Int. J. Metrol. Qual. Eng. 12, 20 (2021)© K. Li et al., published by EDP Sciences, 2021https://doi.org/10.1051/ijmqe/2021017

International Journal ofMetrology and Quality Engineering

Available online at:www.metrology-journal.org

RESEARCH ARTICLE

Development of gear dynamic performance testing machineKe Li, Bo Yu* , Zhaoyao Shi, Zanhui Shu, and Rui Li

Beijing Engineering Research Center of Precision Measurement Technology and Instruments, Faculty of Materialsand Manufacturing, Beijing University of Technology, Beijing 100124, China

* Correspo

This is anO

Received: 16 April 2021 / Accepted: 8 July 2021

Abstract.With the development of gears towards high temperature, high pressure, high speed and high stress,gear measurement, in which only the static geometric accuracy is considered, is unable to meet the currentapplication requirements.While, the low precision and single function gear tester constrains the measurement ofgear dynamic performance. For the resolution of this problem, based on the principle of gear system dynamicsand several precision mechanical design techniques, a gear dynamic testing machine has been developed,providing new instruments for gear testing. On the basis of research of the principle of dynamic performance test,the primary measurement items of the testing machine have been determined. The measuring principles of eachitem and the driving and loading form of the testingmachine have been examined. Themeasurement and controlsystem of the testing machine and its corresponding software have been developed. The instrument can not onlyobtain the static precision index of the gear, but also obtain the dynamic performance index of the gear invariable working conditions. According to the actual test, the uncertainty of instrument is 3.8mm and theexternal disturbance caused by the shaft vibration is less than 0.6mm, which can meet the 5–6 grade precisiongear testing requirement.

Keywords: Gear / transmission error / gear test / gear dynamics / gear vibration and noise

1 Introduction

The advantages of gear transmission are accurate driving,range of application, high bearing capacity, long life, highreliability, high transmission efficiency, compact structureand much more. Gear transmission is the most widely usedof all mechanical transmission parts, which are essential tomanufacturing equipment. It has a wide range ofapplications in various industries [1–3].

The Gear measurement technology has been developedfor over 100 years. It can be classified into three categories:gear single geometric error measurement, gear comprehen-sive measurement and gear integrated error measurement[4]. These measurement techniques mainly focus on themeasurement of geometric accuracy of gears, which belongsto the static performance measurement of gears. However,the modern industrial production takes the development inthe direction of high precision, high speed and multiple-functions, so only inspecting the geometric accuracy of thegear cannot meet the requirements of gears in theconditions of high temperature, high pressure, and highspeed and high stress [5]. In particular, the currentrequirements on gear noise level are becoming stricter.

nding author: yubo@bjut.edu.cn

penAccess article distributed under the terms of the CreativeComwhich permits unrestricted use, distribution, and reproduction

The test of gear static geometric precision cannot reach theincreasing low noise requirement [6,7]. In general, thegeometrical errors of the gears are intertwined with eachother and interact with the operating conditions of thegears [8–10]. Therefore, it is necessary to predict thedynamic performance of gear during the design andmanufacture, and the dynamic performance of gear canbe detected before use.

Eventually the working speed of the gear transmissionsystem will be higher, and the vibration and noise of geartransmission system will be more optimal. Pairing ofinvolute cylindrical gears is the most convenient way tosolve the reasonable use of large amounts of product gears.It is also an effective way to control gear vibration and noisein order to improve gear transmission quality [11,12]. Atpresent, some gearbox manufacturers in Japan use gearsafter appropriate matching, and the comprehensiveevaluation of the gear design and processing is adoptedby detecting the effect of matching. Some gear manufac-turers in the China have similar approaches [13,14].Therefore, there is an urgent need for an instrument totest the dynamic performance of gear pairs before they areassembled in a gearbox [15–17].

Gear testing not only tests the conventional gear staticstrength, gear tooth bending fatigue strength, tooth surfacewear and gear bonding test, but also contains gear parameters

monsAttribution License (https://creativecommons.org/licenses/by/4.0),in any medium, provided the original work is properly cited.

