- Email: [email protected]
Vibration-based updating of wear prediction for spur gears
Wear 426–427 (2019) 1410–1415
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Vibration-based updating of wear prediction for spur gears a,⁎
a
a
a
a
Ke Feng , Pietro Borghesani , Wade A. Smith , Robert B. Randall , Zhan Yie Chin , Jinzhao Renb, Zhongxiao Penga a b
T
School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, Australia School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China
ARTICLE INFO
ABSTRACT
Keywords: Gear wear Dynamic model Wear prediction Vibration Model updating Gear prognostics
Gear wear introduces geometric deviations in gear teeth and alters the load distribution across the tooth surface. Wear also increases the gear transmission error, generally resulting in increased vibration, noise and dynamic loads. This altered loading in turn promotes wear, forming a feedback loop between gear surface wear and vibration. Having the capability to monitor and predict the gear wear process would bring enormous benefits in cost and safety to a wide range of industries, but there are currently no reliable, effective and efficient tools to do so. This paper develops such tools using vibration-based methods. For this purpose, a vibration-based scheme for updating a wear prediction model is proposed. In the proposed scheme, a dynamic model of a spur gear system is firstly developed to generate realistic vibrations, which allows a quantitative study of the effects of gear tooth surface wear on gearbox vibration responses. The sliding velocity and contact forces from the model are used in combination with the well-known Archard wear model to calculate the wear depth at each contact point in mesh. The worn gear tooth profile is then fed back into the dynamic model as a new geometric transmission error, which represents the deviation of the profile from an ideal involute curve and is thus zero for perfect gears. New vibration responses and tooth contact forces are then obtained from the model, and the process repeated to generate realistic gear wear profiles of varying severities. Since the wear coefficient in the model is not constant during the wear process (and in any case is difficult to estimate initially), measured vibrations are compared with those generated by the model, so as to update the coefficient when a deviation from predictions is detected. With the continually updated dynamic wear model, the wear process can be well monitored, and at any particular time the best possible prediction of remaining useful life can be achieved. The paper illustrates the ability and effectiveness of the proposed scheme using measurements from a laboratory gear rig.
1. Introduction A gearbox system, due to its unique mechanical structures, has many distinguished merits such as high transmission efficiency, stable operation quality and accurate transmission ratio. Thanks to those merits, gearboxes are widely used in various mechanical systems, such as wind turbines, ships and helicopters. In real-life working practice, gearboxes often operate in tough working conditions and consequently gearbox failures are inevitable which may lead to unexpected economic losses, even serious accidents. Abrasive gear wear is a common phenomenon during the gear's service lifetime. It can lead to the formation of stress concentrations and initiate other modes of gear wear and failure, such as macro-pitting (spalling), scuffing and gear cracks [1]. Therefore, the monitoring and prediction of the gear wear process is a vital issue for the health management of transmission systems.
⁎
Theoretically, the gear wear process and gear dynamic characteristics will affect each other. This phenomenon is closely related to special nature of involute gears. The reason is that all the points on the tooth flank of a spur gear experience sliding movement, except for the pitch point; at the pitch point, a pure rolling condition occurs because the sliding velocity is zero. The sliding action can lead to abrasive surface wear, which alters the gear tooth profile geometry and thus the distribution of dynamic load. As a consequence, the level of vibration will increase, and other failure mechanisms may also be accelerated [2]. Meanwhile, because of the changes in the tooth working surface caused by wear, the gear transmission ratio would no longer be static [3], especially for the case of spur gears, whose gear transmission error is easily affected by tooth surface wear [4]. In addition, the dynamic responses of the gearbox could also alter the tooth surface wear process. This two-way relationship between the tooth surface wear process and
Corresponding author. E-mail address: [email protected] (K. Feng).
https://doi.org/10.1016/j.wear.2019.01.017 Received 4 September 2018; Received in revised form 3 January 2019; Accepted 3 January 2019 0043-1648/ © 2019 Elsevier B.V. All rights reserved.
Wear 426–427 (2019) 1410–1415
K. Feng et al.
the gear dynamic characteristics will produce more complex gear dynamic features and brings more difficulties in condition monitoring for gear wear compared with other failures. The majority of existing research works mainly focused on studying the effects of surface wear processes on gear system dynamic characteristics such as transmission error and dynamic meshing force [3,5–8]. In contrast, few studies investigated the effect of gear dynamics on surface wear [9–11]. There are some studies on prediction of spur gear wear under quasi-static operating conditions [12–14]. But dynamic response characteristics are quite different from those under quasi-static conditions. The dynamic meshing forces are typically larger than the corresponding quasi-static forces and their magnitudes and waveforms are quite different [9]. Theoretically, dynamic response characteristics of a gear pair are sensitive to deviations of the tooth surface profiles from a perfect involute: geometric deviation and elastic deformation [15]. Gear surface wear is a material removal process, which results in a geometric deviation. And, elastic deformation is determined by contact force and meshing stiffness. Based on this theory, a study [9] used a periodically time-varying meshing stiffness function and an external displacement excitation to represent the effects of dynamic response on the gear wear process. In that research, the authors used a torsional model with single degree-of-freedom (DOF) and then combined it with a quasi-static wear model [13] to develop a dynamic wear model. This model is capable of investigating the interactions between the surface wear and the spur gear system's dynamic characteristics. Later, in order to demonstrate the effect of translational deflection on the wear process, one study [11] introduced a 3-DOFs dynamic model to replace the torsional model in [9], and some differences were observed. It should be noted that both Refs. [9] and [11] only investigated the coupling effects between gear dynamics and the wear process through a set of simulations, while gear wear prediction, which would bring many benefits to a wide range of industries, was not included. The combination of dynamic and wear models, proposed in a few variants in the literature, is theoretically able to predict the evolution of wear, considering its interactions with the gearbox dynamics. This prediction will however likely drift away from the actual wear process due to the imperfect knowledge and modelling of wear, especially considering that the parameters governing wear dynamics vary in time (e.g. with the contamination of the lubricant and the change in surface roughness). Increasing the complexity of the wear models to follow these complex trends is an option, but it is likely to result in additional parameters whose quantification for practical applications could be difficult and carry additional uncertainty in the model predictions. The aim of this work is to develop a scheme for the vibration-based updating of a wear prediction model to monitor and predict gear wear. The paper is organised as follows: Section 2 presents the whole procedure of the proposed vibration-based updating of the wear prediction scheme theoretically; the development of the dynamic model and its calibration are introduced in Section 3; the wear model and its updating methodology are described in Section 4; and Section 5 shows the wear prediction results with an accelerated run-to-failure gear wear test. Conclusions and recommendations for further studies are discussed in Section 6.
