On new types of vibrations of thick cascade junctions with concentrated masses

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1. INTRODUCTIONProblems in thick cascade junction domains havebeen addressed relatively recently (see [1, 2]). A newtransmission condition at the interface was obtainedthat is dictated by the behavior of the short rods. It wasfound in [3] that the concentrated masses on the shortrods affect the transmission conditions. For the spectral problem in such a junction with light and moderate masses, it was proved that the spectral parameter isalso involved in the transmission conditions; i.e., ahomogenized operator function arises. As the massdensity increases, the long rods begin to vibrate, following the vibrations of the short rods. Accordingly,the effect of a volume (thick) skin in a solid arises,which is known as the spatial skin effect.This paper analyzes new effects, such as an analogue of surface waves appearing in the solid, the spatial skin effect, etc. (for more detail, see [3, 4]), whichoccur in thick cascade junctions with concentratedmasses.Concerning vibrations of systems with concentrated masses, we should mention experimental studies in which it was shown that the presence of a concentrated mass on a small set of diameter (e) leads toa great change in the eigenfrequencies of the body.However, a rigorous study of the influence exerted bythe concentrated masses on the behavior of suchobjects relies on mathematical methods andapproaches, such as asymptotic analysis and homogenization theory. The effect of local vibrations wasexamined in SanchezPalencia’s pioneering work [5].Later, this subject was addressed in many other mathematical works. For a detailed bibliography concerning spectral problems with concentrated masses, thereader is referred to [3, 6–9]. For the first time, theskin effect was found in [9].2. STATEMENT OF THE PROBLEMLet