sensors Article

Dynamic Calibration and Verification Device of Measurement System for Dynamic Characteristic Coefficients of Sliding Bearing Runlin Chen *, Yangyang Wei, Zhaoyang Shi and Xiaoyang Yuan Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 41049, China; [email protected] (Y.W.); [email protected] (Z.S.); [email protected] (X.Y.) * Correspondence: [email protected]; Tel.: +86-137-2063-0034 Academic Editor: Gangbing Song Received: 20 May 2016; Accepted: 27 July 2016; Published: 30 July 2016

Abstract: The identification accuracy of dynamic characteristics coefficients is difficult to guarantee because of the errors of the measurement system itself. A novel dynamic calibration method of measurement system for dynamic characteristics coefficients is proposed in this paper to eliminate the errors of the measurement system itself. Compared with the calibration method of suspension quality, this novel calibration method is different because the verification device is a spring-mass system, which can simulate the dynamic characteristics of sliding bearing. The verification device is built, and the calibration experiment is implemented in a wide frequency range, in which the bearing stiffness is simulated by the disc springs. The experimental results show that the amplitude errors of this measurement system are small in the frequency range of 10 Hz–100 Hz, and the phase errors increase along with the increasing of frequency. It is preliminarily verified by the simulated experiment of dynamic characteristics coefficients identification in the frequency range of 10 Hz–30 Hz that the calibration data in this frequency range can support the dynamic characteristics test of sliding bearing in this frequency range well. The bearing experiments in greater frequency ranges need higher manufacturing and installation precision of calibration device. Besides, the processes of calibration experiments should be improved. Keywords: sliding bearing; dynamic characteristics; stiffness and damping coefficients; measurement system; calibration

1. Introduction Research on the dynamic characteristics of sliding bearing became serious in the 1980s [1,2], and it is generally understood that the dynamic characteristics of sliding bearing are important to the stability of the rotor [3]. Under the linear theory, the dynamic relationships between the motivation and response of bearings are usually described by stiffness and damping coefficients [4–7]. The dynamic characteristics measurement of sliding bearing aims to obtain the inner relationships between the motivation and response by analyzing the dynamical behavior data, which is the stiffness and damping coefficients of sliding bearings [8–10]. There are many methods to measure the dynamic characteristics, including time domain methods and frequency domain methods, such as dynamic excitation method, influence coefficient method, hammering method, harmonic scanning method, and so on [11–15]. The measurement systems are different, but they all include force sensors, displacement sensors, data collectors and other auxiliary components. Scholars at home and abroad have done a lot of theoretical and experimental research on the dynamic characteristics measurement of sliding bearing [16–20]. However, the test results are usually very different from the theoretical results [20–23], and the repeated accuracy of the test is not high. Sensors 2016, 16, 1202; doi:10.3390/s16081202

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The system incurs incurs test test errors errors itself. itself. Some shows that that the the 11°˝ The measurement measurement system Some theoretical theoretical research research shows phase error of displacement test will cause over 10% identification errors of stiffness and damping phase error of displacement test will cause over 10% identification errors of stiffness and damping coefficients [24]. So measurement system system is is very very important. important. There coefficients [24]. So the the calibration calibration of of the the measurement There are are two two types of calibration methods: static calibration and dynamic calibration. In the static calibration, types of calibration methods: static calibration and dynamic calibration. In the static calibration, the the functional relationships between input andsignals the output signals are given[25]. by functional relationships between the inputthe signals andsignals the output are given by experiments experiments [25]. The static calibration ofalmost different sensors are almost the same in The static calibration methods of different methods sensors are the same in principle [26–28], and the principle [26–28], and the calibrations of sensors are usually accomplished by the manufacturing calibrations of sensors are usually accomplished by the manufacturing factories. The high quality factories. The high quality testing elements have a smaller test error after calibration. only testing elements have a smaller test error after calibration. Generally, only amplitudeGenerally, data are given amplitude aredata givenare because the phase datatest aredata not are detected or the test data areof not accurate. because thedata phase not detected or the not accurate. This type calibration This type of calibration only aims at the sensor itself, and cannot eliminate the test errors of other only aims at the sensor itself, and cannot eliminate the test errors of other elements and error caused elements and error caused by connection of components [29,30]. In the dynamic calibration, multiple by connection of components [29,30]. In the dynamic calibration, multiple test channels are jointly test channels are jointly calibrated at the same time, and the test errors of the whole measurement calibrated at the same time, and the test errors of the whole measurement system can be eliminated to system can be eliminated to some extent [31–33]. The measurement system is calibrated by using the some extent [31–33]. The measurement system is calibrated by using the vibration of a single quality vibration of a single quality system in the calibration method of suspension quality [34]. However, system in the calibration method of suspension quality [34]. However, the movements of mass point the movements of mass point cannot simulate the bearing vibration, which causes the ranges of cannot simulate the bearing vibration, which causes the ranges of amplitude and frequency after high amplitude and frequency after high accuracy calibration to be small. The measurement system will accuracy calibration to be small. The measurement system will cause large errors when used as the cause large errors when used as the bearing test in a wide range of amplitudes and frequencies. bearing test in a wide range of amplitudes and frequencies. A new device for implementing the dynamic calibration method is proposed in this paper. It A new device for implementing the dynamic calibration method is proposed in this paper. It uses uses the movements of spring-mass system, and the movements can simulate the vibration of the the movements of spring-mass system, and the movements can simulate the vibration of the bearing bearing to some extent. Then the amplitude and frequency ranges of measurement system in to some extent. Then the amplitude and frequency ranges of measurement system in calibration calibration condition are basically the same with that in a working condition. A measurement system condition are basically the same with that in a working condition. A measurement system for dynamic for dynamic characteristic coefficients of the sliding bearing has been calibrated in this new characteristic coefficients of the sliding bearing has been calibrated in this new calibration device, and calibration device, and the calibration data are verified to be effective by simulation experiment. the calibration data are verified to be effective by simulation experiment. 2. 2. Identification IdentificationTheories Theoriesand andMethods Methods for for Dynamic Dynamic Characteristic Characteristic Coefficients Coefficients of of Sliding Bearing Sliding Bearing Taking theinversion inversiontest test bed of sliding bearing an example [35],structure the structure diagram is Taking the bed of sliding bearing as anasexample [35], the diagram is shown shown in 1.Figure 1. The rotor is supported by two rolling bearings, it can rotate the in Figure The rotor is supported by two rolling bearings, and it canand rotate around thearound horizontal horizontal axis. The test bearing a slidingisbearing) installed on It the rotor. only has the axis. The test bearing (which is a(which slidingisbearing) installedison the rotor. only hasItthe freedoms freedoms of translational in and horizontal vertical and directions, and other of of translational motions inmotions horizontal verticaland directions, other freedoms of freedoms motions are motions are by constrained by chains. There an oil film between and test The test constrained chains. There is an oil filmisbetween the rotor andthe testrotor bearing. The bearing. test components components are installed outsidepedestal, the bearing pedestal, through whichexcitation the dynamic excitation forces are installed outside the bearing through which the dynamic forces act on the test act on the testthe bearing. the and bearing vibrates, and so does theforces rotor transferred because ofthrough the forces bearing. Then bearingThen vibrates, so does the rotor because of the the transferred through the oil film. oil film.

