Further Modification of Bolotin Method in Vibration Analysis of Rectangular Plates

Abstract

A further modi cation of the Bolotin method for the determination of the natural frequencies and mode shapes of isotropic and orthotropic rectangular plates with various types of boundary conditions is given. Unlike the Bolotin method (BM) or the modi ed Bolotin method (MBM), the present approach does not postulate the formula for the eigenfrequency, but rather is based on the condition that the frequency obtained from the governing differential equations has to be equal to that yielded by Rayleigh's method. This modi cation is shown to be more straightforward and faster in computation, and the mode shapes derived are valid on a larger portion of the plate. Furthermore, the proposed modi cation easily provides a solution for boundary conditions for which the BM and MBM cannot provide a solution. Problems with two different sets of boundary conditions were solved in this study: a rectangular orthotropic plate with all edges clamped and rectangular isotropic and orthotropic plates clamped along one pair of opposite edges and free along the other pair. The results obtained for the rst set compared favorably with those yielded by the MBM and Rayleigh methods, whereas in the second case the BM and MBM failed to predict the beam-like modes of vibration, while the present modi cation treats the problem satisfactorily. Nomenclature= lengths of the side of the plate along the x and y axes, respectively c 1 , c 2 , C 1 , C 2 = real numbers= exural rigidity from the coupling of x and y directions, D x y = D x m y D y = exural rigidity in y direction, D y = E y h 3 / 12 (1 ¡ m x m y ) D 66 = torsional exural rigidity, D 66 = G h 3 / 12 E x , E y = Young's moduli along the x and y axes, respectively G = rigidity modulus H = D x y + 2D 66 h = plate thickness K = kinetic vibration energy m, n = integer numbers p, q = real numbers s 1 , s 2 , S 1 , S 2 = real numbers t = time w = transverse displacement X = mode shape function in x direction x, y = rectangular coordinates Y = mode shape function in y direction h= real or imaginary numbers, roots of characteristic equation m x , m y = Poisson's ratios for the material where E x m y = E y m x P = potential vibration energy q = mass density s = direction normal to the contour of the plate x = eigenfrequency or cycle frequency

Stability and vibrations of nonclosed circular cylindrical shells reinforced with discrete longitudinal ribs

“…Substituting (12) into the equations of motion (1) (p = q = 0) and following the procedure of the Bubnov-Galerkin method, we obtain an equation for k: …”

“…These solutions are inconvenient when there are either few ribs (they lead to ill-conditioned systems of linear algebraic equations) or many ribs (they lead to very large systems of equations). The finite-element method (see, e.g., [8,12,13]) and the finite-difference method (see, e.g., [10]) make the application of single trigonometric series for shells with discrete ribs noncompetitive, unlike analytic methods based on asymptotic methods [2,6,7,9,16,23] or double trigonometric series [2,3,5,6,16]. The latter allow obtaining, in some cases, exact solutions in convenient form to the equations of motion and, as asymptotic methods, simple approximate formulas suitable for analysis of the deformation of shells without labor-intensive calculations.…”

Respons Dinamik Pelat Beton Akibat Beban Kendaraan yang Bergerak dengan Kecepatan Konstan

“…Metode ini digunakan untuk menyelesaikan masalah pelat dan cangkang (shell) secara numerik dengan menggunakan fungsi trigonometri (Pevzner et al, 2000). MBM akan memberikan hasil yang lebih akurat untuk ragam getar yang lebih tinggi dengan kondisi perletakan yang beragam.…”

“…Untuk masalah pelat dengan perletakan tidak simetris pada keempat sisinya, wave number dinyatakan sebgai pπ/a dan qπ/b dengan p dan q adalah bilangan riil yang perlu diselesaikan dengan dua buah persamaan auxiliary tipe Levy pertama dan kedua. Metode inilah yang dinamakan dengan Modified Bolotin Method (Pevzner et al, 2000). Penyelesaian ini dinilai memberikan hasil frekuensi natural dan ragam getar yang paling akurat dengan cara penyelesaian persamaan transcendental yang paling mudah (Elishakoff, 1974).…”

Dynamic Response of Pavement Plates to the Positive and Negative Phases of the Friedlander Load

“…where p and q are real numbers to be solved from a system of two transcendental equations, obtained from the solution of two auxiliary Levy's type problems, also known as the modified Bolotin method [16].…”