A misaligned formulation for planetary gears with analytical 3D contact characterization

Abstract

In the study of planetary drivetrains the substantially high transmitted loads lead to significant deflections in the supporting gear structures, which directly impact the gear pair alignment and alter the gear meshing. This contribution proposes an analytical formulation to model misaligned planetary gears by means of distributed gear contact forces. The presented model relies on a thin-sliced approach for the computation of misaligned contact forces, where the two main nonlinear phenomena affecting contact force accuracy are: (1) variable contact stiffness, and (2) contact detection and compression assessment. With regard to the former, this contribution proposes an internal gear extension of the analytical contact compliance approach previously developed by the authors for external gears. With regard to the latter, this contribution proposes a novel technique for analytical misaligned contact detection for three-dimensional helical gear contact. The model is validated with quasi-static simulations, including gear microgeometry modifications and misalignments, by comparison to numerical results. The computational efficiency of the presented method is enhanced by two kinds of contributions: a linearization of the compliance model to speed up the contact force computation, and an analytical contact Jacobian formulation. The availability of an efficient planetary gear contact model uncovers many interesting applications in the design and operational phases merged with digital twins frameworks.

1 Introduction

Planetary gear systems, integral to a wide industry spectrum encompassing automotive, aerospace, and renewable energy, must meet demanding performance criteria while attaining compactness, lightweight design, and durability. Extensive research has thoroughly examined crucial aspects such as: mesh stiffness [1], planet load sharing [2], modal properties [3], mesh phasing [4], planet spacing [5], and errors in mounting and manufacturing [6], including the nonlinear dynamics induced by backlash and clearances [7].

The compelling necessity for precise misaligned gear contact models, integrating distributed force characterization to accurately depict planetary gear dynamics, has been recognized through comprehensive research spanning the last few decades [8]. The flexible structures supporting the high transmitted gear loads in planetary drivetrains experience deflections which can alter gear alignment and hence gear meshing performance [9, 10].

In the context of planetary gear dynamics, extensive prior research has given rise to numerous contact models employing varied modeling techniques. These models are systematically classified from three distinct perspectives: (1) the computation of mesh stiffness, intricately tied to requisite software tools and computational time, (2) the behavioral aspects of mesh stiffness, directly impacting model accuracy and limitations, and (3) gear contact detection, intricately linked to the physical characterization of the gear pair and the computation of contact forces.

• (1) In the domain of mesh stiffness computation, models are broadly categorized into numerical and analytical methods. Numerical approaches predominantly employ the Finite Element (FE) method with spatial discretization, offering heightened accuracy but at the cost of increased computational demands [11,12,13,14]. To address this, hybrid models emerge, combining local contact analytical solutions with coarse FE models, striking a balance between computational efficiency and geometric flexibility [15,16,17]. While advantageous for intricate gear designs and lightweight structures, the practical utility of FE-based stiffness models may be limited by their offline pre-computation requirements, particularly when computational time is a critical factor. On the contrary, analytical models present an efficient alternative, proving valuable in design phase for: design optimization [18], sensitivity analysis [19], Design Of Experiments (DOE) [20], and large system-level models [21]; and in the operational phase for: failure prediction [22], condition monitoring [23], and digital twin applications [24]. Analytical gear stiffness models exhibit a broad spectrum of complexities. Foundational formulations involve constant [25] or time-varying [26] prescribed gear mesh stiffness models. More advanced approaches incorporate prescribed tooth pair stiffness models [27, 28] providing tooth load sharing ratios, and individual tooth stiffness models [29] constructing the overall mesh stiffness through spring series and parallel connections. The most sophisticated models integrate separate stiffness terms, considering tooth, body, and local contact contributions, and are extended to helical gears by means of the thin-slice method [30] encompassing slice and tooth coupling effects alongside axial stiffness contributions [31].

• (2) From the mesh stiffness behaviour perspective, models can be classified into linear or nonlinear and into time/angle invariant or variant methods [32].

• (3) From the gear contact detection algorithmic perspective and the physical representation of the contact, models can be classified into numerical FE-based methods and analytical methods exhibiting varying levels of complexity:

Numerical methods, employing FE contact detection achieve high accuracy, capturing extended contact beyond the Line Of Action (LOA) at the cost of high computational demands [11,12,13]. Conversely, analytical methods offer computational efficiency, incurring in certain modeling constraints [31]. These analytical models are categorized into two classifications: the straightforward Lumped-Parameter Models (LPM) and the more intricate yet precise Distributed-Parameter Models (DPM). LPMs span from aligned representations incorporating 1D rotational motions [33], to misaligned representations, encompassing either 2D planar translations due to support flexibility [2, 34, 35] or the inclusion of all 3D rigid body motions [3]. DPMs enhance accuracy by providing a detailed physical representation of the distributed contact force along the gear width, allowing for microgeometry profile modifications, in aligned and fully misaligned gear contacts [36, 37].

The subsequent section provides a refined overview of the State-Of-The-Art (SOTA) in planetary gear analytical models, elucidating the primary contributions within this domain.

Torsional LPM: Torsional LPMs commenced with Kahraman’s seminal work establishing a foundational understanding of planetary gear dynamics and natural modes [33]. Subsequent advancements incorporated time-varying mesh stiffness, backlash and clearance effects, for evaluating gear dynamics [38].

Torsional-translational LPM: The prevailing approach in planetary gear publications involves the utilization of a 2D planar LPM. This model extends the torsional DOF with two additional translational DOFs for each gear. Early works by Cunliffe, Botman, and NASA laid the groundwork for understanding planet load sharing, system vibrations, and natural frequencies [39,40,41]. Kahraman et al.’s pioneering planar LPM [2] served as a pivotal benchmark, prompting extensive investigations into diverse aspects, including floating sun motions [42], ring thickness effects [43, 44], manufacturing mounting errors [45, 46], and load-sharing behavior [47]. Parker et al.’s planar LPM contributions are noteworthy, encompassing studies on vibrational modes, gyroscopic effects [35], parameter sensitivity analysis [48], semi-analytical dynamic gear contact models [49], planet phasing, and vibration suppression [50]. The model was extended by an analytical flexible ring gear formulation [51], and by nonlinear factors like bearing clearance, tooth separation, and time-varying mesh stiffness [52], demonstrating its versatility in wind turbine applications [53]. Eritenel extended this model to a 3D LPM for planetary helical gears, emphasizing the importance of tilting motions [3]. Velex et al. conducted a comprehensive study on planetary gear dynamics using a 2D planar model [34], investigating mesh parametric excitations, planet errors in dynamic load sharing [54], trajectories of floating gears [6], and eccentricity errors [55]. Other significant contributions employing planar models delved into nonlinear dynamics, addressing backlash clearances, shaft-bearing compliance [1, 56], system resonances [57], cracked gear teeth [58] and pitting defects [59].

Three-dimensional DPM: In contrast to the conventional LPM approach, recent advancements involve sophisticated DPMs which employ complex misaligned formulations. Velex et al. introduced a fully misaligned 3D helical gear model using thin slicing technique [37], later augmented with flexible ring and planet carrier components [9], and employed for profile modification optimization [60]. Kahraman et al. developed a widely adopted semi-analytical model for fully misaligned planetary gears, incorporating all 6DOF rigid body motions per gear [36], which was applied to investigate tooth wear dynamics [61] and a flexible three-stage planetary wind turbine gearbox [62].

A great persisting challenge within the literature on planetary gear modeling pertains to the development of a comprehensive stiffness formulation and gear contact methodology, that not only ensures accuracy but also enhances computational efficiency, thus facilitating its applicability within system-level modelling. This endeavor seeks to mitigate the inherent limitations of simplistic LPM approaches, while avoiding the computational burden of FE or hybrid methodologies. From the standpoint of mesh stiffness computation (1) and behavior (2), there exists a critical need for a validated framework that employs a tractable analytical formulation to calculate gear mesh stiffness, not only for sun-planet, but also for planet-ring contacts. From the standpoint of gear contact (3), there is a requirement for a validated analytical 3D-DPM methodology that aptly captures the motions of the gear bodies, arising from both gear and supporting structure deflections, in a straightforward formulation.

This work provides a validated reference misaligned planetary gear model, entirely analytical based prioritizing high efficiency and accuracy. The algorithm’s computational performance enables it to excel in conducting extensive studies, adept at capturing the intricate nonlinear phenomena inherent in gear contact dynamics. The authors believe that this study makes two significant contribution to the research field: firstly, by presenting a novel and efficient linear stiffness formulation for internal gear contacts derived from a thorough critical assessment and combination of existing methodologies; and secondly, by presenting a DPM capable of preserving three-dimensional distributed gear contact force information, including misalignments in a direct formulation, where its uniqueness in combining misalignments (either coming from structural elastic deformations or manufacturing assembly errors) and microgeometry modifications results in a complete analytical model. The combination of these two contributions expands the current state-of-the-art: in achieving a greater computational efficiency while notably preserving the model?s accuracy confirmed through rigorous validation against numerical tools; and in providing detailed implementation guidelines to facilitate a seamless reproduction of the model.