Fig. 1. Gear dynamic performance testing machine.

Fig. 2. Torsional vibration model of gear pair.

2 K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021)

and other performance tests, such as gear efficiency, gearlubrication, gear dynamic load, gear noise, gear toothload distribution, tooth surface and gear body tempera-ture, and much more [18,19]. With different purposes,requirements and methods of the test, there are manydifferent forms of gear testing equipment. Their loadingmethods, operating methods and structural patterns areoften very different. These test equipment are generallydesigned for specific test purposes or meeting a certainrequirement, and the accuracy of some equipment isrelatively low.

In order to solve the problem of the dynamicperformance measurement of the parallel shaft gears, agear dynamic testing machine which can satisfy thestatic transmission error measurement and the geardynamic performance test has been developed accordingto the gear dynamics principle, as shown in Figure 1. Thetest results show that the instrument can meet therequirements of dynamic performance testing of 5–6grade precision gears.

2 Test principle

Gear dynamic excitation is affected not only by thestructure, geometry and error of the gear itself, but also byother components of the system [20].

Since the direction of calculating the transmission erroris along the meshing line, equation (1) mainly highlightsthe torsion vibration in the meshing line direction by using1-DOF model. The cylindrical gear system can besimplified as the torsional vibration system of the gearpair without considering the elastic deformation of thedrive shaft, supporting bearings, box and so on. Thedynamic model is shown in Figure 2.

According to the theory of mechanical dynamics andmechanical vibration, the torsional vibration equation of apair of gear pair is as follows:

Md2x

dt2þ c

dx

dtþ kx ¼ W0 þWd ð1Þ

In which,

x ¼ r1u1 � r2u2

M ¼ I1I2I1r22 þ I2r21

W0 ¼ T 1

r1¼ T 2

r2

Wd ¼ k1e1 þ k2e2 þ c1de1dt

þ c2de2dt

:

ui(i=1, 2) are torsional vibration angular displacementof the drive gear and the driven gear.Ii(i=1, 2) are rotaryinertias of the driving and driven gears. ri(i=1, 2) are baseradiuses of the driving and driven gears. ki(i=1, 2) is gearcomposite meshing stiffness of the ith gear tooth pair.ci(i=1, 2) is the damper coefficient of the ith gear toothpair. k(t) is the gear pair composite meshing stiffness. c isthe gear pair meshing stiffness damper. ei(i=1, 2) is the ithgear tooth pair error. Ti(i=1, 2) are the external torquesloaded on the driving and driven gears. So, the statictransmission error can be derived from equation (1) byomitting the dynamic items (inertial force items anddamping force items), which is shown as equation (2).

xs ¼ W0

kðtÞ þðk1e1 þ k2e2Þ

kðtÞ ð2Þ

And then, the dynamic transmission error xd can beexpressed as equation (3).

xd ¼ W0

kðtÞ þðk1e1 þ k2e2Þ

kðtÞ � c _x

kðtÞ þðc1e1 þ c2e2Þ

kðtÞ ð3Þ

When testing the dynamic performance of the gear pair,it is necessary to know exactly the external load momentapplied to the gear. When testing the static transmissionerror and the dynamic transmission error of the gear pair, itis essential to accurately know the rotation angles of thedriving and driven gear.

Fig. 3. Transmission error measurement principle.

K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021) 3

2.1 Transmission error measurement principle

Gear pair transmission error is the difference between theactual angle of the output end of the gear pair and itstheoretical angle [21], it is a process of continuouscomparison between the actual position and the theoreticalposition. The displacement synchronization comparisonprinciple is adopted during the measurement process,which is using the angle of one end of the part as abenchmark to examine the difference between the actualposition value of the other end and the theoretical value[22]. Gear single flank meshing measurement principle isshown in Figure 3. In order to do single flank meshingrolling with backlash, Gears are installed at the nominalcenter distance between the input and output shafts.Whenthe input shaft rotary angle is u1(t), the theoretical angle ofthe output shaft is u2(t). If the input and output gear teethare Z1 and Z2 respectively, u2(t) can be expressed asu2(t)= (Z1/Z2) u1(t) according to the gear transmission.Assuming the transmission error is u(t), in accordancewith the definition of transmission error, u(t) can beexpressed as:

