Formulas for dynamics, acoustics and vibration

Front Matter
Contents
Preface
1 Definitions, Units, and Geometric Properties
	1.1. Definitions
	1.2 Symbols
	1.3 Units
	1.4 Motion on the Surface of the Earth
	1.5 Geometric Properties of Plane Areas
	1.6 Geometric Properties of Rigid Bodies
	1.7 Geometric Properties Defined by Vectors
	References
2 Dynamics of Particles and Bodies
	2.1 Kinematics and Coordinate Transformations
	2.2 Newton’s Law of Particle Dynamics
		2.2.1 Constant Mass Systems
		2.2.2 Variable Mass Systems
		2.2.3 Particle Trajectories
		2.2.4 Work and Energy
		2.2.5 Impulse
		2.2.6 Armor
		2.2.7 Gravitation and Orbits
	2.3 Rigid Body Rotation
		2.3.1 Rigid Body Rotation Theory
		2.3.2 Single-Axis Rotation
		2.3.3 Multiple-Axis Rotation
		2.3.4 Gyroscopic Effects
	References
3 Natural Frequency of Spring–Mass Systems, Pendulums, Strings, and Membranes
	3.1 Harmonic Motion
	3.2 Spring Constants
	3.3 Natural Frequencies of Spring–Mass Systems
		3.3.1 Single-Degree-of-Freedom
		3.3.2 Two-Degree-of-Freedom System
	3.4 Modeling Discrete Systems with Springs and Masses
		3.4.1 Springs with Mass
		3.4.2 Bellows
	3.5 Pendulum Natural Frequencies
		3.5.1 Mass Properties from Frequency Measurement
	3.6 Tensioned Strings, Cables, and Chain Natural Frequencies
		3.6.1 Equation of Motion
		3.6.2 Cable Sag
	3.7 Membrane Natural Frequencies
		3.7.1 Flat Membranes
		3.7.2 Curved Membranes
	References
4 Natural Frequency of Beams
	4.1 Beam Bending Theory
		4.1.1 Stress, Strain, and Deformation
		4.1.2 Sandwich Beams
		4.1.3 Beam Equation of Motion
		4.1.4 Boundary Conditions and Modal Solution
		4.1.5 Beams on Elastic Foundations
		4.1.6 Simplification for Tubes
	4.2 Natural Frequencies and Mode Shapes of Single-Span and Multiple-Span Beams
		4.2.1 Single-Span Beams
		4.2.2 Orthogonality, Normalization, and Maximum Values
		4.2.3 Beams Stress
		4.2.4 Two-Span Beams
		4.2.5 Multispan Beams
	4.3 Axially Loaded Beam Natural Frequency
		4.3.1 Uniform Axial Load
		4.3.2 Linearly Varying Axial Load
	4.4 Beams with Masses, Tapered Beams, Beams with Spring Supports, and Shear Beams
		4.4.1 Beams with Masses
		4.4.2 Tapered and Stepped Beams
		4.4.3 Spring-Supported Beams
		4.4.4 Shear Beams
		4.4.5 Effect of Shearing Force on the Deflections of Beams
		4.4.6 Rotary Inertia
		4.4.7 Multistory Buildings
	4.5 Torsional and Longitudinal Beam Natural Frequencies
		4.5.1 Longitudinal Vibration of Beams and Springs
		4.5.2 Torsional Vibration of Beams and Shafts
		4.5.3 Circular Cross Section
		4.5.4 Noncircular Cross Sections
	4.6 Wave Propagation in Beams
	4.7 Curved Beams, Rings, and Frames
		4.7.1 Complete Rings
		4.7.2 Stress and Strain of Arcs
		4.7.3 Supported Rings and Helices
		4.7.4 Circular Arcs, Arches, and Bends
		4.7.5 Lowest Frequency In-Plane Natural Frequency of an Arc
		4.7.6 Shallow Arc
		4.7.7 Portal Frames
	References
5 Natural Frequency of Plates and Shells
	5.1 Plate Flexure Theory
		5.1.1 Stress and Strain
		5.1.2 Boundary Conditions
		5.1.3 Plate Equation of Motion
		5.1.4 Simply Supported Rectangular Plate
		5.1.5 Plates on Elastic Foundations
		5.1.6 Sandwich Plates
		5.1.7 Thick Plates and Shear Deformation
		5.1.8 Membrane Analogy and In-Plane Loads
		5.1.9 Orthogonality
	5.2 Plate Natural Frequencies and Mode Shapes
		5.2.1 Plate Natural Frequencies
		5.2.2 Circular and Annular Plates
		5.2.3 Sectorial and Circular Orthotropic Plates
