Multi-objective Optimization of a Two-Stage Helical Gearbox to Improve Efficiency and Reduce Height

The goal of this research is to look at multi-target optimization of a two-stage helical gearbox in order to determine the best key design elements for reducing gearbox height and enhancing gearbox efficiency.To do this,the method known as Taguchi and GRA(Grey Relation Analysis)were used in two stages to address the problem.The single-objective optimization problem was addressed first to close the gap between variable levels,and then the multi-objective optimization problem was solved to determine the best primary design variables.The first and second stage CWFWs(Coefficients of Wheel Face Width),ACS(Permissible Contact Stresses),and first stage gear ratio were also calculated.The study’s findings were utilized to identify the best values for five critical design aspects of a two-stage helical gearbox.

Statistical Modification Analysis of Helical Planetary Gears based on Response Surface Method and Monte Carlo Simulation

Tooth modification technique is widely used in gear industry to improve the meshing performance of gearings. However, few of the present studies on tooth modification considers the influence of inevitable random errors on gear modification effects. In order to investigate the uncertainties of tooth modification amount variations on system＇s dynamic behaviors of a helical planetary gears, an analytical dynamic model including tooth modification parameters is proposed to carry out a deterministic analysis on the dynamics of a helical planetary gear. The dynamic meshing forces as well as the dynamic transmission errors of the sun-planet 1 gear pair with and without tooth modifications are computed and compared to show the effectiveness of tooth modifications on gear dynamics enhancement. By using response surface method, a fitted regression model for the dynamic transmission error（DTE） fluctuations is established to quantify the relationship between modification amounts and DTE fluctuations. By shifting the inevitable random errors arousing from manufacturing and installing process to tooth modification amount variations, a statistical tooth modification model is developed and a methodology combining Monte Carlo simulation and response surface method is presented for uncertainty analysis of tooth modifications. The uncertainly analysis reveals that the system＇s dynamic behaviors do not obey the normal distribution rule even though the design variables are normally distributed. In addition, a deterministic modification amount will not definitely achieve an optimal result for both static and dynamic transmission error fluctuation reduction simultaneously.

Finite element analysis and simulation for cold precision forging of a helical gear

To investigate the effects of billet geometry on the cold precision forging process of a helical gear, six different billet geometries were designed utilizing the relief-hole principle. And the influences of the billet geometry on the forming load and the deformation uniformity were analyzed by three-dimensional （3D） finite element method （FEM） under the commercial software DEFORM 3D. The billet geometry was optimized to meet lower forming load and better deformation uniformity requirement. Deformation mechanism was studied through the distribution of flow velocity field and effective strain field. The forging experiments of the helical gear were successfully performed using lead material as a model material under the same process conditions used in the FE simulations. The results show that the forming load decreases as the diameter of relief-hole do increases, but the effect of do on the deformation uniformity is very complicated. The forming load is lower and the deformation is more uniform when do is 10 mm.

A distributed dynamic mesh model of a helical gear pair with tooth profile errors

A dynamic model of a helical gear rotor system is proposed.Firstly,a generally distributed dynamic model of a helical gear pair with tooth profile errors is developed.The gear mesh is represented by a pair of cylinders connected by a series of springs and the stiffness of each spring is equal to the effective mesh stiffness.Combining the gear dynamic model with the rotor-bearing system model,the gear-rotor-bearing dynamic model is developed.Then three cases are presented to analyze the dynamic responses of gear systems.The results reveal that the gear dynamic model is effective and advanced for general gear systems,narrow-faced gear,wide-faced gear and gear with tooth profile errors.Finally,the responses of an example helical gear system are also studied to demonstrate the influence of the lead crown reliefs and misalignments.The results show that both of the lead crown relief and misalignment soften the gear mesh stiffness and the responses of the gear system increase with the increasing lead crown reliefs and misalignments.

Tooth surface geometry optimization of spiral bevel and hypoid gears generated by duplex helical method with circular profile blade

In order to effectively improve meshing performance of spiral bevel and hypoid gears generated by the duplex helical method, the effects of straight lined and circular cutting edges profile on meshing and contact of spiral bevel and hypoid gears were investigated analytically. Firstly, a mathematical model of spiral bevel and hypoid gears with circular blade profile was established according to the cutting characteristics of the duplex helical method. Based on a hypoid gear drive, the tooth bearings and the functions of transmission errors of four design cases were analyzed respectively by the use of the tooth contact analysis(TCA), and the contact stresses of the four design cases were analyzed and compared using simulation software. Finally, the curvature radius of the circular profile blade was optimized. The results show that the contact stresses are availably reduced, and the areas of edge contact and severe contact stresses can be avoided by selecting appropriate circular blade profile. In addition, the convex and concave sides are separately modified by the use of different curvature radii of inside and outside blades, which can increase the flexibility of the duplex helical method.

