1. Introduction, calculation hypotheses#
This documentation describes the methods used in macro CALC_ESSAI.
1.1. Problem position#
We consider a structure, which constitutes a dynamic system that is assumed to be linear (hypothesis H1) .
It is assumed that this structure is subject to an excitation that is supposed to be random (typical example: turbulent fluid forces), but that can be perfectly described using its DSP (power spectral density, see the document « Modeling of turbulent excitations » R4.07.02, section 2.1). This vibration is measured at a certain number of x-axis points \({x}_{k}(1\le k\le \text{nmes})\). For their part, the efforts applied to the structure are not measurable. We therefore want to regain efforts based on measurement. This is an inverse problem, for the resolution of which it is necessary to take a lot of care in order to obtain a result that, while not accurate, is nevertheless relevant.
1.2. Modal behavior hypothesis#
The structure is assumed to be linear (H1), so its mechanical behavior can be described on a modal basis. Each mode i of the structure is defined by the following parameters:
\({\omega }_{i}\): natural pulsation, \({f}_{i}\) associated natural frequency,
\({\xi }_{i}\): reduced amortization,
\({m}_{i}\): modal mass,
\({\varphi }_{i}(\underline{x})\): modal deformed.
The flow exerts a \(f(\underline{x},t)\) force on the structure. It is assumed that these efforts are applied in one direction. The two measurement directions will be treated separately, assuming that they are not coupled.
Movement on a modal basis: it is assumed that the movement of the structure is fairly well described by its first \(N\) modes:
\(u(\underline{x},t)\mathrm{\approx }\mathrm{\sum }_{i\mathrm{=}1}^{N}{q}_{i}(t)\text{.}{\varphi }_{i}(\underline{x})\)
The generalized coordinates then verify the decoupled equations:
\({m}_{i}({\ddot{q}}_{i}+2{\xi }_{i}{\omega }_{i}{\dot{q}}_{i}+{\omega }_{{i}^{2}}{q}_{i})\mathrm{=}\underset{0}{\overset{L}{\mathrm{\int }}}{\varphi }_{i}(\underline{x})f(\underline{x},t)\text{dx}\mathrm{=}{Q}_{i}(t)\mathrm{\iff }{m}_{i}(\mathrm{-}{\omega }^{2}{\stackrel{}{q}}_{i}+2{\xi }_{i}{\omega }_{i}\omega {\stackrel{}{q}}_{i}+{\omega }_{i}^{2}{q}_{i})\mathrm{=}{Q}_{i}(\omega ),{}^{}1\mathrm{\le }i\mathrm{\le }N\)