Natural frequencies of rectangular plates with free edges

Mizusawa, T.

doi:10.1016/0022-460x(86)90171-9

Cited by 34 publications

(10 citation statements)

References 7 publications

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“…The products of eigenfunctions of vibrating beams [3][4][5][6][7] are the most commonly used two-dimensional trial functions. Other frequently used functions include degenerated beam functions [8], orthogonal characteristic beam polynomials [9][10][11][12][13][14], and spline functions [15]. The use of two-dimensional polynomials as basic functions has attracted special attention in recent years, and noteworthy works in the vibration analysis of general two-dimensional structures of arbitrary boundary conditions include those of Bhat [16], Liew and Lam [17], Liew et al [18] and Liew and Wang [19,20].…”

Section: Introductionmentioning

confidence: 99%

Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method

Zhao

Liew

2004

International Journal of Mechanical Sciences

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Section: Introductionmentioning

confidence: 99%

Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method

Zhao

Liew

2004

International Journal of Mechanical Sciences

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“…A comparison of frequency parameter Ω for homogeneous (α 1 = α 2 = β 1 = β 2 = 0) square (a/b = 1) plate of uniform (γ 1 = γ 2 = 0) thickness obtained by Rayleigh-Ritz method [7,34,37], exact and approximate results using Ritz method [33], Differential quadrature method [11,36], Optimized Kantorovich method [35] and non-uniform thickness obtained by finite element method [6], Rayleigh-Ritz method [7,9], differential quadrature method [11], optimized Kantorovich method [35] has been presented in Table 2. A close agreement of results is obtained.…”

Section: Resultsmentioning

confidence: 99%

Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

Lal

Kumar

2012

Shock and Vibration

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Abstract. The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

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“…, where m and n are the numbers of beam vibration nodes, then the plate mode shape function has to manifest m + n cross straight lines of zero displacement, Fig 3. Such a mesh can be noticed only in the modes 1, 8, and 17, and roughly in the modes 4,5,6,7,11,12,15,16,20,21, and 24 in Fig. 2.…”

Section: Generalmentioning

confidence: 88%

An Analytical Solution to Free Rectangular Plate Natural Vibrations by Beam Modes – Ordinary and Missing Plate Modes

et al. 2016

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SummaryRelatively simple analytical procedures for the estimation of natural frequencies of free thin rectangular plates, based on the Rayleigh quotient and the Rayleigh-Ritz method, are presented. First, natural modes are assumed in the usual form as products of beam natural modes in the longitudinal and transverse directions, satisfying the grillage boundary conditions. Based on a detailed FEM analysis, the missing of some natural modes, defined as a sum and a difference of the cross products of beam modes, is noted. The frequencies of these modes are very similar and identical in some special cases, manifesting in such a way a double frequency phenomenon. These families of natural mode shapes form a complete natural frequency spectrum of a free rectangular plate as a novelty. The reliable approximation of natural modes enables the application of the Rayleigh quotient for the estimation of higher natural frequencies. The application of the developed procedure is illustrated by the case of a free thin square plate. The obtained results are compared with those determined by FEM and also with rigorous ones from the relevant literature based on the Rayleigh-Ritz method. The achieved accuracy is acceptable from the engineering point of view. Furthermore, the same problem is solved by the Rayleigh-Ritz method using approximate natural modes as mathematical ones. Direct and iterative procedures are presented. A small number of mathematical modes and iteration steps are sufficient to achieve reliable results.

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Natural frequencies of rectangular plates with free edges

Cited by 34 publications

References 7 publications

Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method

Free vibration of two-side simply-supported laminated cylindrical panels via the mesh-free kp-Ritz method

Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation

An Analytical Solution to Free Rectangular Plate Natural Vibrations by Beam Modes – Ordinary and Missing Plate Modes

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