inversion testtest bedbed of sliding bearing (getting rid of rid the of excitation device Figure 1. 1. Structure Structurediagram diagramfor for inversion of sliding bearing (getting the excitation for dynamic vibration). device for dynamic vibration).

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The dynamical dynamical model model of of the the inversion inversion test test bed bedsystem systemisisshown shownin inFigure Figure2.2.FF1,, FF2 are the exciting The 1 2 are the exciting forces in in two two directions directions respectively. respectively. X, X, Y Y are are respectively respectively the the absolute absolute displacements displacements of of bearing bearing in in forces two coordinate directions with respect to static balance position. X 1, Y1 are respectively the absolute two coordinate directions with respect to static balance position. X1 , Y1 are respectively the absolute displacements of of the the rotor rotor in in two two coordinate coordinate directions directions with with respect respectto tostatic staticbalance balanceposition. position.XX2,, Y Y2 displacements 2 2 are respectively respectively the the relative relativedisplacements displacementsofofthe thebearing bearinginintwo two coordinate directions with respect are coordinate directions with respect to to the rotor. the rotor.

Figure 2. Dynamical model of the test bearing system. Figure 2. Dynamical model of the test bearing system.

If the vectors and thethe bearing moves in vectors of of exciting excitingforces forcespass passthe thegeometric geometriccenter centerofofthe thebearing, bearing, and bearing moves ainplane, then thethe differential equations of motion forfor thethe system areare as follows: a plane, then differential equations of motion system as follows: $ « .. ff « ff « . ff « ff « . ff « ? ff « ? ff  X  2  2{2   2  ´ 2{2 ’ X X X X ’  2 2         ’  X 2 ` c X 2. .. ` k0 X ` c0 X  . `Xk F2 “ 2 ?   F1 ` 2 ? ’ & m Y m Y   k F2{2 F2 2{2  c    kY2   c  Y2   1     « .. ff  «Y  ff0 Y Y « 0. Yff « ff « ff Y (1)  2  Y2   2.   2 2  ’ X1 X1 X1 ´X2 ´X 22  (1) ’  ’ .. . . ` k1 ` c1 `k `c “0 ’ % m1 Y Y  ´Y ´Y2 Y  1 m  X 11  k  X 1   c1  X 1   k   X 2   c   X 2   20  Y   1 Y  1 Y1  1 Y  Y2  1  1     where k , c , k , c are all the 2 ˆ 2 matrixes, and2 k , c are the coupling stiffness and damping 0

