A novel tooth surface modification method for spiral bevel gears with higher-order transmission error

doi:10.1016/j.mechmachtheory.2018.04.001

Mechanism and Machine Theory

Volume 126, August 2018, Pages 49-60

https://doi.org/10.1016/j.mechmachtheory.2018.04.001 Get rights and content

Highlights

•
The higher-order transmission error for high contact ratio spiral bevel gears.
•
The function-oriented design based on ease-off.
•
The auxiliary tooth surface correction motion with vertical offset and increment of machine centre to back modification.

Abstract

To reduce the running noise and vibration of spiral bevel gears, we present a novel method for designing high-contact-ratio spiral bevel gears with the higher-order transmission error (HTE) based on the function-oriented design. According to the design HTE curve and contact path, the pinion target tooth surface can be acquired by correcting the conjugated tooth surface derived from the mating gear. We establish a mathematical model for the computerized numerically controlled cradle-style pinion generator with the following design parameters: tool parameters, initial machine settings, and polynomial coefficients of the auxiliary tooth surface correction motion. An optimal model was developed to solve for the polynomial coefficients. TCA and LTCA results show that the HTE spiral bevel gears designed with the auxiliary tooth surface correction motion meets the design requirements. The meshing quality of the HTE spiral bevel gears is also better than that of gears designed using the parabolic transmission error (PTE). The tooth surface correction method proposed in this paper can serve as a basis for modifying high-contact-ratio (HCR) spiral bevel gears.

Introduction

As a key component in the aviation industry, spiral bevel gears need to meet many requirements in terms of noise, strength and reliability. However, because the contact path of traditional low-contact-ratio gears is nearly perpendicular to the root cone, they produce significant amounts of running vibration when working under high-speed, heavy-load conditions [1], [2]. Determining the best way to use the relationship between the transverse-contact-ratio and the face-contact-ratio of spur gears to increase the contact-ratio of spiral bevel gears and then to improve their strength and dynamic performance has been a great concern to scholars worldwide [3]. Engineers have expended much effort to solve this problem. Fang and Deng proposed the use of high-contact-ratio (HCR) spiral bevel gears, which were developed by adjusting the direction of the contact pattern and extending the contact path [4], [5].

Transmission error (TE) has a significant impact on gear vibration and noise. To reduce this vibration and noise of spiral bevel gears, the TE is usually designed as a parabola. Artoni proposed an optimization methodology to minimize the loaded transmission error of hypoid gears based on ease-off [6]. Simon analyzed the influences of machine settings on tooth contact pressure and loaded transmission error, and then presented a method to improve gear meshing performance by reducing the loaded transmission error and the maximum tooth contact pressure of hypoid gears [7], [8], [9], [10], [11]. However, as shown in Fig. 1(a), there are three meshing areas in one engagement period involving low-contact-ratio spiral bevel gears, and the PTE can effectively reduce the amplitude of the loaded transmission error (ALTE). There are five meshing zones in one engagement period involving HCR spiral bevel gears, as shown in Fig. 1(b). The loaded transmission error (LTE) curve is a higher-order curve, and the PTE cannot meet the demands of reducing the ALTE. To reduce vibration, Stadtfeld [2] and Su [12] proposed the seventh-order TE for spiral bevel gears. Notably, the ALTE of spiral bevel gears with seventh-order TE is much larger than that of the PTE when the actual contact-ratio is greater than 2, as shown in Fig. 1(c). Therefore, seventh-order TE cannot reduce the ALTE of HCR spiral bevel gears, and it is necessary to design a novel transmission error curve.

Calculating the machine settings for gears is based on the local synthesis method [13], which was applied to spiral bevel gears by Litvin [14], [15], [16]. Tooth surface correction is important for reducing the noise and vibration of spiral bevel gears. In recent years, a new concept for spiral bevel gears was proposed with the application of a CNC gear generator developed by Gleason Works. Fong built a mathematical model for hypoid generator with higher-order tooth surface correction motion [17]. Stadtfeld proposed a higher-order tooth surface correction method with ease-off [18]. Shih proposed an ease-off flank correction method for spiral bevel gears based the auxiliary tooth surface correction motion [19], [20]. Fan proposed a higher-order tooth surface error correction for face-milled spiral bevel and hypoid gears to reduce noise [21], [22]. With the widespread application of CNC technology in spiral bevel gear machine tools, the generative theory of spiral bevel gears is no longer limited by cradle machines. It is possible to perform nonlinear corrective motions during gear processing. Fong proposed the MRM method for spiral bevel gears without causing an increase to the Hertzian contact stress [23], [24]. Mu and Fang proposed the modified curvature motion method for spiral bevel gears with seventh-order TE [25].

