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An efficient method to model the time-domain behavior of viscoelastically damped systems based on an improved fractional derivative model

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Abstract

Viscoelastic materials are widely used as an efficient passive control technique to mitigate undesirable vibration and noise in many engineering fields. However, the damping properties of these materials are strongly influenced by operational and environmental factors, such as excitation frequency and operating temperature, leading to some difficulty during the finite element modelling of engineering systems containing these materials, especially for transient analysis. Thus, it is proposed herein an improved fractional derivative model to be combined with the finite element method in a straightforward way to describe the frequency- and temperature-dependent behavior of viscoelastic materials. As an academic application, the influence of the temperature on the transient responses of a viscoelastic sandwich beam subjected to some types of excitations is addressed. Moreover, the parameters of the viscoelastic model are obtained by curve-fitting for each operating temperature and the resulting equations of motion of the viscoelastic system in the time-domain are solved by using the Newmark integration scheme. Through this improved fractional derivative model, the time-domain responses of the viscoelastic sandwich beam are evaluated for several operating temperatures and compared with the corresponding obtained by the open literature to demonstrate the main features and capabilities of the proposed method.

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Abbreviations

A :

Transversal area

A j+1 :

Grünwald coefficients

a E :

Parameter of the model associated to elongation

aE :

Parameter of the model associated to shear

b :

Width

E 0 :

Low frequency Young’s modulus

E ∞ :

High frequency Young’s modulus

{f} :

External force vector

{f v}:

Viscoelastic force vector

G 0 :

Low frequency shear modulus

G ∞ :

High frequency shear modulus

G ⋆ :

Complex shear modulus

h :

Thickness

k :

Layer i dentification

[K] :

Stiffness matrix

li :

Length of the finite element

[M] :

Mass matrix

N :

Time step number

N L :

Memory size

{q (e)}:

Degrees of freedom vector

t :

Time

T :

Temperature

T e :

Elementary kinetic energy

u (k) :

x axis displacement associated to the k-th layer

V e :

Elementary deformation energy

x :

x axis

w :

z axis displacement

z :

z axis

α :

Fractional derivative order

α T :

Shift factor

β :

Viscoelastic layer shear deformation

β E j+1 :

Recurrence term associated to elongation

β G j+1 :

Recurrence term associated to shear

γ :

Shear deformation

Δt :

Time step

ε :

Normal deformation

ν :

Poisson’s ratio

ρ :

Density

σ :

Normal stress

τ :

Shear stress

ω :

Frequency

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Acknowledgments

This work is supported by the Brazilian National Council for Scientific and Technological Development (CNPq), through the research grants 140072/2021-7 (E. P. Nunes) and 306138/2019-0 (A. M. G. de Lima), the Minas Gerais Research Foundation (FAPEMIG), through the research projects APQ01865 and PPM005818 (A. M. G. de Lima), and the Federal University of Uberlândia (UFU).

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Authors and Affiliations

  1. School of Mechanical Engineering, Federal University of Uberlândia, Uberlândia, MG, 38400-902, Brazil

    Erivaldo P. Nunes, Antônio M. G. de Lima & André G. Cunha Filho

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  1. Erivaldo P. Nunes
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  2. Antônio M. G. de Lima
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  3. André G. Cunha Filho
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Correspondence to Erivaldo P. Nunes.

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Erivaldo P. Nunes received his M.Sc. in Mechanical Engineering from the Federal University of Uberlândia — Brazil. He is currently a Ph.D. candidate at the same university. His research interests include composite and viscoelastic materials, vibration control techniques, damage, stochastic modelling and finite elements reduction methods.

Antônio M. G. de Lima is an Associate Professor of Mechanical Engineering at Federal University of Uberlândia in Brazil. He received his Ph.D. in Sciences pour l’Ingénieur from University of Franche-Comté in Besançon, France, and his Ph.D. in Mechanical Engineering from Federal University of Uberlândia. His research interests include vibration control techniques using viscoelastic materials, shape memory alloy and shunt piezoceramics, uncertainty quantification and stochastic modeling, reliability combined with uncertainties, aeroelasticity control techniques in super- and subsonic regimes, robust reduction methods and multiobjective optimization.

André G. Cunha Filho lectures at the Federal Institute of São Paulo — Brazil. He received his Ph.D. in Sciences at the Federal University of Uberlândia in partnership with the Aeronautics Institute of Technology (ITA). His field of research includes passive vibration control techniques applied to linear and nonlinear aeroelastic systems, sub- and supersonic aeroelastic modeling and finite elements reduction methods.

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Nunes, E.P., de Lima, A.M.G. & Cunha Filho, A.G. An efficient method to model the time-domain behavior of viscoelastically damped systems based on an improved fractional derivative model. J Mech Sci Technol 36, 1645–1653 (2022). https://doi.org/10.1007/s12206-022-0303-7

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  • DOI: https://doi.org/10.1007/s12206-022-0303-7

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