0

1

1

0

0

coefficients. areallthe stiffness damping coefficients of rotor, and can be obtained where k0, c0, kk11, ,cc1 1are thesupporting 2 × 2 matrixes, and kand 0, c0 are the coupling stiffness and damping coefficients. by the stiffness and damping superposition of the two supporting bearings. k, c are stiffness k1, c1 are the supporting stiffness and damping coefficients of rotor, and can be the obtained by and the damping coefficients of bearing oil film. m is the mass of test bearing. m is the mass of rotor. 1 stiffness and damping superposition of the two supporting bearings. k, c are the stiffness and In Equation (1), the masses are and exciting and m displacements can be acquired damping coefficients of bearing oil known, film. m is thethe mass of testforces bearing. 1 is the mass of rotor. by sensors. So the eight stiffness and damping coefficients of the bearing can be solved theoretically by In Equation (1), the masses are known, and the exciting forces and displacements can be acquired only four groups forces and displacements data. However, in practice, experiment system by sensors. So theofeight stiffness and dampingtest coefficients of the bearing can the be solved theoretically is disturbed by environmental active disturbance forces, temperature drift of measurement system, by only four groups of forces and displacements test data. However, in practice, the experiment roundness of journal,byand so on, whichactive cause disturbance it to be difficult to identify the dynamic characteristic system is disturbed environmental forces, temperature drift of measurement confidents of sliding bearing in time domain conditions. system, roundness of journal, and so on, which cause it to be difficult to identify the dynamic Considering the periodicity near time-invariance these influencing factors, they can be characteristic confidents of slidingor bearing in time domain of conditions. eliminated by solving the equationorinnear frequency domain condition the Fourier transform. Considering the periodicity time-invariance of theseafter influencing factors, they can be $ ff equation « ? infffrequency«domain ff « ff eliminated by « solving the after the Fourier transform. ? condition ’ X pωq 2{2 ´ 2{2 Xpωq ’ 2 ’ ? F pωq ` F2 pωq ` pmω 2 ´ k0 ´ jωc0 q ’ & pk ` jωcq Y pωq  X“( ) ?2{2  2 / 21   2{2  X ( )  Ypωq 2 / 2 2 2 2 (k  j c)  «    ff  F1 ( )  «  F2ff( )+(m  k0  j c0 )   ’ ) 1pωq 2 / 2   ’  2X/ 22pωq  Y ( )  Yq2 (X 2` ’ p´m ω k ` jωc “ pk ` jωcq ’ 1 1 (2) %  1 Y 1 pωq Y pωq  X 1 ( )   X22 ( )   2 (2)   (k  j c)   ( m1  k1  j c1 )  Y1 ( )  Y2 ( )       The common methods for applying exciting forces include the single frequency excitation method and multi-excitation method.forIn applying the multi-excitation method, twothe sinusoidal exciting forces with The common methods exciting forces include single frequency excitation different frequencies are simultaneously on the test bearing in two vertical directions. Then the method and multi-excitation method. Inapplied the multi-excitation method, two sinusoidal exciting forces responses are plugged into the equation, and the Equation (2) is transformed as follows: with different frequencies are simultaneously applied on the test bearing in two vertical directions.

Then the responses are plugged into the equation, and the Equation (2) is transformed as follows:

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« « ? ff ff X 2 pω1 q{F1 pω1 q 2{2 Xpω1 q{F1 pω1 q 2 “  ? ` pmω1 ´ k0 ´ jω1 c0 q Y pω q{F Xpω X (1 ) FYpω 2 2 1 (1 )1q{F 1 pω1 q 1 c) 1 2 ( 11q) F1 (1 )  2{2 (k2  j  (m12  k0  j1c 0 )  ff « ?  ff « ff   (3) Y (  ) F (  ) Y (  ) F X 2 pω2 q{F 22pω21 q 1 1  ´ 2{2  2 q{F2 pω2 q 2 2   1 1 ( 1)  Xpω 2 ? (3) “ ` pmω2 ´ k0 ´ jω2 c0 q   X (2 ) Ypω  2 pω2 q F2 (22 )q{F  2 2 Y 2 pω2 q{F2Xpω 2 q2 ) F2 (2 )  2{2 2 ( 2   (m2  k0  j2 c 0 )    (k  j c)  Y2 (2 ) F2 (2 )   2 2  Y (2 ) F2 (2 )   The Equation (3) is the measurement equation of the multi-excitation method. The masses, Equation (3) is the measurement of the method. masses, the coupling The stiffness and damping confidents areequation all known. Themulti-excitation equation can be solvedThe by plugging coupling stiffness and damping confidents are all known. The equation can be solved by plugging amplitude ratio and the phase difference between the forces and displacements into it. $ « ’ ’ ’ ’ & pk ´ jωcq « ’ ’ ’ ’ % pk ´ jωcq

ff

the amplitude ratio and the phase difference between the forces and displacements into it.

3. Constitution and Dynamic Calibration Method of the Measurement System 3. Constitution and Dynamic Calibration Method of the Measurement System

3.1. Constitution of the Measurement System

3.1. Constitution of the Measurement System

According to Equation (3), the measurement system for dynamic characteristic coefficients of According to Equation (3), the measurement system for dynamic characteristic coefficients of sliding bearing in the inversion test bed should contain four displacement test channels and two force sliding bearing in the inversion test bed should contain four displacement test channels and two force test channels at least, which is is shown test channels at least, which shownininFigure Figure 3. 3. The The voltage signals of force sensors and displacement sensors are acquired DAQ voltage signals of force sensors and displacement sensors are acquired by DAQ by Card after Card afterbeing beingamplified amplified modulated, are sent to computer. the computer. Then the voltage signals or or modulated, and and are sent to the Then the voltage signals will be will be transformed signalsand anddisplacement displacement signals by software. test software. transformedinto into force force signals signals by test

Figure Constitutionof ofthe the measurement measurement system forfor sliding bearing. Figure 3. 3. Constitution system sliding bearing.

3.2. Error Propagation Analysis of Test Signals 3.2. Error Propagation Analysis of Test Signals From the Equation (3) and Figure 3, the two physical quantities of force and displacement should From the Equation (3) and Figure 3, the two physical quantities of force and displacement should be tested at first, and the force and displacement signals acquired are analyzed in frequency domain be tested at first, andthe thestiffness force and signalsofacquired arecan analyzed in frequency domain conditions. Then anddisplacement damping coefficients the bearing be identified by plugging conditions. Then the stiffness damping coefficients offorces the bearing identified by plugging the amplitude ratio and the and phase difference between the signals can and be displacements signals the amplitude ratio and the phase difference between the forces signals and displacements signals into Equation (3). So the ratio between the displacement signal and force signal is defined as the into Equation (3).function So the of ratio the displacement signal and force signal is defined as the transfer transfer the between sliding bearing.

function of the sliding bearing.

