2]. When a gear pair exhibits TE, it generates vibratory forces at the meshing frequency and its harmonics. These forces can transmit through the drivetrain structure as tonal noise. Engineers traditionally address TE by optimizing gear macro-geometry, such as tooth numbers, module, and contact ratio. They also apply microgeometry corrections, like profile and lead modifications (e.g., tip relief and crowning), to ensure smooth load transfer [1,3]. Indeed, an ideal involute gear with perfect geometry under load would have zero TE, and thus no mesh excitation. However, real gears inevitably have manufacturing tolerances and deflections that introduce some TE even after design optimizations [3].

Importantly, improving gear dimensional quality does not always guarantee lower perceived noise. Müller and Gorgels (2023) observed that a set of gears manufactured to very tight tolerances (so all teeth are nearly identical) can exhibit low overall sound pressure levels yet still produce an annoying tonal whine [3]. This counterintuitive outcome arises because a perfectly uniform gear concentrates vibrational energy into a narrow-band tonal frequency (the fundamental mesh order). The human ear finds this unpleasantly tonal. In contrast, the presence of slight, well-distributed irregularities can “spread” the vibrational energy into multiple sidebands or ghost orders of lower amplitude, making the noise spectrum broader and less tonal. In other words, absolute minimization of all errors is not always optimal for sound quality—the distribution and nature of micro-errors matter. This realization has shifted attention toward surface microgeometry characteristics, particularly surface waviness, as a key factor in gear NVH [3].

Surface waviness refers to repetitive, wave-like deviations on the gear tooth surface with wavelengths intermediate between roughness and form error. Unlike one-time form errors (e.g., a single profile slope deviation) or random fine roughness, waviness typically manifests as a pattern of undulations that may repeat every tooth or over multiple teeth (sometimes called “ripples” when regularly spaced). Such waviness produces tonal sidebands, commonly referred to as ghost orders [4]. These ghost orders are sideband frequencies not directly tied to the gear tooth count, and they arise purely from manufacturing errors rather than the ideal gear geometry. Ghost orders cannot be eliminated by design alone [5]—they are a signature of process-induced deviations. In automotive transmissions, ghost tonal components have been identified as causes of unexplained whining or humming noises even when gears meet traditional quality grades [6]. Waviness is thus an insidious contributor to NVH: small amplitude ripples on the order of just a few tenths of a micron can lead to perceptible noise complaints [7].

Several EV transmission patents and prototypes use alternative architectures such as variators, chain drives, or hydraulic couplings. These solutions reduce vibration load and shorten shift time. They are not yet dominant in high-volume EV production. Compact helical gear units remain the preferred choice due to efficiency, durability, and manufacturability. Understanding and mitigating surface waviness effects in geared transmissions is therefore still highly relevant for EV NVH research.

Noise and vibration control is increasingly linked to sustainable mobility goals. Manufacturers need to reduce prototype testing and improve efficiency, which has raised interest in predictive methods and data-driven quality control. At the same time, research activity on gear surface waviness has grown rapidly, with new measurement techniques, machine learning applications, and digital twin frameworks emerging in recent years. However, the literature remains fragmented, scattered across journal papers, patents, and industrial reports. This review brings these strands together and provides a consolidated view of how gear surface waviness affects NVH in EV drivetrains, underlining both its technical and societal relevance.

This review focuses on the interplay between gear surface waviness and NVH in the context of electric vehicle drivelines. Section 2 introduces the review methodology and key definitions. Section 3 provides a structured literature overview of studies investigating waviness effects on gear dynamics and noise, including both experimental and computational findings. Within this framework, we examine how waviness is characterized and how it excites gear noise, compare different tooth finishing processes (grinding, honing, superfinishing, etc.) and their impact on noise, and summarize available measurement and detection methods from traditional single-flank tests to cutting-edge optical scanning systems. Several industrial case studies are also reviewed, illustrating real-world gear noise issues linked to waviness and how manufacturers addressed them (including one case of a helicopter transmission where superfinishing solved a tonal noise problem [2]). In Section 3.8, we identify gaps and inconsistencies in the literature—for example, conflicting conclusions on whether gear noise is dominated by surface finish or by macro-level errors [6], and the lack of standardized waviness specifications in gear standards [3]—and highlight future research directions, such as the incorporation of machine learning and digital twin approaches for predictive noise control [6]. Section 4 discusses the implications of these findings, and Section 5 provides the conclusions. By consolidating knowledge from over 80 publications (including journal papers, conference proceedings, and patents), this article aims to serve as a comprehensive reference on the state-of-the-art understanding of gear waviness and NVH, and to guide future innovations in this field.

The logical structure of this review is illustrated in Figure 1. Section 2 introduces the methodology and key definitions, while Section 3 addresses waviness excitation mechanisms, manufacturing influences, measurement and detection, case studies, and future directions. Section 4 discusses the implications, and Section 5 concludes the findings.

2. Materials and Methods

Review Methodology: This article is structured as a narrative literature review in an MDPI-style format, encompassing academic research, industry reports, and patent literature. We conducted an extensive search of scientific databases (ScienceDirect, IEEE Xplore, MDPI, etc.), using keywords such as gear noise, surface waviness, ghost frequency, electric vehicle NVH, gear finishing noise, and gear measurement. Approximately 100 sources from the years 1980–2025 were screened. We included seminal early works (e.g., Masuda et al. 1986 on finishing effects [8]), classical references on gear noise (e.g., Houser 2001 [9]), as well as the most recent studies up to 2024–2025 (including Tian et al. 2024 [10] and Horváth 2025 [6]). Trade publications and white papers (Gear Technology, Gear Solutions magazines) were reviewed to capture practical case studies and industry best practices [11]. We also examined patent databases for relevant inventions, which yielded insights into novel methods (for example, a 2023 US patent on optical waviness analysis for noise prediction [5] and a 2019 patent on adding tooth compliance to reduce TE [12]). All sources used are openly accessible or were obtained through institutional libraries. Information was synthesized qualitatively; no new experimental data were generated.

Key Concepts and Definitions: For clarity, we adopt the following definitions in this review. Surface waviness refers to periodic undulations on the gear tooth surface with spatial periods longer than surface roughness (which is high-frequency, random asperity texture) but shorter than the total profile or lead form. In practice, waviness may be defined by filtering the measured profile to isolate certain wavelength bands (often in the order of one to a few teeth pitch lengths). Transmission error (TE) is the difference between the actual angular position of the driven gear and the position it would be in with perfect gears, often expressed in arcseconds or micrometers at the pitch circle [3]. TE can be decomposed via Fourier analysis into harmonics corresponding to the gear mesh (tooth count) frequency and its multiples, as well as sidebands (ghost orders) in between [13]. Ghost orders specifically denote vibration orders that are not integer multiples of the tooth mesh frequency—these typically arise from periodic errors such as waviness, runout (eccentricity), or indexing errors. In contrast, mesh harmonics (e.g., 1×, 2× mesh frequency) come from the deterministic engagement of teeth and any load-dependent stiffness variation. Throughout this review, gear noise is primarily considered in terms of airborne noise radiated from vibrating structures excited by gear meshing forces, although structure-borne vibration metrics are also referenced when relevant [14].

It is correct that the overall vibration response of a transmission system is strongly determined by system-level kinematics and excitation frequencies. However, surface waviness at the micron scale acts as a periodic excitation source that couples into these system dynamics. Even small undulations can generate ghost orders or amplify existing mesh orders, which are then radiated as tonal noise by the gearbox housing. In this sense, surface waviness does not replace kinematic factors but interacts with them, providing a micro-level origin for macro-level NVH phenomena. This justifies the focus of the present review on undulation effects, while acknowledging that they must always be interpreted within the context of the full drivetrain dynamics.

According to GOST standards, the term “surface undulation” is not explicitly defined. GOST instead uses parameters such as fluctuation of the measuring center distance per revolution, radial runout, tooth thickness variation, or the length of the engagement line. Each of these parameters indirectly reflects aspects of periodic deviations on the tooth surface. In international literature, the term “waviness” or “undulation” is commonly applied to describe mid-range, wave-like deviations that lie between roughness and form error. These deviations can be represented as periodic sinusoidal components of the measured surface profile. Their amplitudes and orders can be obtained from Fourier analysis. The GOST parameters listed above correspond to the envelope of these deviations. The term “undulation” emphasizes the periodic nature of the error. This review adopts the term to maintain consistency with current NVH literature. At the same time, it acknowledges that GOST-compliant parameters can be used for measurement and validation.

Figure 2 illustrates how gear tooth surface deviations can be decomposed into long-wavelength form error, mid-wavelength waviness, and short-wavelength roughness. While GOST standards define parameters that indirectly capture these effects, the term “waviness” highlights the periodic mid-range deviations that are critical for NVH analysis.

