The document discusses constrained impulsive motion and analytical methods for impulse response dynamics. It introduces the Hamiltonian and Lagrangian approaches for modeling constraints. Equations are presented for writing the general equation of constrained impulsive motion using Lagrangian multipliers. Examples are provided to illustrate constraints and solving for post-impact velocities and energies using impulsive constraint equations.

1. Impulse
response, analytical
methods
Dynamics
Iqbal Ali Husni
LOGO

2. LOGO
Hamiltonian approach


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Hamiltonian approach

(2.311)

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Legrangian approach

(2.313)
(2.314)
(2.315)

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Constrained impulsive motion

(2.318)
(2.320)

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Constrained impulsive motion
(2.321)
Finally we can write the general equation for constrained impulsive motion
(2.323)

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Legrangian multiplier form


8. LOGO
Impulsive Constraints

(2.329)

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Example 2.13

The values of coordinates are
The corresponding velocities
just before impact are

10. LOGO
Constrained impulsive motion
Three holonomic constraints which specify the particle separation are
Upon differentiation w.r.t time, these equation take form
These equations are valid continuously before and after the impact

11. LOGO
Constrained impulsive motion


12. LOGO
Constrained impulsive motion


13. LOGO
Energy relations


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Energy relations


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Impulsive constraints


16. LOGO
Impulsive constraints


17. LOGO
Impulsive constraints


18. LOGO
Impulsive constraints


19. LOGO
Example 2.14

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Example 2.14


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Impulsive constraints


22. LOGO
Impulsive constraints

(2.388)
(2.389)
(2.390)
(2.391)
(2.392)

23. LOGO