X  j  H    X pjωq H pωq “ F  j  F pjωq

(4)

(4)

where is the transfer functionofofsliding sliding bearing. bearing. ItItrepresents thethe relationships between the the where H(ω)H(ω) is the transfer function represents relationships between exciting force and displacement response of the bearing. The transfer function of sliding bearing is a exciting force and displacement response of the bearing. The transfer function of sliding bearing is plural flexibility, and its value is related to the frequency and amplitude of the exciting force. F(jω) is a plural flexibility, and its value is related to the frequency and amplitude of the exciting force. F(jω) is the real force signal, X(jω) is the real displacement response signal of the sliding bearing. the real force X(jω) the real response signal of the sliding bearing. into In thesignal, practical test is process, thedisplacement forces acting on the bearing by the exciter are transformed In the practical test process, the forces acting on the bearing by the exciter are transformed voltage signals by force sensors. These signals are usually slight, and are amplified by the signal into voltage signals by force sensors. These signals are usually slight, and are amplified by the signal

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amplifier before being being acquired acquired by by the the DAQ DAQ Card. Card. It It is is assumed assumed that that the the transformation transformation coefficient coefficient 1 pjωq and the real force signal F pjωq is H pjωq, which is defined as the between the test force signal F between the test force signal ( ) and the real force signal ( ) isF ( ), which is defined as transfer function of the test test channel. It is It shown in Figure 4. 4. the transfer function of force the force channel. is shown in Figure

F ( j )

H ( )

X ( j )

H F ( j )

G ( j )

H X ( j )

F '( j )

H '( )

X '( j )

conversion of of test test signals signals in in measurement measurement system. system. Figure 4. Error propagation and conversion

The transfer transfer function function of of the the force force test test channel channel is is related related to to the the materials materials and and structures structures of of the the The sensor itself, and is the intrinsic property of the sensor. So the test signals can be converted to the real sensor itself, and is the intrinsic property of the sensor. So the test signals can be converted to the real signals though though the the following following equation: equation: signals F '( j )  F ( j ) H F ( j ) (5) F1 pjωq “ FpjωqHF pjωq (5) In a similar way, the relationship between the real displacement signal ( ) and the test signal ′( In ) aofsimilar the sliding is as follows: way, bearing the relationship between the real displacement signal X pjωq and the test signal 1 X pjωq of the sliding bearing is as follows: X '( j )  X ( j ) H X ( j ) (6) 1 “ XpjωqHtest (6) X pjωq ( ) is the transfer functionXofpjωq displacement channel. where Then the transfer function of the sliding bearing is deduced by jointing the Equations (4)–(6): where HX pjωq is the transfer function of displacement test channel. X '( j ) H F ( j ) by jointing the Equations (4)–(6): Then the transfer function of the sliding H   bearing is deduced (7) F '( j ) H X ( j ) 1 X pjωq HF pjωq H pωq “ 1 ¨ (7) It is defined as follows: F pjωq HX pjωq

H ( j )  $ G  j  = X jq) & G pjωq “HHFX(p jω HF p jω q  q) X '(1 jjω %H   “ XF1 ppjω H 1' pωq  F '( jq) and the transfer function of sliding bearing is converted to: and the transfer function of sliding bearing is converted to: It is defined as follows:

H 1 pωq H pωq “ H '   H    G pjωq G  j 

(8) (8)

(9) (9)

where G pjωq is the relative transfer function between the force test channel and displacement test where ( ) is the relative transfer function between the force test channel and displacement test channel. H 1 pωq is the ratio between the test displacement signal and test force signal. channel. ′( ) is the ratio between the test displacement signal and test force signal. From Equation (9), if the G pjωq is known, then the transfer function of sliding bearing H pωq From Equation (9), if the ( ) is known, then the transfer function of sliding bearing ( ) can be acquired by the test signals F1 pjωq and X 1 pjωq. Theoretically, the transfer functions HYF pjωq can be acquired by the test signals ′( ) and ′( ). Theoretically, the transfer functions ( ) and HX pjωq of force sensors and displacement sensors in the measurement system are all known. and ( ) of force sensors and displacement sensors in the measurement system are all known. However, in fact, there are many other errors caused by elements and connection of components in the However, in fact, there are many other errors caused by elements and connection of components in test channels except for sensors. The transfer function of each test channel is no longer known, and the test channels except for sensors. The transfer function of each test channel is no longer known, cannot even be detected individually. The purpose of the dynamic calibration for the measurement and cannot even be detected individually. The purpose of the dynamic calibration for the system in this paper is to detect the relative transfer function G pjωq by uniting the force test channel measurement system in this paper is to detect the relative transfer function ( ) by uniting the and displacement test channel synthetically. After the G pjωq is confirmed, the transfer function of force test channel and displacement test channel synthetically. After the ( ) is confirmed, the transfer function of sliding bearing ( ) can be obtained by test signals ′( ) and ′( ) and

