كتاب Vibration Fundamentals and Practice

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Vibration Fundamentals and Practice

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Chapter 1 Vibration Engineering
1.1 Study of Vibration
1.2 Application Areas
1.3 History of Vibration
1.4 Organization of the Book
Problems
References and Further Reading
Author’s Work
Other Useful Publications
Chapter 2 Time Response
2.1 Undamped Oscillator
2.1.1 Energy Storage Elements
Inertia
Spring
Gravitational Potential Energy
2.1.2 Conservation of Energy
System 1 (Translatory)
System 2 (Rotatory)
System 3 (Flexural)
System 4 (Swinging)
System 5 (Liquid Slosh)
System 6 (Electrical)
Capacitor
Inductor
2.1.3 Free Response
Example 2.1
Solution
2.2 Heavy Springs
2.2.1 Kinetic Energy Equivalence
Example 2.2
Solution
2.3 Oscillations in Fluid Systems
Example 2.3
Solution
2.4 Damped Simple Oscillator
2.4.1 Case 1: Underdamped Motion
Initial Conditions
2.4.2 Logarithmic Decrement Method
2.4.3 Case 2: Overdamped Motion
2.4.4 Case 3: Critically Damped Motion
2.4.5 Justification for the Trial Solution
First-Order System
Second-Order System
Repeated Roots
2.4.6 Stability and Speed of Response
Example 2.4
Solution
2.5 Forced Response
2.5.1 Impulse Response Function
2.5.2 Forced Response
2.5.3 Response to a Support Motion
Impulse Response
The Riddle of Zero Initial Conditions
Step Response
Liebnitz’s Rule
Problems
Chapter 3 Frequency Response
3.1 Response to Harmonic Excitations
3.1.1 Response Characteristics
Case 1
Case 2
Case 3
Particular Solution (Method 1)
Particular Solution (Method 2): Complex Function Method
Resonance
3.1.2 Measurement of Damping Ratio (Q-Factor Method)
Example 3.1
Solution
3.2 Transform Techniques
3.2.1 Transfer Function
3.2.2 Frequency-Response Function (Frequency-Transfer Function)
Impulse Response
Case 1 (
ζ < 1)
Case 2 (
ζ > 1)
Case 3 (
ζ = 1)
Step Response
3.2.3 Transfer Function Matrix
Example 3.2
Example 3.3
Example 3.4
Solution
3.3 Mechanical Impedance Approach
Mass Element
Spring Element
Damper Element
3.3.1 Interconnection Laws
Example 3.5
Example 3.6
3.4 Transmissibility Functions
3.4.1 Force Transmissibility
3.4.2 Motion Transmissibility
System Suspended on a Rigid Base (Force Transmissibility)
System with Support Motion (Motion Transmissibility)
3.4.3 General Case
Example 3.7
3.4.4 Peak Values of Frequency-Response Functions
3.5 Receptance Method
3.5.1 Application of Receptance
Undamped Simple Oscillator
Dynamic Absorber
Problems
Chapter 4 Vibration Signal Analysis
4.1 Frequency Spectrum
4.1.1 Frequency
4.1.2 Amplitude Spectrum
4.1.3 Phase Angle
4.1.4 Phasor Representation of Harmonic Signals
4.1.5 RMS Amplitude Spectrum
4.1.6 One-Sided and Two-Sided Spectra
4.1.7 Complex Spectrum
4.2 Signal Types
4.3 Fourier Analysis
4.3.1 Fourier Integral Transform (FIT)
4.3.2 Fourier Series Expansion (FSE)
4.3.3 Discrete Fourier Transform (DFT)
4.3.4 Aliasing Distortion
Sampling Theorem
Aliasing Distortion in the Time Domain
Anti-Aliasing Filter
Example 4.1
4.3.5 Another Illustration of Aliasing
Example 4.2
4.4 Analysis of Random Signals
4.4.1 Ergodic Random Signals
4.4.2 Correlation and Spectral Density
4.4.3 Frequency Response Using Digital Fourier Transform
4.4.4 Leakage (Truncation Error)
4.4.5 Coherence
4.4.6 Parseval’s Theorem
4.4.7 Window Functions
4.4.8 Spectral Approach to Process Monitoring
4.4.9 Cepstrum