Investigation of a Design Modification for Double Helical Gears Reducing Vibration and Noise

To reduce vibration and noise and increase transmission efficiency, a three segment method for modifying the pinion profile was proposed. Cutter surface equations were deduced by changing the shape of the cutter-edge, substituting three segment parabolas for the line. The influence of longitudinal tooth modifications on tooth surface load distributions was discussed. Transmission error minimization and uniformity of tooth surface load distribution were chosen as optimization goals and the modified parameters were obtained by applying the complex method. Finally, an experiment comparing the loaded transmission error, vibration, and noise both before and after modifications was carried out. The results indicate that the modified design is reliable.

Flow-Optimization to Enhance Gas Quenching Efficiency for Helical Gears Specimen

Numerical simulations based on a conjugate heat transfer solver have been carried out to analyze various gas quenching configurations involving a helical gear streamed by an air flow at atmospheric pressure in a gas quenching chamber. In order to optimize the heat transfer coefficient distribution at key positions on the specimen, configurations involving layers of gears and flow ducts comprising single to multiple gears have been simulated and compared to standard batch configurations in gas quenching. Measurements have been performed covering the local heat transfer for single gears and batch of gears. The homogeneity of the heat transfer coefficient is improved when setting up a minimal distance between the gears (batch density) and when introducing flow ducts increasing the blocking grade around the gears. An offset between layers of the batch as well as flow channels around the gears plays a significant role in increasing the intensity and the homogeneity of the heat transfer in gas quenching process.

Effects of parameters on rotational fine blanking of helical gears

The application of fine blanking to the manufacturing of helical gears directly from a strip has been restricted due to the traditional linear cutting stroke of the punch and die.In this work,rotational fine blanking which combined the linear and rotational motion of punch and counterpunch was applied for the forming of helical gears.A three-dimensional(3D) rigid-plastic finite element model was developed on the DEFORM-3D platform.By finite element simulation and analysis,the influences of key parameters on the punch load and cut surface were investigated.It is shown that: 1) with increasing the counterforce or helical angle,the punch load and the depth of die roll increase; 2) with increasing blank holder force,the punch load increases while the depth of die roll decreases; 3) V-ring indenter facilitates an improvement in the quality.The results of this research reveal the deformation mechanism of rotational fine blanking of helical gears,and provide valuable guidelines for further experimental studies.

Analysis of axial force of double circular arc helical gear hydraulic pump and design of its balancing device

In view of the axial force produced in the working process of double arc helical gear hydraulic pump,the theory of differential equation of curve and curved surface was utilized so that the calculation formula of axial force was obtained and the relationship between the axial force and structure parameters of gears was clarified.In order to balance the axial force,the pressure oil in the high pressure area was introduced into the end face of the plunger to press the plunger against the gear shaft,and the hydrostatic bearing whose type is plunger at the end of the shaft was designed.In order to verify the balance effect of axial force,the leakage owing to end clearance and volume efficiency of gear hydraulic pump before and after the balancing process was analyzed.This paper provides a new analysis idea and balance scheme for the axial force produced in the working process of the double arc helical gear hydraulic pump,which can reduce the leakage owing to end clearance caused by the axial force and improve the volume efficiency of the gear hydraulic pump.

Tooth surface error correction of hypoid gears machined by duplex helical method

In this work,synchronous cutting of concave and convex surfaces was achieved using the duplex helical method for the hypoid gear,and the problem of tooth surface error correction was studied.First,the mathematical model of the hypoid gears machined by the duplex helical method was established.Second,the coordinates of discrete points on the tooth surface were obtained by measurement center,and the normal errors of the discrete points were calculated.Third,a tooth surface error correction model is established,and the tooth surface error was corrected using the Levenberg-Marquard algorithm with trust region strategy and least square method.Finally,grinding experiments were carried out on the machining parameters obtained by Levenberg-Marquard algorithm with trust region strategy,which had a better effect on tooth surface error correction than the least square method.After the tooth surface error is corrected,the maximum absolute error is reduced from 30.9μm before correction to 6.8μm,the root mean square of the concave error is reduced from 15.1 to 2.1μm,the root mean square of the convex error is reduced from 10.8 to 1.8μm,and the sum of squared errors of the concave and convex surfaces was reduced from 15471 to 358μm^(2).It is verified that the Levenberg-Marquard algorithm with trust region strategy has a good accuracy for the tooth surface error correction of hypoid gear machined by duplex helical method.

