On free vibrations of thin functionally graded plate bands resting on an elastic foundation

Abstract

In this note free vibrations of plate bands with functionally graded properties, resting on an elastic foundation, are investigated. On the micro-level, these plate bands have a tolerance-periodic structure. It can be shown that in dynamic problems of those objects, the effect of the microstructure size plays a role. This effect can be described in the framework of the tolerance model, which is presented here for these bands. Obtained results are evaluated introducing the asymptotic model. Both fundamental and higher free vibration frequencies of these plate bands are calculated using the Ritz method. The effects of differences of material plate properties in the cell on the microlevel and of the foundation are shown.

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

“…Different thermomechanical problems of functionally graded media with microstructures were also considered using the tolerance modelling approach; e.g., Michalak and Wirowski [ 58 ], Perliński et al [ 59 ] and Wirowski et al [ 60 ] analyzed dynamics and stability of plates with longitudinally graded structures with or without an elastic foundations; free vibrations of transversally graded thin plates on an elastic foundation were presented by Jędrysiak and Kaźmierczak-Sobińska [ 61 ]; the dynamics of a thin-walled structure with a dense system of ribs was considered by Michalak [ 62 ]; Ostrowski and Michalak [ 63 ] showed an application for heat conduction in cylindrical composite conductors with non-uniform microstructure; the effect of microstructure in thermoelastic problems of transversally graded laminates was investigated by Pazera and Jędrysiak [ 64 ]; vibrations of thin plates with the microstructure size of an order of the plate thickness were analyzed by Jędrysiak [ 65 ]; Tomczyk and Szczerba [ 66 ] proposed a new asymptotic-tolerance model of dynamics for thin microstructured transversally graded shells; Jędrysiak [ 67 ] considered free vibrations of medium thickness functionally graded plates with a microstructure.…”

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

“…The tolerance modelling approach can be also successfully applied to consider various problems of thermomechanics for functionally graded structures. For instance: investigations of dynamics for longitudinally graded plates were presented in References [ 75 , 76 , 77 ]; vibrations of thin functionally graded plates with the size of the microstructure of an order of the thickness of the plate were analysed in Reference [ 78 ]; vibrations of thin transversally graded plates with the thickness smaller than the microstructure size were shown in Reference [ 2 , 79 , 80 ]; dynamics of a thin-walled structure with a system of ribs was presented in Reference [ 81 , 82 ]; heat distribution of composite cylindrical conductors having non-uniform microstructure was analysed in Reference [ 83 , 84 ]; problems of thermoelasticity in transversally graded laminates were considered in Reference [ 85 ]; dynamics of thin functionally graded microstructured shells was presented in Reference [ 86 , 87 , 88 ].…”

“…It seems that most of them show models based on the tolerance modelling method, cf. References [ 2 , 60 , 75 , 76 , 77 , 78 , 79 , 80 , 81 , 82 , 86 , 87 , 88 ], and between them References [ 76 , 88 ] present stability problems of functionally graded plates or shells.…”

The effect of uncertain material properties on free vibrations of thin periodic plates

“…This modelling method is applied successfully to analysis some non-stationary and stationary problems of functionally graded structures, e.g. stability of functionally graded structure interacting with elastic heterogeneous subsoil is considered by Perliński et al [35]; transversally graded thin plates are analysed by Jędrysiak and Kaźmierczak [36]; Pazera and Jędrysiak [37] show thermoelastic phenomena in transversally graded laminates; heat conduction phenomenon in micro-heterogeneous solids was investigated by Ostrowski and Michalak [38,39].…”

“…Numerical simulations are made using the Monte Carlo method, in which it is assumed that ratios (36) are calculated around of their expected value 0.5 in 10,000 points. Since formulas (37) are quite complex, we expect that the new random variables of frequencies might be not of normal distributions (nonvanishing skewness and kurtosis).…”