uðtÞ ¼ u2ðtÞ � Z1

Z2u1ðtÞ ð4Þ

In general measurement, the circular grating is used toacquire the angular displacement information of thedriving and driven axis. The accuracy of the gratingsensor determines the accuracy of the transmission errormeasurement results. Considering the effect of eccentricity,the read head and the grating disc should be separated andbe installed within a certain extent, which can improve thespeed of the measuring process and meet requirements ofthe practical working conditions. This project adoptsRenishaw dual read head split-type grating.

2.2 Vibration measurement principle

When measuring vibration by contact, the sensor housing,attached to the vibration surface of the measured object,

will vibrate together with the measured object. In thesensor housing, the mass m is connected with the shellthrough the spring k and the damper c. The vibrationdisplacement of the measured surface together with theshell is recorded as y(t), the vibration displacement of themass m is x(t), and the relative motion between the massand the shell is z(t), which can also be expressed asz(t)= x(t)� y(t). For the whole vibration measurementsystem, the following differential equation can be deter-mined using the vibration system theory.

€zðtÞ þ 2jvn _zðtÞ þ v2nzðtÞ ¼ �€yðtÞ ð5Þ

in which z(t) could be written in oscillation ode,z(t)=A cos (vt+’); vn ¼ ffiffiffiffiffiffiffiffiffiffi

k=mp

; j= c/(2mvn).When the mass m is small and the stiffness k is

relatively large, vn will be very large. So the first twoterms of the equation (5) compared with the third one canbe omitted.

v2nzðtÞ ¼ �€yðtÞ ð6Þ

Thus, in this case, the vibration acceleration measuredby the electrical measurement device is proportional to thevibration acceleration of the measured object. Usually inapplications, the acceleration sensor is used to acquire thespeed of the measured object, and then the vibrationdisplacement of measured object is also known through theintegral method. The advantage of this method is that thequality of sensors is relatively small.

The developed gear dynamic performance testingmachine of this project belongs to precision measuringand testing device. It has high requirements on the staticbalance of the shafting. Therefore, when measuring thevibration displacement of the shafting system, it is requiredthat the quality of the vibration sensor used is as small aspossible so as not to affect the balance of the shafting.Considering the signal characteristics and mass propertiesof the displacement sensor and the accelerometer, thethree-axis piezoelectric acceleration sensor is used asthe transducer element. So, the vibration value of thethree directions can be obtained simultaneously throughone sensor.

The coordinate system of each component in thedriving unit is established as shown in Figure 4. Theabsolute coordinate system Sr(Xr, Yr, Zr) is establishedat the position where the center line of the driving unitguide rail intersects with the driven unit guide rail inorder to simplify the calculation. The coordinate systemSa(Xa, Ya, Za) of the driving unit box is established at thecenter of the end face closest to the gear of driving unitbox, and the directions are the same as the directions ofthe absolute coordinate system. At the same time, three-axis piezoelectric acceleration sensor is fixed according toSa(Xa, Ya, Za).

According to this method the coordinate systemSh(Xh, Yh, Zh) is established on the driving axis; the drivingunitfixture coordinate systemSs(Xs,Ys,Zs) is established onthe fixture end face of the fixture axis; the driving unit gearcoordinate system S1(X1, Y1, Z1) is established at the samepoint as the active unit fixture coordinate system.

Fig. 4. Coordinate systems of driving unit.

Table 1. Specifications of gear dynamics machine.

Items Requirements

Testing object Spur gear/helical gearModules 1–5Face width 0–80 mmPrecision grade 5–6Torque loaded 0–100N mSpeed 0–2000 r/minLubrication ControllableShafting run-out �3mm

Shafting parallelism �3mm

Testing function Static TE, loaded TE,dynamic TE, vibration, noise,efficiency

Working temperature 20±5°C

Fig. 5. The test device of power open.

