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Book ChapterDOI

Nonlinear Vibrations and Stability of Shells and Plates: Vibrations of Circular Cylindrical Panels with Different Boundary Conditions

01 Jan 2008-pp 242-271
TL;DR: In this article, the linear and nonlinear vibrations of simply supported circular cylindrical panels are studied by using the Flugge-Lur'e-Byrne theory.
Abstract: Introduction This chapter addresses the linear (small amplitude) and nonlinear (large amplitude) vibrations of circular cylindrical panels. Simply and doubly curved panels are structural elements largely used in engineering applications, such as aeronautics, aerospace, cars, boats, buildings, trains and many others. The linear vibrations of simply supported circular cylindrical panels are studied by using the Flugge-Lur'e-Byrne theory. Nonlinear vibrations of circular cylindrical panels with different boundary conditions (including flexible rotational constraints) under radial harmonic excitations are studied by using Donnell's shell theory with in-plane inertia, as this gives practically the same results as for other refined classical theories for very thin isotropic shells (Amabili 2005). The solution is obtained by using the Lagrange equations of motion and up to 39 degrees of freedom (dofs). Numerical results show that a simply supported panel for mode (1,1) presents a significant softening nonlinearity, which turns to the hardening type for vibration amplitude larger than the shell thickness; a similar behavior is observed for the panel with fixed edges and free rotations; on the other hand, the same panel with free in-plane edges or clamped edges presents hardening nonlinearity. A peculiar aspect of nonlinear vibrations of curved panels is the asymmetric oscillation with respect to the initial undeformed middle surface. In fact, the oscillation amplitude inward (i.e. in the direction of the center of curvature) is significantly larger than the amplitude outward.
Citations
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01 Oct 2010

38 citations


Cites methods from "Nonlinear Vibrations and Stability ..."

  • ...18 A different form for the equation of motion can be found in [13], using Donnels shell theory applied to a circular cylindrical shell one can find a set of three coupled PDE's describing the displacement:...

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Journal ArticleDOI
TL;DR: In this article, the Galerkin method is used to obtain the non-linear equations of motion which are solved by the Runge-Kutta method and a detailed parametric analysis clarifies the influence of the orthotropic material properties on the nonlinear buckling and vibration characteristics of the shell.

30 citations


Cites background or methods from "Nonlinear Vibrations and Stability ..."

  • ...The basic theory backing the present investigation can be found in Paı̈doussis [22] and Amabili [23]....

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  • ...Following previous studies and using the method of separation of variables [22,23,31], the perturbation pressure on the shell wall is found to be Ph 1⁄4 rF L mp fw @(2)w @t2 þ2U @ (2)w @t@x þU(2) @ (2)w @x2 ! , ð19Þ with fw 1⁄4 1 Inðn 1,mpR=LÞ I 0 nðn,mpR=LÞ n mpR=L , ð20Þ where rF is the fluid density, In the nth order modified Bessel function and I 0 n is its derivative with respect to its argument....

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  • ...On the other hand, excepting Case 5, the resonant branch becomes unstable, as the forcing frequency decreases due to a pitchfork bifurcation [23] and simultaneously the companion mode is excited and participates in the oscillations with nonzero value in the main resonance region around j/O01⁄41....

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  • ...Expansion (13) has been thoroughly tested by Amabili and co-workers [23,25]....

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Journal ArticleDOI
TL;DR: In this paper, the influence of geometric imperfections and an axial fluid flow on the nonlinear vibrations and instabilities of simply supported circular cylindrical shells under axial load was studied.

17 citations


Cites background or methods from "Nonlinear Vibrations and Stability ..."

  • ...This model includes the basic vibration (driven) mode, its companion mode, four axisymmetric modes and the gyroscopic modes (driven and companion) with twice the number of waves in the axial direction as the basic vibration mode, which becomes important at high flow velocities [3,12]....

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  • ...These topics are also present in detail in the books by Paı̈doussis on fluid–structure interactions [3] and Amabili on nonlinear vibrations and stability of shells and plates [12]....

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  • ...However, more refined modal solutions can be found in the literature [12]....

