Natural Vibrations of Ribbed Cylindrical Shells with Low Shear Stiffness

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A polynomial approximation of displacement components is used for the numerical analysis of the natural vibrations of a cylindrical shell reinforced with stringers and rings. It is shown by way of numerical examples that the sign of the eccentricity of ribs has a weak effect on the natural frequencies Introduction. The natural vibrations of rib-reinforced shells have been studied adequately [1, 3, etc.]. However, there are issues still unresolved. Among them is the influence of the eccentricity of ribs (whether they are on the inside or outside surface of the shell) on the natural frequencies. This issue was earlier attacked with the formulas derived from the theory of structurally orthotropic shells and the formulas derived considering the discrete arrangement of ribs and using the monomial approximation of the displacement components [2]. The stiffness characteristics of ribs and the error of the monomial approximation were examined in [1-3].We will analyze the effect of the sign of the eccentricity of ribs on the natural frequencies of cylindrical shells. To this end, we will use equations of motion that allow for the discrete arrangement of ribs and are based on a polynomial approximation of the displacement components. Note that the influence of the discrete arrangement of ribs on the natural frequencies of cylindrical shells and rectangular plates and the wave numbers of harmonic waves propagating in them was studied in [4][5][6][7][8].1. We will use the equations of motion [2, 6] that allow for the discrete arrangement of ribs and are based on the assumptions that the motions of the casing and one-dimensional reinforcement (ribs) are described by the classical theories of shells and rods, respectively, and that the ribs are rigidly fastened to the casing. We will consider shells regularly reinforced with stringers and rings (the geometrical and mechanical parameters of all ribs are equal, the distances between ribs are equal, and the distance from the edge of the shell to the nearest ring rib is equal to the distance between ring ribs). 2.As in similar problems [2], we assume that the shell is hinged at the edges. Then it is convenient to seek the solution of the system of equations of motion in the form of double trigonometric series:

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confidence: 99%

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confidence: 99%

Influence of the eccentricity of ribs on the natural frequencies of ribbed cylindrical shells

Zarutskii

Podil’chuk

2010

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“…The propagation of harmonic waves in plates reinforced with unidirectional ribs was earlier addressed in [2-4, 10, 22, 29]; however, wave propagation along a ribbed plate was studied only in [2,22] without analysis of the influence of discrete ribs on the wavenumbers. The effect of discrete ribs on the wavenumbers of harmonic waves and on the natural frequencies of ribbed cylindrical shells was examined in [33,[35][36][37][38].By solving the second problem, we will study the natural vibrations of rectangular plates hinged along all edges and reinforced with an orthogonal grid of ribs working in out-of-plane and in-plane bending, torsion, and tension/compression. The computational algorithm will be based on double trigonometric series and the Galerkin method, as in [34].…”

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confidence: 99%

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confidence: 99%

Influence of discrete ribs on the vibrations of rectangular plates

Zarutskii

Prokopenko

2006

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A rectangular plate reinforced with longitudinal ribs is considered. The influence of the discrete ribs on the wavenumbers of harmonic waves propagating along the ribs is investigated. The effect of the stiffness parameters of the ribs on the natural frequencies and modes of a cross-ribbed plate is studied Introduction. Ribbed plates and shells are widely used as structural members in shipbuilding, aircraft engineering, civil engineering, and other branches. Methods of design of ribbed shells subject to dynamic loads are developed in a great many studies [1]. Far fewer publications discuss techniques for and results of studying the vibrations of ribbed rectangular plates. Recent works mainly accounted for the discrete arrangement of ribs and their asymmetry about the midplane of the plate. Two problem statements were used: (i) ribs are thin bars rigidly fixed to the plate [5, 8, 11-21, 23-25, 27, 28, 30-32, etc.] and (ii) a ribbed plate is a plate with stepwise varying thickness [7, 8, 26, etc.]. The latter statement appears less general than the former one, since it is incapable of describing all the variety of rib profiles used in practice and of accounting for the twisting and bending of ribs in a plane tangent to the plate surface, which, as will be shown below, affect considerably, among other things, the natural frequencies of vibration.We will further use the former statement for two dynamic problems: (i) propagation of harmonic waves along a plate reinforced with longitudinal ribs and (ii) natural vibrations of a plate reinforced with a rib grid. To solve the former problem, we will use the exact solution of the equations of motion for a plate reinforced with longitudinal ribs and hinged along the edges parallel to the ribs [32]. The natural and forced vibrations of plates reinforced with unidirectional ribs were earlier addressed in [5, 13-14, 16, 20, 28, 31], wherein an exact solution for a ribbed panel expressed in terms of hyperbolic trigonometric functions was employed. Such an approach is inconvenient for thin plates with widely spaced rigid ribs, since in this case hyperbolic trigonometric functions rapidly change with distance from a rib, which involves common computational difficulties. A technique similar to that used here was proposed in [16,20]. However, the representations of solution used by the authors did not allowed them to derive simple relations needed to analyze the influence of discrete ribs. A more convenient general solution was obtained in [32]. It was mainly used to assess the applicability limits of approximate approaches to the stress-strain analysis of ribbed plates. We will use this solution to study, by way of a numerical example, the influence of discrete ribs on the wavenumbers of harmonic waves propagating along a ribbed plate. The propagation of harmonic waves in plates reinforced with unidirectional ribs was earlier addressed in [2-4, 10, 22, 29]; however, wave propagation along a ribbed plate was studied only in [2,22] without analysis of the influence of discrete ribs...

