An analysis of the modified Dahl and Masing models: Application to a belt tensioner

doi:10.1016/j.jsv.2006.12.013

Journal of Sound and Vibration

Volume 302, Issues 4–5, 22 May 2007, Pages 841-864

https://doi.org/10.1016/j.jsv.2006.12.013 Get rights and content

Abstract

The objective of this paper is to describe the modified Dahl and Masing models used for predicting hysteretic behavior, and tested on a belt tensioner for automotive engines. An experimental study with deflection imposed on the tensioner is first carried out to identify hysteresis loop parameters for the two models. The models are implemented in the general motion equations which govern the behavior of a belt–tensioner–mass system. Particular attention is paid to the use of numerical schemes. The numerical and experimental investigations show the reliability of the modified Dahl model.

Introduction

The hysteresis behavior of components permits efficient passive control of mechanical systems but makes response prediction delicate due to their high nonlinearity [1], [2]: basically it is necessary to choose an efficient hysteresis model associated with a numerical integration method to ensure convergence with the scheme. Therefore, there is a much of scientific and technological research devoted to the investigation of such problems.

Vestroni and Noori in Ref. [2] and Visintin in Ref. [3] established an overview of hysteresis models. Among the latter mention can be made of those of Bouc and Wen, Bader and Noori, and Masing. Rheological models and restoring force models are the two main categories widely used in mechanical engineering. The former provide damping and stiffness parameters, while the latter provide a restoring force to be introduced in the second member of the equations.

Let the Masing model (MM) [4], [5] and the modified Dahl model (MDM) proposed by Al Majid [6], [7] be the rheological and restoring force models, respectively, selected for the current analysis. The classical MM is composed of a spring parallel to a spring–dry friction system, but in this study a viscous damping element is added. The MDM originates from the Dahl and Duhem models and is based on a first differential equation that provides the time derivative of the restoring force from the velocity of the deflection and from the envelop curves of the hysteresis loop. The MM is governed by a non-smooth differential equation containing a multi-valued function while the Dahl model is governed by a smooth nonlinear dynamic equation. Consequently, the numerical integration schemes have to take into account these two typical characteristics to obtain a convergence. The reliability of these two hysteresis models have to be tested to predict the hysteretic behavior of a belt tensioner.

Tensioners used in belt drive systems play a predominant role in the dynamic behavior of the belt: they maintain nominal tension in the slack span and reduce transverse vibration levels, see Ref. [8]. Tensioners often require complicated designs in order to satisfy technological challenges, see for example Ref. [9]. This type of design leads to considerably nonlinear behavior mainly due to stick–slip motion [10].

The MDM and MM are described in detail in Section 2 and then applied to a belt tensioner of an automotive engine in Section 3, where an initial experimental setup is used for identifying the model parameters. Section 4 concerns the numerical and experimental investigations performed on a belt–tensioner–mass system in which tensioner behavior is described by the two models studied. This section permits comparing the predicted and measured harmonic responses in order to discuss model reliability.

Section snippets

The models

In this section, two models describing the hysteretic behavior of a one degree of freedom mechanical system are presented. The behavior of the mechanical system studied can be analyzed via the progression of the restoring force versus the deflection, as presented in Fig. 1.

The objective is to find the relation between a restoring force $F$ and a deflection u. It is assumed that after a transient phase $[0, t_{0}]$ , the pair $(u (t), F (t))$ belongs to a periodic curve, called hysteresis loop (see Fig. 1 and

Experimental investigation and parameter identification

The tensioner is composed of three parts, see Fig. 7 (a): Part 1 is a solid (Idler pulley) that rotates around axis $Δ = (AB)$ of part 2; part 2 is the tensioner arm ABC, that rotates around the fixed axis $Δ^{'}$ of part 3, bolted to the reference part 4 (i.e. an engine for automotive applications). All the parts are considered as rigid bodies. The pin joint of axis $Δ^{'}$ between parts 2 and 3 includes a torsion spring and friction components that cause dry and lubricated contact forces, and a moment

Comparison, validation and prediction

In the previous section, the MDM and MM were formulated for the belt tensioner. The tensioner is now a part of a mechanical system subjected to a variable load excitation. The purpose is to test the models efficiency considering a multi-degree of freedom system and an experimental investigation. Each tensioner model is implemented in the system motion equations that are solved numerically. The predicted and measured results are compared.

