Bifurcation and stability properties of periodic solutions to two nonlinear spring-mass systems

Abstract The behavior of a periodically forced, linearly damped mass suspended by a linear spring is well known. In this paper we study the nature of periodic solutions to two nonlinear spring-mass equations; our nonlinear terms are similar to earlier models of motion in suspension bridges. We contrast the multiplicity, bifurcation, and stability of periodic solutions for a piecewise linear and smooth nonlinear restoring force. We find that while many of the qualitative properties are the same for the two models, the nature of the secondary bifurcations (period-doubling and quadrupling) differs significantly.

[1]  Y. Choi,et al.  The structure of the solution set for periodic oscillations in a suspension bridge model , 1991 .

[2]  Stephen John Hogan,et al.  Non-linear dynamics of the extended Lazer-McKenna bridge oscillation model , 2000 .

[3]  P. McKenna.  Large Torsional Oscillations in Suspension Bridges Revisited: Fixing an Old Approximation , 1999 .

[4]  P. J. McKenna,et al.  The global structure of periodic solutions to a suspension bridge mechanical model , 2002 .

[5]  P. McKenna,et al.  Multiple periodic solutions for a nonlinear suspension bridge equation , 1999 .

[6]  Y. Chen,et al.  Travelling waves in a nonlinearly suspended beam: some computational results and four open questions , 1997, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  W. Walter,et al.  Nonlinear oscillations in a suspension bridge , 1987 .

[8]  H. Keller.  Lectures on Numerical Methods in Bifurcation Problems , 1988 .

[9]  Alan R. Champneys,et al.  Localization and solitary waves in solid mechanics , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  Alan C. Lazer,et al.  Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis , 1990, SIAM Rev..

[11]  Kristin S. Moore.  Large Torsional Oscillations in a Suspension Bridge: Multiple Periodic Solutions to a Nonlinear Wave Equation , 2002, SIAM J. Math. Anal..

[12]  A. Scott.  Encyclopedia of nonlinear science , 2006 .

[13]  Alan C. Lazer,et al.  Large scale oscillatory behaviour in loaded asymmetric systems , 1987 .

[14]  Cillian Ó Tuama,et al.  Large Torsional Oscillations in Suspension Bridges Visited Again: Vertical Forcing Creates Torsional Response , 2001, Am. Math. Mon..