Collision response

In the context of classical mechanics simulations and physics engines employed within video games, collision response deals with models and algorithms for simulating the changes in the motion of two solid bodies following collision and other forms of contact.

Rigid body contact

Les phases de compression et d'expansion d'une collision entre deux corps solides — The compression and expansion phases of a collision between two solid bodies

Two rigid bodies in unconstrained motion, potentially under the action of forces, may be modelled by solving their equations of motion using numerical integration techniques. On collision, the kinetic properties of two such bodies seem to undergo an instantaneous change, typically resulting in the bodies rebounding away from each other, sliding, or settling into relative static contact, depending on the elasticity of the materials and the configuration of the collision.

Contact forces

The origin of the rebound phenomenon, or reaction, may be traced to the behaviour of real bodies that, unlike their perfectly rigid idealised counterparts, do undergo minor compression on collision, followed by expansion, prior to separation. The compression phase converts the kinetic energy of the bodies into potential energy and to an extent, heat. The expansion phase converts the potential energy back to kinetic energy.

During the compression and expansion phases of two colliding bodies, each body generates reactive forces on the other at the points of contact, such that the sum reaction forces of one body are equal in magnitude but opposite in direction to the forces of the other, as per the Newtonian principle of action and reaction. If the effects of friction are ignored, a collision is seen as affecting only the component of the velocities that are directed along the contact normal and as leaving the tangential components unaffected

Reaction

The degree of relative kinetic energy retained after a collision, termed the restitution, is dependent on the elasticity of the bodies‟ materials. The coefficient of restitution between two given materials is modeled as the ratio ${\displaystyle e\in [0..1] <semantics><mrow><mstyle><mi>e</mi><mo>∈</mo><mo>[</mo><mn>0..1</mn><mo>]</mo></mstyle></mrow><annotation>{\displaystyle e\in [0..1]}</annotation></semantics></math><img src=$ ${\displaystyle e <semantics><mrow><mstyle><mi>e</mi></mstyle></mrow><annotation>{\displaystyle e}</annotation></semantics></math><img src=$ conservation of momentum.

Friction

Another important contact phenomenon is surface-to-surface friction, a force that impedes the relative motion of two surfaces in contact, or that of a body in a fluid. In this section we discuss surface-to-surface friction of two bodies in relative static contact or sliding contact. In the real world, friction is due to the imperfect microstructure of surfaces whose protrusions interlock into each other, generating reactive forces tangential to the surfaces.

To overcome the friction between two bodies in static contact, the surfaces must somehow lift away from each other. Once in motion, the degree of surface affinity is reduced and hence bodies in sliding motion tend to offer lesser resistance to motion. These two categories of friction are respectively termed static friction and dynamic friction.

Applied force

It is a force which is applied to an object by another object or by a person. The direction of the applied force depends on how the force is applied.

Normal force

It is the support force exerted upon an object which is in contact with another stable object. Normal force is sometimes referred to as the pressing force since its action presses the surface together. Normal force is always directed towards the object and acts perpendicularly with the applied force.

Frictional force

It is the force exerted by a surface as an object moves across it or makes an effort to move across it. The friction force opposes the motion of the object. Friction results when two surfaces are pressed together closely, causing attractive intermolecular forces between the molecules of the two different surface. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. Friction always acts parallel to the surface in contact and opposite the direction of motion. The friction force can be calculated using the equation.

Impulse-based contact model

A force ${\displaystyle \mathbf {f} (t)\in \mathbb {R} ^{3 <semantics><mrow><mstyle><mrow><mi>f</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {f} (t)\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle t\in \mathbb {R} <semantics><mrow><mstyle><mi>t</mi><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle t\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle m\in \mathbb {R} <semantics><mrow><mstyle><mi>m</mi><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle m\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle \lbrack t_{0},t_{1} brack <semantics><mrow><mstyle><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle \lbrack t_{0},t_{1} brack }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {p} (t)=m\mathbf {v} (t) <semantics><mrow><mstyle><mrow><mi>p</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>m</mi><mrow><mi>v</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {p} (t)=m\mathbf {v} (t)}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {v} (t) <semantics><mrow><mstyle><mrow><mi>v</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></mstyle></mrow><annotation>{\displaystyle \mathbf {v} (t)}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} \in \mathbb {R} ^{3 <semantics><mrow><mstyle><mrow><mi>j</mi></mrow><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {j} \in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$

