Nonlinear vibrations and damping of fractional viscoelastic rectangular plates
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ORIGINAL PAPER
Nonlinear vibrations and damping of fractional viscoelastic rectangular plates Marco Amabili
. Prabakaran Balasubramanian . Giovanni Ferrari
Received: 16 January 2020 / Accepted: 12 August 2020 Springer Nature B.V. 2020
Abstract Damping is largely increasing with the vibration amplitude during nonlinear vibrations of rectangular plates. At the same time, soft materials present an increase in their stiffness with the vibration frequency. These two phenomena appear together and are both explained in the framework of the viscoelasticity. While the literature on nonlinear vibrations of plates is very large, these aspects are rarely touched. The present study applies the fractional linear solid model to describe the viscoelastic material behavior. This allows to capture at the same time (i) the increase in the storage modulus with the vibration frequency and (ii) the frequency-dependent nonlinear damping in nonlinear vibrations of rectangular plates. The solution to the nonlinear vibration problems is obtained through Lagrange equations by deriving the potential energy of the plate and the dissipated energy, both geometrically nonlinear and frequency dependent. The model is then applied to a silicone rubber rectangular plate tested experimentally. The plate was glued to a metal frame and harmonically excited by stepped sine testing at different force levels, and the vibration response was measured by a laser Doppler vibrometer. The comparison of numerical and experimental results was very satisfactorily carried out for:
M. Amabili (&) P. Balasubramanian G. Ferrari Department of Mechanical Engineering, McGill University, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, QC H3A 0C3, Canada e-mail: [email protected]
(i) nonlinear vibration responses in the frequency and time domain at different excitation levels, (ii) dissipated energy versus excitation frequency and excitation force, (iii) storage energy and (iv) loss factor, which is particularly interesting to evaluate the plate dissipation versus frequency at different excitation levels. Finally, the linear and nonlinear damping terms are compared. Keywords Nonlinear damping Nonlinear vibrations Fractional viscoelasticity Storage modulus Rectangular plate
1 Introduction Nonlinear vibrations of rectangular plates received significant attention in the literature [1–12]. A rectangular plate is a strongly hardening system, which can be transformed to an initially softening system, turning to hardening for larger vibration amplitudes, in case of geometric imperfections of amplitude comparable with the plate thickness. The geometric nonlinearity gives quadratic and cubic nonlinear stiffness terms in the equations of motion; the cubic terms are largely prevalent on the quadratic ones for perfectly flat plates and plates with small imperfections. Boundary conditions have a strong effect on the nonlinear vibration response of plates [6, 12]. Nonlinear dynamics of
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