The Analysis Of In-Plane Free Vibrations Of Curved Beams

Curved beams are common structural and functional elements in traditional and emerging civil and mechanical engineering fields nowadays.There are also many processing methods,including the currently popular 3D printing technology,to make the curved beams for various applications with periodic feature and various combined structures made of curved beams.For existing and potential applications,certain characteristics of curved beam structures,such as deformation and vibration,are important to ensure safety and achieve various functions in design and applications.The objective of this research is vibrations of curved beams,including curved beams of constant and variable curvatures as well as periodic curved beams and combined curved beams.Firstly,the thesis briefly explained the Rayleigh-Ritz method,the Euler-Bernoulli beam theory,and the Timoshenko beam theory for vibration analysis.Then,based on the variational principle,the energy equations of straight beams with two beam theories are presented,and the in-plane free vibration frequencies of straight beams are calculated by the Rayleigh-Ritz method.The results are verified with the standards results in textbooks.Next,the in-plane free vibration frequencies of beams of constant curvature,variable curvature,and periodic patterns were calculated by the same method,and the results were compared with the results in existing literatures and by finite element method(FEM),which shows that our method is capable and accurate to calculate frequencies of the these types of curved beams.At the same time,we also found out that the Euler-Bernoulli beam theory has its limitations when the curved beam has a large ratio of height to span,but the Timoshenko beam theory can perfectly deal with it.Furthermore,in order to analyze the vibration problem of the combined curved beams,we first use the penalty function method to calculate the free vibration frequency of the curved beam under arbitrary boundary conditions.After obtaining the correct results,the penalty function method is extended to the problems of the combined curved beams.Finally,for the better selection of basis function in the Rayleigh-Ritz method,we studied the convergence speed of three kinds of basis functions satisfying free boundary conditions,and obtained the fastest converging basis functions,which provide practical guidance for the choice of basis functions.Through these studies,we can calculate the in-plane free vibration frequency of curved beams accurately.This research provides a theoretical basis for the analysis of safety problems of curved beam structures due to vibrations in engineering applications.