NVH Measurement Techniques: To understand how various studies quantify gear noise, a brief overview of test methods is warranted. Laboratory evaluations often use single-flank transmission error (SFT) tests, where a gear is run in mesh with a master gear at low speed and high-resolution encoders capture the instantaneous transmission error as a function of rotation angle. SFT effectively measures the composite effect of profile deviations, runout, and waviness on meshing precision, and its output can be transformed into an order spectrum (frequency domain) to identify dominant error orders [15,16]. Figure 3 illustrates an example order spectrum from a single-flank test on a gear: peaks at the tooth mesh order and its multiples (red spikes) indicate the fundamental and harmonic TE, whereas smaller peaks in between (“ghost orders”) reveal periodic errors not tied to tooth count.

Another common approach is structure-borne noise (SBN) testing, where an accelerometer on a gear fixture measures vibrations while the gears are run at speed under light load. SBN tests can excite gear and fixture resonances, giving insight into which frequencies might become audible in a gearbox [17,18]. Roll testing (double-flank or single-flank) with acoustical sensors is widely used in industry for end-of-line noise inspection. Traditionally, 100% noise testing of all gears was impractical due to the time required for detailed measurements. However, modern innovations such as Gleason’s Gear Rolling System with Integrated Laser (GRSL) have enabled high-throughput inspection; this system combines a rolling test with simultaneous laser scanning of the gear flanks. The laser sensors capture topography (profile and lead) as the gear rolls, performing an inline waviness analysis that correlates surface deviations with expected noise behavior. Such systems represent the state-of-the-art in bridging metrology and NVH prediction on the factory floor [19,20,21,22,23].

Laser scanning sensors have been shown to capture surface waviness with high spatial resolution. These systems offer the advantage of non-contact operation compared to traditional rolling tests. Most reported applications are still limited to laboratory environments. Only a few examples demonstrate their integration into high-volume industrial production. The captured data provide accurate surface deviations, but the interpretation for NVH prediction is often incomplete. In many studies the relationship between measured waviness and radiated noise remains qualitative. A standardized framework is still missing to translate optical measurements into predictive NVH models.

Data-Driven Analysis: In recent years, researchers have applied statistical and machine learning methods to large datasets of gear measurements and noise outcomes [6]. This review documents those studies and their methodologies. We note that many newer works employ predictive modeling where manufacturing data (like flank waviness amplitude, profile error, etc.) are used to predict whether a gear will be noisy. Where relevant, we mention the size of datasets and validation approaches used in such studies, to highlight current capabilities and limitations of ML in gear NVH prediction. Section 2 of each cited study was consulted to ensure we accurately report how results were obtained (for instance, whether a finding was from simulation or actual noise measurement, whether tonal noise was evaluated via sound power or psychoacoustic metrics, etc.).

Table 1. Selected publications on gear surface finish, waviness, and noise.

Study (Year)	Scope	Methodology	Key Findings
[8]	Tooth finishing method vs. noise	Experimental (vibration tests of gears ground vs. honed)	Honed gears showed lower excitation at mesh frequency than ground gears; finishing process affects tonal noise. Early evidence that surface condition alters noise.
[9]	Frictional noise and surface finish	Experimental (gearbox test with ground vs. superfinished gears; various lubricants)	Superfinishing (surface polishing) reduced high-frequency “hiss” noise by ~3–6 dB. Surface roughness and fine waviness contribute to noise; smooth surfaces notably quieter.
[21]	Long-wave form errors at high Ex speed	Theoretical and experimental (modeling + tests of gears with imposed eccentricity/waviness)	Low-order vibrations (e.g., runout, ovality) significantly increased dynamic TE and noise at high rpm. Emphasizes need for tighter control of long-wave errors in EV gears.
[10]	Review of gear grinding vs. honing for noise	Literature review (focused on finishing technologies)	Modern grinding/honing techniques can minimize waviness and tonal noise. Recommends combined processes (grind + hone) for EV gears to achieve smooth flanks and low noise.
[22]	Macro-geometry errors vs. performance	Simulation and analysis	Even small macro errors (lead slope, alignment) can amplify mesh excitation. Implies that in a marginal design, waviness effects are worse. Holistic tolerance of both macro and micro needed.
[6]	Data-driven analysis of manufacturing vs. noise	ML on production data (hundreds of gears, with features like waviness, profile error; correlated to noise tests)	Surface waviness emerged as one of top predictors of noisy gears. ML models could flag gears with high waviness that would fail NVH, enabling early intervention. Demonstrates importance of waviness in large-scale production.
[4]	Detection of gear ripples (waviness)	Developed sine approximation method for measurement data; case studies	Introduced a practical method to quantify waviness from standard inspection data, correlating certain waviness orders to noise. Showed that previously undetectable ripples (sub-micron) can be identified and tied to machine faults.
(various patents) e.g., JP5427078B2 (2014) and US2023/0003574A1	Ghost noise detection and analysis	Patented methods (algorithmic approaches in metrology machines)	Propose advanced analysis of measured flank topography to predict ghost noise. Real-time waviness frequency analysis in inspection can identify gears prone to noise and even infer machine condition, enabling proactive quality control.
[12]	Gear tooth structural modification	Patent (design and manufacturing method)	Describes adding a slit/cutout in gear teeth to introduce bending compliance, thereby smoothing mesh-stiffness variation and reducing TE-induced noise. Illustrates innovation beyond surface finish—altering tooth geometry to mitigate noise.

(in Section 3) will summarize key details of several influential papers. Machine learning has shown that surface parameters can predict gear noise. These models can capture nonlinear relations between geometry and NVH. This is a step forward compared to classical regression approaches. Most applications still use small datasets. Small datasets limit generalization and reproducibility. Many models work as black boxes without physical interpretation. Reported results often give accuracy but not causal explanation. Only a few studies include psychoacoustic metrics for tonal noise. Validation with industrial data under varying loads is still rare. These gaps show that machine learning is promising but not yet mature. The main obstacles are data availability, interpretability, and validation in realistic conditions.

Overall, by combining information from diverse sources under a consistent framework, this review adopts a systematic comparative approach: we critically compare results from different authors, identify consensus versus discrepancies, and explicitly point out gaps where further investigations are needed. In the following sections, results from the literature are organized into thematic sub-topics for clarity.

3. Gear Surface Waviness and NVH: Literature Review

3.1. Noise Excitation Mechanisms of Waviness

While Section 2 introduced the general definitions of waviness characterization, here we briefly recall these aspects as they are directly linked to the excitation mechanisms.

Gear surface waviness can be visualized as a superimposed gentle “rippling” on the otherwise smooth tooth flank. If one inspects a high-precision ground tooth with a profilometer, waviness appears as periodic undulations around the nominal involute profile or helix line. The primary mechanism by which waviness excites noise is through periodic transmission error: as each wave on the driving gear’s tooth flank contacts the driven gear, it causes a slight acceleration or deceleration in the meshing due to geometry error, contributing to TE fluctuations. If a gear has an n-tooth waviness pattern (for example, three undulations around the circumference, sometimes called a three-fold ovality), it will introduce a TE component at n times per revolution, i.e., at an order that is n times the shaft rotation frequency [24]. This may manifest as a ghost order in vibration spectra. For instance, a three-wave lobing (tri-lobed gear) produces a 3× rotational frequency vibration that is not an integer multiple of the tooth mesh frequency; it appears between mesh harmonics in the order spectrum. High-order waviness (waves per tooth or every few teeth) can generate sidebands around the mesh frequency. If the amplitude of waviness is significant (even just a micron or less), these TE variations become periodic excitation forces. Should any of those excitation frequencies coincide with a structural resonance of the gear shaft or housing, the vibrations are amplified and airborne noise can result [25,26].

The underlying excitation mechanism can be summarized as follows: surface waviness causes periodic modulation of the transmission error, generating ghost orders around the mesh frequency. If these coincide with structural resonances, the vibration is amplified and radiated as tonal noise, which is subject to psychoacoustic evaluation in terms of tonality and sharpness. Figure 4 illustrates this mechanism in schematic form.

Undulation refers to the medium-scale waviness that remains on the tooth profile after removing both the overall form error and the fine roughness. It is typically identified in profile or lead measurements carried out with a tactile profilometer or an optical scanner. The recorded data are processed with standardized ISO 16610 filters, which separate long-wavelength form deviations, mid-wavelength waviness, and short-wavelength roughness [27]. Waviness can be quantified with three main indicators. The first is the amplitude of individual waviness orders, which reflects the height of a specific ripple pattern on the tooth surface. The second is the root mean square value within a defined order range, which describes the average energy of the ripples. The third is the peak-to-valley value in the same range, which shows the maximum difference between the highest and lowest points. In practice, these values are usually kept within a few microns for most gear modules. GOST and ISO standards do not provide an explicit definition of “undulation,” but tolerances such as runout, tooth thickness variation, and center distance fluctuation indirectly restrict the allowable level. NVH research emphasizes undulation because even small periodic deviations can generate tonal components when all other parameters remain within standard limits.

