1st Edition

Analytical and Experimental Modal Analysis

By Subodh V. Modak Copyright 2024
    544 Pages 348 B/W Illustrations
    by CRC Press

    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.


    • 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
    Review questions

    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
    Review questions

    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
    Review questions

    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)
    Review questions

    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)
    Review questions

    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
    Review questions

    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
    Review questions

    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
    Review questions

    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


    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.