Dynamic analysis of involute spur gears with asymmetric teeth

New gear designs are needed because of the increasing performance requirements, such as high load capacity, high endurance, low cost, long life, and high speed. In some applications, such as in wind turbines, the gears experience only uni-directional loading. In these instances, the geometry of the drive side does not have to be symmetric to the coast side. This allows for the designing of gears with asymmetric teeth. In previous studies related to bending stress and load capacity, high performance has been achieved for gears with asymmetric teeth. These gears provide flexibility to designers due to their non-standard design. If they are correctly designed, they can make important contributions to the improvement of designs in aerospace industry, automobile industry, and wind turbine industry. At high operation speeds, there is always a concern of dynamic loads and vibrations of equipment. Therefore, there is a need to fully understand the dynamic behavior of gears with asymmetric teeth. Thus, the primary objective of this paper is to use dynamic analysis to compare conventional spur gears with symmetric teeth and spur gears with asymmetric teeth. The secondary objective is to optimize the asymmetric tooth design in order to minimize dynamic loads. This study offers preliminary results to designers for understanding dynamic behavior of spur gears with asymmetric teeth. For this study, a dynamic model was developed, using MATLAB, and used for the prediction of the instantaneous dynamic loads of spur gears with symmetric and asymmetric teeth. Furthermore, a 2-D three-tooth model was developed for finite element analysis. Fast Fourier transform was used for the frequency analysis of the static transmission errors. It is shown that generally, the dynamic factor, for spur gears with asymmetric teeth, increases with increasing pressure angles on the drive side. For asymmetric teeth, increasing the addendum leads to a significant decrease in the dynamic factor. The static transmission error, at the center of the single tooth contact zone, decreases with increasing pressure angle. The first two harmonics slightly increase with increasing pressure angle. It is further shown that the amplitudes of harmonics of the static transmission errors are significantly reduced when asymmetric teeth with long addendum providing high gear contact ratio close to 2.0 are used.     

Dynamic loading of spur gears with linear or parabolic tooth profile modifications

A computer simulation was conducted to investigate the effects of both linear and parabolic tooth profile modification on the dynamic response of low-contact-ratio spur gears. The effect of the total amount of modification and the length of the modification zone were studied at various loads and speeds to find the optimal profile modification for minimal dynamic loading.

Design charts consisting of normalized maximum dynamic load curves were generated for gear systems operated at various loads and with different tooth profile modification. An optimum profile modification can be determined from these design charts to minimize the dynamic loads of spur gear systems.     

Analysis of the nonlinear behavior of gear pairs considering hydrodynamic lubrication and sliding friction

This paper describes an analysis of the nonlinear behavior of gear pairs according to the direct contact elastic deformation model over a wide range of speeds, considering the hydrodynamic effects and friction force. The inclusion of the hydrodynamic effect facilitates nonlinearity by increasing the overlap range (i.e., multiple solution regimes) and damping, as well as decreasing elastic deformation and tooth reaction forces. The effects of various lubrication parameters, such as viscosity and film width, on the nonlinear dynamic behavior were analyzed. While the viscosity has a strong effect on the behavior of gear pair systems, friction has very little effect on torsional behavior. Although the model of direct contact without friction has overall nonlinear behavior similar to the model including hydrodynamic effects with friction, the time data of these models are different due to the squeeze effect. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Cheon Gill-Jeong received his B.S. in Mechanical Engineering from Seoul National University (SNU), Korea, in 1981. He then received his M.S. and Ph.D. degrees from SNU in 1983 and 1988, respectively. He served as a senior research engineer at Seoul National University Hospital and Daewoo Heavy Industry for several years. Dr. Cheon is currently a Professor at the Division of Mechanical Engineering at Wonkwang University in Iksan, Korea. His research interests include dynamics, tribology, and design engineering.     