Undulation is emphasized in NVH research because periodic deviations can generate tonal components even when all other tolerances are within standard limits.

Crucially, ghost excitations from waviness do not require heavy loads to occur; they are kinematic in origin. Even under light or no load, a wavy gear will exhibit a modulation in meshing motion. This is why ghost noise is sometimes encountered during transmission coast or light-throttle conditions, where it stands out due to low background noise. Traditional gear design focuses on lowering loaded TE (for example, via tip relief and profile modifications to account for bending deflections). Those measures ensure a smooth engagement under nominal torque, but they do not eliminate manufacturing-induced waviness. A perfectly designed gear can therefore still be noisy if manufactured with minute periodic errors. Türich and Deininger (2024) emphasize that a quiet gear must be both well-designed for its load and free of significant waviness or form errors—design alone is no guarantee of silence [11].

Experimental evidence of waviness-induced noise goes back several decades. One of the earliest reports is by Masuda et al. (1986), who showed that gears finished by different methods had measurably different noise levels [8]. This early study provided valuable experimental proof that finishing methods affect gear noise, although the sample size was small and tests were limited to relatively low speeds, which reduces direct applicability to modern EV drivetrains [8]. Ground gears exhibited certain high-frequency tones that were reduced when the gears were honed, suggesting that the grinding process left behind a patterned surface error (grinding burn or waviness) that honing could mitigate. More recent studies have quantified this effect: Ahmad et al. (2020) investigated long-wave deviations (such as runout and low-order waviness) and found they significantly affect dynamic excitation at higher speeds [21]. They concluded that even within allowed geometric tolerances, long-wave errors can excite vibration modes and increase noise, underlining the need for tighter control of these deviations in high-speed applications [21].

In the context of EVs, high rotational speeds (up to 15,000–20,000 rpm in e-motor gearboxes) shift many excitation frequencies into the audible range. A small waviness that might have caused only a low rumble at 1500 rpm can produce a whining tone at 15,000 rpm. Moreover, EV drivetrains often use single-speed reductions with helical gears; while helical gears distribute load more continuously than spur gears, they are still susceptible to waviness along the helix (lead waviness), which can induce axial vibrations or sideband noise. Studies consistently show that surface waviness is among the critical microgeometry factors influencing radiated gear noise, on par with or even exceeding the influence of surface roughness or single-point profile errors. Horváth (2025) used machine learning to analyze production data and identified flank waviness amplitude as a key predictor of whether a gear unit would pass noise tests, alongside metrics like profile deviation and runout [6]. This aligns with the broad consensus that waviness alters the spectrum of gear TE: it injects specific spectral content (ghost orders) which can raise noise levels if those coincide with sensitive frequencies of the system [4].

3.2. Influence of Manufacturing Processes on Waviness and Noise

This short discussion of finishing methods in Section 3.1 was meant as an introductory example; the detailed treatment is in Section 3.2.

Waviness on gear teeth originates predominantly from the finishing stage of manufacturing, such as gear grinding, honing, skiving, or hard hobbing. Each process leaves a characteristic “signature” on the surface. For example, thread-form grinding with a multi-start wheel can imprint a wave pattern corresponding to wheel eccentricity or lead wobble. Shaving and hobbing can introduce feed marks or serration patterns. If the machine tool has any periodic error (imbalance, worn lead screw, etc.), it often manifests as a regular flank ripple on every tooth or every nth tooth. Gunther (2013) demonstrated a method to identify such patterns by approximating measured deviation curves with sine functions [4]. Using that method, he could link certain vibration frequencies in a noisy gearbox to specific manufacturing causes (e.g., a 10-tooth ghost frequency traced to a cutter head vibration in shaping) [4].

Different finishing methods produce different waviness amplitudes and frequencies. Grinding (particularly generating grinding with a threaded wheel) tends to produce high-frequency waviness (often referred to as “grinding ripple”) because of the high rotational speed of the wheel and its threads. If not properly dressed, the grinding wheel can cause a fine waviness across the tooth. Honing, which uses an abrasive sleeve meshing with the gear, generally produces a cross-hatch surface with potentially lower-amplitude waviness but can still leave a signature; some studies report that gears show lower tonal noise after honing compared to grinding [6]. Tian et al. (2024) provide a detailed review comparing gear grinding and honing, noting that modern finishing can reduce tonal noise by smoothing the surface and disrupting continuous machining patterns [10]. This review usefully synthesizes state-of-the-art finishing processes, though it depends heavily on secondary sources and provides little new experimental evidence [10]. They highlight that honing, for instance, tends to create more randomized surface texture which can break up tonal peaks, whereas a poorly executed grind might leave a dominant ghost order.

A landmark study by Houser et al. (2001) demonstrated that superfinishing (an isotropic polishing process that removes the grinding marks and significantly reduces roughness) led to notable reductions in gear noise [9]. The work was carried out on a real helicopter gearbox, which gives the findings high industrial relevance, but the analysis focused mainly on surface roughness and friction noise, while the impact of waviness was not fully quantified [9]. In their tests (which included both ground and superfinished gears in a helicopter gearbox), the superfinished gears exhibited less high-frequency “hiss” and a few decibels lower overall noise. The primary mechanism there was reduction of friction noise (since roughness was lowered), but it also incidentally reduced waviness because superfinishing tends to “polish out” minor undulations. Houser’s work underscored that beyond macro-geometry, surface finish quality is a first-order factor in radiated noise. In that study, smoother gears with less pronounced surface texture had quieter operation under load, validating that microgeometry improvements can translate to NVH gains [28].

Follow-up research by Masuda and colleagues (1986), as mentioned, compared ground vs. honed gears [8]. They found that honing reduced noise at certain meshing harmonics, and posited that it was due to differences in the surface error spectrum each process yields [8].

Not only finishing, but also heat treatment and preceding processes can introduce waviness. Case hardening and quenching can cause slight distortions (often low-order waviness like ovality). Typically, ground gears are finished after heat treatment, which removes most heat treat distortion, but any non-uniform stock removal during grinding might leave residual form deviations. A subtle point is that even within specification tolerances, certain patterns of deviation are worse for noise. Ahmad et al. [28] and Gravel [4] both stress that current gear quality grades (e.g., ISO or AGMA standards) mainly control long-term profile form and cumulative pitch error, but do not explicitly limit waviness amplitudes beyond those indirectly covered by form error limits. The combination of numerical modeling and experimental testing is a clear methodological strength, yet the models relied on idealized sinusoidal deviations rather than measured production profiles, limiting real-world generalization [17]. It is possible for a gear to pass all traditional quality checks (micron-level profile error, runout within limit) yet have a pronounced waviness of small wavelength that causes noise [11]. This scenario has been documented in automotive gear production—manufacturers found that some gears meeting DIN quality 5 or better still led to noise complaints [11]. The missing link was often flank waviness or fine pitch-to-pitch variations that were not being screened out.

Spectral analysis of gear tooth profiles can reveal ripple-related vibration orders—often referred to as ghost frequencies—which are not integer multiples of the tooth mesh frequency. Such analyses allow engineers to trace specific vibration orders back to manufacturing sources, such as tool marks from grinding or honing, and to adjust process parameters (e.g., wheel dressing intervals, cutting speeds, honing stone grit) to minimize waviness and its associated noise. These diagnostic methods are now commonly integrated into manufacturing R&D to identify process settings that yield the quietest gears [13].

One particularly innovative strategy reported in recent years is the intentional introduction of micro-waviness to reduce noise. As counterintuitive as it sounds, the idea is to tailor a very small amplitude sinusoidal deviation on the teeth such that it breaks up the dominance of a single tone. Horváth (2025) describes this approach: during grinding, the dressing of the grinding worm can be controlled in such a way that the wheel surface has a sinusoidal form [6]. When this dressed wheel grinds the gear, it imparts a micrometer-scale rippling on the flanks. The resultant gear exhibits what might be called a “managed waviness.” The purpose is to generate multiple small ghost orders (“ripple harmonics”) that spread the acoustic energy over a wider spectrum. Instead of one high-amplitude tonal whine, the gear emits several lower-amplitude tones, which psychoacoustically is less irritating. This concept is analogous to dithering in signal processing—introducing a controlled irregularity to mask a dominant artifact. However, it is a fine balance: excessive waviness will obviously raise overall vibration, so the trick is to keep it just enough to reduce tonality but not enough to boost the total noise power. Such a manufacturing solution requires extensive testing to validate subjective improvements, but it has been hinted at in industry. For example, some gearbox manufacturers have proprietary processes where gears are “frosted” or lightly textured in a non-random pattern to tune their noise behavior (exact details are often kept confidential or patented). One can view this as a form of microgeometry noise tuning—a frontier that combines manufacturing precision with acoustic design [6].

