Scholarly article on topic 'Sensor selection of helicopter transmission systems based on physical model and sensitivity analysis'

Sensor selection of helicopter transmission systems based on physical model and sensitivity analysis Academic research paper on "Mechanical engineering"

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Chinese Journal of Aeronautics
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{"Condition indicators" / "Health indicator" / "Helicopter transmission systems" / "Mann–Kendall test" / "Physical model" / "Sensitivity analysis" / "Sen slop estimator" / "Sensor selection"}

Abstract of research paper on Mechanical engineering, author of scientific article — Kehong Lyu, Xiaodong Tan, Guanjun Liu, Chenxu Zhao

Abstract In the helicopter transmission systems, it is important to monitor and track the tooth damage evolution using lots of sensors and detection methods. This paper develops a novel approach for sensor selection based on physical model and sensitivity analysis. Firstly, a physical model of tooth damage and mesh stiffness is built. Secondly, some effective condition indicators (CIs) are presented, and the optimal CIs set is selected by comparing their test statistics according to Mann–Kendall test. Afterwards, the selected CIs are used to generate a health indicator (HI) through sen slop estimator. Then, the sensors are selected according to the monotonic relevance and sensitivity to the damage levels. Finally, the proposed method is verified by the simulation and experimental data. The results show that the approach can provide a guide for health monitoring of helicopter transmission systems, and it is effective to reduce the test cost and improve the system’s reliability.

Academic research paper on topic "Sensor selection of helicopter transmission systems based on physical model and sensitivity analysis"

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Lyu Kehong, Tan Xiaodong, Liu Guanjun, Zhao Chenxu

Sensor selection of helicopter transmission systems based on physical model and sensitivity analysis

S1000-9361(14)00095-8 http://dx.doi.org/10.1016/j.cja.2014.04.025 CJA 298

Received Date: 12 January 2013

Revised Date: 8 July 2013

Accepted Date: 22 August 2013

Please cite this article as: L. Kehong, T. Xiaodong, L. Guanjun, Z. Chenxu, Sensor selection of helicopter transmission systems based on physical model and sensitivity analysis, (2014), doi: http://dx.doi.org/10.1016/j.cja. 2014.04.025

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Sensor selection of helicopter transmission systems based on physical

model and sensitivity analysis

Lyu Kehong1'*, TAN Xiaodong12, LIU Guanjun1, ZHAO Chenxu1

1 Science and Technology on Integrated Logistics Support Laboratory, National University of Defense Technology, Changsha 410073r

2 Department of Electronic Technology, Officer's Colledge of CAPFf Chengdu 610213 ,China Received 15 June 2013; revised 8 July 2013; accepted 22 August 2013

Abstract

In the helicopter transmission systems, it is important to monitor and track the tooth damage evolution using lots of sensors and detection methods. This paper develops a novel approach for sensor selection based on physical model and sensitivity analysis. Firstly, a physical model of tooth damage and mesh stiffness is built. Secondly, some effective condition indicators (CIs) are presented, and the optimal CIs set is selected by comparing their test statistics according to Mann-Kendall test. Afterwards, the selected CIs are used to generate a health indicator (HI) through sen slop estimator. Then, the sensors are selected according to the monotonic relevance and sensitivity to the damage levels. Finally, the proposed method is verified by the simulation and experimental data. The results show that the approach can provide a guide for health monitoring of helicopter transmission systems, and it is effective to reduce the test cost and improve the system's reliability.

Keywords: Condition indicators; Health indicator; Helicopter transmission systems; Mann-Kendall test; Physical model; Sensitivity analysis; Sensor selection; Sen slop estimator

1. Introduction1

In the helicopter systems, transmission systems are the most efficient and compact devices used to transmit torque and change the angular velocity. The operating conditions of gears are very complicated, because they may encounter various problems, such as excessive applied torque, bad lubrication and manufacture or installation problems 1. Local tooth damage (e.g., tooth crack, pitting, breakage, etc.) occurs due to excessive stress conditions 2. As the damage level increases, the function of systems will be affected, and it will result in the final failure of the systems.

To minimize the loss that result from the interruption of production and high machine failure cost, it is necessary to monitor machine condition on-line using an effective condition monitoring system to provide timely information for condition-based maintenance (CBM) decision-making 3. Generally, condition monitoring for CBM involves the observation of machine condition using periodically sampled dynamic response measurements through massive sensors instrumented in the system and detection methods. Obviously, data or information obtained from sensors is the basis of CBM decision-making. In this paper, the meaning of "sensor" is developed, and it represents the available condition variables in physical model of a system (e.g., displacement, velocity, acceleration, temperature, pressure, flow rate, forces, etc.), condition monitoring techniques (e.g., vibration monitoring, acoustic emission4, temperature monitoring, fluid monitoring, corrosion, etc.), and accelerometers, thermocouple, oil

^Corresponding author. Tel.: +86-731-84574329. E-mail address: fhrlkh@ 163.com

sensor in different locations, etc 4-8.

Recently, studies have shown that traditional ways of simply adding sensors are impractical, and it will ultimately reduce systems' reliability and increase the monitoring cost 4-8. However, if the number of sensors is insufficient, the objective of condition monitoring cannot be achieved, and the false alarm and missed detection can be caused. Therefore, careful selection and implementation of sensors is critical to enable high fidelity system health assessment, improve the systems' reliability, and reduce the test cost on the basis of meeting the requirements of CBM 5.