4.5 Other Topics of Signal Analysis
4.5.1 Bandwidth
4.5.2 Transmission Level of a Bandpass Filter
4.5.3 Effective Noise Bandwidth
4.5.4 Half-Power (or 3 dB) Bandwidth
4.5.5 Fourier Analysis Bandwidth
4.6 Resolution in Digital Fourier Results
4.7 Overlapped Processing
Example 4.3
4.7.1 Order Analysis
Speed Spectral Map
Time Spectral Map
Order Tracking
Problems
Chapter 5 Modal Analysis
5.1 Degrees of Freedom and Independent Coordinates
5.1.1 Nonholonomic Constraints
Example 5.1
Example 5.2
5.2 System Representation
5.2.1 Stiffness and Flexibility Matrices
5.2.2 Inertia Matrix
5.2.3 Direct Approach for Equations of Motion
5.3 Modal Vibrations
Example 5.3
5.4 Orthogonality of Natural Modes
5.4.1 Modal Mass and Normalized Modal Vectors
5.5 Static Modes and Rigid Body Modes
5.5.1 Static Modes
5.5.2 Linear Independence of Modal Vectors
5.5.3 Modal Stiffness and Normalized Modal Vectors
5.5.4 Rigid Body Modes
Example 5.4
Equation of Heave Motion
Equation of Pitch Motion
Example 5.5
First Mode (Rigid Body Mode)
Second Mode
5.5.5 Modal Matrix
5.5.6 Configuration Space and State Space
State Vector
5.6 Other Modal Formulations
5.6.1 Non-Symmetric Modal Formulation
5.6.2 Transformed Symmetric Modal Formulation
Example 5.6
Approach 2
Approach 3
5.7 Forced Vibration
Example 5.7
First Mode (Rigid Body Mode)
Second Mode (Oscillatory Mode)
5.8 Damped Systems
5.8.1 Proportional Damping
Example 5.8
5.9 State-Space Approach
5.9.1 Modal Analysis
5.9.2 Mode Shapes of Nonoscillatory Systems
5.9.3 Mode Shapes of Oscillatory Systems
Example 5.9
Problems
Chapter 6 Distributed-Parameter Systems
6.1 Transverse Vibration of Cables
6.1.1 Wave Equation
6.1.2 General (Modal) Solution
6.1.3 Cable with Fixed Ends
6.1.4 Orthogonality of Natural Modes
Example 6.1
Solution
6.1.5 Application of Initial Conditions
Example 6.2
Solution
6.2 Longitudinal Vibration of Rods
6.2.1 Equation of Motion
6.2.2 Boundary Conditions
Example 6.3
Solution
6.3 Torsional Vibration of Shafts
6.3.1 Shaft with Circular Cross Section
6.3.2 Torsional Vibration of Noncircular Shafts
Example 6.4
Solution
Example 6.5
Solution
6.4 Flexural Vibration of Beams
6.4.1 Governing Equation for Thin Beams
Moment-Deflection Relation
Rotatory Dynamics (Equilibrium)
Transverse Dynamics
6.4.2 Modal Analysis
6.4.3 Boundary Conditions
6.4.4 Free Vibration of a Simply Supported Beam
Normalization of Mode Shape Functions
Initial Conditions
6.4.5 Orthogonality of Mode Shapes
Case of Variable Cross Section
6.4.6 Forced Bending Vibration
Example 6.6
Solution
Example 6.7
Solution
6.4.7 Bending Vibration of Beams with Axial Loads
6.4.8 Bending Vibration of Thick Beams
6.4.9 Use of the Energy Approach
6.4.10 Orthogonality with Inertial Boundary Conditions
Rotatory Inertia
6.5 Damped Continuous Systems
6.5.1 Modal Analysis of Damped Beams
Example 6.8
Solution
6.6 Vibration of Membranes and Plates
6.6.1 Transverse Vibration of Membranes
6.6.2 Rectangular Membrane with Fixed Edges
6.6.3 Transverse Vibration of Thin Plates
6.6.4 Rectangular Plate with Simply Supported Edges
Problems
Chapter 7 Damping
7.1 Types of Damping
7.1.1 Material (Internal) Damping
Viscoelastic Damping
Hysteretic Damping
Example 7.1
Solution
7.1.2 Structural Damping
7.1.3 Fluid Damping
Example 7.2
Solution
7.2 Representation of Damping in Vibration Analysis
7.2.1 Equivalent Viscous Damping
7.2.2 Complex Stiffness
Example 7.3
