Numerical Solution of Dynamic Systems with Impacting Elements ME 535 Final Project

1. Numerical Solution of Dynamic Systems with Impacting ElementsME 535 Final Project Sam Wallen PhD Student University of Washington | Mechanical Engineering June 2014

2. Introduction • Impact dynamic systems model devices containing moving parts that hit each other • Can be used to study dynamics, deformation, and wear • Impact dynamics / contact mechanics are still not fully understood or easy to simulate accurately • Typical situations: vehicle collisions, stick and slip conditions in turbines and other machinery, ballistics, impact dampers, etc. • This report proposes and demonstrates techniques to simplify and/or expedite the simulation process.

3. Difficulties • EOM change form during impact • Makes EOM highly nonlinear, even if linear between impacts. • Characteristic time scale of the system is much shorter (orders of magnitude!) during impact • For numerical solutions, time step must be chosen to satisfy the impact time scale

4. Difficulties • Error in the numerical integration (i.e. RK4) can appear immediately or accumulate slowly over many cycles • Solution may not match physical intuition • Is momentum conserved / should it be? • Is energy conserved or dissipated at the correct rate? • Does the solution display the correct amount of overlap between impacting elements (deformation)? • The necessary step size is much smaller than that of the corresponding non-impacting system.

5. A Simple Impact System Analytical solution between impacts!

6. Contact Models • Coefficient of Restitution • Piecewise Linear • Hertzian

7. Alternative Techniques • Impulse – momentum arguments / coefficient of restitution • Useful if impact time is negligible or unimportant • Energy arguments / equivalent harmonic oscillator model • Useful if impact time is important • Allows comparison of different contact models

8. Impulse-Momentum / C of R • Integrate or use analytical solution between impacts (depends on the system) • Apply instantaneous velocity change with coefficient of restitution whenever the position hits 0 from below • Piece together a complete time history

9. Impulse-Momentum / C of R • Numerical results – no dissipation (e = 1) • Is there any overshoot? • Is the rebound distance the same after every impact (energy conserved)? • How does the error behave?

11. Impulse-Momentum / C of R • Significant overshoot has occurred in RK4 solution with this large timestep • Impact time is nonzero • Rebound distance is the same for all impacts • Error accumulates slowly over time • Creates a “false period” – nonzero impact time makes the RK4 solution go in and out of phase with the analytical solution

13. Energy Arguments / SMD Model • Assume contact forces are dominant during impact • Find a linear spring-mass-damper model to approximate the solution during impact • Analytical solution! • Find duration and exit velocity for each impact • Piece together a complete time history as before

14. Energy Arguments / SMD Model • Work done by stiffness force in going from x = 0 to the maximum penetration depth is preserved: • Average dissipation rate between initial impact velocity and zero velocity is preserved:

18. Energy Arguments / SMD Model • For PWL model, the equivalent SMD is exact • This is because the PWL model is already linear, so the equivalent stiffness and damping do not change from impact to impact • For Hertzian model, the equivalent SMD is close, but not exact • The RK4 solution leads the approximation by a small amount of time • This lead becomes less significant at smaller timesteps

19. Conclusion • Proposed and demonstrated alternatives to direct numerical integration of impact systems • Reduces computational cost be eliminating the need for very small time steps • Adaptable to many contact models • Net advantage depends on the number and length of simulations needed • Future work: in-depth study on the computation time of these methods / compare with direct integration

20. Questions?