Fractional Order Derivative Model Research Articles

Applications of fractional calculus to thermodynamics analysis of hydromagnetic convection in a channel

The present analysis signifies the impacts of fraction calculus on the MHD analysis of an incompressible fluid flow with entropy generation, viscous dissipation, and joule heating carried out between two endless vertical plates. The left vicinity of the MHD fluid flow channel has a convectively hot MHD fluid with a specific temperature that produces the heat transfer coefficient inside the flow channel. The MHD fluid flow model has been formulated with the fractional order derivative, which is an exciting area of research. The transient form of the system of dimensionless PDEs with imposed boundary conditions is transformed to a fractional order derivative model, which was formerly solved via the Finite Difference Scheme (FDS). The graphical illustrations signify the significance of the transverse magnetic field, viscous dissipation, and Ohamic heating alongside Biot and Bejan numbers. The velocity and temperature illustrations have been shown graphically to observe the influences within the flow channel. With predetermined values of the corresponding dimensionless parameters, the coefficient of skin friction and Nusselt number accompanied by a range of convection constraints are displayed graphically within the flow channel.

Microscopic double-chain sawtooth model and macroscopic four-parameter fractional-order derivative viscoelastic model based on magnetorheological elastomers

In order to restore the distribution of ferromagnetic particles in magneto-rheological elastomers to the greatest extent, the magnetotropic effect of magneto-rheological elastomers is more accurately represented, and the dynamic mechanical properties of magneto-rheological elastomers are more accurately reflected. In this paper, based on the dipole theory, a new microscopic model of magnetorheological elastomer, the double chain sawtooth model, is proposed based on the chain model and the main chain adsorption model, and a macroscopic viscoelastic mechanical model of magnetorheological elastomer, the four-parameter fractional order derivative improvement model, is proposed based on the classical macroscopic mechanical parametric model and the fractional order derivative model. A microscopic model of magnetorheological elastomer was combined with a macroscopic model to describe its mechanical properties. Isotropic and anisotropic magnetorheological elastomers with different mass fractions and different silicone oil contents were prepared using silicone rubber as the matrix and carbonyl iron powder as the filling phase, and dynamic shear tests were conducted to verify the validity of the model.

Study of Transmission Dynamics of Streptococcus suis Infection Mathematical Model between Pig and Human under ABC Fractional Order Derivative

In this paper, a mathematical model for Streptococcus suis infection is improved by using the fractional order derivative. The modified model also investigates the transmission between pigs and humans. The proposed model can classify the pig population density into four classes, which are pig susceptible class, pig infectious class, pig quarantine class, and pig recovery class. Moreover, the human population density has been separated into three classes, these are human susceptible class, human infectious class, and human recovery class. The spread of the infection is analyzed by considering the contact between humans and pigs. The basic reproduction number (R0), the infectious indicator, is carried out using the next generation matrix. The disease-free equilibrium is locally asymptotically stable if R0<1, and the endemic equilibrium is locally asymptotically stable if R0>1. The theoretical analyses of the fractional order derivative model, existence and uniqueness, have been proposed. The numerical examples were illustrated to support the proposed stability theorems. The results show that the fractional order derivative model provides the various possible solution trajectories with different fractional orders for the same parameters. In addition, transmission between pigs and humans resulted in the spread of Streptococcus suis infection.

Caputo fractional order derivative model of Zika virus transmission dynamics

The Zika Virus (ZIKV) is a highly contagious disease, and several outbreaks have occurred since it emerged. It is transmitted from one to another human via a mosquito Aedes aegypti. There is no vaccine or established medicine available for ZIKV to date. There is an urgent need to enhance an understanding of the progression mechanism of the disease when drugs or vaccines are not available. Mathematical modeling is a tool that might be helpful to understand the progression dynamics of ZIKV which can enable us to make control strategies for invading the progression dynamics of disease. SEIR-SEI is a famous compartmental deterministic modeling based on integer-order derivative calculus. Nowadays, conversion from integer to fractional order-based derivative modeling is in trend, and it is a very effective and high degree of accuracy. In this paper, we proposed a Caputo fractional-order based susceptible-exposed-infected-recovered (SEIR) structure for hosts and a susceptible-exposed-infected (SEI) structure for mosquitoes for transmission dynamics of ZIKV. For this purpose, we modified the classical compartmental model used in the study of progression dynamics of the Zika fever outbreak in El-Salvador during 2015-16. The modified model involves nonlinear differential equations of fractional (non-integer) order which has an advantage over the classical model due to its memory effect property. Our study includes eight regions across the globe where the Zika outbreak has occurred during the year 2013-2016 including six major archipelagos of French Polynesia, i.e., Tahiti, Sous-le-vent, Moorea, Tuamotu, Marquises, and Australes. The other two regions included Costa Rica and Colombia. The outbreak in selected regions was studied first using a classical model and then compared by a fractional-order model. The data of outbreaks are best fitted with the fractional-order model which enables us to estimate the best parameters values for the outbreaks. Using this modeling, the epidemic threshold parameter $R_0$ was computed which is more accurate than the classical one. Hence, the fractional-order model for ZIKV transmission dynamics is much better prediction, analysis, and disease parameters estimation than the classical model. This modeling enhances the knowledge in the field of fractional order and understanding the ZIKV transmission accurately.

