Chaotic vibrations of a beam with non-linear boundary conditions

Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.     

Longitudinal vibrations of a beam with a moving load

International Applied Mechanics -     

Vertical vibrations of a rigid edge inclusion under a harmonic load

The paper considers the problem of vibrations of a rigid edge inclusion, which lies in an elastic half-plane and emerges on the surface perpendicular to that half-plane. The vibrations are initiated by a harmonic force acting on the end of the inclusion, which emerges on the surface. The field of translations in the half-plane is shown to be represented by the superposition of two discontinuous solutions with discontinuities at the boundary between the half-plane and the line of the inclusion. The unknown discontinuities are determined from the boundary conditions and the conditions of the inclusion-medium interaction. The problem is thus reduced to one of solving a singular integral equation with an immobile singularity for the jump in shear stresses on the line of the inclusion. The equation obtained is solved numerically by the method of mechanical quadratures. The amplitudes of the inclusion vibrations and the stressed state of the medium near it are studied.Odessa State Marine Academy, Odessa, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 31, No. 7, pp. 46–55, July, 1995.     

Non-local free and forced vibrations of graded nanobeams resting on a non-linear elastic foundation

We consider in this paper the free and forced vibration response of simply-supported functionally graded (FG) nanobeams resting on a non-linear elastic foundation. The two-constituent Functionally Graded Material (FGM) is assumed to follow a power-law distribution through the beam thickness. Eringen׳s non-local elasticity model with material length scales is used in conjunction with the Euler–Bernoulli beam theory with von Kármán geometric non-linearity that accounts for moderate rotations. Non-linear natural frequencies of non-local FG nanobeams are obtained using He׳s Variational Iteration Method (VIM) and the direct and discretized Method of Multiple Scales (MMS), while the primary resonance analysis of an externally forced non-local FG nanobeam is performed only using the MMS. The effects of the non-local parameter, power-law index, and the parameters of the non-linear elastic foundation on the non-linear frequency-response are investigated.     

Nonlinear forced vibrations of a cylindrical shell with two internal resonances

Forced vibrations of cylindrical shells described by a system of three ordinary differential equations are studied. There are two internal resonances. Standing and traveling waves in the shells are described by a system of six modulation equations derived using the multiple-scales method. These waves are analyzed for stability __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 51–58, February 2006.     

Analysis of forced vibrations by nonlinear modes

The combination of Rausher method and nonlinear modes is suggested to analyze the forced vibrations of nonlinear discrete systems. The basis of the Rausher method is iterative procedure. In this case, the analysis of a nonautonomous dynamical system reduces to the multiple solutions of the autonomous ones. As an example, the forced vibrations of shallow arch close to equilibrium position are considered in this paper. The results of the analysis are shown on the frequency response.     

Flapping flight produced by high-frequency forced vibrations of a wing

Forced vibrations of an elastic wing are investigated. The main attention is devoted to the solution of the problem for Strouhal number k 1. It is shown that an elastic wing makes it possible to obtain high efficiencies and high thrust coefficients in a wide range variation of the Strouhal number. The maximum of the thrust is usually achieved near resonance, while the maximum of the efficiency is achieved in a different regime, which is determined by the parameters of the vibrating system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 154–158, March–April, 1984.     

Steady-state vibrations of an elastic beam on a visco-elastic layer under moving load

Summary  The steady-state response of an elastic beam on a visco-elastic layer to a uniformly moving constant load is investigated. As a method of investigation the concept of “equivalent stiffness” of the layer is used. According to this concept, the layer is replaced by a 1D continuous foundation with a complex stiffness, which depends on the frequency and the wave number of the bending waves in the beam. This stiffness is analyzed as a function of the phase velocity of the waves. It is shown that the real part of the stiffness decreases severely as the phase velocity tends to a critical value, a value determined by the lowest dispersion branch of the layer. As the phase velocity exceeds the critical value, the imaginary part of the equivalent stiffness grows substantially. The dispersion relation for bending waves in the beam is studied to analyze the effect of the layer depth on the critical (resonance) velocity of the load. It is shown that the critical velocity is in the order of the Rayleigh wave velocity. The smaller the layer depth, the higher the critical velocity. The effect of viscosity in the layer on the resonance vibrations is studied. It is shown that the deeper the layer, the smaller this effect. Received 22 March 1999; accepted 26 July 1999     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Subharmonic steady vibrations of a non-linear damper with two degrees of freedom and viscous damping

The steady-state. $\text{1}\text{3}$ subharmonic vibrations of a dynamic damper (or vibration absorber) with two degrees of freedom, sinusoidal forcing function and internal viscous damping. are presented. The study of these oscillations leads to the determination of suitable “form functions” of the solutions, by following a methodology recently introduced by Nocilla for studying the harmonic vibrations of non-linear systems with one and two degrees of freedom. The proposed theory. which is valid even if the non-linearity is large, gives satisfactory results in all the cases in which the subharmonic component is predominant in the steady-state oscillation of the system.     

Free vibrations of a beam with elastic end restraints subject to a constant axial load

A general model for vibration of beams restrained with two transversal and two rotational elastic springs subject to a constant axially load is presented. The frequency equations and the shape functions are derived analytically. The proposed model can be employed for simulating the dynamic responses of elastically supported beams in tension or compression for most classical boundary conditions. Some simplifications in the degenerate cases are deduced to evaluate the effectiveness of the model. Numeric examples are given for engineering applications. This model unifies most of the previous vibration models and provides a convenient tool for the analyses of various beam vibrations in tension and compression conditions.     

Chaotic frictional vibrations excited by a quasiperiodic load

A Duffing oscillator frictionally interacting with a moving belt under a quasiperiodic load is studied. The multiple-scales method is used to derive a system of two nonautonomous equations with small parameters, which describes the modulation of vibrations. It is shown that the system of modulation equations has a heteroclinic structure. Melnikov functions are used to analyze the domain of heteroclinic chaos __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 127–133, September 2006.     

Free and forced vibrations of graphene platelets reinforced composite laminated arches subjected to moving load

Meccanica - In the present investigation, free vibration and also forced vibration response of a graphene platelet reinforced composite (GPLRC) laminated curved beam is investigated. It is assumed...     

On the non-linear vibrations of a circular membrane

Free axisymmetric vibrations of a stretched circular membrane are studied using a membrane theory consisting of a pair of non-linear partial differential equations coupled between the transverse and radial displacements of the membrane. A systematic perturbation method, in which the amplitude of the transverse displacement is taken as the perturbation parameter, is used to obtain periodic solutions of the non-linear equations. The initial membrane strain enters the problem as a parameter which is allowed to vary over a range of values. A case of self-resonance is encountered when the initial membrane strain approaches some critical values. This self-resonance case is also treated through an appropriate modification of the perturbation method.