stabilizer parameters
Length30.0 in
Outer diameter15.0 mm
Wall thickness2.0 mm
m_rod computed from ρ · A · L using material density and cross-section above.
Result: —
Tip weight
— include end cap
6.0 oz
resonant freq. f₀
—
Hz
total mass
—
g
rigid body MOI — J_rigid (from rod base)
—
g·m²
rod (m·L²/3): — + tip (m·L²): —
motion frequencies
Aiming
Frequency1.0 Hz
aiming freq.
—
Hz
Arrow launch
Draw weight40 lb
Draw length30.0 in
Brace height9.0 in
Arrow weight330 gr
launch freq.
—
Hz
Release
Hook depth1.00 cm
release freq.
—
Hz
frequency response — stabilizer behavior at each motion frequency
① pre-resonance damping (0.15–0.70 f₀) — best: J_eff > J_rigid
② rigid body (f < 0.15 f₀) — good: no residual oscillation
③ transparent (f > 3 f₀) — neutral: rod too slow, no effect
④ post-resonance (1.40–3.00 f₀) — bad: J_eff negative, slight amplification
⑤ resonance (0.70–1.40 f₀) — worst: strong amplification, avoid
| Motion | Frequency | r = f / f₀ | Zone | J_eff | J_eff / J_rigid |
|---|
formulas
Cross-section geometry
I = π/4 · (r_o⁴ − r_i⁴) [m⁴]
A = π · (r_o² − r_i²) [m²]
r_o = outer radius, r_i = inner radius (0 for solid). Carbon = hollow tube. Aluminum = solid.
Cantilever stiffness
k = 3·E·I / L³ [N/m]
E = Young's modulus. L = length. Fixed-free cantilever.
Effective mass & resonant frequency
m_rod = ρ · A · L
m_eff = (33/140)·m_rod + m_tip
f₀ = (1/2π) · √(k / m_eff) [Hz]
33/140: Rayleigh effective mass fraction for cantilever with tip mass.
Rigid body MOI (from rod base)
J_rod = m_rod · L² / 3 [kg·m²]
J_tip = m_tip · L²
J_rigid = J_rod + J_tip
Continuous integral ∫ρA·x²dx for uniform rod. Tip mass treated as point mass at x = L.
Arrow launch frequency
stroke = draw_len − brace_ht
a = F / m_arrow (F = draw_weight × 4.448 N)
t_launch = √(2·stroke / a)
f_launch = 1 / t_launch [Hz]
Constant-force approximation (linear limb). Typical: 80–130 Hz.
Release frequency
a = F / m_finger (m_finger ≈ 12 g)
t_release = √(2 · hook_depth / a)
f_release = 1 / t_release [Hz]
Finger slip modeled as point mass accelerated over hook depth. Deeper hook → longer slip time → lower f_release. Typical: 300–1500 Hz.
Frequency response & effective MOI (r = f / f₀)
|H(r)| = 1 / |1 − r²| (undamped SDOF)
J_eff = J_rigid / (1 − r²)
r < 0.15 → rigid body
0.15–0.70 → pre-resonance, J_eff > J_rigid
0.70–1.40 → resonance, J_eff → ±∞
1.40–3.00 → post-resonance, J_eff < 0 (phase reversed)
r ≥ 3.00 → transparent, J_eff → 0
Negative J_eff in post-resonance zone = 180° phase reversal. Magnitude decays rapidly — attenuation remains, but weaker than pre-resonance zone. Transparent zone: f₀ too low (rod too flexible/heavy), rod cannot follow excitation.