Transmission chain error elimination for gear hobbing machines based on classified compensation theory and frequency response identification

Journal Pre-proofs Transmission chain error elimination for gear hobbing machines based on classified compensation theory and frequency response identification Changjiu Xia, Shilong Wang, Tan Long, Chi Ma, Sibao Wang PII: DOI: Reference:

S0263-2241(20)30133-0 https://doi.org/10.1016/j.measurement.2020.107596 MEASUR 107596

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Measurement

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5 November 2019 28 January 2020 5 February 2020

Please cite this article as: C. Xia, S. Wang, T. Long, C. Ma, S. Wang, Transmission chain error elimination for gear hobbing machines based on classified compensation theory and frequency response identification, Measurement (2020), doi: https://doi.org/10.1016/j.measurement.2020.107596

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Transmission chain error elimination for gear hobbing machines based on classified compensation theory and frequency response identification Changjiu Xiaa,b, Shilong Wanga,b,*, Tan Longa,b, Chi Maa,b, Sibao Wanga,b a The

State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, 400044, China b College

of Mechanical Engineering, Chongqing University, Chongqing, 400044, China

Abstract This paper proposes a novel classified compensation method considering frequency response to eliminate the transmission chain error (TCE) of gear hobbing machines. Based on error source analysis and Discrete Fourier Transform (DFT), the measured TCE was firstly decomposed into different error components, including the integer harmonic, the non-integer harmonic and the other. A classified compensation theory was then presented, in which the Measurement System Error Compensation (MSEC) and the software axis were used to compensate the first two types of error components, respectively. Then, the frequency response identification for the C-axis control system was performed, and the final compensation values were calculated and integrated into the classified compensation process. The experimental results on a YS3120-type gear hobbing machine showed that the peak-to-peak value of TCE was reduced by 85.5% and that both the integer and non-integer harmonic errors were effectively compensated. It lays the foundation for gear precision hobbing. Keywords: Transmission chain error; Error compensation; Harmonic error; Frequency response; Gear hobbing machine

1 Introduction Hobbing is a preferred manufacturing method for the batch processing of spur and helical gears before precision grinding because of its high productivity [1-3]. In order to improve gear hobbing accuracy, the geometric inaccuracy, the kinematic error, the thermal error, and the force-induced deformation in the gear hobbing machine have become research hotspots [4, 5]. Among them, the relative kinematic error between the hob and the worktable, which is also described as the transmission chain error (TCE), will not only produce gear machining errors, including the pitch error, the profile error and the helix error, but also cause vibration and noise of gear hobbing machines. Thus, intensive study on the error source, propagation and elimination mechanisms for TCE should be developed to improve machining performance and hobbing accuracy of gear hobbing machines. TCE is closely related to the transmission error of multi-stage gear pairs in the transmission system, and it is generally caused by the working surface deviations and assembly errors of gear pairs. Some scholars have developed extensive previous researches on the analysis, calculation of transmission error in the gear system. Mark [6] provided measuring methods and mathematical representation of working surface deviations of helical gears based on Legendre polynomials, and derived the contribution of working surface deviations on the transmission error using Fourier-series. On this basis, an effective method for computation and diagnosis of the transmission error was then proposed. Lin et al. [7] obtained static transmission error by a finite element model with considering machining errors, assembly errors and modifications simultaneously, and calculated dynamic transmission error by a bending1

torsional-axial discrete dynamic model established by the lumped mass method. Velex et al. [8, 9] provided theoretical basis for the calculation and minimization of transmission error by taking the influence of shafts, bearings, couplings, etc. into consideration. Bruyère et al. [10] derived analytical formulations for profile modifications of narrow-faced spur helical gears to minimize quasi-static transmission error. Obviously, the above researches are helpful for the theoretical derivation for transmission error of gear pairs. However, TCE is not only related to single gear pairs but also determined by the specific machine kinematic chain. Thus, the error source and error component analysis for the TCE with considering the kinematic chain of gear hobbing machines needs to be further developed, and it is the premise of effective elimination of TCE. Besides, various measurement methods for the transmission error have been proposed by using different measurement devices. Representatively, to accurately measure dynamic transmission error between the hob and the worktable in a large gear hobbing machine, Yagishita [11] developed a measurement scheme with two servoaccelerometers. White et al. [12] utilized rotational laser vibrometers for the measurement of transmission error in the gear box. Actually, the encoder [13, 14] is the most widely used measurement device. Remond et al. [15] introduced an improved technique to measure gear transmission error at high speed with low-pulse optical encoders, and Palermo et al. [16] discussed the measurement of transmission error of a gearbox via low-cost digital encoders. To ensure measurement accuracy, deviation analysis [17, 18] and error compensation [19] of encoders were also reported. Moreover, an angular sampling method that could be broadened to phase difference measurements [15] was presented to measure various signals in rotating machines with discrete geometry. As such, this paper designs a measurement scheme by combining the angular sampling method and high-precision absolute encoders to ensure measurement accuracy of TCE in gear hobbing machines, which can be taken as a special type of rotating machines. The main purpose for the accurate measurement of TCE is to enhance the machining performance of machine tools. For usual machine tools, most researchers attempted to optimize machining parameters for better machinability. For example, Mia et al. [20] proposed the Grey-Taguchi method to optimize chip-tool interaction parameters for improved surface finish in MQL-assisted turning, and Kuntoğlu et al. [21] optimized machining parameters in turning to reduce the tool wear and breakage. Öztürk et al. [22] modeled and optimized machining parameters during grinding of flat glass. But for the gear hobbing machine, the gear transmission accuracy should be ensured before optimization of machining parameters in hobbing. Usually, the most widely used methods to improve the gear transmission accuracy can be divided into two categories: control optimization and error compensation. In terms of control optimization, Zuo et al. [23] proposed a novel differentiable deadzone model to aid the design of controller of Gear Transmission Servo (GTS) systems, and then presented a full state feedback controller based on L1 adaptive control law. Wang et al. [24] designed a 2-ADTSMC controller to deal with backlash and inertia variations in gear transmission servo systems, and verified its effectiveness by both simulations and experiments. Tian et al. [25] proposed a simple estimation approach of pitch errors and helix errors of the helical gear, and then designed an electronic gearbox (EGB) cross-coupling controller (ECCC) to eliminate EGB control error. On the other hand, some previous researches focus on error compensation. Considering high control accuracy of the rotary axis, many scholars devotes to measure and identify geometric error of the rotary axis and obtain the corresponding rotation error and compensation value. Yamamoto et al. [26] considered the transmission error caused by nonlinear elastic deformations in micro-displacement region in harmonic drive gearings, and proposed a new modeling and feedforward compensation method. Then, the static positioning error was successfully reduced, proving the effectiveness of modeling and compensation. To some extent, this positioning error compensation method towards the rotary axis provides a theoretical basis for the effective compensation of TCE in gear hobbing machines. However, it only works on the elimination of low-order harmonics of TCE, and may lead to unexpected increasement of high-order harmonics. 2

