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Abstract

By considering the uncertainness of initial measuring position of encoders and signal sidebands caused by the fault gear pair, this paper presented a new comprehensive harmonic analysis method for the transmission error of gear hobbing machine. Based on that, a test platform was established, in which two circle grating encoders were connected to the hob spindle and workpiece spindle respectively. With the help of this new harmonic analysis method as well as the self-developed test platform, a new improved transmission error fault diagnosis method was developed for the gear hobbing machines. To verify its accountability, a case study was conducted on a YS-type gear hobbing machine. According to the spectrum amplitude comparison and the analysis of harmonic frequency distribution, the fault transmission gear pair was successfully located. This improved transmission error source tracing method was very helpful for quantifying both the manufacturing qualities and assembly qualities of parts and locating potential error source for new gear hobbing machines.

Keywords

Gear manufacturing transmission error wet gear hobbing machine error source tracing modulation phenomenon

Introduction

Gear hobbing machines are widely-used specialized machine tools for various gear productions in mechanical manufacturing industries, such as the automobile, ship, and aerospace industries, as shown in Figure 1. To improve the hobbing accuracy, the thermal-induced error,¹ force-induced error,² transmission error³ regarding the gear hobbing machines have become research hotspots. Among them, the transmission error not only leads to the gear hobbing error, but the noise and vibration issue as well. Therefore, the transmission precision is an important indicator to judge the comprehensive performance of a gear hobbing machine. Researches on the transmission error attracts more and more attention worldwide.

Figure 1.

Cutting zone of wet gear hobbing machine.

Generally speaking, there are three major methodologies to evaluate the transmission error of the transmission system, namely, the CAE analysis, the theoretical modeling, and the direction measurement by means of rotary encoder(s). The first method is mainly to develop finite element model of targeted transmission system and then analyze the effects of gear wear, assembly error, or manufacturing error on the transmission behavior thereby evaluating transmission error. Lin and He⁴ adopted a finite element method to simulate the gear transmission train with the consideration of machining errors, assembly errors, modifications, and the static transmission error was obtained. Wang et al.⁵ evaluated the transmission error of a spur gear pair adopting a finite element model, and the results showed the variation pattern of transmission error for a healthy gear pair was generally consistent with the form of time-varying meshing stiffness. Peng et al.⁶ developed a finite element model for the planetary gearbox to investigate the transmission error due to planet cracks and spalls.

Instead of using infinite element method, some researchers tended to build mathematical model and directly calculate the transmission error. Yang et al.⁷ proposed a one-degree-of-freedom model for the spur gear pair system, in which the modeling of the time varying mesh stiffness and the static transmission error were included. Bozca⁸ established a four-degree-of-freedom model of the transmission error of a five-speed gearbox. Litvin et al.⁹ presented a tooth contact analysis method, and accordingly investigated the effects of meshing misalignment on the transmission error of spiral bevel gears. Velex et al¹⁰ developed a transmission error theoretical model by comprehensively considering the influence of shafts, bearings, couplings. Iwasaki et al.¹¹ presented a novel modeling approach for the angular transmission error in harmonic drive gearings, in which the transmission error due to nonlinear elastic deformations and structural errors of gears were both considered. Bruyère et al.¹² established analytical formulations for profile modifications of narrow-faced spur helical gears in order to calculate and reduce static transmission error. To better explain the side band phenomenon appears around the mesh order component. Morikawa et al.¹³ presented a new transmission error modeling method with the periodical variation of the eccentricity and inclination direction of each component considered.