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sliding bearing H pωq can be obtained by test signals F1 pjωq and X 1 pjωq and Equation (9), and then the dynamic characteristic coefficients of sliding bearing can be identified by plugging H pωq into Equation (3). 3.3. Dynamic Calibration Method of Measurement System According the analysis above, the dynamic calibration of the measurement system is to acquire the transfer function G pjωq between the force test channel and displacement test channel, which usually proceeds by creating a calibration device with a known transfer function. It is assumed that the transfer function of the calibration device is H0 pjωq, on which the exciting force F pjωq is acted. The displacement is X pjωq. The force signal and displacement signal obtained by the measurement system are F1 pjωq and X pjωq. Then the transfer function between the test force and displacement is as follows: XpjωqHX pjωq Xpjωq HX pjωq X 1 pjωq “ “ ¨ “ H0 pjωq ¨ Gpjωq (10) H0 1 pjωq “ 1 F pjωq FpjωqHF pjωq Fpjωq HF pjωq where H0 pjωq is the transfer function of calibration device. It is a known physical quantity. H0 1 pjωq can be obtained by the test force and displacement signals. According to Equation (7), the transfer function of measurement system is as follows: Gpjωq “

H0 1 pjωq H0 pjωq

(11)

As the dynamic characteristic confidents test is carried out under frequency domain conditions, the transfer function and test signals are usually expressed by amplitude and phase. It is assumed that the test force and displacement signals obtained in calibration are as follows: #

F1 pjωq “ A F pωqe jpωt` ϕ F pωqq X 1 pjωq “ A X pωqe jpωt` ϕx pωqq

If the transfer function of measurement system is: Gpjωq “ Apωqe jp ϕpωqq , and the transfer function of calibration device is: H0 pjωq “ A H0 pωqe jp ϕ0 pωqq . According to the Equations (10) and (11), the transfer function of measurement system can be expressed as: Gpjωq “

H0 1 pjωq X 1 pjωq 1 A X pωq “ 1 ¨ “ e jp ϕx pωq´ ϕ F pωq´ ϕ0 pωqq H0 pjωq F pjωq H0 pjωq A H0 pωqA F pωq

(12)

So the amplitude and phase of transfer function of measurement system are as follows: #

Apωq “

A X pω q A H0 pω q A F pω q

(13)

ϕpωq “ ϕ X pωq ´ ϕ F pωq ´ ϕ0 pωq 4. Dynamic Calibration Device and Experiment 4.1. Dynamic Calibration Device The scheme of dynamic calibration device proposed in this paper is shown in Figure 5. The embedded loading technique is introduced into this scheme to act the static and dynamic forces on the rotor. The embedded loading technique is implemented by piezo-actuator, which can generate dynamic loads, such as harmonic, square wave and pulse forces.

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Figure 5. Scheme Scheme of of dynamic dynamic calibration device.

A is is installed onon thethe −45° direction of the The ˝ direction A set set of ofloading loadingdevices devicesofofpiezo-actuator piezo-actuator installed ´45 of test the bed. test bed. fixed pad is a slot with two grooves on the ±45° direction to set the disc spring components, of which The fixed pad is a slot with two grooves on the ˘45˝ direction to set the disc spring components, of 7 N/m order of magnitude. The main parameters of the disc spring are as follows: the stiffness is 106 ~ 10 6 ~107 N/m order of magnitude. The main parameters of the disc spring are which the stiffness is 10 the inner diameter is 25.4 mm, the outer diameter is 50 mm andisthe is 3 thickness mm. The is stiffness as follows: the inner diameter is 25.4 mm, the outer diameter 50 thickness mm and the 3 mm. 7 N/m. Two eddy current sensors are respectively set in of disc spring components is 6.2 × 10 7 The stiffness of disc spring components is 6.2 ˆ 10 N/m. Two eddy current sensors are respectively set horizontal and vertical directions. The loading in horizontal and vertical directions. The loadingforces forcesare aremeasured measuredby bythe theforce forcesensor sensor set set between between the tilting pad and the bearing base. The calibration device and the test points placement shown the tilting pad and the bearing base. The calibration device and the test points placement areare shown in in Figure 6. Figure 6. 24V DC Power source

DAQ Card

Terminal board

Piezo-actuator components Displacement sensor on Y direction

Force sensor

PC computer

Signal ampulifier

Signal generator

Displacement sensor on X direction

Magnetic meter base

Figure 6. 6. Calibration Calibration device device and and the the test test points points placement. placement. Figure

The dynamical model of calibration device is shown in in Figure Figure 7. 7. The sinusoidal excitation is shown generated by by piezo-actuator piezo-actuator acts acts on on the the rotor rotor through a loading pad. Because Because of of the the stiffness stiffness of the generated springs and the damping of the junction surfaces, the differential equation of rotor’s motion is disc springs as follows: « .. ff « . ff « ff « ? ff X X X 2{2 . M .. `C `K (14) “ 2 ´?  2{2 F Y  X  Y X   XY   2 M  C   K     (14) F   Y 2 where M is the mass of rotor; K is the Ystiffness on the direction of the exciting force; C is   Yof discsprings    2 the damping on the direction of the exciting force. where M is the mass of rotor; K is the stiffness of disc springs on the direction of the exciting force; C is the damping on the direction of the exciting force.

