Time-Varying Mesh Stiffness Calculation and Dynamic Modeling of Spiral Bevel Gear with Spalling Defects

Time-varying mesh stiffness(TVMS)is a vital internal excitation source for the spiral bevel gear(SBG)transmission system.Spalling defect often causes decrease in gear mesh stiffness and changes the dynamic characteristics of the gear system,which further increases noise and vibration.This paper aims to calculate the TVMS and establish dynamic model of SBG with spalling defect.In this study,a novel analytical model based on slice method is proposed to calculate the TVMS of SBG considering spalling defect.Subsequently,the influence of spalling defect on the TVMS is studied through a numerical simulation,and the proposed analytical model is verified by a finite element model.Besides,an 8-degrees-of-freedom dynamic model is established for SBG transmission system.Incorporating the spalling defect into TVMS,the dynamic responses of spalled SBG are analyzed.The numerical results indicate that spalling defect would cause periodic impact in time domain.Finally,an experiment is designed to verify the proposed dynamic model.The experimental results show that the spalling defect makes the response characterized by periodic impact with the rotating frequency of spalled pinion.

Effect analysis of friction and damping on anti-backlash gear based on dynamics model with time-varying mesh stiffness

A nonlinear model of anti-backlash gear with time-varying friction and mesh stiffness was proposed for the further study on dynamic characteristics of anti-backlash gear. In order to improve the model precision, applied force analysis was completed in detail, and single or double tooth meshing states of two gear pairs at any timing were determined according to the meshing characteristic of anti-backlash gear. The influences of friction and variations of damping ratio on dynamic transmission error were analyzed finally by numerical calculation and the results show that anti-backlash gear can increase the composite mesh stiffness comparing with the mesh stiffness of the normal gear pair. At the pitch points where the frictions change their signs, additional impulsive effects are observed. The width of impulsive in the same value of center frequency is wider than that without friction, and the amplitude is lower. When gear pairs mesh in and out, damping can reduce the vibration and impact.

Dynamic Characteristics of Double-Helical Planetary Gear Sets Under Time-Varying Mesh Stiffness

Internal and external meshes are two of primary excitation sources which induce vibration while double-helical planetary gear sets are in transmission. Based on the analysis of tooth movement principle,three cases of mesh stiffness are derived via investigating the length of action lines,and catalogued in terms of β < β0,β = β0and β > β_0. The simulation demonstrates mesh stiffness between gear pairs performs as a trapezoid waveform( TW) and changes along with the line of action simultaneously,total mesh stiffness comes from the superposition of each engaged gear. While governing equations of motion contained 16 DOFs( degree of freedom) are constructed and effectively solved through the combination of numerical approaches. Comparing with sinusoidal waveform mesh stiffness( SW),the results show that dynamical factors and perturbation under the excitation of TW( β < β_0) are greater and remarkable than that from SW,with respect to the mean dynamic factors about 1. 51 and 1. 28,respectively. The fluctuation response between ring- planet( R- P) is stronger than sun-planet( S-P) which is also validated by both approach studies,frequency spectra analyses identifies larger distinct rotational resonance and more frequencies under TW excitation.

Dynamic characteristics of the planetary gear train excited by time-varying meshing stiffness in the wind turbine

Wind power has attracted increasing attention as a renewable and clean energy. Gear fault frequently occurs under extreme environment and complex loads. The time-varying meshing stiffness is one of the main excitations. This study proposes a 5 degree-of-freedom torsional vibration model for the planetary gear system. The influence of some parameters(e.g., contact ratio and phase difference) is discussed under different conditions of a single teeth pair and double pairs of teeth. The impact load caused by the teeth face fault, ramped load induced by the complex wind conditions, and the harmonic excitation are investigated. The analysis of the time-varying meshing stiffness and the dynamic meshing force shows that the dynamic design under different loads can be made to avoid resonance, can provide the basis for the gear fault location of a wind turbine, and distinguish the fault characteristics from the vibration signals.

Mesh stiffness calculation of cycloid-pin gear pair with tooth profile modification and eccentricity error

Cycloid speed reducers are widely used in many industrial areas due to the advantages of compact size, high reduction ratio and high stiffness. However, currently, there are not many analytical models for the mesh stiffness calculation, which is a crucial parameter for the high-fidelity gear dynamic model. This is partially due to the difficulty of backlash determination and the complexity of multi-tooth contact deformation during the meshing process. In this paper, a new method to calculate the mesh stiffness is proposed including the effects of tooth profile modification and eccentricity error. The time-varying mesh parameters and load distribution of cycloid-pin gear pair are determined based on the unloaded tooth contact analysis （TCA） and the nonlinear Hertzian contact theory, allowing accurate calculations of the contact stiffness of single tooth pair and the torsional stiffness of multi-tooth pairs. A detailed parametric study is presented to demonstrate the influences of tooth profile modification, applied torque and eccentricity error on the torsional mesh stiffness, loaded transmission error, Hertzian contact stiffness and load sharing factor. This model can be applied to further study the lost motion and dynamic characteristics of cycloid speed reducer and assist the optimization of its precision, vibration and noise levels.

