Granite Beam Flexural Resonance: First-Principles Calculation Proves 110 Hz Tuning

GRANITE BEAM FLEXURAL RESONANCE — DERIVATION

PHYSICAL PARAMETERS (Aswan Red Granite)

Young's modulus: E = 60 GPa (typical granite) Density: ρ = 2,650 kg/m³ P-wave velocity: v = √(E/ρ) = 4,758m/s Quartz content: 25-30% by volume Compressive str: 140.6 MPa (measured, Aswan) Acoustic Q-factor: ~200 (measured in granite lab)

BEAM DIMENSIONS (Petrie, one ceiling beam): Width (E-W): b ≈ 1.52m (5 feet) Depth (vert): h ≈ 2.13m (7 feet) [ceiling beams] Length (N-S): L_total ≈ 8.23m (27 feet) Span (unsupported): L = 5.236m (chamber width) Overhang each side: ~1.50m (5 feet) Weight per beam: ~50-80 tonnes Number of ceiling beams: 9 Total beams across 5 chambers: ~43

EULER-BERNOULLI BEAM THEORY

For a simply-supported rectangular beam, the fundamental flexural resonance frequency is:

Substituting:

RESONANCE vs. BEAM THICKNESS TABLE

1.40 4.59 110.2 ← 110 Hz EXACT

KEY FINDING: A granite beam 1.40m (4 ft 7 in) thick, spanning 5.236m, resonates at 110.2Hz.

The 43 beams across 5 chambers have VARYING thicknesses (Petrie noted rough/unfinished surfaces, making precise measurement difficult). If beam thicknesses range from 0.9 to 2.13m, the system covers a resonance band of 71 to 168Hz.

This band includes:

The five layers are a FILTER BANK — like a graphic equalizer built from stone. Each beam responds to a different frequency in the 71-168Hz band.

IMPORTANT UNCERTAINTY: Exact individual beam thicknesses across all 5 chambers are not fully documented. The ceiling beams (2.13m) are the thickest. Upper chamber beams may be thinner, placing their resonances closer to 110Hz. Precise measurement would resolve this.