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HDY1_005 Impact

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Point Particle - 1D Impact

Compliant Contact Model

Consider a particle with a single DOF , where motion is restricted to 1D with a unilateral contact constraint . When the particle collides with the constraint surface, we model the contact using a compliant (spring) model that assumes small deformations and high stiffness.

Figure 5.1: Compliant contact model for 1D impact.

The equation of motion during contact is piecewise-defined:

At the collision time , the initial conditions are and , where the negative sign indicates approach toward the constraint.

For , the equation is a simple harmonic oscillator. Defining the natural frequency , the general solution is:

Applying initial conditions and :

  • From :
  • From : , so

Therefore:

where the maximal deformation is:

This result has a clear physical interpretation: higher mass or velocity leads to deeper penetration, while higher stiffness reduces penetration.

Figure 5.2: Full elastic contact.

The contact ends at the release time when and . From equation (5.2), when , i.e., when for integer . The first release (after collision) occurs at :

The velocity at release is:

For a purely elastic spring with no damping, the particle rebounds with the same speed it had before impact.

The maximum contact force occurs at maximum deformation:

In the limit of rigid bodies, , we observe:

  • Contact duration: - instantaneous collision
  • Deformation: - negligible deformations (rigid)
  • Contact force: - infinite impulse force

However, the velocity jump is finite: and .

For instantaneous collisions, we use the impulse-momentum relation:

where is the impulse (integral of force over time). Even though as , the impulse remains finite:

This is the key insight for rigid-body impact: we work with impulses rather than forces.

Kelvin-Voigt Contact Model

When adding a damper in parallel with the spring, we obtain the Kelvin-Voigt model:

Figure 5.3: Kelvin-Voigt contact model with spring and damper in parallel.

With initial conditions and , the nature of the solution depends on the damping ratio:

For , the characteristic equation has complex roots:

where is the natural frequency and is the damped natural frequency.

The general solution is:

Applying initial conditions:

  • From :
  • From :

Therefore:

The velocity is:

Figure 5.4: Underdamped case.

A subtle issue arises when defining the release time . Two natural definitions are:

  1. Kinematic definition: (contact surface reached)
  2. Dynamic definition: (zero contact force)

For the Kelvin-Voigt model, these two definitions are inconsistent:

  • If we define by , then at that instant:
    $$
    f({t}{r})=-kx({t}{r})-c\dot{x}({t}{r})=-c\dot{x}({t}{r})<0
  • If we define by , then:
    $$
    0=-kx({t}{r})-c\dot{x}({t}{r}) \implies x({t}{r})=-\dfrac{c}{k}\dot{x}({t}{r})<0

Comparison of kinematic and dynamic release conditions for Kelvin-Voigt model.

Using the kinematic definition , from equation (5.9):

Therefore:

The velocity at release from equation (5.10):

The coefficient of restitution is:

This elegant result shows that depends only on the damping ratio :

  • For (no damping): (perfectly elastic)
  • For (critical damping): (perfectly plastic)

Hunt-Crossley Nonlinear Contact Model

The inconsistency in release conditions for the Kelvin-Voigt model motivates Hunt-Crossley’s nonlinear force law:

For Hertzian contact theory, .

This model resolves the release time issue: when , automatically as well, since the force is proportional to .

Advantages:

  • Consistent release conditions: and simultaneously
  • More physically realistic for contact between curved surfaces

Disadvantages:

  • Nonlinear dependence on initial velocity
  • No closed-form analytic expression for (except for , which reduces to Kelvin-Voigt)

For practical applications, numerical integration is typically required to determine the coefficient of restitution.

Lagrangian Formulation of Frictionless Impact

General Framework

Consider a multi-body system with degrees of freedom, described by generalized coordinates . A unilateral contact constraint is given by , with normal velocity:

where is a row vector.

Collision occurs at time when:

During collision, the constrained Lagrange equations are:

Velocity Jump Equation

For an instantaneous collision (), we integrate the equations of motion:

Key assumptions for :

  1. No configuration change: (localized contact deformations)
  2. Constant matrices: and
  3. Impulsive force dominance: In the rigid-body limit ,

Under these assumptions:

where is the contact impulse.

