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The sources of gear rattle in automotive transmission and other components are clearance non-linearities, which include backlashes, multi-valued springs, hysteresis, etc. Periodic vibro-impacts are generated because of the single sided or double sided impacts. It is obviously desirable to develop an appropriate computer simulation model which can aid in the control of transient noise and vibration signatures. Such models can also be used to design experiments and to interpret measured data. Several approaches have been attempted in the past, but most common is the digital simulation of the governing differential equations describing the non-linear, torsional dynamics of the drivetrain. Some progress has been made in understanding the basic rattle phenomenon and in developing suitable mathematical models. This article intends to be a status report on the mathematical or computer models including pre- and post-processing considerations. Various topics that will be briefly discussed include dimensionality and reduced order models, numerical difficulties encountered by solution algorithms, design analysis and optimization, etc. Two specific case histories are analyzed in this paper. The first deals with a reverse-idler gear pair rattle problem, and the second case examines flywheel-clutch-hub spline subassembly.

Department of Mechanical Engineering

World University root node

SAE Technical Papers

Modeling of automotive gear rattle phenomenon: State of the art

The dynamic behavior and frequency response of harmonically excited piecewise linear and/or non-linear systems has been the subject of several recent investigations. Most of the prior studies employed harmonic balance or Galerkin schemes, piecewise linear techniques, analog simulation and/or direct numerical integration (digital simulation). Such techniques are somewhat limited in their ability to predict all of the dynamic characteristics, including bifurcations leading to the occurrence of unstable, subharmonic, quasi-periodic and/or chaotic solutions. To overcome this problem, a parametric continuation scheme, based on the shooting method, is applied specifically to a periodically excited piecewise linear/non-linear system, in order to improve understanding as well as to obtain the complete dynamic response. Parameter regions exhibiting bifurcations to harmonic, subharmonic or quasi-periodic solutions are obtained quite efficiently and systematically. Unlike other techniques, the proposed scheme can follow period-doubling bifurcations, and with some modifications obtain stable quasi-periodic solutions and their bifurcations. This knowledge is essential in establishing conditions for the occurrence of chaotic oscillations in any non-linear system. The method is first validated through the Duffing oscillator example, the solutions to which are also obtained by conventional one-term harmonic balance and perturbation methods. The second example deals with a clearance non-linearity problem for both harmonic and periodic excitations. Predictions from the proposed scheme match well with available analog simulation data as well as with multi-term harmonic balance results. Potential savings in computational time over direct numerical integration is demonstrated for some of the example cases. Also, this work has filled in some of the solution regimes for an impact pair, which were missed previously in the literature. Finally, one main limitation associated with the proposed procedure is discussed. © 1995 by Academic Press, Inc.

Journal of Sound and Vibration

Analysis of periodically excited non-linear systems by a parametric continuation technique

Analysis of brushless D.C. motor (BDCM) torque pulsations is an essential step in the diagnosis and control of vibration and noise generated by many electro-mechanical devices. The broad band spectral content of the torque pulsations, as predicted by a mathematical model which accounts for various complex effects, can often be obtained only by numerical integration which is time consuming while permitting little understanding of the dynamic interactions. Prior analytical approaches, such as the Fourier series technique or the d-q axis theory, are limited by the simplifying assumptions needed to compute the torque spectrum. This paper develops a new semi-analytical formulation for the analysis of nonlinear, time-varying BDCM’s which involve both spatial and temporal domains. A modified multi-term harmonic balance method, based on a transformation of the dual-domain problem to a spatial domain formulation, is developed here specifically to compute the magnitude of several harmonics of the pulsating torque. The interacting effects of key parameters, like dynamic eccentricity, magnetic saturation and open stator slots, on the time-varying inductances and rotor flux density distribution are included explicitly in the formulation. The predicted spectra compare very well with those obtained by direct time domain numerical integration. Yet, the proposed method is computationally efficient especially when the model dimension is reduced. It also provides better insight into the high frequency dynamics of the sample case.

J. Dyn. Sys., Meas., Control.

Dynamic Analysis of a Brushless D.C. Motor Using a Modified Harmonic Balance Method

This paper proposes a new procedure for formulating the gear rattle type problem analytically before attempting a numerical solution. It also outlines appropriate evaluation criteria for direct time domain integration algorithms used to solve such problems. The procedure is necessary due to the non-analytical nature of the mathematical formulation describing vibro-impacts, which can lead to numerical “stiffness” problems. The method is essentially an “intelligent” pre-processing stage and is based on our experience in simulating such systems. Important concepts such as model order reduction, gear or clutch stiffness contact ratio, appropriate choice of non-dimensionalization parameters are illustrated through examples. Several case studies of increasing complexity are solved using various well known numerical algorithms; solutions are compared qualitatively and quantitatively using the proposed evaluation criteria, and specific numerical problems are identified. Some of the simulation models have also been validated by comparing predictions with experimental data. © 1995 by ASME.

Journal of Mechanical Design, Transactions of the ASME

Computational issues associated with gear rattle analysis

This paper extends the multi-body dynamics modeling strategy for a gear pair [6] to multi-mesh transmissions with external, fixed center, helical or spur gears. Each gear is modeled as a rigid body with six degrees of freedom. A multi-dimensional, position-dependent formulation is used to describe the gear mesh stiffness which is assumed to be distributed along the line of action. A simplified model of the shaft-bearing subsystems is included since the focus of this study is on the gear dynamics. Excitation to the system is considered in the form of either external torque pulsation or internal static transmission error. The governing equations are linearized to yield a formulation with position or time-varying coefficients (LTV). Subsequently, three examples of linearized time-invariant (LTI) transmission systems are solved, and eigensolution predictions of the multi-body dynamics model compare very well with finite element calculations. Then the periodic response of a non-unity gear pair system is studied in depth. New results including a comparison between LTI and LTV models are presented. It has been demonstrated that both time and frequency domain solutions can be efficiently and accurately constructed by using the multi-term harmonic balance method, provided that several shaft and gear mesh harmonics are included. © 1995 Academic Press Ltd.

Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears

The dynamic behavior of piecewise linear and/or non-linear vibratory systems subject to a single harmonic excitation has been recently analyzed by using a one- or multi-term harmonic balance (or Galerkin’s) method, piecewise linear techniques, analog simulation and direct numerical integration (digital simulation). Instead, this paper proposes to utilize the technique of parametriccontinuationto study the steady state response and global dynamics of a two-degree-of-freedom piecewise non-linear system with backlash or multi-valued springs and impact damping. The physical system is under the influence of a mean load and is subject to a dual harmonic external excitation, where the second frequency is either twice or three times the fundamental excitation frequency. The effects of various parameters such as impact damping, mean and alternating loads, the piecewise stiffness ratios and the relative phase between the excitation harmonics on the system response are thoroughly examined. System or excitation parameter regimes that exhibit sub-harmonic, super-harmonic quasi-periodic and chaotic solutions are obtained efficiently and systematically by using the proposed scheme. This investigation also clarifies the differences between viscous and non-linear impact damping models, which have been proposed earlier in the literature for studying clearance non-linearities. Limited experimental data validate our modelling strategy. Finally, selected predictions from this technique match very well with a multi-term harmonic balance method and with direct numerical integration, wherever applicable.

Dynamics of a piecewise non-linear system subject to dual harmonic excitation using parametric continuation

Although non-linear systems subject to combined parametric and external excitations have been examined in some depth, very few of these investigations have addressed the influence of a mean (time-invariant) load on the system response. In particular, the importance of the mean load effect has been highlighted in the study of whirling asymmetric shafts, where significant changes in system response and stability occur under the influence of gravity. The present paper intends to characterize the specific effect of mean load on the dynamic behavior of a single degree-of-freedom Hill's oscillator with a clearance type non-linearity while also being subjected to a periodic base displacement excitation. The parametric continuation technique and method of harmonic balance are used for this purpose. Issues discussed include the coupling between the mean load and the dynamic response amplitude, interaction between the parametric excitation effect and the clearance nonlinearity and comparison between time-invariant and time-varying systems.

American Society of Mechanical Engineers, Design Engineering Division (Publication) DE

Influence of mean load on the response of a forced non-linear Hill's oscillator

The main focus of this study is to establish evaluation criteria for direct time domain integration algorithms used to solve gear rattle type problems. Such criteria may be used to identify specific numerical problems encountered. The ultimate goal obviously is to find reasonably accurate and reliable methods of solution for such physical systems. Six case studies of increasing complexity, linear to highly non-linear, are solved using well known algorithms. The solutions to the linear model are verified by using analytical results. Non-linear model solutions as yielded by different algorithms are compared qualitatively and quantitatively. Several non-linear simulation models have been validated by comparing predictions with experimental data and results available in the literature.

Computational issues associated with gear rattle analysis: part II - evaluation criteria for numerical algorithms

In an earlier study [14], the frequency response characteristics of a multi-degree-of-freedom system with clearance non-linearities were presented. The current study is an extension of this prior work and deals specifically with the issue of dynamic interactions between resonances. The harmonic balance method, digital solutions and analog computer simulation are used to investigate a two-degree-of-freedom system under a mean load, when subjected to sinusoidal excitations. The existence of harmonic, periodic and chaotic solutions is demonstrated using digital simulation. The method of harmonic balance is employed to construct approximate solutions at the excitation frequency which are then used to classify weak, moderate and strong non-linear spectral interactions. The effects of parameters such as damping ratio, mean load, alternating load and frequency spacing between the resonances have been quantified. The applicability of the methodology is demonstrated through the following practical examples: (i) neutral gear rattle in an automotive transmission system; and (ii) steady state characteristics of a spur gear pair with backlash. In the second case, measured dynamic transmission error data at the gear mesh frequency are used to investigate spectral interactions. Limitations associated with solution methods and interaction classification schemes are also discussed. © 1992.

Spectral coupling issues in a two-degree-of-freedom system with clearance non-linearities

This paper proposes a new procedure for formulating the gear rattle type problem analytically before attempting a numerical solution. This step is necessary due to the nature of the mathematical formulation with vibro-impacts, which is non-analytical and hence causes numerical ``stiffness''. The procedure is essentially an ``intelligent'' pre-processing stage and is based on our vast experience in simulating such systems. Important concepts such as order reduction, gear contact ratio, appropriate choice of non-dimensionalization parameters are illustrated through several examples.

Computational issues associated with gear rattle analysis: Part I - problem formulation

The dynamics of an Euler–Bernoulli beam, with multiple spring/mass attachments, is investigated in this paper. A novel analytical approach for determining the natural frequencies and mode shapes of the system is presented. The formulation is applicable for any parameter value of the attached mass or spring stiffness. The calculation is performed using a minimal binary set of admissible functions for each spring/mass connection, with extremal values (zero or infinity) of the parameter. For example, to study a beam with n intermediate spring/mass connections, only 2n admissible functions are required. These functions are used in a Rayleigh–Ritz-based energy formulation. The small number of functions used in the formulation leads to an efficient computational procedure. The results from the proposed formulation are compared with those from a finite element simulation, for different boundary conditions. The results obtained by the two methods are in excellent agreement for all boundary conditions as well as for different parameter ranges. Detailed design studies, on the effect of spring stiffness and lumped mass values, as well as their locations, on the natural frequencies and mode shapes, have been carried out. Design charts have also been generated to aid the designer of such structures.