Vibrations Of Shallow Spherical Shells Research Articles

Nonlinear Vibration Behavior of Rapidly Heated Temperature-Dependent FGM Shallow Spherical Shells

In the present research, thermally induced vibration of shallow spherical shells made of functionally graded materials is investigated. The thermomechanical properties of the FGM media are assumed to be temperature dependent. Based on the uncoupled thermoelasticity theory, the one-dimensional heat conduction equation is established. The top and bottom surfaces of the shell are subjected to several types of rapid heating boundary condition. Because of the temperature dependency of the material properties, solution of nonlinear heat conduction equation needs a numerical method. In the first step, generalized differential quadrature method (GDQM) is applied to discretize the heat conduction equation across the shell thickness. Afterward, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson scheme. The obtained thermal force and thermal moment resultants at each time step are applied to the equations of motion. The axisymmetric equations of motion, using the first-order shear deformation theory (FSDT) based on the Kármán type of geometrical nonlinearity, are derived with the aid of the Hamilton principle. Using the harmonic differential quadrature method, shell domain is divided into a number of nodal points, and differential equations are turned into a system of ordinary differential equations. To obtain the displacement components at any time, time marching scheme based on the Newmark method is implemented. The obtained highly nonlinear algebraic equations are solved by means of the Newton–Raphson method. Comparison studies are performed to validate the formulation and solution method of the present research. Various examples are illustrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the shell, temperature dependency, shell opening angle, in-plane boundary conditions, type of thermal boundary conditions, and geometrical nonlinearity on thermally induced response of the FGM shell under thermal shock.

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.

Nonlinear vibrations of moderately thick orthotropic shallow spherical shells

This paper deals with the effects of large amplitude on the flexural vibrations of shallow spherical shells. A shallow shell theory is presented for the geometrically nonlinear dynamic analysis of moderately thick orthotropic spherical shells. Transverse shear deformation and rotatory inertia effects are incorporated into the nonlinear equations of motion by means of tracing constants. The coupled set of nonlinear equations of motion are solved using Galerkin's method and solutions are obtained by employing the numerical Runge-Kutta integration procedure. Numerical results are presented for several cases of shallow spherical shells with various geometric and material properties. While the results indicate the well-known softening type behavior between nonlinear frequency and amplitude, certain interesting conclusions are presented for orthotropic shells. The general governing equations presented here can be readily specialized for isotropic shells, and linear vibration cases and they are in agreement with those available in the literature.

Vibrations of a Shallow Spherical Shell on a Kerr Foundation

Large amplitude free vibrations of a clamped shallow spherical shell on a Kerr-type elastic foundation model are investigated. A detailed parametric study is conducted involving geometric and elastic properties of the shell as well as representative foundation parameters. Influence of these variables on the relation between the nondimensional frequency and amplitude are discussed. Shells with immovably clamped edges are more vulnerable to the changes in the above-referred parameters than those with movable clamped edges.

On axisymmetric vibration of shallow spherical shells with fixed circular cutout

A general solution of the third-order partial differential equations of a thin isotropic shallow spherical shell with a fixed circular cutout subject to axisymmetric free vibration is presented in this paper. The frequency equation in terms of Bessel functions as well as modified Bessel functions is solved for the fundamental vibrational frequency and mode shape. Solutions for different size cutouts are obtained and compared to a corresponding finite element model. The relationship between the vibrational frequency and mode shape and the size of the fixed cutout is discussed.

Subcritical crack development with an extensive prefailure zone in viscoelastic material

i. E. I. Grigolyuk and I. T. Selezov, Mechanics of Solid Deformable Bodies: Vol. 5, Nonclassical Theories of Vibrations of Beams, Plates, and Shells [in Russian], Itogi Nauki Tekh>, Moscow (1973). 2. Yu. E. Senitskii, Investigation of the Elastic Deformation of Construction Elements under Dynamic Actions by the Method of Finite Integral Transforms [in Russian], Izd. Saratov. Univ>, Saratov (1985). 3. Yu. E. Senitskii, "A dynamical problem for an elastically fastened circular plate on the basis of a Timoshenko type theory," Soprotiv. Mater. Teoro Sooruzh.. 31. 122-130 ~1977). 4. L. M. Berkovitch, M. L. Netchaevsky, Yu. E. 5enitskii (Y. E. Senitsky), "The method of factorization of differential operators and its applications," in: Complex Analysis and Applications, Proc. Int. Cong. (Varna, Sept. 20-27,(1981)iBulgarianAcad. Sciences, Sofia (1984), pp. 55-62. 5. A. Kalnins, "On vibrations of shallow spherical shells," J. Acoust. Soc. Am., 33, No. 8, 1102-1107 (1961). 6. Ho Reismann and P. M. Culkowski, "Forced axisymmetric motion of shallow spherical shells," J. Eng. Mech. Div. ASCE, 94, No. 2, 653-670 (1968).

Large amplitude free vibrations of shallow spherical shell and cylindrical shell — A new approach

In this paper, large amplitude free vibrations of thin elastic shallow spherical and cylindrical shells have been investigated following a new approach. Numerical results for movable as well as immovable edge conditions have been presented graphically and compared with other known results.