The last methodology is to measure transmission error using encoders installed at both the input and output shafts, the angular measurement results were combined to directly obtain the value of the transmission error. Noted that this direct measuring method not only can directly quantify the transmission error but can be used as the signal basis of tracing the transmission error source as well. Therefore, it attracts more and more attention. Randall et al.¹⁴ presented how to use encoder(s) to measure the transmission error thereby locating failed gears with the crack issue or the eccentricity, and demonstrated the advantage of this direct measuring method over indirect vibration measuring method. Xia et al.³ proposed a novel classified compensation approach to eliminate the transmission error of gear hobbing machines, in which the measured transmission error was firstly divided into three error components according to the error source analysis based on measured encoder signal, and then respective compensation processes were implemented. Following experimental results verified the effectiveness of this new approach. Palermo et al.¹⁵ demonstrated how to use low-cost digital encoders together with elapsed time method to precisely measure the transmission error of a vehicle gearbox. during the transmission error measurement at high speed, to avoid the distortion of the effects of geometric tooth-profile and meshing stiffness due to the resonances. Lu et al.¹⁶ adopted a cepstrum-based operational modal analysis on the experimentally-obtained encoder signal. Following experiment on a single-stage spur gear pair proved that this encoder signal post treatment method was very useful in the gear condition monitoring. In order to measure the transmission error of a spur gear pair, Park et al.¹⁷ used the ensemble empirical mode decomposition (EEMD) method to analyze the signals collected from the encoders connected to the input and output shafts respectively. Peng et al.^18,19 developed a transmission error analysis approach, by which once the transmission error curve was experimentally attained, the error sources can be located with a series of harmonic analyzes. Mark et al.²⁰ related the tooth-surface damage of transmission gears with the change of transmission error rotational-harmonic spectrum. Based on measured encoder signal, Chin et al.²¹ proposed a new approach to enhance its expression ability to the change of gear wear stage by conducting the rephasing treatment. Zhao et al.²² used built-in encoders to achieve the transmission error curve as the data basis of locating major error source and accordingly reduce the transmission error.

Many scholars had proved that it is possible to locate the transmission error source by correctly conducting harmonic analysis of experimentally-obtained encoder signal. However, to some extent, in its industrial applications, it was still difficult to locate the main one among several sources due to unexpected surges in the amplitude in the low harmonic frequency band during the harmonic analysis. Previous theoretical analysis methods failed to explain the unexpected phenomenon and hence there is a need for a theory to clearly explain it. To fill this gap, in this paper, we aimed to proposed a new transmission error source tracing method including a self-developed in situ test platform and a new comprehensive harmonic analysis method with the effects of both the sideband phenomenon modulated by the rotations of the rotating gears and the uncertainness of initial measuring position of encoders considered simultaneously. With the help of that, before the final assembly of any new gear hobbing machines adopting gear transmission structure, the transmission precisions of newly assembled hob assembly and worktable unit can be quantified and potential source causing the transmission error can be located in advance.

Modeling of gear train transmission error

Without considering the effect of transmission error, the meshing between gears was ideally conjugated. Hence, with a constant transmission ratio i, the relationship between the output gear rotation angle φ₀ and the input gear rotation angle φ_i can be defined by

φ_{0} = φ_{i} / i

(1)

In this way, the gear transmission error was defined as the differential between the theoretical rotating angle and practical one, which can be calculated by

TE = φ_{0} - φ_{0}^{'} = \frac{φ_{i}}{i} - φ_{0}^{'} = \frac{Z_{k}}{Z_{j}} φ_{i} - φ_{0}^{'}

(2)

where, $φ_{0}^{'}$ was the actual rotating angle of the driven gear; Z_j and Z_k were the teeth numbers of the driving gear and driven gear respectively; i was the transmission ratio.

However, due to the existence of the manufacturing error and assembly error, the meshing motion between gears barely can reach this ideal condition. Besides, the tear and wear of parts would cause the rotational angle error between the input and output of gear transmission train as well. Therefore, the equation (2) was not accurate enough to quantify the transmission error in the real working condition. More comprehensive one is needed.

Typically, the gear manufacturing error comprises of two components, that is, the tangential comprehensive error reflecting the inherent position error of gear in the large period, and the inter-teeth tangential comprehensive error reflecting the inherent position error of gear in the small period, as shown in equation (3),²³

E_{1} = \frac{1}{2} (F'_{i} - f'_{i}) \sin θ + \frac{1}{2} f'_{i} \sin n θ

(3)

where, $F'_{i}$ represented the tangential comprehensive error whereas $f'_{i}$ denoted the effects of both the tooth form error and tooth direction error. E_1, $θ$ , n were the gear manufacturing error, the phase angle of gear and the teeth number respectively.

