Wen-jun DENG，Kun YANG，Bing-qing CHEN，Jing-fei XIAO，Wei-gng ZHENG

aSchool of Energy and Power Engineering；bReliability Engineering Institute，School of Energy and Power Engineering；cEngineering Training Center，Wuhan University of Technology，Wuhan 430063，China

Research on the correction of eccentric error based on noncontact shaft torque measurement system*

Wen-jun DENG?a，Kun YANGb，Bing-qing CHENa，Jing-fei XIAOa，Wei-gang ZHENGc

aSchool of Energy and Power Engineering；bReliability Engineering Institute，School of Energy and Power Engineering；cEngineering Training Center，Wuhan University of Technology，Wuhan 430063，China

The main content of the online monitoring of operation statues of shaft is the measurement of shaft torque．For the method that the shaft torque is gotten by the non-contact measurement，the accuracy for measurement of torsional angle determines the correction of measurement result for shaft torque．In this paper，the eccentric error of original system is analyzed，and the effect of eccentric installment of the encoder on the time that the photo-electric sweeps sectorial photoelectric light pass hole is also analyzed．Then the specific mathematical expression of the time offset that the light pass the optical aperture is given by geometric analysis and motion decomposition．And the eccentric error of the measurement system is eliminated according to the analysis．The research of this paper will provide theoretical guidance for the design of torque measuring instrument，and it is helpful to improve measurement accuracy of this kind o f devices．

Sensor technology，Non-contact photo-electrical，Torque monitoring，Eccentricity， Error analysis，Correction

In the system that takes shaft as transmission parts，the working condition of shaft will determine the normal operation of the whole system.The torque is one of the most useful parameters associated with the shaft working condition，and it is also one of the most important performance parameters of power plant and the host［1-2］.By measuring the shaft torque，people can understand and detect the power plant，the host and working condition of other related equipments，then make detection and diagnosis for the related parts.

Many methods for shaft torque measurement derive against this background，including the shaft torque measuring methods about contact and non-contact［3-4］.In this paper，the author mainly discussed the measuring methods based on the photoelectric non-contact rotor torque.Photoelectric non contact rotor torque measurement system determines the rotating shaft torque using two encoders fixed on the rotating shaft with the emergence of the shaft torsion and torsion angle.The measurement system can complete non-contact measurement task effectively.It doesn’t contact with the shaft or affect the rotation of the shaft.Measurement system will not be affected with the rotation of the shaft changing in the environment.At the same time it can also make up for other non-contact measurement system’s influence of temperature drift and time drift.But the measurement has a fatal flaw in the system：The encoder eccentric in the process of assembly affecting the result of the measurement cannot be resolved.This article will analyze the measurement error caused by encoder eccentric in the system and put forward a feasible method［5］.

1．The original system measuring principle

Based on the photoelectric non-contact rotor torque measuring device is mainly composed of computer，controller，photoelectric encoder and optoelectronic switch［6-8］.The structure is shown in Figure 1.When the shaft rotates，due to the presence of load torque，torque，which is the same size but in the reverse direction，will produce on the rotation of the shaft section correspondingly.And with the different distance，the size of the torsion angle is also different.Therefore when the shaft rotates，a certain distance away from the two cross sections on the shaft produces relative rotation.Through the photoelectric switch，the two section’s relative torsion angleθcan be measured.So the rotating shaft torque M can be calculated according to the formula of material mechanics：

Where G is the shear modulus of elasticity of elastic axis（Pa）；θis the two section’s torsion angle（rad）；IPis the polar moment of inertia cross-section to the center of the circle（m4）；L is the distance between two cross sections（m）.

Figure 1.Photoelectric encoder installation diagram

After the shaft rotates，two encoders located on the rotating shaft sweep two photoelectric detectors respectively，and then two columns of phase differences of different square wave will be output.After processing，two relative torsion angle encoders can be gotten［9］.Specific process is as follows：each quadrant area on the encoder against the corresponding central angle isθ，and photoelectric detector outputting signal pulse width which is high level of square wave signal is L（In the original system L is taken for θafter quantifying）.The quantitative value is A．When the shafting is not loading，the initial angular deviation on the two encoders’corresponding shading rack isξ，and the quantitative value is a．When the shaft transmits torque，the relative rotation angle on the two encoders’corresponding shading rack isΔξ． And its quantitative value isΔa．So the encoder is used to calculate the shafting torsional angle such as formula（2）and（3）.

2．The original system of eccentric error analyzes

According to the principle of the design of original system，there are many errors，such as the counting quantization error caused by clock／counter，the error in the encoder installation process，and system inherent error of torque，speed characteristics and nonlinearity.This article mainly studies the error of encoders’installation.

