tailieunhanh - Advances in Vibration Analysis Research Part 8

Tham khảo tài liệu 'advances in vibration analysis research part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The Generalized Finite Element Method Applied to Free Vibration of Framed Structures 199 0 if X e xi-1 Xị Y 1 cos V2 - 1 1 e sin V2 -2 - -1 eT- 2 if x xi 1 u 1 1 - -1 Vj 1 - -1 e- 51 x - xi z2 J I xi 1- xi where Ằj are the eigenvalues obtained by the solution of Eq. 28 . Such partition of unity functions and local approximation space produce the cubic FEM approximation space enriched by functions that represent the local behavior of the differential equation solution. The enriched functions and their first derivatives vanish at element nodes. Hence the imposition of boundary conditions follows the finite element procedure. This C1 element is suited to apply to the free vibration analysis of Euler-Bernoulli beams. Again the increase in the number of elements in the mesh with only one level of enrichment j 1 and a fixed eigenvalue 21 produces the h refinement of GFEM. Otherwise the increase in the number of levels of enrichment each of one with a different frequency Xj produces a hierarchical p refinement. An adaptive GFEM refinement for free vibration analysis of Euler-Bernoulli beams is straight forward as can be easily seen. However it will not be discussed here. 5. Applications Numerical solutions for two bars a beam and a truss are given below to illustrate the application of the GFEM. To check the efficiency of this method the results are compared to those obtained by the h and p-versions of FEM and the c-version of CEM. The number of degrees of freedom ndof considered in each analysis is the total number of effective degrees of freedom after introduction of boundary conditions. As an intrinsic imposition of the adaptive method each target frequency is obtained by a new iterative analysis. The mesh used in each adaptive analysis is the coarser one that is just as coarse as necessary to capture a first approximation of the target frequency. Uniform fixed-free bar The axial free vibration of a fixed-free bar Fig. 6 with length L elasticity modulus E mass .

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