This paper comprises four main sections. Section 2 introduces thThe algorithm excels in extensive studies, adept at capturing the intricate nonlinear phenomena inherent in gear contact dynamics.e analytical stiffness model for internal gears, building upon the prior formulation for external gears from [31] and a comparative assessment literature models, incorporating internal gear’s particularities and key simplifications. Section 3 details the analytical distributed gear contact model for internal–external gears, offering algorithmic implementation guidelines for constructing a fully-coupled misaligned planetary gear set. The model accounts for flexible bearing supports, mounting errors, and microgeometry-modified gear profiles, further optimized by the development of an analytical Jacobian formulation. In Sect. 4, the paper presents numerical validation results for aligned and misaligned planetary gear cases through FE static simulations, evaluating performance, distributed gear contact forces, and highlighting its computational costs in comparison with alternative methods. The exploitation of the model is demonstrated, where its reduced computational time facilitates extensive design optimization and robust design studies. Finally, Sect. 5 delineates the research contributions, formulates conclusions, and elucidates prospective avenues for future research.

2 Extension of analytical stiffness model to internal gears

Analytical modeling techniques for internal gears in the current state of the art can be categorized from simpler to more complex formulations. The widely employed ISO 6336 standard [25] provides an empirically-based constant gear mesh stiffness, equivalently in complexity to earlier researchers’ constant [39,40,41] and time-varying [56, 59] mesh stiffness models. Hidaka et al. presented an advanced formulation that approximates the gear tooth as a trapezoid [63]. Chen and Shao further enhanced the modeling approach by introducing the potential energy method, considering bending, shear, and radial deformations [64, 65]. Building on these principles, Liang et al. derived an equivalent time-varying mesh stiffness formulation applicable to planetary gears [66, 67]. More sophisticated models that incorporate rim flexibility in the stiffness formulation are discussed in Sect. 2.2.

This contribution presents an analytical stiffness formulation for internal gears, building upon the authors’ established formulation for external gears [31]. The investigation delves into the underlying physics, synthesizing various literature formulae, and is systematically validated through numerical analysis. The culmination of this study leads to the proposition of an optimal prescribed method for internal gear stiffness. Initially, the compliance matrix is derived for spur internal gears using 2D linear elasticity theory, which is later extended to helical internal gear, accounting for slice coupling and axial compliance effects.

2.1 Tooth stiffness

Based on Weber and Banaschek’s early contributions [68], and similarly followed by other authors [58, 64, 65, 69,70,71], the potential energy method is used to derive tooth bending, shear and radial compressive stiffness terms, based on the following expressions, where the inner ring tooth geometry and force application point are shown in Fig. 1.

$$\begin{aligned} \frac{1}{k_{T}}= & {} \frac{1}{k_{TB}} + \frac{1}{k_{TS}} + \frac{1}{k_{TR}} \end{aligned}$$

(1)

$$\begin{aligned} \frac{1}{k_{TB}}= & {} \frac{12(1-v^2)\cos ^2{\alpha _{yt}}\cos ^2{\beta _b}}{E_e}\nonumber \\ {}{} & {} \quad \times \int _{y_M}^{y_P} \frac{(y_p-y)^2}{(2x)^3}\,dy \end{aligned}$$

(2)

$$\begin{aligned} \frac{1}{k_{TS}}= & {} \frac{2.4(1+v)\cos ^2{\alpha _{yt}}\cos ^2{\beta _b}}{E_e} \int _{y_M}^{y_P} \frac{1}{2x}\,dy \end{aligned}$$

(3)

$$\begin{aligned} \frac{1}{k_{TR}}= & {} \frac{(1-v^2)\sin ^2{\alpha _{yt}}\cos ^2{\beta _b}}{E_e} \int _{y_M}^{y_P} \frac{1}{2x}\,dy\ \end{aligned}$$

(4)

Fig. 1

Internal gear tooth geometry parameters for analytical tooth bending, shear and radial stiffness

2.2 Body rim stiffness

In the modeling of internal gears, body rim stiffness can be addressed through either rigid or flexible formulations. Rigid rim models, become a reasonable assumption under thick, press-fitted or extensively bolted rims [1, 2, 35, 47, 69, 72]. Conversely, flexible rim formulations impact gear dynamics, modulating TE response [73], gear tooth stress and system modes [44], and planet load distributions [37].

Analytical models addressing flexible rim scenarios exist, but are limited. Chen and Shao extended their model [64] to consider thin-rimmed ring gears, observing TE modulation due to equally spaced supports, adding complexity to the frequency spectrum [73, 74]. Other authors applied analytical curved beam expressions based on Timoshenko’s [75], Euler–Bernoulli’s [76], Castigliano’s [77], and uniformly curved beam [78] theorems.

Hybrid techniques blend FE models for far-field deformations with analytical formulas for local contact deformations, and are the most prevalent choice [37, 43, 44, 49, 79, 80]. While the semi-analytical approach offers design flexibility and greater accuracy, its dependence on FE software limits computational efficiency. Alternatively, analytical formulations for bolted or pinned rim bodies also face challenges, as mesh stiffness becomes non-periodic. The presented model could also synthesize this non-periodic mesh stiffness behaviour, but that would require challenging online evaluations which would jeopardize the efficiency of the code.

The widely accepted method proposed by Sainsot et al. [81], employed in the authors’ prior work on external gears [31], is not applicable to internal gear geometries. Instead, the authors evaluated alternative formulations, such as (a) Weber and Banaschek [68] and (b) O’Donnell [82]. Although commonly applied in external gears, these formulations were derived from semi-infinite plane assumptions, allowing potential adaptation for internal gears. The comparison also encompassed (c) a flexible rim torsional model based on in-plane torsional plate theory [83] and (d) a rigid rim model, where no additional stiffness was included in series with the tooth stiffness term. The key equation parameters are defined in Fig. 2, taking into account the internal gear tooth geometry and rim geometry.

Fig. 2

Internal gear tooth and body geometry parameters for analytical body stiffness computation

In the preliminary study of body stiffness effects, various analytical formulations were compared with reference FE and hybrid models (Sect. 4). The results, illustrated in Fig. 3, demonstrate that treating body foundation stiffness as a rigid body or considering in-plane torsion of flexible thin plates [83] yields favorable outcomes, aligning well with the reference model’s TE predictions in terms of mean value and peak-to-peak. Conversely, Weber and Banaschek’s [68] and O’Donnell’s [82] models exhibit larger TE values, thereby overestimating rim flexibility, particularly the latter being more divergent from the reference model.

Fig. 3

Static TE comparison for spur gear considering different body stiffness formulations. Input torque 250 Nm

Therefore, the rim body stiffness is computed according to (c) Yin’s expression [83] or (d) assuming no body stiffness, rigid rim:

$$\begin{aligned} k_{Yin}= & {} \pi G \frac{d_o^2 d_f^2}{d_o^2 - d_f^2} \end{aligned}$$

(5)

$$\begin{aligned} \frac{1}{k_{Tf}}= & {} \frac{d_b}{2} \left( \frac{d_f}{2}-u \right) \frac{\cos \alpha _{yt}}{k_{Yin}} \end{aligned}$$

(6)

And the global tooth pair compliance can be obtained from the series connection of tooth and body individual tooth terms:

$$\begin{aligned} c_{ii}^m = \frac{1}{k_{T1,i}^m} + \frac{1}{k_{Tf1,i}^m} + \frac{1}{k_{T2,i}^m} + \frac{1}{k_{Tf2,i}^m} \end{aligned}$$

(7)

2.3 Contact stiffness

A comprehensive comparison of linear and nonlinear local contact models [31] identified Weber and Banaschek’s nonlinear model [68] as the most accurate, consistent with findings in [32]. Among linear formulations, Cornell’s model [84] demonstrated exceptional accuracy, closely matching nonlinear FE results across various torque levels. Consequently, the authors adopted the established linear Cornell model [84] for the contact stiffness term, balancing accuracy with computational efficiency, as this approach remains applicable to internal gears.

$$\begin{aligned} k_h = \frac{E}{0.5\cdot 4.55(1-v^2)} \end{aligned}$$

(8)

Overall, the pair contact compliance can be computed from the series connection of individual contact stiffnesses scaled by the slice contact width.

$$\begin{aligned} c_{hi}^m = \Big (\frac{1}{k_{h1,i}^m} + \frac{1}{k_{h2,i}^m}\Big ) \frac{1}{b^m_i} \end{aligned}$$

(9)

