Free vibrations of shallow orthotropic shells with variable thickness and rectangular planform

Abstract

The free vibrations of shallow doubly curved orthotropic shells with rectangular planform and varying thickness is solved using a refined formulation and the spline-approximation method. Various boundary conditions are considered. The effect of the curvature of the mid-surface on the spectrum of natural frequencies is examined. The natural frequencies and modes of orthotropic shells of constant and varying thickness are compared and analyzed Introduction. Anisotropic shells of variable thickness are widely used, as structural members, in industry and construction. Modern design and engineering solutions require high strength and reliability of new structures and mechanisms. In this connection, it is necessary to develop efficient numerical methods for the calculation of the load-bearing capacity of structural members, including shells with different physical and geometrical parameters and for the determination of their resonant frequencies.The natural frequencies of shells are often calculated using the classical Kirchhoff-Love theory, which is characterized by a simple analytical tools and high accuracy. However, development and widespread use of advanced composite materials necessitate using refined theories [1-3, 5, etc.].The stress-strain state of shallow shells was examined in [8, 10, 12] using a nonclassical problem formulation. The free vibrations of plates were studied in [6,7,9,11,15] following a refined approach. Most nonclassical theories allow for transverse shear. Of such theories, we choose the nonclassical Timoshenko-Mindlin model.In designing shallow orthotropic shells of constant thickness with hinged edges, use may be made of the procedure of separation of variables by expanding the unknown functions into double trigonometric series. However, this procedure cannot be applied to other boundary conditions, including clamped ends. Because of this, the posed problem has to be solved by numerical methods.The spline-approximation method has recently become the most popular in determining the dynamic characteristics of plates and shells. It has the following advantages: stability of splines against local perturbations; good convergence of spline-interpolation; and simplicity and convenience of numerical implementation of spline algorithms. When used in various variational, projective, and other discrete-continuous methods, spline-functions yield optimal results, simplify the numerical implementation of these methods, and ensure high accuracy of the solution.The present paper analyzes the free vibrations of shallow doubly curved orthotropic shells with rectangular planform and varying thickness with various boundary conditions using a refined problem formulation and the nonclassical Timoshenko-Mindlin model.The problem will be solved by using spline-approximation with respect to one coordinate to reduce the original system of partial differential equations to an eigenvalue boundary-value problem for systems of ordinary differential equations of high

Free vibration analysis of orthtropic thin cylindrical shells with variable thickness by using spline functions

“…crete method in combination with Green's function to obtain natural frequency solution for flat plates with variable thickness in one direction. Table 7 shows frequency parameters for a shallow shell with rectangular platform that its thickness varies parabolically in one direction (Grigorenko and Parkhomenko (2011) Table 4: Natural frequencies (Hz) for a circular cylindrical shell model (Srinivasan and Bobby (1976) Table 5: Natural frequencies (Rad/sec) for a parabolic cylindrical shell model (Cheung and Cheung (1972) …”

“…Their investigations were limited to the effects of variable thickness in one direction (either axial or circumferential) on vibration behavior of the shells. Later, Grigorenko and Parkhomenko (2011) studied free vibration of shallow shells having parabolically-variable thickness with the aid of spline-collocation approach. The effects of variable thickness on the vibration behavior of closed elliptical cylindrical shells and closed oval cylindrical shells have been studied by Suzuki and Leissa (1985) and Khalifa (2011), respectively.…”