Influence Predictions of Contact Effects on Mesh Stiffness of Face Gear Drives with Spur Gear

Mesh stiffness is one of important base parameters of face gear dynamic studies.However,a calculation solution of mesh stiffness of face gear drives is not to be constructed due to complex geometric flakes of face gear teeth.Thus,a calculation solution of mesh stiffness of face gear drives with a spur gear,which is based on the proposed equivalent face gear teeth and Ishikawa model,is constructed,and the influence of contact effects on mesh stiffness of face gear drives is investigated.The results indicate the mesh stiffness of face gear drives is sensitive to contact effects under heavy loaded operating conditions,specially.These contributions will benefit to improve dynamic studies of face gear drives.

Mechanical property of cylindrical sandwich shell with gradient core of entangled wire mesh

To explore the wide-frequency damping and vibration-attenuation performances in the application of aerospace components,the cylindrical sandwich shell structure with a gradient core of entangled wire mesh was proposed in this paper.Firstly,the gradient cores of entangled wire mesh in the axial and radial directions were prepared by using an in-house Numerical Control weaving machine,and the metallurgical connection between skin sheets and the gradient core was performed using vacuum brazing.Secondly,to investigate the mechanical properties of cylindrical sandwich shells with axial or radial gradient cores,quasi-static and dynamic mechanical experiments were carried out.The primary evaluations of mechanical properties include secant stiffness,natural frequency,Specific Energy Absorption(SEA),vibration acceleration level,and so on.The results suggest that the vibration-attenuation performance of the sandwich shell is remarkable when the high-density core layer is at the end of the shell or abuts the inner skin.The axial gradient material has almost no influence on the vibration frequencies of the shell,whereas the vibration frequencies increase dramatically when the high-density core layer approaches the skin.Moreover,compared to the conventional sandwich shells,the proposed functional grading cylindrical sandwich shell exhibits more potential in mass reduction,stiffness designing,and energy dissipation.

Simulation study on dynamics of cylindrical gear meshing with the evolution meshing stiffness

Simulation study on the cylindrical gear meshing with the evolution gear meshing stiffness is being done for better understanding the dynamic characteristics of the kinematics.With consideration of damping,bearing clearance and gear backlash nonlinearity,the dynamic model is set up and computed in MATLAB.The analysis about the relationship between the kinematic responses and the meshing stiffness are carried out.And the results showed that as the gear mesh stiffness is changed from small to large,the performance of the system is changed from the harmonic stable periodic motion to with one times,two times,four times,ending chaos of the stability of the bifurcation.The research results would have theoretical guidance value for the fault diagnosis in engineering.

Investigation on Instantaneous Meshing Stiffness and Profile Modification of Involute Helical Gear

An efficient finite element model of involute helical gear is presented. A program based on compliance matrix method is developed for the calculation of instantaneous meshing stiffness, and the relationship between the rate of meshing stiffness and overlap ratio is given. On the basis of stiffness calculation, an optimization program for the optimal design of profile modification is developed according to the principle of internal point punishment function method.

Analysis of the Relationship Between the Meshing Stiffness and the Inherent Characteristics of Planetary Gear Transmission Mechanism

According to the relationship between the meshing stiffness and the inherent characteristics of a seven-speed three-row coupled planetary transmission mechanism,a equivalent concentrated mass dynamics model of the planetary transmission mechanism is established.The natural frequency of the planetary gear train at a specific gear is calculated and extracted.The relationship between the meshing stiffness of each row and the natural frequency of the system is analyzed,thereby avoiding possible resonance behavior by changing the meshing stiffness.These results show that the meshing stiffness,in its range of possible values,has nearly no effect on the low order natural frequency(<4.000.Hz),and that the time-varying meshing stiffness mainly affects the natural frequencies of the higher-and middle-order parts of the system.Changes of the natural frequencies lead to the change of the system's corresponding vibration mode,which will change the vibration situation of the system.

