移动集中质量作用下Euler-Bernoulli梁的振动频率分析

研究了移动载荷作用下Euler-Bernoulli梁振动频率的变化规律。首先根据Euler-Bernoulli梁振动的控制方程,引入载荷和位移边界条件,建立梁单元的动力刚度矩阵和形状函数矩阵,然后采用傅里叶变换的思想,建立移动载荷惯性力引起的附加动力刚度矩阵,进而得到系统整体的动力刚度矩阵。通过Wittrick-Williams法进行求解得到移动载荷作用下Euler-Bernoulli梁的振动频率,与有限元法计算结果的比较,验证了此方法的正确性,体现了此方法在精度和计算规模上的优势。根据上述方法,揭示简支梁和悬臂梁振动频率随着移动质量速度与质量的变化规律。     

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads

Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love's thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the generalized differential quadrature (GDQ) method. Hill's method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.     

Parametric instability of a rod reinforcing a beam when subjected to a stepped load

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 1, pp. 118–123, January, 1989.     

Parametric instability analysis of a rotating shaft subjected to a periodic axial force by using discrete singular convolution method

Meccanica - Parametric instability problem of a rotating shaft subjected to a periodically varying axial force has been studied by using a numerical simulation method—discrete singular...     

On the parametric stability of a Timoshenko beam subjected to a periodic axial load

Summary The dynamic buckling of a hinged bar under harmonic axial load is studied using Timoshenko's beam theory and considering the effects of longitudinal vibrations. Several new types of parametric resonances were found. The influences of the more accurate Timoshenko beam theory on the instability regions found with the Bernoulli-Euler theory is also discussed.

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Übersicht Die dynamische Stabilität eines gelenkig gelagerten Balkens mit harmonischer Axiallast wird mit Hilfe der Timoshenkoschen Balkentheorie und unter Berücksichtigung der Längsschwingungen untersucht. Verschiedene neue Arten von parametrischen Instabilitäten konnten gefunden werden. Der Einfluß der genaueren Timoshenkoschen Balkentheorie auf die sich aus der Bernoulli-Eulerschen Theorie ergebenden Instabilitätsbereiche wird diskutiert.

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Visiting Professors at COPPE, Universidade Federal do Rio de Janeiro, Brazil.     

Parametric resonance of steel bridges pylons due to periodic traffic loads

This paper deals with the parametric resonance of steel bridges pylons due to time-depended traffic loads. The analysis follows the basic lines of Bolotin’s technique for the solution of nonlinear problems of dynamic instability. In this work, the cases of forced vibrating pylons with and without damping subjected to periodic external dynamic forces acting axially are investigated. The effect of bridge vibration due to traffic loads has been also taken into account. Through the aforementioned technique, useful results regarding the dynamic stability of pylons are obtained, and illustrative examples for various cases of geometry and loading are presented in the form of plots and diagrams.     

Nonlinear dynamic modeling and periodic vibration of a cantilever beam subjected to axial movement of basement

Dynamic modeling of a cantilever beam under an axial movement of its basement is presented. The dynamic equation of motion for the cantilever beam is established by using Kane's equation first and then simplified through the Rayleigh-Ritz method. Compared with the older modeling method, which linearizes the generalized inertia forces and the generalized active forces, the present modeling takes the coupled cubic nonlinearities of geometrical and inertial types into consideration. The method of multiple scales is used to directly solve the nonlinear differential equations and to derive the nonlinear modulation equation for the principal parametric resonance. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling is clarified through the comparisons of its coefficients with those experimentally verified in previous studies. Project supported by the Fundamental Fund of National Defense of China (No. 10172005).     

Non-linear response of a beam to periodic loading

The steady state response of a non-linear beam under periodic excitation is investigated. The non-linearity is attributed to the membrane tension effect which is induced in the beam when the deflection is not small in comparison to its thickness. The effects of multimode participation are investigated for simply supported and clamped boundary conditions. The finite element technique is used to formulate the non-linear differential equations of the straight beam and the method of averaging is used to obtain an approximate solution to the non-linear equations under harmonic loading. An analog computer was used to simulate the non-linear beam equation which was subjected to harmonic excitation. The agreement between theoretical and experimental values is reasonably good.     