On the other hand, the gear assembly error can be categorized into three components, as shown in equation (4),²³

E_{2} = \sum_{i = 1}^{3} e_{i} \sin θ_{i}

(4)

where, e₁,e₂,e₃ were the clearance between the hole and the shaft, the runout clearance of gear journal, and the radial clearance of bearing respectively. E₂ was the gear assembly error whereas $θ_{i}$ was corresponding phase angle.

Theoretical analysis of the transmission error of gear hobbing machine

Based on the gear train transmission error theory introduced above, a conclusion can be made that the transmission error of any gear hobbing machine adopting the gear-driving structure was mainly derived from the manufacturing error or the mounting error of transmission gear(s). A wet gear hobbing machine usually had two gear transmission systems, that is, the hob assembly transmission train and the worktable transmission train. Following assumptions were used in present transmission error modeling for gear hobbing machines:

The transmission error of each transmission part was a kind of periodic error;

The error period of any transmission part was equal to its revolution period;

The first-order component of the transmission error of any transmission part was the essential part and others can be ignored;

The principal component of any transmission error can be formulated as the sine or cosine function of the rotation angle.

Hence, the transmission error of any transmission part j (j = 1, 2, 3…) in a gear hobbing machine can be defined as

Δ ϕ_{j} = A_{j} \sin (ω_{j} t + φ_{j})

(5)

where, A_j was the amplitude of the transmission error of the transmission part j; w_j was the angular speed of the transmission part j; φ_j denoted the phase position of the transmission error of the transmission part j; t represented the revolution time.

Since the gear meshing was a type driving mode of the transmission train in a gear hobbing machine, the Kalashnikov’s error transfer theory would be suitable for describing the propagation process of transmission error. According to Peng et al.’s^18,19 studies, the transmission error of gear pair can be calculated by:

B_{j} = \frac{z_{j}}{z_{k}} Δ ϕ_{j} = \frac{n_{k}}{n_{j}} Δ ϕ_{j}

(6)

where, B_j was the transmission error between gear j and gear k, n_k, and n_j were the rotation speeds of gear k and gear j respectively. Note that n_j/n_k represented not only the transmission ratio but the order of harmonic component (harmonic order in short).

Accordingly, the total transmission error of the gear hobbing machine was defined as the vector sum of each transmission part,

E = \sum_{j = 1}^{k} \frac{n_{k}}{n_{j}} A_{j} \sin (ω_{j} t + φ_{j})

(7)

By substituting ω_j= 2πn_j and θ_k= 2πn_kt into equation (7), we can get

E (θ_{k}) = \sum_{j = 1}^{k} \frac{n_{k}}{n_{j}} A_{j} \sin (\frac{n_{j}}{n_{k}} θ_{k} + φ_{j})

(8)

where, θ_k was the rotating angle of gear k.

Note that the initial position in the measuring of the transmission error of a gear hobbing machine was of randomness, which would affect the final measurement results. In addition, once the gear meshing motion went wrong, a modulation phenomenon would occur, causing modulation sidebands being generated around the gear meshing frequency. These sidebands carried many effective fault information, that is, the distributions and amplitudes of these sidebands change with different failure modes. Hence, to improve the prediction accuracy, the effects of the randomness of initial measuring position of encoder and the modulation phenomenon should be considered in the modeling.

Accordingly, considering the randomness of the initial measuring position as well as the existence of modulation phenomenon, a new comprehensive equation for the transmission error of gear hobbing machines was proposed as below,

\begin{matrix} E (θ_{k}) = C + \sum_{j = 1}^{k} \frac{n_{k}}{n_{j}} A_{j} \sin (\frac{n_{j}}{n_{k}} θ_{k} + φ_{j}) \\ + \sum_{j} \sum_{m} \frac{n_{k}}{n_{j}} B_{j} (m) ((Z_{j} \frac{n_{j}}{n_{k}} \pm m \frac{n_{j}}{n_{k}}) θ_{k} + φ_{j} (m)) \\ m = 0, 1, 2 \dots \end{matrix}