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Journal ArticleDOI
TL;DR: In this paper, the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simply-supported cylindrical shells is investigated.
Abstract: This work investigates the influence of physical and geometrical system parameters uncertainties and excitation noise on the nonlinear vibrations and stability of simply-supported cylindrical shells. These parameters are composed of both deterministic and random terms. Donnell's non-linear shallow shell theory is used to study the non-linear vibrations of the shell. To discretize the partial differential equations of motion, first, a general expression for the transversal displacement is obtained by a perturbation procedure which identifies all modes that couple with the linear modes through the quadratic and cubic nonlinearities. Then, a particular solution is selected which ensures the convergence of the response up to very large deflections. Finally, the in-plane displacements are obtained as a function of the transversal displacement by solving the in-plane equations analytically and imposing the necessary boundary, continuity and symmetry conditions. Substituting the obtained modal expansions into the equation of motion and applying the Galerkin's method, a discrete system in time domain is obtained. Several numerical strategies are used to study the nonlinear behavior of the shell considering the uncertainties in the physical and geometrical system parameters. Special attention is given to the influence of the uncertainties on the parametric instability and escape boundaries.

12 citations


Cites background or methods or result from "Nonlinear Vibrations and Stability ..."

  • ...A detailed review of this subject was p ublished in 2003 by Amabili and Païdoussis (2003)....

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  • ...Finally, experimental results of dynamic buckling loads of slender structures are rather scarce in literature (Virgin, 2000; Amabili, 2008)....

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  • ...These values are bas d on the experimental results by Amabili and co-workers (Amabili, 2008)....

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Journal ArticleDOI
TL;DR: In this paper, an analytical solution to predict the nonlinear forced vibrations of elastic thin-walled cylindrical shells under suddenly applied loads is presented. But the model is based on the energy criterion due to Lagrange, in which the kinematic nonlinear relations are assumed using Donnell's simplified shell theory.
Abstract: This paper presents an analytical solution to predict the nonlinear forced vibrations of elastic thin-walled cylindrical shells under suddenly applied loads. Interest in this problem is motivated by effects due to explosions on fluid-storage metal tanks. The model is based on the energy criterion due to Lagrange, in which the kinematic nonlinear relations are assumed using Donnell's simplified shell theory. Solution is achieved as a series summation in terms of trigonometric functions in the axial and circumferential directions, whereas the degrees of freedom depend on time. A blast load is assumed to represent effects due to explosions on the shell as time-dependent pressures with a given circumferential distribution (a cosine square distribution in terms of the central angle). The procedure is validated by comparison with a nonlinear finite element model under the same load conditions. The influence of load level and shell geometry on the transient response is investigated by mean of parametric studies. Good accuracy is found in the results for the range of shells which are representative of horizontal, fuel storage tanks in the oil industry.

8 citations


Cites background or methods from "Nonlinear Vibrations and Stability ..."

  • ...This is a good approximation for thin shells, say R/h>20 (see, for example, Amabili, 2008)....

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  • ...Amabili (2008) focused on nonlinear vibrations and stability aspects. Interest in previous studies is mainly motivated by structures of planes, cars, submarines and military applications. But engineering problems for which considerations of blast loads are required are not limited to those areas. A new field of interest involving the nonlinear dynamics of cylindrical shells emerged during the last ten years as a consequence of damage and destruction of oil storage tanks caused by explosions for both, vertical and horizontal tanks. Although most cases of explosions on tanks reported in the literature are caused by accidents, some recent events show that vulnerability with respect to intentional acts should also be of great concern to designers and forensic engineers. The mechanics of the problem and the variables involved in the phenomenon of an explosion can be seen for instance in Glasstone and Dolan (1977) for nuclear explosions and with more detail in UFC-3-340-02 (2008), with the aim of improving structural designs to resist the effects of explosions. The spatial distribution of a blast pressure around cylindrical tanks is described in a limited number of publications. In recently reported work, a group of researchers in France performed experiments on small-scale cantilever cylindrical shells having a scale factor of 1:48 to obtain time dependent pressure distributions (Duong et al., 2012a; Duong et al., 2012b; Noret et al., 2012). The goal of such testing program was to establish a probabilistic analysis about threshold values causing different damage levels. Analytical studies based on Donnell’s approximation were performed using static response to establish performance limits based on plasticity. Testing has been recently performed at University of North Carolina at Charlotte by Weggel and Whelan (2013) for small scale rigid models.. An alternative to blast testing is to employ computational Fluid Dynamics or Fluid-Structure interaction to simulate the process, such as in the work of Trajkovski et al. (2014). The analysis of nonlinear elastic behavior of thin-walled structures subjected to short duration pressures were shown by Ruiz et al....

