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Model check
The science
What is simulated
A round 2.7 m trampoline: a mat held by 64 springs (rate ks, preload F0). A jumper of mass m is released from a height h above the mat. The 3D model (frame, legs, net, springs, mat, jumper) is built in Blender; the mat shape and the bounce are computed in the page.
Mat as a spring-loaded membrane
A load at the centre depresses the mat by y. The mat shape is the point-load membrane profile w(r) = y ln(R/√(r²+a²)) / ln(R/a) (a = foot patch radius). The mat does not stretch, so the extra area π y² / ln(R/a) is taken from the springs: the edge moves in by s = y² / (2 R ln(R/a)). With spring energy U = N (F0 s + ½ ks s²) the force is F(y) = dU/dy = k1 y + k3 y³, where k1 = N F0 / (R ln(R/a)) and k3 = N ks / (2 R² ln²(R/a)). The mat is a hardening spring: it gets stiffer the deeper you go.
Bounce
In contact m y″ = m g - [F(y) + c y′ + u m g H(-y′)] (y positive down; c = 2ζ√(k1 m) is the mat and air damping; u is an optional leg push on the way up). The mat cannot pull, so the jumper leaves when the contact force reaches zero; in flight y″ = g. For a linear undamped mat the contact time is (π + 2 asin(δs/A)) / ω with ω = √(k/m) and static sag δs = m g / k: a heavier jumper or a softer mat stays on the mat longer, and the contact time barely depends on the drop height.
Simplifications
- The jumper is a point mass at the centre; no horizontal motion, no body rotation, and the leg push is a simple fraction of body weight.
- The mat has the static point-load shape at every instant; mat inertia and waves travelling over the mat are neglected.
- Damping is linear viscous; real mats and springs have friction and hysteresis. The hard stop at 82 cm stands for the mat meeting the ground.
- The spring preload and rate are typical round numbers, not a particular product.
Checks
Four boxes run the page's own code against independent results: the linear-limit contact time and depth against the closed form, an energy balance of the full nonlinear damped bounce sequence, the damped linear bounce against the analytic solution with a bisection root, and the standing sag from a Newton root against a heavily damped dynamic relaxation.