1. Introduction

In harmonic problems, the system under study is subject to an excitation that varies as the product of any function of space by a sinusoidal function of time.

Finding the answer consists in calculating the field of quantities represented by the degrees of freedom of the finite element modeling of the system. When the system behaves linearly, the response of the field of the observed quantities tends rapidly (due to the extinction of its transient component by internal dissipation) towards a steady state: the resulting field finally varies harmonically like the excitation. It is this steady state of response that we propose to calculate.

General notes:

\(t\)	:	time
\(P\)	:	Current point of the model
\(\omega\)	:	Pulse (\({\mathit{rad.s}}^{\mathrm{-}1}\))
\(j\)	:	Pure unitary imaginary \(({j}^{2}\mathrm{=}\mathrm{-}1)\)
\(M\)	:	Mass matrix from finite element modeling
\(K\)	:	Stiffness matrix from finite element modeling
\(C\)	:	Damping matrix from finite element modeling
\(q\)	:	Vector of degrees of freedom from finite element modeling
\({\mathrm{f}}_{\mathrm{ext}}\)	:	Vector of forces external to the system
\(\Phi\)	:	Matrix of vectors of the base of substructures
\(\eta\)	:	Vector of generalized degrees of freedom