Exact solutions of axial vibration problems of elastic bars

Abstract

SUMMARYWhile exact solutions for linear static analysis of most frame structures can be obtained by the finite element method, it is very difficult to obtain exact solutions for free vibration and harmonic analyses for non-trivial cases. This paper presents a study on new finite element formulation and algorithms for exact solutions of undamped axial vibration problems of elastic bars. Appropriate shape functions are constructed by using the homogeneous governing equations, and based on the new shape functions, a novel element is formulated. An iterative procedure is proposed for determining both the exact natural frequency values and the corresponding vibration mode shapes. Exact solutions can also be obtained for undamped harmonic response analyses by using the new element, as its stiffness and mass matrices are exact for a specified frequency. Illustrative examples are presented to demonstrate the effectiveness of the proposed element and algorithm.

Exact solution of vibration problems of frame structures

“…The algorithms are described in detail in [12] and will not be presented here. In the following section, numerical results will be presented using the new element and the iterative algorithm to calculate natural frequencies of frame structures.…”

“…The governing equation for the torsional vibration of a beam is similar to that for the axial vibration; therefore, 'exact' stiffness and mass matrices for torsion can be easily formulated as in [12]. With the available formulations for axial, torsional and bending vibration, a new beam element can be developed by combining deformation in axial, torsional and two transverse directions.…”

“…Recently, this author presented a new approach for developing elements for dynamic analysis [12]. By constructing shape functions based on solutions to the homogeneous governing equations of the motion, he developed a new element for axial vibration of elastic bars, which can give exact solutions when employed together with the proposed algorithms for natural frequency and harmonic response analyses.…”

“…The new frame element can be used to get exact solutions for both natural frequency and undamped harmonic analyses of frame structures. Illustrative examples are presented to demonstrate the effectiveness of the new element and the algorithm.The governing equation for the torsional vibration of a beam is similar to that for the axial vibration; therefore, 'exact' stiffness and mass matrices for torsion can be easily formulated as in [12].With the available formulations for axial, torsional and bending vibration, a new beam element can be developed by combining deformation in axial, torsional and two transverse directions.…”

“…The governing equation for the torsional vibration of a beam is similar to that for the axial vibration; therefore, 'exact' stiffness and mass matrices for torsion can be easily formulated as in [12].…”