Conclusion

Two hysteretic models have been analyzed and tested on a mechanical system containing a belt tensioner exhibiting a stick–slip behavior. The first model, which is the Masing model (MM) with viscous damping, provides, by the way of differential inclusion, the rheological parameters of the equation of motion. The second model, which is the modified Dahl model (MDM), recently proposed, provides a restoring force which takes place in the second member of the equation of motion. A primary

References (21)

W. Lacarbonara et al.
Nonclassical responses of oscillators with hysteresis
Nonlinear Dynamics
(2003)
F. Vestroni et al.
Hysteresis in mechanical systems—modeling and dynamic response
International Journal of Non-Linear Mechanics
(2002)
A. Visintin
Differential Models of Hysteresis
(1994)
J. Bastien et al.
Study of some rheological models with a finite number of degrees of freedom
European Journal of Mechanics A—Solids
(2000)
R. Fougeres et al.
The evolutive masing model and its application to cyclic plasticity and ageing
Nuclear Engineering and Design
(1989)
A. Al Majid et al.
Formulation of a hysteretic restoring force model: application to vibration isolation
Nonlinear Dynamics
(2002)
A. Al Majid et al.
Harmonic response of a structure mounted on an isolator modelled with a hysteretic operator: experiment and prediction
Journal of Sound and Vibration
(2004)
R. Beikmann et al.
Design and analysis of automotive belt drive systems for steady state performance
ASME Journal of Mechanical Design
(1997)
R. Parker
Efficient eigensolution, dynamic response, and eigensensitivity of serpentine belt drives
Journal of Sound and Vibration
(2004)
M. Leamy et al.
Nonlinear periodic response of engine accessory drives with dry friction tensioners
ASME Journal of Vibration and Acoustics
(1998)

There are more references available in the full text version of this article.

Cited by (46)