“…The natural vibrations of ribbed shells were theoretically analyzed in [4][5][6]. However, despite a great many studies in this area, the problem remains open, in particular, for shells stiffened with high ribs or with rectangular plates.…”

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Experimental study of the influence of rectangular plates stiffening a cylindrical shell on its natural frequencies and modes

Sivak

2006

The results are presented of studying the influence of rectangular plates on the natural frequencies and modes of a fluid-filled cylindrical shell with different end conditions Experimental data collected so far [1] indicate that the effect of longitudinal ribs on the natural frequencies and vibration modes of shells has been studied well enough. The natural vibrations of ribbed shells were theoretically analyzed in [4][5][6]. However, despite a great many studies in this area, the problem remains open, in particular, for shells stiffened with high ribs or with rectangular plates. A problem similar to that stated below was addressed earlier in [3]. This paper analyzed the stability of an axially compressed elastic system consisting of a ribbed cylindrical shell and rectangular plates inserted into it. The purpose of the present paper is to study experimentally the influence of stiffening plates and end conditions on the natural frequencies and modes of shells.1. Characteristics of the Test Shells. Two cylindrical shells were tested. Shell No. 1 has radius R = 15 cm, length L = 45 cm, and thickness h = 0.05 cm. Shell No. 2 was made from shell No. 1 by fastening six plates with screws to the inside surface of the shell along its meridians. The plates have length L 1 = 43 cm, width b = 15.5 cm, and thickness h 1 = 0.05 cm. Each plate has two 0.8 cm flanges along its entire length. Each flange has five screw holes spaced at 8.7 cm. The shells are stiffened with aluminum bands, 1 cm wide and 0.05 cm thick, on the outside surface near the ends. The shells, plates, and bands are made of AMg-6M aluminum alloy. The ready-assembled shell No. 2 is schematically shown in Fig. 1 without the bands.2. Experimental Results. A schematic of the experimental setup was presented in [2]. Figure 2 shows the arrangement of the shells. Shell No. 3 was vertically inserted into the circular slots of steel disks 2 filled with Wood's metal melt to a depth of 0.5 cm, which, when hardened, rigidly fixed the lower end of the shell. The disks are rigidly attached to massive bedplate 1. The upper end of the shell was left free to provide the first variant of end conditions (Fig. 2à) and was inserted into the circular slot of massive disk 4 and also filled with Wood's metal melt to provide the second variant of end conditions (Fig. 2b). Shells Nos. 1 and 2 were filled with service water. The water level was varied from Í 1 = 0 to Í 1 = 1 (Í 1 = Í/L, H is the water level). The natural frequencies and modes of the shells were determined by the resonance method. The vibrations of the shells were excited with an electrodynamic vibrator. Its mobile coil, 5.42 g in mass, was rigidly fixed to the shell's wall, in its midsection. In Fig. 2, the point of attachment of the coil is indicated by an arrow. The methods of excitation and measurement of natural frequencies and modes were detailed in [2]. The experiments were conducted at fixed levels of excitation. Figure 3 shows the dependence of the experimental minimum natural frequencies f e min on the water...