Fig. 1. Basic procedures of the proposed vibration-based scheme for updating wear predictions.
loop allows a knowledge-based prediction of wear, which however is likely to be reliable only on a limited timeframe, within which the wear model parameters remain unchanged. The main novelty of the proposed approach relies in the updating of the wear model parameters based on vibration measurements (on the right of Fig. 1). The vibrations obtained as a by-product of the gearbox dynamic model are compared to measured vibration levels to track the quality of the wear model predictions and, if necessary update the wear model parameters. The details of each component of the approach, as outlined in Fig. 1, will be discussed in depth in the next chapters. 3. Dynamic model The discussion of the model will be presented, for the sake of brevity and clarity, after a brief description of the actual system that the model is representing. 3.1. Description of the modelled system The single stage spur gearbox test rig which was modelled is shown in Fig. 2. The gearbox itself is composed of two shafts of similar length (about 320 mm) and circular sections varying in diameter from 14 to 20 mm. The input and output shafts are connected by two modular gears (module 2) with 19 and 52 teeth respectively. To achieve an accelerated wear rate, gears of mild steel (JIS S45C) were used. The input shaft is powered by a 4 kW electric motor, whose instantaneous speed is controlled by a variable frequency drive (VFD) and connected to a torque meter which can monitor the instantaneous torque of the system. An electromagnetic particle (EMP) brake is connected to the output shaft and is used to control the torque transmitted by the gears. Two encoders are installed at the remaining free ends of the gearbox, for a reference measurement of transmission error. The connections of the gearbox shafts with motor, brake and encoders is achieved using couplings with high torsional stiffness and low bending stiffness. Vibration sensors (B&K 4396 and B&K 4394 accelerometers) are mounted on the top of the gearbox casing in the positions indicated in Fig. 2(b).
2. Approach In this section, the basic procedure of the proposed vibration-based updating of the wear prediction scheme will be introduced briefly. The overall approach is concisely presented in Fig. 1. The modelling component of this methodology (on the left of Fig. 1) is composed by two interacting simulation models: a dynamic model and a wear model. Based on the input of the tooth profile geometry, the dynamic model predicts tooth contact forces, which are passed on to the Archard-based wear model to estimate wear and consequently modify the tooth profile geometry, which is then fed back into the dynamic model. This iterative
3.2. Dynamic model structure A 21 DOF lumped parameter dynamic model is established to simulate the dynamics of the system as shown in Fig. 3. The major parameters that are included in the model are summarised in Table 1. The motion equations describing the coupled torsional and 1411
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Fig. 2. The spur gear test rig at University of New South Wales (UNSW). (a) photo; (b) schematic diagram.
translational model are described as
matrixes. The absence of bearing torsional DOFs ( 4 , 6, 9, 11) is due to the absence of torsional stiffness at the bearings, whereas the missing translational displacement DOFs (all the nodes outside the gearbox) is a result of the low bending stiffness of the joints, effectively isolating the linear displacements of the gearbox. The force vector f includes the input and output torques Tmot and Tbrk provided by the motor and brake (on 1 and 13, respectively), and the contact forces between the two gears, modelled as Fk and Fc (elastic and viscous components). These are simulated by combining a gear meshing stiffness km , damping cm and geometric transmission error (GTE) et :
(1)
Mx¨ + Cx + Kx = f where
x = [ 1,
2,
3,
5,
7,
8,
x 4 , x5, x 6, x 9 , x10 , x11]T
10 ,
12 ,
13 , y4 , y5 , y6 ,
y9 , y10 , y11 , (2)
represents the translational (xi , yi ) and angular displacements i of the different nodes of the system in the plane perpendicular to the shaft axes, and C , K and f are the corresponding damping, stiffness and force
Fig. 3. Diagram of the 21 DOFs gear dynamic model. 1412
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Table 1 Parameters of the 21 DOF model.
Table 2 Natural frequency comparison results of dynamic model and experimental data.
Parameters
Pinion
Gear type Modulus of elasticity, E (GPa) Poisson's ratio, Face width, W (mm) Module, m Pressure angle, (deg) Addendum (mm) Dedendum (mm) Number of teeth Pitch radius, r (mm)
Standard involute 205 0.29 20 2 20 1.00 1.25 19 19
Fk = km (Rb1
5
Rb2
10
Gear
y5 cos + y10 cos + x5sin
52 52
5
Rb2
10
x10sin + et )
y5 cos + y10 cos + x5 sin
x10 sin + et ) (4)
where Rb1 and Rb2 are the basic radius of pinion and gear, and is the contact pressure angle. The contact force is applied at nodes 5 and 10, considering the radius of the two gears and the contact angle.
f = [Tmor , 0, 0,
Rb1 (Fk + Fc ), 0, 0, Rb2 (Fk + Fc ), 0, cos (Fk
+ Fc ), 0, 0,
sin (Fk + Fc ), 0, 0, sin (Fk + Fc ), 0]T
Difference (%)
21.77 86.91 217.80 430.20 511.60 767.10 1248.00 1927.00
22.88 81.08 200.89 426.68 529.35 740.00 1268.30 1884.50
5.1 −6.7 −7.8 −0.8 3.5 3.5 1.6 2.2
1. 2 Hz rotational speed and 10 N m motor torque and 2. 2 Hz rotational speed and 20 N m motor torque.