Restitution Law

The coefficient of restitution relates pre- and post-impact normal velocities:

In matrix form:

Substituting equation (5.15):

Solving for the impulse:

Since (approaching), and (positive definite mass matrix), we have (compressive impulse).

Impact Map

Substituting back into equation (5.15):

Defining the impact map matrix:

We obtain the compact form:

Example: Falling / Bouncing Rod

Consider a rod of mass , length , and moment of inertia about its center of mass. The rod makes frictionless contact with a horizontal surface at one endpoint.

Figure 5.5: Bouncing rod configuration with contact at endpoint.

Coordinate System: We define as the position of the contact point and as the rod angle. Thus .

Constraint: The contact distance is , giving:

Kinematics: The center of mass position and velocity are:

Energies:

Mass Matrix:

Inverse Mass Matrix (required for impulse calculation):
The relevant quantity is . Using the formula for the inverse of a matrix and extracting the element:

Contact Impulse: From equation (5.17):

Post-Impact Velocities: Applying the impact map:

Conservation of Horizontal COM Velocity:

Looking at the result for (simple restitution in normal direction), we can prove that the horizontal velocity of the center of mass is conserved:

To show this, we substitute the post-impact velocities:

This result follows from the fact that the impulsive force acts purely in the normal direction (frictionless contact), and therefore cannot change the horizontal component of linear momentum. The quantity is precisely the horizontal velocity of the center of mass.

Plastic Collision ():

For , we get , but and . This means not all kinetic energy vanishes in a plastic collision - only the normal component of velocity at the contact point is arrested. The rod continues to move horizontally and rotate.

Hybrid Dynamical Systems

Definition

A hybrid system (HS) for state is defined by:

Often, the jump set lies at the boundary of the flow set .

Solutions of Hybrid Systems

A solution of a hybrid system consists of:

  1. A continuous trajectory
  2. A sequence of jump times

such that:

  • Between jumps: and for
  • At jumps: where

Mechanical Systems as Hybrid Systems

For mechanical systems with state , the flow function may be piecewise-defined:

Guard (jump set):

Reset/jump map for frictionless impact:

For frictional impact, the reset map may have multiple branches:

Example: Zeno's Paradox - The Bouncing Ball

A remarkable property of hybrid systems is that they may have infinitely many jump times , , all accumulating in a finite amount of time. This phenomenon is known as Zeno’s paradox (named after the Greek philosopher Zeno of Elea).

Consider a bouncing ball with coordinates :

Figure 5.6: Bouncing ball undergoing repeated impacts.

Flow dynamics (for ):

Jump (when and ):

where .

Analysis: For initial state and :

First flight ():

Setting :

First impact:

Second flight ():

Setting gives a quadratic with positive root:

At the second impact:

This follows from energy conservation during free flight: the ball reaches the same speed when returning to the ground as it had when leaving.

Second impact:

Recursive Pattern: The recursive law is:

which is a decaying geometric series.

Time gaps between impacts:

Peak heights:

This is because the maximum height is reached when all kinetic energy converts to potential energy: .

Finite Accumulation Time:

Limiting cases:

  • As (plastic): (single impact)
  • As (elastic): (infinite bouncing time)

What happens for ?

At the accumulation time, we have:

The ball comes to rest on the surface. For , the system transitions to constrained motion (persistent contact) with .

Impact with Friction and Slippage

Point Particle Impact in 2D

Consider a point particle colliding with a surface where friction is present. The relative velocity and impulse vectors are decomposed into tangential and normal components:

General impact configuration with friction showing tangential and normal directions.

Relative velocity vector:

Impulse vector:

Impulse-momentum balance:

For given pre-impact velocity , we need to find the impulse and the post-impact velocity (2 scalar unknowns).

In the normal direction, we apply the standard restitution law:

By analogy, one might suggest a tangential restitution law:

But what is the valid range of ?

