The geometrically nonlinear dynamic responses of simply supported beams under moving loads

This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as an Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load–structure system is discussed, and Kelvin−Voigt material model is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by using the Newmark method and Newton−Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability.     

Optimal design of beams for prescribed compliance under alternative loads

A global optimality condition is established for minimum-weight design of sandwich beams with elementwise constant cross section for prescribed compliances in alternative states of loading. This condition requires a nonnegative linear combination of the mean-square curvatures of an element in the considered states of loading to have the same value for all elements. The use of the condition in the determination of the minimum-weight design is illustrated by examples.This research was sponsored by the Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, under Contract No. F33615-69-C-1826. The authors wish to express their thanks to Professor J. B. Martin, Brown University, for helpful discussions.     

Chaotic dynamics of the vibro-impact system under bounded noise perturbation

In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed.     

Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads

The finite element dynamic response of an unsymmetric composite laminated orthotropic beam, subjected to moving loads, has been studied. One-dimensional finite element based on classical lamination theory, first-order shear deformation theory, and higher-order shear deformation theory having 16, 20 and 24 degrees of freedom, respectively, are developed to study the effects of extension, bending, and transverse shear deformation. The theories also account for the Poisson effect, thus, the lateral strains and curvatures can be expressed in terms of the axial and transverse strains and curvatures and the characteristic couplings (bend–stretch, shear–stretch and bend–twist couplings) are not lost. The dynamic response of symmetric cross-ply and unsymmetric angle-ply laminated beams under the action of a moving load have been compared to the results of an isotropic simple beam. The formulation also has been applied to the static and free vibration analysis.     

Elasticity solution of multi-span beams with variable thickness under static loads

This paper studies the stress and displacement distributions of continuously varying thickness multi-span beams simply supported at two ends and under static loads. The intermediate supports of the beam may be elastic and/or rigid in one or two directions. On the basis of the two-dimensional plane elasticity theory, the general solution of stress function, which exactly satisfies the governing differential equations and the simply supported boundary conditions, is deduced. In the present analysis, the reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the beam under consideration. The unknown coefficients in the solutions are determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beam and using the linear relations between reaction forces and displacements of the beam at intermediate supports. The solution obtained is exact and excellent convergence has been confirmed. Comparing the numerical results obtained from the proposed method to those obtained from the Euler beam theory, the Timoshenko beam theory and those obtained from the commercial finite element software ANSYS, high accuracy of the present method is demonstrated.     

On lateral buckling of end-loaded cantilever beams

Summary The classical problem of lateral buckling of cantilever beams with transverse end load is reexamined, as an example of a problem fully governed by theintrinsic equations of Kirchhoff's curved beam theory. It is shown that asuitable non-dimensionalization of the differential equations of the problem leads to astraightforward perturbation solutions, with leading and second-order terms of the expansion having well-defined differences in physical significance. The equations of a recent extension of Kirchhoff's theory, which take account of transverse shear deformation, are used for the purpose of obtaining a numerical result for the influence of shear deformability on the lateral buckling load.

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Zusammenfassung Das klassische Problem der Kippstabilität des Kragträgers wird re-examiniert als Beispiel eines Problems, welches vollständig definiert werden kann mit Hilfe der intrinsischen Gleichungen der Kirchhoffschen Theorie gekrümmter Stäbe. Es wird gezeigt, dass eine spezielle Art die Gleichungen dimensionslos zu machen eine einfache Störungsrechnung ermöglicht, mit der die führenden sowohl als auch die eine Grössenordnung kleineren Glieder in der Reihenentwicklung eine wohldefinierte physikalische Bedeutung haben. Schliesslich werden die Gleichungen einer Erweiterung der Kirchhoffschen Theorie, die den Einfluss von Schubverformungen berücksichtigt, benutzt, um ein zahlenmässiges Resultat bezüglich des Einflusses der Schubverformung auf die Kipplast zu erhalten.

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A report on work supported by the Office of Naval Research.     

On the effect of shear center location on the values of axial and lateral cantilever buckling loads for singly symmetric cross-section beams

Summary A generalization of Kirchhoff's beam equations, in which account is taken of deformations due to cross sectionalforces, in addition to those due to cross-sectionalmoments, is used to obtain results on the effect of cross-sectional shear center location on the values of Euler and Michell-Prandtl buckling loads. The analysis of these problems appears to be outside the scope of the original Kirchhoff equations.

