Harmonic Analysis – Dynamic Response Analysis (FEA)

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Harmonic Analysis is a type of dynamic FEA that is used to determine the steady-state response of a system to sinusoidal (harmonic) loading. It is particularly useful for analyzing systems subjected to periodic forces, such as rotating machinery, vibrating structures, or acoustic phenomena.

Key Points of Harmonic Analysis:

  1. Purpose: To evaluate the response of a system when subjected to sinusoidal loads over a range of frequencies.
  2. Input Data: Requires material properties, geometric data, boundary conditions, and the specifics of the harmonic loads (frequency, amplitude).Harmonic loads (forces, pressures, and imposed displacements) are characterized by known magnitudes and frequencies. These loads may include multiple loads all acting at the same frequency. Forces and displacements can be either in-phase or out-of-phase. However, body loads can only be specified with a phase angle of zero.
  3. Output Data: Provides the amplitude and phase of the response (displacement, velocity, acceleration) at each frequency. Harmonic displacements at each degree of freedom (DOF), often out of phase with the applied loads. Additional derived quantities, such as stresses and strains.
  4. Applications: Used in fields like mechanical engineering, civil engineering, aerospace, and automotive engineering for components such as gears, turbines, bridges, and exhaust systems.

Steps in Harmonic Analysis:

  1. Preprocessing:
    • Define the geometry of the model.
    • Specify material properties (e.g., density, Young’s modulus, damping coefficients).
    • Apply boundary conditions and constraints.
    • Specify the harmonic load (magnitude, phase, frequency range).
  2. Solution:
    • The governing differential equations of motion are solved in the frequency domain.
    • Uses the principle of superposition to apply sinusoidal loads at different frequencies.
    • Solves for steady-state response at each frequency point.
  3. Postprocessing:
    • Analyze the results (displacement, stress, strain, velocity, acceleration) at each frequency.
    • Identify resonance frequencies where the system exhibits maximum response.
    • Evaluate the phase difference between the applied load and the response.

Assumptions and Restrictions for Harmonic Analysis

  1. Linear System:
    • The structure exhibits linear elastic behavior with constant or frequency-dependent stiffness, damping, and mass effects.
  2. Steady-State Response:
    • Transient effects are ignored; analysis focuses solely on steady-state response to harmonic loading.
  3. Sinusoidal Variation:
    • All loads and displacements vary sinusoidally at the same frequency, though not necessarily in phase.
  4. Complex Displacements:
    • Displacements are complex if damping is specified or if the applied load is complex.
  5. Real Loads for Acceleration and Bearing:
    • Acceleration and bearing loads are assumed to be real (in-phase) only.
  6. Boundary Conditions and Constraints:
    • Remain constant and do not vary with frequency.
  7. Damping Assumptions:
    • Modeled as linear and proportional (e.g., Rayleigh or viscous damping).
  8. No Initial Stresses or Preloads:
    • Assumes absence of initial stresses or preloads in the structure.

The governing equation for a linear structure is given by:

𝑀𝑒¨+𝐢𝑒˙+𝐾𝑒=𝐹

Assuming 𝐹 and 𝑒 are harmonic with frequency Ξ©,Β they can be represented as:

𝐹=𝐹maxπ‘’π‘–πœ“π‘’π‘–Ξ©π‘‘Β  𝑒=𝑒maxπ‘’π‘–πœ™π‘’π‘–Ξ©π‘‘

Note: Ω represents the input (imposed) circular frequency, while πœ” represents the output (natural) circular frequency.

Solution Techniques:

  1. Full Harmonic Response Analysis: Solves a system of simultaneous equations directly using a static solver designed for complex arithmetic:

[πΎπ‘βˆ’Ξ©2π‘€π‘–Ξ©πΆβˆ’Ξ©πΆπΎπ‘]{𝑒1𝑒2}={𝐹1𝐹2}

  1. Mode Superposition Response Analysis: Expresses the displacements as a linear combination of mode shapes:

[βˆ’Ξ©2𝑀+𝑖Ω𝐢+𝐾]{𝑒Ω𝑒2}={𝐹1𝐹2}

Analysis Techniques:

  1. Full Harmonic Response Analysis:
    • Involves solving a system of simultaneous equations directly.
    • Gives Exact solution and Supports all types of loads and boundary conditions.
    • The entire system is analyzed together, considering all degrees of freedom.
    • Requires solving for all displacements and forces at each frequency of interest simultaneously.
    • Suitable for relatively small systems with simple geometries.
    • Provides a comprehensive solution for all frequencies of interest but can be computationally expensive for large systems.
  2. Mode Superposition Response Analysis:
    • Utilizes mode shapes to express the displacements as a linear combination of modal contributions.
    • Β Gives Approximate solution because accuracy depends on whether an adequate number of modes have been extracted.
    • Decomposes the problem into a series of simpler modal analyses.
    • Each mode is solved individually, and the responses are combined using modal participation factors.
    • Particularly effective for large systems with many degrees of freedom.
    • Offers significant computational savings compared to full harmonic response analysis, especially for large and complex structures.
    • May provide accurate results for the frequencies dominated by a few significant modes but may not capture the complete system response at all frequencies.