Near-grazing dynamics of base excited cantilevers with nonlinear tip interactions

In this article, nonsmooth dynamics of impacting cantilevers at different scales is explored through a combination of analytical, numerical, and experimental efforts. For off-resonance and harmonic base excitations, period-doubling events close to grazing impacts are experimentally studied in a macroscale system and a microscale system. The macroscale test apparatus consists of a base excited aluminum cantilever with attractive and repulsive tip interactions. The attractive force is generated through a combination of magnets, one located at the cantilever structure’s tip and another attached to a high-resolution translatory stage. The repulsive forces are generated through impacts of the cantilever tip with the compliant material that covers the magnet on the translatory stage. The microscale system is an atomic force microscope cantilever operated in tapping mode. In this mode, this microcantilever experiences a long-range attractive van der Waals force and a repulsive force as the cantilever tip comes close to the sample. The qualitative changes observed in the experiments are further explored through numerical studies, assuming that the system response is dominated by the fundamental cantilever vibratory mode. In both the microscale and macroscale cases, contact is modeled by using a quadratic repulsive force. A reduced-order model, which is developed on the basis of a single mode approximation, is employed to understand the period-doubling phenomenon experimentally observed close to grazing in both the macroscale and microscale systems. The associated near-grazing dynamics is examined by carrying out local analyses with Poincaré map constructions to show that the observed period-doubling events are possible for the considered nonlinear tip interactions. In the corresponding experiments, the stability of the observed grazing periodic orbits has been assessed by constructing the Jacobian matrix from the experimentally obtained Poincaré map. The present study also sheds light on the use of macroscale systems to understand near-grazing dynamics in microscale systems.

[1]  D. Sarid,et al.  Kinetics of lossy grazing impact oscillators , 1998 .

[2]  Balakumar Balachandran,et al.  Utilizing nonlinear phenomena to locate grazing in the constrained motion of a cantilever beam , 2009 .

[3]  J. Molenaar,et al.  Dynamics of vibrating atomic force microscopy , 2000 .

[4]  Arne Nordmark,et al.  Non-periodic motion caused by grazing incidence in an impact oscillator , 1991 .

[5]  Balakumar Balachandran,et al.  Dynamics of an Elastic Structure Excited by Harmonic and Aharmonic Impactor Motions , 2003 .

[6]  Alan R. Champneys,et al.  Corner collision implies border-collision bifurcation , 2001 .

[7]  A. Nordmark,et al.  Experimental investigation of some consequences of low velocity impacts in the chaotic dynamics of a mechanical system , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[8]  Grebogi,et al.  Grazing bifurcations in impact oscillators. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

[10]  Molenaar,et al.  Grazing impact oscillations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Stephen W. Howell,et al.  Nonlinear dynamics of microcantilevers in tapping mode atomic force microscopy: A comparison between theory and experiment , 2002 .

[12]  H. Dankowicz,et al.  On the origin and bifurcations of stick-slip oscillations , 2000 .

[13]  J. Molenaar,et al.  Mappings of grazing-impact oscillators , 2001 .

[14]  Balakumar Balachandran,et al.  Grazing bifurcations in an elastic structure excited by harmonic impactor motions , 2008 .

[15]  Alan R. Champneys,et al.  Normal form maps for grazing bifurcations in n -dimensional piecewise-smooth dynamical systems , 2001 .

[16]  Ekaterina Pavlovskaia,et al.  Experimental study of impact oscillator with one-sided elastic constraint , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  Kazuyuki Yagasaki,et al.  Nonlinear dynamics of vibrating microcantilevers in tapping-mode atomic force microscopy , 2004 .

[18]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[19]  Harry Dankowicz,et al.  The nonlinear dynamics of tapping mode atomic force microscopy with capillary force interactions , 2008 .

[20]  Stephen John Hogan,et al.  Local Analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems , 1999 .

[21]  Balakumar Balachandran,et al.  Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces , 2011 .

[22]  Steven W. Shaw,et al.  Chaotic vibrations of a beam with non-linear boundary conditions , 1983 .

[23]  Balakumar Balachandran,et al.  A Review of Nonlinear Dynamics of Mechanical Systems in Year 2008 , 2008 .

[24]  B. Balachandran,et al.  OFF-RESONANCE CANTILEVER DYNAMICS IN THE PRESENCE OF ATTRACTIVE AND REPULSIVE TIP INTERACTION FORCES , 2011 .

[25]  Harry Dankowicz,et al.  Near-grazing Dynamics in Tapping-mode Atomic-force Microscopy , 2007 .