scholarly journals Analysis of Nonviscous Oscillators Based on the Damping Model Perturbation

2016 ◽  
Vol 2016 ◽  
pp. 1-19
Author(s):  
Mario Lázaro ◽  
César F. Casanova ◽  
Ignacio Ferrer ◽  
Pedro Martín
Keyword(s):  
Fractional Derivatives ◽  
Numerical Approach ◽  
Kernel Functions ◽  
Series Expansions ◽  
Convolution Integrals ◽  
Taylor Series Expansions ◽  
Linear Viscoelastic ◽  
Motion Characteristic ◽  
Independent Variable ◽  
Damping Model

A novel numerical approach to compute the eigenvalues of linear viscoelastic oscillators is developed. The dissipative forces of these systems are characterized by convolution integrals with kernel functions, which in turn contain a set of damping parameters. The free-motion characteristic equation defines implicitly the eigenvalues as functions of such parameters. After choosing one of them as independent variable, the key idea of the current paper is to obtain a differential equation whose solution can be considered, under certain conditions, a good approximation. The method is validated with several numerical examples related to damping models based on exponential kernels, on fractional derivatives, and on the well-known viscous model. Taylor series expansions up to the second order are obtained and in addition analytical solutions for the viscous model are achieved. The numerical results are very close to the exact ones for light and medium levels of damping and also very good for high levels if the chosen parameter is close to initial values that are defined for every case.

Modal Analysis of Nonviscously Damped Beams

2006 ◽  
Vol 74 (5) ◽  
pp. 1026-1030 ◽  
Author(s):  
S. Adhikari ◽  
M. I. Friswell ◽  
Y. Lei
Keyword(s):  
Natural Frequencies ◽  
Past History ◽  
Kernel Functions ◽  
Mode Shapes ◽  
Damping Force ◽  
The Past ◽  
Convolution Integrals ◽  
Special Cases ◽  
History Of ◽  
Damping Model

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Characterization of Real Eigenvalues in Linear Viscoelastic Oscillators and the Nonviscous Set

2013 ◽  
Vol 81 (2) ◽  
Author(s):  
Mario Lázaro ◽  
José L. Pérez-Aparicio
Keyword(s):  
Time History ◽  
Negative Numbers ◽  
Closed Intervals ◽  
Convolution Integrals ◽  
History Of ◽  
Linear Viscoelastic ◽  
One Step ◽  
Damping Model ◽  
Theoretical Results

In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, which is mathematically represented by convolution integrals involving hereditary functions. The widespread Biot damping model assumes that such functions are exponential kernels, which modify the eigenvalues' set so that as many real eigenvalues (named nonviscous eigenvalues) as kernels are added to the system. This paper is focused on the study of a mathematical characterization of the nonviscous eigenvalues. The theoretical results allow the bounding of a set belonging to the real negative numbers, called the nonviscous set, constructed as the union of closed intervals. Exact analytical solutions of the nonviscous set for one and two exponential kernels and approximated solutions for the general case of N kernels are developed. In addition, the nonviscous set is used to build closed-form expressions to compute the nonviscous eigenvalues. The results are validated with numerical examples covering single and multiple degree-of-freedom systems where the proposed method is compared with other existing one-step approaches available in the literature.

scholarly journals A Shortcut to LAD Estimator Asymptotics

1991 ◽  
Vol 7 (4) ◽  
pp. 450-463 ◽  
Author(s):  
P.C.B. Phillips
Keyword(s):  
Asymptotic Expansion ◽  
Regression Model ◽  
Taylor Series ◽  
Asymptotic Theory ◽  
Random Variables ◽  
Generalized Functions ◽  
Infinite Variance ◽  
Series Expansions ◽  
Linear Functionals ◽  
Taylor Series Expansions

Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

Direct parameterization of the pressure-dependent volume by using an inverted approximate Vinet equation of state