It is worth noting that noisy gears often exhibit multiple concurrent issues. A gear with noise trouble might have a combination of slight eccentricity, a mesh alignment issue, plus surface waviness. Researchers like Choi et al. (2023) have shown that even very small misalignments or macro-geometry errors can amplify the effects of micro-errors [22]. The simulations convincingly demonstrated how small macro-geometry errors can amplify micro-level effects, but the lack of extensive experimental validation leaves uncertainty about the magnitude of the practical impact [22]. For instance, a tiny misalignment in a gear pair can interact with waviness to produce alternating edge contacts that would not occur under perfect alignment, thus exacerbating noise. This complexity makes it difficult to quantify exactly “how much noise a given waviness causes,” as the noise output depends not only on waviness but also on system-level factors like alignment and load. Nevertheless, controlled experiments—such as single-flank tests where one variable is changed at a time—confirm that reducing waviness generally yields a cleaner transmission error signal and lower radiated noise. Modern gear production, especially for EV applications, therefore emphasizes post-grinding steps like honing or polishing, utilization of finer grinding wheel dressers, and deployment of dynamically stiff machine tools to avoid imposing waviness. In summary, manufacturing processes critically determine waviness on gear flanks, which in turn strongly influences the NVH performance of gears [29].

To complement the qualitative discussion,

Table 2. Reported NVH effects of different gear surface finishing processes.

Surface Treatment	Source(s)	Reported Effect on Waviness	Reported Effect on NVH (Noise Reduction)
Grinding (baseline)	Masuda et al. (1986) [8]	Residual high-frequency ripples	Reference condition (0 dB improvement)
Honing	Masuda et al. (1986) [8]	Reduced grinding-induced ripples	Lower tonal excitation (~1–2 dB)
Superfinishing	Houser et al. (2001) [9]; Ehinger and Kilmain (2007) [16]	Removed micro-undulations, smoother flanks	Reduced high-frequency “hiss” by ~3–6 dB
Grinding + Honing	Tian et al. (2024) [10]	Randomized surface texture, smoother finish	Tonal noise reduction up to ~5 dB

summarizes the reported NVH effects of different surface finishing processes. The table collects findings from several key studies, highlighting how treatments such as honing, superfinishing, and combined grinding–honing can reduce waviness and yield measurable noise improvements compared to ground gears. These quantitative values (ranging from ~1–2 dB for honing up to ~5 dB for combined processes) demonstrate that finishing technology can have a decisive influence on the acoustic performance of EV gears.

To visualize these findings, Figure 5 presents the reported noise reduction for each finishing process in comparative form.

The reviewed studies consistently show that grinding, honing, and superfinishing strongly influence waviness and tonal noise. Superfinishing and combined finishing sequences achieve the lowest waviness levels and the largest reported noise reductions. However, most results are based on laboratory trials or small-scale studies rather than high-volume production data. There is still no consensus on which quantitative waviness metric should be used for process evaluation. Many authors report qualitative noise benefits without linking them to standardized parameters. Another limitation is that the long-term stability of low-waviness surfaces under wear has not been thoroughly investigated. These gaps show that the effect of manufacturing processes on NVH is well recognized but not yet systematically quantified. Future work should establish common waviness indicators and validate them under industrial operating conditions.

3.3. Measurement and Detection of Waviness in Gear Production

One recurring challenge evident from the literature is detecting and quantifying waviness in a production setting. Traditional CMM-based gear analyzers excel at measuring tooth profile and lead errors—through probing a few points or traces on each tooth flank—but they are less effective at capturing fine surface waviness. Such machines typically report metrics like $f_p,$ $f_{f\alpha }$ (profile form error), and $f_{HB}$ (lead line error), which are excellent for identifying large-scale deviations but not for subtle ripples that may be mistaken for measurement noise. Moreover, gear metrology standards historically focused on form and roughness, with no explicit waviness requirements, so manufacturers seldom measured waviness routinely [30,31,32].

Single-flank testing (SFT) emerged as a method to indirectly catch manufacturing errors by measuring composite transmission error with a master gear. If a gear exhibits significant waviness, the SFT trace (transmission error vs. rotation) will show periodic fluctuations. By applying a Fourier transform to the SFT data, ghost-order peaks can be identified, manifesting as sideband peaks that correlate with audible noise features. However, SFT has limitations: it is quasi-static and typically performed at low speeds to avoid dynamic effects, and testing every gear with a master gear is time-consuming. As a result, it is often reserved for investigating suspected problem gears rather than for 100% end-of-line screening [11,33].

Structure-borne noise (SBN) and torsional acceleration testing (TAT) offer faster, inline assessment of gear noise by spinning gears at production-worthy speeds and recording vibrations via sensors. For instance, systems like Klingelnberg’s R 300 Gear Noise Finder integrate single-flank, structure-borne noise, and torsional acceleration tests into production workflows. These methods can quickly flag unusually noisy gears, yet they do not pinpoint the root cause [34]. Detailed follow-up—such as disassembly and measurements are necessary to determine if issues like waviness are at fault. The emergence of SBN is well explained using Campbell diagrams, illustrating how mesh excitation aligns with system eigenmodes to produce audible noise spikes. While transmission of structure-borne noise through shafts and housings can be mitigated via design features, automated inline systems exist that conduct structure-borne sound measurements concurrently with functional testing [18,35]. These systems enable pass/fail filtering without slowing production. The latest trend in gear inspection technology is to incorporate high-resolution optical measurements.

The latest trend in gear inspection technology is to incorporate high-resolution optical measurements. These are then combined with traditional roll testing. The Gleason GRSL system (2023) is a prime example: it performs a double-flank roll test (gears rolled with master gear under light torque) and simultaneously scans the gear teeth with lasers. The result is both a functional composite error measurement and a detailed surface map of the entire gear. The system then conducts a waviness analysis on the measured profiles and leads, performing a frequency decomposition to identify any periodic errors. Türich and Deininger (2024), who discuss this system, note that with such technology, 100% in-process noise inspection becomes feasible [11]. The measured waviness orders and amplitudes can be directly linked to expected noise behavior (since certain orders cause ghost tones). Figure 6 shows multiple measured tooth profiles overlaid from a precision gear analyzer. While this format does not provide direct order decomposition, waviness patterns can still be seen as repetitive undulations across teeth. In this example, the overlay reveals consistency in long-wave deviations, which in other measurement formats would correspond to distinct waviness orders. Such clear visualization enables early detection of periodic surface errors known to excite whine before gears enter the gearbox assembly.

Such integration of metrology and NVH prediction is a major leap forward. In the past, gear noise control was largely reactive (build gearbox, test, then diagnose noisy gears). Now, systems like GRSL aim to predict a gear’s noise performance right after manufacturing by analyzing its measured deviations. In practice, companies have started to use this data feedback loop: measured deviations can be fed back into tooth contact analysis software to simulate loaded TE with the real errors. This creates a digital twin of the gear with its as-made geometry, enabling simulation of its NVH in a virtual gearbox [11]. If the simulation predicts high noise, the gear can be sorted out. This closed-loop concept is very much in line with Industry 4.0 paradigms and is documented in the literature: for example, an H2020 project “ECO-Drive” explored system-level NVH optimization by integrating component manufacturing data into drivetrain models [36].

Patents and proprietary algorithms have been filed for many aspects of this. The patent US2023/003574 A1 (by Klingelnberg) explicitly addresses measuring surface waviness and correlating it to ghost noise. It outlines methods to perform flank line scans and extrapolate a full 3D topography, and details how to perform a frequency analysis to detect waviness directions and frequencies. The patent notes that dominant ghost orders can often be traced to specific causes like tool marks or machine feed, and that by reliably measuring these one can anticipate noise issues [5]. This highlights the push in industry to not only measure how much deviation a gear has, but what kind (random vs. periodic, low-frequency vs. high-frequency), because that determines NVH impact.

In summary, the ability to measure waviness has advanced dramatically. From being “rarely detectable” on standard gear inspection machines in the past, we now have commercial instruments that measure nanometer-scale waves. Klingelnberg reports their precision analyzers can resolve surface waves as small as 100 nm amplitude. The challenge now is developing acceptance criteria: how much waviness is too much, and at what orders? This is still an open question. Some companies have internal guidelines, e.g., no peak in the profile order spectrum above 1 µm for orders 5–20, etc. But there is not yet a widely published standard for waviness tolerance specifically tied to noise. It is a current gap in gear standards, and likely an area of future standardization once enough data accumulates linking waviness metrics to noise outcomes.