{\displaystyle \mathbf {j} =\int _{t_{0}}^{t_{1}}\mathbf {f} dt <semantics><mrow><mstyle><mrow><mi>j</mi></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi>f</mi></mrow><mi>d</mi><mi>t</mi></mstyle></mrow><annotation>{\displaystyle \mathbf {j} =\int _{t_{0}}^{t_{1}}\mathbf {f} dt}</annotation></semantics></math></span><img src=

Pour une impulsion fixe ${\displaystyle \mathbf {j} <semantics><mrow><mstyle><mrow><mi>j</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {j} }</annotation></semantics></math><img src=$ ${\displaystyle t_{1} ightarrow t_{0}\Rightarrow \left|\mathbf {f} ight| ightarrow \infty <semantics><mrow><mstyle><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>⇒</mo><mrow><mo>|</mo><mrow><mi>f</mi></mrow><mo>|</mo></mrow><mo>→</mo><mi>∞</mi></mstyle></mrow><annotation>{\displaystyle t_{1} ightarrow t_{0}\Rightarrow \left|\mathbf {f} ight| ightarrow \infty }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {f} <semantics><mrow><mstyle><mrow><mi>f</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {f} }</annotation></semantics></math><img src=$ ${\displaystyle 0 <semantics><mrow><mstyle><mn>0</mn></mstyle></mrow><annotation>{\displaystyle 0}</annotation></semantics></math><img src=$ ${\displaystyle t_{0 <semantics><mrow><mstyle><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle t_{0}}</annotation></semantics></math><img src=$

{\displaystyle \lim _{t_{1} ightarrow t_{0}}\int _{t_{0}}^{t_{1}}\mathbf {f} dt <semantics><mrow><mstyle><munder><mo>limite</mo><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mrow></munder><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mrow><mi>f</mi></mrow><mi>d</mi><mi>t</mi></mstyle></mrow><annotation>{\displaystyle \lim _{t_{1} ightarrow t_{0}}\int _{t_{0}}^{t_{1}}\mathbf {f} dt}</annotation></semantics></math></span><img src=

existe et est égal à ${\displaystyle \mathbf {j} <semantics><mrow><mstyle><mrow><mi>j</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {j} }</annotation></semantics></math><img src=$ impulsions instantanées peut être introduite pour simuler un changement instantané de vitesse après une collision.

Modèle de réaction basé sur l'impulsion

L'effet de la force de réaction ${\displaystyle \mathbf {f} _{r}(t)\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {f} _{r}(t)\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle [t_{0},t_{1}] <semantics><mrow><mstyle><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle [t_{0},t_{1}]}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} _{r}(t)\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}(t)\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$

{\displaystyle \mathbf {j} _{r}=\int _{t_{0}}^{t_{1}}\mathbf {f} _{r}dt <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mrow><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub></mrow></msubsup><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mi>d</mi><mi>t</mi></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}=\int _{t_{0}}^{t_{1}}\mathbf {f} _{r}dt}</annotation></semantics></math></span><img src=

Par déduction du principe d'action et de réaction, si l'impulsion de collision appliquée par le premier corps sur le second corps au point de contact ${\displaystyle \mathbf {p} _{r}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {p} _{r}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} _{r <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{ displaystyle mathbf {j} _ {r}}</annotation></semantics></math><img src=$ ${\displaystyle -\mathbf {j} _{r <semantics><mrow><mstyle><mo>−</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle -\mathbf {j} _{r}}</annotation></semantics></math><img src=$ ${\displaystyle \pm \mathbf {j} _{r}=\pm j_{r}\mathbf {\hat {n}} <semantics><mrow><mstyle><mo>±</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><mo>±</mo><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \pm \mathbf {j} _{r}=\pm j_{r}\mathbf {\hat {n}} }</annotation></semantics></math><img src=$ ${\displaystyle j_{r}\in \mathbb {R} <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle j_{r}\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {n}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {n}} }</annotation></semantics></math><img src=$ ${\displaystyle -\mathbf {\hat {n}} <semantics><mrow><mstyle><mo>−</mo><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle -\mathbf {\hat {n}} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {n}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {n}} }</annotation></semantics></math><img src=$

En supposant l'amplitude de l'impulsion de collision ${\displaystyle j_{r <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle j_{r}}</annotation></semantics></math><img src=$