2.4 Tooth axial stiffness

For helical internal gears, the tooth axial stiffness term is considered, but the body axial stiffness term is omitted, as the formulation is designed for solid external gear bodies [70]. Internal gear bodies are expected to exhibit minimal axial deformations, as their rims are smaller and inherently stiffer axially than those of external gears. Hence, axial stiffness primarily influences tooth deformations. Tooth axial stiffness is modeled using the potential energy method [70], accounting for internal tooth geometry, shown in Fig. 4.

$$\begin{aligned} \frac{1}{k_{A}}= & {} \frac{1}{k_{AB}} + \frac{1}{k_{AT}} \end{aligned}$$

(10)

$$\begin{aligned} \frac{1}{k_{AB}}= & {} \frac{6\sin ^2{\beta _b}}{E_e b^3} \int _{y_M}^{y_V} \frac{(y_v-y)^2}{x}\,dy \end{aligned}$$

(11)

$$\begin{aligned} \frac{1}{k_{AT}}= & {} \frac{6 x_v^2 \sin ^2{\beta _b}}{Gb} \int _{y_M}^{y_V} \frac{1}{b^2x+4x^3}\,dy \end{aligned}$$

(12)

Thus, the total tooth pair axial compliance is computed from the series connection of individual tooth stiffnesses.

$$\begin{aligned} c_{Ai}^m = \frac{1}{k_{A1,i}^m} + \frac{1}{k_{A2,i}^m} \end{aligned}$$

(13)

Fig. 4

Internal gear tooth geometry parameters for analytical axial tooth stiffness computation

2.5 Tooth coupling stiffness

Tooth coupling effects were included and proven to be significant in prior research [31] for external gears, thus are included in the sun-planet contact (see Eq. 14). However, for the internal gears case the formulation based on [71] cannot be applied, since it is only valid for external gears, thus for the planet-ring contact only planet tooth coupling effects are considered (see Eq. 15).

$$\begin{aligned} \frac{1}{k^{sp}_{f,i}}= & {} \frac{1}{k^{s}_{f,i}} + \frac{1}{k^{p}_{f,i}}; \quad c^{t2-1}_{ii} = \frac{1}{k^{sp}_{f,i} b_i} \end{aligned}$$

(14)

$$\begin{aligned} \frac{1}{k^{pr}_{f,i}}= & {} \frac{1}{k^{p}_{f,i}}; \quad c^{t2-1}_{ii} = \frac{1}{k^{pr}_{f,i} b_i} \end{aligned}$$

(15)

In a first instance, the authors assumed that the rim body remains relatively stiff, and thus tooth coupling effects in the internal gear can be negligible. The assumption is confirmed by means of FE simulations in Sect. 4, where only tooth coupling effects were considered in the external gears and good correspondence was obtained for the gear pair TE. The more advanced analytical formulations existing in literature which consider the ring flexibility depending on the boundary and clamping conditions [73], could potentially be adopted to include tooth coupling effects. The major shortcoming would be the increase in algorithmic complexity, where the ring stiffness would again be non-periodic and present modulation effects.

2.6 Slice coupling stiffness

In the traditional uncoupled method, each slice only deflects under a force applied on that (see Eq. 16). However, in the proposed coupled method, the deflection of a single slice is computed as the sum of all slice contact forces multiplied by the coupling compliance, per unit width (see Eq. 17).

$$\begin{aligned} \delta ^u_i= & {} \frac{1}{k'^{u}_i} f_i = c'^{u}_i f_i \end{aligned}$$

(16)

$$\begin{aligned} \delta ^c_i= & {} \sum ^N_{j=1} \frac{1}{k'^{c}_{ji}} f_j = \sum ^N_{j=1} c'^{c}_{ji} f_j \end{aligned}$$

(17)

The slice coupling novel technique presented in prior work [31], which uses the Umezawa method [85] and moment-image technique [86] for the axial influence function, is employed in this work for the internal gear case. The good correspondence obtained with FE in Sect. 4 for helical internal–external gear contact, confirms that the slice coupling method is also applicable to internal gears. Therefore, the coupling compliance terms are computed as follows, based on the distance between slices, gear pair geometry parameters, and the already computed tooth pair global, contact and axial stiffnesses.

$$\begin{aligned} c'^{c}_{ji}= & {} f(c_{ii}^m,z_j,z_i,h,\alpha _{wt}) \end{aligned}$$

(18)

$$\begin{aligned} c^{c}_{ji}= & {} c'^{c}_{ji}/b_j + c_{hj}^m + c_{Aj} \quad \forall i,j\in [1, N] \end{aligned}$$

(19)

3 Extension of analytical gear contact model to internal–external gears

In this work the linear (complementarity) problem formulation developed for external–external distributed gear contacts in [31], is extended to an efficient misaligned planetary model through three steps: (1) derivation of the internal–external gear contact under perfect alignment conditions, (2) development of the misaligned contact model for the entire planetary system, and (3) obtaining analytical Jacobian expressions for the planetary gear contact model. The methodological sequence of the model is outlined in the algorithmic workflow diagram depicted in Fig. 5.

Fig. 5

Analytical gear contact model algorithmic implementation diagram

3.1 Internal–external aligned gear contact

As illustrated in Fig. 6, the path of contact of an internal–external gear pair follows the LOA, which is tangent to the base circle of each gear ($d_{b1}$ and $d_{b2}$), defining the points $T_1$ and $T_2$, and is always locally perpendicular to the tooth surface, defining the point Y in the involute profile. The current tooth contact points are defined by $Y_1$ and $Y_2$ for the external and internal gears respectively. The working pitch point is defined by point C, where both gear’s working circumferences are in tangency ($d_{w1}$ and $d_{w2}$). The expected tooth contact happens along the active region of the LOA, which extends from point A to point E. The initial point of contact (A) is characterized by the lowest contact diameter in the external gear, closest to the root ($d_{Nf1}$) and by the highest contact diameter in the internal gear, closest to the tip ($d_{Na2}$). The last point of contact (E) is characterized by the highest contact diameter in the external gear ($d_{Na1}$) and by the lowest contact diameter in the internal gear ($d_{Nf2}$).

Fig. 6

Internal–external gear pair contact parameters

As illustrated in Fig. 7 for the internal gear case the current gear’s roll angle ($\xi _1$, $\xi _2$) can be computed from the instantaneous angular position of each gear ($\theta _1$, $\theta _2$) and the angular position of the tangency points $T_1$ ($\theta _{T1}$) and $T_2$ ($\theta _{T2}$). The binary variable side takes the following values for right contact ($side=1$), and left contact ($side=-1$).

$$\begin{aligned} \theta _1= & {} \theta _{T1}-\xi _{y1}*side \end{aligned}$$

(20)

$$\begin{aligned} \theta _2= & {} \theta _{T2}-\xi _{y2}*side \end{aligned}$$

(21)

Fig. 7

Internal gear contact parameters

The distances $T_1Y_1$ and $T_2Y_2$ can be computed from the roll angle ($\xi _1$, $\xi _2$) and the base diameter of each gear ($d_{b1}$, $d_{b2}$), where a negative sign needs to be included in the internal gear case to compute $T_2Y_2$. Following ISO standard nomenclature [87] the sign change of the internal gear is defined by $z_2/|z_2|$, with $z_2$ being negative for internal gears. The total $T_1T_2$ distance is obtained from the working center distance ($a_w$) of the gear pair and the working transverse pressure angle ($\alpha _{wt}$).

$$\begin{aligned} |T_1Y_1|_i= & {} \xi _{y1,i}(\theta _{1,i})d_{b1}/2 \end{aligned}$$

(22)

$$\begin{aligned} |T_2Y_2|_i= & {} \frac{z_2}{|z_2|}\xi _{y2,i}(\theta _{2,i})d_{b2}/2 \end{aligned}$$

(23)

$$\begin{aligned} |T_1T_2|= & {} a_w \sin {\alpha _{wt}} \end{aligned}$$

(24)

The analytical expression for computing the internal–external gear pair compression, for each individual slice, is obtained as follows where the negative sign needs to be included as well in front of the distance $T_1T_2$.

$$\begin{aligned} \delta _{ti}&=|T_1Y_1|_i + |T_2Y_2|_i - |T_1T_2| = \xi _{y1,i}(\theta _{1,i})d_{b1}/2 \nonumber \\&+ \frac{z_2}{|z_2|}\xi _{y2,i}(\theta _{2,i})d_{b2}/2 - a_w \sin {\alpha _{wt}} \end{aligned}$$

(25)

The compression in the normal direction, of the driving gear’s tooth flank into the mating driven gear’s tooth flank, is:

$$\begin{aligned} \delta _{ni}=\delta _{ti}\cos {\beta _b} \end{aligned}$$

(26)

When microgeometry modifications are present, the amount of material removed in the normal direction for each gear ($\delta _{micro1}$, $\delta _{micro2}$) is directly included in the compression expression:

$$\begin{aligned} \delta _{ni}=(\delta _{ti} - \delta _{micro1} - \delta _{micro2}) \cos {\beta _b} \end{aligned}$$

(27)

The procedure is summarized in the Algorithm 1 implementation guidelines.