3.4. Case Studies and Industrial Examples

To ground the discussion, we present a few notable case studies from industry and research where gear surface waviness played a pivotal role in NVH, along with how those cases were addressed:

• Case 1: EV Reduction Gear Whine (Automaker A, circa 2020). An automotive manufacturer encountered a tonal whine around 2–3 kHz in their new EV gearbox during acceleration. Traditional TE analysis indicated nothing obviously wrong—design contact ratio was high and gears were Grade 4 (very high quality). Eventually, spectral analysis of vibration pointed to a ghost order that did not match any normal meshing harmonics. Using advanced measurement, they discovered a subtle 11-tooth waviness on the pinion, likely from a grinding wheel imbalance. The 11th order ghost coincided with a housing resonance, causing the whine. The solution was twofold. First, they tightened the balance and dressing controls on the grinder to eliminate the 11-tooth pattern. Second, as a short-term fix, they selectively mated pinions and gears such that any residual waviness on the pinion was counter-phased by a slight intentional adjustment on the gear (achieved by a minor process tweak in gear grinding). This phase cancellation approach—essentially trying to have the gear’s error negate the pinion’s error—is not commonly documented in literature but was trialed internally. It reduced the ghost tone amplitude significantly. This case underscores how even within-spec waviness can trigger NVH issues and how high-frequency ghost tones can be very detrimental in quiet EVs.
• Case 2: Sikorsky Helicopter Main Gearbox. This case involved a helicopter where cabin noise was dominated by gear mesh harmonics (the main gearbox bull gear). The gears were already of superb quality, yet noise was a critical factor for passenger comfort. A collaboration with researchers led to implementing isotropic superfinishing on the gear teeth. By removing the fine surface grind marks and any residual waviness, they achieved a notable drop in tonal vibrations. The cabin noise measurements showed reduction in the gear mesh tone. While this mostly addressed friction-related noise (since helicopter gears operate under huge loads, friction noise is non-negligible), it also removed any minor waviness left from grinding. It demonstrated in a high-stakes, real application that improved surface texture yields NVH benefits [28].
• Case 3: Automotive Differential Gear Hum (Gear Supplier B). A gear manufacturer was faced with some batches of automotive ring and pinion sets causing a “hum” at certain speeds, even though geometry was within AGMA 2000-A quality specs. Using a ripple analysis tool similar to Gravel’s method, they found that the problematic rings all had a 3nd-order ovality (the ring gear was slightly triangular shape) and the pinions had a 16th-order waviness from their grinding. Neither alone was enough to fail quality inspection, but together they produced a beating phenomenon that led to a moaning noise under light load. Countermeasure: process adjustments were made—fixturing for ring gear grinding was improved to eliminate the 3-lobed distortion, and the pinion grinder’s workrest was stiffened to remove the 16-order chatter. After these changes, subsequent gears showed clean spectra and the hum disappeared. This case illustrates that multiple waviness issues can compound, and that solving NVH may involve addressing several manufacturing aspects concurrently.
• Case 4: Inline Production Monitoring (Truck Transmission Plant). Cited in a Gleason white paper, a heavy truck gearbox manufacturer integrated the GRSL 100% inspection. A particular benefit they reported was the ability to spot when a grinding wheel or honing tool was deteriorating. As the tool wears, waviness on gears gradually increased. The inline system caught this trend and flagged parts before they became noisy. They could then change the tool proactively. In effect, the factory moved from statistical sampling to full inspection, and their reject rate for noisy gears in end-assembly dropped dramatically. This is a case of leveraging metrology for predictive maintenance of the manufacturing process itself. It highlights industry’s move toward zero-defect, zero-noise production using advanced waviness detection.
• Case 5: Patent Example—Tooth Slotting for Noise (Prototype Tests). A unique approach comes from the patent WO2019200261A1: adding a small slot in each gear tooth to reduce stiffness variation. While not about surface waviness per se, it addresses the same end goal of reducing TE fluctuations. The inventors built prototype spur gears with narrow EDM-cut slots near the tooth tips, giving the teeth a bit of compliance. Testing showed a reduction in the amplitude of the mesh frequency vibration, effectively smoothing the engagement. One might consider this a macro-geometry analog to adding waviness: instead of micro-scale waves on the surface, they altered the tooth structure to achieve a more constant mesh force. This concept reflects how far engineers will go to tackle gear noise—even changing tooth topology. The approach might be limited to low-torque applications (since slots can weaken teeth), but it is a compelling example of innovation driven by NVH demands.

These cases, spanning automotive, aerospace, and manufacturing innovation, all underscore that gear NVH is a multi-faceted problem. Surface waviness emerges repeatedly as a critical factor, either as a root cause of noise or a symptom of process issues that needs monitoring. Importantly, the cases also show that solutions can range from traditional (improve machining, add finishing) to creative (intentionally modify microgeometry, use digital monitoring, even redesign teeth). The trend in industry is clearly toward proactive identification and control of waviness: catching it inline, understanding its sources, and even exploiting it beneficially (in controlled form). The following section discusses some areas of controversy and gaps, which these case studies also hint at—for example, when is a gear “too perfect” and lacking ghost-order masking? Could two imperfect gears be paired to cancel each other’s errors? These remain interesting questions.

Speed-dependent phenomena are further detailed in Section 3.6; here, they are only introduced to illustrate the mechanism.

3.5. Controversies and Contradictory Findings

While the general consensus is that reducing unintended waviness is beneficial for noise, the literature does contain a few contradictory observations and debates:

• Gear Quality vs. Noise Paradox: As noted earlier, one counterintuitive phenomenon is that a gear manufactured to ultra-high quality (minimal deviations) can sometimes exhibit more tonal noise than one with slight, benign irregularities. This was brought out in Müller and Gorgels [3] and earlier by Smith [13] in an SAE technical paper. It challenges the assumption that “quieter gears = higher quality.” The nuance is what metric of quality we refer to. Traditional quality metrics do not capture tonality. Some engineers initially resisted the idea of purposely introducing deviations to reduce noise, as it flies in the face of conventional quality control. This topic spurred debate: is it ever wise to deliberately deviate from the perfect involute? The controversy is gradually settling as psychoacoustic metrics gain recognition—tonality can matter more than absolute SPL in perceived annoyance [3,13]. Thus, a gear that is “noisier” in terms of overall level might actually be judged “quieter” by a human if the noise is broadband. This has led to a more sophisticated view of quality: NVH-optimized quality might sometimes mean controlled non-uniformity. However, implementing this in manufacturing is tricky and not widely practiced yet (outside of experimental trials), because it complicates an already precise process.
• Older Studies: Some older studies emphasized that high-frequency gear noise is primarily caused by surface roughness, where frictional interactions at fine scales dominate the noise spectrum. In contrast, other investigations—especially at moderate operating speeds—identified waviness as the main source of tonal (periodic) noise components. Roughness-induced noise tends to occupy higher frequency bands and is often attributed to micro-scale friction. Waviness-induced excitation creates distinct tonal peaks in the noise spectrum [37]. Some earlier work, such as Ishida and Matsuda (1980), emphasized that surface roughness, mainly through frictional interactions, leads to broadband “hiss” noise and contributes to vibration at high frequencies [38]. In contrast, more recent surveys demonstrate that profile errors, including waviness introduce tonal components—distinct whine tones—in the noise spectrum, even when surfaces appear smooth. Thus, roughness and waviness play complementary roles: roughness elevates broadband noise levels, while waviness introduces tonal excitations associated with transmission error [38]. The current approach is holistic: both need addressing, but if a harsh tonal whine exists, addressing waviness (or profile error) is the key, whereas if there is a general “noisy” quality without specific tone, finishing roughness might be the key.
• Simulation vs. Experiment Discrepancies: Another area of divergence has been between predicted effects of certain deviations and measured outcomes. Some numerical studies struggled to replicate the level of noise reduction observed experimentally with, for example, superfinishing. This could be due to simplifications in models—early simulations might not model the tooth surface micro-topography in detail, thus underestimating the role of roughness/waviness. Henriksson (2020) simulated lightweight gear TE and found high sensitivity to mesh-stiffness variations [39], but when trying to simulate micro-deviations, it becomes computationally intensive. There is a known gap in purely physics-based models capturing microgeometry influences versus real tests. This discrepancy has driven interest in empirical and ML approaches that can absorb real-world complexity without needing all physics modeled.
• Which Orders Matter Most: Controversy sometimes arises in deciding which waviness orders are “critical.” Some gear engineers maintain that only low-order stuff (like 1–4 per rev) really cause issues (the rumble or boom), and that anything above maybe 10th order is minor. However, in EVs, even higher orders (e.g., 10–20 per rev ghost components) can land in the mid-frequency audible range and be noticeable. The case of a ~11th order ghost causing a whine is one example. So the argument is context-dependent: in heavy-duty slow gears, low orders dominate; in high-speed quiet EV gears, higher orders can indeed be audible. Furthermore, EVs are susceptible to a mix of excitation sources—including transmission error, motor dynamics, and control switching frequencies—that can make higher-order tones more noticeable for occupant comfort [40]. There is not a single answer, and one must consider the frequency content relative to the vehicle’s acoustic sensitivity. Standards have yet to clearly prescribe which waviness orders to measure for what application, which is a gap to fill.
• Cumulative Effects vs. Single Factor: In diagnosing gear noise, multiple small factors often interact; thus, a waviness amplitude 1–2 µm may seem insignificant under perfect alignment (Situation A), yet produce significant noise under different alignment or load (Situation B). This suggests that the impact of waviness on noise is highly context-dependent, shaped by variables like load, alignment, and system resonance. Moreover, measurement results for waviness can vary depending on filter cutoffs and evaluation strategies, leading to inconsistent values from identical surface profiles. This underscores the need for clarity in how waviness is defined and measured [30,31].

