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آنالیز مودال و تست مودال


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.


  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)
  3. Undamped
  4. Viscously Damped
  5. Hysterically (Structurally) Damped
  6. FRFs
  7. Alternative Forms of FRF
  8. Relation between Different forms of FRFs
  9. Graphical Display of FRFs
  10. Frequency Response of Mass and stiffness Elements
  11. Orthogonality Properties of MDOF
  12. Mass-normalisation
  13. Multiple modes
  14. Forced Response of MDOF
  15. General Concepts
  16. Proportional damping
    1. special case
    2. general case
  17. Proportional Hysteretic damping
  18. MDOF Systems with hysteretic damping- general case
  19. Numerical Example
    1. Undamped
    2. Proportional Structural Damping
    3. Non-Proportional Structural Damping
  20. FRF Characteristics (Hysteretic Damping)
  21. MDOF Systems with viscous damping- general case
  22. FRF Characteristics - Viscous Damping
  23. Real Modes
  24. Complex Modes
  25. Display of Complex Mode
  26. Measurement of Modal Complexity
  27. Origin of Complex Modes
  28. Receptance and Impedance FRF Parameters
  29. Some Definitions
  30. FRF Plot in MDOF System
  31. Point and Transfer FRFs
  32. Example- FRFs of 6DOF System
  33. Display of FRF Data For Damped Systems
    1. Bode Plots
    2. Nyquist diagrams
    3. Real and Imaginary plots
    4. Three-dimensional plots
  34. Periodic Vibration
  35. Transient Vibration
  36. Random Vibration
  37. Basic Modal Theory – Summary
  38. Incomplete Modal Model
    1. Modal to Response
    2. Modal to Spatial
  39. Incomplete Response Model
  40. Incomplete Spatial
  41. Condensation Techniques
  42. Guyan Reduction
  43. Improved Reduction System (IRS)
  44. Dynamic Reduction
  45. System Equivalent Reduction Expansion Process (SEREP).
  46. Matrix properties
  47. Matrix Norm
  48. Generalised Inverses
  49. Singular value Decomposition
  50. Main Applications
    1. Calculation of the Rank of a Matrix
    2. Calculation of Condition Numbers
    3. Calculation of Generalised Inverses
  51. Other applications of SVD
  52. Definitions
  53. Theory
  55. Optimum Suspension Positions
    1. Example
  56. Optimum Driving Positions
  57. Hammer Testing
  58. Shaker Testing
  59. Non-Optimum Driving Point (NODP) Technique
  60. Effective Independence (EI) method
  61. Objectives
  62. Types of Vibration Measurements
  63. Signal Quality
  64. Signal Fidelity
  65. Measurement Repeatability
  66. Measurement Reliability
  67. Measured Data Consistency, including reciprocity
  68. FRF Measurement Systems
  69. 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
  70. Mass Calibration and Mass Cancelation
  71. Fourier Analysis
    1. Fourier series
    2. Fourier Transform
    3. Discrete Fourier series
    4. Fast Fourier Transform
  72. Aliasing
  73. Leakage
  74. Windowing
  75. Filtering
  76. Zero padding
  77. Zoom
  78. Averaging
  79. Type of Modal Analysis
  80. Preliminary Checks of FRF Data
  81. Basic Skeleton Theory
  82. Mode Indicator Functions (MIFs)
  83. 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
  84. Modal Analysis in Frequency Domain
  85. MDOF Curve-fit Method
  86. Nonlinear Least-Squares
  87. Modal Analysis Using Rational Fractions
  88. MDOF Curve-fits - Light Damping
  89. Global frequency Methods
  90. Multiple Curve Fitting
  91. Global Rational Fraction Polynomial Method (GRFP)
  92. Global SVD Method
  93. Example
  94. Global frequency Methods in the Frequency domain
  95. Strategy for Model Validation
  96. Theory/Experiment Comparison
  97. Comparison of Modal Properties
    1. Comparison of Natural Frequencies
    2. Mode Shapes
  98. MAC Correlation Between Two Sets Of Modes
  99. Normalised Version of the Mac
  100. Error Location - The COMAC
  101. Correlation of Other Parameters- Frequency Response Functions
  102. Modal Analysis of Rotating Structures
  103. Test/Analysis Methods to Identify Non-Linearities Parameters
  104. LDVs as Transducer
  105. Application of MAC in Frequency Domain
  106. Operational Modal Analysis
  107. Undamped MDOF systems
  108. MDOF Systems with Proportional Damping
  109. MDOF Systems With Damping – General Case
  110. Complex Modes
  111. Characteristics and Presentation of MDOF FRF Data
  112. Non-Sinusoidal Vibration and FRF Properties
  113. Complete and Incomplete Models
  114. Generalised Inverses
  115. Test Planning
  116. Basic Measurement Techniques
  117. Introduction to Digital Signal Processing
  118. Modal Parameter Extraction Methods
  119. Comparison of Test and Analysis
  120. Other Dimensions in Modal Analysis Testing
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Fall 2012