Internal Gear Pair Research Articles

Effect of bolt constraint of ring gear on the vibration response of the planetary gearbox

The lumped parameter model (LPM) is a common modeling method for simulating the vibration signal of the planetary gearbox. Generally, the simulation results are not directly used for the subsequent analysis, because two factors need to be considered: the transmission paths of vibration signals and the amplitude modulation effect of the carrier. However, both of these factors are subjectively affected. In order to overcome this problem, a new method for generating the gearbox vibration signal is proposed. Based on LPM, the vibration displacement of each component is solved, and meshing forces of the internal gear pairs are calculated. Then, the ring gear with bolt constraints is simplified to the Euler-Bernoulli beam, and the vibration signal of any point on the ring gear is established by solving the vibration of the beam. Subsequently, based on the new signal model, the transmission mechanism, the amplitude modulation and the overlapping phenomenon of the vibration signal, and the effect of bolt constraint on the vibration of the gearbox are analyzed. Finally, the correctness of the new method and the effect of bolt constraint on the vibration signal are verified by experiments.

Simulation Research on Trapped Oil Pressure of Involute Internal Gear Pump

The main structure of an internal gear pump consisted of an internal gear pair, including an internal gear and an external gear. The internal gear pump had oil trapping phenomenon like other hydraulic gear pumps. In order to solve the oil trapping phenomenon of involute gear pump with internal meshing tooth profile, in this paper, the mathematical equation of gear outer contour is established according to the principle of generation method, and the variation law of the trapped oil area in meshing process is deduced by theoretical instantaneous flow rate obtained by scanning method. Then, the minimum trapped oil volume and unloading area are solved by the graphic method. Finally, based on fluid mechanics and dynamics, the trapped oil pressure model is obtained. The change of the trapped oil area and trapped oil pressure in a meshing cycle is simulated by MATLAB. The results show that the trapped oil area changes in a parabola, and the trapped pressure fluctuates in mountains and valleys. When the trapped area is the smallest, the trapped oil pressure reaches the peak at the corresponding corner. The research results can provide guidance for the development of high-performance internal gear pumps.

Meshing impact analysis of planetary transmission system considering the influence of multiple errors and its effect on the load sharing and dynamic load factor characteristics of the system

The meshing impact on transmission system and internal meshing gear pair and its impact on the load sharing and dynamic characteristics of the system are not well understood yet. In this paper, the meshing impact models of internal gear pairs and planetary transmission system were successfully constructed, and the meshing impact point, meshing impact time and meshing impact force were accurately obtained. Meshing impact in gear transmission system is obviously affected by eccentricity error, installation error, and other errors. Due to the difference in error of each component, the internal and external gears of each branch lead to different meshing positions, which causes the constant change in meshing impact point, meshing impact time and meshing impact force. This creates difficulties in the analysis of meshing impact characteristics of gear transmission system. Load Tooth Contact Analysis (LTCA) method can be used to accurately analyse the change in position of gear tooth under load condition. Through the dynamic model of planetary transmission system, the influence in component errors on the contact position of tooth surface is obtained. Combining the loaded transmission error of the tooth surface under load and the geometric transmission errors under the influence of component errors, the model of meshing impact for accurately solving the system is deduced, and the influence of meshing impact on the system's load sharing coefficient and dynamic load factor coefficient is analysed. By comparing the planetary transmission system before and after considering the meshing impact of the system, it is found that the system's load-sharing coefficient increases slightly, dynamic load factor coefficient fluctuates significantly, and meshing force becomes more clutter after considering meshing impact.

Analysis of lubrication performance for internal meshing gear pair considering vibration

The thermal elasto-hydrodynamic lubrication characteristics of the internal meshing gears in a planetary gear train under vibrations were examined considering the influence of the modification coefficient and time-varying meshing stiffness. Based on dynamic theory of the gear system, a dynamic model of the planetary gear train was established. The lubrication performances of modified gear systems under vibrations and static loads were analyzed. Compared with other transmission types, the best lubrication effect could be produced by the positive transmission. A thicker lubricating oil film could be formed, and the friction coefficient and oil film flash temperature are the smallest. Increasing modification coefficient improves the lubrication performance continuously but intensifies the engage-in and tooth-change impact. For the planetary and inner gears, the increase in the modification coefficient also leads a decrease in the oil film stiffness.