Thus, a new compensation method suitable for both low-order harmonics and high-order harmonics simultaneously remains to be further investigated. Besides, the dynamic response of the transmission system should be revealed because it will also significantly affect compensation effect. Therefore, this paper proposes a novel classified compensation method with considering frequency response for TCE in the gear hobbing machine. Based on error source analysis, the measured TCE is decomposed into three types of error components, including the integer harmonic, the non-integer harmonic and the other, and the classified compensation theory is then presented to compensate different error components by the Measurement System Error Compensation (MSEC) and the software axis. To achieve the effective elimination of both low-order harmonics and high-order harmonics of TCE simultaneously, the frequency response of the C-axis control system is identified and the final compensation value for TCE is calculated and integrated into the classified compensation process.

2 Transmission chain error of gear hobbing machine 2.1 Transmission chain of gear hobbing machine Fig. 1 shows the basic structure of a common gear hobbing machine with 3 translational axes (X-, Y-, Z-axis) and 3 rotary axes (A-, B-, C-axis). The B-axis is the hob spindle, which is attached to the hob, and the C-axis is the worktable where the gear to be machined mounts. Thus, the rotational speeds of the hob and the gear are coincident with that of the B-axis and the C-axis, respectively. z

A

y Hob

Z

x Gear

Y

B

Hob spindle Worktable

X

C

Fig. 1. Configuration of the gear hobbing machine

In the gear hobbing process, a constant speed ratio between the hob and the gear should be ensured. Namely, the speed ratio of the C-axis relative to the B-axis is

ihg 

nC zh  nB z g

(2)

where nC and nB denote the rotational speeds of the C- and B-axis, respectively; zh denotes the thread number of the hob; and . zg . denotes the tooth number of the machined gear. In fact, ihg is ensured by the electronic gearbox in the numerical control (NC) system. Besides, there is a secondary gear reduction structure between the B-axis motor and the hob spindle. Similarly, another secondary gear 3

reduction structure lies between the C-axis motor and the worktable. The detailed transmission chain is displayed in Fig. 2, in which #1~#8 denote the transmission gears of multi-stage gear pairs. #5

#6

#7

#8

Hob spindle

Hob

B-axis motor

Worktable

Gear

#4

#3

#2

#1

B-axis C-axis C-axis motor

Electric gearbox Fig. 2. The transmission chain of the gear hobbing machine

2.2 Transmission chain error (TCE) Actually, the speed ratio between the gear and the hob will deviate from its nominal value because of the transmission error of each gear pair and the servo-control error of motion axes. The former usually derives from the working surface deviations and assembly eccentricity of transmission gears, and the latter is commonly caused by the positioning inaccuracy of motion axes. To describe the deviation of the speed ratio ihg , the difference between the theoretical and real rotational angles of the C-axis, which follows the B-axis rotation, is defined as TCE of the gear hobbing machine, as shown in Eq.(3).

eΣ  C  C  C 

zh B zg

(3)

where eΣ denotes TCE of the gear hobbing machine; C and C denote the real and theoretical rotational angles of the C-axis, respectively; and  B denotes the theoretical rotational angle of the B-axis. To study the characteristics of the two main error sources of TCE, the transmission error of a single gear pair is firstly analyzed. Taking the gear pair with the transmission gears #3 and #4 as an example, the transmission error of this gear pair is caused by the installation error of gears, the working tooth surface error and the elastic deformation in power transmission, and it can be expressed as a function of the rotational angle of the gear #4 with the periods of 2π/i34 and 2π. eTE34    e3  2 / i34   e4  2 