Large-amplitude vibrations of geometrically imperfect shallow spherical shells with structural damping

This paper deals with the effects of geometric imperfections on the large-amplitude vibrations Of shallow spherical shells. The initial geometric imperfection, the vibration mode, and the forcing function are of the same spatial shape. It is found that the presence of geometric imperfections of the order of a fractionof the shell thickness may significantly raises the free tine vibration frequencies. Further, they may even cause the nonlinear hard-spring character of spherical shells to exhibit soft-spring behavior. The effects of structural damping are also considered.

Large Amplitude Vibrations of Shallow Spherical Shell With Concentrated Mass

Large Amplitude Vibrations of Shallow Spherical Shell With Concentrated Mass

Vibration of Shallow Spherical Shells at Large Amplitudes

Technical Briefs Vibration of Shallow Spherical Shells at Large Amplitudes J. Ramachandran J. Ramachandran Department of Applied Mechanics, Indian Institute of Technology, Madras, India Search for other works by this author on: This Site PubMed Google Scholar Author and Article Information J. Ramachandran Department of Applied Mechanics, Indian Institute of Technology, Madras, India J. Appl. Mech. Sep 1974, 41(3): 811-812 (2 pages) https://doi.org/10.1115/1.3423399 Published Online: September 1, 1974 Article history Received: August 1, 1973 Revised: February 1, 1974 Online: July 12, 2010

Free vibrations of shallow spherical shells

Normal modes of the shallow spherical shells have been developed for the real and complex roots of the auxilliary equation. Frequency spectrums are presented for symmetric and asymmetric motions of clamped and supported shells. First four axisymmetric modes are presented for the illustration of the edge effects on deformation.

Nonlinear Vibrations of Shallow Spherical Shells

Based on a system of nonlinear dynamic equations and the associated variational equation of motion derived for elastic spherical shells (deep or shallow), an investigation of the axisymmetric vibrations of spherical caps with various edge conditions is made by carrying out a consistent sequence of approximations with respect to space and time. Numerical results are obtained for both free and forced oscillations involving finite deflection. The effect of curvature is examined, with particular emphasis on the transition from a flat plate to a curved shell. In fact, in such a transition, the nonlinearity of the hardening type gradually reverses into one of softening.

Vibration of Shallow Spherical Shells With Concentrated Mass

Vibration of Shallow Spherical Shells With Concentrated Mass

A Study of the Vibrations of Shallow Spherical Shells

This paper presents a comparison of the theoretically and experimentally determined frequencies and zero displacement lines for the normal mode vibrations of shallow spherical shells. Results are given for the clamped-edge and the momentless-edge boundary conditions. Calculations of frequencies and loci of zero displacement components for various shell sizes were made on an IBM 650 computer for the rotationally symmetric vibrations only, as the theory is not available for the case of asymmetric vibrations. Experiments were performed to determine the vibration characteristics both for the symmetric and the asymmetric vibrations. Several model shells constructed of steel were used for the purpose.

Flexural Vibration of Shallow Orthotropic Spherical Shells

The paper presents the theoretical development of the equations of motion for the flexural vibrations of a shallow spherical shell of orthotropic material. For symmetrical vibrations, the motion is described by two ordinary sixth-order differential equations with variable coefficients. Factorization of the sixth-order operators into second-order operators is discussed, and the details of solution are worked out for the case in which the equations are reduced to those for the isotropic shell. It is shown that equations for the latter case agree exactly with those derived previously by Reissner [Eric Reissner, “On Vibrations of Shallow Spherical Shells,” J. Appl. Phys. 17, 1038–1042 (1946)]. A table of calculated frequencies is given along with some experimental results for steel shells. (This investigation was conducted under Contract No. DA-30-115-509-ORD-912, Department of Army Ordnance Corps, Ballistic Research Laboratory, Aberdeen, Maryland.)

Closure to “Discussion of ‘Frequencies of Vibration of Shallow Spherical Shells’” (1962, ASME J. Appl. Mech., 29, p. 216)

Closure to “Discussion of ‘Frequencies of Vibration of Shallow Spherical Shells’” (1962, ASME J. Appl. Mech., 29, p. 216)

On Vibrations of Shallow Spherical Shells

The vibration analysis of shallow spherical shells is extended to (a) frequencies of the order of magnitude of the first thickness-shear mode in an infinite plate and (b) moderately thick shells. A tenth-order system of three uncoupled differential equations is derived, which govern the nonsymmetric dynamic deformation of a shallow spherical shell subjected to arbitrary time-dependent surface loads, and separable solutions are obtained in terms of Bessel functions. As an example, a frequency equation is deduced for the determination of natural frequencies higher than those accurately predicted by the classical theory for free vibration of a shallow spherical cap with a clamped edge.

Frequencies of Vibration of Shallow Spherical Shells

Frequencies of Vibration of Shallow Spherical Shells

Axisymmetric Vibrations of Shallow Elastic Spherical Shells

This paper contains an exact solution within the scope of the bending theory of shells for axisymmetric vibrations of thin shallow elastic spherical shells. Specifically, the solution is applied to an investigation of the natural frequencies of the free axisymmetric vibrations of spherical shell segments with various edge conditions. Comparison drawn with previously known results for transverse vibrations of shallow spherical shells, where the effect of longitudinal inertia is neglected.

On transverse vibrations of shallow spherical shells

On transverse vibrations of shallow spherical shells