A computationally efficient method for dynamic response analyses of accessory drive systems using harmonic balance method together with alternating frequency/time domain technique
2024, Mechanical Systems and Signal Processing
Serpentine belt accessory drive system (SBADS) is a typical coupled vibration system with rigid body vibrations and continuum vibrations. Vibration behaviors of a SBADS consist of rotational vibrations of pulleys and tensioner arm and transverse vibrations of belt spans. In modeling of a SBADS, transverse vibration of a belt span consisting of time and space coordinates is discretized into multiple degrees of freedom with only time variable. As a result, number of degrees of freedom of a SBADS is large, and computational efficiency of dynamic responses of a SBADS is low. To solve this issue, a calculation method for nonlinear dynamic characteristics of a SBADS is proposed based on the harmonic balance method with alternating frequency/time domain technique in the paper. An engine front end accessory drive system is taken as a study example, the coupling vibration model of the SBADS is established, and hysteretic behavior of the tensioner is considered in the model. In addition, a dimensionality reduction method for the dynamic matrix is proposed based on local nonlinearity of the model. A calculation method for nonlinear dynamic responses of the SBADS is subsequently developed by combining the harmonic balance method and alternating frequency/time domain technique. Nonlinear dynamic responses estimated by the proposed method are validated by comparing with numerical simulations. Finally, amplitude-frequency responses of the SBADS are calculated using arc-length continuation method, and influences of nonlinear tensioner on frequency domain responses and internal resonance phenomena of the SBADS are analyzed.
Modeling of serpentine belt accessory drive system coupled vibrations and utilizing nonlinear tensioner to enhance performances
2022, Mechanical Systems and Signal Processing
Serpentine belt accessory drive system (SBADS) is a complex hybrid discrete-continuous system and is a typical coupled vibration system with rigid body vibrations and belt vibrations, which consists of rotational vibrations of pulleys and tensioner arm and transverse vibrations of continuum belt spans. Taking an engine front end accessory drive system as an example, the coupling vibration model of belt, pulleys and tensioner arm is established. In the model, hysteretic behavior of tensioner is considered, and a calculation method for estimating the output torque versus the imposed angle of tensioner arm is proposed. In solving the coupled motion equations of the SBADS, transverse displacement of a belt span is discretized as the product of time function and space function by using Galerkin method, the steady-state deflections of belt spans are calculated by singular perturbation method, and the nonlinear dynamic responses of a SBADS are solved by numerical method. The nonlinear dynamic responses of the SBADS are calculated by the established method and validated by the experiment results. Benefits of nonlinear hysteretic behavior of a tensioner on the dynamic responses of a SBADS are studied, and the method for how to utilize nonlinear characteristics of tensioner to enhance the performances of a SBADS is proposed in the end of this paper.
Stabilisation effect of strain hysteresis loop for steel 45
2021, International Journal of Fatigue
Citation Excerpt :
Revealing the theoretical and experimental subtleties of the stabilisation regime for various materials improves the quality of the assessment of the strength and resource of a structure subjected by cyclic loading [2]. In the theory of adaptability, the investigated stabilisation mode is not considered temporary and is described by the deformation characteristics of the process, which are considered independent of the loading parameters [3,4]. Nevertheless, the duration of the stabilisation of plastic deformation depends on the strain amplitude and strain rate characteristics of low-cycle loading [5].
The adaptability of steel 45 to low-cycle deformation from 100 to 1000 cycles at different constant strain amplitudes, characterised by the effect of plastic-strain stabilisation, is considered. In this study, the proposed structural–temporal model of cyclic deformation simultaneously predicted the stress-strain dependence and the effect of plastic-strain stabilisation based on hysteresis width. Experiments show that range of cycles for stable plastic deformation decreases with an increase in the strain amplitude and strain rate. The proposed model takes into account deformation history experimental data of low-cycle deformation of steel 45 and the presence of an established stabilisation effect.
Dynamic modeling, simulation and experiment of power transmission belt drives: A systematic review
2021, Journal of Sound and Vibration
Citation Excerpt :
This experiment technique can be used to forecast dangerous parametric resonances and the critical operating conditions. Most of the measurements on frictional contact of belt drives are focused on belt tension variation [203], belt-pulley friction measurement [117,214,216,261,262], wear in the belt-pulley interface [263] and hysteretic damping measurement of tensioner resulted from its dry friction [7,122,177,255,256]. In 2013, Della Pietra and Timpone [203] designed an experiment to examinethe Grashof's model [190] and Firbank's model [197] in capturing belt-pulley contact mechanics.