Vibration analysis of a beam with an internal hinge subjected to a random moving oscillator

In this paper, we investigate the dynamic response of a fixed–fixed beam with an internal hinge on elastic foundation, which is subjected to a moving oscillator with uncertain parameters such as random mass, stiffness, damping, velocity and acceleration. This model can be used to simulate the interaction among the train (vehicle), track and foundation, as well as simulate the bridge–vehicle interaction without considering the elastic foundation. In particular, the distributed parameter system is assumed to be a beam of Bernoulli–Euler type, and the system dynamic response is a random process despite its deterministic characteristics. By utilizing the modal analysis and Galerkin’s method, we can obtain a set of approximate governing equations of motion with time-dependent random coefficients and forcing functions. The improved perturbation technique is adopted to evaluate the statistical characteristics of the deflection of the beam, and the Monte Carlo simulation is used to check the results from the improved perturbation technique. The statistical response of the structure from the proposed approach plays an important role in estimating the structural safety and reliability.     

Effective properties of a composite beam subjected to torsion

The problem of finding the effective characteristics of a rectilinear beam under pure torsion is considered. The problem can be reduced to determining the torsional stress function from the solution of a boundary-value problem in a cross section of the beam for a partial differential equation with variable coefficients. Two special boundary-value problems are formulated to find the effective characteristics. It is shown that the effective coefficients are reciprocal in the case of torsion of a layer with nonuniform thickness. In the two-dimensional case, the problem is solved by a finite element method. The cases of a square beam with single and multiple inclusions are discussed. The dependence of the effective characteristics on the inclusion volume fraction is analyzed.     

Dynamic analysis of double-pipe heat exchangers subjected to periodic inlet temperature disturbances

A mixed lumped-differential formulation is employed to model the transient energy equations for fully developed laminar-laminar or laminar-turbulent flow situations in concurrent or countercurrent double-pipe heat exchangers. The temperature distribution in the outer annular channel is radially lumped, providing a more general boundary condition for the inner channel differential energy equation, coupled through the interface condition. The case of periodically varying inlet temperatures is more closely considered, and the dynamic response of the exchanger is established in terms of the governing dimensionless parameters, such as heat capacity flow rate ratio, dimensionless inlet temperature oscillation frequency, and relative wall thermal resistence. The ideas in the generalized integral transform technique are extended to yield analytical solutions to the related periodic problem defined in the complex domain, and offer highly accurate numerical results for quantities of practical interest, such as fluids bulk temperatures.

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Dynamische Untersuchung von Doppelrohr-Wärmetauschern bei periodisch schwankenden Zulauftemperaturen

Zusammenfassung Die nichtstationäre Energiegleichung für vollausgebildete Laminar-Laminarströmung bzw. Laminar-Turbulentströmung in Gleich- oder Gegenstrom-Doppelrohrwärmetauschern wird mit einem teils diskreten, teils differentiellen Formalismus gelöst. Die Temperaturverteilung im äußeren Ringspalt wird diskretisiert, wodurch sich eine allgemeine Randbedingung für die durch die Trennwandbedingung angekoppelte Energiedifferentialgleichung im Innenrohr vorgeben läßt. Der Fall periodisch veränderlicher Einlauftemperaturen wird näher untersucht und die Systemantwort des Wärmetauschers beschrieben als Funktion der charakteristischen dimensionslosen Kenngrößen, wie Wärmekapazität-Volumenstromverhältnis, dimensionslose Frequenz der Einlauftemperaturschwingungen und bezogener Wärmewiderstand der Wand. Die Grundkonzeptionen der generalisierten Integral-Transformationstechnik werden erweitert, um analytische Lösungen für das zugeordnete, in der komplexen Ebene definierte periodische Problem zu finden. Diese liefern sehr genaue numerische Ergebnisse von praktischem Interesse, wie z. B. die gemittelten Fluidtemperaturen.

