M039 · Discrete mechanics

Granular Impact Chain Designer

Give the first bead a fixed kinetic energy, then follow compression-only Hertz contacts through uniform, graded, preloaded and stiff chains. Place a force probe on the rail, measure threshold arrivals and the reflected wall pulse, and compare 25 graded-mass designs under total-mass and maximum-compression budgets.

Acquired normalized time $0$

Total mechanical energy $0$

Kinetic energy $0$

Bond potential energy $0$

Signed energy defect $0$

Chain momentum $0$

Independent net wall impulse $0$

Momentum minus initial and wall impulse $0$

Half-step displacement difference · first ten time units $0$

Actual normalized integration step $0$

Maximum integration energy defect · full record $0$

Record / model domain $0$

Sampled excess wall-force peak · full record $0$

First ten-percent wall-peak crossing $0$

Two-probe threshold transit speed $0$

Signed excess right-wall impulse $0$

Total normalized chain mass $0$

Maximum sampled contact compression $0$

Graded-mass candidates acquired $0$

Finite design-search status $0$

Best feasible end-mass ratio $0$

RELEASE / ACQUIRE / AUDIT

Can mass grading lower the wall-force peak at the same impact energy?

Number of beads

N

24.00\,

Last-to-first graded mass ratio

m_N/m_1

1.00\,

Normalized Hertz coefficient

\kappa_H

1.00\,

Initial contact compression

\delta_0

0.00\,

Initial kinetic impact energy

E_{in}

0.02\,

Normalized observation window

\tau_{end}

120.00\,

First probe along chain

s_p/L

33.00\,

Mean-mass design ceiling

\bar m_{max}

1.80\,

Design compression ceiling

\delta_{max}

0.35\,

Maximum normalized step

h_{max}

0.02\,

Inspect acquired record

t/t_{end}

0.00\,\%

Normalized quantities; use the rail/profile and keyboard ranges. The record stops at a model boundary or step budget; no clipping is applied.

This normalized one-dimensional point-bead model has elastic Hertz contacts, fixed walls and no friction, plastic loss or rotation. The first bead receives the initial kinetic energy; there is no separate driven striker. Precompression is a geometric compression, not force. Contact forces are compressive or zero; tensile contact is never added. Energy uses the preloaded-equilibrium potential and walls do no work. Probe force is positive excess compression above baseline; the right-wall impulse is signed excess force, which can be negative when a preloaded contact unloads. Arrival uses the first ten-percent crossing of each probe’s full-window peak. Peaks and compression limits use recorded checkpoints. The 25-candidate search keeps impact energy, stiffness, preload and time window fixed; it excludes incomplete runs and pulses that never reach the wall. A lower finite-window peak can result from delay or redistribution, not absorption, and does not certify later safety or a global optimum. Contact compression above one half of the reference spacing stops the model.

Start with the essentials

Focus question: Does a lower wall peak come from delay, reflection or energy absorption?
One-sentence intuition: Compare equal input energy, track the signed wall impulse and retained chain energy, and enforce both mass and compression constraints.

Core mathematical model

Compression-only Hertz contact

\delta_i=[\delta_0-r_i]_+,\qquad F_i=\kappa_H\delta_i^{3/2}

An open contact carries zero physical contact force.

Potential relative to preloaded equilibrium

V_i=\tfrac25\kappa_H(\delta_i^{5/2}-\delta_0^{5/2})+F_0r_i,\qquad F_0=\kappa_H\delta_0^{3/2}

The linear reference term accounts for equilibrium precompression; physical contact remains compression-only.

Fixed input energy

E_{in}=\tfrac12m_1v_1^2,\qquad m_i=m_1R^{(i-1)/(N-1)}

Every graded-mass candidate receives the same first-bead kinetic energy.

Momentum and external walls

P-P_0=\int(F_L-F_R)\,dt

Wall impulses are integrated independently of the momentum readout.

Finite constrained design objective

\min_R\max_{t\in\mathcal T}[F_R(t)-F_0]_+,\qquad \sum_i m_i\le N\bar m_{max},\quad\max_{i,t\in\mathcal T}\delta_i\le\delta_{max}

The recorded time set and 25 graded ratios define a finite search; wall arrival is required.

Common difficulties

No tensile contact

Typical misconceptionA contact spring pulls neighbouring beads back together.

Better mental modelHertz compression is clipped only at zero physical contact; open gaps are allowed.

Reduced peak is not dissipation

Typical misconceptionA smaller wall-force peak proves absorbed impact energy.

Better mental modelThis model is elastic; energy remains in motion and contacts while peaks shift or split.

Delay cannot win by missing arrival

Typical misconceptionA zero wall peak in a short record is the best absorber.

Better mental modelThe search excludes runs without wall arrival and reports feasibility only in the stated time window.

Run the experiment

01
Follow the compression pulse
Acquire the uniform chain and scrub its force probes.
What to observe: The pulse reaches the probes before the fixed wall reflects it.
02
Compare mass grading
Select the heavy-end chain at the same impact energy.
What to observe: Transit time and the wall waveform change without artificial energy loss.
03
Inspect precompression
Compare preloaded contacts and change the probe position directly.
What to observe: The force readout is excess above the preload; unloading can produce signed negative excess impulse.
04
Design under two budgets
Run the candidate search, then tighten the mass or compression ceiling.
What to observe: Feasible candidates satisfy both constraints and have a measured wall arrival.