4 K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021)

2.3 Torque measurement principle

When subjected to torsion, elastic axes are deformed withstress and strain and a shear stress on the cross-section.According to the principle of mechanics of materials, whenthe external deformation of the axis is pure torsion, therelationship between the cross-section of the shear stressand the shaft torque is shown as equation (7).

t ¼ M

Wpð7Þ

M is the torque on the shaft.Wp is the torsional modulusof the shaft section. Strain gauges are attached to theelastic axis to form a strain bridge.When the elastic shaft isdeformed by the torque, the resistance value of the bridge ischanged. After a series of changes, the torque of the elasticaxis can be measured. By amplifying the strain signal andconversing the voltage to frequency, the strain signalbecomes in proportion to the frequency. At present, thestrain sensor technology is the most widely used.

3 Mechanical system design

Through a comprehensive survey of the current gearmanufacture industry, the instrument’s main technicalspecifications is determined from various types of test itemsof the precision and reliability requirements, as shown inTable 1.

3.1 Structure type

As to whether the power transfer process of a gear testerconstitutes a closed system, the gear tester is often dividedinto two types. One is power flow closed and the other ispower flow open [23]. The general structure of the poweropen tester is shown in Figure 5. When the tester is loaded,all the required power is dissipated during the loading

process, and the energy loss is large. Generally, it is onlysuitable for small power and short time working condition.

Generally, when the designed testing machine is innormal work, the power is about 3 Kw and the test time ofsingle test is less than 5 minutes. So structure type of thismachine adopts the flow power open. According to themain technical specifications, related measurement princi-ple and the principles of precision instrument design [24],the basic structure of the testing machine is determined,shown as Figure 6.

In the whole structure, sensors, bearings and guide railsare standard parts, and all the other mechanical structuresare newly designed.

3.2 Loading equipment

After analysis of the gear tester loading method, there arethree ways, such as magnetic brake loading, the electricmotor loading and electro-hydraulic servo brake loading.While the electrically controlled motor loading and

Fig. 6. Structure of the testing machine.

Fig. 7. The relationship between torque and current.

Fig. 8. Precision rotary spindle.

Fig. 9. Grating installation instructions.

K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021) 5

electro-hydraulic servo brake loading are usually applied tolarger power occasions, in the low-power loading system,magnetic powder brake loading is universally used. It hasthe advantages of high performance, low price, fastresponse, simple structure, no impact, no vibration andno noise. It is mainly used in tension automatic controllablesystem and in the system for torque, speed and power test.It can also be used in the systems of buffering start,overload protection, adjusting the load, adjusting thespeed, and so on. Due to the hysteresis of the magnetoresistance, the saturation of the magnetic flux, the non-constant permeability, the input current and the outputtorque do not show an approximate linear relationship.When the input current increases linearly and decreaseslinearly, the actual torque loading characteristic curve isshown in Figure 7.

3.3 Shaft components

Precision shafting is one of the important parts of precisionmachinery. In mechanical design, precision shafting isrequired to have high rotation precision, enough rigidity,smoothly rotation flexibility and no blocking phenomenon.

According to the precision instrument design criteriaand the structure of the electric spindle, the designedprecision rotary spindle system structure is shown inFigure 8.

The precision spindle system consists of spindlehousing, precision bearings, bushings, lock nuts and othercomponents. The assembled shaft system is installed intothe spindle box. Precision shafting and the box areseparated not only to improve the accuracy of the spindlebut also to reduce the difficulty of processing and assembly.A high accuracy circular grating is used to measure theangular displacement. The double reading heads structureis adopted to reduce the measurement error caused byeccentricity of the shaft. The average value of the outputsignals of the two reading heads is regarded as themeasured value. The installation shown as Figure 9 is

Fig. 10. Gear installation instruction.

Fig. 11. Mechanical structure of testing machine.

Fig. 12. Testing machine control system.

6 K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021)

in line with “average error principle” in instrument design.The method of repeating measurement and averaging canimprove the accuracy of the instrument.