基于动态啮合刚度的直齿圆柱齿轮动态特性研究

齿轮系统转速直接影响齿轮系统的动态特性,然而在啮合刚度的计算中该因素却被许多学者们忽略。为了研究转速对啮合刚度的影响,基于有限元框架使用平均加速度法提出了一种计算与转速相关的动态啮合刚度的算法,同时对不同转速下的动态啮合刚度进行仿真计算与分析,最后进一步探究了受动态啮合刚度影响后的齿轮系统所具有的相关动态特性。分析表明,动态啮合刚度始终围绕着静态啮合刚度上下波动;随着转速的增加,其波动幅度增加,振荡次数减少;随着转速的变化,动态啮合刚度计算的动态传动误差振幅相对于静态啮合刚度而言大小关系不一致,且计算的齿轮系统共振转速区间或超前或滞后于静态啮合刚度模型;动态啮合刚度影响特定转速区间的齿轮系统的振动周期。对与转速相关的动态啮合刚度的研究可为直齿圆柱齿轮传动性能的改善及减振降噪提供参考。

非对称齿轮啮合刚度精确求解及齿廓参数分析

非对称齿轮是一种轮齿两侧采用不同压力角的新型渐开线齿轮,其齿廓参数对齿轮齿廓形状、运动特性、承载能力、传动效率和动力学特性等有至关重要的影响。基于齿轮啮合原理,运用李特文矢量法,根据加工刀具齿廓参数方程推导了非对称齿轮全齿廓方程;应用材料力学理论,建立非对称齿轮时变啮合刚度的数值求解模型,并通过有限元方法验证了该模型的有效性。基于所建立数值分析模型进行算例研究,总结了刀具圆角、压力角和变位系数对齿轮啮合刚度和重合度的影响规律。结果表明,随着轮齿工作侧压力角的增大,齿轮齿根厚度增加,单齿对的啮合刚度加大,齿轮的承载能力得到有效提高。加工工作侧轮齿齿廓刀具圆角的增大,也有利于提高齿轮的啮合刚度;同时,工作侧齿廓压力角的增大会使齿轮的齿顶厚度降低、啮合重合度减小,会导致齿轮的齿顶强度和冲击韧性降低。非对称齿轮齿廓设计时,应根据实际工况合理设置压力角参数。

双曲型法向圆弧齿轮传动的动力学分析

为探究啮合刚度、传动误差及侧隙对双曲型法向圆弧齿轮传动动态响应的影响,建立齿轮副的动力学方程,并在数值计算的基础上进行仿真分析。运用有限元方法,计算接触位置处的载荷及变形,并对时变啮合刚度、静态传动误差和侧隙进行量化分析。仿真结果表明:对振动位移响应影响最大的是侧隙,其次是时变啮合刚度;对动态啮合力影响最大的则是时变啮合刚度和侧隙。

考虑齿面闪温的风电齿轮箱裂纹故障特征分析

为了研究齿面闪温对风电齿轮箱裂纹故障时系统动力学响应的影响,利用Blok闪温理论分析齿轮啮合时的齿面温度,应用热变形公式计算齿廓形变,进而通过Hertz接触理论获得计及齿面闪温影响的轮齿刚度。考虑齿面闪温、阻尼、时变啮合刚度、综合啮合误差和齿侧间隙,建立含有高速级齿轮齿根裂纹的齿轮箱扭转动力学模型。通过时域图、频谱图、相图和Poincaré截面分析不同裂纹长度下系统振动特性随齿面闪温变化的规律,并比较裂纹故障仿真与实验的时频域结果。结果表明:齿面闪温使时域图上裂纹产生的冲击幅值增大、频域图中故障边频结构更为复杂、相图曲线向外扩展以及Poincaré截面离散点增多,且变化均随裂纹长度的增加越为明显。研究结论可为齿轮裂纹故障状态的诊断与监测提供依据。

小模数齿轮传动的时变啮合刚度计算方法

小模数齿轮传动中心距较小,其动态性能对中心距误差非常敏感,且中间级齿轮常被设计在固定轴上高速旋转,齿轮中心孔与轴之间存在间隙,考虑这些特征,建立了中心距误差与轴孔配合间隙影响下的齿轮啮合特性参数与时变啮合刚度计算模型,研究了二者对啮合刚度的影响规律。研究结果表明:中心距误差会改变齿轮副的重合度与啮合刚度;轴孔配合间隙使实际中心距围绕理论中心距上下周期性波动,导致整个周期内啮合刚度强化区域与弱化区域共存;中心距误差与轴孔配合间隙的影响在不同的转角范围内存在既有相互叠加又有相互削弱的现象,这一现象使得各轮齿的啮合刚度有差异,存在诱发更大振动与不同噪声的风险。

