Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Effect of nonlinear constitutive equation on vibrations of beams

Summary This paper is concerned with the analytical investigation of static and dynamic nonlinear behaviors of beams with different boundary conditions. While geometric type of nonlinearities on beams have been investigated extensively, material type nonlinearities have received very little attention. Therefore, material nonlinearities of the Ramberg-Osgood type are considered in this analysis. The use of Self-Generating functions for nonlinear beam problems is demonstrated for this type of nonlinearity. Transverse shear and rotatory inertia effects have been included in the formulation to study moderately thick beams. For all the cases investigated here nonlinear frequency ratios are calculated at various amplitudes of vibration and geometric parameters of beams. Numerical results indicate that the Ramberg-Osgood type nonlinearity produces softening-type responses. The study is limited to materials which are nonlinearly elastic and the effect of geometric nonlinearity is not considered in this paper.

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Sommario Questo lavoro riguarda lo studio dei comportamenti non lineari statici e dinamici di travi con diverse condizioni a contorno. Mentre le non linearità di tipo geometrico sono state studiate estesamente, quelle di tipo non lineare hanno ricevuto un'attenzione molto ridotta. Perciò in questa ultima analisi si considerano non linearità materiali del tipo di Ramberg-Osgood. Si dimostra uso delle funzioni autogeneratrici nei problemi non lineari per le travi con questo tipo di non linearità. Nella formulazione dello studio di travi moderatamente spesse si sono inclusi effetti di taglio e inerzia rotatoria. Per tutti i casi qui studiati si calcolano i rapporti di frequenza non lineare per varie ampiezze di vibrazione e parametri geometrici delle travi. I risultati numerici indicano che la non linearità del tipo di Ramberg-Osgood produce risposte del tipo ammorbidimento. Lo studio si limita a materiali con nonlinearità elastica e non si considera nel lavoro l'effetto della non linearitá geometrica.

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List of symbols T _s Transverse shear - k ₁ Shear connection factor - A, B, m Material constants - I Moment of inertia - G Shear modulus - Mass per unit length - w Lateral displacement - h Beam thickness - b Bredth of beam - t Time - R _i Rotatory inertia - a Area of cross-section of beam - q(x) Lateral load on beam - _x Stress - _x Strain - Length of beam - x Beam coordinate - - r Radius of gyration - w Nondimensional maximum deflectionw _max/r - q ₀ ^* Nondimensional load, (q ₀ ³/Al - Thickness parameter,h/ - (T _sK₁/Ga) - ₀ Linear frequency - Nonlinear frequency - w _max w measured at the point of maximum deflection.     

Effect of nonhomogenous material properties on transverse vibrations of elastic cantilevers

In this paper, the method of the influence functions and the method of partial discretization are proposed to solve the boundary-value problem of free transverse vibrations of a nonhomogenous cantilever with a concentrated mass attached to the free end. In order to demonstrate the possibilities of the methods, the case of a cantilever in the form of a sharp cone, a frustum of a cone, and a linear wedge made of two different materials is treated in detail. The general characteristic equations which allow one to take into consideration the nonhomogeneous material properties and the cross-sectional geometry of cantilever are introduced. The expressions for the first three terms of the characteristic series are obtained in closed form using the method of Cauchy influence functions. The results of calculations of the first two frequencies of free transverse vibrations are presented for selected material combinations and various cantilever geometries. There is very good agreement between the numerical results obtained using the method of partial discretization and the analytical results obtained using the method of influence functions. The high accuracy of the proposed methods and agreement with known theoretical data and with the experimental results obtained by the authors in the homogeneous cantilever case are shown. Presented at the International Conference on the Theory of Machines and Mechanisms, Poland, 1996. Published in Prikladnaya Mekhanika, Vol. 35, No. 6, pp. 103–110, June, 1999.     

Flexural vibrations of a nonlinear elastic plate

International Applied Mechanics -     

Stability of nonlinear periodic vibrations of 3D beams

The geometrically nonlinear periodic vibrations of beams with rectangular cross section under harmonic forces are investigated using a p-version finite element method. The beams vibrate in space; hence they experience longitudinal, torsional, and nonplanar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The theory employed is valid for moderate rotations and displacements, and physical phenomena like internal resonances and change of the stability of the solutions can be investigated. Green’s nonlinear strain tensor and Hooke’s law are considered and isotropic and elastic beams are investigated. The equation of motion is derived by the principle of virtual work. The differential equations of motion are converted into a nonlinear algebraic form employing the harmonic balance method, and then solved by the arc-length continuation method. The variation of the amplitude of vibration in space with the excitation frequency of vibration is determined and presented in the form of response curves. The stability of the solution is investigated by Floquet’s theory.     

Nonlinearly coupled vibrations of thin-walled elastic beams of open section

An account of certain subharmonic vibrations as observed during a resonant testing of thin-walled beams of monosymmetric open section for coupled torsional and bending vibrations is presented. The phenomenon can be described in terms of the vibrational modes of the beam. When the beam is excited at the resonant frequency of a higher mode, there is a tendency for the lowest mode to be excited, resulting in a high-order subharmonic oscillation. It is found that when such phenomenon occurs, the high-mode frequency is a multiple or near multiple of the fundamental frequency of the beam. Under such condition, the response of the beam consists of a superposition of the response of the high mode (harmonic response) and that of the fundamental mode (subharmonic response). The amplitude of the subharmonic motion is generally much larger than that of the harmonic response.     

