mathematics Article Vibration Response of a Concrete Structure with Repetitive Parts Used in Civil Engineering Sorin Vlase 1,2 , Marin Marin 3, * , Maria Luminita Scutaru 1 and Catalin Pruncu 4   Citation: Vlase, S.; Marin, M.; Scutaru, M.L.; Pruncu, C. Vibration Response of a Concrete Structure with Repetitive Parts Used in Civil Engineering. Mathematics 2021, 9, 490. https://doi.org/10.3390/math9050490 Academic Editor: Simeon Reich Received: 31 January 2021 Accepted: 22 February 2021 Published: 27 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Department of Mechanical Engineering, Transilvania University of Bras , ov, 500036 Bras , ov, Romania; svlase@unitbv.ro (S.V.); lscutaru@unitbv.ro (M.L.S.) 2 Romanian Academy of Technical Sciences, 700506 Bucharest, Calea Victoriei, Romania 3 Department of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Bras , ov, Romania 4 Department of Mechanical Engineering, Imperial College, London, Exhibit Rd, London SW7 2AZ, UK; c.pruncu@imperial.ac.uk * Correspondence: m.marin@unitbv.ro; Tel.: +40-268-722-643-020 Abstract: The paper studies the vibration behavior of a concrete structure, currently used in civil engineering. The truss structure considered has symmetries that can be used to facilitate both the de- sign and construction of the building. Moreover, the symmetries encountered can be used to simplify the calculation of vibrations of the system. Based on the mechanical model built, eigenvalues and eigenvectors of such mechanical system are determined, and properties, specific to these symmetries, are identified. In this way, the dynamic analysis of the structure can be simplified and also, the design as well as the costs related to this stage. Keywords: vibrations; symmetry; symmetrical parts; repetitive cell; eigenmodes; building; concrete 1. Introduction Symmetries are common in civil engineering, in all types of buildings and construc- tions. They are also common in structures with large roofs such as industrial warehouses, different exhibition halls, sports complexes, covered stadiums, and a large range of pub- lic places and other modern facilities. Using symmetries involves many advantages for the builder. First, the design time is reduced, then the calculation of these mechanical systems is also reduced considering the substructures that can be treated separately [1]. Furthermore, the manufacture of these buildings becomes easier as the number of different elements decreases. Advantages are also offered in the case of vibration calculation where it is found that the number of operations that can be performed for such an analysis can be significantly reduced [2,3]. All these elements can lead to lower costs related to structures that have symmetries. The existence of symmetries and their use has long been observed and are used by designers and users of the finite element method in practical applica- tions [4]. The main idea that helps the builder and designer is that the information needed to describe such a system is reduced, which allows a reduction of the operations required for design and calculation and to facilitate both static and dynamic analysis. Particular cases of systems with different symmetries have been studied by different authors and presented in the papers [5–10]. In general, research has focused more on how these symmetries influence the formal aspects and how to write equations of motion [11–13]. Practical, engineering applications have been less addressed in the literature, although designers and users of computer programs know some properties and use them. Symmetries can occur not only in civil engineering but also in manufacturing industry, as well as in automotive and aerospace engineering where parts of a product or of a machine are repetitive or identical. We note that the existence of repetitive elements has long been observed in the aviation industry and has become a relatively common method to reduce computational time. Parts Mathematics 2021, 9, 490. https://doi.org/10.3390/math9050490 https://www.mdpi.com/journal/mathematics