Invariance properties in deformable flexion-torsion loaded bodies spectra

We present here in the framework of resonnant behaviors an eigenfrequency invariance properties for loaded flexural and torsional deformable bodies. We found that under a preload condition each eigenfrequency is invariant versus the tipload . For the mode 7 and beyong the preload conditions are less than , and the seven first modes (form 0 to 6) can be easily modelled by polynomial functions of in the neighbourhood of useful . To further improve this result, the role of passive system parameters has been studied. We will show that compliant behaviors at mechanical joints create a new order on the system which leads to a large invariance property in the space. Compliant mechanical joint diminishes drastically the preload condition, as shows a lower threshold value for and a lower invariance mode number ( now typically 3 ) moreover the invariance is more precise. These properties open the way to new finite dimension controllers.