Acta Scientiarum Polonorum

The aim of the study is to develop an analytical reformulation the known theory for thin periodically nonhomogeneous
viscoelastic beams, which should to form the basis for the planned research of the full beam
dispersion relation. The methodology applied in the paper is similar to that described in paper Tolerance Modelling
of Vibrations and Stability for Periodic Slender Visco-Elastic Beams on a Foundation will Damping.
Revisiting by Jarosław Jędrysiak, published in scientific journal “Materials” in 2020, but it differs in the way
of the use of tolerance modeling in the beam theory. The mathematical tool of considerations is a special
choice of the micro-macro decomposition of the beam deflection. It is based on a certain regularization of the
displacement field (in the small neighborhood of discontinuity surfaces) and results a certain reformulations
of the classical beams theory. Obtained model equations is an alternative proposal for model equations obtained
in mentioned paper by Professor Jędrysiak as a certain approximation for periodically nonhomogeneous
viscoelastic beams theory as a result of the original tolerance modeling developed. The standardization of
the beam deflection field proposed in the presented paper allows us to use the infinite Fourier expansion for
deflection field in any region occupied by a homogeneous periodic composite material, which can be modeled
in a typical way using the virtual work principle for commonly used beam constrains. The deflection field of
the beam is written in the form of an infinite Fourier series, using periodically distributed region of material
homogeneity of the beam. Applied method can be viewed as an attempt to use an infinite number of shape
functions in the tolerance modeling but at the same time as an equivalent reformulation of the equations of
the beam vibration theory. The obtained system of equations is an infinite system of ordinary differential
equations for infinitely many Fourier coefficients in the mentioned Fourier expansion of the beam deflection
field with respect to the base described in the surface location method