Fractional Derivative Zener Model Research Articles

Investigating nonlinear dynamic properties of a swing arm rubber joint and their effects on the dynamic behavior of high-speed trains

The rubber joints mounted on swing arms provide the main part of the yaw stiffness for wheelsets and thus have a significant influence on the dynamic behaviour of railway vehicles. However, their nonlinear dynamic properties concerning the frequency, amplitude, and temperature were not fully considered in previous studies, most of which adopt simple models such as the Kelvin–Voigt model to represent rubber components. This study aims to investigate the radial nonlinear properties of swing arm rubber joints under various conditions and assess their effects on the dynamic performance of high-speed trains. A nonlinear rubber spring model which combines the fractional derivative Zener model with Berg’s friction model is established. To achieve a better fit to the measurements of a swing arm rubber joint, an optimisation-based method is employed to identify the model parameters. The viscous and frictional effects of the rubber joint are compared at different frequencies, amplitudes, and temperatures, by determining the predominant element of the nonlinear model. Additionally, the nonlinear rubber spring model is integrated into a high-speed vehicle dynamic model to investigate the effects of the nonlinear properties of swing arm rubber joints on the vehicle dynamic behaviour through the MATLAB/SIMPACK co-simulation method.

A high order Newton method to solve vibration problem of composite structures considering fractional derivative Zener model

This paper presents a numerical method to solve the non-linear vibration problem of composite structures made of many unidirectional fibers/matrix layers. To take into account damping properties of structures, fractional derivative Zener model for complex moduli is used in each layer. The frequency dependence of these models leads to a non-linear vibration problem. To solve it, a High Order Newton (HON) solver based on homotopy and perturbation techniques is proposed. Thus, the initial non-linear problem turns into a set of linear systems, easier to solve. Numerical results obtained by the HON solver are compared with experimental and numerical results of literature.

Effect of viscoelastic-plastic dynamic properties of rail pads on curved rail dynamic characteristics based on the modified SEM–SM hybrid method

To accurately and efficiently investigate the influence of the viscoelastic-plastic dynamic properties of rail pads on curved rail vibration characteristics, a spatial model of an infinitely curved track was developed by combining the spectral element method (SEM) and symplectic method (SM). Specifically, a fractional derivative Zener model combined with a Berg friction model was used to describe the dynamic properties of the rail pad. Subsequently, the curved rail mobility and decay rate (DR) were analyzed by accounting for the frequency, preload, and amplitude dependence of the rail pad. In conclusion, the frequency dependence mainly influenced the low- and medium-frequency vibrations of the curved rail and made the vibration within this range to shift to higher frequencies. In terms of the curved rail DRs of different degrees of freedom, because of their coupling effects, the wave modes were converted mutually at the cross frequency of their DR curves, and the torsional (axial) wave had the maximum (minimum) DR. The preload dependence had a significant effect on the vertical and torsional vibrations of the curved rail. As the preload increased, the cut-on frequencies of the first vertical and torsional waves increased; consequently, the curved rail vibrations were transmitted more in the downward.

Investigation on the adhesive contact and electrical performance for triboelectric nanogenerator considering polymer viscoelasticity

The triboelectric nanogenerator (TENG) is a new mechanical energy harvesting technology in which the typical viscoelastic material polydimethylsiloxane (PDMS) is widely used. Micro-/nano-textures are often fabricated on the PDMS surface to enhance the electrical performance of TENG. As the contact region decreases to micro/nano scale, the adhesive forces become dominant. However, there is still a lack of contact mechanics model considering both material viscoelasticity and the adhesive forces to guide the surface texture design. In this paper, the explicit data-fitting formulas based on the fractional derivative Zener model are firstly derived to identify the viscoelastic constitutive parameters, which can not only avoid the influence of the initial contact point, but also ensure the accurate conversion between the creep compliance and the relaxation modulus function. Then a viscoelastic-adhesive contact model based on the fitted constitutive parameters is established, and the numerical algorithms such as bi-conjugate stabilized (Bi-CGSTAB) method and fast Fourier transform (FFT) technique are employed to analyze the effects of material viscoelasticity and texture sizes on the contact and electrical performance. It is shown that, compared with results from the elastic-adhesive contact model, the contact area ratio based on the viscoelastic-adhesive contact model is significantly larger, which is much closer to the experimental results. Among the selected sizes of pyramid texture, the higher electrical performance can be obtained from the textures with a smaller pitch and a larger width under the heavier applied load. This study can provide a theoretical reference for the design of viscoelastic surface texture of TENG.