Tbrk, 0, cos (Fk
+ Fc ), 0, 0,
Dynamic model (Hz)
initial adjustment of the most uncertain model parameters (e.g. stiffness of joints and bearings) with the ultimate aim of obtaining a good match between simulated and experimental natural frequencies in the frequency a range of interest (0–2 kHz). The comparison result is summarised in Table 2. Table 2 shows that the first eight natural frequencies agree between the experiments and the dynamic model within eight percent, which was deemed sufficient to approximate the system response. Two constant speed tests were then executed in order to fine-tune a scale factor for the meshing stiffness function km and the damping coefficient cm :
(3)
Fc = Cm (Rb1
Experiments (Hz)
(5)
The encoder signals on the input and output shaft were used in these tests to calculate the static transmission error (STE) of the gearbox system using the phase demodulation method [17] assuming negligible dynamic effects at this low speed. This STE was then used as an input to the dynamic model to simulate a vibration response. After low-pass filtering in the band of interest (0–2 kHz), the RMS of the simulated vibration signal y6(SIM ) (t ) was then compared with the experimental results and the meshing stiffness and damping were manually adjusted until good agreement between the results was obtained, as shown in Fig. 5. The low pass filtering was necessary to remove high frequency effects which arise in the simulation from the unrealistic steps in the stiffness function, smoothed in practice by a gradual engagement/disengagement of the teeth. Indeed, even in this limited band the stronger weighting of the higher gear mesh harmonics is apparent in the simulated response in Fig. 5(b). Nonetheless, the overall magnitudes of the responses were deemed to match the measurements sufficiently for the purposes of the present study. This calibrated dynamic model was used both to generate dynamic
Whereas the damping coefficient cm is modelled as constant, the meshing stiffness km is considered a function of the angular rotation of the pinion 5 . The shape of the dependency km ( 5) is derived as in Ref. [16] (potential energy method) and is shown in Fig. 4 (the actual amplitude is obtained by experimental calibration and will be discussed in Section 3.2). Note: meshing stiffness change is neglected as it is significantly less important than transmission error effect. The GTE, et (geometric deviation from perfect involute), is used to represent gear surface wear, but could also include initial profile errors, that can be estimated on the basis of manufacturing quality or measured. 3.3. Model calibration A series of initial tests were performed for the calibration of the dynamic model, including a speed ramp, impact tests and constant speed tests. The speed ramp and impact tests were used to provide an
Fig. 4. Meshing stiffness curve.
Fig. 5. Experimental (a) and predicted (b) signals: 00 min. 1413
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contact forces (for specified GTEs) for input into the Archard wear model (Section 4) and to simulate responses for comparison with vibration measurements to enable updating of the wear model. In the next section, the Archard wear model for wear induced GTE evaluation and experimental update methodology will be introduced, which is then integrated with the proposed dynamic model to develop a vibration-based updating of wear prediction scheme for gear wear process monitoring.
of manufacturing precision), or GTE-based estimates. This could speed up the convergence of the updating methodology in the first phases of the wear evolution, but regardless of the choice, the self-adapting scheme is expected to correct for any initial profile estimation error after a few iterations. For this reason, in this study the updating procedure was implemented starting from a null initial geometric transmission error (i.e. perfect involute tooth and zero wear depth). 5. Results
4. Wear prediction
After the initial tests performed for the calibration of the dynamic model, the run-to failure campaign was executed. The gears were run in dry conditions with a rotating speed of 10 Hz and an input shaft torque of 10 N m. Vibration and encoder measurements were collected every 10 min (i.e. every 6000 rotations of the pinion). Three measurements were obtained after calibration and before a catastrophic failure of the pinion teeth, which occurred shortly after 18,000 cycles. At each measurement time the wear model parameter Kwear could be updated (if necessary) with the procedure discussed in Section 4.2. After the first updating (10 min), the wear model parameter Kwear was found to be 2.7×10 6N 1, and the maximum predicted wear depth was estimated at 156 µm (note that this and subsequent wear depth figures represent the combined wear of the two profiles: that of the pinion alone is about 73% of this figure). The trends of RMS and maximum wear depth vs wear cycles (rotations of the pinion) are plotted separately in Figs. 6 and 7. The agreement of the prediction for 12,000 cycles with the corresponding experimental results shows the effectiveness of the model updating procedure. A second updating at 12,000 cycles was deemed not to be necessary, given the already low discrepancy between RMS values. The further projection of RMS and wear depth for a further 6000 cycles shows that this choice was reasonable, giving a prediction error of about 13% on RMS and 7% on wear depth at 18,000 cycles (i.e. 12,000 cycles from the last updating). A sudden failure of the gears (due to the extreme level of wear) then prevented any further updating and prediction, but conceptually, the 13% error of RMS at 18,000 cycles would justify a further updating of Kwear .
In this section, the proposed vibration-based updating of wear prediction scheme is presented by combining the aforementioned dynamic model with the well-known Archard wear model [18] and experimental update methodology. 4.1. Wear model Although many advanced wear models have been proposed using different methodologies and parameter sets, Archard's wear model remains the most commonly used for practical applications, and is chosen in this paper. Neglecting changes in the contact area over the meshing cycle, and differentiating the expression of Ref. [9], we obtain the following form of Archard's law:
dh = Kwear Fv dt
(6)
Herein, h is the wear depth of the gear contact surface, Kwear is the wear coefficient, F is the normal load (i.e., contact force), and v is the sliding velocity at time t (representing the derivative of the sliding distance). The sliding velocity v ( 5) throughout the meshing cycle can be determined by the kinetic formula [19], while the contact force F is obtained by the dynamic model. The Archard model parameter Kwear is affected by a number of factors including material properties, such as hardness and roughness of the two surfaces, and the lubrication condition, which may evolve in the wear process. Since it is very difficult to estimate or directly measure Kwear experimentally, this coefficient is often a major unknown parameter. In this study Kwear is determined experimentally, based on the comparison of simulated and experimental vibrations. The details of this updating methodology will be introduced in the following section.
6. Conclusion and future work In this paper, a new scheme for vibration-based updating is proposed to monitor and predict the gear wear process. The updating procedure is able to track and correct for changes in wear rates, thus allowing analysts to obtain reliable wear predictions with relatively simple modelling tools. Unique to existing studies in gear-wear prediction, the vibration-based updating procedure was successfully tested on a dry run-to-failure test.