Algorithm 1

Computation of gear pair LOA compression in aligned case

3.2 Planetary misaligned gear contact

The presented misaligned planetary gear model defines the system DOFs, encompassing the six rigid body DOFs for the ring, carrier, sun, and N planets, resulting in a planetary gear system with 18+6N DOFs. Radial translational motions (along x, y axis) are relevant for the floating motions of planetary gears. Axial translational motion (along z axis) becomes important in the presence of axially loaded helical gears. Misaligned rotational motions (along x, y axis), also referred as twisting and tilting or yaw and pitch motions, have seldom been included in the planetary models available in literature, but have significant effects in the contact pattern. Torsional rotational motion (along the z axis) is essential in torque transmission and always included in the most simple 1-DOF LPM.

The misaligned formulation presented in this work improves the capabilities of classical LPMs [2, 3, 34, 35] in extending the contact characterization from a single lumped-point contact (LPM) to a distributed force contact (DPM). In respect to other advanced DPMs [36, 37], which commonly use a direct projection of the slice displacements to treat gear misalignments, this methodology instead considers a reduction technique by which obtains the relative mesh misalignments. This method allows to simply and more efficiently incorporate the body misalignment effects, uniquely combining those with assembly errors and microgeometry modifications, in the mesh compression equation, permitting the full analytical derivation of the Jacobian matrix to accelerate static and dynamic analysis.

Generalized coordinates: The generalized coordinates of each gear are expressed as follows, where each planet is numbered by index (n):

$$\begin{aligned} \begin{aligned}&q=\{q_r,q_c,q_s,q_p^n\} \\&q_j=\{x_j,y_j,z_j,\varphi _{j},\psi _{j},\theta _{j}\} \forall j \in r,c,s,p^n \\ \end{aligned} \end{aligned}$$

(28)

Gears are supported by flexible bearings featuring $5 \times 5$ linear stiffness matrices allowing free torsional rotations. Structural body deformations, like ring housing support or carrier structural deformation, can also be incorporated using linear springs in series with bearing stiffness through node-condensation FE techniques. Inertia and mass properties of the bodies are lumped to the gear’s center. The proposed formulation accommodates both left and right side contacts, but for the seek of clarity, the equations presented consider left-flank contact between sun-planet and right-flank contact between planet-ring. The global frame of reference aligns with the X-axis horizontally, the Y-axis vertically, and the Z-axis outward. Each gear pair local frame of reference is attached to the gear center and oriented along the pair center line. The angle of the center distance line ($\theta _{CD}^n$) models the global to local rotations and vice versa (see Fig. 8).

Fig. 8

Generalized coordinates (blue) of the misaligned planetary gear system with 36 DOFs, in the global (black) and local gear pair’s (red) frame of reference

The 3D rotation matrix from local to global frame expressed in Bryan angles is obtained from [13], where c stands for $\cos$ and s stands for $\sin$, and for the particular case of the carrier frame 3D rotation the matrix is denoted by $R_c$.

$$\begin{aligned}{} & {} \begin{aligned}&R(\varphi ,\psi ,\theta ) = \\&\begin{bmatrix} c\psi c\theta &{} \quad -c\psi s\theta &{}\quad s\psi \\ s\varphi s\psi c\theta + c\varphi s\theta &{}\quad -s\varphi s\psi s\theta + c\varphi c\theta &{}\quad -s\varphi c\psi \\ -c\varphi s\psi c\theta + s\varphi s\theta &{}\quad c\varphi s\psi s\theta + s\varphi c\theta &{}\quad c\varphi c\psi \\ \end{bmatrix} \end{aligned}\nonumber \\ \end{aligned}$$

(29)

$$\begin{aligned}{} & {} R_c = R(\varphi _c,\psi _c,\theta _c) \end{aligned}$$

(30)

Planet carrier pin coordinates: The planet carrier pin locations in the global frame can be computed from the pin position in the local carrier frame, given by the center distance ($a_w$) and the axial coordinate of the pin ($z_{pin}^n$). Together with the carrier rotation matrix from local to global, including as well possible pin mounting errors: radial center distance offset ($r_{pin}^n$) and angular orientation errors ($\varphi _{pin}^n$, $\psi _{pin}^n$).

$$\begin{aligned}&q_{pin}^{n(1:3)} = q_c^{(1:3)} + \nonumber \\&R_c \begin{Bmatrix} (a_w + r_{pin}^n)\sin {(2\pi (n-1)/N)} \\ (a_w + r_{pin}^n)\cos {(2\pi (n-1)/N)} \\ z_{pin}^n \\ \end{Bmatrix} \end{aligned}$$

(31)

$$\begin{aligned}&\quad q_{pin}^{n(4:6)} = q_c^{(4:6)} + \{ \varphi _{pin}^n; \psi _{pin}^n; 0\} \end{aligned}$$

(32)

The angle of the center distance line can be computed from the known pin position:

$$\begin{aligned} \theta _{CD}^n = \arctan {(q_{pin}^{n(2)}/q_{pin}^{n(1)})}-\pi /2 \end{aligned}$$

(33)

Equivalent gear pair relative misalignments:

The misalignment behavior of a gear pair is determined solely by their relative misalignment, not their absolute positions. Thus, the reduced formulation employs only relative misalignments, which are fewer than the total DOFs, enhancing computational efficiency (5 relative misalignments vs. 10 absolute DOFs, excluding rotations). Instead, for rotations, both relative (i.e. transmission error) and absolute rotational displacement are crucial in system modelling, so both absolute rotational DoFs are kept in the generalized coordinates vector. This reduction allows for a more compact analytical formulation and helps to conceptualize the definition of relative gear mesh misalignments. Then, the relative misalignments are used to displace slices and project those displacements along the LOA to get the final equivalent mesh compression.

Fig. 9

Equivalent sun-planet mesh misalignment: original aligned configuration (gray), misaligned configuration (red) and theoretical planet location or reference point

The reduction technique consists on computing the relative misalignments between two mating gears (i.e. planet and sun), expressed in a world oriented frame located at the sun center origin, as illustrated in Fig. 9. The relative translational misalignments are computed as the difference between the planet location ($q_p^n$) and the reference point location ($p_{ref}$), later explained in detail. The relative angular misalignments are computed by directly subtracting the sun rotations from the planet in the global frame.

$$\begin{aligned} \delta _{eq} = \delta _{sp}^{n(1:5)} = \begin{Bmatrix} q_p^{n(1:3)}-p_{ref} \\ q_p^{n(4:5)}-q_s^{(4:5)} \\ \end{Bmatrix} \end{aligned}$$

(34)

The reference point is defined as the closest point to the actual planet center, located in the circle of theoretical planet locations, which is a circle of radius ($a_w$) from the sun center (see Fig. 9). The reference point is computed from intersecting the line that unites sun’s center ($q_s^{(1:3)}$) and planet’s center ($q_p^{n(1:3)}$) with a sphere of radius ($a_w$) located at the sun’s center. Therefore, the following quadratic system of equations is obtained by combining the single sphere’s equation with the three parametric equations of a 3D line. The system can be solved either numerically or analytically upon substitution of the parametric line equations into the sphere equation, resulting in a quadratic equation:

$$\begin{aligned} p_{ref} = \left\{ \begin{aligned}&x^2 + y^2 + z^2 = a_w^2 \\&x(t) = q_s^{(1)} + t(q_p^{n(1)}-q_s^{(1)}) \\&y(t) = q_s^{(2)} + t(q_p^{n(2)}-q_s^{(2)}) \\&z(t) = q_s^{(3)} + t(q_p^{n(3)}-q_s^{(3)}) \end{aligned} \right. \end{aligned}$$

(35)

The same reduction technique is applied to the planet-ring contact to obtain the 5 DOFs relative mesh misalignments. Instead of the sun, now the ring local frame is taken as the reference, thus the same set of Eqs. 34 to 35 hold, by replacing sun related variables by ring ones.

Displaced slices misaligned formulation: Upon computing the relative mesh misalignments for both the sun-planet and planet-ring contacts, the contact force algorithm utilizes them to displace the slices and project the displacements along the LOA direction, normal to the tooth flank surface. In the sun-planet case, indexes 1 and 2 correspond to the sun and planet, while in the planet-ring case, indexes 1 and 2 correspond to the planet and ring.