Characterization & evaluation of Al7075 MMCs reinforced with ceramic particulates and influence of age hardening on their tensile behavior

Aluminium has a massive demand in the areas of automobile, aerospace and diverse engineering applications in order to furnish the requirement in those fields. But this technological evolution needs something more than aluminium. Materialogists are struggling hard to find out a material which owns sound mechanical and thermal properties and also superior than aluminium in each extent. Metal matrix composite (MMC) is a solution. Generally, metal matrix composites contain a low density material, i.e. aluminium or magnesium, reinforced with fibers or particulate of a ceramic material, i.e. silicon carbide or graphite. They show greater specific strength, high stiffness, elevated operating temperature, and superior wear resistance, along with the possibility to customize these properties for a specific use. In this study, Al 7075 is taken as a base matrix material, whereas ceramic materials like SiC, Al₂O₃, B4C and TiB₂ are used as reinforcements. There are different methods available for fabricating metal matrix composite materials and in this work, stir casting technique, which is a liquid state process, is used. Four different MMC specimens were produced with 15 % SiC, 15 % Al₂O₃, 15 % B4C and 15 % TiB₂. Mechanical properties i.e. tensile strength, hardness, and impact strength were studied for the prepared specimens. The results were charted and presented graphically to describe these materials characteristics.     

Contact and thermal analysis of an alumina—steel dry sliding friction pair considering the surface roughness

In the present study, finite element transient contact and thermal sliding simulation and temperature measurement of dry sliding friction were performed in order to analyse the real contact area and temperature developed in the contact region. Real 3D surface worn topographies were taken into consideration, at macro and intermediate stages. The calculated real contact area has been changing in time and space in the course of sliding. The sliding components were high purity alumina ceramic palettes and 100Cr6 steel with constant accelerated motion. The calculated temperature results are in good agreement with the temperature data measured. Heat partition was changing in time during sliding. The developed algorithm based on incremental FE technique can characterize real processes.     

Mesh relationship modeling and dynamic characteristic analysis of external spur gears with gear wear

Gear wear is one of the most common gear failures, which changes the mesh relationship of normal gear. A new mesh relationship caused by gear wear affects meshing excitations, such as mesh stiffness and transmission error, and further increases vibration and noise level. This paper aims to establish the model of mesh relationship and reveal the vibration characteristics of external spur gears with gear wear. A geometric model for a new mesh relationship with gear wear is proposed, which is utilized to evaluate the influence of gear wear on mesh stiffness and unloaded static transmission error (USTE). Based on the mesh stiffness and USTE considering gear wear, a gear dynamic model is established, and the vibration characteristics of gear wear are numerically studied. Comparison with the experimental results verifies the proposed dynamic model based on the new mesh relationship. The numerical and experimental results indicate that gear wear does not change the structure of the spectrum, but it alters the amplitude of the meshing frequencies and their sidebands. Several condition indicators, such as root-mean-square, kurtosis, and first-order meshing frequency amplitude, can be regarded as important bases for judging gear wear state.     

Dynamic behavior analysis of reciprocating compressor with subsidence fault considering flexible piston rod

Subsidence wear is inevitable in a reciprocating compressor system, but how to induce body vibration is not clear. We investigated the dynamic behavior of a reciprocating compressor system with subsidence fault considering a flexible piston rod. The flexible piston rod was modeled as a cantilever beam. The dynamic model, in which the influence of the subsidence size, variational cylinder pressure, impact effects and piston rod flexibility are taken into account, was established. Influences of the four parameters including subsidence size, vertical force of flexible piston rod, cylinder pressure and crankshaft speed were analyzed in the dynamic response. Numerical results reveal that with the change of the four parameters, the existence of subsidence fault has a significant effect on the displacement, velocity and acceleration of the crosshead in the transverse direction, but only conspicuously affects the acceleration in the longitudinal direction. In addition to cylinder pressure, the larger the values of the other three parameters, the greater the influences. The cylinder pressure is a time-varying working load. Not only the cylinder pressure but also the open/close time of the valves and change rate of the pressure all affect the dynamic behavior of the system with subsidence. Meanwhile, the change in the vertical force of the flexible piston rod has a remarkable effect on the jump position of the crosshead, but the effect of other parameters is not obvious. In addition, the changes of parameters produce obvious influences on the impact force and the impact range of the crosshead. The stability of the reciprocating compressor system was studied using Poincaré portraits method. The result shows that the system with subsidence fault has chaotic behavior.     

