Tooth Mesh Stiffness Research Articles

The effect of time-varying operating condition on the crack induced impulses and its application to gearbox tooth crack diagnosis

Gearbox tooth crack diagnosis under Time-Varying Operating Conditions (TVOC) is a challenging issue. TVOC induce both Amplitude Modulation (AM) and Frequency Modulation (FM) effects into gearbox vibration signals, which results in difficulties in tooth crack diagnosis. To overcome this problem, the TVOC-induced AM and FM effects on vibration signals of gearboxes with tooth cracks need to be well studied. Many studies have been reported to investigate the influence of TVOC on gearbox vibration signals using dynamic modelling, some of which only focused on healthy gearboxes while others only considered piecewise constant operating conditions, failing to make comprehensive analyses involving both tooth cracks and TVOC. This paper aims to conduct a comprehensive study on how TVOC affect vibration signals of gearboxes with tooth cracks using dynamic simulation. Firstly, gear tooth mesh stiffness is evaluated considering both tooth crack severity progression and operating condition variations, through which gearbox vibration responses are generated under TVOC. A signal analysis procedure is proposed with its focus placed on the Crack Induced Impulses (CII), which are extracted from gearbox vibration signals using the adaptive harmonic decomposition method or its modified version. Envelope analysis is conducted on the CII to study how the CII are affected by TVOC, and a linear dependence of the AM effect of the CII on TVOC is identified. Based on the identified linear dependence, a novel Condition Indicator (CI), which is sensitive to crack growth while insensitive to operating condition variations, is proposed to track tooth crack severity progression under TVOC. The linear dependence of the AM effect of the CII on TVOC and the effectiveness of the proposed CI for tooth crack diagnosis are demonstrated using both simulated gearbox vibration signals and experimental gearbox vibration datasets.

Analysis of spur gearbox dynamics considering tooth lubrication and tooth crack severity progression

A novel tribo-dynamic model of a spur gearbox considering both tooth lubrication and tooth crack is proposed, which integrates an elastohydrodynamic lubrication (EHL) model of a spur gear pair into a gearbox lumped parameter model. The combined stiffness and combined damping of a spur gear pair with a tooth crack are obtained using the oil film stiffness and oil film damping provided by the EHL model and the tooth mesh stiffness and tooth mesh damping calculated by the potential energy method. Effects of tooth lubrication and tooth crack severity progression on combined stiffness, combined damping, and gearbox vibration responses are studied. It is found that considering the tooth lubrication in gearbox dynamic modeling leads to a backward shift of the frequency components induced by tooth crack. A comparison has been made between the proposed tribo-dynamic model and the conventional model of a spur gearbox, and the effectiveness of RMS and Kurtosis for tracking tooth crack severity progression is investigated in the presence of tooth lubrication. The comparison results reveal that the consideration of tooth lubrication overall reflects tooth crack signatures better.

Analysis of vibration signals and detection for multiple tooth cracks in spur gearboxes

Tooth crack diagnosis is important to ensure the reliability of gearboxes. However, most reported studies on tooth crack diagnosis only involved the scenario of one tooth crack in a gearbox, which is not always the case since gearboxes may also suffer from multiple tooth cracks in the industry. To overcome this problem, this study first brings some insights into the vibration characteristics of a spur gearbox with multiple tooth cracks using dynamic modelling. Three scenarios of multiple tooth cracks are considered, including two nonadjacent tooth cracks on the pinion and a healthy gear, two adjacent tooth cracks on the pinion and a healthy gear, and one tooth crack on the pinion and one tooth crack on the gear. Tooth mesh stiffness is analytically evaluated using the potential energy method and is further inserted into the spur gearbox dynamic model to generate vibration responses. Analyses of mesh stiffness and vibration responses are conducted in both time and frequency domains. Besides, a novel method focusing on the crack induced impulses (CII) is proposed to detect the number and locations of multiple tooth cracks. Firstly, the CII are extracted from gearbox vibration signals using a new strategy based on the singular value decomposition. Time synchronous average (TSA) is conducted on the CII to get the TSA signals for both the pinion and the gear. Afterwards, detection of the number and locations of multiple tooth cracks is achieved via analyzing the squared envelopes of the TSA signals for both the pinion and the gear. The obtained insights into vibration characteristics for multiple tooth cracks and the effectiveness of the proposed method for detecting the number and locations of multiple tooth cracks are demonstrated using both simulated gearbox vibration signals and experimental gearbox vibration datasets.