Solution
7.2.3 Loss Factor
7.3 Measurement of Damping
7.3.1 Logarithmic Decrement Method
7.3.2 Step-Response Method
7.3.3 Hysteresis Loop Method
Example 7.4
Solution
7.3.4 Magnification-Factor Method
7.3.5 Bandwidth Method
7.3.6 General Remarks
7.4 Interface Damping
Example 7.5
Solution
7.4.1 Friction In Rotational Interfaces
7.4.2 Instability
Problems
Chapter 8 Vibration Instrumentation
8.1 Vibration Exciters
8.1.1 Shaker Selection
Force Rating
Power Rating
Stroke Rating
Example 8.1
Solution
Hydraulic Shakers
Inertial Shakers
Electromagnetic Shakers
8.1.2 Dynamics of Electromagnetic Shakers
Transient Exciters
8.2 Control System
8.2.1 Components of a Shaker Controller
Compressor
Equalizer (Spectrum Shaper)
Tracking Filter
Excitation Controller (Amplitude Servo-Monitor)
8.2.2 Signal-Generating Equipment
Oscillators
Random Signal Generators
Tape Players
Data Processing
8.3 Performance Specification
8.3.1 Parameters for Performance Specification
Time-Domain Specifications
Frequency-Domain Specifications
8.3.2 Linearity
8.3.3 Instrument Ratings
Rating Parameters
8.3.4 Accuracy and Precision
8.4 Motion Sensors and Transducers
8.4.1 Potentiometer
Potentiometer Resolution
Optical Potentiometer
8.4.2 Variable-Inductance Transducers
Mutual-Induction Transducers
Linear-Variable Differential Transformer (LVDT)
Signal Conditioning
Example 8.2
Solution
8.4.3 Mutual-Induction Proximity Sensor
8.4.4 Self-Induction Transducers
8.4.5 Permanent-Magnet Transducers
8.4.6 AC Permanent-Magnet Tachometer
8.4.7 AC Induction Tachometer
8.4.8 Eddy Current Transducers
8.4.9 Variable-Capacitance Transducers
Capacitive Displacement Sensors
Capacitive Angular Velocity Sensor
Capacitance Bridge Circuit
8.4.10 Piezoelectric Transducers
Sensitivity
Example 8.3
Solution
Piezoelectric Accelerometer
Charge Amplifier
8.5 Torque, Force, and Other Sensors
8.5.1 Strain-Gage Sensors
Equations for Strain-Gage Measurements
Bridge Sensitivity
The Bridge Constant
Example 8.4
Solution
The Calibration Constant
Example 8.5
Solution
Data Acquisition
Accuracy Considerations
Semiconductor Strain Gages
Force and Torque Sensors
Strain-Gage Torque Sensors
Deflection Torque Sensors
Variable-Reluctance Torque Sensor
Reaction Torque Sensors
8.5.2 Miscellaneous Sensors
Stroboscope
Fiber-Optic Sensors and Lasers
Fiber-Optic Gyroscope
Laser Doppler Interferometer
Ultrasonic Sensors
Gyroscopic Sensors
8.6 Component Interconnection
8.6.1 Impedance Characteristics
Cascade Connection of Devices
Impedance-Matching Amplifiers
Operational Amplifiers
Voltage Followers
Charge Amplifiers
8.6.2 Instrumentation Amplifier
Ground Loop Noise
Problems
Chapter 9 Signal Conditioning and Modification
9.1 Amplifiers
9.1.1 Operational Amplifier
Example 9.1
Solution
9.1.2 Use of Feedback in Op-amps
9.1.3 Voltage, Current, and Power Amplifiers
9.1.4 Instrumentation Amplifiers
Differential Amplifier
Common Mode
Amplifier Performance Ratings
Example 9.2
Solution
Common-Mode Rejection Ratio (CMRR)
AC-Coupled Amplifiers
9.2 Analog Filters
9.2.1 Passive Filters and Active Filters
Number of Poles
9.2.2 Low-Pass Filters
Example 9.3
Solution
Low-Pass Butterworth Filter
Example 9.4
Solution
9.2.3 High-Pass Filters
9.2.4 Bandpass Filters
Resonance-Type Bandpass Filters
Example 9.5
Solution
9.2.5 Band-Reject Filters
9.3 Modulators and Demodulators
9.3.1 Amplitude Modulation
Modulation Theorem
