Dynamic response analysis of helical gear pair considering the interaction between friction and mesh stiffness

Meccanica - This work investigates the dynamic responses of a pair of helical gears considering the interaction between friction and time-varying mesh stiffness (TVMS). Firstly, governing equations...     

Nonlinear dynamics of a spur gear pair with force-dependent mesh stiffness

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.     

Semi-analytical solution for a system with clearance nonlinearity and periodic excitation

Many dynamical systems such as gears, tire-pavement, automotive brakes, and cam-follower have clearance nonlinearity and excitation, which are periodic in nature. It is essential to accurately predict the steady-state response of these systems using contact-mechanics-based model for understanding their nonlinear dynamic behavior. Among the methods available to theoretically solve the system’s nonlinear governing equation(s), a semi-analytical technique such as the harmonic balance method (HBM) is preferred over numerical approaches for various reasons, including accuracy. An HBM formulation that can predict the fundamental, sub-, and super-harmonic solutions is presented here. As multiple variants of HBM exist in the literature, this work focuses on comparatively evaluating the most appropriate variant for the system under consideration. Since the system has multiple discontinuities in terms of contact stiffness and damping forces, these have to be smoothed precisely to be utilized in the HBM. Hence, a novel smoothing function was proposed and evaluated against other existing smoothing functions in literature based on various criteria. Next, the most applicable HBM variant was selected with reference to steady-state solutions from numerical methods. The predictions from the selected HBM variant were validated against the results furnished in the literature for a similar system. Finally, the nonlinear frequency response of the system with multiple discontinuities was estimated using the selected HBM and found to be in good agreement with numerical results.

     

Stochastic responses of viscoelastic system with real-power stiffness under randomly disordered periodic excitations

The stochastic dynamic responses of viscoelastic systems with real-power exponents of stiffness term subjects to randomly disordered periodic excitations are studied. The assumed viscoelastic damping depends on the past history of motion via convolution integrals over an exponentially decaying kernel function. The multiple scales method is used to derive the stochastic different equations of modulation of amplitude and phase. The changes of the shape of resonance curves are obtained with real-power exponents of stiffness term and viscoelastic parameters, and then, the numerical simulation method was used to verify the accuracy of the theoretical analysis results. Theoretical analysis and numerical simulations show that as the intensity of the random excitation increases, the steady-state solution changes from a limit cycle to a diffused limited cycle. Under some conditions, the system may have two steady-state solutions and the phenomenon of jumps will happen to them under the random excitations.     

Influence of system parameters on dynamic behavior of gear pair with stochastic backlash

The influence of stochastic backlash on nonlinear dynamic behavior of spur gear pair with stochastic assembling backlash is discussed. Bifurcation diagram and maximum Lyapunov exponent diagram of the system are presented to evaluate the influences of load ratio, damping ratio, and backlash on the dynamic behavior of the system. The results show that backlash has great contribution to light-loaded gear pair. Therefore, dynamic behavior with stochastic assembling backlash is analyzed. Two novel indexes are introduced to evaluate the dynamic behavior of the system, dynamic instability exponent and critical variance. In addition, the relationship between the dynamic instability exponent and variance of the backlash, and the relationship between the critical variance and the mean value of the backlash are studied. The results show that it is not the unique way to improve the dynamic behavior of the system by minimizing the assembling backlash, which provide a novel way to determine the machining precision requirements, tolerance for assembling, and etc.     

Dynamic characteristics and load-sharing performance of concentric face gear split-torque transmission systems with time-varying mesh stiffness,flexible supports and deformable shafts

Meccanica - The concentric face gear split-torque transmission system is a new gear mechanism which integrates the advantages of face gears, split-torque transmissions and concentric transmissions....     

On the frequency bandwidth change of a servo system with a gear reducer due to backlash and motor input voltage

Summary The paper analyzes how the frequency bandwidth of a servo system is affected by the input voltage of its motor and the magnitude of the total backlash. The bandwidth of the system is defined as the set of antiresonance frequencies, which appears in the frequency-response characteristic. The change in the frequency-response characteristics is investigated depending on motor input voltage and the magnitude of the total backlash. It is shown that at infinite motor input voltage, the servo system has a bandwidth equivalent to that of a backlash-free system. The amount of the systems bandwidth reduction due to the backlash changes greatly with the motor input voltage. It becomes possible then to determine the necessary maximum input voltage of the motor and the admissible magnitude of the total backlash to satisfy the desired bandwidth for a servo system with a gear reducer.     

Dynamic modeling and analysis of a spur planetary gear involving tooth wedging and bearing clearance nonlinearity

Tooth wedging, also known as tight mesh, occurs when a gear tooth comes into contact on the drive-side and back-side simultaneously. Tooth wedging risks bearing failures from elevated forces. This work studies the nonlinear tooth wedging behavior and its correlation with planet bearing forces by analyzing the dynamic response of an example planetary gear. This planetary gear is representative of a wind turbine geartrain. A two-dimensional lumped-parameter model is extended to include tooth separation, back-side contact, tooth wedging, and bearing clearances. The results show significant impact of tooth wedging on planet bearing forces for a wide range of operating speeds. To develop a physical understanding of the tooth wedging mechanism, connections between planet bearing forces and tooth forces are studied by investigating physical forces and displacements acting throughout the planetary gear. A method to predict tooth wedging based on geometric interactions is developed and verified. The major causes of tooth wedging relate directly to translational vibrations caused by gravity forces and the presence of clearance-type nonlinearities in the form of backlash and bearing clearance.     

