Modal Analysis

Modal Analysis

1. Equations of Undamped Free Vibrations

   • Orthogonality of Eigenvectors
   • Transformation of the Equations of Motion to a Diagonal Set
   • Modal Damping Idealisation
     • In practice, damping within most engineering structures is much less than either the stiffness or the inertia forces. 
     • Damping becomes important if the structure is excited close to resonance. 
     • Rayleigh Model
       • Linear combination of stiffness and mass distributions.

2. Eigenvalues & Eigenvectors of Unsupported Structures

   • Shift of Origin
     • Remove singularity from the equations.
       • Selecting the value of g
         • Too small - Ill conditioned problem.
         • Too large - Swamps eigenvalues making it difficult to distinguish between distinct modes.
   • Isolating the Singular (Rigid Body) Modes
     • Elastic displacements
     • Rigid body displacements

3. Transient Forced Response of a Damped Single DoF System

   • Integrating the Convolution Integral

4. General Form of Linear Equations of Motion

   • Solution
     • Switch between methods using a Fourier (or Inverse Fourier) transform.
     • Time Domain
       • Force defined by its time history.
     • Frequency Domain
       • Force defined in terms of its frequency content.

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