Introduction

The aim of the course is to understand and determine the magnitude of vibration and modal characteristics of a structural system. There are two routes to achieve this goal:

1. modal analysis (the theoretical approach), and
2. modal testing (the experimental approach).

The single degree of freedom (SDoF) model studied in vibration course enables us to understand the fundamental concepts of free and forced vibration, natural frequency, resonance and damping. However in MDoF systems, resonance may occur at a number of different frequencies, each of which corresponds to a different pattern or shape of the system's motion. These are known as the natural or normal modes of vibration or mode shapes. While only one differential equation is sufficient to describe the motion of an SDoF system, there is a differential equation of motion for each degree of freedom for MDoF systems. In other words, a set of n simultaneous equations is needed to mathematically describe an MDoF system.
In the experimental method, the structure is excited by applying forced vibration and measuring the responses, from which the vibration modes are determined and a structural model developed. This is sometimes called the reverse process to the theoretical method.
This course starts with a review of structural and dynamic theory. Next, methods of measuring frequency response from the structure under test are explained in details. Various methods of input excitation are discussed, such as shaker and impact hammer. Structural preparation and suspension methods are also examined.
A review of transducers, shakers and hammers together with signal processing equipment is made before discussing analysis methods, such as frequency-domain and time-domain curve fitting. Modal test philosophy including the sequence of steps and practical considerations in undertaking the test are discussed. The tabulation of results and derivation of mode shapes and construction of spatial models (mass, stiffness and damping) are covered before discussing the application of the modal test results.

Syllabus

1. Modal testing and analysis
   1. Introduction
   2. Types of tests
   3. Historical developments
   4. Applications of modal testing
   5. Route to vibration analysis
   6. References
      1. Books
      2. Conferences
      3. Journals
2. Single Degree-of-Freedom (SDOF)
1. Undamped
2. Viscously Damped
3. Hysterically (Structurally) Damped
4. FRFs
5. Alternative Forms of FRF
6. Relation between Different forms of FRFs
7. Graphical Display of FRFs
8. Frequency Response of Mass and stiffness Elements
1. Orthogonality Properties of MDOF
2. Mass-normalisation
3. Multiple modes
4. Forced Response of MDOF
1. General Concepts
2. Proportional damping
   1. special case
   2. general case
3. Proportional Hysteretic damping
1. MDOF Systems with hysteretic damping- general case
2. Numerical Example
   1. Undamped
   2. Proportional Structural Damping
   3. Non-Proportional Structural Damping
3. FRF Characteristics (Hysteretic Damping)
4. MDOF Systems with viscous damping- general case
5. FRF Characteristics - Viscous Damping
1. Real Modes
2. Complex Modes
3. Display of Complex Mode
4. Measurement of Modal Complexity
5. Origin of Complex Modes
1. Receptance and Impedance FRF Parameters
2. Some Definitions
3. FRF Plot in MDOF System
4. Point and Transfer FRFs
5. Example- FRFs of 6DOF System
6. Display of FRF Data For Damped Systems
   1. Bode Plots
   2. Nyquist diagrams
   3. Real and Imaginary plots
   4. Three-dimensional plots
1. Periodic Vibration
2. Transient Vibration
3. Random Vibration
1. Basic Modal Theory – Summary
2. Incomplete Modal Model
   1. Modal to Response
   2. Modal to Spatial
3. Incomplete Response Model
4. Incomplete Spatial
5. Condensation Techniques
1. Guyan Reduction
2. Improved Reduction System (IRS)
3. Dynamic Reduction
4. System Equivalent Reduction Expansion Process (SEREP).
1. Matrix properties
2. Matrix Norm
3. Generalised Inverses
4. Singular value Decomposition
5. Main Applications
   1. Calculation of the Rank of a Matrix
   2. Calculation of Condition Numbers
   3. Calculation of Generalised Inverses
6. Other applications of SVD
1. Definitions
2. Theory
3. ADDOFD, ADDOFV and ADDOFA
4. Optimum Suspension Positions
   1. Example
5. Optimum Driving Positions
1. Hammer Testing
2. Shaker Testing
3. Non-Optimum Driving Point (NODP) Technique
4. Effective Independence (EI) method
1. Objectives
2. Types of Vibration Measurements
3. Signal Quality
4. Signal Fidelity
5. Measurement Repeatability
6. Measurement Reliability
7. Measured Data Consistency, including reciprocity
8. FRF Measurement Systems
9. Important Aspects of Measurement Methods
   1. Support of test structure
   2. Attachment of exciter
   3. Push rod or stingers
   4. Transducers
   5. Excitation signal used
   6. Processing of signals
10. Mass Calibration and Mass Cancelation
1. Fourier Analysis
   1. Fourier series
   2. Fourier Transform
   3. Discrete Fourier series
   4. Fast Fourier Transform
2. Aliasing
3. Leakage
4. Windowing
5. Filtering
6. Zero padding
7. Zoom
8. Averaging
1. Type of Modal Analysis
2. Preliminary Checks of FRF Data
3. Basic Skeleton Theory
4. Mode Indicator Functions (MIFs)
5. SDOF Modal Analysis Method
   1. SDOF Curve-fit Method
      1. Peak Amplitude Method
      2. Circle-Fit Method
      3. Inverse or Line-fit Method
   2. Regenerated FRFs
   3. Residuals
6. Modal Analysis in Frequency Domain
1. MDOF Curve-fit Method
2. Nonlinear Least-Squares
3. Modal Analysis Using Rational Fractions
4. MDOF Curve-fits - Light Damping
5. Global frequency Methods
1. Multiple Curve Fitting
2. Global Rational Fraction Polynomial Method (GRFP)
3. Global SVD Method
4. Example
5. Global frequency Methods in the Frequency domain
1. Strategy for Model Validation
2. Theory/Experiment Comparison
3. Comparison of Modal Properties
   1. Comparison of Natural Frequencies
   2. Mode Shapes
4. MAC Correlation Between Two Sets Of Modes
5. Normalised Version of the Mac
6. Error Location - The COMAC
7. Correlation of Other Parameters- Frequency Response Functions
1. Modal Analysis of Rotating Structures
2. Test/Analysis Methods to Identify Non-Linearities Parameters
3. LDVs as Transducer
4. Application of MAC in Frequency Domain
5. Operational Modal Analysis
6. Undamped MDOF systems
7. MDOF Systems with Proportional Damping
8. MDOF Systems With Damping – General Case
9. Complex Modes
10. Characteristics and Presentation of MDOF FRF Data
11. Non-Sinusoidal Vibration and FRF Properties
12. Complete and Incomplete Models
13. Generalised Inverses
14. Test Planning
15. Basic Measurement Techniques
16. Introduction to Digital Signal Processing
17. Modal Parameter Extraction Methods
18. Comparison of Test and Analysis
19. Other Dimensions in Modal Analysis Testing