In recent years, many researchers have paid more attention to sensor selection problems 5-12. National Aeronautics and Space Administration (NASA) has studied sensor optimization configuration technology for engine health management since 2005, and proposed a famous system sensor selection strategy (S4) 5, and the researchers also studied some experimental validation and verification for health monitoring and management of some aerospace systems such as turbo engine, RS-68 rocket engine 6-7. Cheng et al. studied sensor selection optimization for prognostics and health management (PHM) systematically, and proposed the state-of-art sensor systems for PHM and further discussed the emerging trends in technologies of sensor systems 8. Xu et al. proposed a fault tolerant sensor architecture and realized the architecture through the design of dual mode humidity/pressure micro electro mechanical system (MEMS)sensors with an integrated temperature function for health and usage monitoring 9. Novis and Powrie analyzed the characteristics of sensor systems used in real PHM environment in order to improve system diagnostic capability 10. Baer and Lally constructed an open standard smart sensor structure, and designed a sensor system for PHM n. Cheng et al. introduce a novel radio-frequency-based wireless sensor system for PHM,

and it includes a radio frequency identification sensor tag, a wireless reader, and diagnostic-prognostic software 12. Yang et al. proposed a sensor selection model by considering the impacts of sensor actual attributes on fault detectability 13.

The main purposes of above sensor selection methods are to provide data for fault diagnosis, detection or isolation. However, with the development of PHM theory and technique, besides meeting the above requirements, the information obtained from sensors must also provide useful data support for fault prognostics and health state assessment. There are many types of performance measures in fault prognostics and health state assessment, for example health monitoring performance (eg., the monitoring performance for fault growth, time to monitoring, etc.). To better improve those performance levels of fault prognostics and health state assessment, we should select sensors, which maximize sensitivity for crack evolution process, i.e., crack growth in different components should, as soon as possible, be able to be tracked or monitored effectively, and it means that the crack evolution trends described by sensors have a better monotonic characteristic. Previous works have demonstrated that constructing a physical model including damage levels and selecting the better sensitive sensors to damage growth so as to track the damage evolution are essential to improve the performance of PHM7, 8. So, a physical model of crack tooth needs to be build to better analyze the effect of different crack levels on those variables in the model, thus we can obtain the crack evolution process, which is described by those variables that can be monitoring by using corrective sensors. Hence, this paper proposes a sensor selection technique based on physical model and sensitivity analysis. We take the tooth crack as an example, and the relation between the crack levels and the reduction of mesh stiffness is built. Some condition indicators (CIs) are presented to describe the crack evolution trend, and some optimal CIs having better monotonic trend with damage levels are selected using Mann-Kendall test method. The selected CIs can generate a health indicator (HI) indicating the damage level and the HI trend of sensors with damage growth will be derived. The sen slop estimator is used to calculate the sensitivity of each sensor to damage evolution, and then the optimal sensors can be selected.

The rest of this paper is organized as follows. In Section 2, the physical model of tooth crack is developed. In Section 3, the sensitivity of sensors to damage growth is developed to assist in selecting the optimal sensors. In Section 4, a simulation data of a one-stage gearbox and experimental data provided by mechanical diagnosis test bed (MDTB) of Applied Research Laboratory (ARL) at Pennsylvania State University are used to verify the effectiveness of the method proposed in this paper. Finally, this paper concludes with a summary and future research direction in Section 5.

2. The physical model of crack tooth

The stiffness of crack tooth is found to be decreased proportionally to the severity of the crack 2. In order to build the mesh stiffness models of crack tooth, many researches have been carried out. Finite elements method (FEM) is the most popular tool applied to this 14. In FEM, the higher solution accuracy of FEM relies on more mesh refinement, but the FEM model including more mesh refinements is computationally expensive and is very difficult to build in certain applications. Moreover, The FEM does not give precise details about when the stiffness reduction occurs and how much the various damage levels are correlated to the stiffness reduction, while such information is very important to correctly construct the model of damage dynamics of various damage levels. Choy et al. developed an analytical model that the effect of surface pitting and wear of the gear tooth were simulated by phase and magnitude changes in the gear mesh stiffness 15. Liu et al. developed a quasi-static nonlinear mesh gear model that includes effects associated with tooth crack on the vibration response of a one-stage gearbox with spur gears, and a lumped parameter model was used to simulate the vibration response of the pair of meshing gears 16. Chaari et al. developed an analytical approach to quantify the gear mesh stiffness reduction of spalling or tooth breakage 2. The above research results show that the analytical methods offer satisfying results, good agreements and less computational time than FEM. So, through combining the advantages of the analytical approach, we take the tooth crack as an example and introduce an analytical approach to construct its physical models.

2.1. Mesh stiffness model of a healthy tooth

The mesh stiffness model of a healthy tooth is shown in Fig. 1.

Fig. 1 Mesh stiffness model of healthy tooth.

Based on the beam theory, the shear stiffness ks, the axial compressive stiffness ka and the bending stiffness kb are calculated by: 16,17

1 = 1 ^

■ 1.2cos2 am

K J0 y

1 ryFsin2 a„

0 EA„

,[(yF - y )cosam - xF sin am]2 EL,

A-b —y

where an, x, y, dy, xF, yF, are shown in Fig.1. E is the Young modulus, G represents the shear modulus, Iy and Ay are the moment of inertia and area of the section, the distance between the section and the acting point of the applied force is y, and they can be obtained by:

Ay = 2 xW Iy = ^(2x)3w

2 (1 + v)

where W is the whole tooth width, and v the Poisson's ratio.