In this paper, we proposed the tooth surface correction method for HCR spiral bevel gears with the novel HTE to improve the meshing quality. We also present comparisons of the HTE and PTE based on a combination of the TCA and LTCA [26], [27]. Furthermore, the comparisons show that the HTE can effectively reduce the ALTE of HCR spiral bevel gears. The HTE also presents advantages in terms of the load distribution factor, bending stress, tensile stress and contact stresses.

Section snippets

Function-oriented design

The goal of function-oriented design is to generate the pinion target surface meeting the desired function requirements. We obtained the target surface of HCR spiral bevel gears with seventh-order TE based on function-oriented design in our previous work [25]. The principal is that the pinion target tooth surface is acquired by correcting the conjugated tooth surface derived from the mating gear with the design HTE and contact path.

To obtain a gear set meeting the design requirements, we

Mathematical model

The locus of the tool surface is given by the following equations: $r_{p} (s_{p}, θ_{p}) = [\begin{matrix} (R_{p} + s_{p} \sin α_{1}) \cos θ_{p} \\ (R_{p} + s_{p} \sin α_{1}) \sin θ_{p} \\ - s_{p} \cos α_{1} \\ 1 \end{matrix}]$ $n_{p} (θ_{p}) = [\begin{matrix} \cos α_{1} \cos θ_{p} \\ \cos α_{1} \sin θ_{p} \\ - \sin α_{1} \end{matrix}]$ where s_p, θ_p are tool surface parameters, R_p is the cutter radius, and α₁ is the profile angle.

Fig. 3 presents the coordinate systems of gear generator, where S_p and S₁ are connected with the cutter-head and pinion, respectively. Three sets of parameters describe the gear generator: the tool parameters, the polynomial coefficients of the

Tooth contact analysis (TCA)

The meshing quality of gears can be analysed using the TCA method. Fig. 4 shows that there are four misalignment factors: the distance change between axles (ΔE), gear axial misalignment (ΔG), pinion axial misalignment (ΔP), and shaft angle change (ΔΣ).

The position and unit normal vectors for pinion and gear represented in the meshing coordinate system are as follows: ${\begin{matrix} r_{h}^{(1)} = M_{h 1} (φ_{1}, Δ P) r_{1}^{(1)} (θ_{p}, ϕ_{p}) \\ n_{h}^{(1)} = L_{h 1} (φ_{1}, Δ P) n_{1}^{(1)} (θ_{p}, ϕ_{p}) \end{matrix}$ ${\begin{matrix} r_{h}^{(2)} = M_{h 2} (φ_{2}, Δ E, Δ Σ, Δ G) r_{2}^{(2)} (θ_{g}, ϕ_{g}) \\ n_{h}^{(2)} = L_{h 2} (φ_{2}, Δ E, Δ Σ, Δ G) n_{2}^{(2)} (θ_{g}, ϕ_{g}) \end{matrix}$

Numerical example

A numerical example is used to analyse the meshing characteristic of spiral bevel gears with the HTE. The geometric parameters are shown in Table 1. The machine settings for gear and pinion calculated with the local synthesis method are given in Table 2. The parameters for the HTE curve are given in Table 3.

Conclusions

Computer simulation results are provided to compare the meshing performance of HCR spiral bevel gears with the HTE and PTE, and we arrived at the following conclusions:

Based on the function-oriented design, spiral bevel gear with the HTE can be implemented with the auxiliary tooth surface correction motion. Based on the TCA, we noted that the spiral bevel gear designed with the auxiliary tooth surface correction motion satisfies the design requirements.

The tooth surface correction method can be

Acknowledgements

We thank the support from the National Natural Science Foundation of China (No. 51375384). We also thank all reviewers and editors for their valuable comments and suggestions.

