Remarks on Donnell’s Equations

1955 ◽  
Vol 22 (1) ◽  
pp. 117-118
Author(s):  
Joseph Kempner
Keyword(s):  
Differential Equations ◽  
Cylindrical Shells ◽  
Radial Line ◽  
Line Load ◽  
Simply Supported ◽  
Circular Cylindrical Shells ◽  
Line Loads ◽  
Good Agreement

Abstract Flügge’s set of differential equations of equilibrium for circular cylindrical shells is expressed in a form analogous to the Donnell equations. The results of solutions of the two sets of equations for a simply supported cylinder under a centrally applied, uniformly distributed radial line load over a generator segment, as well as under sinusoidally applied line loads, are in very good agreement for the particular geometry investigated.

LONG CIRCULAR-CYLINDRICAL SHELLS SUBJECTED TO CIRCUMFERENTIAL, RADIAL LINE LOADS

AIAA Journal ◽  
1963 ◽  
Vol 1 (5) ◽  
pp. 1223-1224 ◽  
Author(s):  
K. T. SUNDARA RAJA IYENGAR ◽  
C. V. YOGANANDA
Keyword(s):  
Cylindrical Shells ◽  
Radial Line ◽  
Circular Cylindrical Shells ◽  
Line Loads

Prediction of natural frequencies for thin circular cylindrical shells

2000 ◽  
Vol 214 (10) ◽  
pp. 1313-1328 ◽  
Author(s):  
M N Naeem ◽  
C B Sharma
Keyword(s):  
Variational Approach ◽  
Analytical Procedure ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Polynomial Functions ◽  
Simply Supported ◽  
Analytical Results ◽  
General Eigenvalue Problem ◽  
Circular Cylindrical Shells ◽  
Good Agreement

In this paper an analytical procedure is given to study the free-vibration characteristics of thin circular cylindrical shells. Ritz polynomial functions are assumed to model the axial modal dependence and the Rayleigh-Ritz variational approach is employed to formulate the general eigenvalue problem. Influence of some commonly used boundary conditions, viz. simply supported-simply supported, clamped-clamped and clamped-free, and also the effects of variations of shell parameters on the vibration frequencies are examined. Natural frequencies for a number of particular cases are evaluated and compared with some available experimental and other analytical results in the literature on this topic. These results are given in the form of tables and figures. A very good agreement between the results of the present analysis and the corresponding experimental and analytical results available in the literature is obtained to confirm the validity and accuracy as well as the efficiency of the method.

Free vibration analysis of porous laminated rotating circular cylindrical shells

2019 ◽  
Vol 25 (18) ◽  
pp. 2494-2508 ◽  
Author(s):  
Ahmad Reza Ghasemi ◽  
Mohammad Meskini
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Cylindrical Shells ◽  
Free Vibration Analysis ◽  
Equilibrium Equations ◽  
Rotating Speed ◽  
Simply Supported ◽  
Numerical Result ◽  
Love’S Shell Theory ◽  
Circular Cylindrical Shells

In this research, investigations are presented of the free vibration of porous laminated rotating circular cylindrical shells based on Love’s shell theory with simply supported boundary conditions. The equilibrium equations for circular cylindrical shells are obtained using Hamilton’s principle. Also, Navier’s solution is used to solve the equations of the cylindrical shell due to the simply supported boundary conditions. The results are compared with previous results of other researchers. The numerical result of this study indicates that with increase of the porosity coefficient the nondimensional backward and forward frequency decreased. Then the results of the free vibration of rotating cylindrical shells are presented in terms of the effects of porous coefficients, porous type, length to radius ratio, rotating speed, and axial and circumferential wave numbers.

Imperfection Sensitivity of Compressed Circular Cylindrical Shells Under Periodic Axial Loads

2009 ◽  
Author(s):  
Francesco Pellicano
Keyword(s):  
Differential Equations ◽  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Finite Amplitude ◽  
Axial Loads ◽  
Imperfection Sensitivity ◽  
Lagrange Equations ◽  
Circular Cylindrical Shells

In the present paper the dynamic stability of circular cylindrical shells is investigated; the combined effect of compressive static and periodic axial loads is considered. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration; Lagrange equations are used to reduce the nonlinear partial differential equations to a set of ordinary differential equations. The dynamic stability is investigated using direct numerical simulation and a dichotomic algorithm to find the instability boundaries as the excitation frequency is varied; the effect of geometric imperfections is investigated in detail. The accuracy of the approach is checked by means of comparisons with the literature.