The change in kinetic energy is:

For physical consistency, we require (energy cannot be created during impact). This gives .

Is with negative values physical? Perhaps not directly, but it is a convenient mathematical abstraction that captures velocity reversal in the tangential direction.

The friction bound on impulses is:

This is justified because and Coulomb’s law applies at each instant.

From the impulse-momentum relations:

Substituting into the friction bound:

Special case : The friction bound becomes:

This means that the negative of the incoming velocity must lie inside a friction cone.

General case : The friction bound becomes:

This defines a different cone, scaled by the ratio of restitution factors.

Problem: For shallow collision angles (large ), there is a conflict with friction bounds. Should we set and give up the tangential restitution law ?

Naive Impact Law with Friction

A simpler approach assumes:

  • Normal direction: , giving
  • Tangential direction:

Consider and with (nearly normal impact).

Figure 5.7: Almost vertical impact

Impulses:

Post-impact velocities:

Energy change:

Taking :

Unphysical result: For , we get - energy is created during impact!

Why did this happen? We assumed with constant . But notice that for small , meaning . The velocity changes sign during impact!

Corrected Model: Impact as a Process

The key insight is that collision is a process occurring over some fast time interval , with different stages.

Impulses accumulate in time:

Velocities evolve:

Tangential force follows instantaneous Coulomb law:

We analyze the impact process in the plane:

Impact analysis in the impulse plane showing the s-line, t-line, and friction cone constraints.

Initial conditions: , , with .

Key lines in the impulse plane:

  1. S-line (sticking line): The locus where . From :

  2. T-line (termination line): The locus where . From :

Impact trajectory in the impulse plane:

During slipping with , the friction law gives , so:

The trajectory moves along a line of slope (inside the friction cone).

Case 1: Sticking before termination - If the trajectory reaches the s-line before the t-line:

  • Phase 1 (slipping): Follow slope until reaching s-line where
  • Phase 2 (sticking): Move vertically () until reaching t-line

Case 2: Termination before sticking - If the trajectory reaches the t-line first:

  • Only slipping phase along slope

Case 3: Velocity reversal - For shallow angles, the trajectory may:

  • First slip with (slope )
  • Reach
  • Continue slipping with (slope )

Total impulse: The final impulse is found at the intersection of the trajectory with the t-line:

Chatterjee’s Algebraic Law

Chatterjee proposed a simple algebraic law that captures the essential physics without tracking the full impulse trajectory.

Algorithm:

  1. Compute candidate impulse assuming both tangential and normal restitution:

  2. Check friction feasibility: If , the impulse is feasible:

  3. Otherwise, project onto friction cone: If the candidate impulse violates friction bounds:

The result is:

  • (normal restitution always satisfied)
  • when sliding stops at impact, or when friction limits the tangential impulse

Chatterjee’s algebraic law: the candidate impulse is projected onto the friction cone when it violates friction bounds.

Energy consistency: One can verify that this law always satisfies , avoiding the unphysical energy creation of the naive model.

Lagrangian Formulation of Impact with Friction (2D)

General Framework

Consider a multi-body system with coordinates and a unilateral contact . In 2D, we define:

  • Normal velocity:
  • Tangential velocity:

Combining these into vector form:

The impulse vector is:

Impulse-Momentum Relation

The velocity jump is:

The contact velocity jump is:

where the collision matrix is:

This is a symmetric positive semi-definite matrix:

Example: Single Rigid Body Impact

Consider a rigid body with mass , moment of inertia , and radius of gyration (so ).

Figure 5.8: Single rigid body impact configuration. is the contact point, is the center of mass.

Coordinates: where is the position of the center of mass .

The contact point is located at relative to .

Contact velocities: The velocity of point P is:

Expanding the cross product (with frame aligned with contact directions):

Therefore:

Constraint matrix:

Mass matrix: For a rigid body with COM at :

Collision matrix:

Physical interpretation:

  • Diagonal terms and represent “effective inverse masses” for tangential and normal impulse response
  • Off-diagonal term represents coupling between tangential impulse and normal velocity change (and vice versa)

Special case: or - The collision matrix becomes diagonal, and there is no coupling between tangential and normal directions.

