Potential Energy Method Research Articles

A temperature rise calculation model of wind turbine gearbox gear considering crack fault and tooth number difference

AbstractThe crack fault of gear is usually accompanied by temperature rise. Therefore, to master the temperature characteristics of the cracked gear of the wind turbine gearbox, a temperature rise calculation model of wind turbine gearbox gear considering crack fault and tooth number difference is proposed. Firstly, the time‐varying meshing stiffness model of the cracked gear considering the tooth number difference is established based on the potential energy method. Secondly, the calculation method of the meshing surface normal load of the cracked gear is deduced. Thirdly, the temperature rise calculation model of the meshing surface of the cracked gear is constructed based on Blok flash temperature theory. Finally, the data of the high‐speed gear of a wind turbine gearbox in northern China is selected for simulation verification. By comparing with the finite element method, the effectiveness of the proposed method is verified. The simulation results reveal the gear temperature characteristics of the wind turbine gearbox with different crack ratios. The research can provide some theoretical support for the accurate fault diagnosis and maintenance of wind turbine gearbox, and can also be applied to the fault diagnosis of gear cracks in other mechanical structures with a large transmission ratio.

Noncircular Slip Surface Search on Slopes Based on Minimum Potential Energy Method and Improved SA Algorithm

ABSTRACTThe limit equilibrium method has been widely used in the study of searching the slip surface of slopes. However, the method ignores the deformation characteristics of the rock mass and assumes that the shape of the slip surface is circular, which is quite different from the actual situation of the slope. For this reason, this paper proposes a fast search method for noncircular slip surface considering the deformation characteristics of the rock mass. The method is able to calculate the compression and shear deformation energies stored in the slip surface, as well as the virtual displacement generated by the slide mass when the slope is in a critical equilibrium state. The direction of motion of the slide mass is further calculated from the magnitude of the virtual displacement. In addition, this paper improves the generation of new solutions in the simulated annealing (SA) algorithm for the structural characteristics of the slip surface of the slope, thus achieving a fast search of the slip surface. Finally, the method of this paper is compared with the test question of ACADS and the simulation results of the finite difference method (FDM) to verify the effectiveness of the method of this paper.

A modified solution for evaluating the sliding direction and stability of 3D slopes

In this paper, we present an improved and efficient analytical solution for evaluating the stability of 3D slopes based on the minimum potential energy method. The equilibrium equation of the landslide is employed to determine the shear stress on the failure surface, and a new formulation for shear potential energy is introduced. The vector-combined direction of the force acting on the landslide is considered equivalent to the sliding direction (SD). Two classic examples are reanalyzed to validate the accuracy and performance of the proposed method. The analysis demonstrates that the obtained results align well with reference solutions and tend to be conservative in engineering applications. Parameter investigations are conducted to explore the influences of SD, shear strength parameters, slope angle, and soil unit weight on the safety factor (SF) of 3D slopes. The findings indicate that a larger SD corresponds to a less stable slope. Finally, the effects of optimization parameters, landslide volume, and surcharge on the location of the failure surface, SF, and SD are also examined. The developed analytical approach offers a novel and straightforward solution with sufficient accuracy and eliminates the need for iteration in determining the SF, making it a valuable complementary method for assessing the stability of 3D slopes.

Solving time-varying mesh stiffness of spur gears based on improved potential energy method

Meshing stiffness is an important factor in gear dynamics analysis. At present, there are many ways to calculate the meshing stiffness of gears, and we usually use the potential energy method to solve and calculate the meshing stiffness of healthy (faulty) gears. In this paper, the potential energy method, the improved potential energy method and the theoretical calculation method are used to solve the time-varying stiffness of the healthy spur gear pair, and the solution results are compared, which verifies the feasibility of the improved potential energy method to solve the gear meshing stiffness.