Side Frequencies and Side Bands
9.3.2 Application of Amplitude Modulation
Fault Detection and Diagnosis
9.3.3 Demodulation
9.4 Analog/Digital Conversion
9.4.1 Digital-to-Analog Conversion (DAC)
Weighted-Resistor DAC
Ladder DAC
DAC Error Sources
9.4.2 Analog-to-Digital Conversion (ADC)
Successive-Approximation ADC
Dual-Slope ADC
Counter ADC
9.4.3 ADC Performance Characteristics
Resolution and Quantization Error
Monotonicity, Nonlinearity, and Offset Error
ADC Conversion Rate
9.4.4 Sample-and-Hold (S/H) Circuitry
9.4.5 Multiplexers (MUX)
Analog Multiplexers
Digital Multiplexers
9.4.6 Digital Filters
9.5 Bridge Circuits
9.5.1 Wheatstone Bridge
9.5.2 Constant-Current Bridge
9.5.3 Bridge Amplifiers
Half-Bridge Circuits
9.5.4 Impedance Bridges
Owen Bridge
Wien-Bridge Oscillator
9.6 Linearizing Devices
9.6.1 Linearization by Software
9.6.2 Linearization by Hardware Logic
9.6.3 Analog Linearizing Circuitry
9.6.4 Offsetting Circuitry
9.6.5 Proportional-Output Circuitry
Curve-Shaping Circuitry
9.7 Miscellaneous Signal-Modification Circuitry
9.7.1 Phase Shifter
9.7.2 Voltage-to-Frequency Converter (VFC)
9.7.3 Frequency-to-Voltage Converter (FVC)
9.7.4 Voltage-to-Current Converter (VCC)
9.7.5 Peak-Hold Circuit
9.8 Signal Analyzers and Display Devices
9.8.1 Signal Analyzers
9.8.2 Oscilloscopes
Triggering
Lissajous Patterns
Digital Oscilloscopes
Problems
Chapter 10 Vibration Testing
10.1 Representation of a Vibration Environment
10.1.1 Test Signals
Stochastic versus Deterministic Signals
10.1.2 Deterministic Signal Representation
Single-Frequency Signals
Sine Sweep
Sine Dwell
Decaying Sine
Sine Beat
Sine Beat with Pauses
Multifrequency Signals
Actual Excitation Records
Simulated Excitation Signals
10.1.3 Stochastic Signal Representation
Ergodic Random Signals
Stationary Random Signals
Independent and Uncorrelated Signals
Transmission of Random Excitations
10.1.4 Frequency-Domain Representations
Fourier Spectrum Method
Power Spectral Density Method
10.1.5 Response Spectrum
Displacement, Velocity, and Acceleration Spectra
Response-Spectra Plotting Paper
Zero-Period Acceleration
Uses of Response Spectra
10.1.6 Comparison of Various Representations
10.2 Pretest Procedures
10.2.1 Purpose of Testing
10.2.2 Service Functions
Active Equipment
Passive Equipment
Functional Testing
10.2.3 Information Acquisition
Interface Details
Effect of Neglecting Interface Dynamics
Effects of Damping
Effects of Inertia
Effect of Natural Frequency
Effect of Excitation Frequency
Other Effects of Interface
10.2.4 Test-Program Planning
Testing of Cabinet-Mounted Equipment
10.2.5 Pretest Inspection
10.3 Testing Procedures
10.3.1 Resonance Search
10.3.2 Methods of Determining Frequency-Response Functions
Fourier Transform Method
Spectral Density Method
Harmonic Excitation Method
10.3.3 Resonance-Search Test Methods
Hammer (Bump) Test and Drop Test
Pluck Test
Shaker Tests
10.3.4 Mechanical Aging
Equivalence for Mechanical Aging
Excitation-Intensity Equivalence
Dynamic-Excitation Equivalence
Cumulative Damage Theory
10.3.5 TRS Generation
10.3.6 Instrument Calibration
10.3.7 Test-Object Mounting
10.3.8 Test-Input Considerations
Test Nomenclature
Testing with Uncorrelated Excitations
Symmetrical Rectilinear Testing
Geometry versus Dynamics
Some Limitations
Testing of Black Boxes
Phasing of Excitations
Testing a Gray or White Box
Overtesting in Multitest Sequences