De même pour les vitesses angulaires

où, pour le ${\displaystyle i <semantics><mrow><mstyle><mi>je</mi></mstyle></mrow><annotation>{\displaystyle i}</annotation></semantics></math><img src=$ ${\displaystyle {\omega }_{i}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>ω</mi></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle {\omega }_{i}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle {\omega '}_{i}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><msup><mi>ω</mi><mo>′</mo></msup></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle {\omega '}_{i}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {I} _{i}\in \mathbb {R} ^{3 imes 3 <semantics><mrow><mstyle><msub><mrow><mi>je</mi></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {I} _{i}\in \mathbb {R} ^{3 imes 3}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>r</mi></mrow><mrow><mi>je</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {r} _{i}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {p} <semantics><mrow><mstyle><mrow><mi>p</mi></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {p} }</annotation></semantics></math><img src=$

Les vitesses ${\displaystyle v_{p1},v_{p2}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mi>v</mi><mrow><mi>p</mi><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>v</mi><mrow><mi>p</mi><mn>2</mn></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle v_{p1},v_{p2}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$

pour ${\displaystyle i=1,2 <semantics><mrow><mstyle><mi>je</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mstyle></mrow><annotation>{\displaystyle i=1,2}</annotation></semantics></math><img src=$ ${\displaystyle e <semantics><mrow><mstyle><mi>e</mi></mstyle></mrow><annotation>{\displaystyle e}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {v} _{r}=\mathbf {v} _{p2}-\mathbf {v} _{p1 <semantics><mrow><mstyle><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>p</mi><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>p</mi><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v} _{r}=\mathbf {v} _{p2}-\mathbf {v} _{p1}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {v'} _{r}=\mathbf {v'} _{p2}-\mathbf {v'} _{p1 <semantics><mrow><mstyle><msub><mrow><msup><mi>v</mi><mo>′</mo></msup></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msub><mrow><msup><mi>v</mi><mo>′</mo></msup></mrow><mrow><mi>p</mi><mn>2</mn></mrow></msub><mo>−</mo><msub><mrow><msup><mi>v</mi><mo>′</mo></msup></mrow><mrow><mi>p</mi><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v'} _{r}=\mathbf {v'} _{p2}-\mathbf {v'} _{p1}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {n}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {n}} }</annotation></semantics></math><img src=$

En substituant les équations (1a), (1b), (2a), (2b) et (3) dans l'équation (4) et en résolvant pour l'amplitude de l'impulsion de réaction ${\displaystyle j_{r <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle j_{r}}</annotation></semantics></math><img src=$

Calcul de la réaction basée sur l'impulsion

Ainsi, la procédure de calcul des vitesses linéaires après collision ${\displaystyle \mathbf {v'} _{i <semantics><mrow><mstyle><msub><mrow><msup><mi>v</mi><mo>′</mo></msup></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v'} _{i}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\omega '} _{i <semantics><mrow><mstyle><msub><mrow><msup><mi>ω</mi><mo>′</mo></msup></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {\omega '} _{i}}</annotation></semantics></math><img src=$

Calculer l'amplitude de l'impulsion de réaction ${\displaystyle j_{r <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle j_{r}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {v} _{r <semantics><mrow><mstyle><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v} _{r}}</annotation></semantics></math><img src=$ ${\displaystyle m_{1 <semantics><mrow><mstyle><msub><mi>m</mi><mrow><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle m_{1}}</annotation></semantics></math><img src=$ ${\displaystyle m_{2 <semantics><mrow><mstyle><msub><mi>m</mi><mrow><mn>2</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle m_{2}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {I} _{1 <semantics><mrow><mstyle><msub><mrow><mi>je</mi></mrow><mrow><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {I} _{1}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {I} _{2 <semantics><mrow><mstyle><msub><mrow><mi>je</mi></mrow><mrow><mn>2</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {I} _{2}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {r} _{1 <semantics><mrow><mstyle><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {r} _{1}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {r} _{2 <semantics><mrow><mstyle><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {r} _{2}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {n}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {n}} }</annotation></semantics></math><img src=$ ${\displaystyle e <semantics><mrow><mstyle><mi>e</mi></mstyle></mrow><annotation>{\displaystyle e}</annotation></semantics></math><img src=$
Calculer le vecteur d'impulsion de réaction ${\displaystyle \mathbf {j} _{r <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}}</annotation></semantics></math><img src=$ ${\displaystyle j_{r <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle j_{r}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {n}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {n}} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} _{r}=j_{r}\mathbf {\hat {n}} <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>=</mo><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub><mrow><mrow><mover><mi>n</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}=j_{r}\mathbf {\hat {n}} }</annotation></semantics></math><img src=$
Calculer les nouvelles vitesses linéaires ${\displaystyle \mathbf {v'} _{i <semantics><mrow><mstyle><msub><mrow><msup><mi>v</mi><mo>′</mo></msup></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v'} _{i}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {v} _{i <semantics><mrow><mstyle><msub><mrow><mi>v</mi></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {v} _{i}}</annotation></semantics></math><img src=$ ${\displaystyle m_{i <semantics><mrow><mstyle><msub><mi>m</mi><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle m_{i}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} _{r <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}}</annotation></semantics></math><img src=$
Calculer les nouvelles vitesses angulaires ${\displaystyle \mathbf {\omega '} _{i <semantics><mrow><mstyle><msub><mrow><msup><mi>ω</mi><mo>′</mo></msup></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {\omega '} _{i}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\omega } _{i <semantics><mrow><mstyle><msub><mrow><mi>ω</mi></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {\omega } _{i}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {I} _{i <semantics><mrow><mstyle><msub><mrow><mi>je</mi></mrow><mrow><mi>je</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle \mathbf {I} _{i}}</annotation></semantics></math><img src=$ ${\displaystyle j_{r <semantics><mrow><mstyle><msub><mi>j</mi><mrow><mi>r</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle j_{r}}</annotation></semantics></math><img src=$