4.2. Update methodology The update of the wear coefficient Kwear is obtained based on the comparison of the root mean square (RMS) value of simulated and experimental signals. In particular, the simulated vibration at the bearing housing y6(SIM ) (t ) is compared with the experimental vibration signal y6(EXP ) (t ) measured by the accelerometer B&K 4396 in the corresponding position. The updating procedure is executed iteratively based on a proportional model. In particular, if the ratio of predicted vibrations to experimental vibration is D(i ) =
RMS {y6(SIM , i) (t )} RMS {y6(EXP ) (t )}
, the wear
coefficient for the next iteration is computed as K (i + 1) = D(i)1 K (i) . The iterations are stopped when the absolute error between the simulated and experimental RMS values falls below the predefined threshold (in this case 5% difference). It must be highlighted that the operating conditions of the dynamic model (speed and torque) should be set as close as possible to the actual experimental conditions, to avoid potentially large biases on RMS readings. Luckily, speed measurements and torque estimates (e.g. through current measurements in electromechanical drivetrains) are available in most machines with sufficient criticality to justify a sophisticated condition monitoring system. The methodology discussed in this section requires setting an initial profile, for which a series of options are possible, including random tooth profile deviations (appealing in the case of statistical knowledge
Fig. 6. RMS comparison results: experiment and model. 1414
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methodologies (e.g. random profile deviation based on manufacturing quality) on the initial convergence of the model. Acknowledgement This research was supported by the Australian Government through the Australian Research Council (Discovery Project DP160103501) and the China Scholarship Council (Grant no. 201706070120). References [1] P. Fernandes, C. McDuling, Surface contact fatigue failures in gears, Eng. Fail. Anal. 4 (2) (1997) 99–107. [2] F. Zhao, Z. Tian, X. Liang, M. Xie, An integrated prognostics method for failure time prediction of gears subject to the surface wear failure mode, IEEE Trans. Reliab. (2018). [3] J. Wojnarowski, V. Onishchenko, Tooth wear effects on spur gear dynamics, Mech. Mach. Theory 38 (2) (2003) 161–178. [4] T. Osman, P. Velex, Static and dynamic simulations of mild abrasive wear in widefaced solid spur and helical gears, Mech. Mach. Theory 45 (6) (2010) 911–924. [5] F.K. Choy, V. Polyshchuk, J.J. Zakrajsek, R.F. Handschuh, D.P. Townsend, Analysis of the effects of surface pitting and wear on the vibration of a gear transmission system, Tribol. Int. 29 (1) (1996) 77–83. [6] J. Kuang, A. Lin, The effect of tooth wear on the vibration spectrum of a spur gear pair, J. Vib. Acoust. 123 (3) (2001) 311–317. [7] I. Yesilyurt, F. Gu, A.D. Ball, Gear tooth stiffness reduction measurement using modal analysis and its use in wear fault severity assessment of spur gears, NDT & E Int. 36 (5) (2003) 357–372. [8] C. Yuksel, A. Kahraman, Dynamic tooth loads of planetary gear sets having tooth profile wear, Mech. Mach. Theory 39 (7) (2004) 695–715. [9] H.L. Ding, A. Kahraman, Interactions between nonlinear spur gear dynamics and surface wear, J. Sound Vib. 307 (3–5) (2007) 662–679. [10] A. Kahraman, H.L. Ding, A methodology to predict surface wear of planetary gears under dynamic conditions, Mech. Based Des. Struct. Mach. 38 (4) (2010) 493–515. [11] X.Z. Liu, Y.H. Yang, J. Zhang, Investigation on coupling effects between surface wear and dynamics in a spur gear system, Tribol. Int. 101 (2016) 383–394. [12] A. Flodin, S. Andersson, Simulation of mild wear in spur gears, Wear 207 (1–2) (1997) 16–23. [13] P. Bajpai, A. Kahraman, N. Anderson, A surface wear prediction methodology for parallel-axis gear pairs, J. Tribol. 126 (3) (2004) 597–605. [14] M. Masjedi, M. Khonsari, On the prediction of steady-state wear rate in spur gears, Wear 342 (2015) 234–243. [15] W.D. Mark, Performance-based Gear Metrology: Kinematic-transmission-error Computation and Diagnosis, John Wiley & Sons, Ltd, Chichester, UK, 2013. [16] X.H. Liang, M.J. Zuo, M. Pandey, Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set, Mech. Mach. Theory 76 (2014) 20–38. [17] P. Sweeney, R. Randall, Gear transmission error measurement using phase demodulation, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 210 (3) (1996) 201–213. [18] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (8) (1953) 981–988. [19] S. Li, A. Kahraman, A transient mixed elastohydrodynamic lubrication model for spur gear Pairs, J. Tribol.-Trans. ASME 132 (1) (2010).
Fig. 7. Maximum wear depth comparison results: experiment and model.
