Formulas for Dynamics, Acoustics and Vibration

Robert D. Blevins is a Fellow at United Technologies. He directs experimental and analytical investigation of dynamics, acoustic, vibration certification issues for commercial aircraft engine systems. Previously he participated in design of nuclear power plants. He has performed flow-induced vibration analysis and test of aircraft, off shore platforms, and vibration of heat exchangers at refineries, nuclear power plants and chemical process plants. He has taught the Flow-induced Vibration Seminar for the ASME for the past 20 years. He received his Ph.D and MS from the California Institute of Technology and serves the ASME special working from on dynamic analysis on the ASME Boiler and Pressure Vessel Code. He has over 50 published papers.

• Preface xi1 Definitions, Units, and Geometric Properties 11.1 Definitions 11.2 Symbols 81.3 Units 111.4 Motion on the Surface of the Earth 181.5 Geometric Properties of Plane Areas 191.6 Geometric Properties of Rigid Bodies 301.7 Geometric Properties Defined by Vectors 40References 412 Dynamics of Particles and Bodies 432.1 Kinematics and Coordinate Transformations 432.2 Newton’s Law of Particle Dynamics 502.2.1 Constant Mass Systems 502.2.2 Variable Mass Systems 572.2.3 Particle Trajectories 582.2.4 Work and Energy 632.2.5 Impulse 652.2.6 Armor 682.2.7 Gravitation and Orbits 712.3 Rigid Body Rotation 732.3.1 Rigid Body Rotation Theory 732.3.2 Single-Axis Rotation 732.3.3 Multiple-Axis Rotation 842.3.4 Gyroscopic Effects 85References 873 Natural Frequency of Spring–Mass Systems, Pendulums, Strings, and Membranes 893.1 Harmonic Motion 893.2 Spring Constants 913.3 Natural Frequencies of Spring–Mass Systems 993.3.1 Single-Degree-of-Freedom 993.3.2 Two-Degree-of-Freedom System 1133.4 Modeling Discrete Systems with Springs and Masses 1173.4.1 Springs with Mass 1173.4.2 Bellows 1183.5 Pendulum Natural Frequencies 1193.5.1 Mass Properties from Frequency Measurement 1203.6 Tensioned Strings, Cables, and Chain Natural Frequencies 1213.6.1 Equation of Motion 1213.6.2 Cable Sag 1233.7 Membrane Natural Frequencies 1263.7.1 Flat Membranes 1263.7.2 Curved Membranes 131References 1324 Natural Frequency of Beams 1344.1 Beam Bending Theory 1344.1.1 Stress, Strain, and Deformation 1344.1.2 Sandwich Beams 1364.1.3 Beam Equation of Motion 1374.1.4 Boundary Conditions and Modal Solution 1374.1.5 Beams on Elastic Foundations 1414.1.6 Simplification for Tubes 1414.2 Natural Frequencies and Mode Shapes of Single-Span and Multiple-Span Beams 1424.2.1 Single-Span Beams 1424.2.2 Orthogonality, Normalization, and Maximum Values 1504.2.3 Beams Stress 1504.2.4 Two-Span Beams 1514.2.5 Multispan Beams 1514.3 Axially Loaded Beam Natural Frequency 1584.3.1 Uniform Axial Load 1584.3.2 Linearly Varying Axial Load 1594.4 Beams with Masses, Tapered Beams, Beams with Spring Supports, and Shear Beams 1624.4.1 Beams with Masses 1624.4.2 Tapered and Stepped Beams 1624.4.3 Spring-Supported Beams 1674.4.4 Shear Beams 1674.4.5 Effect of Shearing Force on the Deflections of Beams 1704.4.6 Rotary Inertia 1704.4.7 Multistory Buildings 1744.5 Torsional and Longitudinal Beam Natural Frequencies 1764.5.1 Longitudinal Vibration of Beams and Springs 1764.5.2 Torsional Vibration of Beams and Shafts 1794.5.3 Circular Cross Section 1794.5.4 Noncircular Cross Sections 1824.6 Wave Propagation in Beams 1834.7 Curved Beams, Rings, and Frames 1844.7.1 Complete Rings 1844.7.2 Stress and Strain of Arcs 1894.7.3 Supported Rings and Helices 1904.7.4 Circular Arcs, Arches, and Bends 1904.7.5 Lowest Frequency In-Plane Natural Frequency of an Arc 1964.7.6 Shallow Arc 1974.7.7 Portal Frames 198References 1995 Natural Frequency of Plates and Shells 2035.1 Plate Flexure Theory 2035.1.1 Stress and Strain 2035.1.2 Boundary Conditions 2035.1.3 Plate Equation of Motion 2055.1.4 Simply Supported Rectangular Plate 2065.1.5 Plates