Papers by Keyword: Euler Beam

Abstract: Free vibration of functionally graded materials (FGMs) Euler beam with elastically restrained edges is investigated. The material properties of the FGMs beam vary continuously in the thickness direction according to the power law form. The neutral axis site of the FGMs beam is determined by the static equilibrium condition. The governing equation and boundary conditions are found by applying the Hamilton’s principle. The linear combination of a Fourier cosine series and auxiliary Legendre polynomial function is used to obtain the natural frequencies of the FGMs beam. The effects of the rotational spring stiffness, the translational spring stiffness and the gradient index on the natural frequencies are discussed and analyzed for different material properties and different boundary conditions, indicating that the frequencies are sensitive to the gradient variation of material properties and the spring stiffness.

Abstract: Based on the traditional mechanical model of straight beam element, the paper makes a systematic analysis and research on the pre-twisted beam finite element numerical model. Firstly, the paper proposed the pre-twisted Euler beam element mode, the mode uses 2 node and 12 freedom degrees, the element axial and torsion displacements use 2 nodes Lagrange interpolation function, bending displacement still use the cubic displacement. Secondly, the paper studies a new pre-twisted Timoshenko beam element mode, the proposed new Timoshenko beam element takes separate interpolation polynomial functions both flexure bending and rotation displacement. According to the relationship between bending moment and shear, the relationship between of bending displacement and angle displacement is derived, which is more accurate to consider the effects of shear deformation. Finally, by calculating the pre-twisted rectangle cantilever beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Timoshenko beam element mode has good accuracy.

Abstract: Dynamic vibration and static deformation are two main factors that affect the machining quality in a long slender shaft turning operation. By simplifying the turning system of a slender shaft into a one-section beam with a clamped-pinned constraint condition, the direct receptance at any arbitrary cutting point is derived. On the basis, the regenerative stability lobes diagram (SLD) for a long slender shaft turing operation is achieved. With the proposed modeling methodology and simulation algorithm, the effect of cutting position on the direct frequency response function (FRF) and the predicated SLD, as well as the effect of the cutting conditions on the predicated SLD are investigated. The predicated direct FRF at the cutting point is validated by hammer tests.

Abstract: Based on the traditional mechanical model of straight beam, the paper makes a systematic analysis and research on the pre-twisted Euler beam finite element numerical model. The paper uses two-node model of 12 degrees of freedom, axial displacement interpolation function using 2-node Lagrange interpolation function, beam transverse bending displacements (u and υ) still use the cubic displacement, bending with torsion angle displacement function using cubic polynomial displacement function. Firstly, based on the author previous literature on the flexure strain relationship, the paper deduces the element stiffness matrix of the pre-twisted beam. Finally, by calculating the pre-twisted rectangle section beam example, and contrasting three-dimensional solid finite element using ANSYS, the comparative analysis results show that pre-twisted Euler beam element stiffness matrix has good accuracy.

Abstract: Using the theory of nonlinear elastic mechanics and fracture mechanics, the equation of motion governing equation of cracked beam is derived by the energy method, and solved with separation method of variables. Vibration analysis method based on the energy principle in this paper is proved feasible.Through numerical analysis, the effects of structural damping, crack location and depth on natural frequencies of linear vibration is investigated.

Abstract: This paper is concerned with the free vibration problem for micro/nano beams modelled after Eringen’s nonlocal elasticity theory and Euler beam theory. The small scale effect is taken into consideration in the former theory. The natural frequencies are obtained using the Hamilton’s principle and Chebyshev polynomial functions. The present method, which uses Rayleigh–Ritz technique in this paper, provides an efficient and extremely accurate vibration solution of micro/nano beams where the effects of small scale are significant. Numerical results for a variety of some micro/nano beams with various boundary conditions are given and compared with the available results wherever possible. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is promoted.