3.4 Gear mounting device

To be suitable for different tested gears, the spindle used tomount the gear is designed to be selected and replacedaccording to the size of the gear. So the concentricity of thespindle and the precision shafting is the key factor affectingthe performance of the instrument. With reference to thedesign principle of the machine tool spindle standardinterface, the connection between the spindle and theprecision shafting is in the form of a short taper shaft. Theorientation and centering of the short cone can be a goodguarantee of concentric accuracy. Bolted connection can beused to ensure the flexibility of assembly and adjustment ofthe shaft system accuracy. The expansion of the gearaffects the convenience of the test and the accuracy of theshaft. Especially, in the loaded condition, the size of thetightening force seriously limits the accuracy and reliabilityof the transmission error measurement. Considering theprinciple of simple operation and precise control, thedesigned gear mounting device is shown in Figure 10.

3.5 Measuring machine constitution

The testing machine is mainly composed of bottom base,precision rotary shaft, active shafting box, passive shaftingbox, mobile platform, measuring system and so on. Itsstructure is shown in Figure 11. The base is made of castiron. In order to ensure the smoothness of the movement ofthe box, the flat rail and the moving platform are driven incombination by the ball screw. The measuring system ismainly composed of round grating, torque sensor, magneticpowder brake to complete the gear dynamic performancetest on the testing machine.

4 Testing machine measurement and controlsystem

4.1 Principle of measurement and control

The testing machine control system hardware is mainlycomposed of the data collection system and data displaysystem. The motor is controlled through the PLC. Thedata acquisition system comprises circular grating sensors,subdivision devices, DSI cards, torque moment sensors,vibration sensors, noise sensors, data acquisition cards andthe industrial computer. The data display system com-prises a grating display, a torque display and a display ofthe industrial computer. The structure is shown in Figure 12.

Fig. 13. Software architecture.

Fig. 14. Testing machine precision of detection. (a) Parallelism detection. (b) Run-out detection.

K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021) 7

In order to realize the automatic measurement of geardynamic performance (static transmission error, dynamictransmission error, vibration and noise, etc.), the testsystem mainly consists of driving device, tested gear pair,loading device and detection device. The motor acts as adrive unit for the whole testing machine, providing powerto the tester. The power is transmitted to the loadingdevice (magnetic powder brake) through the detectiondevice (various sensor devices) and gears.

The control unit of the motor uses the control structureof combining PLC and servo drive with the speed closed-loop mode. The programmable power supply is used tocontrol the magnetic powder brake by the current controlmode. The round grating signal is collected by the datacollecting card of the Contac. The vibration and noisesignals are collected by the vibration and noise dataacquisition system, and the synchronization between thevarious cards is carried out by using the round grating dataacquisition card to send the triggering signal. When thetester is offline, it can be used as a gear rotation platform.

After entering the testing machine monitoring and controlsystem, the external button will be unavailable and the testand control software will be used to carry out the test.

4.2 Software system

Software is developed in VC++ which is the most popularWindows platform application development environment.The software system is divided into parameter manage-ment module, measurement on-site control module,comprehensive performance analysis module and usermanagement function modules, as shown in Figure 13.

5 Precision analysis and testing

The accuracy of the testing machine is mainly affected bymechanical system error, standard error and signalprocessing and algorithm error. The spindle run-out andinstallation eccentric is tested by using the dial gauge andstandard unit, as shown in Figure 14.

Table 2. Specifications of gear dynamics machine.

Error items References Error Uncertainty

Input axis radial runout 3mm 2.3mm 1.63mmOutput axis radial runout 3mm 2.8mm 1.98mmInput axis axial runout 3mm 2.2mm 0.11mmOutput axis axial runout 3mm 2.5mm 0.13mmInput axis eccentricity 3mm 1.8mm 1.23mmOutput axis eccentricity 3mm 2.3mm 1.67mmAccuracy of circular grating 3.200 3.200 0.89mmAccuracy of circular grating 3.200 3.200 0.89mmSub-division 1.6200 1.6200 0.08mmElectrical system stability 0.500 0.3500 0.25mmAlgorithm error 0.005mm 0.005mm 0.003mm

Table 3. The gear basic parameters.

