ABSTRACT
This equations have highly oscillating coefficients, so they are difficult to solve.
2.2 Tolerance averaging technique
Next we use tolerance averaging technique (TAT) for modeling of dynamic behavior of thin plates. It was presenting by Woz´niak & Wierzbicki (2000). One can find many examples from TAT and bibliography in monograph Woz´niak et al. (2008). This theory defined some operators and lemas. The most important are:
– averaging operator:
where y is local coordinates, – slowly varying function:
where εDF is tolerance parameter,
– -periodic function:
– displacements field disjoined:
where
are basic unknown, qA( · ) are known shape functions. – the most important theorems:
2.3 Averaging equations
We derive the first equation when we average equation of motion:
Next we multiply the equation of motion by test functions and after averaging procedure we get the remaining N equations
After substitution of the constitutive equations, the strain-displacement relations, and the displacement
field, we get averaging equations:
This is the system of N + 1 differential equations. Coefficients in the above system are continuous and slowly-varying functions.
