Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

“…Generated orthogonal polynomials were also used as trial functions in the 3D Ritz method to study threedimensional free vibrations of homogeneous isotropic circular and annular plates with various boundary conditions [11,12], thick rectangular plates with arbitrary combinations of boundary constraints [33], skewed trapezoidal plates of different planforms [34], elliptic bars with various end constraints [13], hollow cantilevered cylinders with an arbitrary cross section [35], solid and hollow cylinders of arbitrary cross section with different end conditions [14], thick and open cylindrical shells [36], and more recently, rectangular plates of arbitrary thickness and boundary conditions [37].…”

Determining the natural frequencies of an elastic parallelepiped by the advanced Kantorovich–Vlasov method

“…The present study was inspired by the paper [15] where the Ritz method with B-splines as basis functions was tested by calculating the natural frequencies of a rectangular parallelepiped. The vibrations of elastic bodies of such shape are addressed in a great number of Ukrainian and foreign publications reviewed in [15] and [3] with reference to isotropic and anisotropic materials, respectively.The spectrum of natural frequencies of a parallelepiped with four sides hinged was analyzed in detail in [1, 11, 16] using trigonometric Fourier series expansion. Other types of boundary conditions can be examined by using analytical and numerical methods.…”

“…The present study was inspired by the paper [15] where the Ritz method with B-splines as basis functions was tested by calculating the natural frequencies of a rectangular parallelepiped. The vibrations of elastic bodies of such shape are addressed in a great number of Ukrainian and foreign publications reviewed in [15] and [3] with reference to isotropic and anisotropic materials, respectively.…”

“…In some special cases, the solution can be found analytically (for example, a two-dimensional problem of plate bending for M = 1). The values of the parameter W * for several lowest frequencies obtained using the AKM, the exact solution [16], the Ritz method with approximation by B-splines [15] and orthogonal polynomials [12] are collected in Table 1 for the boundary conditions (i) and (ii) and in Table 2 for the boundary conditions (iii) and (iv). Note that comparison is made with the exact solution [16] in case (i) and with the results [12] in case (ii).…”

“…Note that comparison is made with the exact solution [16] in case (i) and with the results [12] in case (ii). The last two columns of Table 1 (boundary conditions (ii)) and Table 2 show that the frequencies found by the Ritz method with continuous approximation by orthogonal polynomials [12] and approximation by B-splines [15] differ by less than 0.1%.…”