2D calculation method on global stability safety factor of diaphragm wall slurry trench based on minimum potential energy principle

In order to limit the phenomenon that the global stability safety factor (GSF) of the diaphragm wall slurry trench calculated by the two-dimensional (2D) wedge method is too small, the 2D calculation model on global instability of trench wall is established by assuming the whole critical failure surface (CFS) is a quarter elliptic cylindrical surface, and the determination method of CFS is given too. In view of the successful application of the minimum potential energy principle in slope stability checking calculation and on the basis of the new model established, the 2D minimum potential energy method for calculating the GSF is proposed according to the minimum potential energy principle. The correctness of the 2D calculation model built and the rationality of the 2D calculation method proposed are verified based on ten engineering cases, and the sensitivity of the GSF to various parameters is analyzed on the basis of six design examples. Finally, the new calculation method is applied to the global stability checking calculation of trench wall in practical engineering, and the result shows the new calculation method given in this paper is still conservative, so the result calculated by the new calculation method is advised to be multiplied by a modification factor of 1.67 in practical application.

Effect of tooth surface wear on dynamic characteristics of spur gear transmission system

The meshing stiffness and system dynamic characteristics caused by gear wear fault are studied. Firstly, potential energy method was used to model and solve the meshing stiffness of gears and the effect of wear on the meshing stiffness and static transfer error was analyzed. A 6-dof spur gear dynamics model was established considering various nonlinear factors, and the dynamic responses of the spur gear system under various wear degrees were analyzed by using time domain, spectrum, phase plane, and Poincare mapping. Finally, the visual test platform of the gear box is built, and the wear condition and dynamic response of the gear box under different wear cycles are analyzed. Results show that the meshing stiffness of gear decreases due to wear, and the meshing stiffness of double tooth decreases more than that of single tooth. In addition, the static transmission errors caused by wear changes periodically with meshing frequency. Wear causes the vibration response of gear system to become complex gradually, and the nonlinear of the system is enhanced, which changes from single period motion to quasi period motion. Results can provide theoretical support for gear wear reduction, life extension, vibration reduction, and noise reduction and lubrication improvement.

An investigation into the effects of profile shifts on helical gear mesh stiffness

Time-varying meshing stiffness (TVMS) is a significant nonlinear periodic excitation in helical gears, it poses significant impacts on gear system’s dynamic characteristics and fatigue life. However, the specific impact of profile shift on it remains unclear. This article establishes an analytical model for the TVMS of helical gears by utilizing the slicing method to address this gap based on the potential energy method, and quantitatively analyze the TVMS with different modification coefficients. The research found that the helical gear’s TVMS decreases with the increase of modification coefficient for individual profile shifts. For the equal compound profile shifts, it rises with the increase of modification coefficient of pinion. For the unequal compound profile shifts, the positive drive increases the TVMS, but the negative drive to the contrary. The research achievements provide valuable guidelines for dynamic design and structure optimization for helical gears.

Study on time-varying meshing stiffness of helical gears with root crack faults based on improved potential energy method

Study on time-varying meshing stiffness of helical gears with root crack faults based on improved potential energy method

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

Dynamic characteristics analysis of gear tooth modification in a three-stage gear system

Gears serve as vital elements in high-efficiency mechanical transmissions, playing a pivotal role in vehicles. Employing gear modification has the capacity to significantly enhance the dynamic properties of the Planetary Gear Transmission (PGT) systems. In this study, the investigation has been directed toward a three-stage PGT model system with a comprehensive consideration of various vibration-related factors that affect the dynamic behavior of the system. These factors are encompassed by Time Varying Mesh Stiffness (TVMS), torsion and bending forces experienced by connecting shafts, dynamic Transmission Error (TE), impact of gear meshing eccentric load forces, inherent flexibility in the supporting shaft structure, and influence of torque and power fluctuations originating from a multi-cylinder engine. The estimation of TVMS is carried out for both external-external and external-internal teeth meshing pairs, utilizing the Potential Energy Method (PEM). Subsequently, this study employs the lumped parameter method to establish a model with a specific objective of investigating the lateral-torsional-coupling behavior within adopted PGT model system. The lateral vibration displacement characteristics of the PGT model system is deeply analyzed, revealing the influence of speed on lateral vibration and displacement frequency domain characteristics through a 3D waterfall diagram. Furthermore, a Tooth Profile Modification (TPM) method is applied utilizing two distinct parameters namely [Formula: see text] TPM parameter and [Formula: see text] TPM parameter. The objective is to mitigate vibrations and evaluate the performance of the adopted PGT model after employing each TPM parameter through both TVMS and Root Mean Square (RMS).

