Determining the natural frequencies of an elastic parallelepiped by the advanced Kantorovich–Vlasov method

An approach to calculate the natural frequencies of an elastic parallelepiped with different boundary conditions is proposed. The approach rationally combines the inverse-iteration method of successive approximations and the advanced Kantorovich–Vlasov method. The efficiency of the approach (the accuracy of the results and the number of approximating functions) is demonstrated against the Ritz method with different basis systems, including B-splines. The dependence of the lower frequencies of a three-dimensional cantilever beam on its cross-sectional dimensions is examined     

Special virtual fields for the direct determination of material parameters with the virtual fields method. 2––Application to in-plane properties

This paper deals with the direct identification of mechanical parameters that govern the in-plane constitutive law of orthotropic materials. Those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a Iosipescu fixture. The procedure used is the virtual fields method with special virtual fields. The case of linear elasticity is first addressed. It is shown that the parameters are directly extracted with this method: no iterative calculations are required. The stability is also discussed in different cases. A non-linear shear response is then considered. The parameter that governs this non-linearity is also directly identified with the special virtual fields.     

Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation

Nonlinear Dynamics - Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable...     

A new method to approximate the field of movements of 1-DOF linkages with lower-pairs

This paper presents a new method to obtain an approximation of the field of movements of 1-DOF linkages with lower pairs. The method is based on a linkage representation by natural coordinates and the storage of the constraint equations by means of a sparse cubic matrix. To obtain a discrete approximation of the field of movements, a three-stage process is used. In the first stage, a special Evolution Strategy is applied to make the population converges towards the zones where the constraints’ error is minimal, obtaining, at the same time, a good distribution of individuals. In the second stage, the final individuals of the ES are used as initial points for an optimization algorithm to obtain a greater accuracy. Later, the third stage is a filtering process to eliminate individuals that represent non-desired solutions. This method has been tested on some linkages with well-known fields of movements, generating comprehensive outcomes that justify the validity of the method.     

Application of digital image correlation to the study of planar anisotropy of sheet metals at large strains

The aim of this work is to measure and model the planar anisotropy of thin steel sheets. The experimental data have been collected using the digital image correlation technique. This is a powerful tool to measure the strain field on differently shaped specimens subjected to large plastic deformations. In this manner, it is possible to observe the material behaviour under different stress-strain states, from small to large deformation conditions, on the entire specimen surface. The experimental results on smooth and notched samples have been used to characterize the flow stress curve after necking and a nonassociated plastic flow rule is proposed to describe the anisotropic behaviour of the material. To compare the experimental data with the predictions of the adopted constitutive model, a novel method, based on the image correlation results, has been implemented.     

On the averaging principle for one-frequency systems. An application to satellite motions

This paper is related to our previous works (Morosi and Pizzocchero in J. Phys. A, Math. Gen. 39:3673–3702, 2006; Nonlinear Dyn., 2008), on the error estimate of the averaging technique for systems with one fast angular variable. In the cited references, a general method (of mixed analytical and numerical type) has been introduced to obtain precise, fully quantitative estimates on the averaging error. Here, this procedure is applied to the motion of a satellite in a polar orbit around an oblate planet, retaining only the J ₂ term in the multipole expansion of the gravitational potential. To exemplify the method, the averaging errors are estimated for the data corresponding to two Earth satellites; for a very large number of orbits, computation of our estimators is much less expensive than the direct numerical solution of the equations of motion.     

Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic–plastic formulation

This paper proposes an extension of the SHB8PS solid–shell finite element to large strain anisotropic elasto-plasticity, with application to several non-linear benchmark tests including sheet metal forming simulations. This hexahedral linear element has an arbitrary number of integration points distributed along a single line, defining the “thickness” direction; and to control the hourglass modes inherent to this reduced integration, a physical stabilization technique is used. In addition, the assumed strain method is adopted for the elimination of locking. The implementation of the element in Abaqus/Standard via the UEL user subroutine has been assessed through a variety of benchmark problems involving geometric non-linearities, anisotropic plasticity, large deformation and contact. Initially designed for the efficient simulation of elastic–plastic thin structures, the SHB8PS exhibits interesting potentialities for sheet metal forming applications—both in terms of efficiency and accuracy. The element shows good performance on the selected tests, including springback and earing predictions for Numisheet benchmark problems.     

Nonlinear and parametric coupled vibrations of the rotor-shaft system as fault identification symptom using stochastic methods

In the paper several stochastic methods for detection and identification of cracks in the shafts of rotating machines are proposed. All these methods are based on the Monte Carlo simulations of the rotor-shaft lateral-torsional-longitudinal vibrations mutually coupled by transverse cracks of randomly selected depths and locations on the shaft. For this purpose there is applied a structural hybrid model of a real cracked rotor-shaft. This model is characterized by a high practical reliability and great computational efficiency, so important for hundreds of thousands numerical simulations necessary to build databases used in solving the inverse problem, i.e. crack parameter identifications. In order to ensure a good identification accuracy, for creating the Monte Carlo samples of data points there are proposed special probability density functions for locations and depths of the crack. Such an approach helps in enhancing databases corresponding to the most probable faults of the rotor-shaft system of the considered rotor machine. In the presented study six different database sizes are considered to compare identification efficiency and accuracy of considered methods. A sufficiently large database enables us to estimate almost immediately (usually in less than one second) the crack parameters with precision that is in most of the cases acceptable in practice. Then, as a next stage, one of the proposed fast improvement algorithms can be applied to refine identification results in a reasonable time. The proposed methods seem to provide very convenient diagnostic tools for industrial applications.     

Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.     

Implementation of the exploding wire technique to study blast-wave–structure interaction

The effort invested in improving our understanding of the physics of high-energy explosion events has been steadily increasing since the latter part of the twentieth century. Moreover, the dramatic increase in computer power over the last two decades has made the numerical simulation approach the dominant tool for investigating blast phenomena and their effects. However, field tests, on both large and small scales, are still in use. In the current paper, we present an experimental tool to better resolve and study the blast–structure interaction phenomenon and to help validate the numerical simulations of the same. The experimental tool uses an exploding wire technique to generate small-scale cylindrical and spherical blast waves. This approach permits safe operation, high repeatability, and the use of advanced diagnostic systems. The system was calibrated using an analytical model, an empirical model, and numerical simulation. To insure that spherical blast geometry was achieved, a set of free air blast experiments was done in which high-speed photography was used to monitor the blast structure. A scenario in which an explosion occurred in the vicinity of a structure demonstrated the system’s capabilities. Using this simple but not trivial configuration showed unequivocally the effectiveness of this tool. From this comparison, it was found that at early times of blast–structure interaction, the agreement between the two sets of results was very good, but at later times incongruences appeared. Effort has been made to interpret this observation. Furthermore, by using similitude analysis, the results obtained from the small-scale experiments can be applied to the full-scale problem. We have shown that an exploding wire system offers an inexpensive, safe, easy to operate, and effective tool for studying phenomena related to blast-wave–structure interactions.     

A Ritz’s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space

In this article, Ritz’s method is used to calculate with unprecedented accuracy the displacements related to a deformable rectangular plate resting on the surface of an elastic quarter-space. To achieve this required three basic steps. The first step involved the study of Green’s function describing the vertical displacements of the surface of an elastic quarter-space due to vertical force applied on its surface. For this case, an explicit formula was obtained by analytically resolving a complicated integral that did not previously have an analytical solution. The second step involved the study of the coupled system of a plate and an elastic quarter-space. This portion focused on determining reactive forces in the contact zone based on Hetenyi’s solution. After determination of the reactive forces, certain features were attributed to the plate’s edges. The final step involved the application of Ritz’s method to determine the deflections of the plate resting on the surface of the quarter-space. Finally, an example calculation and validation of results are given. This is the first semi-analytical solution proposed for this type of contact problem.     

Determination of elastic and plastic characteristics of TiC–TiNi alloys by the ultrasonic resonance method

Elastic characteristics and propagation velocities of ultrasonic waves in a TiC–TiNi composite material are determined by the ultrasonic resonance method. The values of the elastic moduli of the solid composite obtained are used to estimate its plastic properties. The effect of various additives on the elastic and plastic properties of the composite is studied.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

Autoparametric interaction of a liquid surface in a rectangular tank with an elastic support structure under 1:1 internal resonance

Autoparametric interaction of a liquid free surface in a rectangular tank with an elastic support structure, which is subjected to vertical excitation, is investigated. When the natural frequency of the structure is equal to the lowest natural frequency of liquid sloshing, this system is categorized as an autoparametric system with an internal resonance ratio 1:1. The structure is elastically supported so there is a higher possibility that the 1:1 internal resonance can be observed. The nonlinear theoretical analysis is conducted for a fluid assumed to be perfect in a tank with a finite liquid depth. The equations of motion for the first three sloshing modes are derived employing Galerkin’s technique and considering both the nonlinearity of the fluid motion, and the viscous damping effect. Then the theoretical frequency response curves for the harmonic oscillations of the structure and sloshing are determined using van der Pol’s method. The frequency response curves show that high amplitudes of the structure’s vibrations facilitate the liquid sloshing. Furthermore, the influence of the internal detuning parameter is investigated by showing the frequency response curves and bifurcation sets. Hopf bifurcations may occur followed by amplitude-modulated motions. The theoretical results are in quantitative agreement with the experimental data.     

Eigenvibrations of curved elastic bars according to the „Method of internal constraints“

Conclusions In the present paper it has been shown how it is possible to discuss the problem of vibrations of curved bars once that the solution of the corresponding problem for the straight bar is known. The method of approach in deriving the equations of motion being completely contained in the constraint equation (5) and in the application of Hamilton Principle, while the method of solving the equations being based on a perturbation method which utilizes as first approximation the solution of the straight case.More general dynamic problems of elongated solids in which the inertia characteristics of the cross-section of the bar are known functions of the variable x can be attacked and solved by the same method which has so far been demonstrated for the simple cases of bars having constant inertia characteristics.The material in the paper is based on an investigation, conducted at the Rensselaer Polytechnic Institute, Troy, New York, under the sponsorship of the Office of Ordnance Research, Contract No. DA-30-115-ORD-709. Project No. 454.13.     

Identification of the parameters of the stress–strain problem for a multicomponent elastic body with an inclusion

Explicit expressions for residual functional gradients are derived. They are used to identify, using gradient methods, the parameters of elastic problems for multicomponent bodies. The method employs the solutions of conjugate problems in the theory (developed by the authors) of optimal control of distributed multicomponent systems