# Vibrations of Shells and Plates

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## Book Description

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.

## Table of Contents

**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**