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Figure 7. 7. Dynamical Dynamical model model of of calibration calibration device. device. Figure

After Fourier transform: ff « ? ff  2 2{2  X pjωq  X  j   “ K ´ Mω 2 ` 2jωC (15) 2?  F jF pjωq  K  M   jC  YYpjωq (15)    ´ 2{2    j    2     2  So the transform functions between the exciting force and two displacements are as follows: So the transform functions $ between the exciting force and two displacements are as follows: 1 & HFX pjωq “ Xp jωq “ ? p jω 2pK´ Mω  X  jF q 1 2 ` jωCq (16)   “ Ypjωq“ ´ ? %H FXH jpjωq 1 F  Fj FY  q 2  K2pKM jCq   2 2` jωC p jω  ´ Mω (16)  Y  j  1  H are Their amplitudes and phases as follows:  j     FY   F  j  2  K  M  2  jC   $ $ 1 1 c ” c ” ’ ’ ı ı & A HY pωq “ & A HX pωq “ 2 2 2 2 2 pK´ Mω 2 q `pωCq2 pK´ Mωare q `p q Their amplitudes and 2phases asωC follows: ; (17) ¯ ¯ ´ ´ ’ ’ % ϕ pωq “ 180˝ ` arctan % ϕ pωq “ arctan ωC ωC 1  HX  HY Mω 21´K Mω 2 ´K  AHX     AHY    2 2 2   K  M  2  +   C  2  2 into 2 phases K M  2  +  C  the  Plugging Equation (17) Equation (13), amplitudes and of the transform       functions ;  (17)  between the force test channel and two displacement test channels of the measurement system for    C   C  arctan   HX   coefficients  HY    180+ arctan  dynamic characteristic  2are as follows: 2  M  K   M  K    $ » c ” ı ` ˘ ’ A X pω q ’ 2 2 ` pωCq2and Plugging Equation (17) into Equation (13), the amplitudes of the transform functions pωq “ A 2 K ´ Mω ¨ Aphases ’ — GX ’ F pω q ’ ’ – ´ ¯ between the force test channel and two displacement test channels of the measurement system for ’ ’ & ϕ as ϕGX pωq “ ´ ϕ F pωq ´ arctan MωωC X pωq 2 ´K dynamic characteristic » coefficients arec follows: (18) ı ”` ˘ ’ 2 A 2 Y pω q ’ 2 ’ — AGY pωq “ 2 K ´ Mω 2 ` pωCq ’ A¨ (A )pωq 2 ’ ’ –   AGX    2  K  M  2  +  C    X F ´ ¯ ’   ’   AF ( ) ωC % ϕGY pωq “ ϕY pωq ´ ϕ F pωq ´ 180˝ ´ arctan 2 Mω ´K   C   GX     X ( )   F ( )  arctan   2 4.2. Analysis of ExperimentResults Analysis  M  K    (18)  4.2.1. Damping Identification  of Calibration Device 2 A (  )   AGY    2  K  M  2  + C 2   Y There is low damping surfaces of disc In order to eliminate the   Asprings.   in the junction F ( )   influence of the damping on the calibration data, the frequency scanning test is implemented on       ( )   ( )  180  arctan  C   GY the calibration device. Theexciting forces 10 Hz–150 Hz act Y of frequency F  rotor by piezo-actuator. 2 on the  M  K   The amplitude-frequency curves of the rotor vibrations on the X and Y directions acquired though the measurement system are shown in the Figure 8. «

´

¯

There is low damping in the junction surfaces of disc springs. In order to eliminate the influence of the damping on the calibration data, the frequency scanning test is implemented on the calibration device. The exciting forces of frequency 10 Hz–150 Hz act on the rotor by piezo-actuator. The Sensors 2016, 16, 1202 9 of 13 amplitude-frequency curves of the rotor vibrations on the X and Y directions acquired though the Sensors 2016, 16, 1202 9 of 13 measurement system are shown in the Figure 8. 4.2. Analysis of Experiment Results Analysis 4.2.1. Damping Identification12of Calibration Device X direction

Amplitude of X/F (μm/N)

Amplitude of X/F (μm/N)

There is low damping in the junctionYsurfaces In order to eliminate the influence direction of disc springs. Half power Half power of the damping on the calibration data, the frequency point scanning testpoint is implemented on the calibration device. The exciting forces of8 frequency 10 Hz–150 Hz act on the rotor by piezo-actuator. The Q amplitude-frequency curves of the rotor vibrations on the X and Y directions acquired though the measurement system are shown in2 the Figure 8. 4 12

0

0

20

X direction Y direction 40 60

8

Half power 80 point100

Half power 120 point 140 160

Excitation frequency / Hz

Q 2 Figure 8. Amplitude-frequency curves of of the the rotor rotor vibrations Figure 8. Amplitude-frequency curves vibrations on on the the X X and and Y Y directions. directions. 4

Because the calibration calibration device device isis aa system systemofofhigh highstiffness stiffnessand andlow lowdamping, damping,there thereexists existsa asignificant significantresonance resonancephenomenon. phenomenon.The Thesystem systemdamping dampingcan canbebeidentified identifiedthough thoughEquation Equation(19): (19): 0

0

20

C  M 2  1  C “ 60 M pω80 2 ´ ω100 1q

40

120

140

160

(19) (19)

Hz the two corresponding frequencies of where M is the mass of rotor, and M = 7.32Excitation kg; ω1 frequency and ω2 /are where M is the mass of rotor, and M = 7.32 kg; ω 1 and ω 2 are the two corresponding frequencies of the half-power points on the curve. The system damping identified on the two directions are equal, the half-power points on the curve. Thecurves system damping identified two directions are equal, 8. –1 Amplitude-frequency of the rotor vibrations on on thethe X and Y directions. and C = 146Figure N/(m∙s ). and C = 146 N/(m¨s–1 ).