Although the model presented in this work has demonstrated its capability for condition-based wear prediction using vibration, there are some limitations in the experimental data collected in the short dry test. In future work, longer and possibly lubricated wear tests will be carried out to fine tune the model so it can be used for wear monitoring and prediction under a range of lubrication conditions (boundary, mixed and full lubrication) and with abrasive wear and/or contact fatigue pitting. In addition, the current updating scheme only corrects for magnitude of wear but does not correct the shape of the profile error. A future area of work will involve the use of more sophisticated vibration features (e.g., gearmesh harmonics) to characterise the wear profile variation over the tooth length. This will also enable the distinction between the effects of different wear mechanisms. Moreover, since the actual operating conditions of the gearbox will experience small fluctuations in real operation, indicators which are more robust than RMS to variable conditions have been included in our future work plan. With varying operating conditions and multiple wear mechanisms, accompanied by more sophisticated vibration features, the updating scheme proposed in this paper will also require further developments. The updating procedure will not only be responsible for the continual adjustment of multiple wear coefficients, but also the tracking of the relative importance of different wear mechanisms and the consideration of time-varying operating conditions. In addition, future studies should include the analysis of the effect of possible profile initialisation
1415
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Vibration-based updating of wear prediction for spur gears a,⁎
a
a
a
a
Ke Feng , Pietro Borghesani , Wade A. Smith , Robert B. Randall , Zhan Yie Chin , Jinzhao Renb, Zhongxiao Penga a b
T
School of Mechanical and Manufacturing Engineering, University of New South Wales, Sydney, Australia School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China
ARTICLE INFO
ABSTRACT
Keywords: Gear wear Dynamic model Wear prediction Vibration Model updating Gear prognostics
Gear wear introduces geometric deviations in gear teeth and alters the load distribution across the tooth surface. Wear also increases the gear transmission error, generally resulting in increased vibration, noise and dynamic loads. This altered loading in turn promotes wear, forming a feedback loop between gear surface wear and vibration. Having the capability to monitor and predict the gear wear process would bring enormous benefits in cost and safety to a wide range of industries, but there are currently no reliable, effective and efficient tools to do so. This paper develops such tools using vibration-based methods. For this purpose, a vibration-based scheme for updating a wear prediction model is proposed. In the proposed scheme, a dynamic model of a spur gear system is firstly developed to generate realistic vibrations, which allows a quantitative study of the effects of gear tooth surface wear on gearbox vibration responses. The sliding velocity and contact forces from the model are used in combination with the well-known Archard wear model to calculate the wear depth at each contact point in mesh. The worn gear tooth profile is then fed back into the dynamic model as a new geometric transmission error, which represents the deviation of the profile from an ideal involute curve and is thus zero for perfect gears. New vibration responses and tooth contact forces are then obtained from the model, and the process repeated to generate realistic gear wear profiles of varying severities. Since the wear coefficient in the model is not constant during the wear process (and in any case is difficult to estimate initially), measured vibrations are compared with those generated by the model, so as to update the coefficient when a deviation from predictions is detected. With the continually updated dynamic wear model, the wear process can be well monitored, and at any particular time the best possible prediction of remaining useful life can be achieved. The paper illustrates the ability and effectiveness of the proposed scheme using measurements from a laboratory gear rig.
1. Introduction A gearbox system, due to its unique mechanical structures, has many distinguished merits such as high transmission efficiency, stable operation quality and accurate transmission ratio. Thanks to those merits, gearboxes are widely used in various mechanical systems, such as wind turbines, ships and helicopters. In real-life working practice, gearboxes often operate in tough working conditions and consequently gearbox failures are inevitable which may lead to unexpected economic losses, even serious accidents. Abrasive gear wear is a common phenomenon during the gear's service lifetime. It can lead to the formation of stress concentrations and initiate other modes of gear wear and failure, such as macro-pitting (spalling), scuffing and gear cracks [1]. Therefore, the monitoring and prediction of the gear wear process is a vital issue for the health management of transmission systems.
⁎
Theoretically, the gear wear process and gear dynamic characteristics will affect each other. This phenomenon is closely related to special nature of involute gears. The reason is that all the points on the tooth flank of a spur gear experience sliding movement, except for the pitch point; at the pitch point, a pure rolling condition occurs because the sliding velocity is zero. The sliding action can lead to abrasive surface wear, which alters the gear tooth profile geometry and thus the distribution of dynamic load. As a consequence, the level of vibration will increase, and other failure mechanisms may also be accelerated [2]. Meanwhile, because of the changes in the tooth working surface caused by wear, the gear transmission ratio would no longer be static [3], especially for the case of spur gears, whose gear transmission error is easily affected by tooth surface wear [4]. In addition, the dynamic responses of the gearbox could also alter the tooth surface wear process. This two-way relationship between the tooth surface wear process and
Corresponding author. E-mail address: [email protected] (K. Feng).
https://doi.org/10.1016/j.wear.2019.01.017 Received 4 September 2018; Received in revised form 3 January 2019; Accepted 3 January 2019 0043-1648/ © 2019 Elsevier B.V. All rights reserved.
Wear 426–427 (2019) 1410–1415
K. Feng et al.
the gear dynamic characteristics will produce more complex gear dynamic features and brings more difficulties in condition monitoring for gear wear compared with other failures. The majority of existing research works mainly focused on studying the effects of surface wear processes on gear system dynamic characteristics such as transmission error and dynamic meshing force [3,5–8]. In contrast, few studies investigated the effect of gear dynamics on surface wear [9–11]. There are some studies on prediction of spur gear wear under quasi-static operating conditions [12–14]. But dynamic response characteristics are quite different from those under quasi-static conditions. The dynamic meshing forces are typically larger than the corresponding quasi-static forces and their magnitudes and waveforms are quite different [9]. Theoretically, dynamic response characteristics of a gear pair are sensitive to deviations of the tooth surface profiles from a perfect involute: geometric deviation and elastic deformation [15]. Gear surface wear is a material removal process, which results in a geometric deviation. And, elastic deformation is determined by contact force and meshing stiffness. Based on this theory, a study [9] used a periodically time-varying meshing stiffness function and an external displacement excitation to represent the effects of dynamic response on the gear wear process. In that research, the authors used a torsional model with single degree-of-freedom (DOF) and then combined it with a quasi-static wear model [13] to develop a dynamic wear model. This model is capable of investigating the interactions between the surface wear and the spur gear system's dynamic characteristics. Later, in order to demonstrate the effect of translational deflection on the wear process, one study [11] introduced a 3-DOFs dynamic model to replace the torsional model in [9], and some differences were observed. It should be noted that both Refs. [9] and [11] only investigated the coupling effects between gear dynamics and the wear process through a set of simulations, while gear wear prediction, which would bring many benefits to a wide range of industries, was not included. The combination of dynamic and wear models, proposed in a few variants in the literature, is theoretically able to predict the evolution of wear, considering its interactions with the gearbox dynamics. This prediction will however likely drift away from the actual wear process due to the imperfect knowledge and modelling of wear, especially considering that the parameters governing wear dynamics vary in time (e.g. with the contamination of the lubricant and the change in surface roughness). Increasing the complexity of the wear models to follow these complex trends is an option, but it is likely to result in additional parameters whose quantification for practical applications could be difficult and carry additional uncertainty in the model predictions. The aim of this work is to develop a scheme for the vibration-based updating of a wear prediction model to monitor and predict gear wear. The paper is organised as follows: Section 2 presents the whole procedure of the proposed vibration-based updating of the wear prediction scheme theoretically; the development of the dynamic model and its calibration are introduced in Section 3; the wear model and its updating methodology are described in Section 4; and Section 5 shows the wear prediction results with an accelerated run-to-failure gear wear test. Conclusions and recommendations for further studies are discussed in Section 6.
