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Fourier series multiscale method, a concise and efficient analytical approachfor multiscale computation, will be developed out of this series of papers. Inthe seventh paper, the usual structural analysis of beams on an elasticfoundation is extended to a thorough multiscale analysis for a fourth orderlinear differential equation for transverse deflection of the beam, wheregeneral boundary conditions and a wide spectrum of model parameters areprescribed. For this purpose, the solution function is expressed as a linearcombination of the boundary function and the internal function, to ensure theseries expression obtained uniformly convergent and termwise differentiable upto fourth order. Meanwhile, the internal function corresponds to the particularsolution, and the boundary function corresponds to the general solution whichsatisfies the homogeneous form of the governing differential equation. Sincethe general solution has appropriately interpreted the meaning of thedifferential equation, the spatial characteristics of the solution of theequation are expected to be better captured. With the boundary function and theinternal function selected specifically as combination of the linearlyindependent homogeneous solutions of the differential equation, andone-dimensional half-range Fourier sine series over the solution interval, theFourier series multiscale solution of the bending problem of a beam on thePasternak foundation is derived. And then the convergence characteristics ofthe Fourier series multiscale solution are investigated with numericalexamples, and the multiscale characteristics of the bending problem of a beamon the Pasternak foundation are demonstrated for a wide spectrum of modelparameters.

Fourier series (based) multiscale method for computational analysis in  science and engineering: VII. Fourier series multiscale solution for elastic  bending of beams on Pasternak foundations