Structure vibration characteristics of two stage double helical tooth planetary gear trains with unequally spaced planets and stars

A computational model of two stage double helical tooth planetary gear set is employed to built the lateraltorsional coupling dynamic governing equations. Base on the dynamic equation, the free vibration properties of the system with unequally spaced planets and stars are analyzed. The vibration modes are classified into three types: star mode, planet mode, and coupling mode. For each of the modes, the relation between inherent frequency and vibration amplitude is investigated in detail by the eigenvalue and mode characteristics.

EFFECT OF GEAR WIDTH AND HELIX ANGLE ON FACTOR OF DYNAMIC LOAD OF DOUBLE CIRCULAR ARC HELICAL GEARING

Based on theory of mechanical dynamics, meshing characteristic as well as thedynamic model of double circular arc helical gearing, an analysis approach and a computer programhave been developed for studying the state of dynamic load and factor of dynamic load of thegearing, the changing situation of dynamic load and dynamic load factor vs some affecting factorssuch as gear width, helix angle and accuracy grade etc are investigated. A series of conclusions areobtained: ①With the increasing in the values of gear width, the dynamic load factor appears slowdecreasing tendency in most region of gear width. ②When the accuracy grades of the gearing areimproved, the values of dynamic load factor decrease. ③The value of dynamic load factor appears adecreasing tendency with the increasing of value of helix angle at the same ratio of criticalrotational speed.

Acoustics Based Monitoring and Diagnostics for the Progressive Deterioration of Helical Gearboxes

Gearbox condition monitoring(CM)plays a significant role in ensuring the operational reliability and efficiency of a wide range of critical industrial systems such as wind turbines and helicopters.Accurate and timely diagnosis of gear faults will improve the maintenance of gearboxes operating under sub-optimal conditions,avoid excessive energy consumption and prevent avoidable damages to systems.This study focuses on developing CM for a multi-stage helical gearbox using airborne sound.Based on signal phase alignments,Modulation Signal Bispectrum(MSB)analysis allows random noise and interrupting events in sound signals to be suppressed greatly and obtains nonlinear modulation features in association with gear dynamics.MSB coherence is evaluated for selecting the reliable bi-spectral peaks for indication of gear deterioration.A run-to-failure test of two industrial gearboxes was tested under various loading conditions.Two omnidirectional microphones were fixed near the gearboxes to sense acoustic information during operation.It has been shown that compared against vibration based CM,acoustics can perceive the responses of vibration in a larger areas and contains more comprehensive and stable information related to gear dynamics variation due to wear.Also,the MSB magnitude peaks at the first three harmonic components of gear mesh and rotation components are demonstrated to be sufficient in characterizing the gradual deterioration of gear transmission.Consequently,the combining of MSB peaks with baseline normalization yields more accurate monitoring trends and diagnostics,allowing the gradual deterioration process and gear wear location to be represented more consistently.

Coupled lateral-torsional-axial vibrations of a helical gear-rotor-bearing system

Considering the axial and radial loads, a math- ematical model of angular contact ball bearing is deduced with Hertz contact theory. With the coupling effects of lateral, torsional and axial vibrations taken into account, a lumped-parameter nonlinear dynamic model of helical gearrotor-bearing system （HGRBS） is established to obtain the transmission system dynamic response to the changes of dif- ferent parameters. The vibration differential equations of the drive system are derived through the Lagrange equation, which considers the kinetic and potential energies, the dis- sipative function and the internal/external excitation. Based on the Runge-Kutta numerical method, the dynamics of the HGRBS is investigated, which describes vibration properties of HGRBS more comprehensively. The results show that the vibration amplitudes have obvious fluctuation, and the frequency multiplication and random frequency components become increasingly obvious with changing rotational speed and eccentricity at gear and bearing positions. Axial vibration of the HGRBS also has some fluctuations. The bearing has self-variable stiffness frequency, which should be avoided in engineering design. In addition, the bearing clearance needs little attention due to its slightly discernible effect on vibration response. It is suggested that a careful examination should be made in modelling the nonlinear dynamic behavior of a helical gear-rotor-bearing system.

