Tolerance Modelling of Vibrations and Stability for Periodic Slender Visco-Elastic Beams on a Foundation with Damping. Revisiting

Abstract

The mathematical modelling of certain problems of vibrations and stability for periodic slender visco-elastic beams is presented in this note. To consider these problems and take into account the effect of the microstructure, the tolerance modelling approach is proposed. Using this technique, the equation with non-continuous, periodic, highly oscillating coefficients is replaced by a system of differential equations with constant coefficients. Moreover, these governing equations describe the effect of the microstructure on the overall behavior of the beams under consideration. The tolerance modelling can lead to equations of two different tolerance models—the standard and the general, under weakened assumptions. This averaging tolerance method was assessed by comparison with the asymptotic homogenization, the governing equations of which omit this effect. My considerations were limited to proposing and presenting only mathematical models describing investigated beams. In a simple analytical example, the application of the presented average models is shown.

A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

“…f 1 , f 2 , h 1 , and h 2 are depicted in Figure 2. By considering the micro-macro decomposition assumption, and the other assumptions and definitions of the tolerance modelling discussed in [23][24][25], Equation (1) describing the heat conduction issue was averaged, leading to the tolerance model equations:…”

“…By considering the micro-macro decomposition assumption, and the other assumptions and definitions of the tolerance modelling discussed in [ 23 , 24 , 25 ], Equation (1) describing the heat conduction issue was averaged, leading to the tolerance model equations: where K stands for the thermal conductivity tensor whose components are k ij , ∇ is a gradient operator defined as (∂ 1 , ∂ 2 , ∂ 3 ), overlined ∇ is a gradient in x 3 direction (0, 0, ∂ 3 ), and ∂ is a gradient operator defined as (∂ 1 , ∂ 2 , 0).…”

Tolerance Modelling of Vibrations of a Sandwich Plate with Honeycomb Core

“…[ 25 ]. Within the literature, one can find multiple applications of this technique in various mechanical issues, such as stability analysis [ 26 , 27 , 28 , 29 ], dynamics [ 30 , 31 , 32 , 33 ] or even heat conduction issues [ 34 , 35 , 36 , 37 ].…”

Critical Temperatures for Vibrations and Buckling of Magneto-Electro-Elastic Nonlocal Strain Gradient Plates

“…In [63] the flexural vibration band gaps in periodic beams is investigated using differential quadrature method, moreover the influence of shear deformation on the gaps is analyzed. Similarly, natural frequencies of structures made of period cells was presented for beams in [64,65] and for plates in [66,67]. In [68] the aim is the problem of vibration of band gaps in periodic Mindlin plates and it is solved using spectral element method.…”