Dynamics of transversally vibrating non-prismatic Timoshenko cantilever beams

• Published in: Engineering structures 2018-07, Vol.166, p.511-525
• Main Authors: Navadeh, N., Hewson, R.W., Fallah, A.S.
• Format: Article
• Language: English
• Subjects: Cantilever beams; Cantilevers; Differential equations; Dynamic response; Eigenfunction expansion method; Eigenvalues; Eigenvectors; Galerkin method; Mathematical analysis; Mathematical models; Non-dimensionalisation; Non-prismatic cantilever; Ordinary differential equations; Partial differential equations; Resonant frequencies; Series expansion; Shape functions; Structural engineering; Timoshenko beam
• ISSN: 0141-0296; 1873-7323
• DOI: 10.1016/j.engstruct.2018.03.088

•Dynamics of a generic non-prismatic Timoshenko cantilever is considered.•Eigenmodes and eigenfrequencies are obtained using a complete set of functions.•Eigenfunction expansion method is used in the evaluation of dynamic response.•Strong correlation between the proposed model and its 2D FE counterpart is observed.•Non-dimensional parameters are introduced to generalise the study conducted. The present study deals with evaluation of the dynamic response in a pulse loaded homogeneous non-prismatic Timoshenko cantilever beam. Subsequent to the derivation of the partial differential equations (PDE’s) of motion using the Lagrange-d’Alembert principle (or extended Hamilton’s principle) the eigenvalue problem has been set up and solved for eigenfrequencies and eigenfunctions. Galerkin’s method of weighted residuals was then applied to obtain governing ordinary differential equations (ODE’s) for the system. The dynamic response under arbitrary pulse loading is obtained using the method of eigenfunction expansion which attributes to displacement and rotation fields generalised coordinates when the exact modes are chosen as shape functions. It has been shown that inclusion of few terms (in this case 5) in the series expansion provides a good correlation between the displacement fields and the truncated series. Dimensionless response parameters are introduced and two methods of non-dimensionalisation are proposed which could be useful in dealing with generic problems of a specified formulation.