THE ANALYSIS OF THIN WALLED COMPOSITE LAMINATED HELICOPTER ROTOR WITH HIERARCHICAL WARPING FUNCTIONS AND FINITE ELEMENT METHOD

In the present paper, a series of hierarchical warping functions is developed to analyze the static and dynamic problems of thin walled composite laminated helicopter rotors composed of several layers with single closed cell. This method is the development and extension of the traditional constrained warping theory of thin walled metallic beams, which had been proved very successful since 1940s. The warping distribution along the perimeter of each layer is expanded into a series of successively corrective warping functions with the traditional warping function caused by free torsion or free beading as the first term, and is assumed to be piecewise linear along the thickness direction of layers. The governing equations are derived based upon the variational principle of minimum potential energy for static analysis and Rayleigh Quotient for free vibration analysis. Then the hierarchical finite element method. is introduced to form a,. numerical algorithm. Both static and natural vibration problems of sample box beams axe analyzed with the present method to show the main mechanical behavior of the thin walled composite laminated helicopter rotor.

HIERARCHICAL STOCHASTIC FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS

In this paper, the hierarchical approach is adopted for series representation of the stochastic nodal displacement vector using the hierarchical basis vectors, while the Karhunen- Loire series expansion technique is employed to discretize the random field into a set of random variables. A set of hierarchical basis vectors are defined to approximate the stochastic response quantities. The stochastic variational principle instead of the projection scheme is adopted to develop a hierarchical stochastic finite element method （HSFEM） for stochastic structures under stochastic loads. Simplified expressions of coefficients of governing equations and the first two statistical moments of the response quantities in the schemes of the HSFEM are developed, so that the time consumed for computation can be greatly reduced. Investigation in this paper suggests that the HSFEM yields a series of stiffness equations with similar dimensionality as the perturbation stochastic finite element method （PSFEM）. Two examples are presented for numerical study on the performance of the HSFEM in elastic structural problems with stochastic Young＇s Modulus and external loads. Results show that the proposed method can achieve higher accuracy than the PSFEM for cases with large coefficients of variation, and yield results agreeing well with those obtained by the Monte Carlo simulation （MCS）.

THE MUTUAL VARIATIONAL PRINCIPLE OF FREE WAVE PROPAGATION IN PERIODIC STRUCTURES

By taking infinite periodic beams as examples,the mutual variational principle for analyzing the free wave propagation in periodic structures is established and demonstrated through the use of the propaga- tion constant in the present paper,and the corresponding hierarchical finite element formulation is then de- rived.Thus,it provides the numerical analysis of that problem with a firm theoretical basis of variational prin- ciples,with which one may conveniently illustrate the mathematical and physical mechanisms of the wave prop- agation in periodic structures and the relationship with the natural vibration.The solution is discussed and ex- amples are given.