(4)

where eTE34 denotes the transmission error of the gear pair;  denotes the rotational angle of the gear #4; i34 denotes the transmission ratio of the gear pair; e3 denotes the transmission error with the period of 2π/i34 induced by the gear #3; and e4 denotes the transmission error with the period of 2π induced by the gear #4. Eq.(4) can be expanded with trigonometric series as 4





k 1

k 1

eTE34   A3k cos  ki34  3k    A4 k cos  k  4 k 

(5)

where  denotes the rotational angle of the gear #4, i.e. the C-axis rotational angle; A3k and A4k denote the amplitudes of the kth harmonic of the gear #3 and the gear #4, respectively; and 1k and 2k denote the initial phases of the kth harmonic of the gear #3 and the gear #4, respectively. Thus, TCE induced by the transmission error of multi-stage gear pairs can be expressed as 

h

eTE  Alk cos  kilc  lk 

(6)

l 1 k 1

where eTE denotes TCE induced by the transmission error of multi-stage gear pairs; h denotes the number of transmission gears; Alk and lk denote the amplitude and the initial phase of the kth harmonic of the lth gear, respectively;  denotes the C-axis rotational angle; and ilc denote the speed ratio of the lth gear relative to the Caxis. Besides, another important error source of TCE, i.e. the servo-control error of motion axes, is then studied. It derives from the positioning measurement deviations of the B- and C-axis with encoders. Therefore, TCE induced by the servo-control error is periodic, and it can be expressed as

eSE 



 A

l  B ,C k 1

lk

cos  kilc  lk 

(7)

where eSE denotes TCE induced by the servo-control error of motion axes; l denotes the motion axis with an encoder, i.e. B-axis and C-axis; Alk and lk denote the amplitude and the initial phase of the kth harmonic of the lth motion axis, respectively;  denotes the C-axis rotational angle; and ilc denotes the speed ratio of the lth motion axis relative to the C-axis, in which iBc  ihg and iCc =1 . Therefore, Eq.(6) and Eq.(7) can be combined and superimposed to obtain the total TCE. h +2 

e  eTE  eSE  Alk cos  kilc  lk 

(8)

l 1 k 1

Generally, in order to ensure uniform wear of gears during gear mesh, the transmission ratio in a gear pair is designed to be non-integer, so TCE contains error components with both integer and non-integer C-axis frequencies. If all non-integer ilc is integrated into a minimal sequence

 fl m

where there is no integer multiple relation

between the m items of elements, e can be decomposed into two parts, including the harmonic with fl =1 and the rest harmonics with

 fl m . Namely, 

m



e  A0 k cos  k  0 k   Alk cos  kfl  lk  k 1

(9)

l 1 k 1

According to Eq.(9), the spectrum of TCE can be obtained by Fourier Transformation. In fact, the main frequency of the spectral lines includes axis rotation frequency and gear meshing frequency. It should be noted that the gear meshing frequency is usually accompanied by the side band phenomenon [27] due to the inconsistency of tooth surface errors, so some special frequencies around the gear meshing frequency will inevitably appear. Based on the spectral analysis, the main error source of TCE can be inferred and identified if the spectral line has a higher amplitude, and this lays the foundation for subsequent error extraction and elimination. 2.3 Error decomposition 5

TCE consists of periodic components with integer and non-integer C-axis frequency at the same time, and they should be compensated in different ways. When TCE is measured with the method presented in Section 4.1 and is converted into a same-length complex-valued sequence by Discrete Fourier Transform (DFT), it can be decomposed into three parts. The first part is the error component with the same fundamental frequency as that of the C-axis rotation, i.e. eI called integer harmonic error. The second part is a series of error components with a fundamental frequency of fl times that of the C-axis rotation, i.e.

el m

called non-integer harmonic error. The rest part is

eother which is randomly generated and can be ignored due to its rather small amplitude of less than 1 arcsec. In the process of error decomposition, el is firstly separated from the total TCE as hl

el  Alkh cos  kfl  lk 

(10)

k 1

where fl denotes the fundamental frequency of el ; hl denotes the number of harmonics with a larger amplitude; and Alkh and lk denote the amplitude and the initial phase of the kth harmonic of el , respectively. Thus,

el m

can be expressed as m

m

hl

 e  A l 1

l

l 1 k 1

h lk

cos  kfl  lk 

(11)

It should be noted that the sampling number of TCE in the measurement process is limited, and only TCE during n-cycle rotation of the C-axis is measured and analyzed. Namely, the corresponding C-axis rotational angle of the measuring point is in [0, 360n] deg in time domain, and the sampling interval in frequency domain is fΔ=1/n. When Discrete Fourier Transform (DFT) is used to obtain the spectrum of el , it is inevitable to cause the spectrum fence effect [28] when fl is between adjacent integer multiples of f  . Thus, Alkh and lk of el can not be calculated accurately. To solve this problem, an error reconstruction method regarding non-integer harmonic error is proposed. When the frequency of the kth point, i.e. kfΔ, is closest to fl , the spectral amplitude of el can be calculated by DFT of the kth point. If el is discretized and expressed as el (n)   cos  nfl      cos  cos nfl   sin  sin nfl ,   0,    0, 2 

