Basic Vibration Analysis- Modal Analysis (FEA)

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Modal Analysis in the context of Finite Element Analysis (FEA) is a technique used to determine the natural frequencies and mode shapes of a structure. It helps in understanding how a structure will respond to dynamic loading. Each mode shape corresponds to a specific natural frequency and describes a specific pattern of motion the structure undergoes at that frequency.

  • It is the most fundamental type of dynamic analysis.
  • This technique enables designs to avoid resonant vibrations or to vibrate at a specified frequency.
  • It provides engineers with insights into how the design will respond to various dynamic loads.
  • It aids in calculating solution controls for other types of dynamic analysis

Steps in Modal Analysis:

  1. Input and Model Preparation: The geometry is simplified and prepared for analysis, the material properties are assigned, and boundary conditions are applied according to the actual model conditions.
  2. Discretization: The structure is divided into finite elements, creating a mesh.
  3. Formulation: The mass () and stiffness () matrices are assembled from the properties of the individual elements.
  4. Eigenvalue Problem: The equation of motion for a system without damping and external forces is given by:
    𝑀𝑒¨(𝑑)+𝐾𝑒(𝑑)=0

    By assuming a harmonic solution of the form 𝑒(𝑑)=π›·π‘’π‘–πœ”π‘‘ , we get:

    𝐾𝛷=πœ”2𝑀𝛷

    This is a generalized eigenvalue problem, where πœ”2Β are the eigenvalues (squares of the natural frequencies), and 𝛷 are the eigenvectors (mode shapes).

  5. Solution: Solve the eigenvalue problem to obtain the natural frequencies and mode shapes.

Simple Spring-Mass-Damper System with Single Degree of Freedom

A simple spring-mass-damper system is a fundamental model in dynamics and vibration analysis. The system consists of a mass π‘š Β connected to a spring with stiffness and a damper with damping coefficient .

Governing Equation:

The equation of motion for the mass is derived from Newton’s second law:

π‘šπ‘₯Β¨(𝑑)+𝑐π‘₯Λ™(𝑑)+π‘˜π‘₯(𝑑)=𝐹(𝑑)

where:

  • π‘₯(𝑑)Β is the displacement of the mass as a function of time.
  • π‘₯Λ™(𝑑)Β is the velocity (first derivative of π‘₯(𝑑)).
  • π‘₯Β¨(𝑑)Β is the acceleration (second derivative of π‘₯(𝑑)).
  • 𝐹(𝑑)Β is the external force applied to the mass.

Analyzing the System:

  1. Free Vibration (No External Force 𝐹(𝑑)=0):
    • Undamped (𝑐=0):
      π‘šπ‘₯Β¨(𝑑)+π‘˜π‘₯(𝑑)=0Β 

      The solution is harmonic:

      π‘₯(𝑑)=𝐴cos⁑(πœ”π‘›π‘‘)+𝐡sin⁑(πœ”π‘›π‘‘)Β 

      where πœ”π‘›=π‘˜π‘š is the natural frequency.

    • Damped (𝑐≠0):
      π‘šπ‘₯Β¨(𝑑)+𝑐π‘₯Λ™(𝑑)+π‘˜π‘₯(𝑑)=0Β 

      The damping ratio 𝜁=𝑐/2*sqrt(π‘˜π‘š) determines the behavior:

      • Underdamped (𝜁<1): Oscillatory motion with exponentially decaying amplitude.
      • Critically damped (𝜁=1): The system returns to equilibrium as quickly as possible without oscillating.
      • Overdamped (𝜁>1): The system returns to equilibrium without oscillating, slower than in the critically damped case.
  2. Forced Vibration (𝐹(𝑑)β‰ 0):
    π‘šπ‘₯Β¨(𝑑)+𝑐π‘₯Λ™(𝑑)+π‘˜π‘₯(𝑑)=𝐹(𝑑)

    The response of the system depends on the form of 𝐹(𝑑). For a harmonic force 𝐹(𝑑)=𝐹0cos⁑(πœ”π‘‘), the steady-state solution can be found using methods like the Fourier transform or Laplace transform.

Eigenfrequencies and Mode Shapes:

The square roots of the eigenvalues are denoted as Ο‰i, representing the structure’s natural circular frequencies in radians per second (rad/s).

  • To find the natural frequencies fi in cycles per second (Hz), use the formula fi = Ο‰i / (2Ο€). These natural frequencies, fi, are the values that the user inputs and Workbench outputs.
  • The eigenvectors {Ο•}i correspond to the mode shapes, describing the specific deformation patterns the structure assumes when vibrating at each natural frequency fi.

Participation Factor, Effective Mass

The participation factors are calculated using the formula: 𝛾𝑖=πœ™π‘–π‘‡ [𝑀] {𝐷}

where {𝐷} represents an assumed unit displacement spectrum in each of the global Cartesian directions and rotation about each axis.

  • These factors measure the amount of mass moving in each direction for each mode.
  • A high value in a particular direction indicates that the mode will be strongly excited by forces in that direction.
  • The “Ratio” is another list of participation factors, normalized to the largest value.

The effective mass is calculated using the formula:

𝑀eff,𝑖=(𝛾𝑖)2/πœ™π‘–π‘‡[𝑀]πœ™π‘–

where 𝛾𝑖 is the participation factor for mode 𝑖, and πœ™π‘–Β is the eigenvector for mode 𝑖.

  • Ideally, the sum of the effective masses in each direction should equal the total mass of the structure, but this depends on the number of modes extracted.
  • The ratio of effective mass to total mass is useful for determining whether a sufficient number of modes have been extracted.