An analytical assessment of finite element and isogeometric analyses of the whole spectrum of Timoshenko beams. Issue 10 (8th March 2016)

• Record Type: Journal Article
• Authors: Cazzani, Antonio; Stochino, Flavio; Turco, Emilio
• Is Part Of: Zeitschrift für angewandte Mathematik und Mechanik. Volume 96:Issue 10(2016)
• Page Start: 1220
• Page End: 1244
• Publication Date: 2016-03-08
• Subjects: Structural dynamics -- vibration analysis -- Timoshenko beam -- frequency spectrum -- isogeometric analysis -- Primary 74K10 -- 74H45 -- 74H05 -- Secondary 70J10 -- 70J30 -- 34L10 -- 34L15 -- 35C05 -- 35L25; Mathematics -- Periodicals; Mechanics, Applied -- Periodicals; Engineering -- Periodicals; 519
• DOI: 10.1002/zamm.201500280
• Languages: English
• ISSNs: 0044-2267

Abstract

Abstract : The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l 2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees‐of‐freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes. Abstract : The theoretical results relevant to the vibration modes of Timoshenko beams are here used as benchmarks for assessing the correctness of the numerical values provided by several finite element models, based on either the traditional Lagrangian interpolation or on the recently developed isogeometric approach. Comparison of results is performed on both spectrum error (in terms of the detected natural frequencies) and on the l 2 relative error (in terms of the computed eigenmodes): this double check allows detecting for each finite element model, and for a discretization based on the same number of degrees‐of‐freedom, N, the frequency threshold above which some prescribed accuracy level is lost, and results become more and more unreliable. Hence a quantitative way of measuring the finite element performance in modeling a Timoshenko beam is proposed. The use of Fast Fourier Transform is finally employed, for a selected set of vibration modes, to explain the reasons of the accuracy decay, mostly linked to a poor separation of the natural frequencies in the spectrum, which is responsible of some aliasing of modes.