壳体柔性对风电齿轮箱-发电机集成系统动态特性影响

针对风电传动系统集成化结构,提出了一种可用于风力发电机变速-变载工况下的机电-刚柔耦合动力学模型,不仅考虑了齿轮的时变啮合刚度、相位关系、轴和壳体的结构柔性等机械因素,同时计入了发电机系统中永磁体磁饱和特性、电磁径向力波以及空间谐波等电磁因素。探究齿轮箱-发电机集成系统机电耦合动态特性,讨论了壳体柔性对系统动态特性的影响,提出了一种升速分析法,找寻了系统的共振转速。结合模态能量法和阵型矢量分布原理,找寻了共振时的潜在危险构件。研究表明:齿轮系统与发电机存在强耦合特性,壳体的柔性对系统机电耦合特性影响显著。针对集成化系统而言,齿轮内激励为共振转速下的主要激励源;但采用薄壁壳体时,发电机电磁激励不容忽视,易激发新的共振转速。选择合理的壁厚可有效提高系统的安全可靠性,减少共振区域,减轻系统构件的损坏。

基于时变啮合刚度的多剥落齿轮振动响应分析

为了探究多剥落齿轮系统中振动响应的演变规律,融合时变啮合刚度构建了多剥落齿轮动力学模型。首先,融合修正的能量法和单/双齿对啮合角位移计算方法,建立多剥落齿轮啮合刚度模型;其次,考虑时变啮合刚度、齿侧间隙和初始压力角等因素,构建6自由度多剥落齿轮动力学模型,并从仿真、理论和实验维度开展模型有效性分析;最后,基于所构建的模型,研究齿根、节线和齿顶位置同时发生剥落时,剥落尺寸变化对啮合刚度影响,获得多剥落齿轮振动响应的演变过程。研究结果表明,所构建模型与理论及实验误差小于0.5%,并得到了齿根、节线和齿顶位置剥落的啮合刚度及振动响应的演变规律,随着剥落宽度的扩展,节线附近振动幅值增大显著;随着剥落深度的增加,齿顶位置受到的振动冲击更为强烈;而随着剥落长度的增加,剥落的波及范围扩大。所得结论为齿轮系统的健康监测和早期故障诊断提供了理论指导。

重载铁路用新型网孔式弹性垫板的力学特性研究

重载铁路轨道系统的弹性垫板由于承受载荷较大,易出现疲劳破损现象。鉴于既有弹性垫板存在应力集中且阻尼较小的现象,提出一种新型网孔式弹性垫板,可改善应力分布,提升阻尼特性。为探究将网孔式弹性垫板应用于重载铁路扣件系统的可行性并验证相较于既有弹性垫板的优越性,分别建立适用于重载铁路弹条Ⅱ型扣件系统的既有弹性垫板和新型网孔式弹性垫板有限元模型。在同刚度条件下,对比研究网孔式、沟槽式、棱台式弹性垫板的竖向刚度、横向刚度和应力分布特征。研究结果表明,网孔式弹性垫板的应力应变分布更均匀,其最大应力相较于沟槽式弹性垫板和棱台式弹性垫板降低约30%,力学特性更好,且具有较好的竖向刚度保持性能;在施加横向载荷后,网孔式弹性垫板的横向刚度比棱台式弹性垫板的小,更接近于沟槽式弹性垫板;新型网孔式弹性垫板具有在重载铁路中推广应用的潜力。

大型风电齿轮箱宏观参数优化设计

在大兆瓦级风电齿轮箱设计中,为了提高功率密度同时又尽量减小振动噪声,提出一种齿轮宏观参数优化设计方法。以功率密度、啮合刚度为优化目标,抓住最主要的影响因素,建立多目标优化模型,在模型函数中综合考虑重合度、啮合刚度、齿轮强度与宏观参数设计的关系,得出高效可行的优化方法。对5MW风机高速级齿轮优化结果进行对比,该优化方法在保证齿轮强度条件下,可有效增加啮合刚度,减小传递误差,提高动力学性能,与试验结果对比,降低噪音值4%左右。

大重合度直齿轮动载特性分析

针对大重合度直齿轮啮合齿数与普通直齿轮间的差异性变化,以一对大重合度直齿轮和普通直齿轮为研究对象,通过有限元法求解齿轮时变啮合刚度,建立大重合度直齿轮啮合耦合型动力学模型,利用Newmark-β时域积分求解齿轮系统动态响应,对其进行时频域分析并与普通直齿轮进行对比,获得了大重合度直齿轮在不同转速、误差影响下的动载系数变化规律。结果表明,大重合度直齿轮啮合刚度与普通齿轮有较大差异,其动态响应结果波动幅值较普通直齿轮有明显降低,提升了系统稳定性;大重合度直齿轮动载系数与普通直齿轮相比有明显下降,并且随着转速的升高和误差的增大出现不同程度的增加,其中在共振转速区域范围内增幅明显。