This book covers the fundamentals and basic concepts of analytical and experimental approaches to modal analysis. In practice, the analytical approach based on lumped parameter and finite element models is widely used for modal analysis and simulation, and experimental modal analysis is widely used for modal identification and model validation. This book is inspired by this consideration and is written to give a complete picture of modal analysis.
Features:
- Presents a systematic development of the relevant concepts and methods of the analytical and experimental modal analyses.
- Covers phase resonance testing and operational modal analysis.
- Provides the relevant signal processing concepts.
- Includes applications like model validation and updating, force identification and structural modification.
- Contains simulations, examples, and MATLAB® programs to enhance understanding.
This book is aimed at senior undergraduates/graduates, researchers, and engineers from mechanical, aerospace, automotive, civil, and structural engineering disciplines.
Chapter 1 Introduction
1.1 What is modal analysis?
1.2 Vibration, its causes, and detrimental effects
1.3 Need for modal analysis
1.4 Analytical modal analysis
1.5 Experimental modal analysis
1.6 Operational modal analysis (OMA)
1.7 Applications of EMA
1.8 Finite element model validation and updating
1.9 Overview of the field of modal analysis
Review questions
Chapter 2 Lumped parameter modeling of vibrating systems
2.1 Introduction
2.2 Mathematical models of vibrating systems
2.3 Lumped parameter modeling
2.4 Derivation of the governing equations
2.5 Newton’s law
2.6 D’Alembert’s principle
2.7 Law of conservation of energy
2.8 Principle of virtual work
2.9 Lagrange’s equations
2.10 Hamilton’s principle
2.11 Flexibility influence coefficients
2.12 Stiffness influence coefficients
2.13 Reciprocity theorem
2.14 Lumped parameter modeling of systems with beam-like members
Examples
Review questions
Problems
Chapter 3 Finite element modeling of vibrating systems
3.1 Introduction
3.2 Important aspects of FEM
3.3 Hamilton's principle
3.4 Hamilton's principle applied to a discrete/MDOF system
3.5 Hamilton's principle applied to a continuous dynamic system
3.6 FEM for dynamics
3.7 Bar element
3.8 Beam element
3.9 Frame element
3.10 Obtaining equations of motion of the system
Examples
Review questions
Problems
Chapter 4 Analytical modal analysis of SDOF systems
4.1 Introduction
4.2 Free vibration without damping
4.3 Free vibration with viscous damping
4.4 Forced vibration of an undamped system under harmonic excitation
4.5 Forced vibration of a viscously damped system under harmonic excitation
4.6 Vibration with structural damping
4.7 Frequency response function
4.8 Response to harmonic excitation using FRF
4.9 Response to periodic excitation using FRF
4.10 Impulse response function
4.11 Response to Transient excitation
4.12 Response analysis by the Laplace transform approach
Examples
Review questions
Problems
Chapter 5 Analytical modal analysis of undamped MDOF systems
5.1 Introduction
5.2 Free vibration of undamped MDOF systems
5.3 Expansion theorem
5.4 Physical space and Modal Space
5.5 Forced vibration response to harmonic Forces
5.6 Spatial model and Modal model
5.7 FRF matrix
5.8 Relationship between the FRF and modal model
5.9 Response model (Frequency domain)
5.10 Response to periodic forces by modal analysis
5.11 Response to transient excitation by modal analysis
5.12 IRF matrix
5.13 Relationship between the IRFs and the modal model
5.14 Response model (Time domain)
Examples
Review questions
Problems
Chapter 6 Analytical modal analysis of damped MDOF systems
6.1 Introduction
6.2 Eigenvalue analysis of systems with structural damping
6.3 Forced vibration response of systems with structural damping to Harmonic Forces
6.4 Free vibration response of MDOF systems with proportional viscous damping
6.5 Forced vibration response of MDOF systems with proportional viscous damping
6.6 Free vibration of systems with nonproportional viscous damping
6.7 Forced vibration response of systems with nonproportional viscous damping to harmonic forces by modal analysis
6.8 Frequency response function (FRF) matrix
6.9 Response model (Frequency domain)
6.10 Response to transient excitation by modal analysis
6.11 Impulse response function (IRF)
6.12 Response model (time domain)
Examples
Review questions
Problems
Chapter 7 Characteristics of FRFs
7.1 Introduction
7.2 FRF types
7.3 Graphical representation of FRFs
7.4 Characteristics of SDOF system FRFs
7.5 Characteristics of undamped MDOF system FRFs
7.6 FRFs of MDOF Systems with structural damping
7.7 FRFs of MDOF systems with viscous damping
Examples
Review questions
Problems
Chapter 8 Signal processing for experimental modal analysis
8.1 Introduction
8.2 Continuous-time signals
8.3 Fourier analysis of periodic continuous signals
8.4 Fourier analysis of aperiodic continuous signals (Fourier transform)