Effects of loading rate, applied shear strain, and magnetic field on stress relaxation behavior of anisotropic magnetorheological elastomer

Experimental research and numerical computation of stress relaxation behavior of an anisotropic magnetorheological elastomer (MRE) have been conducted in this paper. The anisotropic MRE has been formed from silicone matrix and micro-sized carbonyl iron particles under a magnetic field. Stress relaxation response of the anisotropic MRE was examined by single- and multi-step relaxation tests in shear mode using double-lap shear specimens. The effects of loading rate, applied constant strain, and external magnetic field on the stress relaxation behavior of the anisotropic MRE were studied. Experimental results showed that the stress relaxation of the anisotropic MRE was slightly dependent on the loading rate, but strongly depended on the constant strain level and the magnitude of external magnetic field. When increasing the constant strain level the shear stress of the anisotropic MRE in the single-step relaxation enhanced, while the relaxation modulus declined. The shear stress and modulus of the anisotropic MRE in the relaxation periods increased with increasing the magnetic field intensity. The four-parameter fractional derivative Zener model was used to describe the stress relaxation behavior of the anisotropic MRE. The presented model was fitted well to experimental data of the anisotropic MRE in both single- and multi-step relaxation tests. The fittings of relaxation modulus and shear stress with long-term predictions for the anisotropic MRE are in a very good agreement with the experimental ones. The maximal relative error of the fitted curves compared with measured data for both relaxation modulus and shear stress is less than 5.0%. As a result, the presented model is applicable to predict the long-term stress relaxation behavior of the anisotropic MRE.

Analytical higher-order-theories for noise reduction analysis of viscoelastic composite multilayered shells

The noise transmission of aeronautical panels is an important phase of the design process of an airplane. In this work an analytical Navier-type solution, based on higher-order layer-wise shell models, is proposed for the analysis of the sound insulation of laminated panels. The considered multilayered structures are laminated with cross-ply composite layers embedded with interlaminar viscoelastic sheets. The use of the soft interlayers permits to have a passive insulation effect in the study of the sound transmission. In order to take into account the frequency depedent properties of a realistic viscoelastic layer, the damping behavior is modeled through a fractional derivative Zener model. The Rayleigh integral method is used to extrapolate the acoustic indicators for the sound transmission analysis. Some results are presented to validate the efficiency of the present approach, comparing the present solutions with others taken from the literature.

Influence of viscoelastic mechanical properties of rail pads on wheel and corrugated rail rolling contact at high speeds

The rubber tie-plates on rail fasteners display very marked nonlinear stiffness and damping characteristics, and play an important role in railway systems. This paper studies the influence of the viscoelastic mechanical properties of the rubber tie-plate on the transient rolling contact behavior of the wheel and rail. Firstly, the viscoelastic mechanical properties of high-speed rail pads were obtained through laboratory tests and described theoretically by means of a modified fractional derivative Zener model. After this, an explicit finite element model is proposed to investigate the wheel and corrugated rail rolling contact. The numerical study shows that the viscoelastic mechanical properties have a significant effect on the response strength of the wheel-rail force and axle box acceleration at the main frequencies.

Using fractional derivatives for improved viscoelastic modeling of textile composites. Part II: Fabric under different temperatures

An improved modelling of the viscoelastic behavior of fabric composites at different temperatures is presented by a new means of fractional calculus. To this end, a four-parameter fractional derivative Zener model is first used to describe the dynamic behavior of the fabric, where the constitutive relation’s response to a unit step of strain (for determination of the time dependent stress relaxation modulus) can be readily found in the literature. This response function is then fitted to the experimental data of samples of a typical prepreg composite made of E-glass fibers comingled with polypropylene fibers, both at the dry and consolidated conditions. It is shown that the Zener model can better capture the experimental data when compared to the conventional approaches such as integer order modelling and Prony series. The results, particularly, showed that the viscoelastic behavior of the material at lower temperature regimes is far more accurate using the fractional derivatives approach. Finally, the variation of storage and loss moduli of the dry fabric with external cyclic loading were scrutinized, to determine the rubbery and glassy regions of the frequency response.

