Arrow Oscillation & Node Calculator
Enter your arrow's spine, GPI, cut length, and tip weight. The calculator finds the two points along the shaft that move least during oscillation — and the frequency at which the arrow oscillates. Tip weight shifts those points significantly; the default 22% figure assumes no tip mass.
What nodes are and why they matter
When an arrow oscillates, it bends in a standing wave pattern. The two points along the shaft that remain stationary during that bending are the nodes. Everything between them moves; the nodes do not.
For a bare shaft with no tip mass, the first bending mode places nodes at 22.4% from each end — a result of the free-free beam equation. Adding a point weight shifts both nodes toward the nock. A 100 grain tip on an 8.5 GPI, 28" shaft (μ ≈ 0.42) moves the front node forward to roughly 28–30% from the nock and the rear node to roughly 70–72%.
A note on rest contact: The nodes this calculator shows are free-flight nodes — positions where the arrow moves least during oscillation in the air. They are not rest placement targets. Rest contact happens during the shot, while the arrow is still accelerating past a stationary rest. By the time the arrow has traveled far enough to develop its free-flight oscillation pattern, it is already well clear of the rest. The common advice to "place the rest at the node" conflates two different physical events. The free-flight node position is useful for batch matching and understanding in-flight behavior — not for dialing in rest forward position.
What this does not tell you: The oscillation plane on a compound with a release is not reliably known without high-speed video. It depends on d-loop geometry, release jaw angle, and limb movement. These nodes are the positions — but the plane they sit in is unknown. On recurve with finger release, the plane is horizontal (the Archer's Paradox plane) and node placement is directly actionable. On compound, treat node placement as a starting point for rest height, not a precise prescription.
Frequency and batch matching: Two arrows with different oscillation frequencies will exit the bow at different phases, producing groups at distance even with identical spine numbers. Matching frequency — which requires matching both spine and GPI — is more precise than matching spine alone. The frequency shown here can be used to sort a batch: measure each arrow's spine and bare shaft weight, compute frequency (or use the proxy spine × shaft_weight), and group the closest matches.
How it is calculated
Model: Free-free Euler-Bernoulli beam with a concentrated tip mass. This represents the arrow in free flight — the in-flight oscillation mode, not the during-shot mode (which has more complex boundary conditions at the nock).
EI from spine: EI = 2546 / spine N·m² — derived from the AMO 28-inch span spine test using 1.94 lb (8.63 N) center load. This is consistent with the corrected Euler buckling constant used in the ALR calculator (8,757,000).
Characteristic equation: cosh(u)cos(u) − 1 − μu[cosh(u)sin(u) − sinh(u)cos(u)] = 0 — solved numerically by bisection. μ = tip_gr / (gpi × length_in) is the tip-to-shaft mass ratio.
Frequency: f = (u/L)² × √(EI / ρA) / (2π) where ρA = gpi × 2.551×10⁻³ kg/m and L is arrow length in meters.
Mode shape: Y(t) = sinh(t) + sin(t) − R·[cosh(t) + cos(t)] where R = (sinh u − sin u)/(cosh u − cos u) and t = u·(x/L). Nodes found by bisection over Y(t) = 0.
→ ALR calculator: find the right spine for your bow weight and cam
→ Article: arrow nodes, oscillation planes, and what they actually mean