Influence of tooth root cracks on the mesh stiffness of asymmetric spur gear pair with different backup ratios

Gears are critical machine elements that transmit power and motion in diverse implementation fields. Over time, gears may produce a series of faults due to harsh operating conditions, fatigue, manufacturing errors, etc., leading to severe performance degradation. During the meshing process, the stiffness of a single tooth controls the load sharing, vibration, and noise characteristics of a geared system. An undetected fault could decrease the gear stiffness and thus may lead to a fatal breakdown, substantial economic losses, or even human casualties in safety-critical applications such as helicopters, high-speed trains, and wind turbines. Hence, the accurate quantification of the gear stiffness emerges as an important research area to obtain reliable gear designs. With this in mind, the asymmetric tooth concept offers superior bending strength, fatigue propagation life, and the ability to lessen vibration over the standard (symmetric) designs in applications where unidirectional loadings are predominant. This study investigates the influence of tooth root cracks on the single-tooth and meshing stiffness characteristics of the standard and asymmetric involute spur gears. To this end, the numerical crack propagation paths obtained in our previous works were introduced to the created 3D CAD geometries. Subsequently, the single tooth stiffness of both healthy and cracked (2 5%-50%-75%-100%) gears was calculated through the ANSYS® Workbench, and the time-varying mesh stiffness was obtained. The present study evaluated the effects of backup ratio and the tooth asymmetry on the spur gears’ meshing stiffness characteristics simultaneously and further expanded the scope of the research work. The results indicated that the single tooth stiffness and mesh stiffness could be improved by 35% and 22%, respectively, as the drive side pressure angle increased from 20° to 35°. It has been noted that the gear stiffness decreased as the crack level increased, while the increment of the backup ratio further increased the reduction in the stiffness. The findings could provide significant outputs for a better understanding of the influence of tooth asymmetry on the gear dynamics characteristics, life prediction, and early fault diagnosis.

Analytical calculation models for mesh stiffness and backlash of spur gears under temperature effects

The temperature has a great contribution to the mesh stiffness and backlash of the gear pair. Presence of thermal deformation caused by temperature will complicate the gear teeth interaction. In this paper, the thermal time-varying stiffness model and thermal time-varying backlash model are proposed with the consideration of tooth profile error and total thermo-elastic deformation consists of the teeth deformation, teeth contact deformation, and gear body-induced deformation. The key parameters of thermo-elastic coupling deformation affected by temperature are calculated. Based on the proposed models, the influencing mechanism of temperature on the tooth profile error, mesh stiffness, total deformation, and backlash are revealed. The effects of shaft radius and torque load on the thermal stiffness and thermal backlash are studied. The proposed thermal stiffness and backlash calculation model are proven to be more comprehensive and the correctness is validated.

Dynamic modelling of the gear system under non-stationary conditions using the iterative convergence of the tooth mesh stiffness

This paper presents a new gear dynamic model for the transmission system under the non-stationary condition with the consideration of the coupling between the external load and the internal gear excitation. The gear mesh stiffness variation is found to be the main internal excitation and therefore, the effect of the varying external load on the gear mesh stiffness is evaluated firstly, which can then be incorporated into the gear dynamic model. When the gearbox is subjected to a wide range of external load, the mesh stiffness varies correspondingly and an iterative process is proposed to ensure the convergence of the gear dynamic simulation at each step. The dynamic responses from the model with and without the iterative process have been compared. It indicates that the proposed model can successfully incorporate the external-internal coupling effect in gearbox and further shows that the operating condition has a significant influence on the gear response. The proposed model can provide an effective tool to improve the accuracy of gear dynamic modelling and improve the accuracy of gear fault diagnostics under non-stationary conditions.

Dynamic Characteristics Analysis of the Composite Motion of Curve-Face Gear

Abstract The unified model was proposed for the three configurations of the composite motion of curve-face gear pair. A coupled bending-torsional-axial dynamic model with multi degree-of-freedom transmission system was proposed comprehensively considering nonlinear parameters such as tooth mesh stiffness, static transmission error and backlash. The fourth-order Runge–Kutta numerical method was applied to solve the dimensionless dynamic differential equations of the system, and then the dynamic responses of the system were investigated by time history, phase-plane portrait, poincare map, FFT spectrum and bifurcation diagram. It was found that the dynamic responses of the three configurations were complex. It presented quasi-periodic motion only under certain configuration and parameter conditions. With the increase of meshing frequency, the system state was characterized by the alternation between chaotic motion and periodic motion. Eccentricity ratio of displacement curve of curve-face gear pair mainly affected the rotational frequency of the system. As the eccentricity ratio increased, the dominant frequency of the system gradually changed from meshing frequency to rotational frequency. The correctness of the theoretical model was verified through corresponding experimental studies.