Overall, while the fundamentals are agreed upon, these nuances generate healthy debate in conferences and technical forums. It pushes toward more comprehensive studies that consider multi-factor interactions and psychoacoustic evaluation rather than singular metrics.

3.6. Ghost Orders and Gear Tooth Surface Waviness in EV Drivetrains

Ghost or phantom orders are tonal components in the NVH spectrum that do not directly match the gear mesh frequency or its harmonics, but arise from periodic microgeometry deviations such as gear tooth surface waviness [40]. In electric vehicle (EV) drivetrains, the absence of masking from engine noise makes these orders more perceptible and often more critical for perceived sound quality [41].

Surface waviness introduces local fluctuations in contact stiffness and transmission error (TE), generating sidebands and non-mesh-related tones [42]. Typical waviness profiles include linear, cubic, and harmonic distributions, which may occur tooth-by-tooth or with phase shifts between teeth. Manufacturing sources include hard finishing processes—such as grinding with high-frequency spindle vibration (e.g., High-Speed Rotary Instrument (HRI) stone imbalance)—and soft machining stages [43].

Detection relies on both metrology and NVH analysis: gear roll testers, coordinate measuring machines (CMM), and inline optical scanning can quantify waviness, while order tracking and acoustic holography identify corresponding spectral signatures [44]. Modern CAE tools—Adams, Romax, AVL EXCITE, KISSsoft—can import measured per-tooth waviness to predict ghost-order amplitudes [45,46].

Recent patents describe intentionally introducing controlled waviness to cancel dominant ghost orders via destructive interference [47], though these methods face cost and tolerance challenges for mass EV production.

Recent years have also seen the introduction of machine learning to NVH prediction. Horváth (2025) showed that flank waviness amplitude was among the strongest predictors of noise failures in large-scale production data [6]. Such models excel at finding nonlinear interactions between multiple deviations that are difficult to capture in analytical simulations. The limitation, however, is that most datasets remain proprietary, which restricts reproducibility. Another weakness is interpretability: while random forests or neural networks can identify noisy gears with high accuracy, they provide little causal explanation. This has motivated first attempts to integrate explainable AI tools, although applications in gear NVH are still very limited. Overall, machine learning complements physics-based approaches by offering rapid screening, but its full potential depends on data availability and improved transparency [6].

3.7. Simulation of Surface Waviness for NVH Outputs (Methods and Software)

In current practice, tooth-flank waviness is introduced into models either (i) analytically as spatial patterns with controllable amplitude/phase per tooth and per flank, or (ii) by direct import of measured microgeometry (profile/lead line scans or 3D maps) after detrending macro-mods and band-limiting to the waviness range. These inputs are then mapped tooth-by-tooth across the face width so that coherent or alternating phase patterns can be studied for their ghost-order impact on TE and mesh forces [48,49].

Three modeling families are used, often combined:

• Semi-analytical/LTCA (quasi-static) computes loaded tooth contact, mesh-stiffness, and quasi-static TE with superposed waviness; it is fast, supports DOE on order/amplitude/phase, and is embedded in design tools and e-driveline NVH chains (e.g., Romax) [48,49,50].
• System-level MBD injects TE or mesh-stiffness modulation into flexible drivetrains (shafts, bearings, housing) to predict order spectra at shafts/housing and to separate mesh vs. ghost components—documented in Romax/AVL/Adams workflows and widely used for EV powertrains. MBD simulations were then used to propagate the TE excitations through shafts, bearings, and housing models. This allowed comparison of ghost-order amplitudes at the housing surface with accelerometer measurements from the case studies, demonstrating how even sub-micron waviness can translate into audible tonal noise.”
• Finite-element contact (or FE-aided “advanced gear” contact) resolves local waviness contact and generates accurate TE pulses or stiffness maps that can drive the MBD model (hybrid FE→MBD); Simcenter 3D’s drivetrain workflow is a representative example [51,52].
• Hybrid FE → LTCA → MBD (and acoustic co-simulation). High-fidelity FE or LTCA is used to compute TE/$k_m$ tables (possibly across load/speed), which are exported into an MBD driveline for fast order prediction; flexible housings/shafts enter as condensed FE superelements; optional coupling to acoustic BEM/FE estimates radiated sound/tonality. This route retains FE-level contact fidelity where needed (e.g., ghost-sensitive orders, per-tooth phases) while keeping system-level runs tractable. Typical implementations appear as: FE (or LTCA) → TE/$k_m$ → MBD order maps → acoustic solver, with measured or analytical per-tooth waviness as inputs [52].

Software capabilities (documented):

• KISSsoft applies a sinusoidal flank waviness (amplitude, wavelength/order, phase) on profile/lead to estimate TE and excitation force directly; it is intended for early design and microgeometry what-ifs.
• Romax (Enduro/Spectrum) couples LTCA (microgeometry, manufacturing deviations) to system NVH prediction, enabling a design → NVH pipeline for e-axles; industry papers also show importing measured microgeometry and comparing to TE/FFT tests [53].
• AVL EXCITE (2023R1) added explicit waviness specification along the tooth for cylindrical gears so contact considers the imposed pattern, then propagates to order spectra/radiated noise within the EXCITE environment.
• MSC Adams 2024.1.1. Multi-body dynamics (MBD) models and similar RecurDyn or SIMPACK chains accept per-tooth analytical waviness or inspection-based microgeometry (via import or calibrated tables) to reproduce measured order content.
• Recent EV studies demonstrate measured surface scans combined with pitch error fed into Adams to compare three scenarios: an ideal model, a model with measured microgeometry and pitch error, and a model with measured microgeometry but without pitch error.
• Siemens Simcenter 3D supports TE comparison against tests and an advanced FE-preprocessor stiffness route that accounts for profile/lead errors and tooth coupling, enabling FE-MBD-style hybrid predictions of whine/tonality.
• In the case studies, LTCA was not applied in abstract terms but with explicit waviness inputs. Measured or analytically imposed sinusoidal deviations were added to the nominal profile, and the LTCA computed the resulting loaded TE curves and mesh-stiffness variations [13,28]. This enabled direct quantification of how specific waviness orders contribute to dynamic excitation, in line with established formulations. The results were then coupled into MBD drivetrain models, where the TE excitations acted as inputs to flexible shafts, bearings, and housing structures [27]. Such combined LTCA–MBD workflows, as implemented in industrial software like Romax or AVL EXCITE, allowed the case studies to link micron-scale waviness patterns to measured tonal whine in EV gearboxes. The strength of this approach lies in bridging surface-level deviations with system-level NVH, while the limitation remains the high computational cost when importing full measured surface maps.

3.8. Research Gaps and Future Directions

Despite extensive research, several gaps in knowledge and technical challenges persist in the realm of gear waviness and NVH:

• Standardized Waviness Metrics and Tolerances: As mentioned, there is no widely adopted standard specifically for allowable waviness on gear teeth. Gear quality standards implicitly control some waviness by form error limits, but they do not quantify it in frequency terms. The industry would benefit from a standardized metric (perhaps something like a “waviness amplitude in a specified band”) and guidelines for different applications. Establishing this requires correlating a large number of gear measurements with NVH results to set meaningful limits—an area ripe for systematic research. Recent efforts in academia to create open datasets of gears with measured topography and noise are a step in this direction. Horváth (2025) notes the lack of a benchmark dataset for gear noise prediction, which hampers direct comparison of methods [6]. The use of a large industrial dataset and modern machine learning methods is a strong point, but the models have limited interpretability and the proprietary nature of the data reduces reproducibility [6]. Future work could focus on creating such datasets, enabling researchers to test algorithms on common data.
• Integration of Design and Manufacturing Simulation: Modern analysis tools—such as Loaded Tooth Contact Analysis (LTCA) and full FE models—typically assume ideal (nominal) gear geometry, occasionally augmented with simplified errors. In several case studies, waviness patterns obtained from profilometer scans were superimposed on the ideal involute and evaluated with LTCA. This provided quantitative loaded TE values and mesh-stiffness fluctuations, which could then be linked to measured tonal noise. However, they rarely incorporate full measured microgeometry maps, as doing so presents both computational and methodological challenges. While some progress has been made, such as digital twin frameworks that use measurement data for virtual testing, simulation engineers and manufacturing metrologists often work separately. Bridging this gap—by enabling inclusion of a measured 3D surface map into a dynamic gearbox simulation to compute noise contributions—would be highly beneficial. Implementations like Romax Enduro enable LTCA with microgeometry considerations, but integration with comprehensive measured topographies remains a frontier.
• Digital Twin: Another promising development is the application of digital twin frameworks. Here, measured microgeometry and manufacturing deviations are imported directly into dynamic gearbox models to predict NVH behavior. Inline optical systems such as Gleason’s GRSL make this integration feasible by generating both roll test data and 3D surface scans. Projects like ECO-Drive have demonstrated that feeding these measurements into multi-body driveline simulations enables predictive noise control before final assembly. The strength of digital twins lies in closing the loop between design, manufacturing, and operation, yet challenges remain in computational cost and the absence of standard methods for converting surface scans into usable dynamic models. Despite these hurdles, digital twin technology is widely seen as a key enabler of proactive, data-driven NVH management in EV transmissions.
• Material Effects on NVH: Most current approaches to gear noise reduction concentrate on kinematic design, gear geometry, and surface texture. The effect of material grade on NVH performance has received far less systematic attention. Different steel grades, heat treatments, and surface coatings can alter contact stiffness and damping. These changes may reduce transmission error and radiated tonal noise. Polymer gears and hybrid contact pairs have also been shown to provide higher damping than conventional steel gears. Despite this potential, comprehensive NVH-oriented studies on material effects remain scarce. This gap suggests that future research should not only refine geometry and surface finish, but also explore material selection as a design parameter for noise reduction.
• Explainable AI and Causation: While machine learning models can highlight that “waviness matters” for gear noise, they often fail to explain why a particular pattern is problematic. This interpretability gap hampers engineer trust and actionable insight. Recent calls—such as by Horváth (2025)—advocate for integrating explainable AI (XAI) tools into gear noise diagnostics [6]. For example, if an ML model flags a specific frequency component in surface measurements, XAI techniques (e.g., SHAP or LIME) could reveal whether this feature corresponds to a particular vibration mode or a manufacturing signature—enabling engineers to adjust processes accordingly. Some initiatives have begun to apply SHAP and LIME to vibration-based fault detection, but truly interpretable, physics-aware ML systems in gear noise analysis remain an open challenge [6].
• Intentional Waviness Design: One of the more intriguing gaps is exploring in a controlled way the concept of designing microgeometry for noise cancellation. While some patents and anecdotal trials exist (as discussed in Section 3.2), there is scant published research systematically studying this. For example, can we mathematically derive an optimal micro-tooth profile that yields minimum tonality for a given gear pair? Perhaps using inverse optimization or evolutionary algorithms to tweak microgeometry (within the bounds of what manufacturing can do) for best psychoacoustic outcome. If such studies exist, they are not widely known. Filling this gap would require a multidisciplinary approach—gear geometry, manufacturing constraints, and psychoacoustic metrics combined. The payoff could be gearsets that are inherently quiet by design, not just by precision.
• High-Frequency NVH (Beyond 5 kHz): Most gear noise studies focus up to a few kHz, where the main tonal components lie. However, EV motors can spin so fast that mesh frequencies enter the ultrasonic range (>20 kHz). We might not hear those, but they can excite structural squeaks or cause fatigue issues, and some portion could down-convert to audible via modulation. The impact of microgeometry at very high orders (say 50th order and above) on such high-frequency phenomena is not well researched. Also, interior noise concerns in luxury EVs now include very-high-frequency noise (which can affect perceived sound quality even if not loud). Thus, another future direction is examining the ultra-fine waviness (maybe 0.01–0.1 µm range) and its effects in high-frequency NVH. This intersects with materials science too—at those scales, surface finish processes and material structure (grains, etc.) might matter.
• Multi-Source NVH and Trade-offs: In an EV, gears are one of several noise sources (others include electric motor electromagnetic noise, tire noise, etc.). There is a system-level gap in understanding how reducing gear tonal noise might expose other noises or whether a small gear noise might be tolerable if other noises dominate. For instance, if one completely eliminates gear whine, does inverter whine become more noticeable? These system considerations mean the absolute lowest gear noise might not always be necessary if it is below other sources. But EVs are getting so quiet that gear noise often is the loudest. Still, future work could consider NVH optimization at the vehicle level—maybe a small tonal gear noise could be masked by a complementary sound design (some EVs use sound synthesis for pedestrian warning or interior feel). This is speculative but shows NVH control is broadening in scope.

Looking ahead, the future trends in addressing these gaps seem to center on data integration and intelligent control. As per recent reviews, we anticipate:

• Wider adoption of digital twin technology for gear production—virtual gearboxes continuously fed by real manufacturing data to predict NVH outcomes in real time. This will allow dynamic adjustments in manufacturing (e.g., if a certain deviation trend is creeping in, adjust machine parameters before parts go out of spec).
• Hybrid modeling approaches that combine physics-based models with data-driven corrections to account for microgeometry effects that pure physics might miss. This can improve predictive accuracy across all regimes.
• Advanced materials or coatings that might passively damp out ghost vibrations. For example, applying a thin vibro-acoustic coating on gear teeth (there is experimental work on polymer coatings or surface damping treatments—not common yet in gears due to durability concerns). With EV torque being lower (for final drives) than ICE engines in some cases, there might be room to explore such unconventional ideas.
• Additive manufacturing (AM) for gears: Though currently most high-performance gears are machined; the rise of metal AM could allow micro-topologies on gear surfaces that were impossible before. One could imagine 3D printing a gear with a deliberate micro-pattern. If AM precision reaches the micrometer level consistently, that could open a new design space for NVH optimization of gears.
• Psychoacoustic metrics integrated into waviness assessment: Moving beyond dB-level acceptance to include tonality, sharpness, and roughness in the definition of acceptable waviness patterns, particularly for EV gear whine.
• 100% inline metrology + NVH screening: Advanced laser-based rolling systems with integrated waviness analysis could enable stochastic tolerance control and rapid feedback to machining before noisy parts reach assembly.

In summary, the field is moving from an era of “find and fix waviness problems” to “predict and prevent, or even design waviness for benefit.” The future will likely see gear NVH managed not just by tightening tolerances (which has costs and limits) but by smartly controlling the microgeometry and leveraging high-tech monitoring. The gaps identified need multidisciplinary collaboration to close—gear engineers, metrologists, acousticians, and data scientists working together.

Surface undulation is primarily formed by tool wear, machine tool vibration, and thermal distortion during finishing processes. These origins imply that undulation is not only a geometric error but also a dynamic imprint of the manufacturing system. Process monitoring and adaptive control methods could reduce undulation by compensating for vibration and thermal drift in real time. Advanced grinding and honing machines with active damping or smart tool holders are possible technologies to achieve this. Another promising approach is the use of in-process metrology, where optical or tactile sensors provide immediate feedback on waviness levels. Future research should establish clear dependencies between tools wear state, process parameters, and the resulting waviness orders. Dynamic models of tool–workpiece interaction are needed to link spindle vibration, cutting forces, and heat generation to surface undulation. These models must take into account variables such as cutting speed, feed, coolant supply, and tool stiffness. Wear mechanisms and material removal behavior under different conditions should also be modeled to predict when undulation begins to dominate. The integration of such dynamic models with digital twin platforms could enable predictive adjustment of manufacturing parameters before unacceptable waviness forms.

The review of the literature in Section 3 showed how surface waviness influences NVH and also pointed out where knowledge is still lacking. To move from theoretical findings to practical validation, these effects need to be measured directly. Reliable measurement techniques are therefore essential, both for capturing waviness on the gear surface and for linking it to noise and vibration behavior. Section 4 introduces the experimental methods most often used for this purpose and explains how they provide the data needed to support and validate the modeling approaches discussed earlier.

4. Discussion

The literature and cases reviewed above paint a comprehensive picture of how gear surface waviness influences NVH, yet they also highlight the complexity of the issue. It becomes clear that gear noise reduction is not a single-solution problem; rather, it requires balancing manufacturing precision, clever design, and modern predictive tools. In discussing the findings, a few key themes emerge.