To displace the slices, it is essential to determine the instantaneous contact point on the gear’s surface, determined by: (a) the current transverse working pressure angle ($\alpha _{yt1,i}$ and $\alpha _{yt2,i}$)

$$\begin{aligned}&\alpha _{yt1,i} = \arctan (\xi _{y1,i}(\theta _{1,i})) \nonumber \\&\alpha _{yt2,i} = \arctan (\xi _{y2,i}(\theta _{2,i})) \end{aligned}$$

(36)

and (b) the current contact radius ($r_{1,i}$ and $r_{2,i}$) at the contact point, based on Fig. 7.

$$\begin{aligned}&r_{1,i} = d_{b1}/(2\cos {\alpha _{yt1,i}}) \nonumber \\&r_{2,i} = d_{b2}/(2\cos {\alpha _{yt2,i}}) \end{aligned}$$

(37)

(c) The current contact point in the local gear pair frame, with y-axis along the center line, is determined for the first gear:

$$\begin{aligned}&x_{1,i} = side*r_{1,i}\sin (\alpha _{yt1,i}-\alpha _{wt}) \nonumber \\&y_{1,i} = r_{1,i}\cos (\alpha _{yt1,i}-\alpha _{wt}) \nonumber \\&z_{1,i} = b/2-b\frac{i-1}{N_s-1} \end{aligned}$$

(38)

and for the second gear:

$$\begin{aligned}&x_{2,i} = side*r_{2,i}\sin (\alpha _{wt}-\alpha _{yt2,i}) \frac{z_2}{|z_2|} \nonumber \\&y_{2,i} = -r_{2,i}\cos (\alpha _{wt}-\alpha _{yt2,i}) \frac{z_2}{|z_2|} \nonumber \\&z_{2,i} = b/2-b\frac{i-1}{N_s-1} \end{aligned}$$

(39)

(d) The contact points in the global frame are obtained by rotating the vectors as:

$$\begin{aligned} q_{G1,i} = R_{1} \begin{Bmatrix} x_{1,i}\\ y_{1,i}\\ z_{1,i}\\ \end{Bmatrix}; q_{G2,i} = R_{2} \begin{Bmatrix} x_{2,i}\\ y_{2,i}\\ z_{2,i}\\ \end{Bmatrix} \end{aligned}$$

(40)

where the gear rotational matrices ($R_1$ and $R_2$) are computed from:

$$\begin{aligned}&R_{1} = R(\varphi _1,\psi _1,\theta _{CD}) \nonumber \\&R_{2} = R(\varphi _2,\psi _2,\theta _{CD}) \end{aligned}$$

(41)

(e) The individual slice displacements in the global frame, are computed from the equivalent mesh misalignments (Eq. 34) and the contact points (Eq. 40), for the sun-planet contact:

$$\begin{aligned} \begin{Bmatrix} \delta _{x,i}\\ \delta _{y,i}\\ \delta _{z,i}\\ \end{Bmatrix} = \delta _{sp}^{n(1:3)}+ \begin{Bmatrix} \delta _{sp}^{n(4)}\\ \delta _{sp}^{n(5)}\\ 0 \\ \end{Bmatrix} \times q_{G2,i}^{(1:3)} \end{aligned}$$

(42)

and for the planet-ring contact:

$$\begin{aligned} \begin{Bmatrix} \delta _{x,i}\\ \delta _{y,i}\\ \delta _{z,i}\\ \end{Bmatrix} = \delta _{pr}^{n(1:3)}+ \begin{Bmatrix} \delta _{pr}^{n(4)}\\ \delta _{pr}^{n(5)}\\ 0 \\ \end{Bmatrix} \times q_{G1,i}^{(1:3)} \end{aligned}$$

(43)

(f) The surface normal directions in the contact point, expressed in the global frame, are necessary to project the slice displacements along the LOA direction. Those are obtained by rotating the known surface normal vector in the local frame:

$$\begin{aligned} \textbf{n}_{L1}= & {} \begin{Bmatrix} \cos {\alpha _{wt}}\cos {\beta _{b}}*side\\ \sin {\alpha _{wt}}\cos {\beta _{b}}\\ \sin {\beta _{b}}*side\\ \end{Bmatrix} \end{aligned}$$

(44)

$$\begin{aligned} \textbf{n}_{L2}= & {} - \textbf{n}_{L1} \end{aligned}$$

(45)

$$\begin{aligned} \textbf{n}_{G1}= & {} R_{1} \textbf{n}_{L1} \end{aligned}$$

(46)

$$\begin{aligned} \textbf{n}_{G2}= & {} R_{2} \textbf{n}_{L2} \end{aligned}$$

(47)

(g) The projection along the LOA is obtained from the dot product between the slice displacements and the normal directions in the global frame, for sun-planet contact:

$$\begin{aligned} \delta _{mis,i} = \textbf{n}_{G2} \cdot \begin{Bmatrix} \delta _{x,i}\\ \delta _{y,i}\\ \delta _{z,i}\\ \end{Bmatrix} \end{aligned}$$

(48)

and planet-ring contact:

$$\begin{aligned} \delta _{mis,i} = \textbf{n}_{G1} \cdot \begin{Bmatrix} \delta _{x,i}\\ \delta _{y,i}\\ \delta _{z,i}\\ \end{Bmatrix} \end{aligned}$$

(49)

(h) The normal compression along the LOA is finally computed for each slice by adding the misalignment contribution to the compression expression for the aligned case, including also microgoemtry modifications as indicated in Eq. 27:

$$\begin{aligned} \delta _{ni}=(\delta _{ti} - \delta _{micro1} - \delta _{micro2})\cos {\beta _b} + \delta _{mis,i} \end{aligned}$$

(50)

The procedure for misaligned contacts in a planetary gear system is summarized in the Algorithm 2 implementation guidelines.

Algorithm 2

Computation of gear pair LOA compression in misaligned case

Contact force computation: The linear problem formulation computes individual slice contact forces ($f_n$) from the known vector of slice compressions ($\delta _{ni}$) and the global compliance map of the gear pair (C), obtained by interpolating pre-computed gear pair compliance (see Sect. 2)). The normal contact forces in the local frame for each slice are determined by solving either a Linear Problem (LP) in the aligned case with no lead modifications, or a Linear Complementarity Problem (LCP) in the misaligned case or with lead modifications, as explained in detail in the author’s prior work [31].

$$\begin{aligned} \{f_n\}= [C] \{\delta _n\} \end{aligned}$$

(51)

To express the contact forces of each gear in the global frame, it is necessary to project the normal forces along the inward pointing vectors, normal to the surface. These surface normal vectors must be previously rotated from the local to the global frame based on the gear’s Bryant angles representation:

$$\begin{aligned} A= & {} R(\varphi ,\psi ,\theta ^t) \end{aligned}$$

(52)

$$\begin{aligned} \textbf{F}_i= & {} -f_{ni} A \textbf{n}_{L} \end{aligned}$$

(53)

where ($\theta ^t$) represents the angular position of the t-th tooth, in contact. To compute the bending moments and torsional torque contributions of each slice force, a multi-body floating frame of reference formulation is employed by means of the ($\bar{G}^T \tilde{\bar{u}}^i A^T$) transformation according to the rotational matrices (A), ($\bar{G}$), and position matrix ($\tilde{\bar{u}}^i$), as explained in [13]. The rotation matrix ($\bar{G}$) is computed as follows, with the small angle approximation for $\varphi$ and $\psi$:

$$\begin{aligned} \bar{G}(\varphi ,\psi ,\theta ){} & {} = \begin{bmatrix} c\varphi c\theta &{} \quad s\theta &{} \quad 0 \\ -c\psi s\theta &{} \quad c\theta &{} \quad 0 \\ s\psi &{} \quad 0 &{} \quad 1 \\ \end{bmatrix} \nonumber \\{} & {} \approx \begin{bmatrix} c\theta &{} \quad s\theta &{} \quad 0 \\ -s\theta &{} \quad c\theta &{} \quad 0 \\ \psi &{} \quad 0 &{} \quad 1 \\ \end{bmatrix} \end{aligned}$$

(54)

The skewed-symmetric matrix ($\tilde{\bar{u}}^i$) is computed as follows, with vector ($\bar{u}^i$) being the point of contact in the local frame:

$$\begin{aligned} \tilde{\bar{u}}^i \approx \begin{bmatrix} 0 &{}\quad -\bar{u}_3^i &{} \quad \bar{u}_2^i \\ \bar{u}_3^i &{} \quad 0 &{} \quad -\bar{u}_1^i \\ -\bar{u}_2^i &{} \quad \bar{u}_1^i &{} \quad 0 \\ \end{bmatrix} \end{aligned}$$

(55)

The rotation matrix (A) given in Eq. 29, yields the following expression with the small angle approximation for $\varphi$ and $\psi$:

$$\begin{aligned} A(\varphi ,\psi ,\theta ) \approx \begin{bmatrix} c\theta &{} \quad -s\theta &{} \quad \psi \\ s\theta &{} \quad c\theta &{}\quad -\varphi \\ -\psi c\theta + \varphi s\theta &{}\quad \psi s\theta + \varphi c\theta &{}\quad 1 \\ \end{bmatrix} \nonumber \\ \end{aligned}$$