A study of effects of tooth surface wear on time-varying mesh stiffness of external spur gear considering wear evolution process

Gear meshing surfaces are readily exposed to wear due to inadequate lubrication and long-time operation. Accumulative wear causes variations of gear tooth profile, resulting in variations of gear tooth's mesh stiffness and gear system's dynamic characteristics. This study presents an evaluation of influences of tooth surface wear (TSW) on stiffness of the gear mesh considering wear evolution process. Specifically, a wear prediction model for external spur gear (ESG) was established according to the Archard's wear equation. Considering the worn tooth profile, a modified analytical time-varying mesh stiffness (TVMS) model was derived by the potential energy method. The modified TVMS model was then applied for an ESG pair and the numerical results demonstrate that TSW may lead to significant reductions of mesh stiffness during the double tooth contact period. Furthermore, effects of TSW on TVMS were quantified and the results indicate that the mean decrease of mesh stiffness is almost proportional to the maximum tooth wear in the initial wear evolution process.

Influence of simultaneous time-varying bearing and tooth mesh stiffness fluctuations on spur gear pair vibration

This work investigates the influence of time-varying roller bearing stiffness on gear-bearing system vibration. An isolated bearing model established by a finite element/contact mechanics approach is introduced to analyze the variation of bearing stiffness due to the continuous change of roller circumferential locations. Then, dynamic gear-bearing models with constant and time-varying bearing stiffness are built by analytical and finite element methods. Comparisons are conducted among the dynamic response of the different gear-bearing models to verify their accuracy and to indicate the special features introduced by the time-varying bearing stiffness. Additional resonances (or parametric instabilities) are caused by the time-varying bearing stiffness, and modulation between the mesh frequency and the bearing ball pass frequency is observed in the dynamic response. Theoretical explanations of the parametric instabilities of the system with simultaneous mesh and bearing stiffness fluctuations are derived using perturbation analysis and verified with Floquet theory.

Analysis of Gear Wheel Body Influence on Gearing Stiffness

The development of modern machinery and production facilities is characterized by constantly increasing performance parameters at the decreasing weight of the machine. It is also a characteristic of the design of the body shape of larger wheel gears. The lowering the weight of the body of wheel gears affects to the deformation and mesh stiffness of the gearing. This work is dealing with the design of large gears, which is assessed based on tooth deformation and mesh stiffness of gearing. The task of this work is decide the design of a body of big gear wheels made as monolithic unit, relative on tooth deformation in meshing, accordingly on the mesh stiffness of gearing. Tooth deformation is calculated by finite element method.

Gear tooth mesh stiffness: A comparison of calculation approaches

This work compares spur gear tooth mesh stiffness calculations using two approaches. The first is a common approach from the literature that calculates the mesh stiffness by dividing the mesh force by the mesh deflection, which we call the average slope method. The second approach calculates the local slope of the force–deflection curve about a nominal deflection. The two approaches result in meaningfully different mesh stiffness predictions that persist for wide ranges of applied torque and for gear teeth with tooth surface modifications. It is shown that each calculation approach has its own distinct use, broadly divided as average slope mesh stiffness for static analyses and local slope for dynamic analyses. Furthermore, the two stiffness calculation approaches lead to different vibration models. This means that for vibration analyses the choice is not solely which of the two stiffnesses to use but also how to implement that stiffness appropriately in a model. Even though the mesh stiffnesses in this work are calculated using a finite element/contact mechanics approach, the findings are equally valid for mesh stiffnesses obtained from conventional finite element methods, analytical models, and experiments.

