Vibration Analysis

Dynamics and Vibration Analysis Skill

Purpose

The Dynamics and Vibration Analysis skill provides expert capabilities for modal analysis, frequency response, and vibration characterization of mechanical systems, enabling systematic evaluation of dynamic behavior and resonance avoidance.

Capabilities

• Natural frequency and mode shape extraction
• Modal participation factor analysis
• Harmonic response analysis configuration
• Random vibration (PSD) analysis
• Transient dynamic response simulation
• Shock response spectrum (SRS) analysis
• Damping estimation and modeling
• Vibration test correlation and model updating

Usage Guidelines

Modal Analysis

Natural Frequency Extraction

1. Analysis Setup

   • Define mass distribution accurately
   • Include structural stiffness
   • Apply appropriate boundary conditions
   • Request sufficient modes

2. Mode Shape Interpretation

   Mode 1-6: Rigid body modes (if unconstrained)
   Mode 7+: Flexible modes
   
   Modal Effective Mass > 5%: Significant participation
   

3. Frequency Targets

   Application	Typical First Mode Target
   Machine mount	> 2x operating speed
   Building floor	> 8 Hz (human comfort)
   Aerospace	Per environment specification
   Automotive	Avoid engine orders

Modal Participation Factors

1. Effective Mass

   Meff = (phi^T * M * r)^2 / (phi^T * M * phi)
   
   Where:
   phi = mode shape
   M = mass matrix
   r = direction vector
   

2. Mode Selection

   • Include modes with significant effective mass
   • Typically capture 90%+ of total mass
   • Consider all excitation directions

Harmonic Response Analysis

1. Frequency Range

   • Start below first mode
   • Include all modes of interest
   • Extend beyond highest excitation frequency

2. Damping Specification

   Damping Type	Typical Values
   Structural steel	1-3% critical
   Aluminum	0.5-2%
   Bolted joints	2-5%
   Elastomers	5-20%

3. Response Quantities

   • Displacement amplitude and phase
   • Acceleration at critical points
   • Dynamic stress at notches
   • Reaction forces

Random Vibration Analysis

1. PSD Input Definition

   Acceleration PSD: g^2/Hz
   Force PSD: N^2/Hz
   
   Common profiles:
   - MIL-STD-810
   - NAVMAT P-9492
   - Customer specification
   

2. Response Statistics

   1-sigma response: RMS value
   3-sigma response: RMS * 3 (99.7% probability)
   

3. Fatigue from Random Vibration

   • Calculate stress PSD
   • Determine zero-crossing frequency
   • Apply Steinberg or Dirlik methods

Shock Response Spectrum (SRS)

1. SRS Generation

   • Time history to SRS conversion
   • Q factor specification
   • Frequency resolution

2. Design Verification

   • Compare component modes to SRS
   • Calculate maximum response
   • Apply dynamic amplification

Transient Dynamic Analysis

1. Time Step Selection

   dt <= T_min / 20
   
   Where:
   T_min = period of highest frequency of interest
   

2. Integration Methods

   • Implicit (Newmark): Large time steps, unconditionally stable
   • Explicit: Small time steps, high-frequency accuracy
   • Modal superposition: Efficient for linear systems

Process Integration

• ME-009: Dynamics and Vibration Analysis

Input Schema

{
  "structure": "FEA model reference",
  "analysis_type": "modal|harmonic|random|transient|srs",
  "boundary_conditions": {
    "type": "fixed|free|constrained",
    "locations": "array"
  },
  "excitation": {
    "type": "base|force|pressure",
    "frequency_range": [0, 2000],
    "psd_profile": "array (if random)",
    "time_history": "array (if transient)"
  },
  "damping": {
    "type": "modal|Rayleigh|structural",
    "value": "number or array"
  },
  "output_locations": ["node IDs or named selections"]
}

Output Schema

{
  "modal_results": {
    "frequencies": "array (Hz)",
    "mode_shapes": "reference to shapes",
    "effective_mass": {
      "X": "array",
      "Y": "array",
      "Z": "array"
    },
    "cumulative_mass": "object"
  },
  "response_results": {
    "max_displacement": "number (m)",
    "max_acceleration": "number (g)",
    "max_stress": "number (Pa)",
    "location": "string"
  },
  "frequency_response": {
    "frequencies": "array",
    "amplitude": "array",
    "phase": "array"
  },
  "random_statistics": {
    "rms_values": "object",
    "three_sigma_values": "object"
  }
}

Best Practices

1. Verify mass and stiffness before modal analysis
2. Include sufficient modes to capture total effective mass
3. Use measured damping values when available
4. Validate models with experimental modal analysis
5. Consider nonlinear effects for large amplitudes
6. Document all damping assumptions

Integration Points

• Connects with FEA Structural for model input
• Feeds into Test Correlation for validation
• Supports Fatigue Analysis for vibration-induced fatigue
• Integrates with Mechanism Design for rotating machinery