(12)

where λ and φ denote the amplitude and the initial phase of el (n) , respectively; and Δ denotes the angular sampling interval in time domain. Thus, DFT of the kth point can be obtained by N 1  i 2 k  E  k   el  n  exp  n  , k   0, N  1  N  n0

(13)

where N denotes the number of the data in DFT. According to Euler’s formula,

2 kn 2 kn  i 2 k  exp  n   cos  i sin N N  N 

(14)

Then, Eq.(13) is further derived as N 1

E  k   el  n  cos n0

2 kn N 1 2 kn  i el  n  sin N N n0

Divide Eq.(15) into two parts including the real part and the imaginary part, 6

(15)

N 1

Re  el  n  cos n0

N 1

Im  el  n  sin n0

N 1 N 1 2 kn 2 kn 2 kn   cos   cos nfl cos   sin   sin nfl cos N N N n0 n0

(16)

N 1 N 1 2 kn 2 kn 2 kn   cos   cos nfl sin   sin   sin nfl sin N N N n0 n0

(17)

Rewrite Eq.(16) and Eq.(17) in a matrix form,

2 kn  N 1 cos nfl cos   N  Re   n  0  Im    N 1    cos nf sin 2 kn   l N  n0

2 kn  N   cos   n0   N 1 2 kn    sin   sin nfl sin   N  n0 N 1

 sin nfl cos

(18)

Namely,  Re     Im  =A       

(19)

According to the invertibility of matrix A, Eq.(19) can be derived as   1  Re     =A  Im     

(20)

In Eq.(20), Re and Im can be calculated by DFT, and A is only related to the frequency and the number of points. Therefore, α and β can be calculated. This means the amplitude λ and the initial phase φ of el  n  can be obtained by Eq.(21) and Eq.(22). So far, the non-integer harmonic error has been reconstructed successfully.

  2  2

(21)

 1   cos  ,   0  2π  cos 1 α ,   0  

(22)

When the non-integer harmonic error is filtered out from TCE, the integer harmonic error eI can then be extracted and expressed as Eqs.(24)~(23) based on DFT and inverse DFT (IDFT). m  N 1    i 2 kn   eΣ  n   el  n   exp    , k    1, 2   N /   N   l 1   n0  0 , k  other  1,..., N  1 EI  k    m  N 1    i 2 kn   eΣ  n   el  n   exp    , k  N     1 , N   2  1 ,..., N / +1 N   l 1   n0 

N 1  i 2 kn  eI  n   EI  k  exp    N  k 0

(23)

(24)

where eΣ  n  denotes the measured TCE; and E I  k  denotes DFT of the kth point of eI  n  . Finally, the remaining part of TCE is m

eother  n   e  n   eI  n   el  n  l 1

7

(25)

3 Methodology for compensation of TCE 3.1 Classified compensation theory In general, the C-axis rotational speed is much lower than that of the B-axis, and the servo control accuracy of the C-axis is higher than that of the B-axis. As such, the C-axis motion command can be adjusted and controlled to compensate TCE. On the basis of the C-axis feedback control theory, a novel classified compensation method for TCE is proposed in this paper. The C-axis control system, which includes a C-axis motor and a secondary reduction transmission chain, is simplified as a linear system, as shown in Fig. 3. C-axis motion command

C

-

C(s)

+

ec

G(s)

C-axis servocontrol motor

Compensation value

Secondary Transmission chain

φC C-axis rotational angle

C-axis Fig. 3. The control system of the C-axis

In Fig. 3, C(s) and G(s) denote the transfer function of the C-axis motor and the secondary reduction transmission chain, respectively. In essence, the elimination of TCE is to construct compensation movement on the C-axis, making the sum of TCE and compensation movement equal to 0. Namely, m

m

l 1

l 1

C  ihg  B  e  ec  ihg  B  eI  el  eIc  elc m

m

l 1

l 1

eI  el  eIc  elc  0

(26) (27)

where C and  B denote the rotational angles of the C-axis and the B-axis, respectively; ec denotes the total compensation value for TCE; eIc denotes the compensation value for eI ; and elc denotes the compensation value for el . When the dynamic response of the C-axis control system is ignored during compensation, the compensation value is

ee

Ic

lc

 eI  el

(28)

The Measurement System Error Compensation (MSEC) module, also known as pitch compensation, usually takes advantage of a series of uniform compensation points to compensate the axis positioning error which has the same period as that of the measurement. That is to say, it is only suitable for the integer harmonic error compensation. Besides, the software axis is a virtual axis in the NC system and is often used for fault diagnosis or error compensation. In the electronic gearbox of the gear hobbing machine, the drive axis can be considered to be a software axis l when the C-axis is taken as a driven axis. Thus, the coupling relationship between the l-axis and the C-axis can be established, and the superposition of the compensation motion and the original motion can be easily realized. The compensation frequency of the l-axis is determined by the C-axis, and it is expressed as flC  nl / nC

(29)

where flC denotes the compensation frequency of the software axis l; and nl denotes the rotational speed of the software axis. The software axis is suitable for the compensation of error components with arbitrary times of the C-axis rotation 8

frequency, so it can be used to compensate non-integer harmonic error. Once the control NC file of the software axis is given in one cycle of the C-axis, it can be superimposed on the C-axis through the electronic gearbox, generating an additional period-controllable compensation motion. For easy understanding, the schematic diagram of the classified compensation process is displayed in Fig. 4. nB