		5.2.4 Rectangular Plates
		5.2.5 Parallelogram, Triangular and Point-Supported Plates
		5.2.6 Rectangular Orthotropic Plates and Grillages
		5.2.7 Stiffened Plates
		5.2.8 Perforated Plates
	5.3 Cylindrical Shells
		5.3.1 Donnell Thin Shell Theory
		5.3.2 Natural Frequencies of Cylindrical Shells
		5.3.3 Infinitely Long Cylindrical Shell Modes (j = 0)
		5.3.4 Simply Supported Cylindrical Shells without Axial Constraint
		5.3.5 Cylindrical Shells with Other Boundary Conditions
		5.3.6 Free–Free Cylindrical Shell
		5.3.7 Cylindrically Curved Panels
		5.3.8 Effect of Mean Load on Natural Frequencies
	5.4 Spherical and Conical Shells
		5.4.1 Spherical Shells
		5.4.2 Open Shells and Church Bells
		5.4.3 Shallow Spherical Shells
		5.4.4 Conical Shells
	References
6 Acoustics and Fluids
	6.1 Sound Waves and Decibels
		6.1.1 Speed of Sound
		6.1.2 Acoustic Wave Equation
		6.1.3 Decibels and Sound Power Level
		6.1.4 Standards for Measurement
		6.1.5 Attenuation and Transmission Loss (TL)
	6.2 Sound Propagation in Large Spaces
		6.2.1 Acoustic Wave Propagation
		6.2.2 Sound Pressure on Rigid Walls
		6.2.3 Mass Law for Sound Transmission
	6.3 Acoustic Waves in Ducts and Rooms
		6.3.1 Acoustic Waves in Ducts
		6.3.2 Mufflers and Resonators
		6.3.3 Room Acoustics
	6.4 Acoustic Natural Frequencies and Mode Shapes
		6.4.1 Structure-Acoustic Analogy
	6.5 Free Surface Waves and Liquid Sloshing
	6.6 Ships and Floating Systems
		6.6.1 Ship Natural Frequencies (1/Period)
	6.7 Added Mass of Structure in Fluids
		6.7.1 Added Mass Potential Flow Theory
		6.7.2 Added Mass
		6.7.3 Added Mass of Plates and Shells
	References
	Further Reading
7 Forced Vibration
	7.1 Steady-State Forced Vibration
		7.1.1 Single-Degree-of-Freedom Spring–Mass Response
		7.1.2 Multiple-Degree-of-Freedom Spring–Mass System Response
		7.1.3 Forced Harmonic Vibration of Continuous Systems
		7.1.4 General System Response
	7.2 Transient Vibration
		7.2.1 Transient Vibration Theory
		7.2.2 Continuous Systems and Initial Conditions
		7.2.3 Maximum Transient Response and Response Spectra
		7.2.4 Shock Standards and Shock Test Machines
	7.3 Vibration Isolation
		7.3.1 Single-Degree-of-Freedom Vibration Isolation
		7.3.2 Two-Degree-of-Freedom Vibration Isolation
	7.4 Random Vibration Response to Spectral Loads
		7.4.1 Power Spectral Density and Fourier Series
		7.4.2 Complex Fourier Transform and Random Response
	7.5 Approximate Response Solution
		7.5.1 Equivalent Static Loads
		7.5.2 Scaling Mode Shapes to Load
	References
8 Properties of Solids, Liquids, and Gases
	8.1 Solids
	8.2 Liquids
	8.3 Gases
		8.3.1 Ideal Gas Law
	References
A Approximate Methods for Natural Frequency
	A.1 Relationship between Fundamental Natural Frequency and Static Deflection
	A.2 Rayleigh Technique
	A.3 Dunkerley and Southwell Methods
	A.4 Rayleigh–Ritz and Schmidt Approximations
	A.5 Galerkin Procedure for Continuous Structures References
	References
B Numerical Integration of Newton’s Second Law
	References
C Standard Octaves and Sound Pressure
	C.1 Time History and Overall Sound Pressure
	C.2 Peaks and Crest
	C.3 Spectra and Spectral Density
	C.4 Logarithmic Frequency Scales and Musical Tunings
	C.5 Human Perception of Sound (Psychological Acoustics)
	References
D Integrals Containing Mode Shapes of Single-Span Beams
	Reference
E Finite Element Programs
	E.1 Professional/Commercial Programs
	E.2 Open Source /Low-Cost Programs
Index