(9)

where, C was the mean value of the $E (θ_{K})$ in one time period; $A_{j}$ was the amplitude of the transmission error component caused by the tooth pitch accumulated error and the mounting error of the gear j; $φ_{j}$ is the phase angle of the transmission error component caused by the tooth pitch accumulated error and the mounting error of the gear j; $B_{j} (m)$ was the amplitude of the transmission error component caused by the tooth surface error of the gear j; $φ_{j} (m)$ was the phase angle of the transmission error component caused by the tooth surface error of the gear j; m was the order number of harmonic frequency of rotating frequency of gear j.

To sum it up, the transmission error defined in equation (9) was consisted of three error components:

C: the average measured value of $E (θ_{K})$ in a single period (one revolution of worktable or hob), which depended on the initial position of encoder in the transmission error measurement;

$n_{k} A_{j} \sin (n_{j} θ_{k} / n_{k} + φ_{j}) / n_{j}$ : the error components due to the tooth pitch accumulated error and the mounting error of the gear j, and the amplitude $n_{k} A_{j} / n_{j}$ decreased with the increase of the rotational speed ratio $n_{j} / n_{k}$ ;

$n_{k} B_{j} (m) ((Z_{j} n_{j} / n_{k} \pm m n_{j} / n_{k}) θ_{k} + φ_{j} (m)) / n_{j}$ : the error component due to the geometric error of gear j, in which $Z_{j} n_{j} / n_{k} \pm m n_{j} / n_{k}$ was its harmonic frequency and the amplitude $n_{k} B_{j} (m) / n_{j}$ increased with increasing rotational speed ratio.

With the equation (9), once the transmission error curve can be experimentally achieved, the value of error induced by different components can be quantified with in-depth harmonic analyses and both expected and unexpected amplitude surges in the harmonic analyses of the gear train can be reasonably explained as well. Accordingly, the major error source of the transmission train of targeted gear hobbing machine can be located precisely.

Development of in situ test platform and transmission error source tracing procedure

To precisely obtain the transmission error curve, an in situ transmission error measuring platform were built, whose design principle was shown in Figure 2. The hardware structure of this self-developed platform was shown in Figure 3, which was consisted of circle grating encoders, corresponding couplings, signal subdivision device, NI data acquisition system, and PC. Noted that due to the different rotation speeds of the hob and workpiece in the gear hobbing process, two different types of circle grating encoders were used here to record the actual rotational angles of the hob and worktable respectively, that is, RON287-type for the hob and RPN886-type for the worktable. They were connected to the hob spindle and the worktable spindle through two couplings respectively. The specifications of these two circle grating encoders were listed in Table 1. A human-machine-interface, built by using commercial software Labview, was used to execute the working parameter setting, data acquisition, data display, data storage, and data analysis, as shown in Figure 4.

Figure 2.

The design principle of transmission error in situ measuring system for gear hobbing machine.

Figure 3.

Hardware structure of in situ transmission error measuring system of gear hobbing machine.

Table 1.

The specifications of the encoders RON287 and RPN886.

Specification	Measuring method	Output signal	The number of pulses per revolution	Measurement accuracy	Periodic position error
RPN886	Interference	~1 Vpp	18,000	±1″	≤±0.1″
RON287	Imaging	~1 Vpp	180,000	±2.5″	≤±0.7″

Figure 4.

(a) Parameter setting interface of transmission error in situ measuring system for gear hobbing machine. (b) Signal acquisition interface of transmission error in situ measuring system for gear hobbing machine.

Figure 5 depicted the working procedure of this test platform, which can be detail as below: in the signal acquisition process, both the hob spindle and the worktable spindle rotated with a given transmission ratio, the original sine wave signals generated from circle grating encoders was sub-divided and further processed into the rectangular pulses. The output pulse signals were sent back to the NI acquisition system via the junction box. Finally, the output signals were uploaded into the PC for display and storage through the serial communication protocol. After the upper monitor received output pulse numbers for the hob spindle and the worktable spindle respectively, real rotational angles of the hob and worktable spindles were obtained. Subsequently, the angle difference between the hob spindle and the worktable spindle was calculated by

TE = \frac{N_{1} \times 2 π / λ_{1}}{i} - N_{2} \times 2 π / λ_{2}

(10)

Figure 5.

