In this paper, research on transverse vibrations of axially moving strings and their control is thoroughly reviewed. In the last few decades, there have been extensive studies on analysis and control of transverse vibrations of axially moving strings because of the wide applications of many engineering devices that axially moving strings represent. In the investigations adopting linear models of moving strings, the paper summarizes recent studies on modal analysis, complicatedly constrained strings, coupled vibrations, and parametric vibration, as well as some early results. In the investigations adopting nonlinear models of moving strings, the paper presents the governing equations with large amplitude, and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Furthermore, investigations are reported on modeling of damping mechanisms as viscoelastic materials, coupled vibration of power transmission systems, and bifurcation and chaos. The state of the art of active control of moving strings is surveyed on controllability and observability, the Laplace transform domain analysis and the energy analysis, nonlinear vibration control and adaptive vibration control. Finally, future research directions are suggested such as nonlinear vibration of moving strings under complex constraints and couplings, energetics of nonlinear and time-varying strings, bifurcation and chaos in transverse motion of moving strings, control of hybrid systems containing moving strings, robust and adaptive controls of nonlinear moving strings, and experimental investigations. In this review article there are 242 references cited. @DOI: 10.1115/1.1849169#

Analysis and Control of Transverse Vibrations of Axially Moving Strings

The vibration of an automotive serpentine belt drive system greatly affects the perceived quality and the reliability of the system. Accessory drives with unfavorable vibration characteristics transmit excessive noise and vibration to other vehicle structures, to the vehicle occupants, and may also promote the fatigue and failure of system components. Moreover, these characteristics are a consequence of decisions made early on in the design and arrangement of the accessory drive system. The present paper focuses on fundamental modeling issues that are central to predicting accessory drive vibration. To this end, a prototypical drive is evaluated, which is composed of a driven pulley, a driving pulley, and a dynamic tensioner. The coupled equations of free response governing the discrete and continuous elements are presented herein. A closed-form solution method is used to evaluate the natural frequencies and modeshapes. Attention focuses on a key linear mechanism that couples tensioner arm rotation and transverse vibration of the adjacent belt spans. Modal tests on an experimental drive confirm the theoretical predictions.

Free Vibration of Serpentine Belt Drive Systems

In this study, a dynamic finite element model is developed for pulley belt-drive systems and is employed to determine the transient and steady-state response of a prototypical belt-drive. The belt is modeled using standard truss elements, while the pulleys are modeled using rotating circular constraints, for which the driver pulley's angular velocity is prescribed. Frictional contact between the pulleys and the belt is modeled using a penalty formulation with frictional contact governed by a Coulomb-like tri-linear friction law. One-way clutch elements are modeled using a proportional torque law supporting torque transmission in a single direction. The dynamic response of the drive is then studied by incorporating the model into an explicit finite element code, which can maintain time-accuracy for large rotations and for long simulation times. The finite element solution is validated through comparison to an exact analytical solution of a steadily-rotating, two-pulley drive. Several response quantities are compared, including the normal and tangential (friction) force distributions between the pulleys and the belt, the driven pulley angular velocity and the belt span tensions. Excellent agreement is found. Transient response results for a second belt-drive example involving a one-way clutch are used to demonstrate the utility and flexibility of the finite element solution approach.

Transient and Steady-State Dynamic Finite Element Modeling of Belt-Drives

Belt drives have long been utilized in engine applications to power accessories such as alternators, pumps, compressors and fans. Drives employing a single, flat, serpentine belt tensioned by an automatic tensioner are now common in automotive engine applications. The automatic tensioner helps maintain constant belt tension and to dissipate unwanted belt drive vibration through dry friction. The objective of this study is to predict the periodic rotational response of the entire drive to harmonic excitation from the crankshaft. To this end, a multi-degree of freedom incremental harmonic balance method (IHB) is utilized to compute periodic solutions to the nonlinear equations of motion over a wide range of engine speeds. Computed results illustrate primary and secondary resonances of the accessory drive and tensioner stick-slip motions.

Nonlinear Periodic Response of Engine Accessory Drives With Dry Friction Tensioners

Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model ofa parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general nth-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly nonlinear equation of motion leads to an analytical expression for the amplitudes (and stability) of nontrivial limit cycles that exist around the nth-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.

Stability and limit cycles of parametrically excited, axially moving strings

A nonlinear model is developed which describes the rotational response of automotive serpentine belt drive systems. Serpentine drives utilize a single {long) belt to drive all engine accessories from the crankshaft. An equilibrium analysis leads to a closed-form procedure for determining steady-state tensions in each belt span. The equations of motion are linearized about the equilibrium state and rotational mode vibration characteristics are determined from the eigenvalue problem governing free response. Numerical solutions of the nonlinear equations of motion indicate that, under certain engine operating conditions, the dynamic tension fluctuations may be sufficient to cause the belt to slip on particular accessory pulleys. Experimental measurements of dynamic response are in good agreement with theoretical results and confirm theoretical predictions of system vibration, tension fluctuations, and slip. 1 Introduction The trend in automotive accessory drives has been to replace multiple V-belt drives with a single flat belt drive to power all the accessories. Such systems are termed "serpentine" belt drives and include a spring loaded tensioner, as well as multiple accessory pulleys (see Fig. 1). These systems can exhibit com­plex dynamic behavior, including rotational vibrations of the pulleys with the belt spans serving as coupling springs, and transverse belt vibrations in the various spans. The transverse vibrations of translating belts [1, 2] belongs to the broad cat­egory of systems referred to as axially moving materials. Other examples of axially moving materials include chain drives [3, 4], band saws [5, 6, 7], V-belts [8], moving threadlines [9], translating cables [10], etc. The recent literature on axially moving materials is reviewed in [11]. Several recent studies have focused on serpentine belt drive systems (see Fig. 1). Gasper etal. [12] and Hawker [13] consider longitudinal deflection of the belt and rotational vibrations of the pulleys in the analysis of free and forced response. These studies, however, do not consider the effect of the tensioner on either the equilibrium state (steady-state tensions) or the dynamic response. The influence of the tensioner on longi­tudinal belt deflection is examined by Barker et al. [14] who focus on transient response due to drive pulley acceleration. Beikmann et al. [15] develop a prototypical model (two pulleys with a tensioner) to calculate the steady-state tensions and the tensioner position as functions of steady operating conditions. Ulsoy et al. [16] consider the coupling between the transverse

Rotational Response and Slip Prediction of Serpentine Belt Drive

A nonlinear model is developed which describes the rotational response of automotive serpentine belt drive systems. Serpentine drives utilize a single {long) belt to drive all engine accessories from the crankshaft. An equilibrium analysis leads to a closedform procedure for determining steady-state tensions in each belt span. The equations of motion are linearized about the equilibrium state and rotational mode vibration characteristics are determined from the eigenvalue problem governing free response. Numerical solutions of the nonlinear equations of motion indicate that, under certain engine operating conditions, the dynamic tension fluctuations may be sufficient to cause the belt to slip on particular accessory pulleys. Experimental measurements of dynamic response are in good agreement with theoretical results and confirm theoretical predictions of system vibration, tension fluctuations, and slip.

R. J. Meckstroth

Papers

Rotational Response and Slip Prediction of Serpentine Belt Drive

Rotational response and slip prediction of serpentine belt drive systems