Ugo Andreaus, Luca Placidi, Giuseppe Rega

1. Soft contact dynamics of an impacting bilinear oscillator: numerical simulation and hints for describing an impacted cantilever beam Ugo Andreaus, Luca Placidi, Giuseppe Rega Department of Structural and Geotechnical Engineering, “La Sapienza”, University of Rome Research Workshop on Bifurcation in Oscillators with Elastic and Impact Constraints 4 November – 6 November, 2009, Imperial College London, United Kingdom

2. Introduction Among the wide range of nonlinear dynamical systems, Piecewise Smooth Systems (PSS) play an important role and can be classified as continuous or discontinuous PSS. An example of discontinuous PSS: - The (hard) impact oscillator; see, e.g., Akashi (1958) An example of continuous PSS: - The (soft) bilinear oscillator; see, e.g., Shaw and Holmes (1983) In this work, features of soft impact non-trivial dynamics of a bilinear oscillator are analyzed via a return map approach. Moreover, in order to grossly distinguish between a periodic and a chaotic trajectory, we use Lyapunov coefficients, that measure the average divergence of nearby trajectories, Ott (1993).

3. For dynamical systems described by smooth differential equations and for discrete maps, the calculation of Lyapunov coefficients is well developed: Eckmann and Ruelle(1985), Müller (1995), Stefanski (2000), Galvanetto (2000). Discontinuity (or impact) maps are defined in such a way the event of a certain discontinuity is given in terms of the foregoing discontinuity. Discontinuity maps are used by de Souza and Caldas (2004), that apply the standard method (see, e.g., Eckmann and Ruelle, 1985) for the calculation of Lyapunov coefficients for discrete maps to the impact maps related to an impact oscillator and to an impact pair system. A novelty of the present work lies in the application of this technique to the case of an impacting bilinear oscillator. Finally, a reduced order model for describing single-mode dynamics of an impacting cantilever beam is identified (on going).

4. Index of the presentation • Differential equations of the SDOF model • The map approach • Numerical simulations of the SDOF model • Bifurcation analysis • Lyapunov coefficients • Hints for describing an impacting cantilever beam • Conclusions

5. The Impacting Bilinear Oscillator

7. Non dimensionalization

8. Non dimensional equations

9. Parameters for non-dimensionalization Parameters for bifurcation-analysis Parameters to be fixed on the basis of technical and experimental reasons

10. Attenuation coefficient cs is chosen in such a way the oscillation amplitude is attenuated at the rate of 10% every 10 cycles. The ratio between the velocity after and before the contact must be compatible with a reasonable restitution coefficient The ratio between the interval of time during which the mass is inside the obstacle and the interval of time between successive impacts (flight-time) must be small As a consequence,

11. Phase portrait Numerical simulations Return or discontinuity map no grazing Discontinuity maps and phase portraits are evaluated as follows …

12. The analytic solution inside the obstacle as a function of the initial time and velocity: The analytic solution outside the obstacle (in the system) as a function of the initial time and velocity: The initial time and velocity can be evaluated only numerically ! How? The initial time and velocity for the solution inside the obstacle is the final time and velocity of the solution outside the obstacle and viceversa.

13. t , t , t , t , t ,... Define the time series of impacts 0 1 2 3 4 Define the velocity series at impacts x , x , x , x , x ,...      0 1 2 3 4 A numerical evaluation of these equations in terms of t allows us to

14. The evaluation of the functions Fo and Fs can be done only numerically! Let us remark, however, that their derivatives can be obtained in the analytic form

16. Discontinuity map or impact map

17. Numerical simulations: sample motions Period-1 orbit Return map Phase portrait

18. Period-1 orbit: Phase portrait

19. Period-8 orbit Return map Phase portrait

20. Period-8 orbit: Phase portrait

21. Chaotic case Return map Phase portrait

22. Chaotic case: Phase portrait

23. Continuous bifurcation Discontinuous bifurcation Analysed by Lyapunov coefficients The point does not touch the obstacle Enlarged in the following Numerical simulations: Bifurcation analysis Nondimensional time instants of contacts vs nondimensional external force frequency

24. Discontinuous bifurcation Continuous bifurcation Chaotic phase range Intermittent chaos Nondimensional time instants of contacts vs nondimensional external force frequency Enlargements

25. Continuous bifurcation Discontinuous bifurcation The point does not touch the obstacle Portions of contact duration vs nondimensional external force frequency