Amplitudes of transfer function AG Amplitudes of transfer function AG

the Data calibration device is a system of under high stiffness low damping, there exists a 4.2.2.Because Calibration of the Measurement System Differentand Frequencies 4.2.2. Calibration Data of the Measurement System under Different Frequencies significant resonance phenomenon. The system damping can be identified though Equation (19): The exciting forces of 10 Hz–100 Hz frequencies act on the calibration device, which is repeated The exciting forces of 10 Hz–100 Hz frequencies act on the calibration device, which is repeated  M 2 are  plugged into Equation (18) along with(19) five times. The values of mass, stiffness andCdamping the five times. The values of mass, stiffness and damping are1 plugged into Equation (18) along with the test forces and displacements. Taking an average of the five tests, the calibration data of measurement test forces average the five tests, calibration data offrequencies measurement where M isand thedisplacements. mass of rotor, Taking and M an = 7.32 kg; ωof1 and ω2 are thethe two corresponding of system under different frequencies are shown in Figures 9 and 10. system under different frequencies in damping Figures 9 identified and 10. on the two directions are equal, the half-power points on the curve.are Theshown system and CIn=the 146excitation N/(m∙s–1). frequency range of 10 Hz–100 Hz, the amplitudes of the transfer function are in the range of 1% ˘ 15%, and 1.3 the phases increase along with the increasing of excitation frequency. Force and under displacment on X direction The of the transfer function between force test channel and two 4.2.2.amplitudes Calibrationand Dataphase of thedifferences Measurement System Different Frequencies Force and displacment on Y direction 1.2 displacement test channels are small (amplitudes difference 5%, phases difference 5˝ ). The exciting forces of 10 Hz–100 Hz frequencies act on the calibration device, which is repeated In the frequency range of 10 Hz–30 Hz, the amplitude and phase of test results using this five times. The values of mass, 1.1 stiffness and damping are plugged into Equation (18) along with the measurement system need to be amended respectively less than 10% and 3˝ . The calibration data in test forces and displacements. Taking an average of the five tests, the calibration data of measurement this frequency range can support 1.0 the dynamic characteristics test of sliding bearing in this frequency system under different frequencies are shown in Figures 9 and 10. range well. 0.9 1.3 0.8 1.2 0

20

Force and displacment on X direction Force40and displacment on Y direction 60 80

100

Excitation frequency f / Hz 1.1

Figure 9. Amplitudes of transfer function of measurement system under different frequencies. 1.0

0.9

0.8 0

20

40

60

80

100

Excitation frequency f / Hz

Figure 9. 9. Amplitudes Amplitudes of of transfer transfer function function of of measurement measurement system system under under different different frequencies. Figure frequencies.

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Phases of transfer function G / °

15

Force and displacment on X direction Force and displacment on Y direction

10

5

0

-5 0

20

40

60

80

100

Excitation frequency f / Hz

Figure frequencies. Figure 10. 10. Phases Phases of of transfer transfer function function of of measurement measurement system system under under different different frequencies.

In the excitation frequency range of 10 Hz–100 Hz, the amplitudes of the transfer function are in It is important toand notethe that the excitation frequencies be equal to the frequency. corresponding the range of 1% ± 15%, phases increase along with theshould increasing of excitation The frequencies of the data points in Figures 9 and 10, when this measurement system is used to identify amplitudes and phase differences of the transfer function between force test channel and two the dynamic characteristic of a slidingdifference bearing. 5%, Moreover, test results displacement test channels coefficients are small (amplitudes phases the difference 5°). are amended by the corresponding amplitudes and phases in calibration data. Otherwise, other errors willthis be In the frequency range of 10 Hz–30 Hz, the amplitude and phase of test results using introduced by interpolation of calibration data, and the reliability of analysis results will decline. measurement system need to be amended respectively less than 10% and 3°. The calibration data in this frequency rangeExperiment can supportofthe dynamic characteristics test of sliding bearing in this frequency 4.2.3. Identification Dynamic Characteristic Coefficients Using the Calibration Data range well. In order to verify data of the measurement system, the simulated experiment It is important to the notecalibration that the excitation frequencies should be equal to the corresponding is implemented thepoints calibration device. of the sliding frequencies of theon data in Figures 9 andThe 10, dynamic when thischaracteristic measurementcoefficients system is used to identify bearing are simulated by the stiffness of disc springs and the damping of junction surfaces, and the the dynamic characteristic coefficients of a sliding bearing. Moreover, the test results are amended exciting forces act on the rotor by piezo-actuator, which forms a simulated erected test bed for the by the corresponding amplitudes and phases in calibration data. Otherwise, other errors will be dynamic characteristic coefficients of the sliding The dynamic model is shown in Figure 7, introduced by interpolation of calibration data, bearing and the[36]. reliability of analysis results will decline. and the differential equations of motion are as follows: 4.2.3. Identification Experiment the « .. ff «of Dynamic « Characteristic « ? Coefficients ff « Using ff Calibration Data . ff ? . ff X X 2{2 ´ 2{2 X ?the simulated .. calibration . m the `c ` k of the measurement “ ? F1system, ` F2 (20) In order to verify data experiment is 2{2 2{2 Y Y Y implemented on the calibration device. The dynamic characteristic coefficients of the sliding bearing are simulated by the the multi-excitation stiffness of disc springs damping ofequations junction are surfaces, and the exciting When using method,and the the measurement as follows: forces act on the rotor$by piezo-actuator, which forms a simulated erected test bed for the dynamic « ff « ? ff characteristic coefficients bearing [36]. The dynamic model is shown in Figure 7, and ’ `of the sliding ˘ X pω q{F pω q 2{2 ’ 1 1 1 2 ’ k ´motion mω1 `are jω1as c follows: “ ? ’ & of the differential equations Y pω1 q{F1 pω1 q 2{2 « ff « ? ff (21) ’ ` q{F pω q 2{2 X pω ´ .. 2 . ˘ ’ 2  2 2   2 ’     ? “ ’ X 2 ` jωX2 c  X  q{F % k ´ mω  2 2 q   2 2{2 2  2 pω m  ..   c  .   k Y pω F1   (20)    F2     Y 2 2    2   2  Y  Y  Acting the exciting forces of frequency 20 Hz and 30 Hz on the test bed respectively, the exciting forcesWhen and the displacements acquiredmethod, by the measurement system are shown 1. using the multi-excitation the measurement equations arein asTable follows:  forces and the displacements test by multi-excitation  2  Table 1. Exciting method. Physical Quantities Units Frequencies Amplitudes Phases

