Approximate formulas for natural frequencies of rectangular plates with linearly varying thickness

1977 ◽  
Vol 61 (4) ◽  
pp. 982-985 ◽  
Author(s):  
Toshiyuki Sakata ◽  
Yoshiko Sakata
Keyword(s):  
Natural Frequencies ◽  
Rectangular Plates ◽  
Varying Thickness ◽  
Approximate Formulas

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

Some Results on Axisymmetric Vibration of Spherical Shells of Variable Thickness

1990 ◽  
Vol 112 (4) ◽  
pp. 432-437 ◽  
Author(s):  
A. V. Singh ◽  
S. Mirza
Keyword(s):  
Variable Thickness ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Axisymmetric Vibration ◽  
Spherical Shells ◽  
Varying Thickness ◽  
Wide Range

Natural frequencies and mode shapes are presented for the free axisymmetric vibration of spherical shells with linearly varying thickness along the meridian. Clamped and hinged edges corresponding to opening angles 30, 45, 60 and 90 deg have been considered in this technical brief to cover a wide range from shallow to deep spherical shells. Variations in thickness are seen to have very pronounced effects on the frequencies and mode shapes.

Layout of Plate Structures for Improved Dynamic Response Using a Homogenization Method

1993 ◽  
Author(s):  
Ciro A. Soto ◽  
Alejandro R. Diaz
Keyword(s):  
Dynamic Response ◽  
Natural Frequencies ◽  
Homogenization Method ◽  
Automotive Application ◽  
Optimum Shape ◽  
Design Practice ◽  
Plate Structures ◽  
Simply Supported ◽  
Varying Thickness ◽  
Square Plate

Abstract A model to compute average properties for Mindlin plates of rapidly varying thickness was introduced in [SOT93]. The model was designed to be of use in computations of the optimum shape and layout of plates using the technique introduced by Bendsøe and Kikuchi [BEN88]. In this paper we discuss the utilization of the model to determine the optimum layout of plate structures that maximizes a function of the structure’s natural frequencies. A simply supported square plate is used to illustrate the problem of optimization in the presence of repeated natural frequencies. An automotive application is presented to illustrate the usefulness in design practice.

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

The Maximum Transient Resonance Response of Rotating Blades With Regard to Centrifugal Force and Non-Linear Damping Effects

1997 ◽  
Author(s):  
Horst D. Irretier
Keyword(s):  
Centrifugal Force ◽  
Natural Frequencies ◽  
Linear Change ◽  
Rotating Blades ◽  
Operation Process ◽  
Non Linear ◽  
Linear Damping ◽  
Pass Through ◽  
Run Up ◽  
Approximate Formulas

During the operation process in many types of fluid flow machines the rotating blades pass through various resonances e.g. during run-up or run-down or other transient conditions. Therefore, for the high cycle fatigue problem of the blades it might be important to consider the transient vibratory response of the blades during these passages through resonance and to get knowledge about the occuring maximum vibratory stresses. In the paper, approximate formulas are presented which allow the estimation of the maximum transient response of the blades. Thereby, the influence of the change of the natural frequencies due to the increasing or decreasing centrifugal force field during the run-up or run-down, respectively, is taken into consideration. Basically, the approximate formulas are based on a linear change of the natural frequencies versus time and on a linear viscous type of damping. Extensions to account for parabolic changes which are more realistic for centrifugal effects and for non-linear damping models e.g. friction damping or turbulence damping are discussed.

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