Flutter analysis of rotating beams with elastic restraints

The aeroelastic stability of rotating beams with elastic restraints is investigated. The coupled bending-torsional Euler-Bernoulli beam and Timoshenko beam models are adopted for the structural modeling. The Greenberg aerodynamic model is used to describe the unsteady aerodynamic forces. The additional centrifugal stiffness effect and elastic boundary conditions are considered in the form of potential energy. A modified Fourier series method is used to assume the displacement field function and solve the governing equation. The convergence and accuracy of the method are verified by comparison of numerical results. Then, the flutter analysis of the rotating beam structure is carried out, and the critical rotational velocity of the flutter is predicted. The results show that the elastic boundary reduces the critical flutter velocity of the rotating beam, and the elastic range of torsional spring is larger than the elastic range of linear spring.     

The photoelastic determination of the stresses in vibrating cantilevered beams

A method of determining the dynamic stresses in vibrating cantilevered beams using photoelasticity is presented. The method uses the basic principles of photomechanics and the optic-stress laws. A high-intensity strobe light is timed with the frequency of vibration so that the beam image appears to be stationary. Data are recorded with a camera and analyzed to provide an experimental solution. The theoretical solution is derived from the Bernoulli-Euler equation of motion. Two basic types of beams were investigated, an aluminum beam coated with a birefringent material to simulate an actual structural member and a birefringent model that was dynamically similar to the aluminum beam. The feasibility of extending the technique to more complicated shapes is suggested by this investigation. The experimental results and feasibility of the concept are verified by close correlation with the analytical solution.     

变截面梁弯曲切应力分析

从一般情况出发在继承经典的弯曲正应力公式前提下,应用静力边界条件与微体平衡方程导出变截面梁的弯曲切应力公式.结果与有限元解基本吻合,而传统材料力学方法与之相差甚远.     

活动铰支座倾斜放置的简支梁弯曲变形

<正>题目图1为活动铰支座倾斜放置的等截面简支梁,承受均布横向载荷.设载荷集度g=32N/m,梁的长度l=1 m,截面惯性矩I=4.5×10~(-11)m~4,弹性模量E=2.01×10~(11)Pa.试确定在活动铰支座的倾斜角φ分别为15。,30。,45。和60。的情况下简支梁的各挠曲线形状.     

Elasticity solution of clamped-simply supported beams with variable thickness

This paper studies the stress and displacement distributions of continuously varying thickness beams with one end clamped and the other end simply supported under static loads. By introducing the unit pulse functions and Dirac functions, the clamped edge can be made equivalent to the simply supported one by adding the unknown horizontal reactions. According to the governing equations of the plane stress problem, the general expressions of displacements, which satisfy the governing differefitial equations and the boundary conditions attwo ends of the beam, can be deduced. The unknown coefficients in the general expressions are then determined by using Fourier sinusoidal series expansion along the upper and lower boundaries of the beams and using the condition of zero displacements at the clamped edge. The solution obtained has excellent convergence properties. Comparing the numerical results to those obtained from the commercial software ANSYS, excellent accuracy of the present method is demonstrated.     

Flexural vibrations and stability of beams with variable parameters

Bialystok Polytechnic Institute, Poland. Translated from Prikladnaya Mekhanika, Vol. 30, No. 9, pp. 75–81, September, 1994.     

Dynamic bifurcation and sensitivity analysis of non-linear non-planar vibrations of geometrically imperfect cantilevered beams

The non-linear non-planar dynamic responses of a near-square cantilevered (a special case of inextensional beams) geometrically imperfect (i.e., slightly curved) and perfect beam under harmonic primary resonant base excitation with a one-to-one internal resonance is investigated. The sensitivity of limit-cycles predicted by the perfect beam model to small geometric imperfections is analyzed and the importance of taking into account the small geometric imperfections is investigated. This was carried out by assuming two different geometric imperfection shapes, fixing the corresponding frequency detuning parameters and continuation of sample limit-cycles versus the imperfection parameter. The branches of periodic responses for perfect and imperfect (i.e. small geometric imperfection) beams are determined and compared. It is shown that branches of periodic solutions associated with similar limit-cycles of the imperfect and perfect beams have a frequency shift with respect to each other and may undergo different bifurcations which results in different dynamic responses. Furthermore, the imperfect beam model predicts more dynamic attractors than the perfect one. Also, it is shown that depending on the magnitude of geometric imperfection, some of the attractors predicted by the perfect beam model may collapse. Ignoring the small geometric imperfections and applying the perfect beam model is shown to contribute to erroneous results.     