(56)

Thus the bending moment and torsional torque can be computed by the following transformation:

$$\begin{aligned} \textbf{M}_i = \bar{G}^T \tilde{\bar{u}}^i A^T \textbf{F}_i \end{aligned}$$

(57)

The overall gear contact reactions in global frame are determined by summing the contributions of individual slice forces and moments over all slices and teeth in contact, repeating the procedure for both gears:

$$\begin{aligned} Q_c = \sum _i \begin{Bmatrix} \textbf{F}_i \\ \textbf{M}_i \\ \end{Bmatrix}_{6x1} \end{aligned}$$

(58)

The same procedure is employed to calculate the sun-planet ($Q_{sp}^n, Q_{ps}^n$) and planet-ring ($Q_{pr}^n, Q_{rp}^n$) contact forces for each gear. The generalized contact forces on each gear body are determined by summing the contributions from each pair:

$$\begin{aligned} Q_{cp}^n= & {} Q_{ps}^n + Q_{pr}^n \end{aligned}$$

(59)

$$\begin{aligned} Q_{cr}= & {} \sum _n Q_{rp}^n \end{aligned}$$

(60)

$$\begin{aligned} Q_{cs}= & {} \sum _n Q_{sp}^n \end{aligned}$$

(61)

The generalized contact forces vector is:

$$\begin{aligned} Q_c=\{Q_{cr},0^{6\times 1},Q_{cs},Q_{cp}^n\} \end{aligned}$$

(62)

Bearing reaction forces computation: The reaction forces of the flexible bearings or equivalent bushings are computed for each gear (sun, planets, ring) and structures (carrier) supports. The ring bearing reaction forces are determined based on the current position of the ring ($q_r$), the bearing linearized stiffness matrix ($K_{br}$) with 6x6 dimensions, considering also a torsional spring, and the linearization point reactions ($Q_{0br}$) and displacement ($q_{0br}$), typically assumed to be null, at the center of the ring gear:

$$\begin{aligned} Q_{br}=Q_{0br}+K_{br}(q_r-q_{0br}) \end{aligned}$$

(63)

The sun bearing reactions are computed similarly:

$$\begin{aligned} Q_{bs}=Q_{0bs}+K_{bs}(q_s-q_{0bs}) \end{aligned}$$

(64)

The forces and moments acting on the carrier ($Q_{bc}$) are a combination of: the (1) carrier bearing forces ($Q_{bcc}$) occurring due to carrier motions, and the (2) reactions of the planet-pin bearing forces ($Q_{bcp}^n$) supported by the carrier:

$$\begin{aligned} Q_{bc}=Q_{bcc}+\sum _n Q_{bcp}^n \end{aligned}$$

(65)

The (1) carrier bearing forces are:

$$\begin{aligned} Q_{bcc} = Q_{0bc}+K_{bc} (q_c - q_{0bc}) \end{aligned}$$

(66)

To acquire the (2) planet-pin bearing forces, it is necessary to determine the planet-pin relative displacements (defined as the difference between planet and pin positions) in the global frame, and rotate them to the local frame based on the pin rotation matrix:

$$\begin{aligned} R_{pin}^n= & {} R(\varphi _{pin}^n,\psi _{pin}^n,\theta _{pin}^n) \end{aligned}$$

(67)

$$\begin{aligned} q_{Gppin}^n= & {} q_{p}^n - q_{pin}^n \end{aligned}$$

(68)

$$\begin{aligned} q_{Lppin}^n= & {} \begin{Bmatrix} (R_{pin}^n)^T q_{Gppin}^{n(1:3)} \\ q_{Gppin}^{n(4:5)} \end{Bmatrix} \end{aligned}$$

(69)

The planet-pin bearing forces in the local frame are computed as:

$$\begin{aligned} Q_{Lbp}^n = K_{bp} q_{Lppin}^n \end{aligned}$$

(70)

and are later rotated from the pin local frame to the global frame as:

$$\begin{aligned} Q_{bp}^n = \begin{Bmatrix} R_{pin}^n \times Q_{Lbp}^{n(1:3)} \\ R_{pin}^n \times Q_{Lbp}^{n(4:5)} \\ \end{Bmatrix} \end{aligned}$$

(71)

Thus, the planet-pin bearing reaction forces and moments on the carrier are computed as:

$$\begin{aligned}&Q_{bcp}^n = \nonumber \\&\begin{Bmatrix} -Q_{bp}^{n(1:3)} \\ -Q_{bp}^{n(4:6)} - \left[ R_c \left( q_{pin}^{n(1:3)} - q_c^{(1:3)} \right) \times Q_{bp}^{n(1:3)} \right] \\ \end{Bmatrix} \end{aligned}$$

(72)

The total bearing reaction vector is:

$$\begin{aligned} Q_b=\{Q_{br},Q_{bc},Q_{bs},Q_{bp}^n\} \end{aligned}$$

(73)

Overall the planetary gear system reactions are determined through the load balance equation, incorporating contributions from contact reactions, externally applied loads, gravitational forces, and bearing forces.

$$\begin{aligned} Q = Q_c+Q_{ext}+Q_g-Q_b \end{aligned}$$

(74)

3.3 Analytical Jacobian formulation

To enhance the computational efficiency of the proposed method, analytical Jacobian expressions for the planetary gear model are derived, and integrated into the Newton search algorithm for solving each quasi-static equilibrium configuration. Unlike computing a numerical Jacobian through finite differences, the use of easily-to-evaluate analytical Jacobian expressions significantly improves computational performance. The overall Jacobian is computed based on the contact and bearing related terms:

$$\begin{aligned} J = \frac{\partial Q}{\partial q} = J_c - J_b = \frac{\partial Q_c}{\partial q} - \frac{\partial Q_b}{\partial q} \end{aligned}$$

(75)

The contact related Jacobian matrix is:

$$\begin{aligned} \frac{\partial Q_c}{\partial q} = \begin{bmatrix} \frac{\partial Q_{cr}}{\partial q_r} &{} \quad 0 &{} \quad 0 &{}\quad \frac{\partial Q_{cr}}{\partial q_p^n} \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad \frac{\partial Q_{cs}}{\partial q_s} &{}\quad \frac{\partial Q_{cs}}{\partial q_p^n} \\ \frac{\partial Q_{cp}^n}{\partial q_r} &{}\quad 0 &{}\quad \frac{\partial Q_{cp}^n}{\partial q_s} &{}\quad \frac{\partial Q_{cp}^n}{\partial q_p^n} \\ \end{bmatrix} \end{aligned}$$

(76)

where upon expansion of each contact force term the contact related Jacobian matrix yields:

$$\begin{aligned}{} & {} \frac{\partial Q_c}{\partial q}\nonumber \\{} & {} = \begin{bmatrix} \sum \frac{\partial Q_{rp}^n}{\partial q_r} &{}\quad 0 &{}\quad 0 &{}\quad \sum \frac{\partial Q_{rp}^n}{\partial q_p^n} \\ 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{}\quad \sum \frac{\partial Q_{sp}^n}{\partial q_s} &{}\quad \sum \frac{\partial Q_{sp}^n}{\partial q_p^n} \\ \frac{\partial Q_{ps}^n}{\partial q_r} + \frac{\partial Q_{pr}^n}{\partial q_r} &{}\quad 0 &{}\quad \frac{\partial Q_{ps}^n}{\partial q_s} + \frac{\partial Q_{pr}^n}{\partial q_s} &{}\quad \frac{\partial Q_{ps}^n}{\partial q_p^n} + \frac{\partial Q_{pr}^n}{\partial q_p^n} \\ \end{bmatrix} \nonumber \\ \end{aligned}$$

(77)

and the bearing related Jacobian matrix is:

$$\begin{aligned} \frac{\partial Q_b}{\partial q}= \begin{bmatrix} \frac{\partial Q_{br}}{\partial q_r} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \frac{\partial Q_{bc}}{\partial q_c} &{}\quad 0 &{}\quad \frac{\partial Q_{bc}}{\partial q_p^n} \\ 0 &{}\quad 0 &{}\quad \frac{\partial Q_{bs}}{\partial q_s} &{}\quad 0 \\ 0 &{}\quad \frac{\partial Q_{bp}^n}{\partial q_c} &{}\quad 0 &{}\quad \frac{\partial Q_{bp}^n}{\partial q_p^n} \\ \end{bmatrix} \end{aligned}$$

(78)

4 Numerical validation and results

The accuracy of the presented analytical misaligned planetary model is verified against two primary reference models developed in-house in The Mathworks Matlab [13]: (1) a nonlinear FE model (2) a hybrid model that combines FE-based and analytical techniques, as elaborated in [31]. The validation process encompasses both spur and helical planetary gear sets, with three planets considered (see Fig. 10), the geometrical parameters of these are defined in Table 1, where the material properties correspond to those of steel ($E=207$ GPa, $\nu =0.3$, $\rho =7800 \,\text {kg/m}^3$). The planet carrier dimensions are: 30 mm width, 60 mm inner diameter and 100 mm outer diameter.