on Elastic Foundations 2075.1.6 Sandwich Plates 2075.1.7 Thick Plates and Shear Deformation 2075.1.8 Membrane Analogy and In-Plane Loads 2085.1.9 Orthogonality 2085.2 Plate Natural Frequencies and Mode Shapes 2095.2.1 Plate Natural Frequencies 2095.2.2 Circular and Annular Plates 2095.2.3 Sectorial and Circular Orthotropic Plates 2145.2.4 Rectangular Plates 2145.2.5 Parallelogram, Triangular and Point-Supported Plates 2155.2.6 Rectangular Orthotropic Plates and Grillages 2155.2.7 Stiffened Plates 2315.2.8 Perforated Plates 2325.3 Cylindrical Shells 2345.3.1 Donnell Thin Shell Theory 2355.3.2 Natural Frequencies of Cylindrical Shells 2375.3.3 Infinitely Long Cylindrical Shell Modes (j = 0) 2415.3.4 Simply Supported Cylindrical Shells without Axial Constraint 2435.3.5 Cylindrical Shells with Other Boundary Conditions 2465.3.6 Free–Free Cylindrical Shell 2485.3.7 Cylindrically Curved Panels 2495.3.8 Effect of Mean Load on Natural Frequencies 2505.4 Spherical and Conical Shells 2505.4.1 Spherical Shells 2505.4.2 Open Shells and Church Bells 2525.4.3 Shallow Spherical Shells 2525.4.4 Conical Shells 254References 2546 Acoustics and Fluids 2606.1 Sound Waves and Decibels 2606.1.1 Speed of Sound 2606.1.2 Acoustic Wave Equation 2646.1.3 Decibels and Sound Power Level 2766.1.4 Standards for Measurement 2776.1.5 Attenuation and Transmission Loss (TL) 2786.2 Sound Propagation in Large Spaces 2856.2.1 Acoustic Wave Propagation 2856.2.2 Sound Pressure on Rigid Walls 2886.2.3 Mass Law for Sound Transmission 2896.3 Acoustic Waves in Ducts and Rooms 2896.3.1 Acoustic Waves in Ducts 2896.3.2 Mufflers and Resonators 2986.3.3 Room Acoustics 3026.4 Acoustic Natural Frequencies and Mode Shapes 3056.4.1 Structure-Acoustic Analogy 3066.5 Free Surface Waves and Liquid Sloshing 3106.6 Ships and Floating Systems 3196.6.1 Ship Natural Frequencies (1/Period) 3196.7 Added Mass of Structure in Fluids 3216.7.1 Added Mass Potential Flow Theory 3286.7.2 Added Mass 3296.7.3 Added Mass of Plates and Shells 330References 331Further Reading 3357 Forced Vibration 3367.1 Steady-State Forced Vibration 3367.1.1 Single-Degree-of-Freedom Spring–Mass Response 3367.1.2 Multiple-Degree-of-Freedom Spring–Mass System Response 3447.1.3 Forced Harmonic Vibration of Continuous Systems 3477.1.4 General System Response 3577.2 Transient Vibration 3597.2.1 Transient Vibration Theory 3597.2.2 Continuous Systems and Initial Conditions 3657.2.3 Maximum Transient Response and Response Spectra 3717.2.4 Shock Standards and Shock Test Machines 3747.3 Vibration Isolation 3747.3.1 Single-Degree-of-Freedom Vibration Isolation 3747.3.2 Two-Degree-of-Freedom Vibration Isolation 3777.4 Random Vibration Response to Spectral Loads 3797.4.1 Power Spectral Density and Fourier Series 3807.4.2 Complex Fourier Transform and Random Response 3817.5 Approximate Response Solution 3857.5.1 Equivalent Static Loads 3897.5.2 Scaling Mode Shapes to Load 389References 3918 Properties of Solids, Liquids, and Gases 3928.1 Solids 3928.2 Liquids 4028.3 Gases 4058.3.1 Ideal Gas Law 405References 409A Approximate Methods for Natural Frequency 410A. 1 Relationship between Fundamental Natural Frequency and Static Deflection 410A. 2 Rayleigh Technique 413A. 3 Dunkerley and Southwell Methods 415A. 4 Rayleigh–Ritz and Schmidt Approximations 415A. 5 Galerkin Procedure for Continuous Structures 416References 417B Numerical Integration of Newton’s Second Law 418References 421C Standard Octaves and Sound Pressure 422C. 1 Time History and Overall Sound Pressure 422C. 2 Peaks and Crest 423C. 3 Spectra and Spectral Density 424C. 4 Logarithmic Frequency Scales and Musical Tunings 424C. 5 Human Perception of Sound (Psychological Acoustics) 426References 427D Integrals Containing Mode Shapes of Single-Span Beams 429Reference 429E Finite Element Programs 435E. 1 Professional/Commercial Programs 435E. 2 Open Source /Low-Cost Programs 436Index 439