The fillet-foundation deflection is computed by using the theory of Muskhelishvili applied to circular elastic rings, which assumes linear and constant stress variations

at root circle F cos am

. This analytical expression is given by [L*(f2 + M•(f) + P*(1 + Q* tan2 am)]

where ¿>f is the fillet-foundation deflection of a tooth, and B the tooth thickness, uf and Sf are defined in Fig. 1. The coefficients L*, M*, P*, Q* approached by polynomial functions, and the related parameters can be found in Ref.19: X*(hfi ,0f) =

A / Bf2 + Bihf2 + Cf I Bf + Dt I Of + Eihfi + F where the values of Ai,Bi,CiDi,Ei and Ft are given in Table 1, hfi=rfIrint and Of are defined in Fig. 2.

Geometrical parameters for the fillet-foundation deflection.

The corresponding fillet-foundation stiffness can be

From the results derived by Yang and Lin 2o, the stiffness of Hertzian contact of two meshing teeth (commonly nonlinear) is practically a constant along the entire line of action independent from both the position of contact and the depth of interpenetration. The stiffness of Hertzian contact of two meshing teeth kh can be approximated by a constant value depending on the tooth width and the mechanical properties of the gear material:

nEB (10)

4 (1 -u2)

For a pair of health teeth in contact the mesh stiffness ko can be written as

ko = 1I

t {- + - + - + - ] + -

< i, i, i, i, i,

v i=1 1

where kbi, ksi, k,ii, kfi are the bending stiffness, shear stiffness, axial compressive stiffness and fillet-foundation stiffness of ith tooth, respectively.

The value of stiffness kd corresponding to two pairs in contact is as follows:

can be related

(8) is d

k (t) =

b2,i s2,i a2,i f2,i

In one mesh cycle, the time-varying mesh stiffness k(t) is defined as follows:

ko (n - 1)Te < t < (Cr -1)nTe kd (Cr - 1)nTe < t <(2- Cr )nTe

where Te is the mesh cycle, n=0,1,2,...., and Cr is the average number of pairs of health tooth in contact between two gears.

2.2. Mesh stiffness model of tooth crack

Tooth crack is one of the main failure modes in helicopter transmission systems, and it will result in various damages of a tooth. Unlike those that occur within individual gear tooth, fractures that expand through the gear rim may lead to the catastrophic loss of the transmission systems, and seriously compromise aircraft safety. Fig. 3 shows the model of tooth root crack. In the figure, bc is the crack length along tooth thickness. Here, we assume that the crack goes through the whole tooth width. Obviously, the tooth crack will result in the reduction of tooth thickness, thus reduces the total mesh stiffness.

obtained by:

Table 1 Values of the coefficients of Eq. (8)

A, (10-5) Bi (10-3) Ci (10-4) Dt (10-3) E, Fi

L*(hfi,B) -5.574 -1.9986 -2.3015 4.77021 0.0271 6.8045

M*(hfi,O) 60.111 28.100 -83.431 -9.9256 0.1624 0.9086

P*(hfi,B) -50.952 185.50 0.0538 53.300 0.2895 0.9236

Q*( hfi,Bf) -6.2042 9.0889 -4.0964 7.8297 -0.1472 0.6904

Addendum circle

líase circle

D.xL'J'i circlt:

Fig. 3 Model of tooth root crack.

The damage level of tooth crack X is defined as: X-bj B (14)

For tooth crack, the Iy and Ay of teeth can be obtained

(X + xc -bc)W 2 xW

0 < y < yc y > yc

1 3 —(X + Xc -bc)3W 0 < y < yc

1 3 -(2x)3W y > yc

The bending stiffness, shear stiffness, compressive stiffness are affected by the crack level and the bending stiffness of a crack tooth kcb, the shear stiffness of a crack tooth kcs, and the axial compressive stiffness kca of a crack tooth can be calculated as follows:

r yF [(yF - y) cos a - Xf sin am ]2

-1' Jo'

. , . fF 1.2cos' kcs - 1' J(

kca = J

r yF sin a

The fillet-foundation stiffness and the stiffness of Hertzian contact do not vary with the increase of crack level according to Eqs. (7)-(10).

k c is defined as the reduction of mesh stiffness due to tooth crack, and it can be obtained by: 1 -X

th crack, ; starts to

1'kcb + 1'kcs + 1'kca

ed on the above analysis, once the crack tooth to mesh, the magnitude of mesh stiffness will reduce with the increasing crack level according to Eq. (17). The mesh stiffness due to tooth crack will be incorporated into the overall system model for simulation analysis. Then, the dynamic response with various damage levels can be obtained, which will provide the simulation data including crack level information for case study.

3.1. Optimal selection of condition indicator set u sing Mann-Kendall test

In a real system, the original signal measured by sensors installed on the transmission systems includes complex background noise. Researchers have developed many CIs to extract the characteristics related to damage levels from the original signal. These indicators process the vibration signal and return a single value indicating its overall health, such as root mean square (RMS) 22, peak ratio (ER), a new statistical moment, Sa 23, zero-order figure of merit (FM0), FM4 24, NA4 25, and so on.

Among the above CIs, not all CIs contain useful information on the gear damage condition. In order to detect incipient tooth damage in time and track the tooth damage evolution process, an effective CI must have the following characteristics: it could change monotonically with the increase of the tooth damage level; and it is supposed to be sensitive enough to track tooth damage growth.

