A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

• Published in: Applied mathematics and mechanics Vol. 44; no. 12; pp. 2243 - 2252
• Main Authors: Hołubowski, R., Jarczewska, K.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023; Springer Nature B.V; Faculty of Civil Engineering,Wroc?aw University of Science and Technology,Wroc?aw 50-370,Poland
• Edition: English ed.
• Subjects: Algorithms; Applications of Mathematics; Axial forces; Boundary conditions; Classical Mechanics; Differential equations; Euler-Bernoulli beams; Fluid- and Aerodynamics; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Ordinary differential equations; Parameters; Partial Differential Equations; Resonant frequencies; Segments; 74H15; O302; multi-segment beam; natural vibration; two-point boundary value problem; two-parameter multiple shooting method; non-prismatic beam; 34B60; two-point boundary value prob-lem
• ISSN: 0253-4827; 1573-2754
• DOI: 10.1007/s10483-023-3062-5

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved. A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams. The proposed algorithm, named as two-parameter multiple shooting method, is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions. The impact of the axial force and additional point masses is also taken into account. Due to the fact that the method is based directly on the fourth-order ordinary differential equation, the structures do not have to be divided into many small elements to obtain an accurate enough solution, even though the geometry is very complex. To verify the proposed method, three different examples are considered, i.e., a three-segment non-prismatic beam, a prismatic column subject to non-uniformly distributed compressive loads, and a two-segment beam with an additional point mass. Numerical analyses are carried out with the software MATHEMATICA. The results are compared with the solutions computed by the commercial finite element program SOFiSTiK. Good agreement is achieved, which confirms the correctness and high effectiveness of the formulated algorithm.