Modèle de frottement basé sur l'impulsion

L'un des modèles les plus utilisés pour décrire le frottement est le modèle de Coulomb . Ce modèle définit les coefficients de frottement statique. ${\displaystyle {\mu }_{s}\in \mathbb {R} <semantics><mrow><mstyle><msub><mrow><mi>μ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle {\mu }_{s}\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle {\mu }_{d}\in \mathbb {R} <semantics><mrow><mstyle><msub><mrow><mi>μ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle {\mu }_{d}\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\mu }_{d μ_{s} > μ_{d}$ {\mu }_{d Ces coefficients décrivent les deux types de forces de frottement en fonction des forces de réaction agissant sur les corps. Plus précisément, les amplitudes des forces de frottement statique et dynamique. ${\displaystyle f_{s},f_{d}\in \mathbb {R} <semantics><mrow><mstyle><msub><mi>f</mi><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mi>f</mi><mrow><mi>d</mi></mrow></msub><mo>∈</mo><mrow><mi>R</mi></mrow></mstyle></mrow><annotation>{\displaystyle f_{s},f_{d}\in \mathbb {R} }</annotation></semantics></math><img src=$ ${\displaystyle f_{r}=|\mathbf {f} _{r}| <semantics><mrow><mstyle><msub><mi>f</mi><mrow><mi>r</mi></mrow></msub><mo>=</mo><mrow><mo>|</mo></mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>|</mo></mrow></mstyle></mrow><annotation>{\displaystyle f_{r}=|\mathbf {f} _{r}|}</annotation></semantics></math><img src=$

La valeur ${\displaystyle f_{s <semantics><mrow><mstyle><msub><mi>f</mi><mrow><mi>s</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle f_{s}}</annotation></semantics></math><img src=$ ${\displaystyle f_{d <semantics><mrow><mstyle><msub><mi>f</mi><mrow><mi>d</mi></mrow></msub></mstyle></mrow><annotation>{\displaystyle f_{d}}</annotation></semantics></math><img src=$

Le modèle de frottement de Coulomb définit un

Les équations (6a), (6b), (7) et (8) décrivent le modèle de frottement de Coulomb en termes de forces. En adaptant le raisonnement pour les impulsions instantanées, on peut dériver une version du modèle de frottement de Coulomb basée sur les impulsions, reliant une impulsion de frottement ${\displaystyle \mathbf {j} _{f}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{f}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {\hat {t}} <semantics><mrow><mstyle><mrow><mrow><mover><mi>t</mi><mo>^</mo></mover></mrow></mrow></mstyle></mrow><annotation>{\displaystyle \mathbf {\hat {t}} }</annotation></semantics></math><img src=$ ${\displaystyle \mathbf {j} _{r}\in \mathbb {R} ^{3 <semantics><mrow><mstyle><msub><mrow><mi>j</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></mstyle></mrow><annotation>{\displaystyle \mathbf {j} _{r}\in \mathbb {R} ^{3}}</annotation></semantics></math><img src=$ ${\displaystyle [t_{0}..t_{1}] <semantics><mrow><mstyle><mo>[</mo><msub><mi>t</mi><mrow><mn>0</mn></mrow></msub><mo>.</mo><mo>.</mo><msub><mi>t</mi><mrow><mn>1</mn></mrow></msub><mo>]</mo></mstyle></mrow><annotation>{\displaystyle [t_{0}..t_{1}]}</annotation></semantics></math><img src=$