Static and dynamic behaviors of belt-drive dynamic systems with a one-way clutch

This paper focuses on the nontrivial equilibrium and the steady-state periodic response of belt-drive system with a one-way clutch and belt flexural rigidity. A nonlinear piecewise discrete–continuous dynamic model is established by modeling the motions of the translating belt spans as transverse vibrations of axially moving viscoelastic beams. The rotations of the pulleys and the accessory are also considered. Furthermore, the transverse vibrations and the rotation motions are coupled by nonlinear dynamic tension. The nontrivial equilibriums of the belt-drive system are obtained by an iterative scheme via the differential and integral quadrature methods (DQM and IQM). Moreover, the periodic fluctuation of the driving pulley is modeled as the excitation of the belt-drive system. The steady-state periodic responses of the dynamic system are, respectively, studied via the high-order Galerkin truncation as well as the DQM and IQM. The time histories of the system are numerically calculated based on the 4th Runge–Kutta time discretization method. Furthermore, the frequency–response curves are presented from the numerical solutions. Based on the steady-state periodic response, the resonance areas of the dynamic system are obtained by using the frequency sweep. Moreover, the influences of the truncation terms of the Galerkin method, such as 6-term, 8-term, 10-term, 12-term, and 16-term, are investigated by comparing with the DQM and IQM. Numerical results demonstrate that the one-way clutch reduces the resonance responses of the belt-drive system via the torque-transmitting directional function. Furthermore, the comparisons in numerical examples show that the investigation on steady-state responses of the belt-drive system with a one-way clutch and belt flexural rigidity needs 16-term truncation     

Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam

This paper investigates the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam. For the first time, the nonlinear dynamic behaviors in the transverse parametric vibration of an axially moving Timoshenko beam are studied. The axial speed of the system is assumed as a harmonic variation over a constant mean speed. The transverse motion of the beam is governed by nonlinear integro-partial-differential equations, including the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation is applied to discretize the governing equations into a set of nonlinear ordinary differential equations. Based on the solutions obtained by the fourth-order Runge–Kutta algorithm, the stable steady-state periodic response is examined. Besides, the bifurcation diagrams of different bifurcation parameters are presented in the subcritical and supercritical regime. Furthermore, the nonlinear dynamical behaviors are identified in the forms of time histories, phase portraits, Poincaré maps, amplitude spectra, and sensitivity to initial conditions. Moreover, numerical examples reveal the effects of various terms Galerkin truncation on the amplitude–frequency responses, as well as bifurcation diagrams.     

Nonlinear dynamics of three-dimensional belt drives using the finite-element method

In this paper, new nonlinear dynamic formulations for belt drives based on the three-dimensional absolute nodal coordinate formulation are developed. Two large deformation three-dimensional finite elements are used to develop two different belt-drive models that have different numbers of degrees of freedom and different modes of deformation. Both three-dimensional finite elements are based on a nonlinear elasticity theory that accounts for geometric nonlinearities due to large deformation and rotations. The first element is a thin-plate element that is based on the Kirchhoff plate assumptions and captures both membrane and bending stiffness effects. The other three-dimensional element used in this investigation is a cable element obtained from a more general three-dimensional beam element by eliminating degrees of freedom which are not significant in some cable and belt applications. Both finite elements used in this investigation allow for systematic inclusion or exclusion of the bending stiffness, thereby enabling systematic examination of the effect of bending on the nonlinear dynamics of belt drives. The finite-element formulations developed in this paper are implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing more detailed models of mechanical systems that include belt drives subject to general loading conditions, nonlinear algebraic constraints, and arbitrary large displacements. The use of the formulations developed in this investigation is demonstrated using two-roller belt-drive system. The results obtained using the two finite-element formulations are compared and the convergence of the two finite-element solutions is examined.     

粘弹性轴向运动梁的非线性动力学行为

本文研究了带有小脉动的轴向运动粘弹性梁的分岔及混沌现象。建立了系统的动力学模型。通过二阶Galerkin截断，把描述系统运动的偏微分方程离散化。利用数值方法分别分析了几种运动脉动频率时，梁随轴向运动脉动幅值，平均速度及粘弹性系数等几个参数变化时的运动分岔行为。利用Lyapunov指数识别系统的动力学行为，区分准周期振动和混沌运动。     

Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed

Non-linear vibration of viscoelastic pipes conveying fluid around curved equilibrium due to the supercritical flow is investigated with the emphasis on steady-state response in external and internal resonances. The governing equation, a non-linear integro-partial-differential equation, is truncated into a perturbed gyroscopic system via the Galerkin method. The method of multiple scales is applied to establish the solvability condition in the first primary resonance and the 2:1 internal resonance. The approximate analytical expressions are derived for the frequency–amplitude curves of the steady-state responses. The stabilities of the steady-state responses are determined. The generation and the vanishing of a double-jumping phenomenon on the frequency–amplitude curves are examined. The analytical results are supported by the numerical integration results.     

Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation

The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are studied for the response of a Timoshenko beam supported by a nonlinear foundation. The nonlinear Pasternak foundation is assumed to be cubic. Therefore, the efects of the shear deformable beams and the shear deformation of foundations are considered at the same time. The Galerkin method is utilized for discretizing the nonlinear partial differential governing equations of the forced vibration. The dynamic responses of Timoshenko beams are determined via the fourth-order Runge–Kutta method. Moreover, the efects of diferent truncation terms on the dynamic responses of a Timoshenko beam resting on a complex foundation are discussed. The numerical investigations shows that the dynamic response of Timoshenko beams supported by elastic foundations needs super high-order modes. Furthermore, the system parameters are compared to determine the dependence of the convergences of the Galerkin method.     

Finite element analysis of V-ribbed belts using neural network based hyperelastic material model

A three-dimensional finite element model was built to study V-ribbed belt pulley contact mechanics. The model consists of a pulley and a segment of V-ribbed belt in contact with the pulley. A material model for the belt, including the rubber compound and the reinforcing cord is developed. Rubber is modeled as hyperelastic material. The hyperelastic strain energy function is approximated by neural network trained by rubber test data. Reinforcing cord is modeled as elastic rebar. The material model developed is implemented in the commercial finite element code ABAQUS to simulate the V-ribbed belt-pulley system. A study is then conducted to investigate the effect of belt pulley system parameters on the contact mechanics. The effects of temperature and aging on belt materials are also investigated. The information gained from the analysis can be applied to optimize V-ribbed belt and pulley design.     

Dynamical behavior of nonlinear viscoelastic beams

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轴向变速运动正交各向异性板的动态稳定性

研究面内载荷作用下轴向变速运动正交各向异性薄板的横向振动及其稳定性。利用Galerkin法与平均法,在激励频率为2倍固有频率或为两阶固有频率之和附近时得到自治的常微分方程组;在参数激励频率和激励振幅平面上,分析由共振引发的失稳区域。数值算例验证了面内载荷、轴向速度、加速度参数对失稳区域的影响。     

DYNAMICAL ANALYSIS OF AUTOMOBILE HYSTERESIS NONLINEAR SYSTEM

This paper adopts free interface modal synthesis method to divide the whole automobile model into many sub-structures and establish dynamical equations of automobile nonlinear coupled system. The Monte Carlo method is used to simulate the spectrum of the random excitation of the road and the engine. Based on the automobile dynamical equations, a simulation is carried out within time domain and frequency domain on the characteristic of vibration due to the excitation of automobile wheel and the engine. The results are verified by bench experiment to make the research more practicable. In order to do research of rubber hysteresis’ influence on automobile dynamic property, Poincare diagrams and amplitude frequency characteristic curves were drawn with automobile linear and nonlinear models. The results show that the nonlinear dynamical model concerning rubber hysteresis not only can improve the simulation accuracy, but also is beneficial to find some complex nonlinear dynamical behaviors of vehicles.     

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

Low-Dimensional Description of Oscillatory Thermal Convection: The Small Prandtl Number Limit

 Low-dimensional models are derived for transitional, buoyancy-driven flow in a vertical channel with prescribed spatially periodic heating. Stationary characteristic structures (empirical eigenfunctions) are identified by applying proper orthogonal decomposition to numerical solutions of the governing partial differential equations. A Galerkin procedure is then employed to obtain suitable low-order dynamical models. Stability analysis of the fixed points of the low-order systems predicts conditions at the primary flow instability that are in very good agreement with direct numerical solutions of the full model. This agreement is found to hold as long as the low-order system possesses a possible Hopf bifurcation (minimum two-equation) mechanism. The effect of the number of retained eigenmodes on amplitude predictions is examined. Received 25 January 1996 accepted 19 July 1996     

Nonlinear vibration analysis of harmonically excited cracked beams on viscoelastic foundations

The frequency response of a cracked beam supported by a nonlinear viscoelastic foundation has been investigated in this study. The Galerkin method in conjunction with the multiple scales method (MSM) is employed to solve the nonlinear governing equations of motion. The steady-state solutions are derived for the two different resonant conditions. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of crack, loading position and the linear and nonlinear foundation parameters, on the frequency-response solution are examined.     

Primary resonance of traveling viscoelastic beam under internal resonance

Under the 3:1 internal resonance condition,the steady-state periodic response of the forced vibration of a traveling viscoelastic beam is studied.The viscoelastic behaviors of the traveling beam are described by the standard linear solid model,and the material time derivative is adopted in the viscoelastic constitutive relation.The direct multi-scale method is used to derive the relationships between the excitation frequency and the response amplitudes.For the first time,the real modal functions are employed to analytically investigate the periodic response of the axially traveling beam.The undetermined coefficient method is used to approximately establish the real modal functions.The approximate analytical results are confirmed by the Galerkin truncation.Numerical examples are presented to highlight the effects of the viscoelastic behaviors on the steady-state periodic responses.To illustrate the effect of the internal resonance,the energy transfer between the internal resonance modes and the saturation-like phenomena in the steady-state responses is presented.     