The Mann-Kendall test can be applied to evaluating whether a CI tends to increase or decrease over time through what is essentially a nonparametric form of monotonic trend regression analysis. The Mann-Kendall test analyzes the sign of the difference between later-measured data and earlier-measured data. Each later-measured value is compared to all values measured earlier, resulting in a total of n(n-1)/2 possible pairs of data, where n is the total number of observations. Missing values are allowed and the data do not necessarily conform to any particular distribution. The Mann-Kendall test assumes that a value can always be declared less than, greater than, or equal to another value; that data are independent; and that the distribution of data remains constant with either the original units or transformed units 26. Because the Mann-Kendall test statistics are invariant to transformations such as logs (i.e., the test statistics will be the same value for both raw and log-transformed data), it is applicable in many situations.

To perform a Mann-Kendall test, we should compute the difference between the later-measured value and all earlier-measured values, CIy-CI;, where j>i, and assign the integer value of +1, 0, -1 to positive difference, no difference, and negative difference, respectively. The test statistic (TS) is then computed as the sum of the integers:

TS - XZ sign(CI j - CI, )

i -1 j-i+1

where sign(CI;-CI;) is equal to +1, 0, or -1 as indicated above.

When TS is a large positive number, later-measured values tend to be larger than earlier values and an upward trend is indicated. When TS is a large negative number, later values tend to be smaller than earlier values and a downward trend is indicated. When the absolute value of TS is small, no trend is indicated. The test statistic T can be computed as

3. Sensor selection based on sensitivity analysis

T =-—--(22)

n(n -1) 12

T has a range of -1 to +1 and is analogous to the correlation coefficient in regression analysis and represents the monotonic level of a trend. When T is equal to -1, the monotonic level of the trend is most significant and the total trend is downward. When T is equal to 1, its monotonic level is also most significant while the total trend is upward. And when T is equal to 0, no trend is indicated.

By calculating the test statistics of each CIs' trend, we can select the optimal CI set, which can indicate the monotonic level with tooth crack growth.

Generally, the monotonic characteristics of CIs in crack evolution process for simulation study are clear without the influence of external environment (e.g., noise, vibration, temperature, etc.), and most of the test statistics of CIs are greater than 0.5, so we choose the CIs whose test statistics are greater than 0.5 in the simulation study in a trade-off among those test statistics. However, the crack characteristics in actual application will be affected by noise, vibration, temperature, etc, and most of CIs' test statistics are less than 0.5, so we choose the CIs whose test statistics are greater than 0.2 in practice in a trade-off among those test statistics.

3.2. An HI calculation based on optimal CI set

In condition monitoring, generating an indicator that varies monotonically with damage levels is desired. This indicator represents the health information of a system or component.

The selected CIs can generate a health indicator (HI) so that the operator has confidence that an alarm indicator requires maintenance 26. This is the first step into a condition-based maintenance practice. The component HI is calculated from n number of CIs using norm energy. If CIs represent a metric such as shaft order acceleration, then one can construct an HI, which is the square of normalized power (e.g., square root of the acceleration squared) 27. This can be defined as normalized energy, where the health index is:

HI = V#cov(#)-1#T (23)

where <P is the vector of the selected CIs.

By calculating the HIs of tooth damage evolution process, we can obtain the HI trend with the development of tooth damage levels.

3.3. Sen

For each sensor, we can obtain the optimal CI set including useful damage growth information and calculate its HI indicating the damage of a system or component. Then, a measure is thus needed to evaluate the sensitivity of a sensor to damage evolution. In this paper, we adopted the rate of change of HI trend to measure the sensitivity. If a significant trend is found, the rate of change can be calculated using the sen slope estimator 26:

P = median

HI, - HI,

3.3. Sensitivity analysis

where t1 and t2 represent the discrete observation time in the damage evolution process, for all t1< t2 and t1 = 1, 2, ..., n—1 and t2 = 2, 3,..., n; dt1 and dt2 represent the damage levels of a system or component at t1 and t2, respectively; HIt1 and HIt2 represent the health indicator of a system or component at t1 and t2, respectively. The slopes for all pairs of data are used to compute TS. The median of those slopes is the sen slope estimator. Here, fi can be used to measure the sensitivity of the sensor to tooth crack.

3.4. Scheme y analysis

of sensor selection

n based on

on sensitivit

The flow of sensor optimization selection based on sensitivity analysis is shown in Fig. 4.

(1) The simulation or experimental crack data including various crack levels can be got from available monitoring parameters in the simulation model or sensors instrumented in test bed, respectively. The model can support to better analyze the effect of different crack levels on those variables in the model, and thus we can obtain the crack evolution process, which is described by those variables that can be monitored by using corrective sensors.

(2) For each sensor or monitoring parameter, we calculate the trends of CIs with damage growth and select the useful CIs having the better monotonic trend with damage levels using Mann-Kedall test.

(3) By normalized energy, a health indicator (HI) indicating the damage of a system or component from the selected CIs can be generated, and its trend with damage growth can be obtained. Fourthly, the sensitivity of each sensor is calculated from its HI trend using sen slop estimator.

(4) We can select the effective sensors having the better monotonic relevance with damage growth and more sensitive to damage evolution.

Fig. 4 Flow of sensor optimization selection based on sensitivity analysis.

