Dynamic Viscoelastic Rod Stability Modeling by Fractional Differential Operator

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
D. Ingman ◽  
J. Suzdalnitsky
Keyword(s):  
Damping Ratio ◽  
Time Derivative ◽  
Additional Term ◽  
Bifurcation Line ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
The Stability ◽  
Fractional Time Derivative ◽  
Stability Modeling

The need to take into account the oscillation of a system is a special feature in the linear problem of the stability of a cantilevered rod under a follower force. Involvement of viscoelastic materials leads to damping of the oscillation hence to overestimation of critical loads. This new problem is solved here by means of an additional term introduced into the constitutive equation and proportional to the fractional time derivative with complex order—besides the inertial one. The effects contributed by the damping ratio, the real part of the order and the corrective role of its imaginary part on the shape of the bifurcation line, on its maximum and on the disposition of the inflection and maximal deflection points on the centerline of the deformed rod during the secondary loss of stability, are discussed.

scholarly journals The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

2017 ◽  
Vol 5 ◽  
Author(s):  
Angel A. Tateishi ◽  
Haroldo V. Ribeiro ◽  
Ervin K. Lenzi
Keyword(s):  
Anomalous Diffusion ◽  
Time Derivative ◽  
Fractional Time Derivative

scholarly journals Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 914
Author(s):  
Oana Brandibur ◽  
Roberto Garrappa ◽  
Eva Kaslik
Keyword(s):  
Differential Equations ◽  
Detailed Analysis ◽  
Linear Systems ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Limit Case ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Stability

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY

2012 ◽  
Vol 22 (04) ◽  
pp. 1250071 ◽  
Author(s):  
SACHIN BHALEKAR ◽  
VARSHA DAFTARDAR-GEJJI ◽  
DUMITRU BALEANU ◽  
RICHARD MAGIN
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Time Derivative ◽  
Bloch Equation ◽  
Extended Model ◽  
Order Ordinary Differential Equation ◽  
Stability Behavior ◽  
System Memory ◽  
The Stability ◽  
Fractional Time Derivative

The fundamental description of relaxation (T1 and T2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T1 and T2 relaxation. The T1 decay is stable for the range of delays tested (1 μsec to 200 μsec), while the T2 relaxation in this extended model exhibits a critical delay (typically 100 μsec to 200 μsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation.

Chaos and bifurcations in a discretized fractional model of quasi-periodic plasma perturbations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ahmed Ezzat Matouk
Keyword(s):  
Homoclinic Orbits ◽  
Flip Bifurcation ◽  
Invariant Curves ◽  
Bifurcation Diagrams ◽  
Homoclinic Connections ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Simulation Results ◽  
The Stability ◽  
Theoretical Results

Abstract The nonlinear dynamics of a discretized form of quasi-periodic plasma perturbations model (Q-PPP) with nonlocal fractional differential operator possessing singular kernel are investigated. For example, the conditions for the stability and occurrence of Neimark–Sacker (NS) and flip bifurcations in the proposed discretized equations are provided. Moreover, analysis of nonlinearities such as the existence of chaos in this map is proved numerically via bifurcation diagrams, Lyapunov exponents and analytically via Marotto’s Theorem. Also, some simulation results are utilized to confirm the theoretical results and to show that the obtained map exhibits double routes to chaos: one is via flip bifurcation and the other is via NS bifurcation. Furthermore, more complex dynamical phenomena such as existence of closed invariant curves, homoclinic orbits, homoclinic connections, period 3 and period 4 attractors are shown. This kind of research is useful for physicists who work with tokamak models.

scholarly journals Fractional-Differential Model of Heat Conductivity Process in Ferroelectrics under the Intensive Heating Conditions

2019 ◽  
pp. 29-47
Author(s):  
L. I. Moroz ◽  
A. G. Maslovskaya
Keyword(s):  
Finite Difference ◽  
Heat Conductivity ◽  
Time Derivative ◽  
Model Verification ◽  
Total System ◽  
Ferroelectric Crystal ◽  
Differential Model ◽  
Linear Algebraic Equations ◽  
Fractional Differential ◽  
Fractional Time Derivative

Ferroelectrics, due a number of characteristics, behave as hereditary materials with fractal structure. To model mathematically the systems with so-called memory effects one can use the fractional time-derivatives. The pyro-electric properties of ferroelectrics arouse interest in developing the fractional-differential approach to simulating heat conductivity process.The present study deals with development and numerical implementation of fractal heat conductivity model for hereditary materials using the concepts of fractional-differential calculus applied to the simulation of intensive heating processes in ferroelectrics.The paper proposes a mathematical model governed through mixed initial-boundary value problem for partial differential equation containing a fractional time-derivative as well as nonlinear temperature dependence on the heat capacity. To solve the problem the computational algorithm was designed which is based on an analog of the Crank – Nicolson finite difference scheme combining with the Grunwald – Letnikov formula for fractional time-derivative approximation. The approximation of Neumann boundary condition is included into the finite difference problem statement using scheme of fictitious mesh points. The total system of linear algebraic equations is solved by sweep method.The designed application program allows one to perform the computer simulation of heat conductivity process in hereditary materials. The model verification was performed for numerical solving test problem with known analytical solution. The results of computational experiments are demonstrated for the example of estimating heat distribution in a typical ferroelectric crystal of TGS (triglycine sulfate) near the temperature of phase transition. The fractional derivative order was approximately evaluated to be ~0.7 at variation of this parameter. We applied the comparison of fractal model implementation results with experimental data related to the time when the ferroelectric crystal is heated to Curie temperature. These findings demonstrate that one needs to use the modified models at the analysis of the field effects arising in hereditary materials.