Dynamic stability analysis and DQM for beams with variable cross-section

The present paper deals with the dynamic behaviour of a clamped beam subjected to a sub-tangential follower force at the free end. The aim of this work is to obtain the frequency–axial load relationship for a beam with a variable circular cross-section. In this way, one can identify both divergence critical loads – where the frequency goes to zero – and the flutter critical load – in correspondence with two frequencies coalescence. The numerical approach adopted for solving the partial differential equation of motion is the differential quadrature method (henceforth DQM). This method was proposed by Bellmann and Casti [Bellmann, R.E., Casti, J., 1971. Differential quadrature and long-term integration. J. Math. Anal. 34, 235–238] and has been employed recently in the solution of solid mechanics problems by Bert and Malik [Bert, C.W., Malik, M., 1996. Differential quadrature method in computational mechanics: a review. Appl. Mech. Rev., ASME, 49 (1), 1–28] and Chen et al. [Chen, W., Stritz, A.G., Bert, C.W., 1997. A new approach to the differential quadrature method for fourth-order equations. Int. J. Numer. Method Eng. 40, 1941–1956]. More precisely, a modified version of this method has been used, as proposed by De Rosa and Franciosi [De Rosa, M.A., Franciosi, C., 1998a. On natural boundary conditions and DQM. Mech. Res. Commun. 25 (3), 279–286; De Rosa, M.A., Franciosi, C., 1998b. Non classical boundary conditions and DQM. J. Sound Vibrat. 212(4), 743–748] to satisfy all the boundary conditions.Some frequencies–axial loads relationships are reported in order to show the influence of tapering on the critical loads.     

Non-linear vibration of variable speed rotating viscoelastic beams

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

General analytic solution of dynamic response of beams with nonhomogeneity and variable cross-section

In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements. In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration, a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations with only     

Free vibration and spatial stability of non-symmetric thin-walled curved beams with variable curvatures

An improved formulation for free vibration and spatial stability of non-symmetric thin-walled curved beams is presented based on the displacement field considering variable curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov’s assumptions and integrating over the non-symmetric cross-section, the total potential energy is consistently derived from the principle of virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis and also thickness-curvature effects and Wagner effect are accurately taken into account. For F.E. analysis, a thin-walled curved beam element is developed using the third-order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, numerical solutions by this study are presented with the results analyzed by ABAQUS’ shell elements. Particularly, the effect of arch rise to span length ratio is investigated on vibrational and buckling behaviour of non-symmetric curved beams.     

一种超静定变截面梁的力法计算技巧

力法计算变截面超静定梁的弯矩图时,在选取基本结构后,用图乘法计算相关系数的过程中,分割图形数目多,计算量大容易出错.针对这个问题,提出了一种变截面处铰化分解杆件的技巧,用来减少图乘次数,降低计算量,提高计算效率.该技巧可广泛应用于变截面超静定梁弯矩图为折线的情况,在授课过程中使学生概念清晰,易于接受,且有助于提高计算正确率,值得推广.     

变截面欧拉梁自由振动分析的重采样微分求积法

采用重采样微分求积法求解了变截面欧拉梁的自由振动问题。推导了变截面梁的控制方程离散格式,采用重采样矩阵方法对边界条件进行处理,给出了变截面梁自由振动算法。采用本文方法对不同类型截面形式和不同边界条件的变截面梁进行自由振动分析,并和其他解法进行比较。计算结果表明,本文方法可以适用于不同变截面类型和不同边界条件,计算精度与解析解吻合良好,具有良好的收敛性能。在同等精度条件下网格点数少于现有计算方法。重采样转换矩阵边界处理方法相比于传统边界处理方法具有更快的收敛性能。     

Applying the generalized step-function to solve the bending problems of nonuniform beams with variable cross section

The bending problems of nonuniform beams with variable cross-section can be approximated by that of a step beam under sectionally uniform load (including both concentrated forces and couples). In this paper, the concept of Heaviside function {x-a}⁰ will be generalized, and a new function {x-a}⁰, n=0,1,2…,will be defined, which may be named as a generalized step function. The rules of operation will also be given to {x-a}ⁿ{x-b}⁰. The reciprocal of the flexural of rigidity 1/EJ and the bending moment M(x) can all be expressed in terms of {x-a}ⁿ,and substituted into the differential equation of the elastic curve of the beam respectively. Thus we may establish a set of unified method to solve various types of bending problems of straight beams. The general solution of the deflection equation will be given.