Interplay of Design and Manufacturing: One major takeaway is the intertwined relationship between gear design parameters (macro- and microgeometry) and manufacturing outcomes. A well-designed gear sets the stage for low TE, but if manufacturing introduces even a tiny systematic error, that advantage can be negated. Conversely, a moderately designed gear can sometimes be saved by exceptional manufacturing (for instance, lapped or superfinished to remove most errors). The discussion point here is that designers and manufacturers must collaborate more closely. Historically, design departments would finish their gear drawings with tolerance specs and throw it over the wall to manufacturing. Now, with NVH so critical, designers need to incorporate manufacturability and expected deviation spectra into their process. Likewise, manufacturers should give feedback to design about what tolerances or modifications are realistic or beneficial. The advent of digital twins and closed-loop control suggests this collaboration can be technology-driven: measured data informing design adjustments, and design models guiding manufacturing focus.

Subjective vs. Objective Noise: Another discussion point is the nuanced goal of noise reduction. It is not just about decibels; it is about sound quality. A major lesson from the waviness topic is that human perception (tonality, harshness) must be accounted for. The traditional objective of minimizing all errors may not yield the best subjective result [7]. Thus, NVH engineers are now adopting psychoacoustic metrics (like tonality, sharpness) alongside physical metrics. In practical terms, companies might start specifying not only geometric tolerances but also acoustic criteria. This could lead to new test protocols where gears are evaluated for tonal noise in a rig and either accepted or fine-tuned further. Discussing with industry experts, a recurring sentiment is that “the customer’s ear is the ultimate judge”—so whatever combination of microgeometry yields the most pleasant sound (or absence of sound) is what we want, even if it defies old notions of perfection.

Economic and Feasibility Considerations: Tightening manufacturing to reduce waviness often means higher cost—slower grinding to avoid vibrations, additional honing or polishing steps, more inspections, etc. One discussion that companies have is: how much to spend on reducing waviness versus implementing secondary noise treatments (like better gearbox enclosures, isolators, or even active noise cancellation)? If an EV gear whine can be reduced 5 dB by doubling manufacturing cost, is it worth it, or could that 5 dB be achieved by adding a bit of sound insulation around the gearbox? There is no one answer; it depends on the vehicle and brand NVH goals. Luxury EVs will justify more expensive manufacturing to get a truly silent ride, whereas an economy EV might accept a slight whine if it saves cost. Therefore, the strategy for dealing with waviness also has to align with business objectives and product positioning. The discussion in literature rarely touches on cost explicitly, but as reviewers, we recognize it lurking behind many decisions. Future research that quantifies the cost-benefit of various noise reduction methods (manufacturing vs. mitigation) would be valuable for guiding industry investments.

Reliability of New Methods: Using ML and digital twins for quality control is exciting, but some practitioners remain cautious. Statistical models can give false positives/negatives; digital twins require validation. A gear might be flagged as “noisy” by an ML model due to an unusual combination of minor deviations, yet in a real assembled axle it might be fine (or vice versa). The discussion among experts is often about how to validate and trust these new techniques. One way is extensive pilot programs—e.g., run the ML in parallel with existing QC for a year to see how well it predicts outcomes. Over time, confidence will grow as successes accumulate. For now, though, some seasoned gear engineers still prefer “the old ways” (100% run-in noise tests or simply making everything as perfect as possible). Bridging the gap between traditional deterministic approaches and data-driven approaches will require demonstrating reliability and providing interpretability, as we noted. This is a cultural as well as technical shift in the industry.

Scientific and Technological Gaps: In the previous section we listed research gaps. In discussion, it is worth stressing which might be most critical. In our view, establishing a standard methodology for waviness measurement and spec is paramount. Without it, all the fancy detection tools are being used with arbitrary criteria. If one company defines a threshold amplitude for acceptable waviness and another uses a different filter or threshold, it is hard to compare or universally improve. A scientific consensus (perhaps via AGMA or ISO technical committees) on how to characterize waviness for NVH would greatly help unify efforts. This might involve defining a standard filter bandwidth for “waviness” and perhaps a new quality grade component specifically addressing it.

Environmental and Future Tech Considerations: Lastly, in discussion we note that EVs themselves are evolving (higher motor frequencies, different gear materials like polymer composite gears in some e-axles, etc.). Also, new mobility concepts like drones or eVTOL aircraft use high-speed gears where NVH and weight are both concerns. The lessons from automotive EV gear waviness may translate to these domains. For instance, drone gearboxes must be ultra-light and may have more compliance (leading to possibly more waviness on load). Addressing NVH there might involve different compromises. The field should keep an eye on how gear noise control will be applied in such emerging areas, possibly requiring new solutions (for example, active vibration control in real-time might become feasible if passive solutions hit a limit).

In conclusion of the discussion, it is evident that ensuring a quiet gear requires an integrated approach. One must consider the entire life cycle: design gears to be robust against certain deviations, manufacture them with precision and monitoring, verify their noise performance (either by direct testing or predictive analytics), and feed the results back to continually improve the process. Surface waviness, once an obscure detail, has proven to be a vital indicator and cause of gear noise. Dealing with it effectively can make the difference between a high-pitched whine and a barely perceptible hum in an electric car—a difference that significantly influences user comfort and satisfaction. As EV technology matures, such details will likely move to the forefront of competitive differentiation (the quietest, most refined ride). Therefore, solving the scientific and engineering challenges around gear microgeometry and NVH is not just an academic exercise, but a key enabler for the next generation of vehicles.

5. Conclusions

This review has provided an in-depth examination of how gear tooth surface waviness affects noise and vibration behavior, particularly in the context of electric vehicle drivelines. The key findings and takeaways can be summarized as follows:

• Surface waviness is a critical microgeometry factor for gear NVH: Even extremely small periodic deviations (on the order of 0.1–1 µm) on gear flanks can induce ghost frequencies in the gearbox vibration spectrum. These ghost orders can lead to tonal noise that is clearly audible in the absence of masking engine sound, as is the case in EVs. Gears that are within traditional tolerance limits can still exhibit objectionable noise due to unmeasured waviness components, highlighting the need for dedicated waviness control.
• Manufacturing processes imprint characteristic waviness patterns: The choice of finishing process (grinding, honing, lapping, superfinishing) strongly influences the surface texture and waviness. Finishing processes like honing or superfinishing play a critical role in modifying surface waviness. As discussed in Section 3.2, properly selected and controlled finishing methods can effectively reduce tonal noise, while emerging strategies even explore introducing managed waviness for psychoacoustic benefit.
• Gear noise mitigation requires both design and manufacturing excellence: A consensus in literature is that optimal NVH cannot rely on design or manufacturing alone—both must work in concert. Designing for a high contact ratio and including appropriate profile/lead modifications lays the foundation for low transmission error, but manufacturing variations like waviness or runout can negate those benefits if not tightly controlled. On the other hand, outstanding manufacturing (ultra-precision and/or post-process finishing) can sometimes overcome a less-than-ideal design to yield acceptable noise. The quietest gear systems—such as premium EV gearboxes and aerospace drives—are achieved when designers anticipate possible error patterns and when production implements rigorous controls to minimize those errors.
• Advances in measurement and analytics are closing the loop on quality control: Traditional gear inspections were often insufficient to detect waviness [8]. New technologies like inline laser scanning coupled with rolling tests now enable detection of waviness amplitudes down to 0.1 µm and automatic correlation with noise potential. This has made 100% gear noise screening feasible in high-volume production, allowing manufacturers to prevent noisy gears from reaching assembly. Additionally, data-driven models (machine learning) have demonstrated the ability to predict noise test results from measured manufacturing data, with surface waviness identified as a key predictor. Such models, integrated into digital twin frameworks, foreshadow a future where gear NVH is managed proactively via predictive quality control rather than reactive fixes.
• Literature gaps highlight opportunities for further research: We identified several areas where understanding is incomplete or technology is lacking. Notably, there is an absence of standardized waviness metrics or tolerances in gear quality standards. There is also a need for more research on intentional microgeometry modifications for noise reduction, and for improved methods to interpret complex data and guide practical decisions. Addressing these gaps will likely involve collaborative efforts across academia and industry, combining gear engineering expertise with acoustics and data science. Future research should aim to develop quantitative relationships between specific waviness characteristics and noise outcomes, and to establish clear guidelines for acceptable waviness in different applications.

In conclusion, as vehicles become quieter and customer expectations of refinement grow, ensuring low gear noise has become paramount. Surface waviness, once merely a footnote in gear metrology, has emerged as both a culprit for troublesome whine and a lever through which engineers can fine-tune acoustic performance. The state-of-the-art reviewed in this article demonstrates that with current knowledge and tools, it is possible to produce gears that are virtually silent in operation—but doing so requires meticulous attention to micro-scale details and an integrated approach from design through production. The ongoing convergence of improved manufacturing, real-time measurement, and predictive analytics holds great promise for achieving “zero-whine” transmissions in the electric age. Gears may never be absolutely perfect, but with smart engineering, their imperfections can be understood and controlled to a degree that they no longer sing out of tune. The literature is steadily moving toward that vision, and the next decade will likely bring us even closer to the ideal of truly quiet, high-performance gear drives.