Dynamic modelling of flexibly supported gears using iterative convergence of tooth mesh stiffness

This paper presents a new gear dynamic model for flexibly supported gear sets aiming to improve the accuracy of gear fault diagnostic methods. In the model, the operating gear centre distance, which can affect the gear design parameters, like the gear mesh stiffness, has been selected as the iteration criteria because it will significantly deviate from its nominal value for a flexible supported gearset when it is operating. The FEA method was developed for calculation of the gear mesh stiffnesses with varying gear centre distance, which can then be incorporated by iteration into the gear dynamic model. The dynamic simulation results from previous models that neglect the operating gear centre distance change and those from the new model that incorporate the operating gear centre distance change were obtained by numerical integration of the differential equations of motion using the Newmark method. Some common diagnostic tools were utilized to investigate the difference and comparison of the fault diagnostic results between the two models. The results of this paper indicate that the major difference between the two diagnostic results for the cracked tooth exists in the extended duration of the crack event and in changes to the phase modulation of the coherent time synchronous averaged signal even though other notable differences from other diagnostic results can also be observed.

The spur planetary gear torsional stiffness and its crack sensitivity under quasi-static conditions

Abstract The sun–planet and ring–planet tooth mesh stiffness variations and the resulting transmission errors are the main internal vibration generation mechanisms for planetary gear systems. This paper presents the results of torsional stiffness analysis of involute spur planetary gear systems in mesh using finite element methods. A planetary gear model with three planet gears and fixed ring gear and its subsystem models have been developed to study the subsystem and overall torsional stiffnesses. Based on the analysis of torsional mesh stiffness, predictive models for single branch sun–planet–ring and overall planetary gear torsional stiffnesses have been proposed. A crack coefficient was introduced to the sun–planet and ring–planet meshes to predict the effect and sensitivity of changes to the overall torsional mesh stiffness. The resulting mesh stiffness crack sensitivity of the overall gear system was analysed under quasi-static conditions. It was found that the carrier arm stiffness has great influence on the crack sensitivity while the overall stiffness was most sensitive to the crack on the sun–planet mesh.

Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears

In the last research [S. Li, Effects of machining errors, assembly errors and tooth modifications on load-carrying capacity, load-sharing rate and transmission error of a pair of spur gear, Mech. Mach. Theory 42 (2007) 698–726], effects of machining errors, assembly errors and lead crowning on tooth surface contact stresses (CS), root bending stresses, load-sharing ratios (LSR) and transmission errors of a pair of spur gears were investigated through performing loaded tooth contact analysis (LTCA) with developed finite element method (FEM) programs. But this research couldn't investigate the effects of tooth profile modification and lead relieving on tooth engagements. Also, the effects of machining errors, assembly errors and tooth modifications on tooth mesh stiffness (MS) couldn't be investigated. So, as a continuous study of the last research, this paper investigates the effects of tooth profile modification and lead relieving on tooth engagements of a pair of spur gears and the effects of misalignment error of gear shafts on the plane of action, tooth lead crowing and transmitted torque on tooth MS. An arc curve is used to modify tooth profiles of a pair of spur gears in this paper. This is because this method is used very popularly for the spur gears. Methods used in the last research are also used here to investigate the effects of the tooth profile modification, lead relieving and transmitted torque on tooth engagements. Based on the results, it is found that the tooth profile modification and lead relieving have significant effects on tooth CS, LSR and MS. It is also found that transmitted torque has a little effect on tooth MS, but has no effect on LSR of the gears. For the lead-relieved gears, calculation results show that edge-loads happened at the joint parts of the relieved part and the non-relieved part of tooth lead when the lead is relieved with straight lines. Since the edge-loads resulted in greater contact stresses at the joint parts and weakened tooth contact strength, attention must be paid to the lead relieving. It is necessary to reduce the edge-loads as small as possible through making the joint part smooth when the lead relieving is made.

Dynamic analysis of wind turbine drive train subjected to nonstationary wind load excitation

The dynamic analysis of wind turbine drive train is presented in this paper. A typical wind turbine drive train consists of a rotor, gearbox and generator. The dynamic modelling of epicyclic gearbox that exists in wind turbine is challenging due to the fact that it has both rotating and orbiting gears. The dynamic equations of motion are obtained based on the rigid multibody modelling with discrete flexibility approach by Lagrange’s formulation. The dynamic model accounts for the time varying gear tooth mesh stiffness, linear stiffness of bearings and torsional shaft stiffness. The aerodynamic torque that a wind turbine drive train subjected to, is modelled based on the simplified method for load calculation in wind turbine, Danish Standard DS472. The characteristic load value acting per unit length at the two-thirds length of the blade is used for calculating the total load of the torsional moment. The vibration signals that are obtained from wind turbine drive train are nonlinear and nonstationary in nature. This is due to the fact that the applied torque load on drive train is nonlinear and nonstationary in nature. The coupled dynamic model of 18 degrees of freedom is solved for responses in time and frequency domains for some nonstationary wind load realizations. The dynamic responses of the system, contact forces between gear tooth pairs in time and frequency domains are obtained numerically. The study envisages that this dynamic model of wind turbine drive train is very useful for subsequent studies on condition monitoring.