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In the wide application of multi-axis CNC spiral bevel gear machining machine tools, complex tooth surface correction and accurate manufacturing can be achieved [26]. Mu and Fang raised a tooth surface modification method combining ease-off with tooth surface high-order transmission error, and established a tooth surface high-order transmission error design method through tooth surface curvature correction to realize accurate tooth surface design [27–30]. Zhang, et al. established the relationship between modification of conjugate differential surface of spiral bevel gear and relative curvature, conjugate contact line and contact path, and designed ease-off surface [31].
In order to realize the global integrated control of spiral bevel gear tooth surface, this paper presents a direct preset method for solving ease-off tooth surface of spiral bevel gear. This method integrates the preset contact path (CP), geometric transmission error (TE) and curvature of difference curve at the contact point (CDC), which is called CPTE-CDC. Based on the relationship between ease-off and tooth surface planning grid, CPTE-CDC is used to establish a nonlinear overdetermined equations to solve ease-off surface equation. According to the ease-off surface obtained from the solution, the ease-off target tooth surface of pinion related to cuttting parameters is established, and its optimal machining parameters are solved based on Levenberg–Marquardt (LM) algorithm. Finally, a numerical example is created using TCA and LTCA techniques. The results show that the tooth surface obtained by CPTE-CDC method not only meets the preset contact characteristics, but also meets the design requirements, which proves the correctness of the proposed method. CPTE-CDC method can be used as a theoretical basis for improving the global control of spiral bevel gear tooth surface.
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To reduce the running vibration of high-contact-ratio (HCR) spiral bevel gear transmissions, an innovative higher-order tooth surface modification (HTSM) method is proposed to reduce the vibration excitation of gear transmission. First, according to the meshing characteristic of HCR spiral bevel gear transmissions, an innovative modification curve is proposed. Second, the auxiliary tooth surface modification (ATSM) method is used to generate the higher-order modified spiral bevel gear. Third, the loaded transmission error (LTE) and meshing impact (MI) of higher-order modified spiral bevel gear transmissions are calculated based on the LTCA method and the impact theory while considering tooth deformation. Last but not least, based on the ease-off technology, an optimization model is built to reduce the LTE and MI of HCR spiral bevel gear transmissions. Through simulation analysis, it is found that compared with the second-order modified spiral bevel gear, the high-order tooth surface modification method can not only effectively reduce the LTE, MI and dynamic load factor (DLF) of HCR spiral bevel gear, but also improve its dynamic performance within full speed range. This study provides an effective way for improving the dynamic performance of HCR spiral bevel gear transmissions within full speed range.
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Therefore, the main goal of this study is to systematically define optimal tool geometry and machine tool settings to simultaneously minimize tooth contact pressure, angular displacement error of the driven gear and average temperature in the gear mesh, and to maximize the efficiency of the gear pair. During the last decades many research works have been directed towards the design and manufacture of spiral bevel and hypoid gears with optimal tooth surface modifications to reduce the maximum tooth contact pressure and transmission error [1–43]. The most relevant papers to the present work are as follows: Ding et al. [34] propose a novel multi-objective correction of machine settings correlating to the loaded tooth contact performance using nonlinear interval optimization algorithm for spiral bevel gears.
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A novel tooth surface modification method for spiral bevel gears with higher-order transmission error

Highlights

Abstract

Introduction

Section snippets

Function-oriented design

Mathematical model

Tooth contact analysis (TCA)

Numerical example

Conclusions

Acknowledgements

Mech. Mach. Theory

Chin. J. Aeronaut.

Mech. Mach. Theory

Chin. J. Aeronaut.

Mech. Mach. Theory

Mech. Mach. Theory

Math. Comput. Modell.

Mech. Mach. Theory

Mech. Mach. Theory

The ultimate motion graph

ASME J. Mech. Des.

Effect of involute contact ratio on spur gear dynamics

ASME J. Mech. Des.

Analysis of meshing behavior and experiments of spiral bevel gears with high contact ratio

J. Aerosp. Power

Strength analysis of spiral bevel gears with high contact ratio

J. Aerosp. Power

An ease-off based optimization of the loaded transmission error of hypoid gears

ASME J. Mech. Des.