Influence of the temperature- and frequency-dependent dynamic properties of rail pads on the vibration characteristics of rails at low temperature

The dynamic properties of railway tracks and rail pads are significant for accurately predicting both the wheel–rail system vibration and rolling noise. The effect that the dynamic properties of rail pads have on the dynamic characteristics of railway tracks at extremely low temperatures was not adequately studied in previous research. In order to better predict the attenuation of the rail vibration, the viscoelastic dynamic properties of rail pads varying nonlinearly with temperature and frequency were first tested, in a wide temperature range (−60 ℃ to 20 ℃), and represented by the fractional derivative Zener model. Then, rail vibration and its attenuation characteristics were investigated by accounting for the frequency-dependent, temperature-dependent and frequency- and temperature-dependent properties, respectively. To be more specific, the frequency and amplitude of the rail first-order bending resonance and pinned–pinned resonance, as well as the rail decay rate were analyzed for the three cases. In conclusion, the study shows that the temperature/frequency-dependent properties of the rail pad have a significant effect on the first-order bending resonance of the rail, but no influence on the pinned–pinned resonance frequency. The rail decay rate indicates a clear increasing trend in the entire frequency domain with the decrease of temperature (especially below about −20 ℃). The frequency dependence mainly affects the vibration and its attenuation characteristics of the rail below about 400 Hz, which should not be ignored in the track dynamics modelling. Therefore, when the analyzed environmental temperature is below −20 ℃, the temperature dependence of rail pads should be considered.

System dynamics modeling and experimental study of railway track with thermoelectric heater/generator in extreme weather conditions

This paper develops a novel railway fastener system with thermoelectric heater/generator to heat the rail pads with the clean and sustainable electric energy harvested from the ambient thermal radiation. Due to the extremely cold temperature in some areas, the varying dynamic properties of rail pads result in a potential threat to the safe operation of the railway track system. The proposed thermoelectric generator can be used to continuously supply sufficient clean electric power to heat the rail pad, ensuring the railway fastener work within a proper temperature range. The experimental results show that the thermoelectric generator can provide an electric power of 5.8 mW with a low temperature gradient of 8 °C, and the working temperature of the rail pads can be improved by an average of 9.5 °C with 4 heaters of 2.5 W, which is consistent with the theoretical results of the finite volume method by FloTHERM. A fractional derivative Zener model of the coupled vehicle-track system is established to investigate the random vibration of the railway track structure with and without the proposed intelligent fastener system, and the calculated results show that the peak amplitude of wheel-rail contact force changes from 1.37 kN/Hz to 2.34 kN/Hz; whereas the first dominant frequency varies from 56.8 Hz to 68.0 Hz. It indicates that the vibration of the coupled vehicle-track system can be reduced by 41.5% with the proposed fastener system in extreme cold weather conditions.

High-speed vehicle–slab track coupled vibration analysis of the viscoelastic-plastic dynamic properties of rail pads under different preloads and temperatures

The viscoelastic-plastic dynamic properties of rail pads were are not adequately accounted for in the traditional vehicle–track coupled dynamic model. To investigate the influence of the viscoelastic-plastic dynamic properties of rail pads on the high-speed vehicle–track coupled vibrations, we obtained the viscoelastic-plastic dynamic properties of rail pads first by a series of laboratory tests, after which we described these properties theoretically by a modified Fractional Derivative Zener (FDZ) model and a traditional Berg friction model. A modified vertical vehicle–slab track coupled dynamic model was established by introduction of the modified FDZ and traditional Berg models for rail pads to investigate the influence of the viscoelastic-plastic dynamic properties of the rail pads on the high-speed vehicle–slab track coupled vibration responses under various preloads and temperatures. The FDZ and Berg models for rail pads improved the calculation accuracy of the high-speed vehicle–slab track coupled vibration responses, compared with the traditional KV model for rail pads, especially in the dominant high frequency range. The dynamic plastic properties of the rail pads effectively suppressed the high-frequency wheel-rail forces and yet increased the high-frequency vibration responses of the track substructures under the rail. Low temperatures and increased preloads on rail pads had a similar effect on the vehicle–track coupled vibrations. Increasing the preload on rail pads or decreasing the temperature increased the wheel-rail random excitation energy and its dominant frequencies, increased the vibration responses of the wheel-set and the track substructure under the rail, and decreased the vibrations of the rail at nearly all frequencies. Highlights The dynamic viscoelasticity and plasticity of rail pad under preloads and temperatures were tested. A modified FDZ model for the dynamic viscoelastic properties of rail pads was proposed. The dynamic plastic properties of rail pads can decrease the high-frequency wheel-rail forces. The dynamic plastic properties of rail pads can increase the high-frequency vibration under the rail. Low temperatures and increased preloads on rail pads had a similar effect on vehicle/track vibration.