B-axis C-axis control system

Z-axis Σ

Y-axis nl

Software axis l

elc

C-axis motor

Gearbox

nC

MSEC Classified compensation

eIc

Fig. 4. The schematic diagram of classified error compensation

3.2 Frequency response identification The classified compensation method regards the compensation value as the input of the C-axis control system to generate the compensation motion. There are two conventional ways to obtain the compensation value: (i) inverting the measured error directly as the compensation value and (ii) extracting the low-order harmonic of the measured error as the compensation value. The former only works on reducing the contribution of static transmission error such as the positioning error of a rotary axis while the latter can suppress the low-order harmonic with an integer reduction ratio in the gear system. However, both of them can not effectively deal with high-order harmonics with non-integer frequency in the TCE of the gear hobbing machine. Therefore, to effectively compensate TCE, the frequency response of the C-axis control system should be analyzed. The C-axis control system will show different frequency responses under different rotational frequency. Actually, the C-axis control system can be deemed as a linear time-invariant system, and the total input can be obtained by the linear superimposition of the original motion command and the compensation value. Thus, the final output, i.e. the C-axis rotational angle, is the sum of the nominal rotational angle and the additional rotational angle caused by the compensation value. If the new TCE e is measured after the first compensation with ecf, which denotes the first compensation value, i.e. eIc or elc in Eq.(28), the additional output induced by ecf through the C-axis control system is deemed as the difference of the original TCE relative to that after compensation. eout  eΣ  e

(30)

The frequency response of the C-axis control system after DFT can be identified and expressed as

H  f   eout  f  / ecf  f 

(31)

where eout  f  denotes the output in frequency domain; ecf  f  denotes the first compensation value in frequency domain; and H  f



denotes the frequency response function. 9

On this basis, the required compensation value in frequency domain for additional compensation motion can be calculated by

ec  f   eC  f  / H  f   eC  f  ecf  f  / eout  f  where ec  f  denotes the compensation value in frequency domain; and eC  f



(32) denotes additional compensation

motion caused by ec  f  in frequency domain. Then, the required compensation value in time domain can be obtained by IDFT of ec  f  . As such, the frequency response and required compensation values at different rotational frequency can be determined by the above derivation process. 3.3 Classified compensation based on frequency response identification The flow chart of compensating the two types of error components of TCE, i.e. eI and

el  , is displayed in

Fig. 5. First error measurement eΣ Decomposition

eI Input

{el} -eI

{-el}

Input

Software axis compensation

MSEC

Second error measurement e'Σ

eΣ - e'Σ Decomposition

e'I

{e'l} Frequency response identification

HI(f) Final

{Hl(f)}

eIc

Final {elc} Software axis compensation

MSEC End

Fig. 5. The flow chart of classified compensation based on frequency response identification

As shown in Fig. 5, the classified compensation based on frequency response identification will go through five steps. Firstly, TCE is measured for the first time and decomposed into the integer hormonic error and the non-integer harmonic error, i.e. eI and

el  . Secondly, the two types of error components are considered to be compensated 10

with eI in the MSEC module and

el 

in the software axis module, respectively. TCE after compensation is

measured for the second time and it is similarly decomposed into eI and

el . Thirdly, by combining

eI , the frequency response H I ( f ) is calculated according to Eq.(31), and the frequency response similarly obtained by combining

el 

with

eI with

H l ( f )

is

el . Fourthly, according to Eq.(32) and IDFT, the final compensation

values considering frequency response in time domain, i.e. eIc and

elc  , are computed for obtaining additional

compensation motion. Finally, by generating a pitch compensation file in the MSEC module, and obtaining NC codes with considering motion relationship between the software axis l and the C-axis in the software axis module, the final compensation is carried out. It should be noted that the above compensation process can be directly used to measure and compensate TCE of gear hobbing machines under machining conditions.

4 Experiments for error measurement and decomposition 4.1 Measurement experiment of TCE TCE of the hobbing machine is a kind of dynamic error between the B-axis and the C-axis, and its amplitude is closely related to the B-axis rotational speed nB and the reduction ratio between the B-axis and the C-axis. Generally, different rotational speeds of the rotary axes will form different TCE, so the laser interferometer or the double ball bar, which functions well in a low-speed condition, is not suitable to measure TCE in a high-speed hobbing condition. Therefore, two high-precision absolute encoders are used to measure rotational angles of the B-axis and the C-axis simultaneously, and then TCE can be calculated by Eq.(3). In the measurement of TCE, one encoder is mounted on the worktable (C-axis), named as Encoder-C, and the other one is installed on the hob spindle (B-axis), designated by Encoder-B. To improve measurement resolution, two digital interpolators designated by Interpolator-C and Interpolator-B are also used. Their interpolation factors are 5 and 25 folds, respectively. The detailed parameters are shown in Tab. 1. Tab. 1. Parameters of measurement equipment Equipment