ON THE FUNDAMENTAL FREQUENCIES AND MODES OF FREE VIBRATION OF CYLINDRICAL SHELLS

1991 ◽  
Vol 15 (2) ◽  
pp. 147-159
Author(s):  
J.L. Urrutia-Galicia ◽  
L.J. Arango
Keyword(s):  
Free Vibration ◽  
Elastic Material ◽  
Cylindrical Shells ◽  
Simply Supported ◽  
Fundamental Frequencies ◽  
Circular Cylindrical Shells

The fundamental frequencies and modes of free vibration of simply supported circular cylindrical shells are explored. The results include the fundamental frequencies ωmn and the modes (m,n) of steel cylindrical shells which are presented in the form of a nomogram, see Figure 6. Besides, single more general formulas are given for cylindrical shells made out of any elastic material which turn out to be very suitable for design and analysis purposes.

Stability of Empty and Fluid-Filled Circular Cylindrical Shells Subjected to Dynamic Axial Loads

2002 ◽  
Author(s):  
F. Pellicano ◽  
M. Amabili ◽  
M. P. Pai¨doussis
Keyword(s):  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Axial Loads ◽  
Simply Supported ◽  
Mean Oscillation ◽  
Shell Motion ◽  
Circular Cylindrical Shells ◽  
The Mean

In the present study the dynamic stability of simply supported, circular cylindrical shells subjected to dynamic axial loads is analyzed. Geometric nonlinearities due to finite-amplitude shell motion are considered by using the Donnell’s nonlinear shallow-shell theory. The effect of structural damping is taken into account. A discretization method based on a series expansion involving a large number of linear modes, including axisymmetric and asymmetric modes, and on the Galerkin procedure is developed. Both driven and companion modes are included allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward deflection of the mean oscillation with respect to the equilibrium position. The shell is simply supported and presents a finite length. Boundary conditions are considered in the model, which includes also the contribution of the external axial loads acting at the shell edges. The effect of a contained liquid is also considered. The linear dynamic stability and nonlinear response are analysed by using continuation techniques.

Comparison of Different Shell Theories for Large-Amplitude Vibrations of Circular Cylindrical Shells

2002 ◽  
Author(s):  
M. Amabili
Keyword(s):  
Large Amplitude ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Continuation Method ◽  
Elastic Strain Energy ◽  
Harmonic Excitation ◽  
Lagrange Equations ◽  
Simply Supported ◽  
Nonlinear Shell ◽  
Circular Cylindrical Shells

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh’s dissipation function. Four different nonlinear shell theories, namely Donnell’s, Sanders-Koiter, Flu¨gge-Lur’e-Byrne and Novozhilov’s theories, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four nonlinear shell theories are also compared to results obtained by Galerkin approach, used to discretise Donnell’s nonlinear shallow-shell equation of motion. A validation of calculations by comparison to experimental results is also performed. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized coordinates, associated to natural modes of simply supported shells, are used. The nonlinear equations of motion are studied by using a code based on arclength continuation method that allows bifurcation analysis.

Energy Expressions and Differential Equations …. for Stress and Displacement Analyses of Arbitrary Cylindrical Shells

1958 ◽  
Vol 2 (02) ◽  
pp. 8-19
Author(s):  
Joseph Kempner
Keyword(s):  
Differential Equations ◽  
Cylindrical Shells ◽  
Arbitrary Cross Section ◽  
Natural Boundary ◽  
Equilibrium Equations ◽  
Small Deflection ◽  
Circular Cylindrical Shells ◽  
Natural Boundary Conditions ◽  
Theory Of Shells

Energy expressions and the related equilibrium equations and natural boundary conditions for the determination of the stresses in and displacements of uniform, thin-walled cylinders of arbitrary cross section loaded in an arbitrary manner by surface and edge forces and moments are presented. The derivations are based upon the Kirchhoff-Love assumptions of the classical theory of shells and are performed to within a degree of accuracy employed by Flügge in his derivation of the equilibrium equations applicable to circular cylindrical shells; hence, in terms of stress resultants, the exact, small-deflection equilibrium equations are obtained. Methods of simplification of the relations derived and of solution of the differential equations presented are indicated.

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