Nonlinear dynamics simulation of the helical gear transmission system for high-speed train

Considering the bending-torsional-axial-coupled vibration of the helical gear transmission system, a 10 degrees of freedom high-speed train gear transmission system model, including motor and load, is established based on the lumped parameter method. In addition, the potential energy method based on the slicing principle is used to accurately calculate the time-varying meshing stiffness of helical gears. Furthermore, the influence of geometric parameters of helical gears on the time-varying meshing stiffness is studied, and the impact of time-varying meshing stiffness on the system is further investigated. Meanwhile, by analyzing the dynamic phenomena of the system under different parameter excitations under different parameter excitations, the influence of internal and external excitation parameters such as gear speed, wheel rail creep force, support stiffness, and support damping on the stability of high-speed train gear transmission system is deeply studied.

Stiffness Modeling of Tooth Interior Fatigue and Tooth Flank Fracture and Their Fault Characteristics in Electromechanical Coupling Systems

Tooth flank fracture (TFF) and tooth interior fatigue fracture (TIFF) usually occur on case-harden gears in electromechanical coupling systems, both starting from the core caused by metal fatigue. The cracks propagate rapidly compared with the total life of a gear, so that it is necessary to detect and identify the faults in time when they take place in order to avoid danger. In this paper, the time-varying mesh stiffness model of two types of faults is established by the potential energy method and validating by the finite element method. Based on the stiffness model, the influence of faults on the electromechanical system and their fault characteristics are analyzed, which provides a theoretical basis for the health detection of electromechanical coupling systems. The results indicate that in the early stage of the three faults, i.e., tooth root fracture, tooth interior fatigue fracture, and tooth flank fracture, it is hard to discriminate the faults. But after the second stage, the faults can be distinguished by the vibration amplitude, frequency band components, and phase diagrams. The TFF can be discerned into different stages by frequency domains and phase diagrams. When the fracture occurs completely, three faults can be easily distinguished by the time domain and phase diagram. In the frequency domain, the TRF can also be distinguished from the other two faults by referring to the sideband component near the meshing frequency.

Establishment of the dynamic BTAP model and load-sharing analysis of herringbone gear planetary system with cracks

To reveal the influence of component cracks on the dynamic load-sharing characteristics of the herringbone gear planetary system (HGPS), a 55-DOF bending-torsional-axial-pendular (BTAP) nonlinear dynamic model of the system was constructed, taking into account parameters of different crack degrees (depth, angle, length), as well as factors such as manufacturing error (ME) and installation error (IE), time-varying mesh stiffness (TVMS), time-varying support stiffness, damping, and friction. The TVMS model of components with cracks was established using the cumulative potential energy method, and the mathematical quantification relationship between different crack parameters and TVMS was studied. Based on the theory of elastohydrodynamic lubrication (EHL), calculate the time-varying friction coefficient and friction force of each gear pair. Using the nonlinear elastic theory of rolling elements and raceways, the time-varying support stiffness of each component is obtained. Using the Runge-Kutta numerical integration method to solve the dynamic differential equations, and analyzed the load-sharing characteristics and sensitivity influence law of the system when the sun gear, planetary gear, and internal gear ring contain cracks respectively. Built a vibration test bench for the HGPS with cracks, and the experimental results were compared with the theoretical results. The results show that as the crack intensifies, the TVMS value decreases at the crack tooth. When the sun gear contains cracks, there will be obvious protruding excitation in the system’s load-sharing curve, and the internal meshing load-sharing characteristics are better than the external meshing load-sharing characteristics. At the same time, as the degree of cracks increases, the system load-sharing characteristics first improve and then deteriorate. At the same crack, as manufacturing and installation errors increase, the load-sharing characteristics of the internal and external meshing pairs deteriorate. When the internal gear ring contains cracks, the dynamic load-sharing coefficients of the internal and external meshing pairs of the system are 7.96 and 4.10, respectively, which have the highest impact on the dynamic load-sharing coefficient and sensitivity of the internal and external meshing pairs of the system. The experimental results are consistent with the theoretical results, verifying the correctness of the model.