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Zusammenfassung Eine Verallgemeinerung der Kirchhoffschen Stabgleichungen, welche neben dem Einfluß der Schnittmomente auf die Verformung auch denjenigen der Schnittkräfte berücksichtigt, wird benutzt, um den Einfluß der Lage des Schubmittelpunkts im Querschnitt auf die Werte der Eulerschen und Michell-Prandtlschen Knicklasten zu untersuchen. Die Lösung dieser Aufgaben scheint außerhalb des Anwendungsbereichs der ursprünglichen Kirchhoffschen Gleichungen zu liegen.

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Supported by the Office of Naval Research.     

Cusped elastic beams under the action of stresses and concentrated forces

A dynamical problem in the (0, 0) approximation of elastic cusped prismatic beams is investigated when stresses are applied at the face surfaces and the ends of the beam. Two types of cusped ends are considered when the beam cross-section turns into either a point or a straight line segment. Correspondingly, at the cusped end either a force concentrated at the point or forces concentrated along the straight line segment is applied. We prove the exists and uniqueness theorems in appropriate weighted Sobolev spaces.     

On the dynamics of Rayleigh beams resting on fractional-order viscoelastic Pasternak foundations subjected to moving loads

The standard averaging method is used to provide an analytical explanation on the effects of spacing loads, load velocity, order of the fractional viscoelastic property of shear layer material on the amplitude of the beam. The geometric nonlinearity is taken into account in the model. The analysis shows that, when the moving loads are uniformly distributed upon all the length of the structure, it vibrates the least possible. Moreover, as the order of the derivative increases, the resonant amplitude of the beam vibration decreases. In other hand, by means of Melnikov technique, a necessary condition for onset of horseshoes chaos resulting from heteroclinic bifurcation is derived analytically. We point out the critical weight of moving loads and order of the fractional derivative above which the system becomes unstable.     

Dynamical optimization of ribbed cylindrical shells under the action of periodic nonharmonic loads

Ribbed cylindrical shells that are under the action of a nonharmonic cylindrical load are considered. The problem of optimal (with respect to the criterion of minimum mass) projection is formulated in terms of nonlinear programming. Some results of a numerical experiment about the choice of optimal parameters are shown, which was carried out on an IBM with the help of the method of random search.Translated from Dinamicheskie Sistemy, No. 9, pp. 22–27, 1990.     

Investigation of stressed-strained state of three-layer cylindrical shells under the action of localized loads

The problem of the determination of the stressed-strained state of three-layer cylindrical shells with a light filler under the action of localized loads is solved by means of semimomentless hypotheses. By using single trigonometric series, the solution is reduced to integration of ordinary differential equation of the fourth order, similar to the Vlasov's equation. Decisive relationships were obtained for the components of the stressed-strained state of an asymmetric three-layer packet. A solution is given for a series of problems of the action of differently applied radial forces on a long cylindrical shell.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 300–305, March–April, 1975.     

Bending of a nonlinear orthotropic cantilever beam

The method of elastic solutions is employed to investigate the plane problem of the deformation of a cantilever beam of orthotropic glass-reinforced plastic under a concentrated load with allowance for the non-linear properties of the material. The first approximation of the stress function is given and the stress distribution over the cross section is calculated for a specific GRP.Mekhanika Polimerov, Vol. 2, No. 5, pp. 773–778, 1966     

Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation

This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincaré map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.     

Chaotic dynamics of respiratory sounds

There is a growing interest in nonlinear analysis of respiratory sounds (RS), but little has been done to justify the use of nonlinear tools on such data. The aim of this paper is to investigate the stationarity, linearity and chaotic dynamics of recorded RS. Two independent data sets from 8 + 8 healthy subjects were recorded and investigated. The first set consisted of lung sounds (LS) recorded with an electronic stethoscope and the other of tracheal sounds (TS) recorded with a contact accelerometer. Recurrence plot analysis revealed that both LS and TS are quasistationary, with the parts corresponding to inspiratory and expiratory flow plateaus being stationary. Surrogate data tests could not provide statistically sufficient evidence regarding the nonlinearity of the data. The null hypothesis could not be rejected in 4 out of 32 LS cases and in 15 out of 32 TS cases. However, the Lyapunov spectra, the correlation dimension (D₂) and the Kaplan–Yorke dimension (D_KY) all indicate chaotic behavior. The Lyapunov analysis showed that the sum of the exponents was negative in all cases and that the largest exponent was found to be positive.

The results are partly ambiguous, but provide some evidence of chaotic dynamics of RS, both concerning LS and TS. The results motivate continuous use of nonlinear tools for analysing RS data.     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.