M037 · Stopping dynamics

Crash Pulse & Impulse Engineering

If the same impulse stops an object, why does it matter when the force acts? Because the impulse fixes how much speed is lost, not how far the object travels while losing it. Late force acts while the object is still fast, so it needs more distance. For a given distance, the lowest peak comes from a force that rises at once and then stays flat. If the absorber runs out of room first, the object hits the end still moving, and that kinetic energy is not absorbed.

Shape the force pulse that stops a moving mass, compare the peak force with the stroke it needs, then record force and speed to check the impulse and work. If the absorber runs out of stroke, the mass keeps the kinetic energy that was not absorbed.

Force pulse family $\text{Ready}$

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

Available stroke $\text{Ready}$

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

Endpoint 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-momentum residual $\text{Ready}$

Raw integrated net work $\text{Ready}$

Raw work-energy residual $\text{Ready}$

Acquired stations $\text{Ready}$

Stopping with impulse and work

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\,

The absorber applies a prescribed force along one line. The force follows one of four pulse shapes, each with unit area: rectangle, triangle, half sine and raised cosine, numbered from zero. The peak-time control only affects the triangle.

Every complete pulse stops the calibrated mass from the same starting speed. The pulse duration and the timing of the force decide how much stroke is needed: a later force peak needs more travel even when the total impulse is the same.

If the stroke runs out, the record ends there with whatever speed and energy remain. The model does not include the harder impact at bottom-out, rebound, injury to the body or the material behaviour of a real airbag.

The piston head follows calculated travel; the folds and cart dimensions are schematic. Blue shows the retarding force and gold the velocity, each drawn at a fixed scale. The dashed force is the complete design reference; a stroke-limited run ends before it is complete.

An acquisition samples the whole time window actually reached, at the chosen sensor rate. The noise is the noise gain times one millimetre, one hundredth of a metre per second and one newton. Calibrated time and mass are exact, and raw values are never clipped.

Raw impulse and work are computed only from measured force, speed and time. Sampling can miss area under a sudden jump or a narrow peak in force. Whether the model's numbers agree is shown separately from whether the acquired record's numbers agree.

The dynamics are idealized and the sensors are simulated. The Lab states its stopping conditions and the limits of its approximations explicitly.

Start with the essentials

Focus question: Can the absorber finish its pulse?
One-sentence intuition: The impulse fixes how much the speed changes, but when the force acts still changes how far the mass travels.

Core mathematical model

Impulse sets the area under the force curve

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

A nonnegative pulse shape with unit area delivers exactly the impulse needed to stop. It is a prescribed force history, not a constitutive law for any material.

Speed and travel follow the accumulated 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 drawing, the speed and the stroke all come from the same accumulated pulse.

When the force acts changes the travel

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

The travel depends on the centroid of the force in time. For a triangular pulse that peaks at some fraction of its duration, the centroid lies at one third of one plus that fraction.

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]

When the stroke runs out, the actual pulse is cut short, and the energy left over stays as kinetic energy. No hard stop is added to soak it up.

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 estimate uses only the raw time, force and speed plus the calibrated mass. Coarse sampling and noise can keep the impulse and work from matching the change in motion.

Common difficulties

An ideal model is not a safety rating

Typical misconceptionIf the ideal model stops in time, a real cushion or road distance must be safe.

Better mental modelThis is a teaching model with prescribed force laws and simulated sensors. Real materials, tyres and differences between drivers have to be checked against measurements.

Run the experiment

01
Compare pulse shapes
Switch between constant force, half sine and triangular peaks. Keep mass, speed and duration fixed.
What to observe: The same total impulse can come with different peak forces.
02
Move the peak
Compare the early and late triangular presets. Drag the gold vehicle to replay the motion.
What to observe: Both complete pulses stop the mass, but the one with the later peak needs more travel.
03
Run out of stroke
Select Stroke exhausted, drag the vehicle to its endpoint, and open Sensors and diagnostics for the remaining kinetic energy.
What to observe: The recorded data end while the mass is still moving. The dashed design curve continues only as a reference.
04
Check the record
Open Sensors and diagnostics to acquire, clear or export raw sensors; sensor rate and noise are in Vehicle and numerical settings.
What to observe: Whether the raw integrals balance shows the cost of finite sensor resolution.