ISSN 10645624, Doklady Mathematics, 2013, Vol. 87, No. 1, pp. 102–106. © Pleiades Publishing, Ltd., 2013. Original Russian Text © T.A. Mel’nik, G.A. Chechkin, 2013, published in Doklady Akademii Nauk, 2013, Vol. 448, No. 6, pp. 642–647. 102 1. INTRODUCTION Problems in thick cascade junction domains have been addressed relatively recently (see [1, 2]). A new transmission condition at the interface was obtained that is dictated by the behavior of the short rods. It was found in [3] that the concentrated masses on the short rods affect the transmission conditions. For the spec tral problem in such a junction with light and moder ate masses, it was proved that the spectral parameter is also involved in the transmission conditions; i.e., a homogenized operator function arises. As the mass density increases, the long rods begin to vibrate, fol lowing 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 ana logue of surface waves appearing in the solid, the spa tial skin effect, etc. (for more detail, see [3, 4]), which occur in thick cascade junctions with concentrated masses. Concerning vibrations of systems with concen trated masses, we should mention experimental stud ies in which it was shown that the presence of a con centrated mass on a small set of diameter (ε) leads to a great change in the eigenfrequencies of the body. However, a rigorous study of the influence exerted by the concentrated masses on the behavior of such objects relies on mathematical methods and approaches, such as asymptotic analysis and homoge nization theory. The effect of local vibrations was examined in SanchezPalencia’s pioneering work [5]. Later, this subject was addressed in many other math ematical works. For a detailed bibliography concern ing spectral problems with concentrated masses, the reader is referred to [3, 6–9]. For the first time, the skin effect was found in [9]. 2. STATEMENT OF THE PROBLEM Let a, b 1 , b 2 , h 1 , and h 2 be positive numbers satisfy ing the inequalities 0 < b 1 < b 2 < , 0 < b 1 – , b 1 + < b 2 – , and b 2 + < – . It is easy to see that the intervals b 1 – , b 1 + , b 2 – , b 2 + , , , 1 – b 2 – , 1 – b 2 + , and 1 – b 1 – , 1 – b 1 + do not intersect and belong to (0, 1). The interval [0, a] is divided into N identical sub intervals [εj, ε(j + 1)], j = 0, 1,…, N – 1. Here, N is a large positive integer. Consequently, ε = is a small discrete parameter. The model thick cascade junction Ω ε (see Fig. 1) consists of a junction body where γ ∈ C 1 ([0, a]) and =: γ 0 > 0, and a large number of thin rectangles 1 2  h 1 2  h 1 2  h 1 2  h 1 2  1 2  h 2 2  ⎝ ⎛ h 1 2  h 1 2  ⎠ ⎞ ⎝ ⎛ h 1 2  h 1 2  ⎠ ⎞ 1 h 2 – 2  ⎝ ⎛ 1 h 2 + 2  ⎠ ⎞ ⎝ ⎛ h 1 2  h 1 2  ⎠ ⎞ ⎝ ⎛ h 1 2  h 1 2  ⎠ ⎞ a N  Ω 0 x  2 : 0 x 1 a , 0 x 2 γ x 1 ( ) < < < < ∈ { } , = γ 0 a , [ ] min G j 1 () d k ε , ( ) x  2 : x 1 ε j d k + ( ) – ∈ ε h 1 2  , < ⎩ ⎨ ⎧ =  x 2 ε l 1 – 0 , ] ( ∈ ⎭ ⎬ ⎫ , k 1234 , , , , = On New Types of Vibrations of Thick Cascade Junctions with Concentrated Masses T. A. Mel’nik a and G. A. Chechkin b Presented by Academician V.V. Kozlov August 15, 2012 Received August 29, 2012 DOI: 10.1134/S1064562413010389 a Taras Shevchenko National University of Kyiv, Kyiv, Ukraine b Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119992 Russia MATHEMATICS