Semianalytical Solution for Thermal Consolidation of Viscoelastic Marine Clay with the Fractional Order Derivative

The deformation property of marine clay under a heat source has received considerable attention in the geotechnical literature. In this paper, a three‐parameter fractional order derivative model is introduced into the thermo‐hydro‐mechanical coupling governing equations with thermal filtration and thermo‐osmosis to simulate viscoelastic characteristics of marine clay. The excess pore pressure, temperature increment, and displacement of marine clay are derived by using the Laplace transform method, and the semianalytical solution for the one‐dimensional thermal consolidation in the time domain is derived by using a numerical inversion of the inverse Laplace transform. The influence of the order of the fractional derivative, material parameters, and phenomenological coefficient on thermal consolidation is investigated based on the present solutions. It is shown that the influence of the fractional derivative parameter on the excess pore pressure and displacement of marine clay depends on the properties of soil mass, and the temperature increment has an obvious effect on the thermal filtration and thermo‐osmosis process.

GPU Based Modelling and Analysis for Parallel Fractional Order Derivative Model of the Spiral-Plate Heat Exchanger

Heat exchangers are commonly used in various industries. A spiral-plate heat exchanger with two fluids is a compact plant that only requires a small space and is excellent in high heat transfer efficiency. However, the spiral-plate heat exchanger is a nonlinear plant with uncertainties, considering the difference between the heat fluid, the heated fluid, and other complex factors. The fractional order derivation model is more accurate than the traditional integer order model. In this paper, a parallel fractional order derivation model is proposed by considering the merit of the graphics processing unit (GPU). Then, the parallel fractional order derivation model for the spiral-plate heat exchanger is constructed. Simulations show the relationships between the output temperature of heated fluid and the orders of fractional order derivatives with two directional fluids impacted by complex factors, namely, the volume flow rate in hot fluid, and the volume flow rate in cold fluid, respectively.

Active damping of geometrically nonlinear vibrations of smart composite shells using elliptical smart constrained layer damping treatment with fractional derivative viscoelastic layer

In this article, the performance of elliptical smart constrained layer damping treatments on active damping of geometrically nonlinear vibrations of doubly curved smart laminated composite shells is analyzed. The constraining layers of the smart constrained layer damping treatments comprised vertically/obliquely reinforced 1–3 piezoelectric composites, while the constrained layers of isotropic viscoelastic materials are modeled using the three-dimensional fractional order derivative model. A mesh-free model of the smart composite shells is developed for analyzing their nonlinear transient responses within the framework of a layerwise shear and normal deformation theory considering the von Kármán–type geometric nonlinearity. Thin, doubly curved laminated composite shells integrated with elliptical/rectangular smart constrained layer damping patches with different stacking sequences and boundary conditions are considered for presenting the numerical results. The numerical analyses demonstrate the higher effectiveness of the elliptical smart constrained layer damping treatments over the rectangular ones in attenuating the nonlinear vibrations of laminated composite shells.

Active Control of Nonlinear Transient Vibration of Laminated Composite Beams Using Triangular SCLD Treatment With Fractional Order Derivative Viscoelastic Model

Abstract This paper is concerned with the analysis of the effectiveness of triangular shaped smart constrained layer damping (SCLD) treatment in attenuating geometrically nonlinear transient vibrations of laminated composite beams. The SCLD treatment is comprised of an advanced vertically reinforced 1–3 piezoelectric composite (PZC) as the constraining layer and an isotropic viscoelastic layer as the constrained layer, which is modeled using a two-dimensional fractional order derivative (FOD) model with Grünwald definition of the FODs. A nonlinear meshfree model of the smart composite beam is developed for analyzing its nonlinear transient response within the framework of a layerwise shear and normal deformation theory considering von Kármán type geometric nonlinearity. Cantilever type composite beams having different lamination sequences integrated with regular rectangular/triangular type of SCLD treatments are considered for presenting the numerical results. For comparison purpose, a geometrical constraint has been imposed such that both the rectangular and triangular shaped SCLD treatments will cover the equal area on the top surface of the beam. The numerical analyses demonstrate the effectiveness of the triangular shaped SCLD patches over the rectangular SCLD treatment in controlling the nonlinear vibration of laminated composite beams. The two-dimensional FOD model of the viscoelastic material has been efficiently implemented for the active damping analysis of smart composite beam.

