592 Pages
by
CRC Press
592 Pages
by
CRC Press
Also available as eBook on:
With increasingly sophisticated structures involved in modern engineering, knowledge of the complex vibration behavior of plates, shells, curved membranes, rings, and other complex structures is essential for today’s engineering students, since the behavior is fundamentally different than that of simple structures such as rods and beams. Now in its third edition, Vibrations of Shells and Plates continues to lay an analytical and computational foundation for the study of vibration in these structures.
Vibrations of Shells and Plates, Third Edition is updated with substantial new material reflecting advances made over the past decade since publication of the second edition. The author demonstrates how the vibration behavior of shells and plates differs from that of beams through theoretical development and examples. He also explains complicating effects on vibration such as the influence of rotation, shear, rotatory inertia, moment loading, residual stresses, and composite layers. New material includes the parabolic cylindrical shell, natural frequencies and modes, power series method, and explicit strain energy equations for many standard cases.
Intended for graduate and post-graduate study in vibration, acoustics, noise control, and stress analysis, this textbook provides a strong foundation in vibration theory, offers analytical solutions that illustrate actual behavior of structures, and prepares students to perform finite element and finite difference analysis.
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
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
Biography
Werner Soedel
