Non-linear vibrations of beams and membranes

Summary Non-linear vibrations of beams and membranes are considered using a modified perturbation method based on the Linstedt-Poincare technique, which avoids the secular terms.

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Zusammenfassung Mittels einer abgeänderten Störungsmethode, welche auf der Linstedt-Poincaré-Technik basiert und die säkularen Glieder vermeidet, werden nichtlineare Schwingungen von Trägern und Membranen untersucht.

     

Non-linear axial vibrations of a cylindrical liquid column in zero-gravity

The non-linear response of an axially excited cylindrical liquid column consisting of frictionless liquid has been investigated for harmonic excitation with the excitation amplitude¯z ₀ and the forcing frequency . Free non-linear surface displacement-response has been determined and shows softening liquid behavior. The results shall be used for the efficient planning of the LICORE (liquid columns' resonances) experiments in the second German Spacelab mission D-2.

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Zusammenfassung Es wurde das nichtlineare Antwortverhalten einer harmonisch axial angeregten zylindrischen Flüssigkeitssäule behandelt. Dabei wurde die Flüssigkeit als reibungsfrei betrachtet und mit einer Anregungsamplitude¯z ₀ und der erzwungenen Frequenz erregt. Die Vergröeßrungsfunktionen der nichtlinearen freien Oberflächenauslenkung wurde bestimmt und unterlineares Verhalten festgestellt. Die Ergebnisse sind für die erfolgreiche Planung des LICORE (liquid columns' resonances)-Experiment der Zweiten Deutschen Spacelab-Mission D-2 von Bedeutung.

     

The vibrations of a free and loaded tyre

Using the model of a wheel with a reinforced tyre [Vil’ke VG, Kozhevnikov IF. A model of a wheel with a reinforced tyre. Vestnik MGU. Ser.1. Matematika Mekhanika 2004;4:37–45], the natural frequencies and natural forms of vibrations of a free or loaded tyre in the neighbourhood of the equilibrium position are determined. The spectrum of natural frequencies and natural forms of vibration are found analytically for a free (unloaded) tyre with a fixed disc. A similar problem is solved numerically in the case of a loaded tyre. The results of this analysis can be used to estimate the level of noise which occurs when a vehicle moves on an uneven surface.     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

A new approach to analysis of free beam vibrations

Based on the linear theory of elasticity and the method of integrodifferential relations a countable system of ordinary differential equations is derived to describe longitudinal and lateral free vibrations of rectilinear beams. In the approach under consideration the stress-strain relation is specified by an integral equality instead of the local Hooke's law. After expansion of unknown stress and displacement functions in polynomial series with respect to transversal Cartesian coordinates and system decomposition due to domain symmetry the consistent boundary value problems are solved. The influence of geometrical and elastic characteristics of rectilinear beams on eigenfrequencies is investigated. The various kinds of longitudinal and lateral free motions are found out in the models under study. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear second-order periodic systems with non-smooth potential

In this paper we study second order non-linear periodic systems driven by the ordinary vectorp-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by thep-Laplacian. In the last section of the paper we examine the scalar non-linear and semilinear problem. Our approach uses a generalized Landesman-Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue.     

Theory of nonlinear vibrations of free and loaded polymer molecules

The contributions of three- and four-phonon processes to the halfwidth of the valence vibration band of the linear chain are numerically estimated. It is concluded that the experimentally observed halfwidths of the skeletal vibration bands of polymers are attributable to anharmonic phonon interactions. The effect of chain loading on the halfwidth is also investigated.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 579–585, July–August, 1971.     

Analysis of free damped vibrations of laminated composite cylindrical shells

Damped free vibrations of multilayered composite cylindrical shells are investigated. Vibration and damping analysis of cylindrical shells is performed by using the first-order shear deformation theory (FOSDT). Based on other researchers' works, two damping models are developed, i.e., the energy method (EM), and the method of complex eigenvalues (MCE). Several numerical examples of the damped free vibration problem of laminated composite cylindrical shells have been solved and comparison has been made with the results of other authors.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 646–659, September–October, 1995.     