Investigation on nonlinear and fractional derivative Zener model of coupled vehicle-track system

ABSTRACTThe nonlinear and fractional derivative Zener model of viscoelastic material considering Berg’s friction force model is established, which can be used to describe the frequency-dependent and amplitude-dependent behaviour of rubber isolator. The simplified Berg’s friction force scheme was proposed to improve computational efficiency in the frequency domain. The frequency response of the single degree of freedom system modelled with the fractional derivative Zener model and the proposed simplified Berg’s friction force scheme was investigated, and it agrees well with the results calculated by discrete Fourier transform. In addition, the fractional derivative Zener model and the proposed simplified Berg’s friction force scheme are implemented to model the rail pads and the primary suspension rubber spring in a railway vehicle, and then the nonlinear and fractional derivative Zener model of the coupled vehicle-track system was established. The frequency response calculated by the proposed nonlinear and fractional derivative Zener model of coupled vehicle-track system was compared to the calculation using ordinary Kelvin–Voigt model, and the results have shown that there is little difference in the calculation results between the two models, but the vibration responses of bogies, wheelsets and rails are quite different in time domain and frequency domain. Furthermore, the comparison between the proposed model and the fractional derivative Kelvin–Voigt model were analysed. The difference between the two models is small in the frequency range from 1 to 200 Hz and in the whole time domain, but the difference is large in the high frequency region. The proposed nonlinear and fractional derivative Zener model of the coupled vehicle-track system has high efficiency and accuracy both in time domain and frequency domain.Abbreviations: FDZ: fractional derivative Zener; SBFF: simplified Berg’s friction force; CVTS: coupled vehicle-track system; DFT: discrete Fourier transform; SDOF: single degree of freedom; FDKV: fractional derivative Kelvin–Voigt.

Applying the fractional derivative Zener model to fitting the time‐dependent material viscoelasticity tested by nanoindentation

Material viscoelasticity is of particular importance for polymers, biotissues and many others, due to the significant influence not only on the elastic response of structures, but also on their failure mechanism in actual applications. In this study, the experimental tests by nanoindentations are categorized according to the objective parameter combined with the indenter shape into four distinct cases, and the Fractional Derivative Zener model (FDZM) is employed to characterize this time-dependent property. The explicit data-fitting formulas are derived to determine the relaxation modulus and creep compliance tested with the conical and spherical indenters. Besides, a fitting scheme is provided to facilitate the identification of these viscoelastic parameters. In comparison with the general methods adopting the Prony series or empirical models, two silent merits present: (1) fewer model coefficients necessitate being determined based on the experimental data; (2) the relaxation modulus G(t) and the creep compliance J(t) can be obtained through one indentation test, and their exact interconversion is guaranteed as well. Several sets of experimental data are applied to examine the validity of the developed approach, and three more examples are adopted to demonstrate the exact interconversion between G(t) and J(t). A few concluding remarks are drawn eventually.

Characterization of the dynamic behaviour of flax fibre reinforced composites using vibration measurements

Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced epoxy (FFRE) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited on the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine.

Theoretical modelling and experimental analysis of the vertical stiffness of a convoluted air spring including the effect of the stiffness of the bellows

A new refined model including a model of the bellows to describe the vertical stiffness of a convoluted air spring is developed. The new model of a convoluted air spring incorporates the structural parameters including the effective area, the rate of change in the effective area, the effective volume, the rate of change in the effective volume and the stiffness of the bellows. Analytical models of the structural parameters of a convoluted air spring are established using a geometrical analysis approach. Two convoluted air springs (type 1B5002 and type 2B5281) are designed and manufactured for testing purposes. An experimental set-up is designed in order to carry out identification of the structural parameters and to test the vertical static stiffness and the vertical dynamic stiffness. The analytical models of the structural parameters are validated by experiments, which provide good design guidance and improvement in the convoluted air spring at the design stage. Tests on the vertical static stiffness and the dynamic stiffness indicate that the stiffness of the bellows should be considered; furthermore, the stiffness of the bellows exhibits amplitude-dependent chacteristics and frequency-dependent characteristics. In a different approach from the previous studies, a model of the bellows is developed in this paper and is composed of a smooth friction model and a fractional derivative Zener model in parallel to represent the viscoelastic properties of the bellows. It is concluded that the simulation results of the proposed new model are in good agreement with the experimental data. The new refined model may be an effective tool for predicting the mechanical behaviours of a convoluted air spring before a prototype is constructed. The proposed refined model can also be practically utilized as a guide for designing the parameters and for matching the air suspension of the vehicle.