Diagram of transmission error measuring principle.

where, N₁, N₂ were the pulse numbers of the high-speed end and low-speed end respectively; λ₁, λ₂ were the circle grating encoder subdivision numbers for the hob spindle and the worktable spindle respectively; i was the gear ratio of the hob to the workpiece.

Based on the newly-established theoretical model and the self-developed test platform, a new transmission error source tracing method were proposed for gear hobbing machines, by which once the spectrogram of transmission error curve was experimentally obtained, the error source can be precisely located by analyzing the amplitude of harmonic frequency of each gear and corresponding sidebands. This error source tracing procedure were detailed as below:

Placing the hob assembly and worktable unit on the self-developed test platform respectively;

Conducting inspection experiments on these two components;

Experimentally obtaining the transmission error curve;

Calculating the rotating frequency of each transmission gears;

According to transmission relationship, selecting a gear as the reference and treating the rotating frequency of this gear as the fundamental frequency;

Calculating respective harmonic order of each gear;

Executing low-pass filtering on the transmission error curve based on the harmonic order of gear j (j = 1, 2, 3, …n) being treated as the cutoff frequency and accordingly obtaining harmonic amplitudes.

Calculating sidebands of all gear j;

Repeating (5) and (6) and acquiring respective harmonic amplitudes and sideband distributions of all involved gears;

Preliminarily locating potential fault gear(s) by conducting amplitude comparison;

Running time-frequency transform on the experimentally-obtained transmission error curve;

By checking whether the surge in the spectrum amplitude of original transmission error curve occurred at calculated harmonic order and whether sidebands occurred nearby (according to equation (9)) to finally locate the error source;

Note that if the surge in harmonic amplitude occurs at the harmonic order of 1, it does not necessarily mean the reference gear is a major error source. This amplitude surge may be due to the randomness of initial measuring position of encoders and may no long exist in another experiment. In that case, extra experiments should be done to eliminate the random error.

Case study

To verify the accountabilities of newly-proposed transmission error source tracing method and clearly demonstrate how to use it, the transmission error and error source of a second-hand YS-type wet gear hobbing machine were investigated as a case study. The spindle assembly and worktable assembly were installed on the test platform respectively, and the working parameters of the YS-type gear hobbing machine used in the case study was detailed in Table 2.

Table 2.

Major experimental parameters used in the case study.

Parameter	Value
Rotating speed of hob	1150 r/min
Rotating speed of worktable	15 r/min

The transmission error curve of the YS-type gear hobbing machine was shown in Figure 6. By implementing FIR filtering (cutoff frequency: 20,000 Hz) and fast Fourier transform, the spectrum of transmission error can be obtained, as shown in Figure 7.

Figure 6.

Experimentally obtained transmission error curve.

Figure 7.

Low-pass filtered transmission error curve.

Figure 8 was the diagram of transmission trains of YS-type gear hobbing machine, in which B, C, D, E, G, H, I, J were gears inside the transmission system. By analyzing the transmission relations of gears involved (i.e. the tooth number ratio and the rotating speed ratio), a corresponding harmonic analysis was conducted and the results were showed in the Figure 9. Noted that in the harmonic analysis, the rotating frequency of gear B was regarded as the fundamental frequency and the harmonic orders of respective gears were defined as the ratios of respective harmonic frequency to the fundamental frequency. Accordingly, the harmonic order of respective gears and corresponding sidebands can be calculated, as listed in Table 3.

Figure 8.

Diagram of gear transmission system of YS-type gear hobbing machine.

Figure 9.

Spectrum of transmission error curve.

Table 3.

Harmonic frequencies of transmission gears in the YS-type gear hobbing machine.