In this paper, studies on dynamic modeling, simulation and experiment of power transmission belt drives are comprehensively reviewed. In the past few decades, many investigations are conducted on dynamic modeling, simulation and experiment of different kinds of power transmission belt drive systems. In the dynamic modeling and simulation of the belt drive systems, surveys are focused on vibrations of a single axially moving belt span, rotational vibrations of pulley components, coupled belt-pulley vibrations and contact mechanics between the belt and pulley as well as some experimental investigations. Influences of tensioner dry friction and one-way clutch on dynamics of the belt drive systems including system rotational vibrations and coupled belt-pulley vibrations are separately reported. The investigations are also surveyed on modeling and predicting complicated belt-pulley contact behaviors like belt creeps and slips on pulleys, contact force distributions of the belt and pulley, and variation of wrap angles of a belt around pulleys, etc., which are categorized by different approaches including the creep theory, shear theory, multi-body dynamics and finite element methods. Fatigue life estimation and failure analysis of power transmission belt drives are discussed in detail as well. In addition, experimental techniques are reviewed on parameters identifications, and measurements of static and dynamic performances including energy/power loss, system vibration, dynamic belt tension, belt deformation, stress and strain distribution in the belt and pulley, and contact friction force, etc. Finally, conclusion of this work is summarized and topics of future potential studies on the power transmission belt drive systems are suggested.
Modeling and validation of dynamic performances of timing belt driving systems
2020, Mechanical Systems and Signal Processing
Citation Excerpt :
The model for tensioner torque was validated and the parameters of the model are identified by the measurements. Bastien et al. [30] and Michon et al. [31] proposed a modified Dahl and Masing model to describe the hysteretic behavior of a tensioner. The tensioner was modeled as a torsional spring stiffness, an equivalent friction stiffness and an equivalent viscous damping.
A modeling method for a generic layout of timing belt driving system (TBDS) is presented, and the formulas for modeling a meshed teeth belt, an automatic tensioner, and rotational pulleys are established. A numerical method consisting of an iterative algorithm for calculating the dynamic responses of system is proposed. One engine TBDS with a crankshaft, a camshaft and a tensioner is taken as a studying example. The method and the procedure for obtaining the dynamic responses are described and studied. The dynamic performance of the TBDS, such as oscillation angles of pulleys and tensioner arm, and the transmission error between pulleys, are calculated and compared with the measurement, which validates the presented method. The influence of tensioner performance (stiffness and damping), meshed teeth model and different belt pre-tensions on the dynamic responses of a TBDS system are investigated. The tensioner behavior and belt tension are also analyzed based on the calculated dynamic responses. The developed method presented in this paper can be used for predicting the dynamic responses, optimizing the parameters of an engine TBDS, and reducing the design period and the cost for prototype validation.
Modeling and compensation of asymmetric hysteresis for pneumatic artificial muscles with a modified generalized Prandtl–Ishlinskii model
2018, Mechatronics
This paper presents a modified generalized Prandtl–Ishlinskii (MGPI) model for the asymmetric hysteresis characterization of pneumatic artificial muscles (PAMs), which can be considered as a cascade of the superposition of weighted generalized play operators and the superposition of weighted dead-zone operators. Compared with the established hysteresis models, the significance of the MGPI model is that it has a simple expression including a small number of parameters to be identified. Besides, the analytical form of its inverse model is easy to be obtained, which can be applied to the compensation of asymmetric hysteresis of the PAM in real-time. The wiping-out and congruency properties of the proposed model are verified by simulations. Meanwhile, by carrying out the experimental study on length–pressure hysteresis of a PAM, the parameters in the MGPI model are identified from measured data using the Levenberg–Marquardt method. Then, a feedforward hysteresis compensator based on the inverse MGPI model is designed for trajectory tracking control of the PAM. The simulation and experimental results indicate that the proposed model and its inversion are effective to characterize and compensate the length–pressure hysteresis of the PAM.

View all citing articles on Scopus

View full text

An analysis of the modified Dahl and Masing models: Application to a belt tensioner

Abstract

Introduction

Section snippets

The models

Experimental investigation and parameter identification

Comparison, validation and prediction

Conclusion

Nonclassical responses of oscillators with hysteresis

Nonlinear Dynamics

Hysteresis in mechanical systems—modeling and dynamic response

International Journal of Non-Linear Mechanics

Differential Models of Hysteresis

Study of some rheological models with a finite number of degrees of freedom

European Journal of Mechanics A—Solids

The evolutive masing model and its application to cyclic plasticity and ageing

Nuclear Engineering and Design

Formulation of a hysteretic restoring force model: application to vibration isolation

Nonlinear Dynamics

Harmonic response of a structure mounted on an isolator modelled with a hysteretic operator: experiment and prediction

Journal of Sound and Vibration

Design and analysis of automotive belt drive systems for steady state performance

ASME Journal of Mechanical Design

Efficient eigensolution, dynamic response, and eigensensitivity of serpentine belt drives

Journal of Sound and Vibration

Nonlinear periodic response of engine accessory drives with dry friction tensioners

ASME Journal of Vibration and Acoustics