On the transient response of plates on fractionally damped viscoelastic foundation
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On the transient response of plates on fractionally damped viscoelastic foundation R. K. Praharaj1 · N. Datta1 Received: 12 April 2020 / Revised: 23 June 2020 / Accepted: 6 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This work underlines the importance of the application of fractional-order derivative damping model in the modelling of the viscoelastic foundation, by demonstrating the effect of various orders of the fractional derivative on the dynamic response of plates resting on the viscoelastic foundation, subjected to concentrated step load. The foundation of the plate is modelled as a fractionally-damped Kelvin–Voigt model. Modal superposition method and Triangular strip matrix approach are used to solve the partial fractional differential equations of motion. The influence of (a) fractional-order derivative, (b) foundation stiffness, and (c) foundation damping viscosity parameter on the dynamic response of the plate are investigated. Theoretical results show that with the increase in the order of derivative, the damping of the system increases, which leads to decreased dynamic response. The results obtained from the fractional-order damping model and integer-order damping model are compared. The results are verified with literature and numerical results (ANSYS). Keywords Fractional viscoelasticity · Plate vibration · Fractional damping · Viscoelastic foundation · Step load Abbreviations x, y, z Axis of the reference system t Time [s] L Length of the plate [m] B Width of the plate [m] D Flexural rigidity of the plate [Pa.m3] h Thickness of the plate [m] E Elastic modulus [ N∕m2 ] 𝜐 Poisson’s ratio C0 Viscous damping coefficient [Ns𝛼 ∕m3 ] K Stiffness of the foundation [N∕m3 ] 𝜌 Density of the beam material [ kg∕m3 ] Communicated by José Tenreiro Machado. * R. K. Praharaj [email protected] 1
Indian Institute of Technology, Kharagpur, India
123 Vol.:(0123456789)
R. K. Praharaj, N. Datta
F Magnitude of force [N] 𝛼 Order of derivative 𝛿 Dirac delta function Γ Gamma function [GM] Generalized mass matrix [GC] Generalized damping [GK] Generalized stiffness matrix {GF} Generalized force vector N Number of modal superposition terms w Dynamic displacement [m] Mx , My Dynamic bending moment [Nm] 𝜔r Rth complex natural frequency 𝜎r Rth damping coefficient Ωr Rth damped natural frequency [rad/s] 𝜁r (t) Rth generalized coordinate 𝜑i (x) Ith beam eigenfunction
1 Introduction Plates resting on a viscoelastic foundation is a universal model for various real-life engineering applications like concrete pavements, building foundations, airport runways, and pier platforms, which are often analyzed by modelling them as plates on a viscoelastic foundation. Such structures are often subject to impulsive and step loads, leading to transient vibrations. To enhance the fatigue life of the structure, excessive vibration can be controlled by both internal and external damping. The internal damping of the structure and its rheological proper
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