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  • ...Amabili (2008) focused on nonlinear vibrations and stability aspects....

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  • ...Amabili (2008) focused on nonlinear vibrations and stability aspects. Interest in previous studies is mainly motivated by structures of planes, cars, submarines and military applications. But engineering problems for which considerations of blast loads are required are not limited to those areas. A new field of interest involving the nonlinear dynamics of cylindrical shells emerged during the last ten years as a consequence of damage and destruction of oil storage tanks caused by explosions for both, vertical and horizontal tanks. Although most cases of explosions on tanks reported in the literature are caused by accidents, some recent events show that vulnerability with respect to intentional acts should also be of great concern to designers and forensic engineers. The mechanics of the problem and the variables involved in the phenomenon of an explosion can be seen for instance in Glasstone and Dolan (1977) for nuclear explosions and with more detail in UFC-3-340-02 (2008), with the aim of improving structural designs to resist the effects of explosions. The spatial distribution of a blast pressure around cylindrical tanks is described in a limited number of publications. In recently reported work, a group of researchers in France performed experiments on small-scale cantilever cylindrical shells having a scale factor of 1:48 to obtain time dependent pressure distributions (Duong et al., 2012a; Duong et al., 2012b; Noret et al., 2012). The goal of such testing program was to establish a probabilistic analysis about threshold values causing different damage levels. Analytical studies based on Donnell’s approximation were performed using static response to establish performance limits based on plasticity. Testing has been recently performed at University of North Carolina at Charlotte by Weggel and Whelan (2013) for small scale rigid models.. An alternative to blast testing is to employ computational Fluid Dynamics or Fluid-Structure interaction to simulate the process, such as in the work of Trajkovski et al. (2014). The analysis of nonlinear elastic behavior of thin-walled structures subjected to short duration pressures were shown by Ruiz et al. (1989) and Hoo-Fatt and Pothula (2010) for cylindrical shells, Kowal-Michalska et al. (2011) for conical and spherical shells, Gao and Hoo-Fatt (2012) for a cylindrical sector, Goel et al....

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  • ...Use of these equations has been illustrated, for example in the texts by Brush and Almroth (1975) for buckling problems and by Amabili (2008) for vibrations of shells....