Fig. 1. Basic procedures of the proposed vibration-based scheme for updating wear predictions.
loop allows a knowledge-based prediction of wear, which however is likely to be reliable only on a limited timeframe, within which the wear model parameters remain unchanged. The main novelty of the proposed approach relies in the updating of the wear model parameters based on vibration measurements (on the right of Fig. 1). The vibrations obtained as a by-product of the gearbox dynamic model are compared to measured vibration levels to track the quality of the wear model predictions and, if necessary update the wear model parameters. The details of each component of the approach, as outlined in Fig. 1, will be discussed in depth in the next chapters. 3. Dynamic model The discussion of the model will be presented, for the sake of brevity and clarity, after a brief description of the actual system that the model is representing. 3.1. Description of the modelled system The single stage spur gearbox test rig which was modelled is shown in Fig. 2. The gearbox itself is composed of two shafts of similar length (about 320 mm) and circular sections varying in diameter from 14 to 20 mm. The input and output shafts are connected by two modular gears (module 2) with 19 and 52 teeth respectively. To achieve an accelerated wear rate, gears of mild steel (JIS S45C) were used. The input shaft is powered by a 4 kW electric motor, whose instantaneous speed is controlled by a variable frequency drive (VFD) and connected to a torque meter which can monitor the instantaneous torque of the system. An electromagnetic particle (EMP) brake is connected to the output shaft and is used to control the torque transmitted by the gears. Two encoders are installed at the remaining free ends of the gearbox, for a reference measurement of transmission error. The connections of the gearbox shafts with motor, brake and encoders is achieved using couplings with high torsional stiffness and low bending stiffness. Vibration sensors (B&K 4396 and B&K 4394 accelerometers) are mounted on the top of the gearbox casing in the positions indicated in Fig. 2(b).
2. Approach In this section, the basic procedure of the proposed vibration-based updating of the wear prediction scheme will be introduced briefly. The overall approach is concisely presented in Fig. 1. The modelling component of this methodology (on the left of Fig. 1) is composed by two interacting simulation models: a dynamic model and a wear model. Based on the input of the tooth profile geometry, the dynamic model predicts tooth contact forces, which are passed on to the Archard-based wear model to estimate wear and consequently modify the tooth profile geometry, which is then fed back into the dynamic model. This iterative
3.2. Dynamic model structure A 21 DOF lumped parameter dynamic model is established to simulate the dynamics of the system as shown in Fig. 3. The major parameters that are included in the model are summarised in Table 1. The motion equations describing the coupled torsional and 1411
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Fig. 2. The spur gear test rig at University of New South Wales (UNSW). (a) photo; (b) schematic diagram.
translational model are described as
matrixes. The absence of bearing torsional DOFs ( 4 , 6, 9, 11) is due to the absence of torsional stiffness at the bearings, whereas the missing translational displacement DOFs (all the nodes outside the gearbox) is a result of the low bending stiffness of the joints, effectively isolating the linear displacements of the gearbox. The force vector f includes the input and output torques Tmot and Tbrk provided by the motor and brake (on 1 and 13, respectively), and the contact forces between the two gears, modelled as Fk and Fc (elastic and viscous components). These are simulated by combining a gear meshing stiffness km , damping cm and geometric transmission error (GTE) et :
(1)
Mx¨ + Cx + Kx = f where
x = [ 1,
2,
3,
5,
7,
8,
x 4 , x5, x 6, x 9 , x10 , x11]T
10 ,
12 ,
13 , y4 , y5 , y6 ,
y9 , y10 , y11 , (2)
represents the translational (xi , yi ) and angular displacements i of the different nodes of the system in the plane perpendicular to the shaft axes, and C , K and f are the corresponding damping, stiffness and force
Fig. 3. Diagram of the 21 DOFs gear dynamic model. 1412
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Table 1 Parameters of the 21 DOF model.
Table 2 Natural frequency comparison results of dynamic model and experimental data.
Parameters
Pinion
Gear type Modulus of elasticity, E (GPa) Poisson's ratio, Face width, W (mm) Module, m Pressure angle, (deg) Addendum (mm) Dedendum (mm) Number of teeth Pitch radius, r (mm)
Standard involute 205 0.29 20 2 20 1.00 1.25 19 19
Fk = km (Rb1
5
Rb2
10
Gear
y5 cos + y10 cos + x5sin
52 52
5
Rb2
10
x10sin + et )
y5 cos + y10 cos + x5 sin
x10 sin + et ) (4)
where Rb1 and Rb2 are the basic radius of pinion and gear, and is the contact pressure angle. The contact force is applied at nodes 5 and 10, considering the radius of the two gears and the contact angle.
f = [Tmor , 0, 0,
Rb1 (Fk + Fc ), 0, 0, Rb2 (Fk + Fc ), 0, cos (Fk
+ Fc ), 0, 0,
sin (Fk + Fc ), 0, 0, sin (Fk + Fc ), 0]T
Difference (%)
21.77 86.91 217.80 430.20 511.60 767.10 1248.00 1927.00
22.88 81.08 200.89 426.68 529.35 740.00 1268.30 1884.50
5.1 −6.7 −7.8 −0.8 3.5 3.5 1.6 2.2
1. 2 Hz rotational speed and 10 N m motor torque and 2. 2 Hz rotational speed and 20 N m motor torque.