Determination of dynamic properties of flax fibres reinforced laminate using vibration measurements

Experimental and numerical methods to identify the linear viscoelastic properties of flax fibre reinforced polymer (FFRP) composite are presented in this study. The method relies on the evolution of storage modulus and loss factor as observed through the frequency response. Free-free symmetrically guided beams were excited in the dynamic range of 10 Hz to 4 kHz with a swept sine excitation focused around their first modes. A fractional derivative Zener model has been identified to predict the complex moduli. A modified ply constitutive law has been then implemented in a classical laminates theory calculation (CLT) routine. Overall, the Zener model fitted the experimental results well. The storage modulus was not frequency dependant, while the loss factor increased with frequency and reached a maximum value for a fibre orientation of 70°. The damping of FFRP was, respectively, 5 and 2 times higher than for equivalent carbon and glass fibres reinforced epoxy composites.

Application of a fractional model for simulation of the viscoelastic functions of polymers

ABSTRACTIn the present work the dynamic behavior of two representative polymeric materials, experimentally studied in previous works, has been analyzed by a fractional derivative model. It is shown that the well‐known fractional derivative Zener model, in its simplest form as a four‐parameter model is capable of capturing the main features of the dynamic moduli of the polymeric structures examined. Furthermore, the time dependent viscoelastic functions, namely the compliance and the relaxation modulus could be simulated with the same model parameter values, indicating this way that the fractional model can provide a method of interconversion between viscoelastic material functions. The model's inadequacy of describing the loss modulus peak asymmetry, exhibited by the materials, has been encountered by the five‐parameter version of the fractional Zener model. © 2016 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2016, 133, 43505.

Prediction of Dissipative Properties of Flax Fibers Reinforced Laminates by Vibration Analysis

This study proposes an experimental-numeric method to identify the viscoelastic properties of flax fibres reinforced composite laminate (flax/epoxide). The used method consists in identifying the evolutions of both loss factor and stiffness when vibrational frequency changes. In this way, several free-free symmetrically guided beams are excited on a dynamic range of 10 to 4000 Hz with sweep sine excitation focused around the 4-first’s modes. Fractional derivative Zener model is used to identify the on-axis ply complex moduli and describe the laminate dissipative linear behavior with the classical laminate theory. Results obtained on a quasi-isotropic laminate show that this model adequately predicts the vibrational behavior of the tested laminates.

A fractional derivative Zener model for the numerical simulation of base isolated structures

A fractional derivative Zener (FDZ) model connected in parallel with a linear viscous damper and a Coulomb friction slider is used to numerically simulate the mechanical behavior of a base isolated (BI) building tested under free vibration conditions in Solarino, Sicily. This hybrid BI system comprises high damping rubber bearings in combination with low friction sliding bearings. A comparison study of the present model with previous ones appearing in the literature, namely the bi-linear and the tri-linear models defined in the time domain, is carried out here. Furthermore, the linear viscoelastic solid, namely the classical Zener model, is also implemented and evaluated. The rheological models representing all the above BI systems are analyzed and, for the first time, the rheological formulation for the tri-linear model is presented. The present comparison study shows that the FDZ model is capable of describing the complex nonlinear response of BI systems.

LOSS FACTOR PEAK OF VISCOELASTIC MATERIALS: MAGNITUDE TO WIDTH RELATIONS

The loss factor of viscoelastic materials as a function of frequency has at least one peak. The relation between the magnitude and width of the loss factor peak is investigated in this paper by means of the fractional derivative Zener model with special reference to polymeric materials used for sound and vibration damping. It is shown that the magnitude and width are interrelated through the dispersion of dynamic modulus and the rate of frequency variation of loss factor. Moreover, it is proved that the relation between the magnitude and width of the loss factor peak is not unequivocal; either proportionality or inverse proportionality may exist between them. The important consequence of prediction on the proportionality is that, in contrast to the common belief concerning polymers, it is physically possible to increase the loss factor while simultaneously broadening the peak. The validity of model predictions is discussed and it is proved that the predictions are of a general nature, because they obey fundamental physical principles. Experimental data supporting the theoretical predictions are presented.