A Galerkin method for distributed systems with non-local damping

Abstract

In this paper, a non-local damping model including time and spatial hysteresis effects is used for the dynamic analysis of structures consisting of Euler-Bernoulli beams and Kirchoff plates. Unlike ordinary local damping models, the damping force in a non-local model is obtained as a weighted average of the velocity field over the spatial domain, determined by a kernel function based on distance measures. The resulting equation of motion for the beam or plate structures is an integro-partial-differential equation, rather than the partial-differential equation obtained for a local damping model. Approximate solutions for the complex eigenvalues and modes with non-local damping are obtained using the Galerkin method. Numerical examples demonstrate the efficiency of the proposed method for beam and plate structures with simple boundary conditions, for non-local and non-viscous damping models, and different kernel functions.

A new displacement-based framework for non-local Timoshenko beams

“…Depending on the formulation, the non-local terms always involve a number of unknown parameters/functions that only experimental evidence may allow to select. For instance, examples of enriched continua with five additional parameters for non-local terms exist in the literature [12,50], or non-local beam models with different potential choices of the attenuation functions [49,51]. Obviously, any optimization procedure to fit experimental evidence shall be generally preceded by numerical simulations, that may serve to determine the expected order of magnitude of the non-local terms, with respect to the local ones.…”

Direct Time-Domain Integration Approach for Viscoelastic Systems Involving Various Damping Models

“…Some authors developed exact state-space approaches [4][5][6][7][8][9][10][11][12] based on internal variables. Others proposed approximated approaches [13][14][15][16][17][18][19][20] and model reduction methods [21][22][23][24][25][26] to solve the eigenproblem for the exponential-like damping models. The previously mentioned methods are devoted to efficiently solve the nonviscous eigenproblem under different damping models.…”

On the Damped Beams with Hysteresis

“…A specific feature exhibeted by auxetic materials in comparison with other foams is their significant damping capacity at various loading levels, with increase up to 16 times compared to conventional foams [11][12][13]. The size-dependent theories of the damping capacity of auxetic foams have received increasing attention in recent years due to the need to model and analyse very small-sized mechanical interactions inn the rapid developments of micro-nanotechnologies [14][15][16]. The nonlocal theory becomes significant when dealing with such problems.…”

“…Such models represent a generalization of the viscous damping, and examples include the beams and nanobeams with external patches [22,23], the auxetic composites [24][25][26], the micropaddles with the periodic auxetic cores [27], the multifunctional nanofoils based on the carbon nanotubes and the auxetic foams [14]. A nonlocal damping model including the time and spatial hysteresis effects for beams and plates is analysed in [14] by using the Galerkin method.…”

“…A nonlocal damping model including the time and spatial hysteresis effects for beams and plates is analysed in [14] by using the Galerkin method. The hysteresis in connection with PDEs and applicative problems are analyzed in [28][29][30][31][32][33][34].…”