Active damping of geometrically nonlinear vibrations of smart composite plates using elliptical SCLD treatment with fractional derivative viscoelastic layer

In this paper, the performance of elliptical smart constrained layer damping treatments on active damping of geometrical nonlinear vibrations of smart laminated composite plates are analyzed. The constraining layers of the damping treatments are comprised of vertically/obliquely reinforced 1–3 piezoelectric composites, while the constrained layers are of isotropic viscoelastic material, modelled using the three dimensional fractional order derivative model. A meshfree model of the smart composite plate is developed for analyzing its nonlinear transient responses within the framework of a layerwise shear and normal deformation theory considering the von-Kármán type geometric nonlinearity. Thin composite plates integrated with elliptical/rectangular patches of the smart constrained layer damping treatments and having different stacking sequences and boundary conditions are considered for presenting the numerical results. The numerical analyses demonstrate the higher effectiveness of the elliptical patches over the rectangular ones in attenuating the nonlinear vibrations of laminated composite plates.

Smart damping of geometrically nonlinear vibrations of composite shells using fractional order derivative viscoelastic constitutive relations

ABSTRACTIn this article, a three-dimensional fractional order derivative model has been developed for the constrained viscoelastic layer of the active constrained layer damping (ACLD) treatment of laminated composite shells undergoing geometrically nonlinear vibrations. The constraining layer of the ACLD treatment is made of vertically/obliquely reinforced 1–3 piezoelectric composites and acts as the distributed actuator. A three-dimensional smart nonlinear finite element model has been developed. Several numerical results are presented to check the accuracy of the present three-dimensional fractional derivative model of the constrained viscoelastic layer for smart damping of geometrically nonlinear vibrations of laminated composite shells.

Fractional order spring/spring-pot/actuator element in a multibody system: Application of an expansion formula

Fractional order models of a spring/spring-pot and spring/spring-pot/actuator element connected into a multibody system are proposed in order to represent smart materials and components in adaptronic systems by introducing new tuning parameter. The models are introduced into dynamic equations via generalized forces and using the Lagrange's equations of the second kind in covariant form. Generalized forces are derived by taking into account fractional order derivatives in force–displacement relations and by using the principle of virtual work. The numerical scheme for solving fractional order differential equations proposed in Atanacković and Stanković (2008) is used in order to approximate fractional order derivative of a composite function appearing in the presented fractional order model. Numerical example for the multibody system with three degrees of freedom is presented. The results obtained for generalized forces are compared for different values of parameters in the fractional order derivative model.

On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: modeling and experiments.

The nonlinear, preload-dependent dynamic stiffness of a cylindrical vibration isolator is examined via measurements and modeling within an audible frequency range covering 50 to 1000 Hz at various preloads. The stiffness is found to depend strongly on frequency-resulting in peaks and troughs, and on preload-particularly above 500 Hz. The problems of simultaneously modeling the rubber prestrain dependence and its audible short-term response are removed by adopting a nearly incompressible material model, being elastic in dilatation while displaying viscoelasticity in deviation. The latter exhibits a time strain separable relaxation tensor with a single function embodying its time dependence. This function is based on a continuous fractional order derivative model, the main advantage being the minimum number of parameters required to successfully model the rubber properties over a broad structure-borne sound frequency domain, while embodying a continuous distribution of relaxation time. The weak formulations corresponding to the stiffness problem are solved by an updated Lagrangian nonlinear finite-element procedure. The model and measurement results agree strikingly well with static and dynamic measurements throughout the whole frequency domain for the examined preloads.

ON THE WAVEGUIDE MODELLING OF DYNAMIC STIFFNESS OF CYLINDRICAL VIBRATION ISOLATORS. PART I: THE MODEL, SOLUTION AND EXPERIMENTAL COMPARISON

A waveguide model of the axial dynamic stiffness for cylindrical vibration isolators in the audible frequency range is presented. The problems of satisfying the cylinder boundary conditions simultaneously are removed, by adopting the mode-matching technique, using the dispersion relation for an infinite cylinder and approximately satisfying the boundary conditions at the lateral surfaces by a circle-wise fulfilment or a subregion method. The rubber material is assumed to be nearly incompressible with deviatoric viscoelasticity based on a fractional order derivative model. The main advantage of the viscoelastic model is the minimum parameter number required to model the material properties successfully over a broad structure-borne sound frequency domain. The work is verified by experiments on a rubber cylinder, equipped with bonded circular steel plates, in the frequency range 100–5000 Hz. The model and the measurements are shown to agree strikingly well within the whole frequency range. Comparisons with alternative material models, known as the Kelvin–Voigt and frequency-independent or ‘hysteric’ material models, are made. The results are shown to diverge substantially from the presented material model; in particular, the Kelvin–Voigt model overestimates the material damping in the high-frequency region, while the frequency-independent model underestimates it. In addition, the resonance and anti-resonance frequencies are incorrectly predicted. In a companion paper the dispersion relation solution, convergence analysis and comparison with simple models are addressed.