Dynamic response of a rub-impact rotor system under axial thrust

A model of a rigid rotor system under axial thrust with rotor-to-stator is developed based on the classic impact theory and is analyzed by the Lagrangian dynamics. The rubbing condition is modeled using the elastic impact-contact idealization, which consists of normal and tangential forces at the rotor-to-stator contact point. Mass eccentricity and rotating speed are used as control parameters to simulate the response of rotor system. The motions of periodic, quasi-periodic and chaotic are found in the rotor system response. Mass eccentricity plays an important role in creating chaotic phenomena.     

Analytical models of mesh stiffness for cracked spur gears considering gear body deflection and dynamic simulation

Meccanica - Accurate mesh stiffness models for cracked spur gears are critical to obtain gearbox failure characteristics. However, the influence of crack on gear body deflection is often ignored by...     

Drift response of a bilinear hysteretic system to periodic excitation under sustained load effects

This paper presents an investigation of response characteristics for hysteretic systems idealized as a bilinear hysteretic model subjected to period excitations composed of a harmonic function and a sustained load. It is shown that the displacement solution can exhibit a drift sequence persistently repeated at a frequency identical to the excitation frequency in the case of zero post-yielding stiffness. The periodic-like drift sequence is further classified into three major types according to their different hysteretic looping behaviors. An approximate solution approach based on the method of weighted residuals is proposed to analyze the drift amplitude per response cycle. The assumed response shape is composed of two concatenated harmonic functions each with a frequency slightly detuned from the excitation frequency. The method is accompanied with a subsequent first-order analysis to obtain a closed-form approximation for the drift response. Good response predictions of the proposed solution method are demonstrated through both undamped and damped drift-frequency analyses.     

The dynamical nature of a backlash system with and without fluid friction

We study the dynamics of a simple system with backlash and impacts. Both the presence or the absence of fluid friction is considered. The fluid friction is modeled by a fractional derivative, but it is also shown how an inhomogeneous time scale, although not arising from a fractional differential equation, may lead to some features similar to fractional solutions.     

Effect of spalling or tooth breakage on gearmesh stiffness and dynamic response of a one-stage spur gear transmission

Tooth faults affecting gear transmission are always accompanied by a stiffness reduction. In this article an analytical method is proposed to quantify the reduction of gearmesh stiffness due to two common tooth faults: spalling and breakage. Bending, fillet-foundation and contact deflection are taken into account. The dynamic response of a single stage spur gear transmission is computed by using analytical gearmesh issued from analytical modelling and the vibration signatures of each tooth fault is identified.     

考虑弹流润滑作用的斜齿轮啮合分析方法研究

油膜弹流润滑在齿轮传动中有着非常重要的作用,为得到油膜润滑作用下的齿轮啮合响应,基于ABAQUS/STANDARD的静态计算结果,首先提取了仅有齿轮啮合的齿面接触刚度,再结合油膜刚度得到了齿轮和油膜的综合接触刚度,并以此综合刚度作为接触属性关系进行齿轮的静动态运动响应计算。此外,对齿轮啮合时出现的接触区域(接触斑)不连续现象也进行了分析。最终结果表明考虑油膜润滑作用时,齿轮面的最大接触应力比无润滑作用时下降了30%左右,而齿根处最大拉应力则下降了6.14%。本方法为齿轮动力学分析和齿轮的优化设计提供了基础条件。     

齿轮箱非线性耦合系统的动力学分析

分别建立了啮合齿轮子系统弯扭耦合振动模型、弹性轴子系统的弯曲振动和扭转振动模型以及齿轮箱弹性箱板子系统的横向振动模型,通过轴承多维刚度矩阵和子结构之间的受力变形协调关系将各子结构的振动模型联立起来,从而得到整个齿轮箱耦合系统的动力学模型。结合相图和庞加莱映射图分析了系统的运动特性,采用功率流方法研究了系统振动能量传递特性及其影响因素,揭示了外部激励和轴承刚度等因素对系统特性的影响,为齿轮箱系统的振动控制与结构设计提供了新的分析方法和理论依据。     

Dynamic stability for a simply supported beam under periodic axial excitation

This paper studies the dynamic stability for a simply supported straight beam under periodic axial excitation by using the averaging method and the Routh-Hurwitz stability criteria. By considering the first two modes coupled, we discuss the effect of the stability-instability region and the amplitudes of vibration. Furthermore, by studying the principal parametric resonance i.e. subharmonic order $\text{1}\text{2}$, we investigate the effect of the amplitude of the main system by various kinds of non-linearities of the subsystem. Finally, by obtaining the transient results, we describe the beat phenomenon, and harmonic oscillation.     

Dynamic analysis of the variable stiffness support rotor system with elastic rings

Nonlinear Dynamics - Elastic rings are common rotor supporting structure, which have been widely used in aeroengine rotor support system. However, large inertia force and gyroscopic moment may...     

Nonlinear dynamic response of an isolation system with superelastic hysteresis and negative stiffness

Nonlinear Dynamics - The negative stiffness exhibited by bi-stable mechanisms together with the tunable superelasticity offered by shape memory alloy (SMA) wires can enhance the dynamic resilience...     

高冲击载荷作用下弹载记录仪防护系统动力学响应特性

为了给弹载记录仪的防护设计提供依据,从机械振动的角度揭示了高冲击载荷作用下弹载记录仪防护系统的动力学响应机理。在分析弹载记录仪内部载荷传递关系的基础上,基于双自由度弹簧-质量-阻尼系统建立了一种简化的防护系统动力学响应模型,并开展了数值模拟,通过脉冲响应分析和谐响应分析辨识了模型参数。理论计算与数值模拟的对比分析结果表明：建立的动力学响应模型能较准确地预测高冲击载荷作用下弹载记录仪防护系统的动力学响应特性。在此基础上,以模型的幅频响应特性为依据,分析了防护系统动力学响应特性随各种参数的变化规律。研究结果可为更有效地指导弹载记录仪的防护设计提供依据。