     

Stochastic jump and bifurcation of a slender cantilever beam carrying a lumped mass under narrow-band principal parametric excitation

A detailed theoretical investigation into the single-mode approximate response of a slender cantilever beam carrying a lumped mass subjected to base narrow-band random excitation is presented for the first time. The method of multiple scales is used and the stochastic jump and bifurcation have been investigated for the principal parametric resonance of the system using the stationary joint probability. Results show that stochastic jump occurs mainly in the region of triple-valued solution. For the frequency-response domain, if the excitation central frequency is a variable and others keep constant, the basic phenomena imply that the higher the frequency, the more probable the jump from the stationary non-trivial branch to the stationary trivial one once the frequency exceeds a certain value. If the bandwidth is a variable and others keep constant, the basic phenomena indicate that the most probable motion is around the non-trivial branch when the bandwidth is smaller, whereas the most probable motion gradually approaches the trivial one when the bandwidth becomes higher. For the force-response domain, there is a region of excitation acceleration within which the joint probability density has two peaks: an outer flabellate peak and a central volcano peak. Results show that the outer flabellate peak decreases while the central volcano peak increases as the value of the excitation acceleration decreases.     

Homoclinic orbits in a shallow arch subjected to periodic excitation

Homoclinic orbits in a shallow arch subjected to periodic excitation are investigated in the presence of 1:1 internal resonance and external resonance. The method of multiple scales is used to obtain a set of near-integrable systems. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect equilibria in a resonance band of the slow manifold. These orbits arise from singular homoclinic orbits and are composed of alternating slow and fast pieces. The result obtained imply the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the class of shallow arch systems.     

Active feedback control of a beam subjected to a nonconservative force

This study examines the possibility of controlling through feedback a thin cantilevered beam subjected to a nonconservative follower force. A converging frequency flutter instability which occurs in this model is similar to classical bending-torsion flutter of an aircraft wing. Because of the similar nature of the instabilities, the beam under the follower force can be a useful vehicle for investigating the fundamental aspects of stabilization of wing flutter by feedback control. A modal approach is used for obtaining the mathematical model and control laws. A standard root locus technique for simple analytical models is also used to understand and explain the control of the beam. Experiments are carried out to verify the validity of this theoretical model. Good correlation is shown between theoretically and experimentally determined stability boundaries as well as for modal frequency and damping variation with follower force.     

Postbuckling of elastic beam subjected to a concentrated moment within the span length of beam

This paper aims to present the exact closed form solutions and postbuckling behavior of the beam under a concentrated moment within the span length of beam. Two approaches are used in this paper. The nonlinear governing differential equations based on elastica theory are derived and solved analytically for the exact closed form solutions in terms of elliptic integral of the first and second kinds. The results are presented in graphical diagram of equilibrium paths, equilibrium configurations and critical loads. For validation of the results from the first approach, the shooting method is employed to solve a set of nonlinear differential equations with boundary conditions. The set of nonlinear governing differential equations are integrated by using Runge–Kutta method fifth order with adaptive step size scheme. The error norms of the end conditions are minimized within prescribed tolerance (10⁻⁵). The results from both approaches are in good agreement. From the results, it is found that the stability of this type of beam exhibits both stable and unstable configurations. The limit load point existed. The roller support can move through the hinged support in some cases of β and leads to the more complex of the configuration shapes of the beam.     

Nonlinear dynamic instability analysis of laminated composite thin plates subjected to periodic in-plane loads

In this paper, the dynamic instability of thin laminated composite plates subjected to harmonic in-plane loading is studied based on nonlinear analysis. The equations of motion of the plate are developed using von Karman-type of plate equation including geometric nonlinearity. The nonlinear large deflection plate equations of motion are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu-Hill equations. Dynamically unstable regions, and both stable- and unstable-solution amplitudes of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both antisymmetric and symmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the orthotropy, magnitude of both tensile and compressive longitudinal loads, aspect ratios of the plate including length-to-width and length-to-thickness ratios, and in-plane transverse wave number on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.