On dry sliding friction and wear behavior of PPESK filled with PTFE and graphite

Novel poly(phthalazinone ether sulfone ketone) (PPESK) resins have become of great interest in applications such as bearing and slider materials. In this paper, dry sliding wear of polytetrafluoroethylene (PTFE) and graphite-filled PPESK composites against polished steel counterparts were investigated on a block-on-ring apparatus at the same sliding velocities and different loads. The results indicated that the addition of 5–25 wt% PTFE and 5–30 wt% graphite contribute to an obvious improvement of tribological performance of PPESK at room temperature. Worn surfaces were investigated using a scanning electron microscope (SEM). As a result, the friction coefficient and wear rate of the PPESK composites decreased gradually with addition of fillers. A moderately low friction coefficient and specific wear rate were reached when the filler contents were above 20 wt%. The mechanical properties of PPESK composites were also investigated. The tensile and impact strength of PPESK composites decrease slightly as the addition of fillers contents were below 15 wt%.     

Dynamic behavior of the spur gear system with time varying stiffness by gear positions in the backlash

Gear backlash is a nonlinear effect of the gear system. In a spur gear system with the backlash, the initial position of gears with the backlash affects the impact force. This work conducted a dynamic analysis of the spur gear system with time-varying mesh stiffness and bearing stiffness with a focus on the initial gear position within the backlash. For this purpose, the time-varying stiffness of the gears and rolling bearings were calculated. Mesh force with the time-varying stiffness and the gear backlash was applied to four DOF equations of motion. The equations of motion were solved using the Newmark beta method and Newton-Raphson method. The dynamic characteristics of the spur gear system by the initial position of gears within the backlash were investigated along with the magnitude of the backlash. The results showed that as the backlash increased, the mesh and bearing forces increased as well. The mesh and bearing forces were highly dependent on the initial gear position within the backlash. Significant initial mesh and bearing forces by the initial gear position within the backlash can lead to cumulative damages to the gear system.     

Wear and friction behavior of sand cast brake rotor made of A359-20 vol% SiC particle composites sliding against automobile friction material

The effect of load range of 30-100 N and speed range of 3-12 m/s on the wear and friction behavior of sand cast brake rotor made of A359-20 vol% SiC particle composites sliding against automobile friction material was investigated. Dry sliding frictional and wear behavior were investigated in a pin-on-disc type apparatus. Automobile friction material was used as pin, while the A359-20 vol% SiC particle composites formed the rotating disc. For comparison, the wear and friction behavior of commercially used cast iron brake rotor were studied. The results showed that the wear rate of the composite disc decreased with increasing the applied load from 30 to 50 N and increased with increasing the load from 50 to100 N. However, the wear rate of the composite disc decreased with increasing the sliding speed at all levels of load applied in the present work. For all sliding speeds, the friction coefficient of the composite disc decreased with applied load. The worn surfaces as well as wear debris were studied using scanning electron microscopy (SEM), energy dispersive X-ray (EDX) analyzer and X-ray diffraction (XRD) technique. At load of 50 N and speed range of 3-12 m/s, the worn surface of the composite disc showed a dark adherent layer, which mostly consisted of constituents of the friction material. This layer acted as a protective coating and lubricant, resulting in an improvement in the wear resistance of the composite.     

