Flutter Analysis Of Slender Beams With Variable Cross Sections Based On Integral Equation Formulation (original) (raw)
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Journal of Sound and Vibration, 2006
This paper presents a mathematical model based on integral equations for numerical investigations of stability analyses of damped beams subjected to subtangential follower forces. A mathematical formulation based on Euler-Bernoulli beam theory is presented for beams with variable cross sections on a viscoelastic foundation and subjected to lateral excitation, conservative and non-conservative axial loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebro-differential system related to internal and boundary unknowns. Generalized formulations for the deflection, the slope, the moment and the shear force are presented. The free vibration of loaded beams is formulated in a compact matrix form and all necessary matrices are explicitly given. The load-frequency dependence is extensively investigated for various parameters of non-conservative loads, of internal and viscous dampings and for various positions of the concentrated foundation. For an undamped beam, a dynamic stability analysis is illustrated numerically based on the coalescence criterion. The flutter load and instability regions with respect to various parameters are identified. The effects of internal and viscous dampings on the critical flutter load are examined separately and relative effects are evaluated. The dynamic responses, before, near and after the flutter are investigated. A simple and quite general methodological approach is presented. Comprehensive numerical tests for flutter analysis are reported and discussed.
World journal of engineering and technology, 2024
The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with con
Non-linear dynamic analysis of beams with variable stiffness
Journal of Sound and Vibration, 2004
In this paper the Analog Equation method (AEM), a BEM-based method, is employed to the nonlinear dynamic analysis of an initially straight Bernoulli-Euler beam with variable stiffness undergoing large deflections. In this case the cross-sectional properties of the beam vary along its axis and consequently the coefficients of the differential equations governing the dynamic equilibrium of the beam are variable. The formulation is in terms of the displacement components. Using the concept of the analog equation, the two coupled nonlinear hyperbolic differential equations are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious time dependent load distributions. A significant advantage of this method is that the time history of the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Beams with constant and varying stiffness are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.
Journal of applied science and environmental management, 2024
In this paper, the effect of coefficient of viscous damping on the dynamic analysis of Euler-Bernoulli beam resting on elastic foundation was investigated using Integral-Numerical method which reduces to an ordinary differential equation with series representation of Heaviside function. The dynamic responses of the beam in terms of normalized deflection and bending moment has been investigated for different velocity ratios under moving load and moving mass conditions. Generally, closed-form solution to the generalized mathematical model for prismatic beam was computed by means of symbolic programming approach through MAPLE 18. Results obtain revealed that the presence of an elastic foundation and the provision of sufficient reinforcement in beams and beam-like structure reduces vibration intensity and ensure safe passage of load and prolong the beam life.
INCAS BULLETIN, 2017
This work presents a synthesis of the use of an integral approximate method based on structural influence functions (Green's functions) concerning the behavior of beam-like structures. This integral method is used in the areas of static, dynamic, aeroelasticity and stability analysis. The method starts from the differential equations governing the bending or/and torsional behavior of a beam. These equations are put in integral form by using appropriate Green's functions, according to the boundary conditions. Choosing a number of n collocation points on the beam axis, each integral are then computed by a summation using weighting numbers. This approach is suitable for conventional Euler-Bernoulli beams and also for the thin-walled open or closed cross-section beams which can have bending-torsion coupling. Generally, for a static analysis this approach leads to a linear system of equations (the case of the lift aeroelastic distribution analysis) or to an eigenvalues and eigenvectors problem in the case of dynamic, stability or divergence analysis.
Nonlinear Behaviour Analysis of Composite Thin Beams and Plates
2020
In engineering applications, it is well known that the minimum weight criteria with high performance is essential in the design of certain structures like aircraft components, aerospace vehicles and civil structures.. etc. This task could be a challenge especially when the design of wing structures such as aircraft wings, rotor blades, robotic arms is the subject. The behavior of such structures is highly nonlinear due to the deformation of their geometry. The solution of such problems becomes very complex, especially with the use of composite materials. The effects of large displacements may play a primary role in the correct prediction of the behavior of these structural members, which continue to be modeled as a flexible beams. In this way, another difficult task can be imposed here when some structural elements like plates and shells which can undergo inplane thermo-mechanical stresses that affect their dynamic and static behaviors. This problem has stimulated several researcher...
Computers & Structures, 2018
Free vibration problem of an arbitrary cross-sectional homogeneous or composite beam is investigated using a refined 1D beam theory (RBT). This theory includes a set of 3D displacement modes of the cross-section (CS) which reflects its mechanical behavior: the main part of these sectional modes is extracted from the 3D Saint Venant's solution and another part is related to the CS dynamic behavior. These sectional modes, which are first derived from a CS analysis, lead to a consistent 1D beam model which really fits the section nature (shape and materials), and hence the beam problem. The numerical strategy to apply such general approach, is based on a first set of CS problems solved by 2D-FEM computations to get the sectional modes, and then the dynamic beam problem is solved by 1D-FEM computation according to RBT displacement model to provide (in fine) the first natural frequencies and 3D vibration mode shapes of the beam. To do so and in order to easily apply such method, a user friendly Matlab numerical tool named CSB (Cross-Section and Beam analysis) has been developed. To illustrate the capabilities and the accuracy of the method to catch the main 3D-effects, such as elastic/inertial coupling effects and 3D local/global mode shapes, a significant set of beam cross-section configurations with isotropic and anisotropic materials are analyzed. The first ten natural frequencies and 3D mode shapes are systematically compared to those obtained by full 3D-FEM computations, and some of them to literature.
International Journal of Non-Linear Mechanics
We propose a new non-linear method for the static analysis of an infinite non-uniform beam resting on a non-linear elastic foundation under localized external loads. To this end, an integral operator equation is newly formulated, which is equivalent to the original differential equation of non-uniform beam. By using the integral operator equation, we propose a new functional iterative method for static beam analysis as a general approach to a variable beam cross-section. The method proposed is fairly simple as well as straightforward to apply. An illustrative example is presented to examine the validity of the proposed method. It shows that just a few iterations are required for an accurate solution.
An Analytical Approach to Vibration Analysis of Beams with Variable Properties
Arabian Journal for Science and Engineering, 2013
In this paper, an approach is proposed for determination of vibration frequencies of variable cross-section Timoshenko beams and of non-prismatic Euler beams under variable axial loads. In each case, the governing differential equation is first obtained and, according to a harmonic vibration, is converted into a single-variable equation in terms of location. Integral equations for the weak form of governing equations are derived through repetitive integrations. Mode shape functions are approximated by a power series. A system of linear algebraic equations is obtained by substitution of the power series into the integral equation. Using a non-trivial solution for system of equations, natural frequencies are determined. The presented method is formulated for beams having various end conditions and is extended for determination of the buckling load of non-prismatic Euler beams. The effectiveness and accuracy of the method is shown through comparison of the numerical results to those obtained using available finite element software.
Large deflection analysis of beams with variable stiffness
Acta Mechanica, 2003
In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.