Type

Parameters

Encoder-C

HeiDenHain-RPN 886

180000 counts per revolution

Interpolator-C

HeiDenHain-IBV101

5 folds

Encoder-B

HeiDenHain-ERA 420C

20000 counts per revolution

Interpolator-B

HeiDenHain-IBV102

25 folds

Digital signal acquisition board

PXI-7842R

Clock frequency up to 40MHz

Host computer

PXIe-1071

-

Fig. 6 shows the measurement process of TCE. Sinusoidal signals with a peak-to-peak value of 1 Vpp are firstly sampled by the Encoder-C and Encoder-B, and then they are subdivided into TTL signals by digital interpolators and synchronously acquired by the PXI-7842R digital signal acquisition board. As such, a finite sampling sequence of TCE varying with equally-spaced angles can be calculated by

11

  

 z NC NB  h  e  3600  360    180000  5 180000  5 z g 

(33)

where NC and NB denote the count values of Encoder-C and Encoder-B, respectively. B-axis (Hob spindle) Encoder-B Gear hobbing machine

1Vpp

Interpolator-B

1Vpp

Encoder-C

Interpolator-B

PC(PXIe-1071)

Digital signal acquisition board

TTL

TTL

Calculating program for transmission chain error

Counting program for TTL

C-axis (Worktable)

Fig. 6. Measurement process of TCE

The measurement experiment was carried out on a YS3120-type hobbing machine, whose transmission chain was shown in Fig. 2. The thread number of the hob was 3, and the B-axis rotational speed was 200 rpm. When the tooth number of the machined gear was set as 121, 91, 61 and 31 in turn, the transmission ratios between the B-axis and the C-axis were 121:3, 91:3, 61:3, and 31:3, respectively. The measurement results of TCE in time domain under the four working conditions are shown in Fig. 7. 20

Y 15 10

-20

low-oder harmonics

5

Tooth number of 121

-40

high-oder harmonics

0 15

20

Tooth number of 91

0

X: 156 Y: 7.884

10

-20

Amplitude (arcsec)

Transmission chain error (arcsec)

X: 156 Y: 13.64

Tooth number of 121

0

Tooth number of 91

-40 20 0 -20

0 15 Tooth number of 61

10 X: 156 Y: 2.445

5

Tooth number of 61

-40

5

0 15

20

Tooth number of 31

0

10

-20

0

60

120

180

240

300

X: 156 Y: 0.544

5

Tooth number of 31

-40

0

360

C-axis rotational angle (deg)

0

20

40

60

80

100

120

140

Order of C-axis (f/fc)

(a)

160

180

200 X

(b)

Fig. 7. TCE under four working conditions: (a) in time domain; (b) in frequency domain

Fig. 7(a) shows the similar change of TCE with the C-axis rotational angle in time domain under four working conditions. It should be noted that when the B-axis rotational speed and the thread number of the hob keep unchanged, the C-axis rotational speed is inversely proportional to the tooth number according to Eq.(2). From the spectrum diagram of TCE in Fig. 7(b), it can be seen that the amplitudes of low-order harmonics are not sensitive to the change 12

of the C-axis rotational speed while the amplitudes of high-order harmonics show a dramatical drop trend with the increasement of the C-axis rotational speed. Namely, the dynamic characteristics of the C-axis control system will strongly affect TCE of the gear hobbing machine, and this must be taken into consideration in the compensation process. 4.2 Decomposition experiment of TCE When TCE has been measured accurately, error decomposition should be carried out for subsequent error compensation. Considering the similarity of decomposition of TCE at different hobbing conditions, the following analysis proceeds when the transmission ratio between the B-axis and the C-axis is 121:3, which means the thread number of the hob is 3 and the tooth number of the machined gear is 121. The tooth numbers of the 8 transmission gears in Fig. 2 and their speed ratios relative to the C-axis are listed in Tab. 2. Tab. 2. Tooth numbers of the transmission gears and their speed ratios relative to the C-axis Gear

#1

#2

#3

#4

#5

#6

#7

#8

Tooth number

34

85

26

156

60

75

22

88

Speed ratio

15

6

6

1

605/3

484/3

484/3

121/3

According to Fig. 7(a), the minimum value and the maximum value of TCE are -55 arcsec and 21 arcsec, respectively, and the peak-to-peak value of TCE is 76 arcsec. Besides, seen from Fig. 7(b), the main frequencies of TCE are 1, 2, 6, 12, 15, 150, 156, 162, 168, 180 times that of the C-axis. Among them, the 1st, 6th, 15th order frequencies come from the assembly eccentricity of the transmission gears #1~#4, and the special 2nd and 12th order frequencies are possibly generated by the working surface deviations of these gears. Besides, the 150th, 156th, 162th, 168th and 180th order frequencies possibly comes from the diffusion of the 484/3th order frequency caused by the inconsistent tooth surface errors of the transmission gears #6~#7. As the decomposition process presented in Section 2.3, TCE can be decomposed into three parts, including eI ,

e1 with the fundamental frequency of 484/3 times of fc, and eother . Obviously, eI comes from the transmission chain of the worktable, accounting for the vast majority of TCE, and e1 derives from the transmission chain of the hob. eother shows a relatively small amplitude and is ignored in the compensation process.