Items Gear 1 Gear 2

Tooth number/z 38 56Module/m 2.25 2.25Helical angle/b 0 0Pressure angle/a 20 20Face width/b 25 30Variation Coefficient/x 0 0

Table 4. Gear tooth profile deviation.

No. fpt (mm) Rp (mm) Fp (mm) Q

1 4.6 22 18.5 52 1.8 0.7 26.2 5

Table 5. Test conditions parameters

Speed (r/min) Torque (Nm) Lubrication

10 0 no30 0 no60 0 no90 0 no

8 K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021)

The detection accuracy indexes, grating accuracy andsoftware and electrical indicators are acquired by compre-hensive testing, as shown in Table 2. The influencingfactors are independent of each other, and the correlationcoefficient is zero. Therefore, the influence on the totaldeviation of tangential synthesis is:

DF 0i ¼ ±

X11i¼1

ffiffiffiffiffiffiD2i

q¼ ± 3:54mm ð8Þ

The allowable value of the total deviation of thetangential integration of the 5-class accuracy (indexingdiameter 200mm, modulus 2.5, and pressure angle 20°) is30mm. The measured error is less than 1/3 (10mm) of thetotal allowable tangential total deviation. So the testingmachine can meet the requirements of five precision gearmeasurements.

6 Analysis of the test resultsA series of measurement tests are carried out on the geardynamics testing machine. The parameters of a pair ofgears are shown in Table 3, the tooth profile deviationvalues are shown in Table 4, and the test conditionparameters are shown in Table 5. The measured transmis-sion error curves are shown in Figure 15. The gear pairworking center distance is 153.14mm, and the transmissionratio is 1.47.

Three-axis piezoelectric acceleration sensor is fixedwith the screw on the front end of the bearing seat near thegear. The acceleration signal of the gear vibration isobtained through the vibration data acquisition system.When the test speed is 10 r/min, the signals of theacceleration of the driving gear in the x, y and z directionsare shown in Figure 16.

The time-domain statistical index of the vibrationsignal can be used to judge the operation state of theequipment, such as hidden trouble, the extent and thedevelopment trend of the trouble. Major statistical indi-cators used are average value, root-mean-square quantity,peak value pulse value margin value and kurtosis value.

In this paper, the average value, square value and kurtosisvalue are selected as the evaluation index for the operationcondition of gear testing machine.

The average value X is the first order momentstatistical average of the signal and is used to describethe stable component of the signal, which is the equilibriumposition of the mechanical vibration.

X ¼ 1

N

XNi¼1

xi tð Þ ð9Þ

Fig. 15. Gear transmission error curve. (a) TE at 10 r/min (b) TE at 30 r/min (c) TE at 60 r/min (d) TE at 90 r/min.

Fig. 16. Acceleration curve at n=10 r/min. (a) n = 10 r/min, x direction (b) n = 10 r/min, y direction (c) n = 10 r/min, z direction.

Table 6. Vibration acceleration signal statistical indica-tors.

x y z

Average value –0.01074 –0.03292 0.01819Square value 0.03448 0.02481 0.11403Kurtosis value 0.00277 0.00097 0.02518

K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021) 9

Square value is the second order moment statisticalaverage of the signal, and root-mean-square quantity isused to describe the vibration signal energy, which is oftenused to determine whether the state of operation isnormal.

X2rms ¼

1

N

XNi¼1

x2i tð Þ ð10Þ

Xrms is root-mean-square quantity. Kurtosis Cq is thefourth-order moment average of the signal, which is oftenused to characterize the impact vibration of the mechanicalduring the operation. The characteristic of index issensitive. Normally the value is 3. If the value is close to4 or more than 4, it means that there is impact vibrationduring working. Generally speaking, the reason is that thegap is too large, or the sliding surface is broken.

Cq ¼1

N

Xjxij � xð Þ4

X4rms

ð11Þ

According to the equations (9) to (11), the statisticalanalysis of the obtained signal is shown in Table 6. Amongthe indicators in the table, the mean value of the averagevalue of the vibration signal is 0.0206, the mean squarevalue is 0.0577, and the average kurtosis is 0.0096.Therefore, it can be shown that the shaft of the geartesting machine is stable, the vibration signal is very smalland the disturbance of vibration energy is very small. Sothere is no obvious impact during gear in normal meshingoperation.