Fully-Plastic Impact Law

For fully-plastic impact (), the post-impact normal velocity is zero: .

From :

The generalized velocity jump is:

Substituting :

Energy Balance with Friction

The kinetic energy change is:

Substituting :

Using :

For fully-plastic impact with :

Since is positive definite, is also positive definite, confirming (energy is dissipated).

What about friction constraints ? And nonzero restitution with ?

These require more sophisticated treatment - either the impulse plane analysis described earlier, or advanced methods like Routh’s graphical method, which handles the interplay between friction and restitution during the impact process.

Detailed Algebraic Impact Law (Chatterjee)

Consider a general Lagrangian system where the velocity jump is governed by the collision matrix :

where .

We define two reference impulse vectors:

  1. Frictionless impulse : The impulse that results in zero normal velocity () assuming no friction ().

  2. Completely plastic impulse : The impulse that results in zero contact velocity ().

We construct a candidate impulse that combines these reference impulses with restitution coefficients:

This candidate impulse satisfies the kinematic restitution laws:

Note:

The tangential part holds strictly only if is diagonal or ).

Friction-Bounded Impulse

To satisfy Coulomb’s friction bound , Chatterjee proposes a modified impulse:

where the coefficient is determined by:

Geometric Interpretation:

  • If the candidate impulse is inside the friction cone, we use it directly ().
  • If not, we project onto the “closest” edge of the friction cone along the direction of the vector .

This formulation ensures energy consistency () for any and .

Routh’s Graphical Method for General Systems

Routh’s method treats impact as a continuous process in the impulse plane . For a general system with coupling (), the analysis is more complex than for a point mass.

Process Evolution

During the collision interval , the accumulated impulse and contact velocity evolve as:

or in components:

Key Lines in the Impulse Plane

We identify specific lines in the plane that define flow regions and boundaries:

  1. Line of Full Compression (-line): Defined by .

    Crossing this line marks the transition from the compression phase () to the restitution phase (). The accumulated normal impulse at this point is .

  2. Line of Sticking (-line): Defined by .

    On this line, the contact point sticks.

  3. Friction Cone Edges: defined by .

  4. Termination Line (-line): Defines when the collision ends (). Its position depends on the definition of :

    • Newton’s (Kinematic) Restitution: . The line is parallel to the -line:
      • Poisson’s (Impulse) Restitution: Defined by the ratio of impulses during restitution and compression phases: . This gives a horizontal termination line:
      • Stronge’s (Energetic) Restitution: Ratio of work done during restitution to work done during compression. Used when energy consistency is critical in complex systems.
      ^eq-5-63

Newton vs. Poisson

For a point particle (diagonal ), Newton and Poisson definitions are equivalent. For coupled systems (), they differ. Poisson’s definition guarantees energy consistency (), while Newton’s may occasionally violate it for certain parameters.

Algorithm for Impulse Accumulation

To solve the impact problem graphically or numerically:

  1. Start at origin: .
  2. Determine initial slip direction: Check .
  3. March upwards (increasing ):
    • If slipping (): Move along the friction cone edge .
    • If sticking (): If the -line lies inside the friction cone, move along the -line. If not, slip along the appropriate cone edge.
  4. Monitor crossings:
    • If -line is crossed: Velocity reverses sign. Slip direction changes (move to opposite cone edge).
    • If -line is crossed (): Record current normal impulse .
  5. Terminate:
    • Stop when the Total Normal Impulse reaches the value defined by the termination line (e.g., for Poisson).
  6. Result: The point reached in the plane gives the final impulse , determining the post-impact velocity .

This method resolves paradoxes like Painlevé’s paradox (where rigid body dynamics equations have no solution or non-unique solutions), by treating impact as a proper evolution process.

Figure 5.9: Routh method for rigid body impact.

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