Spatially localized free vibrations for a non-linear string model

We prove the existence of infinitely many non-zero time-periodic solutions (breathers) to the dispersive wave equation of the form which are localized in the spatial variable, that is The main tool employed is the concentration compactness principle of P. L. Lions.     

Mixed finite element models for free vibrations of thin-walled beams

Simple mixed finite element models are developed for the free vibration analysis of curved thin-walled beams with arbitrary open cross section. The analytical formulation is based on a Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling, and the additional effects of transverse shear deformation and rotary inertia. The fundamental unknowns consist of seven internal forces and seven generalized displacements of the beam. The element characteristic arrays are obtained by using a perturbed Lagrangian-mixed variational principle. Only C⁰ continuity is required for the generalized displacements. The internal forces and the Lagrange multiplier are allowed to be discontinuous at interelement boundaries.

Numerical results are presented to demonstrate the high accuracy and effectiveness of the elements developed. The standard of comparison is taken to be the solutions obtained by using two-dimensional plate/shell models for the beams.     

Features of the construction of correction functions in the analysis of the free vibrations of mechanical systems by Bubnov''s method

Free vibrations of mechanical systems are investigated by Bubnov's method, using coordinate functions that do not satisfy all the boundary conditions of the problem. To eliminate discrepancies in the boundary conditions, correction functions are introduced, orthogonal to all coordinate functions involved in the analysis of the natural vibrations of mechanical systems. This method of constructing correction functions reveals the existence of differences between nearly equal numbers in the frequency equation and in the expressions for the vibrational modes. Moreover, for a given number N of coordinate functions, the method guarantees increased precision in computing all the natural frequencies of the mechanical system up to and including the (N + 1)th partial frequency as the number of terms of the power series in the frequency parameter representing the correction functions is increased. In the limit, as the number of terms of the power series tends to infinity, one is assured of obtaining exact values of the frequencies and vibrational modes in the indicated frequency range.     

Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Numerical solution of the plate bending and free vibrations problems

The problem of the bending and free vibrations of a clamped and edge-supported plate is considered. The proposed algorithm is the algorithm described in /1/, made specific for the case of the biharmonic equation. It does not have saturation /2/, i.e., its accuracy will be the higher, the smoother the solution. The program is constructed in such a manner that if the plate boundary is sufficiently smooth and given parametrically, then several of the first eigenvalues can be calculated and the bending problem can be solved. An illustration is presented of the eigenfrequency computation for an edge-supported plate whose boundary (an epitrochoid) has a curvature of the order of 10³ at twelve points (the curvatures enter explicitly in the appropriate boundary condition). The first eigenfrequencies are calculated with 7–8 places after the decimal point. The solution is obtained because of the accurate method of discretization and the study of the structure of the appropriate finitedimensional problem. This would permit execution of computations with a large number of points (up to 1230). A comparison is given with the results of computations of other authors for a circle and an ellipse /3–5/.     

Analysis of vibrations in large flexible hybrid systems

The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions makes it impossible to find exact analytical solutions. To address the problem, we use the asymptotical method of small parameters together with the numerical method of normal fundamental systems of solutions. These methods allow us to investigate vibrations, and a technique for determination of complex eigenvalues of the considered boundary value problem is developed. The conditions, at which vibration processes of different characteristics take place, are defined. The dependence of the vibration frequencies on the physical parameters of the hybrid system is studied. We show that introduction of different feedbacks into the system allows one to control the frequency spectrum, in which excitation of vibrations is possible.     

Theoretical-experimental investigation of frequencies of free vibrations of circular cylindrical shells

We consider a holographic interferometry technique for the determination of frequencies of free vibrations of isotropic circular cylindrical shells and compare the experimental results with results of calculation of frequencies of these shells on the basis of the spline collocation technique.