Tbrk, 0, cos (Fk
+ Fc ), 0, 0,
Dynamic model (Hz)
initial adjustment of the most uncertain model parameters (e.g. stiffness of joints and bearings) with the ultimate aim of obtaining a good match between simulated and experimental natural frequencies in the frequency a range of interest (0–2 kHz). The comparison result is summarised in Table 2. Table 2 shows that the first eight natural frequencies agree between the experiments and the dynamic model within eight percent, which was deemed sufficient to approximate the system response. Two constant speed tests were then executed in order to fine-tune a scale factor for the meshing stiffness function km and the damping coefficient cm :
(3)
Fc = Cm (Rb1
Experiments (Hz)
(5)
The encoder signals on the input and output shaft were used in these tests to calculate the static transmission error (STE) of the gearbox system using the phase demodulation method [17] assuming negligible dynamic effects at this low speed. This STE was then used as an input to the dynamic model to simulate a vibration response. After low-pass filtering in the band of interest (0–2 kHz), the RMS of the simulated vibration signal y6(SIM ) (t ) was then compared with the experimental results and the meshing stiffness and damping were manually adjusted until good agreement between the results was obtained, as shown in Fig. 5. The low pass filtering was necessary to remove high frequency effects which arise in the simulation from the unrealistic steps in the stiffness function, smoothed in practice by a gradual engagement/disengagement of the teeth. Indeed, even in this limited band the stronger weighting of the higher gear mesh harmonics is apparent in the simulated response in Fig. 5(b). Nonetheless, the overall magnitudes of the responses were deemed to match the measurements sufficiently for the purposes of the present study. This calibrated dynamic model was used both to generate dynamic
Whereas the damping coefficient cm is modelled as constant, the meshing stiffness km is considered a function of the angular rotation of the pinion 5 . The shape of the dependency km ( 5) is derived as in Ref. [16] (potential energy method) and is shown in Fig. 4 (the actual amplitude is obtained by experimental calibration and will be discussed in Section 3.2). Note: meshing stiffness change is neglected as it is significantly less important than transmission error effect. The GTE, et (geometric deviation from perfect involute), is used to represent gear surface wear, but could also include initial profile errors, that can be estimated on the basis of manufacturing quality or measured. 3.3. Model calibration A series of initial tests were performed for the calibration of the dynamic model, including a speed ramp, impact tests and constant speed tests. The speed ramp and impact tests were used to provide an
Fig. 4. Meshing stiffness curve.
Fig. 5. Experimental (a) and predicted (b) signals: 00 min. 1413
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contact forces (for specified GTEs) for input into the Archard wear model (Section 4) and to simulate responses for comparison with vibration measurements to enable updating of the wear model. In the next section, the Archard wear model for wear induced GTE evaluation and experimental update methodology will be introduced, which is then integrated with the proposed dynamic model to develop a vibration-based updating of wear prediction scheme for gear wear process monitoring.
of manufacturing precision), or GTE-based estimates. This could speed up the convergence of the updating methodology in the first phases of the wear evolution, but regardless of the choice, the self-adapting scheme is expected to correct for any initial profile estimation error after a few iterations. For this reason, in this study the updating procedure was implemented starting from a null initial geometric transmission error (i.e. perfect involute tooth and zero wear depth). 5. Results
4. Wear prediction
After the initial tests performed for the calibration of the dynamic model, the run-to failure campaign was executed. The gears were run in dry conditions with a rotating speed of 10 Hz and an input shaft torque of 10 N m. Vibration and encoder measurements were collected every 10 min (i.e. every 6000 rotations of the pinion). Three measurements were obtained after calibration and before a catastrophic failure of the pinion teeth, which occurred shortly after 18,000 cycles. At each measurement time the wear model parameter Kwear could be updated (if necessary) with the procedure discussed in Section 4.2. After the first updating (10 min), the wear model parameter Kwear was found to be 2.7×10 6N 1, and the maximum predicted wear depth was estimated at 156 µm (note that this and subsequent wear depth figures represent the combined wear of the two profiles: that of the pinion alone is about 73% of this figure). The trends of RMS and maximum wear depth vs wear cycles (rotations of the pinion) are plotted separately in Figs. 6 and 7. The agreement of the prediction for 12,000 cycles with the corresponding experimental results shows the effectiveness of the model updating procedure. A second updating at 12,000 cycles was deemed not to be necessary, given the already low discrepancy between RMS values. The further projection of RMS and wear depth for a further 6000 cycles shows that this choice was reasonable, giving a prediction error of about 13% on RMS and 7% on wear depth at 18,000 cycles (i.e. 12,000 cycles from the last updating). A sudden failure of the gears (due to the extreme level of wear) then prevented any further updating and prediction, but conceptually, the 13% error of RMS at 18,000 cycles would justify a further updating of Kwear .
In this section, the proposed vibration-based updating of wear prediction scheme is presented by combining the aforementioned dynamic model with the well-known Archard wear model [18] and experimental update methodology. 4.1. Wear model Although many advanced wear models have been proposed using different methodologies and parameter sets, Archard's wear model remains the most commonly used for practical applications, and is chosen in this paper. Neglecting changes in the contact area over the meshing cycle, and differentiating the expression of Ref. [9], we obtain the following form of Archard's law:
dh = Kwear Fv dt
(6)
Herein, h is the wear depth of the gear contact surface, Kwear is the wear coefficient, F is the normal load (i.e., contact force), and v is the sliding velocity at time t (representing the derivative of the sliding distance). The sliding velocity v ( 5) throughout the meshing cycle can be determined by the kinetic formula [19], while the contact force F is obtained by the dynamic model. The Archard model parameter Kwear is affected by a number of factors including material properties, such as hardness and roughness of the two surfaces, and the lubrication condition, which may evolve in the wear process. Since it is very difficult to estimate or directly measure Kwear experimentally, this coefficient is often a major unknown parameter. In this study Kwear is determined experimentally, based on the comparison of simulated and experimental vibrations. The details of this updating methodology will be introduced in the following section.
6. Conclusion and future work In this paper, a new scheme for vibration-based updating is proposed to monitor and predict the gear wear process. The updating procedure is able to track and correct for changes in wear rates, thus allowing analysts to obtain reliable wear predictions with relatively simple modelling tools. Unique to existing studies in gear-wear prediction, the vibration-based updating procedure was successfully tested on a dry run-to-failure test.