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

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.

Analytical determination of back-side contact gear mesh stiffness

Back-side gear tooth contact happens when anti-backlash (or scissor) gears are used, tooth wedging or tight mesh occurs, or vibration amplitudes are high enough that teeth separate and pass the backlash zone. An accurate description of the back-side gear tooth mesh stiffness is needed to study gear mechanics in such cases. This work studies the time-varying back-side mesh stiffness and its correlation with backlash by analyzing the relationship between the drive-side and back-side mesh stiffnesses. Results of this work yield the general form of the back-side mesh stiffness in terms of the known drive-side mesh stiffness for an arbitrary gear pair. The analytical results are confirmed by simulation results from gear contact analysis software that precisely tracks drive- and back-side gear tooth contact.

Experimental Investigations of Noise Control in Planetary Gear Set by Phasing

Now a days reduction of gear noise and resulting vibrations has received much attention of the researchers. The internal excitation caused by the variation in tooth mesh stiffness is a key factor in causing vibration. Therefore to reduce gear noise and vibrations several techniques have been proposed in recent years. In this research the experimental work is carried out to study the effect of planet phasing on noise and subsequent resulting vibrations of Nylon-6 planetary gear drive. For this purpose experimental set-up was built and trials were conducted for two different arrangements (i.e., with phasing and without phasing) and it is observed that the noise level and resulting vibrations were reduced by planet phasing arrangement. So from the experimental results it is observed that by applying the meshing phase difference one can reduce planetary gear set noise and vibrations.

Experimental investigation of spur gear tooth mesh stiffness in the presence of crack using photoelasticity technique

Gear mesh stiffness plays a very important role in gear dynamics and it varies in the presence of gear fault such as crack. The measurement of stress intensity factor can lead to the determination of gear tooth mesh stiffness variation in the presence of crack in a spur gear system. In this paper, the technique of conventional photoelasticity has been revisited to explore the possibility of using it as a supplementary technique to experimentally measure the variation of gear mesh stiffness. An attempt has been made to calculate the variation of mesh stiffness for a pinion having a cracked tooth and a gear tooth with no crack of a spur gear pair. An analytical methodology based on elastic strain energy method in conjunction with total potential energy model has been adopted and implemented within the mesh stiffness calculations. To visualize the state of stress in a structure using finite element and other currently available methods, photoelasticity is considered to be one of the oldest and most developed experimental technique. An experimental methodology based on conventional photo-elasticity technique for computing stress intensity factor (SIF) for cracked spur gear tooth is presented for different single tooth contact position and crack length. The relation between contact position, crack length, crack configuration, SIF and the variation of total effective mesh stiffness have been quantified. Finally, a comparison has been made and the results obtained from finite element method (FEM) based on linear elastic fracture mechanics (LEFM), analytical method and proposed experimental method has been outlined.

S111016 平歯車の歯のかみあい剛性に及ぼす歯車加工精度及び歯面修整の影響

Three-dimensional, finite element method and a face-contact model of the gear teeth are combined with a mathematical programming method to do loaded tooth contact analysis and mesh stiffness calculations of a pair of spur gears with machining errors and tooth modifications. Effects of the tooth profile deviations, tooth pitch errors, tooth profile modification, tooth lead crowning and relieving on the mesh stiffness are investigated. It is found that the tooth profile deviations, tooth pitch errors and tooth profile modifications have significant effects on the mesh stiffness. It is also found that the lead crowning and relieving make the mesh stiffness smaller, but have no effects on load sharing ratios of the gears. Since the lead crowning makes the maximum contact stress of the teeth much greater and the lead relieving make the teeth have edge loads, it is necessary to pay a special attention to the quantity of the lead crowning and the length of the lead relieving, not to make the quantity of the crowning and the length of the relieving too larger.