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References
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Book
01 Sep 1981
TL;DR: In this article, the authors discuss the development of Vibration Analysis of Continuous Structural Elements (SSA) and their application in the field of deep shell physics, including the following:
Abstract: Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Historical Development of Vibration Analysis of Continuous Structural Elements References Deep Shell Equations Shell Coordinates and Infinitesimal Distances in Shell Layers Stress-Strain Relationships Strain-Displacement Relationships Love Simplifications Membrane Forces and Bending Moments Energy Expressions Love's Equations by Way of Hamilton's Principle Boundary Conditions Hamilton's Principle Other Deep Shell Theories Shells of Nonuniform Thickness References Radii of Curvature References Equations of Motion for Commonly Occurring Geometries Shells of Revolution Circular Conical Shell Circular Cylindrical Shell Spherical Shell Other Geometries References Nonshell Structures Arch Beam and Rod Circular Ring Plate Torsional Vibration of Circular Cylindrical Shell and Reduction to a Torsion Bar References Natural Frequencies and Modes General Approach Transversely Vibrating Beams Circular Ring Rectangular Plates That are Simply Supported Along Two Opposing Edges Circular Cylindrical Shell Simply Supported Circular Plates Vibrating Transversely Examples: Plate Clamped at Boundary Orthogonality Property of Natural Modes Superposition Modes Orthogonal Modes from Nonorthogonal Superposition Modes Distortion of Experimental Modes Because of Damping Separating Time Formally Uncoupling of Equations of Motion In-Plane Vibrations of Rectangular Plates In-Plane Vibration of Circular Plates Deep Circular Cylindrical Panel Simply Supported at All Edges Natural Mode Solutions by Power Series On Regularities Concerning Nodelines References Simplified Shell Equations Membrane Approximations Axisymmetric Eigenvalues of a Spherical Shell Bending Approximation Circular Cylindrical Shell Zero In-Plane Deflection Approximation Example: Curved Fan Blade Donnell-Mushtari-Vlasov Equations Natural Frequencies and Modes Circular Cylindrical Shell Circular Duct Clamped at Both Ends Vibrations of a Freestanding Smokestack Special Cases of the Simply Supported Closed Shell and Curved Panel Barrel-Shaped Shell Spherical Cap Inextensional Approximation: Ring Toroidal Shell The Barrel-Shaped Shell Using Modified Love Equations Doubly Curved Rectangular Plate References Approximate Solution Techniques Approximate Solutions by Way of the Variational Integral Use of Beam Functions Galerkin's Method Applied to Shell Equations Rayleigh-Ritz Method Southwell's Principle Dunkerley's Principle Strain Energy Expressions References Forced Vibrations of Shells by Modal Expansion Model Participation Factor Initial Conditions Solution of the Modal Participation Factor Equation Reduced Systems Steady-State Harmonic Response Step and Impulse Response Influence of Load Distribution Point Loads Line Loads Point Impact Impulsive Forces and Point Forces Described by Dirac Delta Functions Definitions and Integration Property of the Dirac Delta Function Selection of Mode Phase Angles for Shells of Revolution Steady-State Circular Cylindrical Shell Response to Harmonic Point Load with All Mode Components Considered Initial Velocity Excitation of a Simply Supported Cylindrical Shell Static Deflections Rectangular Plate Response to Initial Displacement Caused by Static Sag The Concept of Modal Mass, Stiffness Damping, and Forcing Steady State Response of Shells to Periodic Forcing Plate Response to a Periodic Square Wave Forcing Beating Response to Steady State Harmonic Forcing References Dynamic Influence (Green's) Function Formulation of the Influence Function Solution to General Forcing Using the Dynamic Influence Function Reduced Systems Dynamic Influence Function for the Simply Supported Shell Dynamic Influence Function for the Closed Circular Ring Traveling Point Load on a Simply Supported Cylindrical Shell Point Load Traveling Around a Closed Circular Cylindrical Shell in Circumferential Direction Steady-State Harmonic Green's Function Rectangular Plate Examples Floating Ring Impacted by a Point Mass References Moment Loading Formulation of Shell Equations That Include Moment Loading Modal Expansion Solution Rotating Point Moment on a Plate Rotating Point Moment on a Shell Rectangular Plate Excited by a Line Moment Response of a Ring on an Elastic Foundation to a Harmonic Point Moment Moment Green's Function References Vibration of Shells and Membranes Under the Influence of Initial Stresses Strain-Displacement Relationships Equations of Motion Pure Membranes Example: The Circular Membrane Spinning Saw Blade Donnell-Mushtari-Vlasov Equations Extended to Include Initial Stresses References Shell Equations with Shear Deformation and Rotary Inertia Equations of Motion