Research of Time-varying Mesh Characteristics of Cycloid Internal Gear Pair with High Contact Ratio Based on Finite Element Method

Research of Time-varying Mesh Characteristics of Cycloid Internal Gear Pair with High Contact Ratio Based on Finite Element Method

Research and Comparative Analysis of Flow Field Characteristics and Load-Independent Power Losses of Internal and External Gear Pairs

Power loss analysis of gear transmission in a transmission system is of great significance to improve the efficiency of the power system, and load-independent power losses are an important part of the power losses of gear transmission. Based on the computational fluid dynamics (CFD) method, the hydrodynamic models of internal and external gear pairs are established. By analyzing the pressure field and the velocity field, the windage and squeezing power losses and the pressure and viscous power losses, the influence of rotation speed and tooth width on flow field characteristics, and load-independent power losses of internal gear pair are studied. In addition, we compare the flow field characteristics and the load-independent power losses between external and internal gear pairs and discuss the difference between them. The results show that the pressure and fluid velocity in the meshing area of the gear pair are greatly affected by rotation speed and tooth width, and the load-independent power losses increase with the increase of rotation speed and tooth width. At the same rotation speed, the transmission ratio and number of teeth, windage, and squeezing power losses of the external gear pair are smaller than those of the internal gear pair. Compared with the internal gear pair, the external gear pair has more advantages in controlling the load-independent power losses. The difference of the load-independent power losses of the two meshing modes mainly comes from the viscous power losses of the wheel gear of internal gear pair. This paper provides a basis for the selection of the gear meshing mode and the analysis of load-independent power losses of the transmission system.

Mathematical model and meshing analysis of internal meshing gear transmission with curve element

A new internal meshing gear transmission with curve element is put forward in this paper. The mathematical principle of tooth profile generation is described based on conjugate curves theory. For a given spatial curve, the meshing equation and its conjugated spatial curve under the motion law were derived. Considering the equidistant kinematic method, general internal tooth profiles models were established by the conjugate-curve pair. Numerical example of the internal gear pair was developed according to gear parameters and gear solid models were established by MATLAB and UG software. Motion simulation result shows that the gear pair satisfies point contact condition and design requirements. Meshing analysis of tooth profiles using FEA method was carried out. Stress analysis results of tooth profiles with single point contact and two points contact were, respectively, obtained. The conclusions lay the foundation for multi-point contact generation and tooth profile design. Also, further studies on transmission characteristics and manufacturing technology of the new gear drive will be carried out.

A hybrid finite element and analytical model for determining the mesh stiffness of internal gear pairs

This work developed an efficient model for calculating the mesh stiffness of spur/helical internal gear pairs by combining the finite element method (FEM) and analytical formula. The tooth global deformation is obtained by separation of the deformation of a full finite element model and a partial model, and the local contact deformation is derived by an analytical line contact formula based on Hertz contact theory. The transmission error and mesh stiffness of the gear pair can be acquired after solving the nonlinear contact equilibrium equations. Compared with the conventional FEM, the proposed method has much smaller computational consumption. Furthermore, it also overcomes the disadvantage that the analytical method is difficult to consider different ring gear structures. Then the influences of ring thicknesses and the number of support pins of the ring gear on the mesh stiffness are discussed. The results show that the ring flexibility will change the amplitude-frequency components of the mesh stiffness a lot.

Tooth Surface Contact Analysis of Involute Rotate Vector Reducer Based on a Finite Element Linear Programming Method

The deformation compatibility relationship of the general contact object under external normal force and the single secondary internal gear pairs of involute RV reducer under a load torque is derived. By transforming the deformation coordination relationship into a linear programming form, a mathematical model for the analysis of the single secondary internal gear pairs with the tooth surface contact of the involute RV reducer is established. An improved simplex method is used to solve the linear programming problem. The calculation results are compared with the ANSYS simulation values from the three aspects of contact tooth pairs, gear tooth load and tooth surface contact force. The results show that the values obtained by the two methods are almost the same on the contact tooth pairs and the tooth load. In terms of tooth surface contact force, the contact area calculated using the finite element linear programming method is slightly smaller than the ANSYS results, but most of the errors are below 12%, indicating that the tooth surface contact conditions obtained using the two calculation methods are consistent. The accuracy and effectiveness of the finite element-linear programming method are thus verified.