Fig. 10

Geometry of both planetary spur and helical gears, positioned at the initial configuration of the static analysis

Table 1 Planetary gear geometry data for spur/helical cases. (RHS = Right Hand Side)

The in-house FE model [13] employs a mixed-mesh topology, finely discretizing the teeth pairs involved in the contact and coarsely discretizing the rest, in combination with solid hexahedral elements and a Node-To-Surface penalty-based contact algorithm.

The in-house hybrid model [13] combines the FE-based global gear teeth compliance with analytical expressions for local contact compliance. Following the three-step approach of [17]: first, a unit load normal to the tooth surface is applied at each grid point, computing normal displacements other grid points. The second step removes erroneous local contact displacements induced from point forces and discretization, using another FE model with teeth nodal DOFs clamped at a specific depth and considering the same unit load applied, to subtract those displacements. The third step complements these with local contact displacements from analytical expressions, typically being load-dependent, but where linear formulae can be equally used.

Static simulations are performed by rotating the gears over one entire meshing period of the planetary set, equivalent to a carrier rotation over one angular pitch of the ring. The simulation is discretized into multiple angular configurations, for each the torque balance equilibrium is achieved by adjusting the gear’s angles by means of a Newton–Raphson iteration on the residuals. Additionally, in presence of misalignments the force and moment balance equilibrium are achieved by adjusting the gear’s position. An external torque is applied to the carrier, the sun is fixed at different angular configuration, and the planet and carrier are free to move torsionally, while the ring is clamped. In misaligned cases, all gears can freely misalign and are supported by flexible bearings.

The Static Transmission Error (STE) is employed as the main validation metric, given that under quasi-static conditions the STE evolution is directly linked to the variation of the gear pair mesh stiffness, dependent also on the contact detection algorithm. Therefore, comparing STE values allows for the validation of both the analytical stiffness and contact models. In the validation process: firstly the internal–external gear contact model is validated in Sect. 4.1. Then the planetary gear model is validated for perfectly aligned conditions in Sect. 4.2 anf for misaligned conditions in Sect. 4.3. Lastly, the fully misaligned planetary model is simulated in Sect. 4.4, and the model’s efficient performance is evaluated in Sect. 4.5.

Fig. 11

STE comparison for internal–external spur gear contact, between analytical and hybrid models without microgeometry modifications

4.1 Internal–external gear pair model validation

Firstly, a spur internal–external gear pair, with equivalent geometry to the previous planet-ring pair Table 1, is simulated by both the analytical and hybrid models for stiffness validation. In Fig. 11 both models exhibit matching STE trends and comparable peak-to-peak across various torques, a slight difference in absolute STE value suggests that the analytical rim-stiffness formulation is not flexible enough. Secondly, the analytical model is compared to the hybrid and FE reference models to validate the contact model. Microgeometry modifications are introduced to eliminate corner contact in the internal–external pair, outlined in Table 2.

Table 2 Microgeometry modifications applied to spur internal–external gear pair

In Fig. 12 the various contact models exhibit good agreement, predicting comparable contact ratios and STE transition curves with similar peak-to-peak values. The FE model appears slightly more rounded and smooth, potentially indicating the occurrence of Out-of-Line-Of-Action (OLOA) contact.

Fig. 12

STE comparison for internal–external spur gear contact, between analytical, hybrid and FE models with microgeometry modifications to avoid corner contact

The same methodology is applied to helical gears. As illustrated in Fig. 13 analytical, hybrid and FE methods demonstrate consistent agreement in terms of STE characteristics, with only a slight discrepancy in absolute TE values as previously discussed. In terms of TE peak-to-peak values, Fig. 14 further indicates close agreement between the hybrid and analytical models, demonstrating notable consistency with FE results across various torque levels. Therefore, minor deviations in STE shape and rounding are attributed to corner contact and local contact nonlinear effects.

Fig. 13

STE comparison for internal–external helical gear contact, between analytical, hybrid and FE models without microgeometry modifications

Fig. 14

STE peak-to-peak comparison at 400 Nm and 100 Nm for internal–external helical gear contact, between analytical, hybrid and FE models

4.2 Planetary gear model validation

With a validated gear contact model for internal–external pairs, established in the previous section, and for external–external pairs, developed in a prior publication [31], this section focuses on validating the planetary gear model. The analytical model is compared with the two reference models, first for the spur planetary set and subsequently for the helical configuration, as detailed in Table 1.

For the spur case, the individual tooth and overall mesh stiffness evolution predicted by the analytical model for the sun-planet and for the planet-ring gear pair contact are shown respectively in Figs. 15 and 16.

Fig. 15

Tooth and mesh stiffness evolution for sun-planet spur gear contacts, predicted by analytical model

Fig. 16

Tooth and mesh stiffness evolution for planet-ring spur gear contacts, predicted by analytical model

Great correspondence between the proposed analytical model and both the reference FE and hybrid models is observed in terms of trend, peak-to-peak value, and average STE values, as illustrated in Fig. 17. Incorporating tooth coupling effects clearly enhances the accuracy, this becomes evident when contrasting the results with those obtained through the traditional method, which neglects slice and teeth coupling effects. A detailed examination of the sun-planet and planet-ring individual STE, reinforced this necessity, as depicted in Figs. 18 and 19.

Fig. 17

Static transmission error for the planetary spur gear set at multiple torques, comparison between FE versus hybrid versus analytical

Fig. 18

Static transmission error for sun-planet at multiple torques, comparison between FE versus hybrid versus analytical

Fig. 19

Static transmission error for planet-ring at multiple torques, comparison between FE versus hybrid versus analytical

The validation process is extended to the helical planetary configuration, employing a consistent approach. Figures 20 and 21 display the analytical model’s predictions for the evolving individual tooth and overall mesh stiffness in the sun-planet and planet-ring helical gear pair contacts.

Fig. 20

Tooth and mesh stiffness evolution for spur sun-planet helical gear contacts, predicted by analytical model

Fig. 21

Tooth and mesh stiffness evolution for spur planet-ring helical gear contacts, predicted by analytical model

The analytical method aligns closely with reference FE and hybrid models for the helical planetary case, showcasing similar trends, peak-to-peak values, and average STE as depicted in Fig. 22. Inclusion of slice and tooth coupling effects improves accuracy, this becomes evident when contrasting the results with the traditional method, lacking such considerations. Comparable conclusions emerge from detailed analyses of sun-planet and planet-ring individual TE, illustrated in Figs. 23 and 24.

Fig. 22

Static transmission error for the planetary helical gear set at multiple torques, comparison between FE versus hybrid versus analytical

Fig. 23

Static transmission error for helical sun-planet at multiple torques, comparison between FE versus hybrid versus analytical

Fig. 24

Static transmission error for helical planet-ring at multiple torques, comparison between FE versus hybrid versus analytical

4.3 Misaligned internal–external gear pair model validation

A comprehensive validation of the misaligned gear contact formulation is undertaken for the internal–external case (planet-ring pair), which complements the prior author’s validation of the misaligned external–external case (sun-planet pair) [31]. The proposed analytical contact model is compared with a nonlinear FE contact model developed in ABAQUS to validate its performance under misaligned conditions.

Fig. 25

FE mixed-mesh topology used in the planet-ring contact reference simulations in ABAQUS software

Fig. 26

FE equivalent Von Mises stress representation in the planet-ring contact under aligned conditions in ABAQUS software

Fig. 27

FE equivalent Von Mises stress representation in the planet-ring contact under twisting misaligned conditions in ABAQUS software

The ABAQUS FE model adopts a mixed-mesh topology with finer discretization focused on the teeth near the contact zone, with a precision of 24 $\times$ 20 elements in the involute and width directions, and coarser discretization for the remaining teeth resulting in a total of 708,360 DOFs for the planet-ring pair (see Fig. 25). Hexahedral (8-node linear) solid elements are employed with a Surface-To-Surface contact algorithm. In each quasi-static simulation, the rotational angle of the ring gear is fixed, while the planet remains torsionally unconstrained. A fixed angular misalignment is applied by rotating the planet along the x-axis, focusing on twisting-tilting rotations, recognized as the most influential misalignments, particularly along the OLOA [31, 88, 89]. An external torsional torque is applied to the planet as a load boundary condition to achieve static equilibrium with the contact forces. Contact stress maps at the planet gear are depicted in Figs. 26 and 27 for aligned and misaligned conditions, respectively, verifying the accuracy of the FE simulations.