13

20

15

0

eΣ

5

20

0 15 eI

Amplitude (arcsec)

Transmission chain error (arcsec)

-40

-20 -40 5 e1

0 -5

6

eΣ

156

10

-20

0

1 2

12

150

15

162 168 180

eI

10 5 0

e1

2

0

5 e ot h er

e ot h er

0

1

-5 0

60

120

180

240

300

0

360

C-axis rotational angle

0

20

40

60

80

100

120

140

Order of C-axis (f/fc)

(a)

160

180

200

X

(b)

Fig. 8. Decomposition of TCE: (a) in time domain; (b) in frequency domain

5 Results and discussions for error compensation 5.1 Compensation of integer harmonic error based on frequency response identification When the integer harmonic error eI is directly compensated with eI , which is equivalent to the positioning error compensation of the rotary axis, the TCE e is measured again and decomposed to obtain eI , as shown in Fig. 9. The amplitudes of low-order harmonics of eI , such as the 1st, 6th and 15th harmonic, are reduced significantly. However, the amplitudes of high-order harmonics, such as the 150th, 156th and 162th harmonic, show an unexpected increasement. The peak-to-peak of eI is up to 85 arcsec, and it is bigger than that of eI (73 arcsec). In a word, the result shows that the traditional positioning error compensation of the rotary axis only works on eliminating the loworder harmonics of TCE while it is not suitable for compensating the high-order harmonics. Therefore, a novel compensation method based on frequency response identification is proposed and applied. The specific process of frequency response identification is presented in Section 3.2, and the final compensation values are calculated. eI is compensated in the MSEC module as shown in Fig. 5. Thus, the TCE e after compensation with considering frequency response is again measured and decomposed to obtain eI , which is also displayed in Fig. 9.

14

Amplitude (arcsec)

40 eI

20

e'I e''I

0 -20 -40

0

30

60

90

120

150

180

210

240

270

300

330

360

C-axis rotational angle (deg) (a)

Amplitude (arcsec)

25 20 15

e'I(f)

X=1

e''I(f)

X=6

10

X=162

X=15

5 0

eI(f)

X=156

0

X=150

20

40

60

80

100

120

140

160

180

200

Order of C-axis (f/fc) (b)

X

Fig. 9. The integer harmonic errors after the positioning error compensation and the compensation with considering frequency response: (a) in time domain; (b) in frequency domain

Seen from Fig. 9, the amplitudes of the 1st, 6th, 15th, 150th, 156th and 162th harmonic of eI are all greatly reduced. The peak-to-peak of eI is down to 9 arcsec, which means the compensation with considering frequency response can reduce the original eI by 87.7%. Besides, by comparing eI and eI , it can be seen that the novel compensation method with considering frequency response is better than the positioning compensation method, and it can at least accurately compensate all the harmonics whose order is below 200. It should be noted that the maximum compensable order of the proposed method is closely related to the available pitch compensation points, the interpolation period, and the C-axis rotational speed. The more compensation points, the shorter the interpolation period, and the lower the C-axis rotational speed, and the higher the compensable order. It can be seen that most of the TCE in the YS3120-type hobbing machine has been eliminated when eI is compensated, and the peak-to-peak value of e decreases by 85.5% from 76 arcsec to 11 arcsec, as shown in Fig. 10.

Amplitude (arcsec)

20 eΣ e''Σ

0 -20

-40 -60

0

30

60

90

120

150

180

210

240

C-axis rotational angle (deg)

Fig. 10. TCE before and after compensation 15

270

300

330

360

In order to further illustrate the feasibility and effectiveness of the proposed compensation method, TCE after compensation under the four working conditions are all measured again, as shown in Fig. 11. The peak-to-peak values of e are all below 12 arcsec, satisfying the premise of gear precision hobbing. Thus, the hobbing accuracy can be improved. 5

Tooth number of 121

0 -5

Amplitude (arcsec)

5

Tooth number of 91

0 -5 5

Tooth number of 61

0 -5 5

Tooth number of 31

0 -5 0

60

120

180

240

300

360

C-axis rotational angle (deg)

Fig. 11. TCE after compensation under four working conditions

5.2 Compensation of non-integer harmonic error It can be seen that the non-integer harmonic error el is quite small in this hobbing machine, and it seems unnecessary to compensate el because the compensation of eI could reduce TCE by more than 80%. It is possibly because none of the speed ratio of the transmission gear relative to the C-axis is non-integer in the transmission chain of the worktable, and the speed ratios of the transmission gears in the transmission chain of the hob are non-integer but too large, even more than 40. Nevertheless, considering that el might significantly affect TCE in other machines, a further compensation test for el is still carried out. In the test, the software axis is used to compensate el . Firstly, an ideal objective error e2i , as shown in Eq.(34), is additionally generated compared to the original TCE in Fig. 10.

e2i  A2i cos  f 2iC 

(34)

where A2i denotes the amplitude of e2i and is equal to 10 arcsec; f 2i denotes the frequency of e2i with the value of 121/6; and C denotes the C-axis rotational angle in [0, 360] deg. Then, according to the proposed compensation method with considering frequency response, the final compensation value regarding e2i can be calculated as Eq.(35) according to Eqs.(13)~(22). The amplitude and initial 16

phase of e2c are computed as 9.706 arcsec and -0.051, respectively.

 121  e2c  9.706  cos  C  0.051 6  

(35)

TCE before and after compensation of e2i can be measured and illustrated in Fig. 12, in which e and e c

Amplitude (arcsec)

denote TCE before and after compensation, respectively.

