scholarly journals Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Meccanica ◽  
2021 ◽  
Author(s):  
Jan Freundlich
Keyword(s):  
Fractional Derivative ◽  
Cantilever Beam ◽  
Viscoelastic Material ◽  
Material Model ◽  
Mixed System ◽  
Zener Model ◽  
Time Histories ◽  
Fractional Differential ◽  
Fractional Zener Model ◽  
Tip Mass

AbstractThe presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the fractional Zener model of the beam material are assumed. A fractional Caputo derivative is used to formulate a viscoelastic material law. A characteristic equation, modal frequencies, eigenfunction and orthogonality conditions are achieved for the beam considered. The equations of motion of the system are solved numerically. A numerical solution of a multi-term fractional differential equation is obtained by means of a conversion to a mixed system of ordinary and fractional differential equations, each of the order of $$0 < \gamma \le 1$$ 0 < γ ≤ 1 . The transient time histories of the beam vibrations during the passage through resonance are calculated. A comparison between the beam responses obtained with a fractional and an integer viscoelastic material model is presented. The calculations performed reveal that use of the fractional damping affects on the time histories of the system. The calculated beam responses show that for some values of the order of the fractional derivative $$\gamma$$ γ , the amplitudes occurring in the area of the second resonance are greater than those obtained in the area of the first resonance, which does not occur in the case of the integer order of the fractional derivative. Moreover, an evaluation is made of the difference between the results obtained for the calculations using the fractional Zener model and the fractional Kelvin model. It is shown that for some physical beam parameters, the calculation results obtained using both models are virtually the same for both models, which means that the the simpler, fractional Kelvin–Voigt material can be used instead of the fractional Zener material model. This simplifies the solution and decreases the time needed to make the numerical calculations.

scholarly journals Free vibration of frame structures made of Zener type viscoelastic material

2019 ◽  
Vol 285 ◽  
pp. 00009
Author(s):  
Roman Lewandowski ◽  
Przemysław Wielentejczyk
Keyword(s):  
Rheological Properties ◽  
Viscoelastic Material ◽  
Dynamic Characteristics ◽  
Frame Structures ◽  
Response Functions ◽  
Zener Model ◽  
Linear Eigenvalue Problem ◽  
Viscoelastic Structure ◽  
Properties Of Materials ◽  
Fractional Zener Model

A method for determining the dynamic characteristics of structures made of viscoelastic material is presented. The fractional Zener model is used to the describe the rheological properties of materials. All of the elements of a structure must be built of material with identical rheological properties. The solution to the linear eigenvalue problem for some elastic structure and the solution to a single nonlinear algebraic equation are needed to obtain the dynamic characteristics of a viscoelastic structure. Moreover, the frequency response functions are determined in a very efficient way. The results of a representative calculation are presented and briefly discussed.

scholarly journals Identification of the Fractional Zener Model Parameters for a Viscoelastic Material over a Wide Range of Frequencies and Temperatures

Materials ◽  
2021 ◽  
Vol 14 (22) ◽  
pp. 7024
Author(s):  
Zdzisław M. Pawlak ◽  
Arkadiusz Denisiewicz
Keyword(s):  
Viscoelastic Material ◽  
Shift Factor ◽  
Model Parameters ◽  
Master Curves ◽  
Zener Model ◽  
Horizontal Shift ◽  
Wide Range ◽  
Different Temperatures ◽  
Feasible Solutions ◽  
Fractional Zener Model

The paper presents an analysis of the rheological properties of a selected viscoelastic material, which is dedicated to the reduction of vibrations in structures subjected to dynamic loads. A four-parameter, fractional Zener model was used to describe the dynamic behavior of the tested material. The model parameters were identified on the basis of laboratory tests performed at different temperatures and for different vibration frequencies. After proving that the material is thermoreologically simple, the so-called master curves were created using a horizontal shift factor. The Williams–Landel–Ferry formula was applied to create graphs of the master curves, the constants of which were determined for the selected temperature. The resulting storage and loss module functions spanned several decades in the frequency domain. The parameters of the fractional Zener model were identified by fitting the entire range of the master curves with the gradientless method (i.e., Particle Swarm Optimization), consisting in searching for the best-fitted solution in a set of feasible solutions. The parametric analysis of the obtained solutions allowed for the formulation of conclusions regarding the effectiveness of the applied rheological model.

scholarly journals Vibrations of a Simply Supported Beam with a Fractional Viscoelastic Material Model – Supports Movement Excitation

2013 ◽  
Vol 20 (6) ◽  
pp. 1103-1112 ◽  
Author(s):  
Jan Freundlich
Keyword(s):  
Fractional Derivative ◽  
Viscoelastic Material ◽  
Material Model ◽  
Model Parameters ◽  
Beam Model ◽  
Euler Beam ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Liouville Fractional Derivative ◽  
Damping Model

The paper presents vibration analysis of a simply supported beam with a fractional order viscoelastic material model. The Bernoulli-Euler beam model is considered. The beam is excited by the supports movement. The Riemann – Liouville fractional derivative of order 0 α &les; 1 is applied. In the first stage, the steady-state vibrations of the beam are analyzed and therefore the Riemann – Liouville fractional derivative with lower terminal at −∞ is assumed. This assumption simplifies solution of the fractional differential equations and enables us to directly obtain amplitude-frequency characteristics of the examined system. The characteristics are obtained for various values of fractional derivative of order α and values of the Voigt material model parameters. The studies show that the selection of appropriate damping coefficients and fractional derivative order of damping model enables us to fit more accurately dynamic characteristic of the beam in comparison with using integer order derivative damping model.

scholarly journals On Λ-Fractional Viscoelastic Models

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 22
Author(s):  
Anastassios K. Lazopoulos ◽  
Dimitrios Karaoulanis
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Present Article ◽  
Fractional Differential Equations ◽  
Zener Model ◽  
Viscoelastic Models ◽  
Non Local ◽  
Fractional Differential ◽  
Local Derivative

Λ-Fractional Derivative (Λ-FD) is a new groundbreaking Fractional Derivative (FD) introduced recently in mechanics. This derivative, along with Λ-Transform (Λ-T), provides a reliable alternative to fractional differential equations’ current solving. To put it straightforwardly, Λ-Fractional Derivative might be the only authentic non-local derivative that exists. In the present article, Λ-Fractional Derivative is used to describe the phenomenon of viscoelasticity, while the whole methodology is demonstrated meticulously. The fractional viscoelastic Zener model is studied, for relaxation as well as for creep. Interesting results are extracted and compared to other methodologies showing the value of the pre-mentioned method.

Wave propagation dynamics in a fractional Zener model with stochastic excitation

2020 ◽  
Vol 23 (6) ◽  
pp. 1570-1604
Author(s):  
Teodor Atanacković ◽  
Stevan Pilipović ◽  
Dora Seleši
Keyword(s):  
Wave Propagation ◽  
Constitutive Equation ◽  
Equations Of Motion ◽  
Weak Form ◽  
Stochastic Excitation ◽  
Expansion Theorem ◽  
Zener Model ◽  
Dissipation Inequality ◽  
Regularity Properties ◽  
Fractional Zener Model

Abstract Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

scholarly journals Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

Sign in / Sign up

Export Citation Format

Share Document