4. Case studies

4.1. Simulation case

The simulation data including crack information can be derived by a simulation example of a one-stage gearbox system 28. The model is given in Fig. 5. It is a two-parameter (stiffness and damping) model with torsional and lateral vibration. That is to say, it includes both the linear and rotational equations of the system's motion. This model represents a system with six degrees of freedom, which is driven by electric motor moment and loaded with external moment. In this paper, we assume that all gears are perfectly mounted rigid bodies with ideal geometries. Inter-tooth friction is ignored here for simplicity.

I i k I t

p 1 "2

J»— J№-/ft.

1 1 1 1,

rtt -33—33- —33— -33—33-

Co C1 c c, c

(b) A model of the gear system Fig. 5 A one-stage gearbox systei

The following notations of this mod

I are used in

paper. Six general displacement vectors are: q=[y1, y2, 0, 02, 0n, 0>, yi/y2] is the linear displacements of pinion/gear in the y direction, 0/0 the angular displacements of pinion/gear, 0/0 the angular displacements of motor/load, T1/T2 the input motor torque and output torque from load, mi/m2 the masses of the pinion/gear, k(t) the total mesh stiffness of health tooth, kp/kg the torsional stiffness of input/output flexible coupling, k1/k2 the vertical radial stiffness of input/output bearings, c(t) the mesh damping coefficient, cp/cg the damping coefficients of input/output flexible coupling, c1/c2 the vertical radial viscous damping coefficients of input/output bearings, /m//b the mass moment of inertia of motor/load, /1//2 the mass moment of inertia of pinion/gear, ^b1/^b2 the base circle radius of pinion/gear.

The mesh stiffness models due to crack tooth are incorporated into the global dynamic model. The dynamic damage model of the gearbox system can be obtained by

Mq + Cd(t )q + Kd(t )q = F (t )

where M is the mass matrix, Kd(t) and Cd(t) represent the mesh stiffness and damping coefficient of damage tooth, respectively, and F(t) is the external force vector.

Cd(t) =

K (t) =

Cd(t ) + Ci -Cd(t) -Cd(t )Rbi Cd(t )Rb2 0 0

Cd(t) C2 - Cd(t ) ► -Cd(t )Rbi Cd(t )Rb2 0 0

- Cd(t )*bl Cd(t )Rbi Cp + Cd(t )Rb2i -Cd(t )RbiRb2 -cp 0

Cd(t ) Rb2 -Cd(t )Rb2 -Cd(t)RbiRb2 Cg + Cd(t )Rb22 0 - c

0 0 -Cp 0 cp 0

0 0 0 -Cg 0 cg

kd(t ) + ki -kd(t) -kd(t )Rbi kd(t )Rb2 0 0

kd(t) k2 — kd(t) -kd(t )Rbi kd(t )Rb2 0 0

- kd(t )Rbi kd(t ) Rbi kp + kd(t )Rb2i -kd(t ) RbiRb2 -kp 0

kd(t ) Rb2 -kd(t)Rb2 -kd(t)RbiRb2 kg + kd(t )Rb22 0 -kg

0 0 -kp 0 kp 0

0 0 0 —kg 0 kg

F =[0,0, 0, 0, Ti -T2] (29)

For a gearbox system, the mesh damping coefficient c(t) and mesh stiffness are time-varying. For simplicity, the c(t) is set to be proportional to the damage mesh stiffness 28, kd(t), so

cd(t ) = Md(t ) (30)

where ^ is the scale constant measured in seconds, and its value is defined 3.99x10-6 s in this simulation.

The main parameters of this system can be found in Ref. 29. In Eq. (22), we define 18 generalized parameters as available sensor set S={y1, y2, 0, 0, 0i, 0, v1, v2, o, O2, om, Ob, a1, a2, a1, a2, am, «b}, and each parameter corresponds to a sensor. The objective of this paper is to

calculate the sensitivity of 18 available sensors in S to damage growth, then select optimal sensors to monitor the development of tooth damage. According to Eqs. (1)-(17), the model of time-varying mesh stiffness due to tooth crack is built. Fig. 6 represents the time-varying mesh stiffness due to tooth crack. In this figure, the real and dotted lines represent the mesh stiffness of healthy tooth and crack level 0.20, respectively. It shows that the total stiffness will reduce proportionally with the increase of crack levels.

S g Eg

E S É-

-Health toolh .........Crack level 0.20

0 0.Í 1.0 15 2.0 2.5 3.0 3.5 Time (ms)

Fig. 6 Time-varying mesh stiffness due to tooth crack.

The changes in mesh stiffness due to tooth crack are incorporated into Eq. (22) for dynamic response analysis. In this paper, The MATLAB/Simulink toolbox is used to conduct experimental investigations about the effects of tooth crack on the vibration signature of the system. The block diagram of the gearbox system with time-varying stiffness and damping coefficient is shown in Fig. 7. We use MATLAB's ODE15s solver to solve the model, then the dynamic responses of 18 general parameters in the model with damage growth can be generated.

Here, we define the available CI set /={RMS, PV, CF, Kurtosis, Skewness, ER, Sa, FM0, FM4, NA4}, and calculate their trends with tooth crack evolution. To make them comparable, we express all CIs as a percentage of change from the healthy tooth 16. The changes of ten CIs in / from healthy tooth for y2 are shown in Fig. 8. As seen in Fig. 8(a), in ten CIs, FM0 presents a total monotonic increasing with crack growth, and the other CIs change indistinctively. For clarity, we shorten the y coordinate size in Fig. 8(b), and it is also clear from this graph that FM4 shows a monotonic increasing with crack growth. By using the Eqs. (18) and (19), the test statistics of 10 CIs for the sensors in the S

are calculated, and their results are listed in Table 2.