The encoder installation error usually displays in two forms.One is that the encoder’s circle is not on the measured shaft’s centerline.It is called eccentric error.The other is that the plane the encoder existing is out of vertical of the measured shaft centerline.It is called no vertical error［10］.A wide tooling can be added to reduce the vertical error in the process of encoder installation.Here the encoder’s vertical error is no longer discussed.Then a further analysis will be carried out on the encoder eccentric error in the process of installation.Also a solution to its eccentric error will be discussed.

As shown in Figure 2，the encoders，which are installed according to the above methods，are considered to be installed vertically.Thus，the vertical error can be ignored，but the eccentric error is still exists.Moreover，based on the simplified model，the eccentric’s condition of one encoder installation is only considered.With the rotation of the encoder which exists eccentric error，the location that its sectorial light pass hole sweeps the specification of the photoelectric detector will vary with the encoder’s center relative to the changes of the shaft centerline’s position.The conditions for both two encoders existing eccentric error can be analyzed based on this case.

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Figure 2.Encoder eccentric error of installation diagram

With the encoder circle jumping up and down，the trajectory formed when photoelectric detector sweeps through the sector light pass hole of encoder changes constantly.If the distance between the photoelectric detector’s light and the shaft centerline is taken as radius，and the intersection point where this radius and the centerline of the rotating shaft intersecting at is taken as the center of a circle，a circle will be gotten.And the trajectory formed when the circle intersects the encoder’s sectorial light pass hole.Its real trajectory is shown in Figure 3：O1is the center of the shaft；O2is the center of the encoder；θis the angle of the sectorial light pass hole related to the center of the shaft.

Figure 3.The diagram of the trajectory formed when photoelectric detector sweeps through the sectorial light pass hole of encoder

3．Quantization of eccentric error of the system

First，in order to quantify the eccentric error，the authors put forward a set of basic program，namely：first，find out the relationship between the angle that the shaft turned and the distance which is between the shaft centerline and a fixed point on encoder which rotates with the rotating shaft；second，after the above relationship has been obtained，measure the distance between the shaft centerline and the fix point mentioned above in some way when the sectorial light pass hole of the encoder begins sweeping through the photo detector，then measure the distance between the shaft centerline and the fix point mentioned above when sectorial light pass hole of the encoder cannot pass light；third，calculate the amount of error based on the relationship obtained in the first step and the data in second step.

Due to the existence of the eccentric encoder，the distance between the shaft centerline and a fixed point on the encoder will change cyclically with the rotation of the shaft，now a point on the vertical axis O1O2（Figure 3）of the encoder is selected as the fixed point，then the authors will give the relationship between the angle that the shaft turned and the distance between the shaft centerline and the fixed point mentioned above.The relationship is as follow：

Where，δis the eccentricity of the encoder（mm）；R is the distance between the fixed point on the encoder and the center of the encoder（mm）；βis the rotated angle of the shaft（rad）.

Now in order to make the relationship more intuitive，specific value will be put to the parameters. Take the value of R to 93 mm，δto 1 mm，andβ ranges from 0 to 3π，and then use matlab to fit the result as the curve（Figure 4）.

Figure 4.The fitting curve

According to the relevant analysis，it is understood that this curve fits well with the actual rotation of the eccentric encoder.Next，the authors will explain the formula（4）further.

1）On the derivation ofδ，by observing Figure 4 it is found thatδis half of the difference between the peaks and troughs，thusδcan be gotten by measuring the correlation curve of the distance between the fixed point on encoder and the centerline of shaft through a certain way.

3）About analysis of the correlation between the time that the light passes through the sectorial light pass hole of encoder and the formula，first，formula（4）contains a well encoder eccentricity information，and the curve in Figure 4 reflects the beating of the fixed point on encoder when shaft is rotating well.And it is known that when photoelectric detector sweeps through the sectorial light pass hole of encoder，if it is assumed that the encoder does not turn but the light emitted by photoelectric detector cuts stationary photoelectric encoder，the cutting motion of the light can be substituted by synthesis of a rotational movement rotating around the encoder with a constant radius and a motion cutting the light pass hole of encoder from the top down，thus the trajectory formed when the photoelectric detector sweeps through the sectorial light pass hole of encoder is not a section of arc around the shaft centerline，but the situation shown in Figure 3.O2A and O2B in Figure 3 are the two sides of the sectorial light pass hole of encoder，arc AB is the trajectory formed when light sweeps through the sectorial light pass hole of encoder.According to formula（4），it is gotten that the beating of the fixed point on encoder reflects the angle information turned by the shaft，and there is a mathematical relationship of these two values，by decomposing the motion，as long as the time that photo detector cuts down the light pass hole of encoder is able to known（namely，if encoder is not rotating，the time that the light emitted by photodetector cuts the light pass hole of encoder from the top down），it will be able to get the time of another decomposition motion（namely the rotational movement rotating around the encoder with a constant radius）.And this time is the real time that light sweeps through the sectorial light pass hole of encoder.Figure 3 indicates that it is needed to obtain the time light from photodetector jumps from point A to point B．Back to formula（4），use the relationship between the angle that the shaft turned and the distance which is between the shaft centerline and the fixed point on encoder，at this point，as long as the distance between the fixed point on encoder and shaft centerline at point A，and also a distance at point B，is gotten，the angle that the shaft turned when encoder turned from A to B according to formula（4）can be gotten.And then it is able to know the time when light gets through the sectorial light pass hole according to the rotational speed of the shaft.