Excitation Prediction by Dynamic Transmission Error under Sliding Friction in Helical Gear System

Monte Carlo method was adopted to calculate the meshing error considering the manufacture error and assembly error of the meshing point along the time-varying contact line for helical gear pair. The flexural-torsion-axis dynamic model coupled was established under the tooth friction force and solved by the perturbation method to compute real dynamic tooth load. The change laws of the friction force and friction torque were obtained in a meshing period. The transmission error formulation was analyzed to introduce meshing excitations. The maximum dynamic transmission error, the maximum meshing force and the maximum dynamic factor were calculated under different speeds, external loads and damping factors. The conclusions can provide theoretical basis for the gear design especially in tooth profile correction.

THE SELECTION OF CROWNING VALUES FOR HELICAL GEAR DRIVES

Gear crowning takes an important part in alleviating the concentration of local stresses when misalignment of the flanks exists. It is known that an excessive crowning will lead to a high local stress while an insufficient value will not be able to compensate the alignment error resulting in concentration. The selection of the crowning value is always based on empirical estimation. This paper deals with a theoretical method to determine the crowning value. The distribution forms of contact stresses on the flanks of a crowned pinion and a gear are treated as a series of complete or portions of semi-ellipsoids with the same size. The total volume of which equals the whole contact load. From this point of view a series of equations for the computation of the maximum stress is obtained. The calculation of contact stresses considering the existence of alignment errors gives insight into the selection ora proper crowning value for a gear drive.

Faults Analysis and Diagnosis of DRJ-460 Dish Centrifugal Separator′s Helical Gear

The main faults of dish centrifugal separator's helical gear are described inthis paper. In order to diagnose the DRJ-460 dish centrifugal separator correctly, the vibration istested with a helical gear under both normal and abnormal conditions. After comparing severalgeneral methods of the gear's fault feature extraction, a new convenient and effective method ispresented on the basis of analyzing the vibration spectrum under different rotary velocities.

Analysis of Helical Gear System Dynamic Response Based on Fuzzy Numbers

A non-linear dynamic model with the single degree of freedom of a helical gear pair introducing frzzy numbers is developed. In this proposed model, time-variant mesh stiffness, which is a non-linear parameter, mesh damping and composite error of a pair of meshing tooth of the gear pair are all included. Mesh stiffness is calculated by expressing Bo (r) as a Fourier series. Ⅱshape function is introduced as the membership function to characterize the fuzziness of the error. Fuzzy displacement dynamic response of the geared system at A- level, which is a closed interval, is ohtained by removing the fuzziness of the fuzzy differential equations and using Runge-Kutta numerical method. In fact, the fuzzy dynamic response and dynamic loading factor are aH the interval functions related λ. The result obtained here can be used to the fuzzy dynamic optimization design course of the helical gear system. The main advantage of this method is to introduce the concept of fuzzy number for the first time to the analysis of the gear system dynamics.

Experimental Investigation of the Deformation of Helical Gears with Double－Circular－Arc Tooth Profile Using Linear Variable Differential Transformer

A small hole of 0.9mm in diameter is drilled at the theoretical contact point of the convex tooth flank of the measured gear, and the hole leads throughout to the non-working flank. A stylus glued to the core of transformer is put into the hole, and the stylus can freely contact with the meshed concave tooth flank. The transformer is installed on the body of gear to be measured. Rotate the positioning worm slowly after loading, and locate the contact point at the hole of convex tooth flank, the displacement value measured is considered as the deformation of convex tooth. The deformations at the middle and the two ends of tooth breadth for the helical gears with double-circular-cra tooth profile whose modules are 3mm and 4mm respectively are measured in the paper.

Quasi-static-model-based wear analysis of helical gears

A quasi-static wear model for helical gears is proposed based on finite element method and Archard's formula to investigate the wear characteristics of tooth surface in real operation conditions. The numerical simulation reveals that the wear depth distributes unevenly along the tooth lead and varies along the tooth profile. The surface wear depth claims its minimal value at the pitch circle and reaches its maximal value near the tooth root of the pinion. The effects of misalignment on wear depth are further investigated to provide useful information for gear designers. The numerical simulation reveals that the vertical misalignment is more influential than the parallel misalignment on the gear wear depth. Therefore, to eliminate the negative effects of misalignment, the misalignment should be strictly controlled.