The frequency criterion for thermally induced vibrations in elastic beams

Summary Thermally induced vibration in elastic beams, also known as thermal flutter, has not yet found a satisfactory treatment. The sheer weight of complications has so far rendered, and may indeed permanently render, a unified treatment impossible. In the present paper a frequency criterion is established which essentially defines the strength of the thermal excitation. Since damping is inevitably present, the thermal excitation must be large enough to overcome damping. After introducing a thermal time constant it is shown that at a critical frequency, the thermal excitation reaches a maximum. For practical cases, this critical frequency is very small, as typical structural frequencies go. Thus only systems with very low eigenfrequencies can be expected to tend to flutter thermally.

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Übersicht Durch Wärmeeinstrahlung verursachtes Schwingungsverhalten von Tubularbalken, sogenanntes Wärmeflattern, hat bisher noch keine zufriedenstellende allgemeingültige Beschreibung gefunden. Die Zahl der verschiedenen Einflüsse, die berücksichtigt werden müßten, ist in der Tat so groß, das es bisher noch nicht gelungen ist, den Problemkreis in seiner Gesamtheit zu erfassen und es sieht ganz so aus, als ob dies vielleicht nie gelingen wird. In der vorliegenden Arbeit wird ein Frequenzkriterium erstellt, das auf der Stärke der Schwingungserregung durch die Wärmeeinstrahlung beruht. Da Dämpfung bestimmt existiert, muß die Schwingungserregung durch die Wärmeeinstrahlung nicht nur vorhanden, sondern auch größer als die Dämpfung sein. Nach Einführung einer Wärmezeitkonstanten wird gezeigt, daß die Schwingungserregung ihren Höchstwert erreicht, wenn die Eigenfrequenz des Tubularbalkens einen gewissen kritischen Wert annimmt. In allen Fällen von praktischem Interesse ist diese kritische Frequenz übrigens so gering, daß nur bei Balken mit extrem niedriger Eigenfrequenz mit Wärmeflattern gerechnet werden kann.

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Dedicated to Prof. Dr.-Ing. Th. Lehmann on the occasion of his sixtieth birthday     

Free transverse oscillations of elastic beams with locally concentrated inhomogeneities

The eigenvalue problem for the fourth-order differential equation describing the free transverse oscillations of a strongly inhomogeneous elastic beam is studied numerically and analytically. The lowest modes of oscillations that are of interest in applications are found.     

Three-dimensional nonlinear vibrations of composite beams — II. flapwise excitations

The nonlinear equations of motion derived in Part I are used to investigate the response of an inextensional, symmetric angle-ply graphite-epoxy beam to a harmonic base-excitation along the flapwise direction. The equations contain bending-twisting couplings and quadratic and cubic nonlinearities due to curvature and inertia. The analysis focuses on the case of primary resonance of the first flexural-torsional (flapwise-torsional) mode when its frequency is approximately one-half the frequency of the first out-of-plane flexural (chordwide) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations to describe the time variation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability and bifurcations of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic solutions of the modulation equations are studied. Chaotic solutions are identified from their frequency spectra, Poincaré sections, and Lyapunov's exponents. The results show that the beam motion may be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations

Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

A quasilinearization approach to the solution of elastic beams on nonlinear foundations

In this paper, the solution of a beam on nonlinear elastic foundation whose deflection satisfies the nonlinear boundary value problem (1, 2), is studied by means of the theory of quasilinearization. The problem is formulated in Section 2 where conditions for the existence and uniqueness of the solution are stated. In Section 3, the idea of quasilinearization is introduced and the positivity of an associated linear differential operator is investigated. In Section 4 the usual version of quasilinearization, i.e. The Newton-Raphson-Kantorovich sequence, is presented and conditions under which this sequence is monotonically convergent, are established. In Section 5, an alternative successive approximation scheme whose derivation relies on ideas of quasilinearization, is presented. Finally, an example is solved by numerical procedures based in the methods discussed in previous sections.     

Identification of a cavity in an elastic rod in the analysis of transverse vibrations

The direct and inverse problems of the steady-state transverse vibrations of a cylindrical rod with a defect in the form of a cavity of small relative size are considered. An approach to determining the location and volume of the cavity of arbitrary shape is proposed. Results of computational experiments are analyzed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 152–158, November–December, 2008     

Equation of small transverse vibrations of an elastic pipeline filled with a transported fluid

The integro-differential equation of small transverse vibrations of a rectilinear elastic pipeline filled with a transported fluid is obtained. The pipeline vibrations are described in the linear setting in the beam approximation. The mutual dynamic influence of motions of the pipeline and the filling fluid is taken into account. A complete trigonometric series method is presented for solving problems with various boundary and initial conditions for the pipeline deflection.     

非线性弹性大变形材料梁的抗爆特性

从理论上探讨了非线性弹性大变形材料应用于抗爆结构的可行性,为此,基于等效结构体系的分析原理,将两端固定铰支梁的横向和纵向位移表示为三角级数形式,应用第二类Lagrange方程建立了非线性大变形材料梁的非线性分析方法,并且用ABAQUS有限元软件中的超弹性材料模型验证了所提出的方法的有效性。对典型的爆炸荷载作用下非线性弹性大变形材料梁的抗爆特性进行了分析,讨论了动力放大系数和材料性质及动荷载之间的关系。结果表明:与线弹性小变形材料相比,非线性弹性大变形材料具有优良的抗爆特性,结构的抗爆能力随结构变形的增大而显著提高。