4.2. Update methodology The update of the wear coefficient Kwear is obtained based on the comparison of the root mean square (RMS) value of simulated and experimental signals. In particular, the simulated vibration at the bearing housing y6(SIM ) (t ) is compared with the experimental vibration signal y6(EXP ) (t ) measured by the accelerometer B&K 4396 in the corresponding position. The updating procedure is executed iteratively based on a proportional model. In particular, if the ratio of predicted vibrations to experimental vibration is D(i ) =
RMS {y6(SIM , i) (t )} RMS {y6(EXP ) (t )}
, the wear
coefficient for the next iteration is computed as K (i + 1) = D(i)1 K (i) . The iterations are stopped when the absolute error between the simulated and experimental RMS values falls below the predefined threshold (in this case 5% difference). It must be highlighted that the operating conditions of the dynamic model (speed and torque) should be set as close as possible to the actual experimental conditions, to avoid potentially large biases on RMS readings. Luckily, speed measurements and torque estimates (e.g. through current measurements in electromechanical drivetrains) are available in most machines with sufficient criticality to justify a sophisticated condition monitoring system. The methodology discussed in this section requires setting an initial profile, for which a series of options are possible, including random tooth profile deviations (appealing in the case of statistical knowledge
Fig. 6. RMS comparison results: experiment and model. 1414
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methodologies (e.g. random profile deviation based on manufacturing quality) on the initial convergence of the model. Acknowledgement This research was supported by the Australian Government through the Australian Research Council (Discovery Project DP160103501) and the China Scholarship Council (Grant no. 201706070120). References [1] P. Fernandes, C. McDuling, Surface contact fatigue failures in gears, Eng. Fail. Anal. 4 (2) (1997) 99–107. [2] F. Zhao, Z. Tian, X. Liang, M. Xie, An integrated prognostics method for failure time prediction of gears subject to the surface wear failure mode, IEEE Trans. Reliab. (2018). [3] J. Wojnarowski, V. Onishchenko, Tooth wear effects on spur gear dynamics, Mech. Mach. Theory 38 (2) (2003) 161–178. [4] T. Osman, P. Velex, Static and dynamic simulations of mild abrasive wear in widefaced solid spur and helical gears, Mech. Mach. Theory 45 (6) (2010) 911–924. [5] F.K. Choy, V. Polyshchuk, J.J. Zakrajsek, R.F. Handschuh, D.P. Townsend, Analysis of the effects of surface pitting and wear on the vibration of a gear transmission system, Tribol. Int. 29 (1) (1996) 77–83. [6] J. Kuang, A. Lin, The effect of tooth wear on the vibration spectrum of a spur gear pair, J. Vib. Acoust. 123 (3) (2001) 311–317. [7] I. Yesilyurt, F. Gu, A.D. Ball, Gear tooth stiffness reduction measurement using modal analysis and its use in wear fault severity assessment of spur gears, NDT & E Int. 36 (5) (2003) 357–372. [8] C. Yuksel, A. Kahraman, Dynamic tooth loads of planetary gear sets having tooth profile wear, Mech. Mach. Theory 39 (7) (2004) 695–715. [9] H.L. Ding, A. Kahraman, Interactions between nonlinear spur gear dynamics and surface wear, J. Sound Vib. 307 (3–5) (2007) 662–679. [10] A. Kahraman, H.L. Ding, A methodology to predict surface wear of planetary gears under dynamic conditions, Mech. Based Des. Struct. Mach. 38 (4) (2010) 493–515. [11] X.Z. Liu, Y.H. Yang, J. Zhang, Investigation on coupling effects between surface wear and dynamics in a spur gear system, Tribol. Int. 101 (2016) 383–394. [12] A. Flodin, S. Andersson, Simulation of mild wear in spur gears, Wear 207 (1–2) (1997) 16–23. [13] P. Bajpai, A. Kahraman, N. Anderson, A surface wear prediction methodology for parallel-axis gear pairs, J. Tribol. 126 (3) (2004) 597–605. [14] M. Masjedi, M. Khonsari, On the prediction of steady-state wear rate in spur gears, Wear 342 (2015) 234–243. [15] W.D. Mark, Performance-based Gear Metrology: Kinematic-transmission-error Computation and Diagnosis, John Wiley & Sons, Ltd, Chichester, UK, 2013. [16] X.H. Liang, M.J. Zuo, M. Pandey, Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set, Mech. Mach. Theory 76 (2014) 20–38. [17] P. Sweeney, R. Randall, Gear transmission error measurement using phase demodulation, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 210 (3) (1996) 201–213. [18] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (8) (1953) 981–988. [19] S. Li, A. Kahraman, A transient mixed elastohydrodynamic lubrication model for spur gear Pairs, J. Tribol.-Trans. ASME 132 (1) (2010).
Fig. 7. Maximum wear depth comparison results: experiment and model.
Although the model presented in this work has demonstrated its capability for condition-based wear prediction using vibration, there are some limitations in the experimental data collected in the short dry test. In future work, longer and possibly lubricated wear tests will be carried out to fine tune the model so it can be used for wear monitoring and prediction under a range of lubrication conditions (boundary, mixed and full lubrication) and with abrasive wear and/or contact fatigue pitting. In addition, the current updating scheme only corrects for magnitude of wear but does not correct the shape of the profile error. A future area of work will involve the use of more sophisticated vibration features (e.g., gearmesh harmonics) to characterise the wear profile variation over the tooth length. This will also enable the distinction between the effects of different wear mechanisms. Moreover, since the actual operating conditions of the gearbox will experience small fluctuations in real operation, indicators which are more robust than RMS to variable conditions have been included in our future work plan. With varying operating conditions and multiple wear mechanisms, accompanied by more sophisticated vibration features, the updating scheme proposed in this paper will also require further developments. The updating procedure will not only be responsible for the continual adjustment of multiple wear coefficients, but also the tracking of the relative importance of different wear mechanisms and the consideration of time-varying operating conditions. In addition, future studies should include the analysis of the effect of possible profile initialisation
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