A Statistical Study of Process Variables to Optimize a High Speed Curtain Coater: Part I

TAPPI Journal ◽  
2009 ◽  
Vol 8 (1) ◽  
pp. 20-26 ◽  
Author(s):  
PEEYUSH TRIPATHI ◽  
MARGARET JOYCE ◽  
PAUL D. FLEMING ◽  
MASAHIRO SUGIHARA
Keyword(s):  
Boundary Layer ◽  
Experimental Design ◽  
High Speed ◽  
Statistical Study ◽  
Process Variables ◽  
High Speeds ◽  
The Stability ◽  
Air Removal ◽  
Removal System

Using an experimental design approach, researchers altered process parameters and material prop-erties to stabilize the curtain of a pilot curtain coater at high speeds. Part I of this paper identifies the four significant variables that influence curtain stability. The boundary layer air removal system was critical to the stability of the curtain and base sheet roughness was found to be very important. A shear thinning coating rheology and higher curtain heights improved the curtain stability at high speeds. The sizing of the base sheet affected coverage and cur-tain stability because of its effect on base sheet wettability. The role of surfactant was inconclusive. Part II of this paper will report on further optimization of curtain stability with these four variables using a D-optimal partial-facto-rial design.

The Role of Hydrophobicity in the Stability and pH-Switchability of (RXDX)4 and Coumarin-RXDX Conjugate β-Sheets

2020 ◽  
Author(s):  
Ryan Weber ◽  
Martin McCullagh
Keyword(s):  
Biological Imaging ◽  
Ph Sensing ◽  
Hydrophobic Residue ◽  
Ideal Candidate ◽  
Self Assembling ◽  
Sensing Applications ◽  
Β Sheets ◽  
The Stability ◽  
Dynamics Simulations

<p>pH-switchable, self-assembling materials are of interest in biological imaging and sensing applications. Here we propose that combining the pH-switchability of RXDX (X=Ala, Val, Leu, Ile, Phe) peptides and the optical properties of coumarin creates an ideal candidate for these materials. This suggestion is tested with a thorough set of all-atom molecular dynamics simulations. We first investigate the dependence of pH-switchabiliy on the identity of the hydrophobic residue, X, in the bare (RXDX)<sub>4</sub> systems. Increasing the hydrophobicity stabilizes the fiber which, in turn, reduces the pH-switchabilty of the system. This behavior is found to be somewhat transferable to systems in which a single hydrophobic residue is replaced with a coumarin containing amino acid. In this case, conjugates with X=Ala are found to be unstable and both pHs while conjugates with X=Val, Leu, Ile and Phe are found to form stable β-sheets at least at neutral pH. The (RFDF)<sub>4</sub>-coumarin conjugate is found to have the largest relative entropy value of 0.884 +/- 0.001 between neutral and acidic coumarin ordering distributions. Thus, we posit that coumarin-(RFDF)<sub>4</sub> containing peptide sequences are ideal candidates for pH-sensing bioelectronic materials.</p>

Metal-Metal Cooperative Bond Activation by Heterobimetallic Alkyl, Aryl, and Acetylide PtII/CuI Complexes

2020 ◽  
Author(s):  
Shubham Deolka ◽  
Orestes Rivada Wheelaghan ◽  
Sandra Aristizábal ◽  
Robert Fayzullin ◽  
Shrinwantu Pal ◽  
...  
Keyword(s):  
Alkyl Group ◽  
Bond Activation ◽  
Bond Cleavage ◽  
Terminal Alkyne ◽  
Alkyl Aryl ◽  
Metal Metal ◽  
The Stability ◽  
Selective Formation ◽  
Organoplatinum Compounds

We report selective formation of heterobimetallic PtII/CuI complexes that demonstrate how facile bond activation processes can be achieved by altering reactivity of common organoplatinum compounds through their interaction with another metal center. The interaction of the Cu center with Pt center and with a Pt-bound alkyl group increases the stability of PtMe2 towards undesired rollover cyclometalation. The presence of the CuI center also enables facile transmetalation from electron-deficient tetraarylborate [B(ArF)4]- anion and mild C-H bond cleavage of a terminal alkyne, which was not observed in the absence of an electrophilic Cu center. The DFT study indicates that the role of Cu center acts as a binding site for alkyne substrate, while activating its terminal C-H bond.

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