静载荷对夹层圆板超谐波共振的影响

研究静载荷作用下夹层圆板的超谐波共振问题．基于Hoff型夹层板理论,给出了静载荷作用下夹层圆板的非线性动力学方程．应用Galerkin法推导了静载荷作用下夹层圆板的轴对称非线性振动方程．运用多尺度法分别对系统的三次超谐波问题和二次超谐波问题进行了求解,并依据Lyapunov稳定性理论得到了系统稳态运动的稳定性判据．通过算例,得到了周边简支约束下夹层圆板三次超谐波共振和二次超谐波共振的幅频响应曲线图、振幅-静载荷响应曲线图、振幅-激励力幅值响应曲线图;研究了不同参数对系统振幅的影响规律,并对解的稳定性进行了分析．     

Dynamic characteristic analysis for clutch engagement process of series–parallel hybrid electric vehicle

Variable driving conditions can cause an integrated starter generator hybrid powertrain to switch between multiple drive modes. The addition of a permanent magnet synchronous motor (PMSM) gives hybrid powertrains complex electromechanical coupling characteristics. The effects of excitation sources, such as the engine and PMSM, may cause unstable behavior in the drive system, such as speed fluctuations during mode switches due to electromechanical coupling characteristics. Although traditional mode switch strategies and methods have achieved measurable results, they are difficult to improve. To solve this problem, we first considered the combination and separation of the clutch and establish a nonlinear model of mode switches for series–parallel hybrid electric vehicles. Then, we predicted the instability boundary of the drive system during mode switches. Experimental results indicated that the proposed instability boundary has higher accuracy. Numerical results showed that the three-mode switches have different thresholds of instability for the clutch structure gap. The decrease in electromagnetic torque and the increase in load excitation amplitude will improve the critical value of the clutch structure gap. The increase in load excitation frequency causes the critical value of the clutch structure gap to drop first and then rise.

     

Asymptotic Nonlinear Behaviors in Transverse Vibration of an Axially Accelerating Viscoelastic String

This paper investigates longtime dynamical behaviors of an axially accelerating viscoelastic string with geometric nonlinearity. Application of Newton's second law leads to a nonlinear partial-differential equation governing transverse motion of the string. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. By use of the Poincare maps, the dynamical behaviors are presented based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented for varying one of the following parameter: the mean transport speed, the amplitude and the frequency of transport speed fluctuation, the string stiffness or the string dynamic viscosity, while other parameters are fixed.     

On the effect of multiple parallel nonlinear absorbers in palliation of torsional response of automotive drivetrain

Torsional vibrations transmitted from the engine to the drivetrain system induce a plethora of noise, vibration and harshness (NVH) concerns, such a transmission gear rattle and clutch in-cycle vibration, to name but a few. The main elements of these oscillations are variations in the inertial imbalance and the constituents of combustion power torque, collectively referred to as engine order vibration. To attenuate the effect of these transmitted vibrations and their oscillatory effects in the drive train system, a host of palliative measures are employed in practice, such as clutch pre-dampers, slipping discs, dual mass flywheel and others, all of which operate effectively over a narrow band of frequencies and have various unintended repercussions. These include increased powertrain inertia, installation package space and cost. This paper presents a numerical study of the use of multiple Nonlinear Energy Sinks (NES) as a means of attenuating the torsional oscillations for an extended frequency range and under transient vehicle manoeuvres. Frequency–Energy Plots (FEP) are used to obtain the nonlinear absorber parameters for multiple NES coupled in parallel to the clutch disc of a typical drivetrain configuration. The results obtained show significant reduction in the oscillations of the transmission input shaft, effective over a broad range of response frequencies. It is also noted that the targeted reduction of the acceleration amplitude of the input shaft requires significantly lower NES inertia, compared with the existing palliative measures.     

四角点支撑纳米板稳态受迫振动解析解

本文基于非局部弹性理论及辛叠加方法,得到放置在黏弹性介质上四角点支撑矩形纳米板稳态受迫振动问题的解析解．将纳米板受迫振动问题导入哈密顿体系,得到哈密顿控制方程,在无需任何预设函数的情况下可直接对哈密顿控制方程进行求解,得到简支纳米板稳态受迫振动问题在辛空间展开形式的解析解．进而通过边界叠加,可求出四角点支撑纳米板稳态受迫振动的解析解．数值算例中验证了本文应用辛叠加方法得到解析解的准确性,并以石墨烯纳米板为例,分析了非局部参数和黏弹性介质参数对四角点支撑石墨烯纳米板稳态受迫振动的影响．结果表明,非局部参数和黏弹性介质参数的变化会影响石墨烯纳米板的共振频率及共振幅值．