Engineering Applications of Non-smooth Dynamics

This chapter introduces and discusses practically important concept of non-smooth dynamical systems, which are very common in engineering applications. Mathematically, such systems can be considered as piecewise smooth and therefore their global solutions are obtained by stitching local solutions, which are easy to develop by standard methods. If a dynamical system is piecewise linear then an implicit global analytical solution can be given, however the times when non-smoothness occurs have to be determined first. This leads to a set of nonlinear algebraic equations. To illustrate the non-smooth dynamical systems and the methodology of solving them, three mechanical engineering problems were studied. Firstly, a vibro-impact system in a form of moling device was modelled and analysed to understand how the progression rates can be maximised. For this system, periodic trajectories can be reconstructed as they go through three linear subspaces (no contact, contact with progression and contact without progression), and using combination of analytical and numerical methods the optimal range of the system parameters can be identified. In the second application the influence of opening and closing of a fatigue crack on the system dynamics was investigated. Specifically, a novel apparatus to induce aperiodic loading to a specimen with a fatigue crack was studied. It was shown experimentally that fatigue life can be reduced few times if the sample is loaded aperiodically. The analysis of the developed mathematical model shown that as a crack grows linearly before reaching its critical value, the response of the system remains periodic. When its size exceeds the critical value, the system behaviour becomes chaotic and then the crack growth increases exponentially. This phenomenon can be used in structural health monitoring. The last problem comes from rotordynamics, where nonlinear interactions between the rotor and the snubber ring were studied. The influence of the preloading of the snubber ring on the system behaviour was investigated and the range of the system parameters where chaotic vibrations occur was identified. The results obtained from the developed mathematical model confronted with the experiments shown a good degree of correlation.

[1]  Marian Wiercigroch,et al.  Penetration rate prediction for percussive drilling via dry friction model , 2000 .

[2]  M. Elices,et al.  Stress Intensity factor, compliance and CMOD for a General Three-Point-Bend Beam , 1998 .

[3]  M Wiercigroch,et al.  A symmetrically piecewise linear oscillator: Design and measurement , 1999 .

[4]  Marian Wiercigroch,et al.  Nonlinear vibration caused by fatigue , 2007 .

[5]  Marian Wiercigroch,et al.  Applied nonlinear dynamics and chaos of mechanical systems with discontinuities , 2000 .

[6]  I. Grabec Chaotic dynamics of the cutting process , 1988 .

[7]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[8]  Ekaterina Pavlovskaia,et al.  Nonlinear Dynamics of Vibro-Impact Systems: Theory and Experiments , 2003 .

[9]  M. Spektor,et al.  Principles of soil-tool interaction , 1981 .

[10]  Chee-Hoe Foong Influence of fatigue crack growth on the dynamics of engineering components and structures , 2004 .

[11]  Y. C. Chu,et al.  Vibrations of beams with a fatigue crack , 1992 .

[12]  M. Wiercigroch,et al.  Non-linear dynamic interactions of a Jeffcott rotor with preloaded snubber ring , 2004 .

[13]  G. S. Littlejohn,et al.  A study of vibratory driving in granular soils , 1980 .

[14]  Ekaterina Pavlovskaia,et al.  Low-dimensional maps for piecewise smooth oscillators , 2007 .

[15]  Ekaterina Pavlovskaia,et al.  Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring , 2002 .

[16]  F. Peterka,et al.  Transition to chaotic motion in mechanical systems with impacts , 1992 .

[17]  Marian Wiercigroch,et al.  Novel dynamic fatigue-testing device: Design and measurements , 2006 .

[18]  M. Kunze Non-Smooth Dynamical Systems , 2000 .

[19]  Ekaterina Pavlovskaia,et al.  Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes , 2004 .

[20]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[21]  Dara W. Childs Fractional-frequency rotor motion due to nonsymmetric clearance effects , 1982 .

[22]  C. Hsu,et al.  Cell-To-Cell Mapping A Method of Global Analysis for Nonlinear Systems , 1987 .

[23]  F. F. Ehrich Spontaneous Sidebanding in High Speed Rotordynamics , 1992 .

[24]  R. Ganesan Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances , 1996 .