Beams with Shear Deflection and Rotary Inertia Plates with Transverse Shear Deflection and Rotary Inertia Circular Cylindrical Shells with Transverse Shear Deflection and Rotary Inertia References Combinations of Structures Receptance Method Mass Attached to Cylindrical Panel Spring Attached to Shallow Cylindrical Panel Harmonic Response of a System in Terms of Its Component Receptances Dynamic Absorber Harmonic Force Applied Through a Spring Steady-State Response to Harmonic Displacement Excitation Complex Receptances Stiffening of Shells Two Systems Joined by Two or More Displacement Suspension of an Instrument Package in a Shell Subtracting Structural Subsystems Three and More Systems Connected Examples of Three Systems Connected to Each Other References Hysteresis Damping Equivalent Viscous Damping Coefficient Hysteresis Damping Direct Utilization of Hysteresis Model in Analysis Hysteretically Damped Plate Excited by Shaker Steady State Response to Periodic Forcing References Shells Made of Composite Material Nature of Composites Lamina-Constitutive Relationship Laminated Composite Equation of Motion Orthotropic Plate Circular Cylindrical Shell Orthotropic Nets or Textiles Under Tension Hanging Net or Curtain Shells Made of Homogeneous and Isotropic Lamina Simply Supported Sandwich Plates and Beams Composed of Three Homogeneous and Isotropic Lamina References Rotating Structures String Parallel to Axis of Rotation Beam Parallel to Axis of Rotation Rotating Ring Rotating Ring Using Inextensional Approximation Cylindrical Shell Rotating with Constant Spin About Its Axis General Rotations of Elastic Systems Shells of Revolution with Constant Spin About Their Axes of Rotation Spinning Disk References Thermal Effects Stress Resultants Equations of Motion Plate Arch, Ring, Beam, and Rod Limitations Elastic Foundations Equations of Motion for Shells on Elastic Foundations Natural Frequencies and Modes Plates on Elastic Foundations Ring on Elastic Foundation Donnell-Mushtari-Vlasov Equations with Transverse Elastic Foundation Forces Transmitted Into the Base of the Elastic Foundation Vertical Force Transmission Through the Elastic Foundation of a Ring on a Rigid Wheel Response of a Shell on an Elastic Foundation to Base Excitation Plate Examples of Base Excitation and Force Transmission Natural Frequencies and Modes of a Ring on an Elastic Foundation in Ground Contact at a Point Response of a Ring on an Elastic Foundation to a Harmonic Point Displacement References Similitude General Similitude Derivation of Exact Similitude Relationships for Natural Frequencies of Thin Shells Plates Shallow Spherical Panels of Arbitrary Contours (Influence of Curvature) Forced Response Approximate Scaling of Shells Controlled by Membrane Stiffness Approximate Scaling of Shells Controlled by Bending Stiffness References Interactions with Liquids and Gases Fundamental Form in Three-Dimensional Curvilinear Coordinates Stress-Strain-Displacement Relationships Energy Expressions Equations of Motion of Vibroelasticity with Shear Example: Cylindrical Coordinates Example: Cartesian Coordinates One-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Solids Three-Dimensional Wave Equations for Inviscid Compressible Liquids and Gases (Acoustics) Interface Boundary Conditions Example: Acoustic Radiation Incompressible Liquids Example: Liquid on a Plate Orthogonality of Natural Modes for Three-Dimensional Solids, Liquids, and Gases References Discretizing Approaches Finite Differences Finite Elements Free and Forced Vibration Solutions References Index

1,166 citations

Journal ArticleDOI
TL;DR: Curvature effects on shallow shell free vibration frequencies, solving linear eigenvalue problem as discussed by the authors, solving linear Eigenvalue Problem (LEP) and solving linear value problem.

133 citations

Journal ArticleDOI
TL;DR: In this paper, the effect of thickness and curvature on the large amplitude vibration of shallow shells is studied, and the non-linear governing equations for thick shallow shells which have principle curvatures and rectangular planform are derived by Hamilton's principle using first order shear deformation theory.
Abstract: The effect of thickness and curvature upon the large amplitude vibration of shallow shells is studied in this paper. For this purpose, the non-linear governing equations for thick shallow shells which have principle curvatures and rectangular planform are derived by Hamilton's principle using first order shear deformation theory. Applying Galerkin's procedure and eliminating variables except for transverse displacement, the governing equations are reduced to an elliptic ordinary differential equation in time. The period of vibration for the shell is calculated by integrating the equation using a Gauss-Legendre integration method. The present method is applied to a shallow shell which has a rectangular boundary supported by shear diaphragms.

89 citations

Book ChapterDOI
24 Aug 2001

82 citations