Fractional Derivative Elements Research Articles

Modeling of a quasi-zero static stiffness mount fabricated with TPU materials using fractional derivative model

Static behaviors of quasi-zero static stiffness (QZSS) mounts have been widely studied to isolate low frequency vibrations. However, Dynamic behaviors contributed by the damping of QZSS mounts are simplified as a linear viscous damping, leading to a misestimate of isolation performances at low frequencies. In this paper, dynamic behaviors of a novel QZSS mount are modeled and analyzed. Firstly, a mount with QZSS characteristic is designed using finite element method and then fabricated with Thermoplastic polyurethanes (TPU). Secondly, reacted forces of the fabricated mount under static, harmonic and random excitations are measured through the MTS device. Thirdly, a mechanical model consisting of nonlinear elastic, friction and fractional derivative force elements arranged in parallel is presented to capture the measured mount’s nonlinear static, frequency-dependent and amplitude-dependent behaviors. Finally, a single degree of freedom system with a QZSS mount is used to investigate the influences of friction and fractional derivative models on the mount isolation performances.

The Melnikov criterion of instability for random rocking of a rigid block with a fractional derivative element

This paper discusses the problem of the cumulative influence of fractional dissipation and external noise on escape of a rigid block from a prescribed set of rocking oscillations. For simplicity, the dynamics of a weakly dissipative slender block affected by low energy noise is considered. Under these assumptions, the generic motion is depicted by the linearized equation of an undamped block with the well-known periodic vibro-impact solution. The stochastic Melnikov criterion based on the generic solution allows calculating the upper bound of escape probability of the structure in the presence of fractional dissipation of the Caputo type. The analytical results demonstrate an explicit dependence of escape probability on the fractional dissipation parameters.

Transition behaviors of system energy in a bi-stable van Ver Pol oscillator with fractional derivative element driven by multiplicative Gaussian white noise

The stochastic P-bifurcation behavior of system energy in a bi-stable Van der Pol oscillator with fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the non-linear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary probability density function is obtained by stochastic averaging. Then, according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary probability density function curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte-Carlo simulation verifies the theoretical analysis in this paper, and the method used in this paper can directly guide the design of the fractional-order controller to adjust the response of the system.

Approximate stochastic response of hysteretic system with fractional element and subjected to combined stochastic and periodic excitation

A method based on statistical linearization is proposed, for determining the response of single-degree-of-freedom hysteretic systems endowed with fractional derivative element and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two coupled fractional-order nonlinear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method for the fractional-order deterministic equation and the statistical linearization for the stochastic equation are used, to obtain the Fourier coefficients of the deterministic response component and the variance of the stochastic response component, respectively. This yields two sets of coupled nonlinear algebraic equations which can be solved by appropriate standard numerical method. Pertinent numerical examples, including both softening and hardening Bouc–Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

Survival probability of nonlinear oscillators endowed with fractional derivative element and subjected to evolutionary excitation: A stochastic averaging treatment with path integral concepts

An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.

Nonlinear stochastic vibration of a variable cross-section rod with a fractional derivative element

The article deals with the problem of computing efficiently the nonlinear response of a rod involving a fractional constitutive model, and exposed to random excitation. The constitutive model is a three-parameter model comprising an instant elasticity modulus, a prolonged elasticity modulus, and a relaxation parameter. The nonlinear term is a linear-plus-cubic force of the Winkler kind. The resulting nonlinear fractional partial differential equation governing the rod displacement has no known exact solution. Thus, the article proposes an approximate analytical solution by relying on the statistical linearization technique. Further, it develops a Boundary Element Method (BEM)-based approach to estimate numerically the rod response statistics. The statistical linearization solution is obtained by representing the rod displacement as the superposition of linear modes of vibration having time-dependent coefficients. In this context, it is shown that the equation governing the time variation of the mode coefficients is a nonlinear fractional ordinary differential equation, whose solution is computed by a surrogate linear system identified by minimizing the response error between the linear system and the nonlinear one in a mean square sense. Relevant Monte Carlo studies pertaining to rods with fixed-fixed, and fixed-free ends show that the proposed analytical solution is in a good agreement with data obtained by the numerical (BEM) approach.