[25]  Richard David Neilson,et al.  Experimental verification of Jeffcott rotor model with preloaded snubber ring , 2006 .

[26]  Agnes Muszynska,et al.  Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rubs , 1995 .

[27]  Ekaterina Pavlovskaia,et al.  Modelling of Ground Moling Dynamics by an Impact Oscillator with a Frictional Slider , 2003 .

[28]  S. Natsiavas,et al.  Periodic response and stability of oscillators with symmetric trilinear restoring force , 1989 .

[29]  Marian Wiercigroch,et al.  Modelling of dynamical systems with motion dependent discontinuities , 2000 .

[30]  Raymond H. Plaut,et al.  Free and Forced Longitudinal Vibrations of a Cantilevered Bar With a Crack , 1992 .

[31]  D. H. Gonsalves,et al.  A study of the response of a discontinuously nonlinear rotor system , 1995 .

[32]  Marian Wiercigroch,et al.  Experimental Study of a Symmetrical Piecewise Base-Excited Oscillator , 1998 .

[33]  Kurt Wiesenfeld,et al.  Suppression of period doubling in the dynamics of a bouncing ball , 1987 .

[34]  Brian F. Feeny,et al.  A nonsmooth Coulomb friction oscillator , 1992 .

[35]  Celso Grebogi,et al.  Two-dimensional map for impact oscillator with drift. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Marian Wiercigroch,et al.  APPLIED NONLINEAR DYNAMICS OF NON-SMOOTH MECHANICAL SYSTEMS , 2006 .

[37]  Marian Wiercigroch,et al.  Measurement of Chaotic Vibration in a Symmetrically Piecewise Linear Oscillator , 1998 .

[38]  Peter Eberhard,et al.  Rigid-elastic modeling of meshing gear wheels in multibody systems , 2006 .

[39]  Ekaterina Pavlovskaia,et al.  Chaos caused by fatigue crack growth , 2003 .

[40]  Grzegorz Litak,et al.  Dynamics of a Gear System with Faults in Meshing Stiffness , 2004, nlin/0405053.

[41]  Matthew P. Cartmell,et al.  Regular and chaotic dynamics of a discontinuously nonlinear rotor system , 2002 .

[42]  Y. Chu,et al.  Analysis of forced bilinear oscillators and the application to cracked beam dynamics , 1992 .

[43]  Fulei Chu,et al.  Periodic, quasi-periodic and chaotic vibrations of a rub-impact rotor system supported on oil film bearings , 1997 .

[44]  Marian Wiercigroch,et al.  Chaotic Vibration of a Simple Model of the Machine Tool-Cutting Process System , 1997 .

[45]  Marian Wiercigroch,et al.  Dry Friction Model of Percussive Drilling , 1999 .

[46]  M. Wiercigroch,et al.  Comments On The Study Of A Harmonically Excited Linear Oscillator With A Coulomb Damper , 1993 .

[47]  J. Padovan,et al.  Non-linear transient analysis of rotor-casing rub events , 1987 .

[48]  Evgueni Karpenko,et al.  Nonlinear dynamics of a Jeffcott Rotor with imperfections , 2003 .

[49]  Marian Wiercigroch,et al.  Parameter identification of the fatigue-testing rig , 2008 .

[50]  F. Ismail,et al.  Identification of fatigue cracks from vibration testing , 1990 .

[51]  Raymond H. Plaut,et al.  Detection of Cracks in Rotating Timoshenko Shafts Using Axial Impulses , 1991 .

[52]  C Grebogi,et al.  Modeling of an impact system with a drift. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  P. Gudmundson The dynamic behaviour of slender structures with cross-sectional cracks , 1983 .

[54]  F. Chu,et al.  BIFURCATION AND CHAOS IN A RUB-IMPACT JEFFCOTT ROTOR SYSTEM , 1998 .

[55]  Rajendra Singh,et al.  Non-linear dynamics of a spur gear pair , 1990 .

[56]  Ekaterina Pavlovskaia,et al.  Periodic solution finder for an impact oscillator with a drift , 2003 .

[57]  P. Holmes,et al.  A periodically forced piecewise linear oscillator , 1983 .

[58]  M. Wiercigroch,et al.  Bifurcation analysis of a preloaded Jeffcott rotor , 2003 .