In the simulation study, most of the test statistics of CIs are greater than 0.5, so we choose the CIs whose test statistics are greater than 0.5 in the simulation study in a trade-off among those test statistics. For y2, the test statistics of ER, FM0 and FM4 are 0.8871, 1 and 1, respectively. Therefore, the optimal CIs are {ER, FM0, FM4}. Similarly, the optimal CIs for all sensors in the S can be obtained, then, they can generate an HI using Eq. (20), and the test statistics and sensitivity of HIs can be calculated by Eqs. (19) and (21), respectively. The optimal CIs, test statistics and sensitivity of HIs for all sensors in the S are listed in Table 3. In the table, the test statistics of HIs in sensor set {yi, y2, 66, 62, 6b, a1, <%} are 1, which represent that the HIs of these sensors can describe the tooth damage evolution with a total monotonic increasing trend. What's more, the sensitivity of y2 is 9.9483 and it is the maximum value among them. The change of HI from healthy tooth for y2 is shown in Fig. 9. We can see that the HI of sensor y2 presents the best monotonic trend and sensitive level for crack growth. So the corresponding sensor of y2 can be used to monitor the development of crack damage.

4.2. Experimental case

The experimental data are provided by the Applied Research Laboratory at Pennsylvania State University on three test runs of single reduction helical gearboxes 30, which are named as TR#5, TR#10 and TR#12, respectively. The mechanical diagnostics test bed (MDTB) (shown in Fig. 10) was built as an experimental research station for the study of fault evolution in mechanical gearbox power transmission components. It consists of a motor, gearbox, and generator on a steel platform. Gearboxes are instrumented with accelerometers, thermocouples, acoustic emission sensors, and oil debris sensors. The vibration-based sensors in the test rig include six single axis accelerometer sensors (A02-A07) and three triaxial accelerometers (A10-A12), and the different locations of accelerometers are shown in Fig. 11.

Fig. 7 Block diagram of gearbox system with time-varying stiffness and damping coefficient.

Crack length (mm) Fig. 9 Change of HI from healthy tooth for y2.

In real application or run condition of helical gearboxes, redundant sensors will increase the test cost and reduce the reliability of the system, so the purpose of this paper is to select the optimal sensors from A02 to A07 and A10 to A12 according to their sensitivity to tooth damage growth in order to track and monitor the tooth damage evolution process effectively.

Fig. 8 Changes of ten CIs from healthy tooth for y2.

Table 2 Test statistics of CIs for sensors in S

Sensor RMS PV CF Kurtosis Skewness ER Sa FM0 FM4 NA4

yi 0.0808 0.1098 0.0776 -0.7333 -0.5106 0.8871 -0.8024 1.0000 1.0000 -1.0000

y2 -0.0431 -0.0251 -0.0384 -0.7945 -0.5906 0.8871 -0.9012 1.0000 0.9325 -1.0000

( -1.0000 0.1616 0.3349 -0.4729 0.9169 0.3176 -0.8165 -0.9984 0.5216 0.7788

<h 1.0000 0.5890 0.3569 1.0000 1.0000 -1.0000 1.0000 -0.9827 0.3710 1.0000

(m -1.0000 0.1616 0.3349 -0.4729 0.9169 0.3176 -0.8165 -0.9984 0.5216 0.7788

(, i.0000 0.5890 0.3569 1.0000 1.0000 -1.0000 1.0000 -0.9827 0.3710 1.0000

V1 0.7161 -0.0973 -0.6737 0.4902 0.9969 -0.4227 0.4792 -0.9608 1.0000 0.6439

V2 0.8039 -0.0329 -0.6612 0.5012 1.0000 -0.2800 0.5843 -0.3710 1.0000 1.0000

« 0.1075 -0.0643 -0.1843 -0.6110 -0.1608 -1.0000 -0.3867 0.9969 0.2094 -0.2141

«22 -0.1075 0.2259 0.2125 -0.9341 0.8541 -1.0000 -0.9576 0.9122 -0.8306 -0.9200

«m 0.1075 -0.0643 -0.1843 -0.6110 -0.1608 -1.0000 -0.3867 0.9969 0.2110 -0.2141

« -0.1075 0.2259 0.2110 -0.9341 0.8541 -1.0000 -0.9576 0.9122 -0.8306 -0.9200

ai 0.4949 -0.3537 -0.4996 0.7459 -0.3773 -1.0000 0.6031 1.0000 0.8525 0.8573

a2 0.4416 0.6690 -0.1992 -0.3945 0.6925 -0.3553 -1.0000 0.4824 0.9545 0.8494 0.8651

ai -0.0925 -0.6612 -0.6235 0.0604 -1.0000 -0.5843 -0.9608 -1.0000 -0.6345

CC2 -0.6690 -0.0847 0.7412 -0.9545 0.5325 -1.0000 -0.6659 -0.9545 -1.0000 -0.9875

am -0.6675 0.0643 0.1969 -0.4086 -0.4337 -1.0000 -0.4086 -0.9122 -1.0000 -0.2094

Cb -0.6675 0.2439 0.2878 -1.0000 1.0000 -1.0000 -1.0000 -0.9937 -1.0000 -0.3271

Table 3 Optimal CIs, test statistics and sensitivity of HIs for all sensors in the S

