M. Boscolo

Dynamic Stiffness Laminate Elements for Exact Modal Analysis of Aeronautical Structures

The dynamic stifiness method for composite plates based on flrst order shear deformation theory is used to carry out exact free vibration analyses of typical aeronautical structures. Hamilton’s principle is applied to obtain the equations of motion which are solved in Levy’s form. Subsequently, by using general boundary conditions to relate the harmonically varying forces to the harmonically varying displacement, the dynamic stifiness matrix of a laminated plate element based on the flrst order shear deformation theory is obtained. The dynamic stifiness laminate elements can be rotated, ofiset and assembled to model complex geometries and yet the exactness of the solution can be retained. These elements are validated against published results flrst and subsequently used to model a typical stringer panel for which exact solutions were not available before. The results are compared to the ones obtained by the FEM (NASTRAN) to demonstrate the potential of the method in terms of accuracy and e‐ciency when computing the natural frequencies of structures modelled as plate assemblies. Finally a typical aircraft composite wing box is investigated and both natural frequencies and mode shapes are compared with the ones obtained by the FEM.

Dynamic Stiffness Formulation for Plates Using First Order Shear Deformation Theory

The dynamic stiffness method for plates is developed to carry out an exact free vibration analysis by using both classical theory and first order shear deformation theory. Hamiltonian mechanics is used to provide a systematic general procedure for the development of the method. Explicit expressions for the elements of the dynamic stiffness matrices have been derived with the help of symbolic computation. Details of the assembly procedure and application of boundary conditions using the dynamic stiffness elements have been explained when investigating the free vibration characteristics of complex structures modelled by plate assemblies. The usually adopted Wittrick-Williams algorithm has been modified to avoid the requirement of computing the clamped-clamped natural frequencies of individual plates and yet converging upon any number of natural frequencies of the overall structure within any desired accuracy. The results using both classical and first order shear deformation theories are rigorously validated by published results for both uniform and stepped plates with various boundary conditions. Representative mode shapes are presented and the numerical accuracy and computational efficiency of the method are demonstrated. Significant plate parameters are varied and their subsequent effects on the accuracy of classical plate theory when compared to the first order shear deformation theory are investigated. For both uniform and stepped plates, the circumstances when the classical theory leads to inaccurate results are identified and discussed. The investigation offers the prospects for dynamic stiffness development of anisotropic plates using Hamiltonian mechanics and symbolic algebra.