Analysis of vibration and load-sharing characteristics of two-stage planetary gear transmission system with cracks

To reveal the influence of cracks on the dynamic characteristics of a two-stage planetary gear transmission system (TSPGTS), a nonlinear dynamic model with 30-DOF in bending and torsion was constructed, taking into account parameters of different crack degrees (depth, angle, length), as well as factors such as error, time-varying meshing stiffness (TVMS), and damping. In the model, the TVMS model of components with cracks was established using the potential energy method. By incorporating the crack factor into the model, time domain, frequency domain, phase diagram, and Poincare diagram characterizing the dynamic load characteristics were obtained, as well as the curve of dynamic load-sharing characteristics changing with cracks. The influence of different crack degrees on the displacement vibration and dynamic load characteristics of the system was analyzed, and vibration testing experiments were conducted on the system with planetary gear cracks. The results indicate that when the gear contains cracks, it will lead to a decrease in the stiffness of the system, when cracks appear on the II-stage planetary gear, the system will experience impact components with intervals of the rotation period of the II-stage planetary gear, and obvious sidebands will appear near the meshing frequency doubling. These situations will become more pronounced as the degree of cracks intensifies. As the degree of cracks in the II-stage planetary gear increases, the dynamic load-sharing characteristics of the TSPGTS will first improve and then deteriorate, and as the degree of cracks increases, the external and internal dynamic load coefficients continue to increase. Through experiments, the influence of normal gears and cracked gears on system vibration changes was compared and analyzed, and the theoretical results were in good agreement with experimental results, verifying the correctness of the theoretical model. This provides a theoretical basis for fault diagnosis and reliability research of the system.

A novel framework for ascertaining the safety factor and critical slip surface of 3D slopes reinforced with piles

The pile is a widely used reinforcement device in geotechnical engineering. For the present paper, we propose a new solution for calculating the safety factor (SF) of the 3D slope reinforced with piles and determining the critical slip surface (CSS) in the framework of minimum potential energy method. In particular, the mobilized shear stress on failure surface is determined by the equilibrium equation of forces operating on slip body. Meanwhile, the moving direction of 3D slope is equivalent to the direction of combined force acting on slip body. Additionally, the present method is combined with the genetic algorithm to identify CSS, and a novel criterion is proposed to locate the shape and position of CSS of a 3D slope using the total potential energy of landslide system as the objective function. The validation and comparison of results indicate that the SF obtained by the introduced method are close to reference solutions, and SF shows an obvious negative correlation with sliding direction (SD). It is also found that the support effect of pile is related to the pile location and the pile density. Meanwhile, the optimal pile position depends on the shape of failure surface and shear strength parameter of soil. Moreover, the effects of some parameters on SD of 3D slopes reinforced with piles are also studied.

Structural morphology and surface recrystallization properties of GaN nanoparticles with different sizes during sintering

GaN is a promising third-generation semiconductor material for fabricating nanodevices with nanoscale dimensions. In particular, thick-film sensors fabricated with GaN nanoparticles (NPs) have good gas-sensing characteristics. This study used molecular dynamics simulation to investigate the effect of temperature and size on the structural morphology and surface recrystallization of GaN NPs during sintering. The diffusion behavior and changes in the structural morphology of atoms were analyzed based on the shrinkage ratio, mean square displacement, potential energy, and other methods. The results show that during sintering, the surface of two GaN NPs experienced recrystallization, which expanded with increasing temperature; however, the obvious sintering phenomenon occurred at 3000 K. Interestingly, the surface recrystallization of NPs exhibited a lamellar structure formed through a specific stacking method of the crystal structure. However, when the size of one of the two NPs was reduced, the small NP rapidly melted and was induced to crystallize into a complete NP (having the same crystal orientation as the large NP) by the large NP. Moreover, NPs underwent rigid-body rotation and relative rotation during sintering; however, plastic deformation and dislocations resulting from rotation were only found in the neck of NPs of the same size. The study results provide valuable insights at the atomic scale regarding the sintering of GaN NPs and a useful reference for preparing and processing nanodevices.