Rigid-plastic finite element analysis of three-dimensional forging by considering friction on continuous curved dies with initial guess generation

A three-dimensional finite element analysis of forging of blocks with continuous curved dies is carried out on the basis of a rigid-plastic material model. A method of imposing velocity boundary conditions on a three-dimensionally curved surface is proposed using a skew boundary condition. The frictional boundary condition is considered on a continuous curved surface. The initial velocity field is generated by introducing the assumption of linear viscous material. Forging of square blocks with hemispherical punches of various radii is simulated. Experiments are carried out to check the validity of the formulation. The comparison between computation and experiment shows good agreements in forging load as well as in deformed configuration.     

Dynamic response analysis of block foundations with nonlinear dry friction mounting system to impact loads

It is essential to establish a dynamic model to predict and evaluate the dynamic performance of a nonlinear dry friction mounting system during design procedure, when it is impossible to carry out the test of prototype. Unlike the conventional ideal dry friction model where the direction of dry friction force is always considered to be opposite to that of relative velocity, a new equivalent resistance model of dry friction force is proposed based on the bilinear hysteretic model by introducing a parameterg γ in this work. The equivalent resistance contains spring force and damping force, whose direction is not opposite to that of relative velocity. Then, a dynamic model of the block foundation with nonlinear dry friction mounting system is established. When the equivalent resistance is applied to the dynamic model, its dynamic responses are obtained under common practical forms of press loads: rectangular pulse, half-sine pulse, and triangular pulse. Compared to experimental results, the dynamic responses based on the equivalent resistance model are more consistent with the simulation results based on the ideal dry friction model and the validity of the equivalent resistance model for the bilinear hysteretic model in this work is verified. Furthermore, the effect of the pulse shape and pulse duration on the dynamic responses of the block foundation with nonlinear dry friction mounting system is investigated.     

恒温型热线风速测量系统动态特性分析及试验验证

为测量不同工况下流场的脉动流速,恒温型热线风速测量系统在满足稳定性要求的同时,必须具备宽频带的动态特性。分别对热线探头工作过热比、放大器增益、惠斯通桥臂电阻、流场稳定流速、补偿电感和偏置电压等参量对热线测量系统动态特性的影响效果进行了仿真分析,结果表明:增大热线探头工作过热比、放大器增益和流场稳定流速及减小桥臂电阻均能提高系统的频响;补偿电感与热线探头连接电缆电感及电路寄生电感匹配时,系统频响最佳;在上述参量配置不当时,系统频响降低,系统振荡加剧甚至不能正常工作,而提高偏置电压能消除振荡并使系统恢复稳定,同时获得平稳的动态频响。因此,提出了基于偏置电压的动态特性调节方法。方波试验结果表明,基于偏置电压的频响调节方法能使系统在不同的脉动流速测量中均能获得较为平坦的频响,且调节过程简便。     

Dynamic modeling and analysis of a four-bar mechanism coupled with a CVT for obtaining variable input speeds

This paper proposes a novel concept to obtain variable input speeds for a four-bar mechanism driven by an A.C. motor. In such a system, a continuously variable transmission (CVT) is proposed to use instead of an inverter-controlled motor. By this way, the speed variability which is crucial for obtaining different manufacturing and transportation processes can be provided by mechanically. Thus, more efficient system can be achieved than classical motor controlled systems since the CVT allows electrical motor to run at its nominal point in all conditions. For analyzing the dynamics of this new system, a theoretical model is established regarding that a half toroidal CVT is added between the motor and the mechanism. The simulation results associated with the systems with and without a CVT are presented by comparing to each other. The results indicate that the four-bar mechanism equipped with a CVT is feasible for the entire range of possible reduction ratios of the CVT.     