Fig. 28

Angular STE peak-to-peak of planet-ring contact, under angular twisting misalignments (applied to the planet along x-direction)

Fig. 29

Order-based representation of STE peak-to-peak of planet-ring contact, under angular twisting misalignments (applied to the planet along x-direction)

Fig. 30

Contact force distributions over the planet’s face width at multiple angular positions, with $0.1^{\circ }$ of angular twist misalignments, FE simulations

Fig. 31

Contact force distributions over the planet’s face width at multiple angular positions, with $0.1^{\circ }$ of angular twist misalignments, the analytical model simulations

Various incremental angular twisting misalignments are imposed on the planet gear, ranging from 0 to 0.1 degrees along the x-direction, to validate the analytical contact formulation against FE. Results depicted in Fig. 28, affirm that twisting misalignments significantly alter the STE shape, escalating the peak-to-peak value and thereby affecting the STE order spectrum. The analytical model is able to correctly predict the trends in comparison to FE. Analyzing the STE order content in Fig. 29, it is evident that angular twist misalignments lead to an absolute increase in first and second orders. The analytical models, while slightly overestimating the order magnitudes, qualitatively capture the TE order distribution and relative order changes in good agreement with FE.

One of the primary advantages of the proposed analytical DPM over classical LPMs is its capability to accurately capture distributed contact forces across the face width and among different teeth pairs, accounting for load, microgeometry, and misalignment effects. Misaligned contact force distributions, obtained from both the reference FE and analytical simulations at various angular positions over an entire tooth pair meshing period, are presented respectively in Figs. 30 and 31, utilizing a grid of 20 slices. The results demonstrate a notable agreement between FE and analytical models, revealing how severe angular tilt misalignment in the planet gear shifts the distributed contact force towards one tooth’s edge, partly relieving the load at the other end. The predicted trends and maximum slice contact forces are in close agreement.

A crucial metric indicating the severity of non-uniform distributed loads across the gear’s width is the face load factor $(K_{H\beta })$ defined in ISO-6336 standard [25], which holds important implications for gear surface and tooth stresses, and consequently in Remaining Useful Life (RUL) predictions. The computed face load factor for contact stress at each angular configuration reveals significantly higher values in the misaligned case compared to the perfectly aligned scenario (rising from values of 1–1.5 to 2). Figure 32 illustrates this trend, with the analytical model closely aligning with the results obtained from FE-based simulations.

Fig. 32

Face load factor for contact stress $(K_{H\beta })$ at multiple angular positions, under aligned and misaligned conditions, based on FE and analytical model simulations

4.4 Misaligned planetary gear model validation

In the construction of the fully misaligned planetary model, all gears are supported by flexible bearing-shafts, represented through equivalent $5 \times 5$ stiffness matrices, allowing misalignments in all degrees of freedom. The specific three-planet configuration outlined in Table 1, results in a 36 DOF quasi-static system. Results indicate that accounting for bearing stiffness, particularly when exhibiting increased flexibility, significantly influences the absolute value of the STE. More importantly, it affects the STE peak-to-peak and shape, as depicted in Fig. 33, which are indicative of load distribution alterations.

Fig. 33

Angular STE peak-to-peak of the planetary helical gear set, under aligned and misaligned configurations with flexible bearing supports

As a consequence, the amplitude of the primary TE orders is modified, impacting the system’s vibration response, as depicted in Fig. 34.

Fig. 34

Order domain decomposition of the angular STE peak-to-peak of the planetary helical gear set, under aligned and misaligned configurations

Results in Fig. 35 demonstrate that misalignment affects both sun-planet and planet-ring contacts, exhibiting similar sensitivities for this case.

Fig. 35

Angular STE peak-to-peak of the planetary helical gear set, for each individual sun-planet and planet-ring contacts under aligned and misaligned conditions

Variations in contact force distribution due to misalignments are depicted in Figs. 36 and 37 for the planet-ring pair contact. The twisting-tilting motion of the planet results in a more uneven, skewed distribution, causing increased load at the gear’s width edge. To assess gear tooth load distributions, the ISO-6336 face load factor ($K_{H\beta }$) [25] is employed, defined as the ratio of the maximum load intensity (load per unit length of face width) to the mean load:

$$\begin{aligned} K_{H\beta }= \frac{(f_{ni}/b)_{max}}{\sum {f_{ni}/b}} \end{aligned}$$

(79)

In the specific case depicted in Fig. 37, the face load factor is $K_{H\beta }=1.33$ under aligned conditions, while under misaligned conditions the increased load at the face width edges results in a higher face load factor of $K_{H\beta }=1.79$, significantly reducing the gear’s useful life.

Fig. 36

Planet gear 3D contact force distribution along teeth and face width, at the planet-ring contact. Comparison between aligned and misaligned conditions

Fig. 37

Planet gear 2D contact force distribution, at the planet-ring contact. Comparison between aligned and misaligned conditions

4.5 Computational performance metrics

The misaligned planetary model presented in this work was utilized in a sensitivity analysis investigation to determine the main contributing factors to variation in transmission error due to manufacturing tolerances on gear flanks, following a detailed DOE. For that purpose, a total of 15,140 quasi-static simulations were launched, each with 20 evaluations per simulation, executed in parallel on 15 CPU cores. All gears and planet carrier were supported by flexible bearings, microgeometry modifications where applied to gear flanks and external misalignment forces were applied on the planet carrier. Despite the substantial workload, the simulations were completed in 15 h, averaging 3.6 s per simulation, which run in a mobile workstation powered by an Intel Core i9-12950HX and 128 GB of DDR5 RAM running at 3600 MHz, and a limited CPU power of 85 W for thermal considerations. A qualitative comparison in terms of CPU cost between the four main gear modelling techniques (LPM, DPM, Hybrid, FE) is given in Table 3 based on the values reported in prior published works. The CPU times are unified to a single loaded gear contact calculation. However, establishing a equivalent CPU time comparison is an intricate task, since those highly depend on the machine performance, parallelization schemes, definition of the mesh in FEM, etc; thus the table aims to be a qualitative assessment. The proposed DPM achieves great CPU savings in respect to other broadly used modelling techniques: being 2–3 orders of magnitude faster than the hybrid (semi-analytical) methods [90] and 4–5 orders of magnitude faster than the FEM [13, 14]. Moreover, the proposed DPM positions well in respect to state of the art analytical-LPM [91] and analytical-DPM [92, 93] published methods, while offering enhanced 3D contact characterization features.

Table 3 Comparison of CPU time in seconds for different methods reported in literature

Thus, the proposed planetary misaligned model, aided with the analytical Jacobian formulation presented here, provides a powerful solution achieving minimal computational times, specially in comparison with other modelling techniques or similar available software.

5 Conclusions

This study presents an analytical formulation for modeling misaligned planetary gears, emphasizing the three-dimensional characterization of distributed gear contact and addressing key sources of contact force nonlinearity.

Leveraging the authors’ external gear compliance approach, this study meticulously evaluates existing analytical formulas, discusses body rim stiffness models for internal gears ans critically assesses the feasibility of a linear local contact model. The work culminates in a rigorously validated prescribed linear formulation for the mesh stiffness of helical internal gears. The proposed model demonstrates excellent correspondence with FE simulations, highlighting the importance of considering tooth coupling effects in spur gears’ peak-to-peak TE, and slice coupling effects in helical gears’ TE order response. Coupling stiffnesses are found to be also crucial in spur and helical misaligned gear contacts, impacting TE response and, consequently, NVH metrics.

Furthermore, a novel misaligned contact detection method is introduced for planetary gears, employing an analytically efficient technique. This method relies on misaligned slice displacements to preserve distributed gear contact load information. The formulation encompasses all gear bodies DOFs supported by flexible bearings, thereby considering translational, axial, twisting, tilting, and rotational misalignments, along with mounting errors and microgeometry modifications to gear profiles.

The numerical validation of the proposed model involves nonlinear FE static simulations for both spur and helical gears, encompassing aligned and misaligned conditions. The results demonstrate excellent correspondence in terms of static TE, revealing that misalignments notably influence TE meshing orders and contact force distribution, in particular twisting rotations along the OLOA. This leads to undesirable concentrations of contact forces at the gear’s face width edge, quantified from a remaining useful life perspective. Resulting in an 50% increase in the severity of concentrated load, thus reducing the remaining useful life.

Significant computational time savings are achieved with the proposed method, particularly enhanced by the analytical Jacobian and linear stiffness formulation, when compared to other hybrid and FE techniques. The practical utility of the model is highlighted in applications such as Design of Experiments for optimization and robust design studies, as well as in digital twin frameworks, suggesting promising avenues for future research.