What Is the Rayleigh Damping Matrix Coefficients (α & β) Calculator?
Rayleigh damping is the most common classical damping model used in non-linear dynamic FEA codes (SAP2000, ANSYS, Abaqus, OpenSees).
How Does the Rayleigh Damping Matrix Coefficients (α & β) Calculator Work?
Converts cyclic frequencies to angular frequencies ω₁ and ω₂.
Solves the 2×2 linear system to determine α (mass multiplier) and β (stiffness multiplier).
Rayleigh Damping Matrix Coefficients (α & β) Calculator Formula & Variables
The core mathematical equation utilized by this calculator is expressed as:
[C] = α·[M] + β·[K], ζ_i = (α / 2ω_i) + (β·ω_i / 2)
Two-point Rayleigh damping matrix equations solved simultaneously for modal frequencies.
How to Use the Rayleigh Damping Matrix Coefficients (α & β) Calculator
- Select two significant vibration frequencies bounding the response spectrum.
- Input desired damping ratios for both modes.
Step-by-Step Example Calculation
Finite Element Bridge Model Damping
Input Values:
freq1:1
freq2:10
zeta1:0.05
zeta2:0.05
Worked Steps: Matching 5% damping across 1 Hz and 10 Hz frequency band.
Understanding Your Result
Mass damping α dominates low frequencies.
Stiffness damping β dominates high frequencies.
Factors That Affect the Result
- Frequency band selection (spurious over-damping can occur outside the f₁-f₂ window).
When Should You Use This Calculator?
- Time-history non-linear dynamic analysis and seismic earthquake simulations.
Assumptions & Limitations
- Assumes proportional orthogonal damping modes.
Frequently Asked Questions
Calculation Accuracy & Reference Note
Standard structural finite element formulation.