Name of gear	Tooth number: N_i	Harmonic order of each gear based on the rotating frequency of gear B (n_i)	Gear meshing frequency (Hz)	Number of (m)	Sidebands (Hz)
					N_in_i+ mn_i	N_in_i − mn_i
B	156	1	156	1	157	155
				2	158	154
C	26	6	156	1	162	150
				2	168	144
D	85	6	510	1	516	504
				2	522	498
E	34	15	510	1	525	495
				2	540	480
G	88	46	4048	1	4094	4002
				2	4140	3956
H	22	184	4048	1	4232	3864
				2	4416	3680
I	75	184	13800	1	13,984	13,616
				2	14,168	13,432
J	60	230	13800	1	14,030	13,570
				2	14,260	13,340

Based on the different harmonic orders of involved gears listed in Table 3, different low-pass filterings were implemented on the transmission error curve (i.e. 1, 6, 6, 15, 46, 184, 184, 230 Hz respectively) and hence corresponding amplitudes of processed curves were attained. The harmonic amplitudes related to transmission gears B, C, D, E were 7.245, 10.56, 10.56, and 3.51 arc-seconds, as shown in Figure 10. Obviously, the amplitude for gear C–D (same low-pass filtering frequency) was the largest one. Hence, the gear C or D was preliminarily judged as the fault one(s) that caused the transmission error. Noted that as suggested by Figure 10, the harmonic amplitude of gear B at harmonic order of 1 was also relatively large, but that did not necessarily mean gear B was also a major error source. It was because that the rotating frequency of gear B was treated as the fundamental frequency and this surge in the amplitude was caused by the uncertainness of initial measuring position of encoders. Then, by analyzing the sideband positions, it was observed that the largest amplitude of the transmission error spectrum occurred at the harmonic order of 156 and sidebands occurred at the harmonic frequencies of 150, 162, and 168. Those harmonic frequencies were consistent with the theoretically calculated ones based on the harmonic analysis results of gear C, as listed in Table 3. Therefore, a conclusion was made that the gear C was dominant transmission error source.

Figure 10.

About 0–270 order of harmonic frequency of involved gears.

To check whether our conclusion was accountable, the worktable assembly of the YS-type gear hobbing machine was disassembled and subsequently all transmission gears were removed out and measured, as shown in Figure 11. The measurement result showed there were severe wear marks on the tooth surface of gear C, which proved our judgments. Note that if previous harmonic analysis methods (such as Peng et al.^18,19) were used here, the gear B would be regarded as the main error source as well, which was not right since the gear profile measurement results showed that the gear C maintained at a light wear condition.

Figure 11.

Tooth profile measurement on the gear C.

Conclusion

To precisely quantify the transmission error and locate the source of assembly error, this study proposed a new improved transmission error diagnosis method for gear hobbing machines. It comprised a comprehensive theoretical analysis method of transmission error with the consideration of the uncertainness of initial measuring position of encoder and signal sidebands caused by the fault gear pair, and a coupled in situ test platform with two circle grating encoders being connected to free ends of the hob spindle and workpiece spindle respectively. To verify the accountability of this new transmission error diagnosis method, a case study was conducted on a YS-type gear hobbing machine. According to the experiment and analysis results, the fault transmission gear pair was successfully located.

With the help of this improved transmission error source tracing strategy, before the final assembly of any new gear hobbing machines adopting gear transmission structures, the transmission precisions of both newly-assembled hob, and worktable units can be quantified and potential source causing the transmission error can be located in advance. Their applications would be very helpful for gear hobbing machine manufacturers to further improve products’ comprehensive performances.

Footnotes

Handling Editor: Chenhui Liang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was supported by National Natural Foundation of China (Grant No. 51905064/51905060); the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201801146, KJZDM201801101); the Natural Science Foundation of Chongqing (Grant No. Cstc2018jcyjAX0505).

ORCID iDs

Zheng Zou

Ruizhi Shu

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Improving the transmission error source tracing method for gear hobbing machines

Abstract

Keywords

Introduction

Modeling of gear train transmission error

Theoretical analysis of the transmission error of gear hobbing machine

Development of in situ test platform and transmission error source tracing procedure

Case study

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References