Hysteretic behavior of a belt tensioner: modeling and experimental investigation

In this paper we describe the modeling of the hysteretic behavior of belt tensioners. An initial experimental device is composed only of the tensioner by using forcing frequencies, preloads and deflection amplitudes. It permits the identification of the parameters of the restoring force model used. Comparison of the measured and predicted force deflection loops of the tensioner subjected to large deflections permits preliminary validation of the model.The second experimental device consists of a belt-tensioner system. Its non-linear modeling includes the above hysteretic model and the belt’s longitudinal characteristics. Validation of the belt-tensioner model is completed by comparing the measured and predicted belt tension. Finally, it is shown by using a parametric investigation and phase-plane portrait that the response of the belt-tensioner system increases with the frequency and the amplitude of the excitation

Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending

A mathematical model incorporating the higher order deformations in bending is devel- oped and analyzed to investigate the nonlinear dynamics of rotors. The rotor system con- sidered for the present work consists of a flexible shaft and a rigid disk. The shaft is modeled as a beam with a circular cross section and the Euler Bernoulli beam theory is applied with added effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. The kinetic and strain (defor- mation) energies of the rotor system are derived and the Rayleigh–Ritz method is used to discretize these energy expressions. Hamilton’s principle is then applied to obtain the mathematical model consisting of second order coupled nonlinear differential equations of motion. In order to solve these equations and hence obtain the nonlinear dynamic response of the rotor system, the method of multiple scales is applied. Furthermore, this response is examined for different possible resonant conditions and resonant curves are plotted and discussed. It is concluded that nonlinearity due to higher order deformations significantly affects the dynamic behavior of the rotor system leading to resonant hard spring type curves. It is also observed that variations in the values of different parameters like mass unbalance and shaft diameter greatly influence dynamic response. These influences are also presented graphically and discussed

Couplings in parametrically excited inclined cables systems

Cables in stayed bridges are subjected to important dynamic solicitations for which dynamic model are now well established. Due to their design, such structures highlight resonance phenomena and instabilities frequently observed. Nevertheless, some structures exhibit important vibration amplitudes that can not be explained simply. Measurement recently performed on a bridge point a coupling of the cable with the deck or the pillar. The present paper suggests to consider the deck flexibility coupled to the nonlinear dynamic of the inclined cable. Results of previous study are used. The retained nonlinear model of the cable include two degrees of freedom for the in-plane motion. Considering the bridge mass and deck rigidity adds one DOF, assumed linear in a first approach. The excitation is created on the deck, which produce an external force(such as the wind or the car traffic for example). An experimental set-up uses a specific device in order to highlight expected coupling phenomena on the parametric instabilities. It is composed of a flexible blade which represents the deck, and an inclined cable. Both elements are linked to a mass forced to move vertically, and which represent the anchor point and the equivalent mass of a section of the deck. Therefore, the cable has a given initial static tension. An electrodynamic shaker applies a force close to blade clumping. The transmitted force from the shaker to the structure is measured thanks to a piezo-electric sensor. The instantaneous cable tension is measured via a Shape force sensor. And a high resolution laser sensor captures without contact the in-plane motion of the cable. Analytically, the multiple scales method is applied to solve the nonlinear equations of motion. In-plane vibration of the cable and stability in the vicinity of the primary resonance w1 and sub-harmonic resonance 2w1 are computed. The competition between the behaviour at 2w1 and w2 are of particular interest, as it is observed experimentally

Rheological and restoring force models regarding belt tensioner dymamic behavior: prediction and experiment

The objective of this paper is to compare the Masing and modified Dahl model efficiency regarding the prediction of the hysteretic behavior of a belt tensioner used for automotive engines. A first experimental study with deflection imposed on the tensioner is carried out to identify hysteresis loop parameters for the two models. The models are then implemented in the general motion equations modeling the behavior of a belt - tensioner - mass system. The comparison beteen numerical and experimental results show that these two models perform satisfactorily and that the modified Dahl model is a little more efficient

