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# Modal Analysis and Modal Testing

**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:

- modal analysis (the theoretical approach), and
- 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**

# Preliminary

- Introduction to modal analysis and modal testing
- Applications of modal analysis and modal testing
- Philosophy of modal analysis and modal testing
- Terminology and notations
- Use of complex algebra to describe harmonic vibration
- Matrix Decomposition and SVD
- Fourier Analysis

# Theoretical Basis

- Theory of SDOF systems
- Presentation and properties of FRF data for SDOF system
- Theory of undamped MDOF system
- MDOF systems with proportional damping
- MDOF with structural damping
- MDOF with viscous damping
- Presentation and properties of FRF data for MDOF systems
- Complex modes
- Non-sinusoidal vibration and FRF properties

# Response Function Measurement Techniques

- Basic measurement system
- Structure preparation
- Excitation of the structure
- Transducer and amplifier
- Analyzer
- Digital signal processing
- Random signal analysis
- Calibration and mass cancellation
- Multi-point excitation method

# Modal Parameter Extraction Methods

- Preliminary check of FRF data
- SDOF modal analysis method
- MDOF modal analysis methods in frequency domain
- Global modal analysis method in frequency domain
- MDOF modal analysis in time domain
- Modal analysis of nonlinear structure
- Modal analysis of rotating structure

# Derivation of Mathematical Model

- Model identification
- Modal models
- Refinement of modal models
- Response models
- Spatial models
- Mobility skeleton and system models
- Sensitivity of models

# Applications and Advanced Topics

- Comparison and correlation of experiment and prediction
- Comparison of modal properties
- Comparison of natural frequencies
- Comparison of mode shapes
- Comparison of response properties

- Adjustment or updating of models
- Coupled and modified structure analysis
- Test planning

**References:**

- D. J. Ewins, “Modal testing, theory, practice and application”, Second edition, Research studies press Ltd, 2000.
- “Premier on best practice in dynamic testing”, The dynamic testing agency, 1996.
- K. C. McConnell, “Vibration Testing, theory and practice”, John Wiley & Sons, 1995.
- N. Maia, Silva, J. “Theoretical and experimental modal analysis”, Research studies press Ltd, 1997.
- R. B. Randall, ”Frequency analysis”, Bruel & Kjaer.
- D. J. Inman, “Vibration with control measurement and stability”, Prentice Hall Inc., 1989.
- آناليز مودال نوشته جيمني هي ، ترجمه سعيد ضيايي راد و مهدي صالحي، انتشارات دانش پژوهان برين

Vibration, Finite Element Method

30% Mid Term (Date: Saturday 22/8/1395)

50% Final

20% Homworks and Projects

8:00-9:30 Saturday (Location: Class 5)

8:00-9:30 Monday (Location: Class 5)