Tooth Surface Contact Strength of Cycloid Internal Gear Pair with High Contact Ratio

Tooth Surface Contact Strength of Cycloid Internal Gear Pair with High Contact Ratio

Strength model for bending and pitting calculations of internal spur gears

An enhanced model for strength calculations of internal spur gear pairs is presented. The load sharing among couples of teeth in simultaneous contact is determined from the meshing stiffness, whose value has been calculated at any point of the path of contact, considering bending, compressive, shear and Hertzian deflections. This curve of variation of the meshing stiffness has been expressed by means of an approximate accurate equation, as a function of the effective contact ratio exclusively, allowing to express the load at each couple of teeth in a closed form. The values of the bending and pitting stresses along the path of contact have been accordingly computed, as well as their nominal values and their corresponding determinant load conditions. An adapted formulation of the meshing stiffness curve for non-standard tooth dimensions–including reduced tooth height, frequently required to avoid secondary interference–has been also developed. Calculation methods for the nominal stresses for these non-standard internal gears have been presented as well.

Mathematical Model and Tooth Contact Analysis of an Internal Helical Gear Pair with Selectable Contact Path

An internal helical gear pair with point contact pattern based on the space meshing theory is proposed which consists of an involute internal gear and a pinion with quadratic curve profile. Particularly, the contact path of the tooth surface is selectable, which can inherit the characteristic of the surface. Moreover, this type of internal gear pair has lower sensitivity to assembly errors. The generation principle and mathematical model is presented. The motion simulation and adaptability to center distance error of the internal gear pair are discussed. The gear pair is manufactured. Based on the mathematical model and tooth contact analysis (TCA) method, the locations of contact points and kinematic errors are determined under different center distance variation and axial misalignments. The simulated results reveal that the locations of contact points of the gear and pinion change under assembly errors. Little kinematic error occurs under axial misalignment. No edge contact occurs under any assembly error condition. Experiment study is performed and the result is consistent with the theoretical analysis.

Theoretical and experimental investigation on internal gear pair with small sliding ratio

Aiming at the issue of sliding ratio, an internal gear pair is proposed which consists of an involute internal gear and a pinion with quadratic curve teeth. Particularly, the contact pattern is point contact and the pinion is generated based on an involute gear. The generation method and mathematical models of the gear pair are presented. The sliding ratio is calculated and the general calculation formulas of sliding ratios are developed. Also, the comparison between the involute gear and proposed gear is made. The adaptability of center distance and contact stress are also discussed. In addition, the gear pair was manufactured and inspected according to the exactitude solid model of the gear pair. In order to confirm this model to be effective, the efficiency experiment and the contrast experiment with the involute gear pair were performed. Furthermore, these two types of pinions were analyzed by scanning electron microscope and wear depths were measured by measuring center. The experiment results show that the efficiency of the internal gear pair is stable at a range about 97.1% to 98.6% and wear depth is less than 50% of the involute gear pair. The internal gear pair is expected to have excellent transmission performance.

Design principle and modeling method of asymmetric involute internal helical gears

In the field of mechanical engineering, involute helical gears are widely used. Compared with the involute spur gear, helical gears have a high bearing strength, more smooth transmission, less impact and less noise. The internal gear pairs have the features of large transmission ratio, low vibration, low noise and low wear and hence are widely used in planetary gear transmission systems. In order to meet the requirements of high strength, high speed of the modern gear transmission systems, a new type of asymmetric involute internal helical gears is designed based on conventional involute gears. This paper discusses the gear shaper cutter modeling for machining this new gear, analyzes the formation principle of asymmetric tooth profile and establishes a three-dimensional modeling by SolidWorks. Through MATLAB simulation, pressure angle, tooth number, coefficient of displacement and contact ratio of conventional and asymmetric gear are compared and analyzed. Using ANSYS, two types of gears are compared on strength in order to demonstrate superiority of asymmetric tooth and further study about the asymmetric internal helical gears.