10.4 Product Qualification Testing
10.4.1 Distribution Qualification
Drive-Signal Generation
Distribution Spectra
Test Procedures
10.4.2 Seismic Qualification
Stages of Seismic Qualification
10.4.3 Test Preliminaries
Single-Frequency Testing
Multifrequency Testing
10.4.4 Generation of RRS Specifications
Problems
Chapter 11 Experimental Modal Analysis
11.1 Frequency-Domain Formulation
11.1.1 Transfer Function Matrix
11.1.2 Principle of Reciprocity
Example 11.1
11.2 Experimental Model Development
11.2.1 Extraction of the Time-Domain Model
11.3 Curve-Fitting of Transfer Functions
11.3.1 Problem Identification
11.3.2 Single-Degree-of-Freedom and Multi-Degree-of-Freedom Techniques
11.3.3 Single-Degree-of-Freedom Parameter Extraction in the Frequency Domain
Circle-Fit Method
Peak Picking Method
11.3.4 Multi-Degree-of-Freedom Curve Fitting
Formulation of the Method
11.3.5 A Comment on Static Modes and Rigid Body Modes
11.3.6 Residue Extraction
11.4 Laboratory Experiments
11.4.1 Lumped-Parameter System
Frequency-Domain Test
Time-Domain Test
11.4.2 Distributed-Parameter System
11.5 Commercial EMA Systems
11.5.1 System Configuration
FFT Analysis Options
Modal Analysis Components
Problems
Chapter 12 Vibration Design and Control
Shock and Vibration
12.1 Specification of Vibration Limits
12.1.1 Peak Level Specification
12.1.2 RMS Value Specification
12.1.3 Frequency-Domain Specification
12.2 Vibration Isolation
Example 12.1
Solution
12.2.1 Design Considerations
Example 12.2
Solution
12.2.2 Vibration Isolation of Flexible Systems
12.3 Balancing of Rotating Machinery
12.3.1 Static Balancing
Balancing Approach
12.3.2 Complex Number/Vector Approach
Example 12.3
Solution
12.3.3 Dynamic (Two-Plane) Balancing
Example 12.4
Solution
12.3.4 Experimental Procedure of Balancing
12.4 Balancing of Reciprocating Machines
12.4.1 Single-Cylinder Engine
12.4.2 Balancing the Inertia Load of the Piston
12.4.3 Multicylinder Engines
Two-Cylinder Engine
Six-Cylinder Engine
Example 12.5
Solution
12.4.4 Combustion/Pressure Load
12.5 Whirling of Shafts
12.5.1 Equations of Motion
12.5.2 Steady-State Whirling
Example 12.6
Solution
12.5.3 Self-Excited Vibrations
12.6 Design Through Modal Testing
12.6.1 Component Modification
Example 12.7
Solution
12.6.2 Substructuring
12.7 Passive Control of Vibration
12.7.1 Undamped Vibration Absorber
Example 12.8
Solution
12.7.2 Damped Vibration Absorber
Optimal Absorber Design
Example 12.9
Solution
12.7.3 Vibration Dampers
12.8 Active Control of Vibration
12.8.1 Active Control System
12.8.2 Control Techniques
State-Space Models
Example 12.10
Solution
Position and Velocity Feedback
Linear Quadratic Regulator (LQR) Control
Modal Control
12.8.3 Active Control of Saw Blade Vibration
12.9 Control of Beam Vibrations
12.9.1 State-Space Model of Beam Dynamics
12.9.2 Control Problem
12.9.3 Use of Linear Dampers
Design Example
Problems
Appendix A Dynamic Models and Analogies
A.1 Model Development
A.2 Analogies
A.3 Mechanical Elements
A.3.1 Mass (Inertia) Element
A.3.2 Spring (Stiffness) Element
A.4 Electrical Elements
A.4.1 Capacitor Element
A.4.2 Inductor Element
A.5 Thermal Elements
A.5.1 Thermal Capacitor
A.5.2 Thermal Resistance
A.6 Fluid Elements
A.6.1 Fluid Capacitor
A.6.2 Fluid Inertor
A.6.3 Fluid Resistance
A.6.4 Natural Oscillations
A.7 State-Space Models
A.7.1 Linearization
A.7.2 Time Response
A.7.3 Some Formal Definitions
