Abstract
In this paper, an exact dynamic stiffness formulation using one-dimensional (1D) higher-order theories is presented and subsequently used to investigate the free vibration characteristics of solid and thin-walled structures. Higher-order kinematic fields are developed using the Carrera Unified Formulation, which allows for straightforward implementation of any-order theory without the need for ad hoc formulations. Classical beam theories (Euler–Bernoulli and Timoshenko) are also captured from the formulation as degenerate cases. The Principle of Virtual Displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick–Williams algorithm to carry out the free vibration analysis of solid and thin-walled structures. The accuracy of the theory is confirmed both by published literature and by extensive finite element solutions using the commercial code MSC/NASTRAN®.
Highlights
Beam models are widely used to analyze the mechanical behavior of slender bodies, such as columns, rotor-blades, aircraft wings, towers, antennae and bridges amongst others
The accuracy and computational efficiency of the present exact, higher-order dynamic stiffness (DS) elements are demonstrate by carrying out the free vibration analysis of both solid and thin-walled structures and the results are presented
Exact dynamic stiffness method (DSM) solutions are compared with approximate results based on higher-order type expansions (TE) models built using finite element method (FEM)
Summary
Beam models are widely used to analyze the mechanical behavior of slender bodies, such as columns, rotor-blades, aircraft wings, towers, antennae and bridges amongst others. The investigation is carried out in the following steps: (i) first CUF is introduced and higher-order models are formulated, (ii) secondly, the Principle of Virtual Displacements (PVD) is used to derive the differential governing equations and the associated natural boundary conditions for the generic N -order model, (iii) by assuming harmonic oscillation, the equilibrium equations and the natural boundary conditions are formulated in the frequency domain by making extensive use of symbolic computation, (iv) the resulting system of ordinary differential equations of second order with constant coefficients is solved in closed analytical form, (v) subsequently, the frequency dependent DS matrix of the system is derived by relating the amplitudes of the harmonically varying nodal generalised forces to those of the nodal generalized displacements, and (vi) the well-known algorithm of Wittrick and Williams [57] is applied to the resulting DS matrix for free vibration analysis of compact and thin-walled structures
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