26. Enlarged in the following Nondimensional time instants of contacts vs nondimensional external force amplitude

27. Discontinuous bifurcation Continuous bifurcation The point does not touch the obstacle Nondimensional time instants of contacts vs nondimensional external force amplitude Enlargements

28. Enlarged in the following Portions of contact duration vs nondimensional external force amplitude

29. Continuous bifurcation Discontinuous bifurcation The point does not touch the obstacle Portions of contact duration vs nondimensional external force amplitude Enlargements

30. Lyapunov coefficients To distinguish between periodic and chaotic orbits we evaluate the two Lyapunov coefficients h-th eigenvalue of the matrix Where J is the Jacobian matrix of the impact map

31. The evaluation of the functions F and G can be done only numerically! However, their derivative can be obtained in the analytic form Because of the chain rule derivative, the jacobian of the discontinuity map is the following: Discontinuity map or impact map

32. The jacobian matrix must be evaluated on the time series of impacts, that can be calculated only numerically.

33. Intermittent chaos The point does not touch the obstacle Enlarged in the following Lyapunov coefficients vs nondimensional external force frequency

34. Chaotic phase ranges Periodic phase range Due to intermittent chaos Lyapunov coefficients vs nondimensional external force frequency Enlargements

35. Lyapunov coefficients vs nondimensional external force amplitude Enlarged in the following

36. The point does not touch the obstacle Lyapunov coefficients vs nondimensional external force amplitude Enlargements Chaotic phase range

37. Relation between Lyapunov coefficients and previous bifurcation diagrams

38. Relation between Lyapunov coefficients and previous bifurcation diagrams

39. For a sinusoidally forced bilinear oscillator (with the discontinuity displaced by a gap from the unstressed configuration of one spring) it is implemented a numerical iterative method to derive the solution and discuss some relevant aspects. The dynamics of the soft impact oscillator has been investigated based on the discontinuity map obtained by slicing the three-dimensional phase space at the discontinuity. The definition of this transcendental map has been used to evaluate the Lyapunov coefficients. Phase portraits and return maps for three different response samples have been shown: periodic, high-periodic and chaotic. Besides classical bifurcation diagrams in terms of Lyapunov coefficients, more characterizing diagrams have been given in terms of the time instants of contact when the mass recovers the gap and impacts the obstacle, as well as of the percentage portion of contact duration. A very complex scenario has been highlighted, with a rich structured pattern that includes continuous and discontinuous bifurcations, flip bifurcations and alternating regular and chaotic regimes.

40. F0 Sinωt EI  kb Hints for describing an impacting cantilever beam The considered bilinear oscillator can also represent a simple SDOF model for effectively describing the impacting behavior of a cantilevered beam. It can be useful to explore a wide range of problem parameters with low computational efforts, to capture interesting features of the dynamic response, and to get hints for further investigations of the beam via more complex tools, e.g. 1-D FEM models.

41. An example of phase portrait, using a nonlinear Finite Element tool • F0=100N • ω=66,82 Hz (resonance of free beam) • kb=106 N/m

42. F0 Sinωt  kb Towards a reduced order SDOF model EI, 

43. A strategy for the identification of m: the first mode dynamic identification The beam first mode of vibration equivalent to the mode of the mass-spring system A strategy for the identification of ks: the static analogy is used The tip displacement of the beam Equivalent to the displacement of the spring A strategy for the identification of k0: The resonance of the impacted cantilever beam equivalent to the resonance of the bilinear oscillator...

44. Resonance curves for the beam model Maximum and minimum displacement vs external force frequency Each colour is for a given spring rigidity of the obstacle The higher the spring rigidity of the obstacle, the higher the resonance frequency

45. Resonance curves for the SDOF model Maximum and minimum displacement vs external force frequency Each colour is for a given spring rigidity of the obstacle The higher the spring rigidity of the obstacle, the higher the resonance frequency

46. beam sdof The identification is done in terms of the same resonance: The two spring rigidities related to the two models will be called identified when the resonances of the two nonlinear models will be equivalent. Resonance frequencies vs spring rigidities of the obtacle in the two models

47. Identification of the SDOF spring rigidity

48. First results for this kind of identification Attachment velocity vs external force frequency (far from resonance!)‏

49. Conclusions Forced motion of a bilinear oscillator (SDOF) has been analysed by means of return maps and Lyapunov coefficients, in order to highlight periodic and chaotic regimes. A problem of technical interest was considered: An elastic cantilever beam (Euler model!) impacting a deformable obstacle The SDOFbilinear oscillator has been identified in order to simulate the beam behaviour at low computational cost.