8.5 Discrete-time signals
8.6 Fourier analysis of discrete periodic signals
8.7 Fourier analysis of aperiodic discrete signals
8.8 Relationship between DFS and DTFT
8.9 Discrete Fourier Transform (DFT)
8.10 Which one to use in practice, DFS, DTFT, or DFT?
8.11 Relationships between the parameters of a discrete signal and its DFT
8.12 Single-sided spectrum
8.13 Fast Fourier Transform (FFT)
8.14 Effect of time sampling on frequency domain representation
8.15 Aliasing and sampling theorem
8.16 Anti-aliasing filter
8.17 Quantisation
8.18 Windowing
8.19 Fourier analysis of Random signals
Examples
Review questions
Problems
Chapter 9 FRF measurement using an impact hammer
9.1 Introduction
9.2 Basic principle of EMA
9.3 Setup for EMA using impact testing
9.4 FRF estimation using transient input and output
9.5 Impact hammer
9.6 Role of the hammer tip
9.7 Response measurement
9.8 Charge amplifier
9.9 IEPE Piezoelectric accelerometers
9.10 Mounting of accelerometers
9.11 Accelerometer selection
9.12 Laser Doppler vibrometer (LDV)
9.13 Estimation of FRF from auto and cross spectrums
9.14 Coherence function
9.15 FRF measurement using impact testing
9.16 FRF validity checks
9.17 Boundary conditions for modal testing
9.18 Calibration
Review questions
Chapter 10 FRF measurement using shaker excitation
10.1 Introduction
10.2 Experimental setup for FRF measurement using a shaker
10.3 Vibration exciter/shaker
10.4 Electromagnetic shaker
10.5 Attachment and support of shaker
10.6 Shaker-structure interaction model
10.7 Dynamic effects of shaker-structure interaction
10.8 Shaker specifications
10.9 Force transducer
10.10 Mass cancellation
10.11 Impedance head
10.12 FRF measurement using shaker testing
10.13 FRF measurement using other random excitation signals
10.14 FRF measurement using sine excitation
Review questions
Chapter 11 Modal parameter estimation methods
11.1 Introduction
11.2 Classification of the curve fitting methods
11.3 Analytical forms of FRF for curve fitting
11.4 Peak-picking method
11.5 Circle fit method
11.6 Line fit method
11.7 Modified line fit method
11.8 Residuals
11.9 Rational fraction polynomial method
11.10 Global rational fraction polynomial method
11.11 Complex exponential method (CEM)
11.12 Stabilisation diagram
11.13 Least square complex exponential method
11.14 Singular value decomposition
11.15 Ibrahim time-domain method
11.16 Eigensystem realization algorithm
11.17 Complex mode indicator function
Review questions
Chapter 12 Phase resonance testing
12.1 Introduction
12.2 Objective of phase resonance testing
12.3 Phase resonance condition
12.4 Excitation of normal modes
12.5 Extended Asher’s method of force vector estimation
12.6 Real mode indicator function (RMIF)
12.7 Multivariate mode indicator function (MMIF)
12.8 Estimation of other modal properties
12.9 Basic steps in phase resonance testing
12.10 Phase resonance testing simulation
12.11 Comparison of phase resonance and phase separation techniques
Review questions
Chapter 13 Operational modal analysis
13.1 Introduction
13.2 Basic philosophy of OMA
13.3 Cross-correlation of the system output with white noise excitation
13.4 Estimation of free decays for OMA
13.5 OMA in the time domain
13.6 OMA in the frequency domain
13.7 Harmonics detection in OMA
13.8 Advantages and limitations of OMA
Review questions
Chapter 14 Applications of experimental modal analysis
14.1 Introduction
14.2 Incomplete model
14.3 Can we identify the spatial model from the modal model?
14.4 Response simulation
14.5 Dynamic design
14.6 Local structural modification
14.7 Coupled structural analysis
14.8 Force identification
Examples
Review questions
Problems
Chapter 15 Finite element model validation and updating
15.1 Introduction
15.2 Need for FE model validation
15.3 Reference data for FE model validation
15.4 FE model correlation
15.5 Correlation of mode shapes
15.6 Correlation of the natural frequencies
15.7 Correlation of the FRFs
15.8 Model reduction
15.9 Mode shape expansion
15.10 FE model updating
15.11 Updating parameters selection
15.12 Classification of model updating methods
15.13 Direct matrix updating
15.14 Iterative method of model updating using modal data
15.15 Iterative method of model updating using FRF data
15.16 Model updating using normal FRFs
15.17 Regularisation
15.18 Stochastic FE model updating
Review questions
Problems
Biography
Subodh V. Modak is a Professor in the Department of Mechanical Engineering at the Indian Institute of Technology Delhi. He received his bachelor's degree in Mechanical Engineering from Shri G.S. Institute of Technology and Science, Indore, master’s degree in Design Engineering from Indian Institute of Technology, Bombay, and PhD from Indian Institute of Technology, Delhi. His research and teaching interests include experimental modal analysis, model updating, vibration and acoustics, control engineering, and active noise control. His research is published in many leading and reputed journals.
