Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

Nonlinear Dynamics - In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF...     

One-dimensional nonlinear induced dynamics of acoustic waves in a finite three-dimensional domain

The problem of the time-dependent viscous compressible gas flow excited by a small external time-dependent space-and time-periodic force is considered within the framework of the Navier-Stokes equations on a finite interval with periodic boundary conditions. The investigation is carried out numerically for a periodicity interval L, divided by the viscous length, from 10² to 2 × 10³ and external force amplitudes from 10^?4 to 0.1. The nonlinear dynamics of the wave processes are investigated within the framework of this problem. It is shown that nonlinear steady-state oscillations with sharp variation of the quantities in space and time develop when L is greater than or of the order of 10³. This leads to the onset of a continuous spectrum.     

Effects of inhomogeneity of flow in nonlinear nonstationary filtration

A method of numerical solution of two-dimensional nonstationary problems of the theory of elastic regime in nonlinear filtration is developed. The nonstationary processes in an element of five-point grid of wells and in the neighborhood of a well, which occur in the external flow, are investigated. The effects of anisotropy of propagation of perturbations related to the nonlinearity of the problem is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 74–78, November–December, 1977.The authors are grateful to L. A. Chudov for advice and constant attention to the work.     

Stationary mode of a nonlinear elastically hereditary oscillator

The steady-state forced oscillations of a single-mass system subject to an external pure harmonic force are considered. The role of restoring force is played by a nonlinear function, which takes account of hereditary effects as a sum of multiple integrals in accordance with the Volterra theory. The problem is solved by the method of equivalent linearization, the discussion being confined to a triple integral of the hereditary type. The influence of the hereditary nonlinearity on the system dynamic characteristics, namely, its amplitude, phase, dynamic rigidity, hysteresis loop area, and Q, is investigated. In particular, the reciprocal of the Q, which can be taken as a measure of the internal friction, is shown to be independent of the amplitude of oscillation and to be the same as that obtained by linear theory. The other dynamic characteristics prove sensitive to the nonlinear properties. The exponential rational fractions proposed by Yu. N. Rabotnov are used as concrete hereditary functions.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 111–116, May–June, 1970.     

Stationary waves in a bimodular rod of finite radius

Theoretical study of the propagation of longitudinal elastic waves in a bimodular rod of finite radius with dispersion is carried out. The exact analytical solutions are derived for the stationary waves propagating in such a rod without changing in their shape.     

Stationary variational estimates for the effective response and field fluctuations in nonlinear composites

This paper presents a variational method for estimating the effective constitutive response of composite materials with nonlinear constitutive behavior. The method is based on a stationary variational principle for the macroscopic potential in terms of the corresponding potential of a linear comparison composite (LCC) whose properties are the trial fields in the variational principle. When used in combination with estimates for the LCC that are exact to second order in the heterogeneity contrast, the resulting estimates for the nonlinear composite are also guaranteed to be exact to second-order in the contrast. In addition, the new method allows full optimization with respect to the properties of the LCC, leading to estimates that are fully stationary and exhibit no duality gaps. As a result, the effective response and field statistics of the nonlinear composite can be estimated directly from the appropriately optimized linear comparison composite. By way of illustration, the method is applied to a porous, isotropic, power-law material, and the results are found to compare favorably with earlier bounds and estimates. However, the basic ideas of the method are expected to work for broad classes of composites materials, whose effective response can be given appropriate variational representations, including more general elasto-plastic and soft hyperelastic composites and polycrystals.     

Numerical simulation of the dynamics of a reinforced shell subject to nonstationary load

A numerical algorithm to determine the stress-strain state of a reinforced cylindrical shell subject to impulsive loading is elaborated. Results from a numerical analysis of the dynamic behavior of a reinforced elastoplastic shell under explosive loading are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 83–90, July 2008.     

Calculation of nonstationary drainage in the case of a finite number of closed horizontal drains

In the hydrodynamic formulation, a study is made of nonstationary drainage in a water-bearing stratum in the presence of a finite number of closed drains and variable infiltration. The drains are simulated by linear sinks with known time-dependent flow rates per unit length of the drains. The dependences found in the paper enable one to predict, and if necessary also regulate the dynamics of the level of ground water in a drained region, which is of practical interest in the study of salinization and swamping of land.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 165–170, July–August, 1979.     

Linear and nonlinear dynamics of reciprocating engines

In the first part of this study, dynamic models of single- and multi-cylinder reciprocating machines are presented, which may involve torsional flexibility in the crankshaft. These models take into account the dependence of the engine moment of inertia on the crankshaft rotation. In addition, both the driving and the resisting moments are expressed as functions of the crankshaft motion. This leads to dynamic models with equations of motion appearing in a strongly nonlinear form. Initially, an approximate analytical solution is presented for a linearized version of the equations of motion, by applying a suitable asymptotic method. This provides valuable insight into some aspects of the system dynamics. In the second part, numerical results are presented for linear and nonlinear engine models by applying appropriate methodologies, leading to direct determination of complete branches of steady-state response. These results illustrate the effect of the system parameters on its dynamics. Finally, some results are also presented in an effort to investigate the effect of engine misfire.     

Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations

This paper studies a low order mixed finite element method (FEM) for nonstationary incompressible Navier-Stokes equations. The velocity and pressure are approximated by the nonconforming constrained Q ₁ ^rot element and the piecewise constant, respectively. The superconvergent error estimates of the velocity in the broken H ¹-norm and the pressure in the L ²-norm are obtained respectively when the exact solutions are reasonably smooth. A numerical experiment is carried out to confirm the theoretical results.     

Stationary points iteration method for periodic solution to nonlinear system

The value method which is used to obtain the periodic solution to nonlinear system is mentioned in this article. Different point reflection is defined in the nonlinear autonomous and nonautonomous system firstly and then that linear reflection obtained from the inserting value of nonlinear reflection is asymptotic to original nonlinear reflection. The stationary points obtained by linear reflection are regarded as the asymptotic solution of the stationary points of original system. If this asymptotic solution of the stationary points is not satisfactorily accurate it can be used as the initial point of the next reflection. In addition, a corresponding method of researching the stability of periodic solution is put forward in this article.     

On the dynamics of randomly excited nonlinear systems

Summary The problem of characterising the dynamics of randomly excited systems is examined. It is shown that the probability approach, though conceptually more rigorous, is difficult to apply to statistics other than normal ones. The direct method of Axelby is recalled and applied to a nonlinear system with random excitation characterised by a statistic of great interest in real physical systems. The application is developed parametrically with reference to a second order system for which the calculations are developed and the quantitative results discussed.

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Sommario Nella disamina del problema di caratterizzare il comportamento dinamico di sistemi eccitati da segnali casuali, si mette in rilievo come l'approccio probabilistico, sebbene concettualmente più rigoroso, sia di difficile applicazione ai casi statistici oltre che a quelli normali. Si richiama quindi il metodo diretto di Axelby che viene applicato ad un sistema non lineare eccitato da un segnale casuale. Lo sviluppo dell'applicazione, in termini parametrici, fa riferimento ad un sistema di secondo grado e i relativi risultati numerici vengono discussi nel loro significato quantitativo.

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