mathematics Article Transient Dynamic Analysis of Unconstrained Layer Damping Beams Characterized by a Fractional Derivative Model Mikel Brun * , Fernando Cortés and María Jesús Elejabarrieta   Citation: Brun, M.; Cortés, F.; Elejabarrieta, M.J. Transient Dynamic Analysis of Unconstrained Layer Damping Beams Characterized by a Fractional Derivative Model. Mathematics 2021, 9, 1731. https:// doi.org/10.3390/math9151731 Academic Editor: Michael Booty Received: 27 May 2021 Accepted: 20 July 2021 Published: 22 July 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/). Department of Mechanics, Design and Industrial Management, University of Deusto, Avda. de las Universidades 24, 48007 Bilbao, Spain; fernando.cortes@deusto.es (F.C.); maria.elejabarrieta@deusto.es (M.J.E.) * Correspondence: mikel.brunm@deusto.es Abstract: This paper presents a numerical analysis of the influence of mechanical properties and the thickness of viscoelastic materials on the transient dynamic behavior of free layer damping beams. Specifically, the beams consist of cantilever metal sheets with surface viscoelastic treatment, and two different configurations are analyzed: symmetric and asymmetric. The viscoelastic material is characterized by a five-parameter fractional derivative model, which requires specific numerical methods to solve for the transverse displacement of the free edge of the beam when a load is applied. Concretely, a homogenized finite element formulation is performed to reduce computation time, and the Newmark method is applied together with the Grünwald–Letnikov method to accomplish the time discretization of the fractional derivative equations. Amplitudes and response time are evaluated to study the transient dynamic behavior and results indicate that, in general, asymmetrical configurations present more vibration attenuation than the symmetrical ones. Additionally, it is deduced that a compromise between response time and amplitudes has to be reached, and in addition, the most influential parameters have been determined to achieve greater vibration reduction. Keywords: transient analysis; finite element method; unconstrained damping beam; viscoelastic material; fractional derivative model 1. Introduction Vibration reduction is a recurrent topic in mechanical engineering. Vibration generates noise, unpleasant motions and dynamic stress that may cause the failure of a system, and may result in energy losses and structure degradation [1]. Among the different approaches that can be employed to reduce structural vibration, surface treatments are widely used by means of passive damping techniques [2–5]. Some of the most frequently used here are free layer damping (FLD) and constrained layer damping (CLD) configurations. It is known that CLD configurations provide better damping results with respect to the mass-damping relation. Numerous studies have been carried out employing CLD configurations. For example, in [6,7], a CLD beam is analyzed in order to minimize the vibration energy in the frequency domain. In [8,9], an optimization in the frequency domain is made to different parameters such as the density or thickness of different viscoelastic materials applied in a CLD plate configuration in order to maximize the modal loss factor and minimizing the displacement response. A new formulation to reduce computation time, such as an homogenized formulation for plates with a thick constrained viscoelastic core, is carried out in [10] in the frequency domain. However, FLD configurations are still a common occurrence due to being easier to procure and they have a lower economic value [11]. Therefore, this paper is focused on these configurations. One way to mathematically characterize a viscoelastic material is with fractional models, because they allow to reproduce the damping behavior with a reduced number of parameters that, in addition, have a physical interpretation [12,13]. The definition and study of fractional calculus and its applications can be found widely in literature, for example in [13–17]. Several studies have been carried out using fractional derivative models in Mathematics 2021, 9, 1731. https://doi.org/10.3390/math9151731 https://www.mdpi.com/journal/mathematics