Fig. 17. Displacement curve of time domain. (a) n = 10 r/min, x direction (b) n = 10 r/min, y direction (c) n = 10 r/min, z direction.

Fig. 18. Displacement curve of frequency domain. (a) n = 10 r/min, x direction (b) n = 10 r/min, y direction (c) n = 10 r/min,z direction.

10 K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021)

From the acceleration signal of the vibration, we cannotsee a significant amount of vibration displacement. So, it isunobvious whether the disturbance caused by vibrationaffect the normal detection. Therefore, using the numericalintegration method of discrete points, the accelerationsignal of vibration can be integrated twice and thevibration displacement signal can be obtained. Thevibration displacement signals in the three directionsare shown in Figure 17, which are integrated by using theequation (12). The integral formula shows that the integralof the discrete vibration signal can be the vibrationdisplacement signal.

y kð Þ ¼ DtXki¼1

xði� 1Þ þ xðiÞ2

ðk ¼ 1; 2; 3; ⋯ ;NÞ ð12Þ

From Figure 17, it can be seen that the maximum valueof vibration displacement is less than 0.6mm, while themeasured total deviation of the tangential direction of thegear is 30mm, so the effect of vibration on themeasurementresult is relatively small. The signal is shown as a cyclesignal, which is similar to the process of gear meshing.Further, the time domain signal is transformed into afrequency domain signal by using a spectrum analysistechnique, as shown in Figure 18.

In Figure 18, each frequency component of the signalhas a sharp peak at 50Hz or less, and the amplitude ofvibration is particularly obvious near 20Hz. In the geardynamic test system, the gear meshing is the majorexcitation source of the external vibration. When themeasuring speed is 10 r/min, the corresponding shaftrotation frequency is 0.17Hz and the meshing frequency is9Hz. So the testing process can be good to avoid theresonance effect caused by the external excitation.

As the measurement speed increases and the number ofgear teeth increases, the external excitation frequency maycause the resonance of the system. So it is needed to use areasonable speed to avoid the impact of resonance.

In order to analyze the influence of rotational speed onthe measurement results, the displacement curves of thevibration signals are shown in Figure 19 at the speed of 30,60, 90 and 120. In Figure 19, it can be seen that themaximum displacement of the vibration under variousrotational speeds is all below 0.6mm, which can meet theevaluation requirement of the test result.

7 Conclusions

–

From the previous analysis, the developed gear dynamicperformance testing machine based on the gear dynamicsprinciple can meet the requirements of gear dynamicexperiments, such as static transmission error, dynamictransmission error, vibration and noise. This providesnew equipment for gear test.

–

Through the analysis of the accuracy of the machine, theuncertainty is 3.54mm, which can meet the requirementsof 5–6 gear dynamic test.

–

The vibration amplitude of the shaft system is below0.6mm during the testing, which will not affect theaccuracy of test results. The increase of rotational speeddoes not cause the increase of vibration displacement,which indicates that the machine has good anti-vibrationperformance.

–

The testing machine’s low-order resonance frequency isin the vicinity of 20Hz. During the normal working speeddetection, the tester resonance will not happen. Byselecting and detecting the rotational speed, it is possibleto improve the dynamic accuracy.

Fig. 19. Vibration displacement curves at different speeds. (a) n = 30 r/min, x direction (b) n= 60 r/min, x direction (c) n = 90 r/min,x direction (d) n = 120 r/min, x direction.

K. Li et al.: Int. J. Metrol. Qual. Eng. 12, 20 (2021) 11

Thiswork is supportedby theNatural ScienceFoundationofChina(No. 51635001).

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Cite this article as: Ke Li, Bo Yu, Zhaoyao Shi, Zanhui Shu, Rui Li, Development of gear dynamic performance testing machine,Int. J. Metrol. Qual. Eng. 12, 20 (2021)