M037 · Stopping dynamics

Crash Pulse & Impulse Engineering

Shape a stopping-force pulse, compare peak force and required stroke, then acquire force and speed records to audit impulse and work. Stroke exhaustion retains the unabsorbed kinetic energy.

Force pulse family $\text{Ready}$

Required full-pulse stroke $\text{Ready}$

Available stroke $\text{Ready}$

Designed full-pulse peak force $\text{Ready}$

Outcome $\text{Ready}$

Selected time $\text{Ready}$

Selected travel $\text{Ready}$

Selected speed $\text{Ready}$

Selected retarding force $\text{Ready}$

Attained peak retarding force $\text{Ready}$

Selected brake / absorber work $\text{Ready}$

Endpoint kinetic energy $\text{Ready}$

Attained observation duration $\text{Ready}$

Attained total travel $\text{Ready}$

Model momentum balance residual $\text{Ready}$

Model energy balance residual $\text{Ready}$

Raw integrated net impulse $\text{Ready}$

Raw impulse closure residual $\text{Ready}$

Raw integrated net work $\text{Ready}$

Raw work closure residual $\text{Ready}$

Acquired stations $\text{Ready}$

IMPULSE / WORK / LIMITS

Can the absorber finish its pulse?

Calibrated mass

m

80.00\,kg

Initial forward speed

v_0

8.00\,\frac{\mathrm m}{\mathrm s}

Designed pulse duration

T

0.25\,s

Available absorber stroke

S

1.50\,m

Pulse family

q

2.00\,

Triangular peak time fraction

a

0.50\,

Selected time fraction

t/t_{\mathrm{end}}

0.50\,

Sensor sampling rate

f_s

400.00\,Hz

Sensor noise gain

\eta

0.00\,

One-dimensional prescribed absorber force; four unit-area pulse families. Rectangle, triangle, half sine and raised cosine are numbered from zero. The peak-time control applies only to the triangle.

All complete pulses stop the calibrated mass from the same initial speed. Duration and force timing determine required stroke; a later force peak increases travel even when total impulse is unchanged.

If the available stroke is exhausted, the record ends there with its residual speed and energy. No harder bottom-out collision, rebound, tissue injury or real airbag material law is assumed.

The piston head follows acquired travel; folds and cart dimensions are schematic. Blue is retarding force, orange is velocity; each arrow uses a fixed gain. The dashed force is the uncompleted design reference.

Acquire the full attained window at the selected sensor rate. Noise gain scales one millimetre, one hundredth metre per second and one newton. Calibrated time and mass are exact; raw values are never clipped.

Raw impulse and work use only measured forces, speed and time. Sampling a force jump or narrow peak can miss area; agreement of the model is separate from agreement of the acquired record.

Idealized dynamics and simulated sensors; explicit stopping and approximation boundaries.

Start with the essentials

Focus question: Can the absorber finish its pulse?
One-sentence intuition: Impulse fixes the velocity change; force timing still changes travel.

Core mathematical model

Impulse sets the force area

B(t)=\frac{mv_0}{T}f(t/T),\qquad\int_0^1f(u)\,du=1

A nonnegative normalized pulse provides the full stopping impulse. It is a prescribed force history, not a material constitutive law.

Motion follows cumulative impulse

\begin{aligned}C(u)&=\int_0^uf(q)\,dq\\v(t)&=v_0[1-C(u)]\\x(t)&=v_0T\int_0^u[1-C(q)]\,dq\end{aligned}

The apparatus, speed and stroke use the same cumulative pulse.

Force timing changes travel

x_{\mathrm{stop}}=v_0T\int_0^1u f(u)\,du

Travel depends on the force centroid. For a triangular pulse peaking at a fraction a of the duration, the centroid is one third of one plus a.

Absorbed energy and remaining motion

W_B(t)=\int_0^tB(\tau)v(\tau)\,d\tau=\frac{m}{2}[v_0^2-v(t)^2]

A stroke limit truncates the actual pulse. The remainder is kinetic energy, not silently dissipated by an invented hard stop.

Estimate from sensor data

\begin{aligned}J_{\mathrm{raw}}&\approx-\sum_i\frac{B_i+B_{i+1}}2\Delta t_i\\W_{\mathrm{raw}}&\approx-\sum_i\frac{B_i v_i+B_{i+1}v_{i+1}}2\Delta t_i\end{aligned}

The estimator receives raw time, force and speed plus calibrated mass. Finite sampling and noise can spoil closure.

Common difficulties

Scope of the idealization

Typical misconceptionAn ideal stopping model certifies a real cushion or safe road distance.

Better mental modelThis is a teaching model with prescribed laws and simulated sensors; real materials, tyres and driver variation require measured validation.

Run the experiment

01
Compare pulse families
Switch between constant force, half sine and triangular peaks. Keep mass, speed and duration fixed.
What to observe: Equal full impulse does not imply equal peak force.
02
Move the peak
Compare the early and late triangular presets. Scrub the gold time handle.
What to observe: A later peak demands more travel although both complete pulses stop the mass.
03
Exhaust the stroke
Select Stroke exhausted and inspect the final piston position and kinetic energy.
What to observe: The acquired record ends with residual motion. The dashed design continues only as a reference.
04
Audit the record
Acquire, lower the sample rate, add noise, and export raw CSV.
What to observe: Raw integral closure exposes finite sensor resolution.