Sensor Optimal CIs Test statistic of HI Sensitivity of HI

y1 {ER, FM4, NA4} 1 5.6437

y2 {ER, FM4, NA4} 1 9.9483

( {Skewness, FM4, NA4} 1 0.0226

(2 {RMS, PV, Kurtosis, Skewness, Sa, NA4} 1 0.0404

(m {Skewness, FM4, NA4} 0.6204 0.0105

( {RMS, PV, Kurtosis, Skewness, Sa, NA4} 1 0.0404

V1 {RMS, Skewness, FM4, NA4} 0.7788 0.2323

V2 {RMS, Kurtosis, Skewness, Sa, FM4, NA4} 0.6863 0.2531

«1 {FM0} 0.9969 10.1345

«2 {Skewness, FM0 } 0.9122 22.2173

«m {FM0, NA4} 0.9969 10.1373

{Skewness, FM0} 0.9122 22.2456

ai {RMS, S„ FM0, FM4, NA4}

a2 {Kurtosis, FM0, FM4, «.J

ai {RMS}

a2 {CF, Skewness}

a_{Skewness }_

For each test run, the test rig is in a brand-new state at the beginning of each experiment and runs until gear tooth failure. Each inspection generates a piece of signal collected in a 10 s window at sampling rate of 20 kHz. The signal is saved in a computer as a data file and numbered consequently. Thus, it provides full lifetime vibration data for our study. Here we use file number as the time index because it is consistent with the progression of the test run and it is a consecutive integer series compared to time stamp provided by ARL 3. In addition, each experiment includes two kinds of work load conditions, i.e., it starts at normal workload (Conditional) and the work load is doubled or tripled (Condition#2) after some time to accelerate the experiment.

Fig. 10 Mechanical diagnostics test bed.

Fig. 11 Locations of accelerators.

In three test runs, only the gears of TR#5 are subject to obvious damage, i.e., two adjacent broken teeth (40,41) and one root crack (44) of the driven gear are found after test rig shutdown. So, TR#5 is used to validate the proposed method in this paper. Gearbox information and the test run time specifications of TR#5 are provided in Tables 4 and 5, respectively. The total number of running hours is 127.4 h, which includes 83 data files, and the relationship between the timestamp report and the file number is introduced by Wang et al.3.

For each sensor, we obtain the trend of CIs in I with the crack growth, then, their test statistics can be obtained by using Eqs. (18) and (19). The test statistics of CIs for all sensors in MDTB are listed in Table 6. We can find that all values in this table are smaller than those in the Table 1, and the maximum value is 0.5657, which represents that the sensors do not indicate the total monotonic increasing trend. In the experimental study,

1 9.0814

0.9545 17.8494

0.6690 52.007

0.7412 58.0691

_1_0.0262_

the crack characteristics will be affected by noise, vibration, temperature, etc., and most of the test statistics of CIs are less than 0.5, so we choose the CIs whose test statistics are greater than 0.2 in the experimental study in a trade-off among those test statistics. The corresponding HIs and their sensitivity to tooth damage growth can be calculated by Eqs. (20) and (21), respectively. The optimal CIs, test statistics of HIs and sensitivity of HIs for all sensors in MDTB are listed in Table 7. In this Table, we can find that A03 has the maximum test statistic 0.5375 and its sensitivity is 0.0480. The change of HI from healthy tooth for A03 is shown in Fig.12(a), and we can see that its HI trend shows a strong monotonic increasing. Therefore, A03 presents the best monotonic relevance and a good sensitivity with tooth damage of the helical gearbox. Moreover, in Table 6, we also find that the sensitivity of A07 is 0.6856, which is the maximum value in all sensitivity; its corresponding test statistic is 0.4176, and the change of HI from healthy tooth for A07 is shown in Fig. 12(b). The above results show that A07 presents a good monotonic relevance and the best sensitivity with tooth damage growth. Therefore, we can select A03 and A07 sensors to monitor the

development of tooth damage.

| 0.6 i 0.3

9lWl 6/22/97 V2VH 6/23/97 6/24/97 #24/97 6/24/97 6/24/97 G/24/97

14:4)0 1-1 (HI 15;30 22:» 01» 06.30 I HHj |5:30 19:30 20:56

Date/time (a) For A03 sensor

a 0.5 -

60UVt 6/23/97 6/23/97 6/24/97 6/24/97 6/24/97 6/24/97 6/24/97 6/24/97 14:00 14.<10 15:30 22:30 02:30 06:30 11:00 1530 1*30 20:56

Date/time (b) For A07 sensor Fig. 12 Change of HI from healthy tooth.

_Table 4 Gearbox information of TR#5_

Parameter Information

Gearbox ID# DS3S0150XX

Make Dodge APG

Model R86001

Gear ratio 1.533

Contact ratio 2.3B8

Number of teeth (driven gear) 46

Number of teeth (pinion gear) 3о

Meshing frequency B75.33 Hz

Rated input speed 175о rad/ms

Table 5 Time specification of workload change in TR#5

Condition

_Time period

6/19/97 13:35 (GMT)-6/23/97 13:35 (GMT) 6/23/97 13:35 (GMT)-6/24/97 Ю:56 (GMT)

Time stamp

File number

ооо-176 192-262

1-12 (1оо% max) 13-B3 (3оо% max)