A.7.4 Illustrative Example
A.7.5 Causality and Physical Realizability
Appendix B Newtonian and Lagrangian Mechanics
B.1 Vector Kinematics
B.1.1 Euler’s Theorem
Important Corollary
Proof
B.1.2 Angular Velocity and Velocity at a Point of a Rigid Body
Theorem
Proof
B.1.3 Rates of Unit Vectors Along Axes of Rotating Frames
General Result
Cartesian Coordinates
Polar Coordinates (2-D)
Spherical Polar Coordinates
Tangential-Normal (Intrinsive) Coordinates (2-D)
B.1.4 Acceleration Expressed in Rotating Frames
Spherical Polar Coordinates
Tangential-Normal Coordinates (2-D)
B.2 Newtonian (Vector) Mechanics
B.2.1 Frames of Reference Rotating at Angular Velocity
w
B.2.2 Newton’s Second Law for a Particle of Mass m
B.2.3 Second Law for a System of Particles — Rigidly or Flexibly Connected
B.2.4 Rigid Body Dynamics — Inertia Matrix and Angular Momentum
B.2.5 Manipulation of Inertia Matrix
Parallel Axis Theorem— Translational Transformation of [I]
Rotational Transformation of [I]
Principal Directions (Eigenvalue Problem)
Mohr’s Circle
B.2.6 Euler’s Equations (for a Rigid Body Rotating at
w)
B.2.7 Euler’s Angles
B.3 Lagrangian Mechanics
B.3.1 Kinetic Energy and Kinetic Coenergy
B.3.2 Work and Potential Energy
Examples
B.3.3 Holonomic Systems, Generalized Coordinates, and Degrees of Freedom
B.3.4 Hamilton’s Principle
B.3.5 Lagrange’s Equations
Example
Generalized Coordinates
Generalized Nonconservative Forces
Lagrangian
Lagrange’s Equations
Appendix C Review of Linear Algebra
C.1 Vectors and Matrices
C.2 Vector-Matrix Algebra
C.2.1 Matrix Addition and Subtraction
C.2.2 Null Matrix
C.2.3 Matrix Multiplication
C.2.4 Identity Matrix
C.3 Matrix Inverse
C.3.1 Matrix Transpose
C.3.2 Trace of a Matrix
C.3.3 Determinant of a Matrix
C.3.4 Adjoint of a Matrix
C.3.5 Inverse of a Matrix
C.4 Vector Spaces
C.4.1 Field (
)
C.4.2 Vector Space (
)
Properties
Special Case
C.4.3 Subspace
of

C.4.4 Linear Dependence
C.4.5 Basis and Dimension of a Vector Space
C.4.6 Inner Product
C.4.7 Norm
Properties
C.4.8 Gram-Schmidt Orthogonalization
C.4.9 Modified Gram-Schmidt Procedure
C.5 Determinants
C.5.1 Properties of Determinant of a Matrix
C.5.2 Rank of a Matrix
C.6 System of Linear Equations
References
Appendix D Digital Fourier Analysis and FFT
D.1 Unification of the Three Fourier Transform Types
D.1.1 Relationship Between DFT and FIT
D.1.2 Relationship Between DFT and FSE
D.2 Fast Fourier Transform (FFT)
D.2.1 Development of the Radix-Two FFT Algorithm
D.2.2 The Radix-Two FFT Procedure
D.2.3 Illustrative Example
D.3 Discrete Correlation and Convolution
D.3.1 Discrete Correlation
Discrete Correlation Theorem
Discrete Convolution Theorem
D.4 Digital Fourier Analysis Procedures
D.4.1 Fourier Transform Using DFT
D.4.2 Inverse DFT Using DFT
D.4.3 Simultaneous DFT of Two Real Data Records
D.4.4 Reduction of Computation Time for a Real Data Record
D.4.5 Convolution of Finite Duration Signals Using DFT
Wraparound Error
Data-Record Sectioning in Convolution
Appendix E Reliability Considerations for Multicomponent Units
E.1 Failure Analysis
E.1.1 Reliability
E.1.2 Unreliability
E.1.3 Inclusion–Exclusion Formula
Example
E.2 Bayes’ Theorem
E.2.1 Product Rule for Independent Events
E.2.2 Failure Rate
E.2.3 Product Rule for Reliability
Answers to Numerical Problems

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كتاب Vibration Fundamentals and Practice 912775