Influence of test parameters on friction and wear results obtained in oscillating sliding tests with 100cr6 steel against sic‐based materials

In order to characterise materials for tribological applications, model tests with simple contact geometry are in widespread use. Friction and wear behaviour can be determined with high accuracy from this type of test. Tests with oscillating sliding motion have the advantage that only small‐sized specimens are required. However, the fact that the results of model tests are affected considerably by the choice of test parameters is often overlooked. In order to check the influence of test parameters on friction and wear results, tests were performed in which the relevant parameters (stroke, frequency, load, relative humidity of the surrounding air) were varied. Comparative studies with different SiC‐based materials (SiC, SiSiC, and SiC‐TiC) against steel (100Cr6) under unlubricated conditions at room temperature show that in all cases the relative humidity is an important parameter, influencing the friction and wear results substantially. Additionally, several parameters can modify friction and wear behaviour more or less significantly. The effects of test parameters on friction and wear are discussed here on the basis of the wear mechanism. Some of the consequences of this for planning test series and for the practical use of test results are noted.     

Determination of hob offset required to generate nonstandard spur gears with teeth of equal strengthLe calcul du deport d'outil pour generer des dents d'engrenages de meme resistance pour le pignon et la roue

A method has been developed for determining the hob offset e₁ of the pinion from the value of e₁ + e₂ for a pair of spur gears designed to operate at a nonstandard center distance. This method is based on adjusting e₁ and e₂ so that the static stress in the pinion teeth is approximately equal to that in the gear teeth. Charts were plotted so that the ratio of e₁/(e₁ + e₂) could be determined for various velocity ratios and for several changes in center distance. The results of calculating static tooth stresses by the method developed in this paper are compared with those using the Lewis equation and those following the AGMA Standards. Curves were also plotted so that tooth stresses in the form of stress factors could be determined for various values of τC, diametral pitch, and the ratio N₂/(N₁ + N₂).     

Dynamic plastic behavior of a free–free beam striking the mid-span of a clamped beam with shear and membrane effects considered

Recently Yu et al. (Int. J. Solids Struct. 38 (2001) 261) made a study on the dynamic behavior of a flying free–free beam striking the tip of a cantilever beam using the rigid, perfectly plastic (r-p-p) material model. Later, also based on the r-p-p material model Yang and Yu (Mech. Struct. Mach. 29 (2001) 391) analyzed another impact problem of a free rotating hinge beam striking a cantilever beam. Both of these studies ignored the finite deflection effects on the plastic behavior of the colliding beams. However if the free–free beam strikes a clamped beam, the influence of finite-deflections, or, geometric changes, must be retained in the governing equation if the maximum permanent transverse displacement of the clamped beam exceeds the corresponding beam thickness. The problem becomes more interesting since the deformation mechanisms of the beam system and the partitioning of energy dissipation in the beams are significantly different from those predicted by ignoring the influence of membrane forces. Accordingly the failure modes of the structure are different.In the present paper, a theoretical model based on the r-p-p material idealization is proposed to simulate the dynamic behavior when the mid-point of a translating free–free beam impinging on the mid-span of a clamped beam with the beam axes perpendicular to each other. The plastic behavior of the beam system is explored with shear sliding and finite deflection effects taken into account. The final deflection, the dissipation of energy within the two beams after impact and the influence of the structural and material parameters are discussed. It is shown that membrane force plays an important role during the response process, especially when the deflection is of the same order as the thickness of the clamped beam.     

The influence of variation of viscosity with temperature on the steady state characteristics of journal bearings — simplified analysis

Investigation of the influence of temperature on journal bearing characteristics, through the strong dependence of viscosity on temperature, is rendered difficult by the complicated nature of the thermal boundary conditions, the flow in the cavitated region, etc. and by the computational requirements for the simultaneous solution of the Reynolds equation and the energy equation. It was found that the judicious use of the experimentally observed fact that the variation of temperature in the axial direction is negligible can drastically simplify the investigation without impairing its accuracy. The analysis is presented briefly, in dimensionless form, together with a note of caution on the erroneous use of isoviscous bearing characteristics for the estimation of the effective temperature of lubricant film.