Based on the principle of non-contact shaft torque measurement system，it is needed to know the time that light sweeps through the sectorial light pass hole of encoder in the case of that the encoder is not eccentric.Thus on the basis of the original system，it is necessary to remove the eccentric error.According to the analysis above，the eccentric error needed to quantify is actual represented by formula（5）：

Where，Δt—the amount of the offset time caused by the eccentricity（s）；θ—angle of Photoelectric encoder sectorial light pass hole（rad）；α—angle between the vertical axis O1O2and O1A（rad）；φ—angle between the vertical axis O1O2and O1B（rad）；ω—The angular velocity of the shaft（rad／s）.

4．Conclusion

In this paper，the correction of eccentric error based on non-contact shaft torque measurement system has been researched，and the main reason of the cause of eccentric error，and the trajectory formed when photoelectric detector sweeps through the sectorial light pass hole of encoder，which is caused by the eccentric installation of encoder，have been presented.An indirect method is proposed on the measurement of the time that the light passes through the sectorial light pass hole of encoder，and the eccentric error is also eliminating based on the indirect method.Based on the results of the paper，the principles and methods elaborated in this paper can be specifically applied to eliminate the eccentric error，and improve the measurement accuracy of the device.Mainly note the following aspect：it must try to be ensured that the distance between the light emitted by the photoelectric detector and the shaft is measured with sufficient accuracy.

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摘要：針對模具的磨損問題，介紹了模具的磨料磨損機(jī)理，詳細(xì)討論了Archard磨損模型。建立了基于磨損計算的反擠壓三維有限元模型并進(jìn)行計算，得到不同初始硬度凸模的最大磨損深度規(guī)律。結(jié)果表明：模具最大磨損深度隨模具材料的初始硬度升高具有下降的趨勢，在變形區(qū)和速度穩(wěn)定的條件下，模具的最大磨損深度和初始硬度函數(shù)（1／H2）成正比關(guān)系。

關(guān)鍵詞：Archard；仿真；磨損；模具壽命；穩(wěn)態(tài)變形

中圖分類號：TQ330.4+1

非接觸式轉(zhuǎn)軸扭矩測量系統(tǒng)偏心誤差的修正研究*

鄧文軍?a，楊 琨b，陳冰清a，肖勁飛a，鄭衛(wèi)剛c

a.能源與動力工程學(xué)院
b.能源與動力工程學(xué)院可靠性工程研究所
c.工程訓(xùn)練中心，武漢理工大學(xué)，武漢 430063

軸系運(yùn)轉(zhuǎn)狀態(tài)在線監(jiān)測的主要內(nèi)容為轉(zhuǎn)軸扭矩的測量。對于基于非接觸式轉(zhuǎn)軸扭矩的測量方法，能否精確地測量軸系扭轉(zhuǎn)角度將決定扭矩和功率測量結(jié)果的準(zhǔn)確性。分析了原系統(tǒng)的偏心誤差，以及碼盤安裝偏心對光電檢測器掃過碼盤扇形通光孔的時間的影響，并通過幾何分析以及運(yùn)動分解給出了碼盤扇形通光孔的通光時間的偏移的具體數(shù)學(xué)表達(dá)式，以此消除測量系統(tǒng)的偏心誤差。研究結(jié)果對轉(zhuǎn)軸扭矩測量儀的設(shè)計提供了理論指導(dǎo)，有利于提高此類設(shè)備的測量精度。

傳感器技術(shù)；光電非接觸；扭矩測量；偏心；誤差分析；修正

U467.4+4

穩(wěn)態(tài)變形下Archard模型模具磨損數(shù)值分析*

胡建軍?1，2，侯天鳳1

1.重慶理工大學(xué)材料科學(xué)與工程學(xué)院，重慶 400054
2.汽車零部件先進(jìn)制造技術(shù)教育部重點(diǎn)實(shí)驗室，重慶 400054

10.3969／j.issn.1001-3881.2014.18.015

2014-04-15

*Project supported by the Students Innovation and Entrepreneurship Training Program（No.20131049705004）and the National Natural Sciences Foundation of China（NSFC）（No. 51309185）

?Wen-jun DENG，E-mail：2276386760＠qq.com