Non-stationary response power spectrum determination of linear/non-linear systems endowed with fractional derivative elements via harmonic wavelet

This paper presents a wavelet-based method for determining response evolutionary power spectrum density (EPSD) of linear/non-linear systems endowed with fractional derivative elements. Specifically, first, the generalized harmonic wavelets (GHWs) based Galerkin approximation of the stochastic processes are utilized to transform the fractional-order stochastic linear/non-linear differential equations into a set of linear/non-linear algebraic equations with unknown response wavelet coefficients. Next, the linear algebraic equations are solved in a closed-form, while the non-linear ones are treated by the gradient-based standard numerical methods. Further, an analytical relationship between the EPSD of the excitation and of the response for a linear system is derived by considering the wavelet representation of stochastic processes. For a non-linear system, the response EPSD is estimated by repeated solving of sample algebraic equations. Pertinent numerical examples demonstrate the applicability and accuracy of the proposed method.

Stochastic Response Analysis and Reliability-Based Design Optimization of Nonlinear Electromechanical Energy Harvesters With Fractional Derivative Elements

Abstract A methodology based on the Wiener path integral (WPI) technique is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the satisfactory performance of the methodology.

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable transformation and the stochastic averaging technique are applied to derive the expression of probability density function (PDF) of the system response. Critical conditions of the stochastic bifurcation induced by system parameters are presented based on the change in the number of extreme points of the probability density function. Numerical results are given to show the effectiveness of the proposed approach. Stochastic P-bifurcations for additive and multiplicative noise are studied in detail according to the critical conditions. Stochastic P-bifurcation (SPB) for additive noise can be investigated by the critical conditions. SPB induced by the fractional order is described in the bifurcation plane, see Fig. a. Figure b shows that the probability density function (PDF) for (0.1, 0.005) ∈ R2 has one peak, while the PDF for (0.8, 0.005) ∈ R2 has two peaks. It implies that the change of the fractional order leads to the variation of the structure of the PDF, which indicates the occurrence of the SPB.

Time-response functions of fractional derivative rheological models

In view of the increasing attention to the time responses of complex fluids described by power-laws in association with the need to capture inertia effects that manifest in high-frequency microrheology, we compute the five basic time-response functions of in-series or in-parallel connections of two elementary fractional derivative elements known as the Scott-Blair (springpot) element. The order of fractional differentiation in each Scott-Blair element is allowed to exceed unity reaching values up to 2 and at this limit-case the Scott-Blair element becomes an inerter--a mechanical analogue of the electric capacitor that its output force is proportional only to the relative acceleration of its end-nodes. With this generalization, inertia effects may be captured beyond the traditional viscoelastic behavior. In addition to the relaxation moduli and the creep compliances, we compute closed form expressions of the memory functions, impulse fluidities (impulse response functions) and impulse strain-rate response functions of the generalized fractional derivative Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and their special cases that have been implemented in the literature. Central to these calculations is the fractional derivative of the Dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the two-parameter Mittag-Leffler function that emerges invariably in the time-response functions of fractional derivative rheological modes.

A new model of shear creep and its experimental verification

Creep exerts a significant role in rock engineering safety. In engineering practice, it is helpful to develop a mathematical model representing rock creep behaviors. We used a shear rheometer to carry out shear creep tests of cubic mudstone specimens. The experimental results indicate that the mudstone creep can be divided into steady and accelerated stages when the applied shear stress is greater than the long-term shear strength. To capture the creep characteristics of the mudstone samples, in this paper, we propose a variable-parameter fractional derivative element based on fractional theory. A viscosity coefficient is considered to be a variable. We develop a mathematical model describing rock shear creep. The new model shows a visco-elastic-plastic property, which can effectively characterize two stages of rock creep process. The shear creep results show that the new model of shear creep is more accurate than that of the Nishihara model and the constant-parameter fractional creep model. Sensitivity analysis demonstrates that derivative orders have the greatest effect on shear creep.