Workload 54о in-lbs _162о in-lbs_

Table 6 Test statistics of CIs for all sensors in MDTB

Sensor RMS PV CF Kurtosis Skewness ER Sa FMG FM4 NA4

A02 о.иов о.263о -о.о267 о.159о -о.о585 -о.6292 -о.о8о8 -о.3694 о.2313 о.1719

A03 о.оо91 о.5657 о.о467 -о.1В95 о.3376 -о.427о -о.оо76 -о.3453 -о.1419 -о.2оо1

A04 о.о714 о.о691 -о.о591 -о.о932 -о.2383 -о.3547 о.1оо2 -о.36оо -о.1413 -о.1842

A05 -о.Ю73 о.1314 о.Ю96 о.4оо5 о.3947 -о.5469 о.о42о -о.531о о.4934 о.3оо6

A06 о.о297 -о.23о1 -о.о461 -о.2В3о -о.28о6 -о.2213 -о.о655 -о.2659 -о.3112 -о.2971

A07 о.ооо9 о.Ю48 о.о5о2 о.о638 -о.Ю84 о.3212 -о.оо65 о.424о о.о532 о.1372

AW -о.о273 о. 1643 о.о356 -о.о22о о.3488 -о.1837 -о.оо24 -о.26о7 -о.о826 -о.о244

A11 -о.о126 о.о926 о.о356 о.2177 о.о432 -о.2842 о.о855 -о.2о54 о. 1643 о.1972

A12 о.Ю61 -о.о967 -о.Ю61 -о.2о66 о.119о -о.5о1о о.о529 -о.4о7о -о.2о48 -о.24о7

Table 7 Optimal CIs, test statistics of HIs and sensitivity of HIs for all sensors in MTDB

Sensor

Optimal CIs

Test statistic of HI

Sensitivity

A02 {RMS, PV, Kurtosis, FM4, NA4}

A03 {PV, Skewness}

AO 4 {PV}

A05 {Kurtosis, Skewness, FM4, NA4}

A07 {PV, CF, Kurtosis, ER, FM0, FM4, NA4}

A10 {Skewness}

A11 {PV, CF, Kurtosis, Skewness, S„ FM4,

A12 {Skewness}

FM4, NA4} -

о.253о о.5375 о.о691 о.4о7о

о.4176 о.3523 о.2177 о. 119о

о.о147 о.о4Во о.оо13 о.оо67 о

о.6В56

о.оо22 о.оо25 о.оон

5. Conclusions

(1) The physical model of tooth crack is developed to describe the dependencies between tooth damage and the reduction of mesh stiffness.

(2) The Mann-Kendall test is used to evaluate the condition indicators in terms of the monotonic relevance with damage levels and assist in selecting useful CIs including damage growth information.

(3) The trend of selected CIs with damage evolution can generate an HI trend for each sensor using normalized energy, then the sensitivity of sensors to damage growth can be calculated by the sen slop estimator.

It provides a monitoring and damage.

The proposed method is verified by the simulation data of a one-stage gearbox and the experimental data provided by the ARL. Results show that the physical model of tooth crack can correctly describe the relation between tooth damage and the reduction of mesh stiffness. For the simulation model including crack damage, the linear displacements of gear in the y direction, y2, is the most sensitive parameter to crack levels, so that the corresponding sensor can be used to

guide to select optimal sensors for tracking the development of tooth

track the development of tooth crack. For the helical gearbox in APL, A03 and A07 sensors instrumented in the MDTB are the two most effective sensors to monitor the tooth damage, and the results accord with objective status.

In the future, we plan to investigate more complex models including more degrees of freedom, meshing friction, and multiple faults, etc, and apply the proposed method to assist in selecting optimal sensors in order to monitor the development of a system's damage.

Acknowledgments

This research is partially supported by the National Natural Science Foundation of China (No. 51175502).

Qiang Miao obtained the permission to use the gearbox vibration data from the Applied Research Laboratory at the Pennsylvania State University. The authors are most grateful that the Applied Research Laboratory and the Department of the Navy, Office of the Chief of Naval Research (ONR) have provided the gearbox vibration data for verifying our proposed theories. Finally, we appreciate the valuable comments from anonymous referees who help us to improve our work.

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Lyu Kehong received B.S. degree in 2001 from Xi'an Jiaotong University, Xi'an, China, M.S. and Ph.D. degrees from National University of Defense Technology (NUDT) Changsha, China, in 2003 and 2008, respectively. His research interests include condition monitoring, fault diagnosis techniques and the technology of decreasing false alarms. E-mail: fhrlkh@163.com

Tan Xiaodong received his B.S. degree in mechanical engineer from Northeast University, Shenyang, China, in 2006, and M.S. degree from NUDT in 2009. Since 2009, he is working toward the Ph.D. degree in the College of Mechatronics Engineering and Automation, NUDT. His research interests include design for testability, prognositcs and health management, etc.

Liu Guanjun received B.S. and Ph.D. degrees in mechanical engineer from NUDT in 1994 and 2000, respectively. Since 2000, he has been with NUDT, where he is an associate professor of mechanical engineering. He has published over 100 articles, primarily in the condition monitoring, fault diagnosis techniques and the technology of decreasing false alarms. His research interests include fault diagnosis, testability, maintenance and etc.

Zhao Chenxu is now a Ph.D. candidate in NUDT. He received B.S. degree in thermal energy and power engineering from Hunan University in 2009, and the M.S. degree in mechanical engineer from NUDT in 2011. His main research interests are testability design and verification, virtual test, prognostics and

health management.