2014 ◽  
Vol 47 (1) ◽  
pp. 384-390 ◽  
Author(s):  
Martin Etter ◽  
Robert E. Dinnebier
Keyword(s):  
High Pressure ◽  
Equation Of State ◽  
Diffraction Data ◽  
Series Expansions ◽  
Third Order ◽  
Wide Range ◽  
Taylor Series Expansions ◽  
Simultaneous Modeling ◽  
Inverse Functions ◽  
Independent Parameters

Parametric refinement is used for the simultaneous modeling of a series of diffraction data, replacing single independent parameters with physical or empirical equations that are valid for the full sequence of data. For the parametric treatment of diffraction data at high pressure, pressure-dependent constraints can be introduced in the form of an equation of state (EoS). However, the parameterization needs inverse functions of the EoS and most of them are not analytically invertible. In order to overcome this drawback, Taylor series expansions of different orders of the Vinet EoS were calculated and analytically inverted. It is shown that the inverted third-order Vinet EoS approximation, in its volume and linearized version, is applicable to a wide range of materials under high pressure.

scholarly journals A Method for the Generation of Various Ghost Correction Algorithms—the Example of the Positivity Method and the Exponential Method

1991 ◽  
Vol 13 (4) ◽  
pp. 199-212 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Pole Figure ◽  
Taylor Series ◽  
The Other ◽  
New Method ◽  
Series Expansions ◽  
Theoretical Scheme ◽  
Taylor Series Expansions ◽  
Exponential Method ◽  
Pole Figure Data

A theoretical strategy is presented that can derive the algorithms of several existing ghost correction methods. The examples of the positivity method and the “GHOST” method are elaborated. A new method is derived as well: the “exponential” method. It can successfully replace the quadratic method as a method that yields an exactly non-negative complete C.O.D.F. from pole figure data. The theoretical scheme that can generate all these algorithms makes use of the fact, that several parameter sets can be defined in order to describe a C.O.D.F. The parameters of one set are then functions of those of the other. The algorithms are derived from Taylor series expansions of these functions.

Estimating the ‘In-Service’ Modulus of Elasticity and Length of Polyester Mooring Lines via a Non Linear Viscoelastic Model Governed by Fractional Derivatives

2012 ◽  
Author(s):  
Georgios I. Evangelatos ◽  
Pol D. Spanos
Keyword(s):  
Modulus Of Elasticity ◽  
Field Data ◽  
Fractional Derivatives ◽  
Viscoelastic Model ◽  
Mooring Lines ◽  
Static Modulus ◽  
Proposed Model ◽  
Non Linear ◽  
Linear Viscoelastic ◽  
Linear Viscoelastic Model

In this paper a non linear viscoelastic model governed by fractional derivatives is presented for modeling the in-service behavior of polyester mooring lines. In the formulation an iterative approach utilizing the Gauss-Newton minimization algorithm in conjunction with the catenary equations used to determine the static modulus of elasticity and the effective length of polyester mooring lines corresponding to calm sea conditions. Upon establishing the accuracy of the static modulus via comparison with field data, the catenary equations and the offshore platform’s position versus time are used to identify the polyester strain under developed-sea conditions. In this manner, time histories of stress and strain for polyester ropes in service conditions are obtained. Then, a non linear viscoelastic model involving fractional derivative terms is used to capture the in service polyester line behavior. For this, the tension of the proposed model corresponding to the actual polyester strain is compared at each time step to the tension obtained from the field data. Finally, the parameters of the proposed model are derived by minimizing the error in the least-squares sense over a large number of data points using the Levenberg-Marquardt algorithm. The numerically derived force-strain relationship is found to be in reasonable agreement with supplementary field and laboratory experimental data, the field data pertain to an offshore structure moored in position using polyester mooring lines operated in the Gulf of Mexico during Hurricane Katrina (August of 2005).

scholarly journals Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

2019 ◽  
Vol 3 (2) ◽  
pp. 27 ◽  
Author(s):  
Ayşegül Keten ◽  
Mehmet Yavuz ◽  
Dumitru Baleanu
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Kernel Functions ◽  
Banach Contraction ◽  
Exponential Decay Law ◽  
Nonlocal Cauchy Problem ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

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