Harmonic response of the organ of corti: results for wave dispersion

Inner ear is a remarkable multiphysical system and its modelling is a great challenge. The approach used in this paper aims to reproduce physic with a realistic description of the radial cross section of the cochlea. A 2D‐section of the organ of Corti is fully described. Wavenumbers and corresponding modes of propagation are calculated taking into account passive structural responses. The study is extended to six cross sections of the organ of Corti and a large frequency bandwidth from 100 Hz to 3 kHz. Dispersion curves reveal the influence of fluid structure interactions with a dispersive behavior at high frequencies. Longitudinal mechanical coupling provides new interacting modes of propagation

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

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

Experimental investigation on the dynamic characteristics and transverse vibration instabilities of transmission belts

Serpentine belt drives are often used in front end accessory drive of automotive engine. Dynamic characteristics of belts play an important role in the behavior of such transmission and are inputs of simulation software. Moreover, free belt spans exhibit transverse vibration non linear instabilities. An experimental investigation is conducted on multi-ribbed belts, first for the determination of longitudinal stiffness and damping, bending rigidity, then to highlight the belt span transverse instabilities. An experimental set-up has been designed, it enables ,the observation and analysis of instabilities of belt spans axially excited, the determination of instability chart

From transmission error measurement to Pulley-Belt slip determination in serpentine belt drives: influence of tensioner and belt characteristics

Serpentine belt drives are often used in front end accessory drive of automotive engine. The accessories resistant torques are getting higher within new technological innovations as stater-alternator, and belt transmissions are always asked for higher capacity. Two kind of tensioners are used to maintain minimum tension that insure power transmission and minimize slip: dry friction or hydraulic tensioners. An experimental device and a specific transmission error measurement method have been used to evaluate the performances of a generic transmission by determining the pulley-belt slip for these two kinds of tensioner. A data acquisition technique using optical encoders and based on the angular sampling method is used with success for the first time on a non synchronous belt transmission. Transmission error between pulleys, pulley/belt slip are deduced from pulley rotation angle measurements.Results obtained show that: the use of tensioner limits belt slip on pulleys, pulley-belt slip is reachable from transmission error measurement, belt non uniform characteristics are responsible of low frequency modulations of transmission error

Instability zones for isotropic and anisotropic multibladed rotor configurations.

Helicopter ground resonance is an unstable dynamic phenomenon which can lead to the total destruction of the aircraft during take-off or landing phases. The earliest research in this domain was carried out by Coleman and Feingold during the decade of 60s. The instability was predicted by using classical procedures once the rotor was considered as isotropic, consequently, the periodic equations of motion could be simplified to a system with constant coefficients by introducing a change of variables, known as the Coleman Variable Transformation. The goal of the present work is to further comprehend the phenomenon and the influence of the anisotropic properties of rotors by analyzing the periodic set of equations of motion. For this, Floquet's Theory (Floquet's Method — FM) is used. The analysis for predicting the ground resonance phenomenon in isotropic and anisotropic rotor configurations is explored. The conclusions lead to verify the appearance of bifurcation points depending on the anisotropic characteristic present in the rotor. The temporal response analysis in the motion of helicopter with one asymmetric blade at unstable regions highlighted the presence of non symmetric rotor deformation shapes

Operational modal analysis with non stationnary inputs

Operational modal analysis (OMA) techniques enable the use of in-situ and uncontrolled vibrations to be used to lead modal analysis of structures. In reality operational vibrations are a combination of numerous excitations sources that are much more complex than a random white noise or a harmonic. Numerous OMA techniques exist like SSI, NExT, FDD and BSS. All these methods are based on the fundamental hypothesis that the input or force applied to the structure to be analyzed is a stationary white noise. For some applications this hypothesis is reasonable. However in numerous situations, the analyzed structure is subject to harmonic and transient forces. Numerous methods and research has enabled to develop methods that are robust to such harmonic contributions. To enable OMA during pressure oscillations in solid rocket boosters, the authors propose to consider transient and harmonic inputs no longer as parasites but as the main force applied to the structure that must be analyzed. This is the case during pressure oscillations in rocket boosters. We propose the use of phase analysis adapted to a transient context to conduct operational modal analysis under a harmonic transient input. This time-based novel OMA method will be exposed. The theoretical developments and algorithmic implementations are exposed. First tests have been conducted on laboratory single degree of freedom setup to validate this new OMA technique and are reported here