M039 · Discrete mechanics

Granular Impact Chain Designer

For the same impact energy, can grading the bead masses in a chain lower the peak force on the wall? Over the time you watch, yes. Grading the bead masses spreads the pulse out and makes it arrive later, so the wall feels a lower peak. An elastic chain absorbs nothing, though: it only redistributes the energy, and later reflections can still carry it to the wall. A real design also has to stay within limits on total mass and bead compression.

Give the first bead a fixed kinetic energy and follow the pulse through uniform, graded, preloaded and stiff chains, shown in three dimensions with beads moving along the chain and Hertz contacts that only push. Place a force probe on the rail, time when the pulse crosses the arrival threshold, measure the pulse reflected from the wall, and compare 25 graded-mass designs within limits on total mass and maximum compression.

Left physical contact force $0$

Right physical contact force $0$

Inspected bead acceleration $0$

Inspected right contact $0$

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 the chain and record its motion

For the same impact energy, can grading the bead masses in a chain lower the peak force on the wall?

Compressed contactsAmber diagnostic bars vanish when a contact opens.Release outlinesGold marks the starting state.Physical contact forcesIncludes preload; fixed scale for the trial.VelocityBlue, with a separate fixed scale.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\,\%

All quantities are normalized. You can work the rail and the displacement profile directly, or set the sliders from the keyboard. The record stops when the motion leaves the model's valid range or the step limit is reached; the motion itself is never clipped.

The model is a normalized one-dimensional chain of point beads with elastic Hertz contacts and fixed walls; there is no friction, plastic loss or rotation. The first bead starts with the initial kinetic energy; no separate striker drives it. The preload is set as a compression distance, not a force. Contacts push or carry no force; they never pull. Energy is measured from the preloaded equilibrium, and the walls do no work. The probe force is the positive part of the compression above the baseline. The right-wall impulse adds up the force above the preload with its sign, so it can be negative when a preloaded contact unloads. The arrival time is when a probe first reaches ten percent of its peak over the whole window. Peaks and the compression limit are checked only at recorded time points. The 25-design search keeps impact energy, stiffness, preload and time window fixed, and it drops incomplete runs and pulses that never reach the wall. A lower peak within the window can come from delay or from energy being redistributed rather than absorbed, and it does not show that the wall stays safe later or that the design is the best overall. The run stops if any contact is compressed by more than one half of the reference spacing. The three-dimensional force arrows show the full compression force, preload included; the plots still show only the force above the preload. The bead markers are equal-size schematic points; grading stands for a change in density with the contact coefficient fixed. Amber bars mark closed contacts and disappear where a gap opens. The Hertz compression is not the distance between marker surfaces, and the model does not draw the deformed shapes of the beads.

Start with the essentials

Focus question: Does a lower wall peak come from delay, reflection or energy absorption?
One-sentence intuition: Give every design the same input energy, track the signed wall impulse and the energy left in the chain, and keep both the total mass and the compression within their limits.

Core mathematical model

Compression-only Hertz contact

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

When two beads are apart, the contact force between them is zero.

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 term measures energy from the precompressed equilibrium. The contact itself can still only push.

Fixed input energy

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

Every graded-mass design starts with the same kinetic energy in the first bead.

Momentum and external walls

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

The wall impulses are integrated on their own, separately from the momentum readout.

The design goal and its limits

\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 search compares 25 grading ratios at the recorded times. A design counts only if the pulse reaches the wall.

Common difficulties

Contacts never pull

Typical misconceptionA contact spring pulls neighbouring beads back together.

Better mental modelThe Hertz force drops to zero once the beads lose contact, and gaps can open between them.

A lower peak is not lost energy

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

Better mental modelThis model is elastic. The peaks move or split, but the energy stays in the beads’ motion and in the squeezed contacts.

A pulse that never arrives does not win

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

Better mental modelThe search drops runs where the pulse never reaches the wall, and it judges each design only within the recorded 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: The travel time and the force pattern at the wall change, and no energy is artificially removed.
03
Inspect precompression
Compare preloaded contacts and change the probe position directly.
What to observe: The force readout shows only the force above the preload. When a contact unloads, the excess impulse can be negative.
04
Design within two limits
Run the candidate search, then tighten the mass or compression ceiling.
What to observe: A design passes only if it stays within both limits and the pulse is measured reaching the wall.