0

-40 eΣ eΣc

-80

0

60

120

180

240

300

360

C-axis rotational angle (deg)

(a)

Amplitude (arcsec)

12 10

e2c ( f )

eΣ(f)

8

eΣc (f)

6 4 2 0

10

20

(b)

30

Order of C-axis(f/fc)

Fig. 12. TCE before and after compensation: (a) in time domain; (b) in frequency domain

Seen from Fig. 12(b), there is one more frequency component e2c ( f ) in the compensated TCE e c ( f ) compared to e ( f ) . It comes from the error compensation with the software axis. Define the difference between e2i and e2c as the residual error (RE) of the proposed compensation method.

 121  RE =e2c  e2i  0.582  cos  C  1.016   6 

(36)

Fig. 13 shows the objective error e2i , the final compensation data e2c , and the residual error of the proposed compensation method at the same time. It is observed that e2c fits well with e2i , and the amplitude of RE is only 0.582 arcsec, which is only 5.82% of that of e2i . Thus, the compensation effectiveness for el is proved. 17

10 ei2 e c2

RE (arcsec)

5

RE

0

-5

-10

0

50

100

150 200 C-axis rotational angle (deg)

250

300

350

Fig. 13. RE after compensation

So far, the proposed classified compensation method based on frequency response identification has been proved to be effective in compensating both the integer harmonic error and the non-integer harmonic error.

5 Conclusion To improve hobbing accuracy and reduce noise and vibration of gear hobbing machines, a novel classified compensation method with considering frequency response was proposed to eliminate TCE. The MSEC and the software axis were used to compensate integer harmonic error and non-integer harmonic error of TCE, respectively. Moreover, the feasibility and effectiveness of the proposed method was validated on a YS3120-type gear hobbing machine. The contributions of this paper can be summarized as follows. (1) The error source of TCE was analyzed based on the specific kinematic chain of gear hobbing machines and the speed relationship between the gear reduction structures and the electric gearbox. The analytical expressions of TCE induced by the transmission error of multi-stage gear pairs and the servo-control error of motion axes were derived with trigonometric series, respectively. It can provide a basis for the diagnosis and identification of the main frequency component of TCE. (2) The error reconstruction method based on DFT and IDFT was presented to eliminate the spectrum fence effect and calculate the amplitude and initial phase of the non-integer harmonic error. Then, the measured TCE was decomposed into three parts, including the integer harmonic error, the non-integer harmonic error and other error. The method can help to accurately extract out different error components of TCE for subsequent error classified compensation. (3) The classified compensation theory was proposed to compensate the integer harmonic error and the noninteger harmonic error by the MSEC module and the software axis module, respectively. The dynamic characteristic of the C-axis control system was then analyzed based on the frequency response identification. On this basis, the final compensation value with considering frequency response was calculated and integrated into the classified compensation process. Subsequently, the error measurement, decomposition and compensation experiments of TCE under four working conditions were conducted on a YS3120-type gear hobbing machine to validate the effectiveness of the proposed method. Compared to the traditional positioning error compensation method, which only works on compensating low-order harmonics of TCE, the proposed method has the advantage to effectively eliminate loworder harmonics and high-order harmonics at the same time. This paper presents a novel effective method to improve kinematic accuracy of the gear hobbing machine. The 18

experimental results show that TCE can be reduced by more than 80%, and it lays the foundation for gear precision hobbing. On this basis, further work will focus on the elimination of the thermal error and force-induced deformation error to further improve gear hobbing accuracy.

Acknowledgments This work was supported by the Key Project of National Natural Science Foundation of China (Grant No.51635003); the National Natural Science Foundation of China (Grant No.51905057); the Natural Science Foundation Project of Chongqing, Chongqing Science and Technology Commission (Grant No.cstc2019jcyjmsxmX0050); the National Key Research and Development Program (Grant No.2018YFB1701203); the Fundamental Research Funds for the Central Universities (Grant No.2018CDXYJX0019) and the China Scholarship Council (Grant No.201906050173).

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Credit Author Statement Changjiu Xia: Methodology; Formal Analysis; Roles/Writing–Original Draft; Visualization. Shilong Wang: Conceptualization; Supervision; Funding Acquisition; Project Administration. Tan Long: Data Curation; Formal Analysis; Resources; Validation; Software. Chi Ma: Methodology; Writing–Review&Editing. Sibao Wang: Investigation; Writing–Review&Editing.

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Declaration of Interest Statement Article Title: Transmission chain error elimination for gear hobbing machine based on classified compensation theory and frequency response identification Authors: Changjiu Xia, Shilong Wang*,Tan Long, Chi Ma, Sibao Wang Corresponding author: Shilong Wang We declare that there are no financial or other relationships that might lead to a conflict of interest in the present article. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us. We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property. We understand that the Corresponding Author is the sole contact for the Editorial process (including Editorial Manager and direct communications with the office). He is responsible for communicating with the other authors about progress, submissions of revisions and final approval of proofs. We confirm that we have provided a current, correct email address which is accessible by the Corresponding Author. Professor Shilong Wang

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Highlights 

A new compensation method is proposed to eliminate TCE of gear hobbing machines.



TCE is decomposed into three types of error components for compensation.



Feasibility and effectiveness is verified on a YS3120-type gear hobbing machine.



TCE can be reduced by more than 80%, laying the foundation for precision hobbing.

23

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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