A visco-elastic-plastic constitutive model of isotropic magneto-sensitive rubber with amplitude, frequency and magnetic dependency

A three-dimensional visco-elastic-plastic constitutive model of isotropic magneto-sensitive (MS) rubber with amplitude, frequency and magnetic dependency under a continuum constitutive framework is developed. MS rubber is a kind of smart material mainly composed of a rubber matrix and magnetizable particles. Under a magnetic field, there is an increase of its dynamic modulus due to the interaction between the magnetic field and MS rubber, which is often referred to as the magnetic dependency of MS rubber. Experimental results reveal that besides the magnetic dependency, there is a frequency and amplitude dependency of its dynamic modulus. In specific, the modulus of MS rubber increases with increasing frequency and dereases with increasing strain amplitude. To depict the above properties and to consider the balance of energy in continuum mechanics framework, a new constitutive model consisting of a viscoelastic fractional derivative element, a bounding surface model in series with a neo-Hookean elastic model with magnetic sensitivity and a magnetic stress tensor term for MS rubber is proposed. The contribution of this constitutive model is that by using a free energy based method, with only eight material parameters, the amplitude, frequency and magnetic dependency of MS rubber can be reflected. After parameter identification, the simulation results show a good agreement with those of measurements. Therefore, by utilizing the model proposed, the ability of approaching the dynamic behavior of MS rubber-based vibration reduction devices in the design phase is possible which contributes to the application of MS rubber in noise and vibration reduction area.

One dimensional constitutive model of isotropic magneto-sensitive rubber under shear deformation with amplitude, frequency and magnetic dependency

Modelling the amplitude, frequency and magnetic dependency of magneto-sensitive (MS) rubber in one dimensional case under shear deformation is proposed in this paper. MS rubber is a kind of smart materials which mainly consists of magnetizable particles and rubber matrix. The modulus of MS rubber changes rapidly after the applying of magnetic field which is referred as the magnetic dependency of MS rubber. Besides the magnetic dependency, there is an amplitude and frequency dependency of MS rubber. Specifically, the shear modulus of MS rubber under larger strain magnitude is smaller than the corresponding modulus under small strain and the shear modulus of MS rubber increases with increasing of frequency. A one-dimensional constitutive model consists of a fractional derivative element in parallel with a nonlinear kinematic elastoplastic model to depict the amplitude, frequency and magnetic dependency of MS rubber is developed in this paper. After comparison with the test result, it found that with eight parameters the amplitude, frequency and magnetic dependency can be depict precisely by the model developed. It is helpful for the predicting the dynamic behaviours of MS rubber-based vibration devices in the design phase and promotes MS rubber’s application in sound and vibration area.

Implicit analytic solutions for a nonlinear fractional partial differential beam equation

Analytic solutions in implicit form are derived for a nonlinear partial differential equation (PDE) with fractional derivative elements, which can model the dynamics of a deterministically excited Euler-Bernoulli beam resting on a viscoelastic foundation. Specifically, the initial-boundary value problem for the corresponding PDE is reduced to an initial value problem for a nonlinear ordinary differential equation in a Hilbert space. Next, by employing the cosine and sine families of operators, a variation of parameters representation of the solution map is introduced. Due to the presence of a nonlinear term, a local fixed point theorem is employed to prove the local existence and uniqueness of the solution. Relying on the regularity properties of cosine and sine families, taking into account the form of the nonlinear term, and considering the properties of the fractional derivative, the solution map of the abstract problem is cast into a derivative-free analytic solution in implicit-form for the initial boundary value problem. Results corresponding to the limiting purely elastic and purely viscous cases are also provided. The herein developed technique and derived implicit form solutions can be construed as generalizations of available results in the literature to account for fractional derivative elements. This is of significant importance given the vast utilization of fractional calculus modeling in modern engineering mechanics, and in viscoelastic material behavior in particular.

Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam

Abstract In this paper, an effective numerical algorithm is proposed for the first time to solve the fractional visco-elastic rotating beam in the time domain. On the basis of fractional derivative Kelvin–Voigt and fractional derivative element constitutive models, the two governing equations of fractional visco-elastic rotating beams are established. According to the approximation technique of shifted Chebyshev polynomials, the integer and fractional differential operator matrices of polynomials are derived. By means of the collocation method and matrix technique, the operator matrices of governing equations can be transformed into the algebraic equations. In addition, the convergence analysis is performed. In particular, unlike the existing results, we can get the displacement and the stress numerical solution of the governing equation directly in the time domain. Finally, the sensitivity of the algorithm is verified by numerical examples.

Nonlinear vibrations of beams and plates with fractional derivative elements subject to combined harmonic and random excitations

This paper proposes an efficient approach for estimating reliably the second order statistics of the response of continua excited by combinations of harmonic and random loads. The problem is relevant in several engineering applications, where, for instance, the harmonic load is influenced by significant noise that cannot be neglected when computing the response statistics. The considered problems pertain to the vibration of beams and of plates endowed with fractional derivative elements. In both cases, it is shown that by representing the system response by the linear modes of vibration, systems of nonlinear fractional ordinary differential equations describing the time-dependent variation of the modes amplitudes are obtained. These equations are coupled and are treated by combining the harmonic balance and statistical linearization techniques, leading to the determination of the second-order statistics of the response. Relevant Monte Carlo data demonstrate the reliability of the proposed solution approach. The specific numerical examples considered pertain to simply supported beams, and plates with simply supported stress-free edges conditions.

Stochastic Averaging for the Piezoelectric Energy Harvesting System With Fractional Derivative Element

Combining the stochastic averaging method and generalized harmonic function, we propose a procedure to investigate the fractional-order energy harvesting system. First, we can obtain the approximately equivalent system by means of the generalized harmonic transformation; Second, the stochastic averaging method is utilized to get the approximate stationary solutions of the simplified system. Finally, In order to illustrate the accuracy of the developed scheme, two examples are carried out. The results show that the developed procedure has a satisfactory reliability. The influences of the intensity of noise excitation, the fractional order, the fractional coefficient and the nonlinear restoring force on the mean-square voltage (MSV) and mean output power (MOP) are also investigated.

Response spectral density determination for nonlinear systems endowed with fractional derivatives and subject to colored noise

A computationally efficient method for determining the response of non-linear stochastic dynamic systems endowed with fractional derivative elements subject to stochastic excitation is presented. The method relies on a spectral representation both for the system excitation and its response. Specifically, first the ordinary non-linear differential equation of motion is transferred into a set of non-linear algebra equations by employing the harmonic balance method. Next, the response Fourier coefficients are determined by solving these non-linear equations. Finally, repeated use of the proposed procedure yields the response power spectral density. Pertinent numerical examples, including a fractional Duffing and a bilinear oscillator, demonstrate the accuracy of the proposed method.

Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements

In this paper, the steady-state dynamic response of hysteretic oscillators comprising fractional derivative elements and subjected to harmonic excitation is examined. Notably, this problem may arise in several circumstances, as for instance, when structures which inherently exhibit hysteretic behavior are supplemented with dampers or isolators often modeled by employing fractional terms. The amplitude of the steady-state response is determined analytically by using an equivalent linearization approach. The procedure yields an equivalent linear system with stiffness and damping coefficients which are related to the amplitude of the response, but also, to the order of the fractional derivative. Various models of hysteresis, well established in the literature, are considered. Specifically, applications to oscillators with bilinear, Bouc–Wen, and Preisach hysteretic models are reported, and related parameter studies are presented. The derived results are juxtaposed with pertinent numerical data obtained by integrating the original nonlinear fractional order equation of motion. The analytical and the numerical results are found in good agreement, establishing, thus, the accuracy and efficiency of the considered approach.