 English
 فارسی
آنالیز مودال و تست مودال
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 frequencydomain and timedomain 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

Modal testing and analysis
 Introduction
 Types of tests
 Historical developments
 Applications of modal testing
 Route to vibration analysis

References
 Books
 Conferences
 Journals
 Single DegreeofFreedom (SDOF)
 Undamped
 Viscously Damped
 Hysterically (Structurally) Damped
 FRFs
 Alternative Forms of FRF
 Relation between Different forms of FRFs
 Graphical Display of FRFs
 Frequency Response of Mass and stiffness Elements
 Orthogonality Properties of MDOF
 Massnormalisation
 Multiple modes
 Forced Response of MDOF
 General Concepts

Proportional damping
 special case
 general case
 Proportional Hysteretic damping
 MDOF Systems with hysteretic damping general case

Numerical Example
 Undamped
 Proportional Structural Damping
 NonProportional Structural Damping
 FRF Characteristics (Hysteretic Damping)
 MDOF Systems with viscous damping general case
 FRF Characteristics  Viscous Damping
 Real Modes
 Complex Modes
 Display of Complex Mode
 Measurement of Modal Complexity
 Origin of Complex Modes
 Receptance and Impedance FRF Parameters
 Some Definitions
 FRF Plot in MDOF System
 Point and Transfer FRFs
 Example FRFs of 6DOF System

Display of FRF Data For Damped Systems
 Bode Plots
 Nyquist diagrams
 Real and Imaginary plots
 Threedimensional plots
 Periodic Vibration
 Transient Vibration
 Random Vibration
 Basic Modal Theory – Summary

Incomplete Modal Model
 Modal to Response
 Modal to Spatial
 Incomplete Response Model
 Incomplete Spatial
 Condensation Techniques
 Guyan Reduction
 Improved Reduction System (IRS)
 Dynamic Reduction
 System Equivalent Reduction Expansion Process (SEREP).
 Matrix properties
 Matrix Norm
 Generalised Inverses
 Singular value Decomposition

Main Applications
 Calculation of the Rank of a Matrix
 Calculation of Condition Numbers
 Calculation of Generalised Inverses
 Other applications of SVD
 Definitions
 Theory
 ADDOFD, ADDOFV and ADDOFA

Optimum Suspension Positions
 Example
 Optimum Driving Positions
 Hammer Testing
 Shaker Testing
 NonOptimum Driving Point (NODP) Technique
 Effective Independence (EI) method
 Objectives
 Types of Vibration Measurements
 Signal Quality
 Signal Fidelity
 Measurement Repeatability
 Measurement Reliability
 Measured Data Consistency, including reciprocity
 FRF Measurement Systems

Important Aspects of Measurement Methods
 Support of test structure
 Attachment of exciter
 Push rod or stingers
 Transducers
 Excitation signal used
 Processing of signals
 Mass Calibration and Mass Cancelation

Fourier Analysis
 Fourier series
 Fourier Transform
 Discrete Fourier series
 Fast Fourier Transform
 Aliasing
 Leakage
 Windowing
 Filtering
 Zero padding
 Zoom
 Averaging
 Type of Modal Analysis
 Preliminary Checks of FRF Data
 Basic Skeleton Theory
 Mode Indicator Functions (MIFs)

SDOF Modal Analysis Method

SDOF Curvefit Method
 Peak Amplitude Method
 CircleFit Method
 Inverse or Linefit Method
 Regenerated FRFs
 Residuals

SDOF Curvefit Method
 Modal Analysis in Frequency Domain
 MDOF Curvefit Method
 Nonlinear LeastSquares
 Modal Analysis Using Rational Fractions
 MDOF Curvefits  Light Damping
 Global frequency Methods
 Multiple Curve Fitting
 Global Rational Fraction Polynomial Method (GRFP)
 Global SVD Method
 Example
 Global frequency Methods in the Frequency domain
 Strategy for Model Validation
 Theory/Experiment Comparison

Comparison of Modal Properties
 Comparison of Natural Frequencies
 Mode Shapes
 MAC Correlation Between Two Sets Of Modes
 Normalised Version of the Mac
 Error Location  The COMAC
 Correlation of Other Parameters Frequency Response Functions
 Modal Analysis of Rotating Structures
 Test/Analysis Methods to Identify NonLinearities Parameters
 LDVs as Transducer
 Application of MAC in Frequency Domain
 Operational Modal Analysis
 Undamped MDOF systems
 MDOF Systems with Proportional Damping
 MDOF Systems With Damping – General Case
 Complex Modes
 Characteristics and Presentation of MDOF FRF Data
 NonSinusoidal Vibration and FRF Properties
 Complete and Incomplete Models
 Generalised Inverses
 Test Planning
 Basic Measurement Techniques
 Introduction to Digital Signal Processing
 Modal Parameter Extraction Methods
 Comparison of Test and Analysis
 Other Dimensions in Modal Analysis Testing


