--- title: "Granular Impact Chain Designer Interactive Lab | Sandbox Physics" description: "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." canonical_url: "https://sandboxphysics.com/labs/granular-impact-chain/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/granular-impact-chain/" last_modified: "2026-10-22" --- 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. > Interactive simulation: open this page in a web browser to change the parameters and run the model. 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. Physics tutorial ## A compression pulse meets a graded chain **Background** Two beads in contact can push on each other, but once their surfaces separate the force is gone. A Hertz contact gets stiffer as it is squeezed, so the speed and shape of the pulse depend on the impact. **Why it matters** Studies of beads in Hertz contact motivate this finite chain model. The model is purely elastic, and its design goal looks only at a finite time window. 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 misconception** A contact spring pulls neighbouring beads back together. **Better mental model** The Hertz force drops to zero once the beads lose contact, and gaps can open between them. #### A lower peak is not lost energy **Typical misconception** A smaller wall-force peak proves absorbed impact energy. **Better mental model** This 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 misconception** A zero wall peak in a short record is the best absorber. **Better mental model** The search drops runs where the pulse never reaches the wall, and it judges each design only within the recorded time window. ### Run the experiment 1. 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. 2. 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. 3. 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. 4. 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. Physical conclusion An elastic chain moves energy around. A lower peak within the recorded window does not show that the wall is safe afterwards. Motion & dynamics · Starter path ## From a falling ball to an orbit Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit. [Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/) 1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/) 2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/) 3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/) 4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/) 5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/) 6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/) [View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)