Time-varying Mesh Stiffness Calculation and Load Distribution among Teeth of Cycloid Internal Gear Pair with High Contact Ratio

Time-varying Mesh Stiffness Calculation and Load Distribution among Teeth of Cycloid Internal Gear Pair with High Contact Ratio

Manufacturing and contact characteristics analysis of internal straight beveloid gear pair

This paper presents a manufacturing method for straight internal beveloid gear pair by shaper cutter with parallel axes between the cutter and the gear blank. The cutting motions are represented using three degrees of freedom that are the rotation along its axes, translations along the axial and radial directions. With the proposed method, the beveloid gear with a linear or nonlinear variable profile shift coefficient along the tooth width direction can be generated and the tilt deviations by the inclined cutting tool shaft can be avoided. The mathematical description of shaper cutter and the coordinate systems for the cutting procedure were defined. And the mathematical model of external and internal beveloid gears were derived and the loaded tooth contact analysis were conducted. Results show that the teeth mesh in at the toe and mesh out at the heel. The length of the simultaneous contact line on tooth surface increases first and then decreases due to the cone angle. As the load increases, the mesh area, contact pressure, mean value of mesh stiffness and the peak-peak value of angular transmission error are increased due to the elastic deformation. However, the speeds of the increase tendency slow down.

Mesh stiffness evaluation of an internal spur gear pair with tooth profile shift

Tooth profile shift will change the thickness of gear teeth and a part of geometrical parameters of a gear pair, thus influencing its mesh stiffness and consequently the dynamic performances. In this paper, an analytical mesh stiffness calculation model for an internal gear pair in mesh considering the tooth profile shift is developed based on the potential energy principle. Geometrical representations of the tooth profile shift are firstly derived, and then fitted into the analytical tooth stiffness model of gears. This model could supply a convenient way for mesh stiffness calculation of profile shifted spur gears. Then, simulation studies are conducted based on the developed model to demonstrate the effects of tooth profile shift coefficient on the tooth compliances and the mesh stiffness of the internal spur gear pair. The results show that tooth profile shift has an obvious influence on the mean value, amplitude variation and phase of the mesh stiffness, from which it can be predicted that the dynamic response of an internal gear transmission system will be affected by the tooth profile shift.

Synthesis of the steepest rotation pitch curve design for noncircular gear

A steepest rotation method for designing pitch curve with fixed boundaries of noncircular gear is proposed by resorting to calculus of variations. A general mathematical model is established to design the steepest rotation pitch curves with integral constraint for noncircular gear pair, including external and internal meshing gear pair. In particular, to achieve the design of pitch curve with epicyclic constraint for driven noncircular gear, the epicyclic constraint conditions are also established. In addition, the unified design algorithm of pitch curves of noncircular gear pair with this steepest rotation characteristic is given in this paper, the pitch curves of desired geometrical and transmission properties can be solved easily by using the proposed algorithm. Examples presented in this paper are implemented in MATLAB, and feasibility and validity of above algorithm are verified.

A Dynamic Model for Double-Planet Planetary Gearsets

In this study, a two-dimensional (2D), steady-state, discrete dynamic model of a double-planet planetary gearset is proposed. The dynamic model is generalized such that it can consist of N number of planet branches and can operate under any operating conditions (load and speed). The contact between each external to external and external to internal gear pair is modeled to obtain stiffnesses and mesh displacement excitations using a generalized load distribution model. The natural modes are computed by solving the corresponding eigenvalue problem. The forced vibration response to gear mesh excitations is obtained by applying the modal summation technique. The model is capable of predicting gear mesh dynamic load and dynamic transmission error spectra for each gear mesh, dynamic bearing load spectra for each bearing as well as gear body dynamic displacements. Forced vibration response of an example system that consists of three double-planet branches is studied to demonstrate the influence of some of the key design parameters.

Study on the Multi-Tooth Meshing Bending Strength for Involute Gear Drive with Small Tooth Number Difference

This article has established numerous entity assembly models of internal gear pair with small tooth number difference of stub gear, then calculates the bending stress of multi-tooth meshing by means of FEM, determines bending strength factor of multi-tooth meshing, and forms the calculation method of bending strength based on influence by multi-tooth meshing.