Variable stiffness-based vibration prediction for full coupling model of gearbox

Vibration is a problem worthy of attention in gearbox design. In the case of high speed and heavy load, vibration may be more significant, so some methods are needed to predict the vibration of the gearbox. At present, most vibration prediction methods have the problems of few research parameters, low fidelity and a large amount of calculation. In this paper, a fully coupled finite method model of variable stiffness gearbox based on lumped parameter method (LPM) is proposed, and the vibration prediction of gearbox housing is carried out considering time-varying meshing stiffness (TVMS), transmission error and other factors. Firstly, a new stiffness calculation method was proposed based on finite element theory and potential energy method, which can accurately solve the TVMS of fault gear. At the same time, the comprehensive meshing error of gear was solved. At then, the TVMS and comprehensive meshing error of the healthy gear and fault gear were brought into the dynamic equation of two-stage spur gear to obtain meshing force. After that, the meshing force was used as an excitation to solve the vibration of the gearbox housing. Finally, the vibration regularity of the health gearbox and fault gearbox was compared. The sensitivity of each gear shaft to the vibration caused by fault and the vibration regularity of different bearing seats of the same gear shaft was explored. The effectiveness of the simulation results was verified by experiments.

Measurement and evaluation of time-varying meshing stiffness of faulty gears under different types

When operating under complex conditions, gear failure is inevitable, and gears are often accompanied by multiple failures. In view of this phenomenon, this paper establishes a pitting crack coupling failure model and sets four different fault levels, from healthy to severe. Time-varying meshing stiffness (TVMS) decreases with the occurrence of faults, such that it could be employed as a gauge to assess the level of failure. Firstly, considering that the base circle and the tooth root circle do not coincide, the improved potential energy method is used to solve the TVMS under different fault degrees. Secondly, the effects of crack depth, angle, pitting radius, depth and number on meshing stiffness are studied. Then, the result of the potential energy method is verified by the finite element method, and the consequence shows that the difference between the two is within 5%, which indicates the effectiveness of the proposed model. Finally, when the crack degree is certain, the influence of pitting evolution on the meshing stiffness of the coupling model is studied.

Investigation of vibration characteristics of gear-bearing transmission system considering the effect of profile shifted coefficient

In this paper, a dynamics model of the gear-bearing transmission system with bending and torsion coupling is established, and the influence of gear shifted on the vibration characteristics of the gear system is considered. First, the time-varying mesh stiffness of the profile shifted gear is derived by using the potential energy method. Then, based on the Lagrange equation numerical method, the vibration differential equation of the model is obtained, and the differential equation is solved by the fourth-order Runge-Kutta method. Finally, the influence of the profile shifted coefficient and input torque on the vibration characteristics of the gear transmission system is analyzed by using the time domain graph and the frequency domain graph. The conclusions drawn from this study can help engineers better understand the running performance and vibration characteristics of profile shifted gear.

BTAP dynamic model establishment and influence analysis of marine two-stage herringbone gear transmission system considering cracks

To reveal the influence of cracks on the vibration characteristics of marine two-stage herringbone gear transmission system (TSHBGTS), a 48 degree of freedom bending-torsional-axial-pendular (BTAP) dynamic model of the system was established, which took into account the influence factors such as error, time-varying meshing stiffness (TVMS), torsional stiffness, support stiffness, tooth surface friction, and undercut. In the model, the TVMS of each gear pair in the Two-stage herringbone gear system with cracks was calculated by using the potential energy method, and the TVMS variation rule under different lengths, depths, and angle parameters of cracks was analyzed. The Runge-Kutta numerical integration method was used to solve the problem. The influence of different crack degrees on the vibration of each position of the TSHBGTS was analyzed. The influence of different crack degrees on the vibration response of was tested through tests. The difference in mechanical vibration between the normal gear and cracked gear was analyzed by using the frequency domain diagram and time domain diagram, and the influence mechanism of cracks on vibration characteristics was revealed. The results show that when the gear cracks, the TVMS of the gear decreases, and with the increase of the crack depth and length, the TVMS of the gear pair will gradually decrease. With the increase of crack depth and length, the TVMS of the gear pair is greatly reduced, and there is a periodic impact phenomenon in the time domain response of the system. With the continuous expansion of the crack, the impact phenomenon becomes more and more serious. In the frequency domain response, a modulated sideband centered on the meshing frequency appears. When the crack is generated and the crack degree increases, a small noise frequency is generated near the frequency doubling and increases with the growth of the crack. The correctness of the theoretical analysis model is verified through experiments, and the research results can provide a theoretical basis for the fault diagnosis of the herringbone gear transmission systems with cracks.