Exciting Force F 1 N 19.54 22.1 ´83.82

 X 1  F1 1    2   2  k  m1  j1c  Y  F      First   1  1  1    2  Second Excitation  Excitation   2 Displacement Displacement Exciting Displacement Displacement (21)   Direction on X on Y Direction Force F2 2on X on Y Direction  Direction X  F  2  k  m 2  j c    2  2  2     µm N  µm 2 2  µm  µm   Y F     2   2 2 2 29.33   19.6 19.53 29.32 29.32     2  16.95 4.54 4.72 90.87 17.52 ´83.43

´82.72

150.9

´27.85

154.56

Acting the exciting forces of frequency 20 Hz and 30 Hz on the test bed respectively, the exciting forces and the displacements acquired by the measurement system are shown in Table 1.

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Without considering the test errors of the measurement system, the test data of forces and displacements are plugged into the measurement Equation (18) of the erected test bed. The stiffness and damping coefficients of the simulated bearing are identified. After the measurement system is calibrated, the test errors of the measurement system are separated from the test data using the calibration data in Figures 9 and 10. Then the stiffness and damping coefficients of the simulated bearing are identified though the amended data. The comparisons between identification results and the given values are shown in Table 2. Table 2. Comparisons between identification results of dynamic characteristic coefficients and the given values. Simulated Dynamic Characteristic Coefficients

Units

Given Values

kxx kxy kyx kyy cxx cxy cyx cyy

(ˆ106 ) N/m (ˆ106 ) N/m (ˆ106 ) N/m (ˆ106 ) N/m N/(m¨s´1 ) N/(m¨s´1 ) N/(m¨s´1 ) N/(m¨s´1 )

4.09 0 0 4.09 146 0 0 146

Identification Results Not Calibrated

Error %

After Calibrated

Error %

3.724 ´0.055 ´0.075 3.598 ´347.463 167.316 343.945 ´866.497

´8.95 ´12.02 ´337.99 ´693.49

3.993 ´0.071 ´0.103 3.944 173.126 68.165 87.227 156.923

´2.36 ´3.58 18.58 7.48

According to the comparisons in Table 2, the identification results of dynamic characteristic coefficients are obviously different form the given values when the measurement system is not calibrated. The identification errors of principal stiffness are respectively ´8.95% and ´12.02%. The identified values of principal damping are negative, and the absolute value is 3–7 times the given values. These errors are too large for the research on dynamic behavior of sliding bearing, because they may cause a very big deviation of the research result. However, the identification results of principal stiffness using the calibration data of the measurement system are close to the given values, of which the identification errors are respectively ´2.36% and ´3.58%. The identification errors of principal damping are slightly larger than the given values, and the errors are 18.58% and 7.48%. These identification errors are in the permitted ranges. So it is verified that the calibration data of the measurement system in the frequency range of 10 Hz–30 Hz can support the dynamic characteristics test of sliding bearing well. 5. Conclusions (1)

For the measurement system for dynamic characteristics coefficients of sliding bearing, a novel dynamic calibration method by jointly calibrating multiple test channels is proposed in this paper. The calibration device contains a spring-mass system, which can simulate the dynamical characteristics of the sliding bearing. (2) The dynamic calibration device, including the piezo-actuator, force sensor and eddy current displacement sensor is designed and built. The dynamic calibration experiment in a wide frequency range simulating the bearing stiffness by disc springs is implemented. The experimental results show that the amplitude errors of this measurement system are small (less than ˘15%) in the frequency range of 10 Hz–100 Hz, and the phase errors increase along with the increasing of frequency. (3) The simulated experiment of dynamic characteristics coefficients identification is implemented on this calibration device using the calibration data of this measurement system in the frequency range of 10 Hz–30 Hz. The identification errors of principal stiffnesses are respectively ´2.36% and ´3.58%, and the identification errors of principal dampings are 18.58% and 7.48%, which are

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all far smaller than the identification errors without calibration. It is preliminarily verified that the calibration data in this frequency range can support the dynamic characteristics test of sliding bearing in this frequency range well. Acknowledgments: This work is supported by the National Basic Research Program of China (No.: 2015CB057303-2) and the National Natural Science Foundation of China (Grant No. 51275395). Author Contributions: Each co-author made important contributions to this research. Runlin Chen was responsible for the organization of the research and paper writing. Yangyang Wei joined in the design of calibration device and the experiment. Zhaoyang Shi joined in the experiment and collected the test data. Xiaoyang Yuan provided technical support and guidance throughout the project. All authors approved the final version of this paper. Conflicts of Interest: The authors declare no conflict of interest.

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