Central Impact

1. Central Impact • Direct Central Impact • Oblique Central Impact MER045: Dynamics of Mechanisms

2. Central Impact of Two Particles • Impact • Collision between two bodies which occurs in a very small interval of time • Bodies exert relatively large forces on each other • Line of Impact • The common normal to thesurface in contact during the impact • Central Impact • When the mass centers of the two colliding bodies are located on the IMPACT LINE MER045: Dynamics of Mechanisms

3. Impact • Direct Central Impact • Velocities of the two particles are directed along the line of impact • Oblique Central Impact • Velocities of the two particles are directed along lines other than the line of impact MER045: Dynamics of Mechanisms

4. Direct Central Impact • va is larger than vb • “A” will Strike “B” • Under impact the two bodies will deform • at maximum deformation the velocity of the two bodies will be the same • Period of RESTITUTION then takes place • particles can retain original shape • particles may stay deformed • Velocity of particles after impact need to be determined MER045: Dynamics of Mechanisms

5. Determination of Final Velocity • Consider the particles during the impact • There are no external impulsive forces to the system • MOMENTUM IS CONSERVED • Since velocities are along the line of impact • Another equation is needed MER045: Dynamics of Mechanisms

6. Particle A During Deformation • The only impulsive force on “A” is the force P exerted by “B’ • Principle of Impulse and Momentum MER045: Dynamics of Mechanisms

7. Particle A During Restitution • R denotes the force exerted by “B” on “A” • Principle of Impulse and Momentum MER045: Dynamics of Mechanisms

8. Review: Deformation and Restitution • Deformation • Restitution • In General R≠P MER045: Dynamics of Mechanisms

9. Coefficient of Restitution • In general • Coefficient of Restitution Defined as MER045: Dynamics of Mechanisms

10. Considering Particle B • Using a similar analysis MER045: Dynamics of Mechanisms

11. Eliminating u • The Quotient of 1 and 2 are Equal • The Quotient after adding the numerator and denominator of 1 and 2 are also equal MER045: Dynamics of Mechanisms

12. Two Special Cases • e=0 • no period of restitution • particles stay together • e=1 • total Energy and Momentum of the particles is conserved • In general e≠1 MER045: Dynamics of Mechanisms

13. Oblique Central Impacts • The velocities of the two colliding particles are NOT along the line of impact • Assume particles are frictionless MER045: Dynamics of Mechanisms

14. General Approach • y component of the momentum of particle A is conserved • y component of the momentum of particle B is conserved • x component of the total momentum of the particles are conserved • x component of the relative velocities after impact are obtained in the same manner as center line impacts MER045: Dynamics of Mechanisms

15. Example 1 The magnitude and direction of the velocities of two identical frictionless balls before they strike each other are as shown. Assume e=.90. Determine the magnitude and direction of the velocity of each ball after the impact. MER045: Dynamics of Mechanisms

16. Example 1 - Diagram MER045: Dynamics of Mechanisms

17. Example 2 A 30kg block is dropped from a height of 2m onto a 10kg pan of a spring scale. Assume the impact to be perfectly plastic. The constant of the spring is k=20kN/m. Determine the maximum deflection of the pan. MER045: Dynamics of Mechanisms

18. Example 2 - Diagram MER045: Dynamics of Mechanisms