Swinging Atwood Machine: Experimental and numerical results, and a theoretical study

A Swinging Atwood Machine (SAM) is built and some experimental results concerning its dynamic behaviour are presented. Experiments clearly show that pulleys play a role in the motion of the pendulum, since they can rotate and have non-negligible radii and masses. Equations of motion must therefore take into account the moment of inertia of the pulleys, as well as the winding of the rope around them. Their influence is compared to previous studies. A preliminary discussion of the role of dissipation is included. The theoretical behaviour of the system with pulleys is illustrated numerically, and the relevance of different parameters is highlighted. Finally, the integrability of the dynamic system is studied, the main result being that the machine with pulleys is non-integrable. The status of the results on integrability of the pulley-less machine is also recalled.

[1]  C. Simó,et al.  NON-INTEGRABILITY CRITERIA FOR HAMILTONIANS IN THE CASE OF LAME NORMAL VARIATIONAL EQUATIONS , 1996 .

[2]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[3]  F. Fauvet,et al.  Application of J.-J. Morales and J.-P. Ramis’ theorem to test the non-complete integrability of the planar three-body problem , 2003 .

[4]  Nicholas B. Tufillaro,et al.  Swinging Atwood's Machine : integrability and dynamics , 1990 .

[5]  J. Ramis,et al.  GALOISIAN OBSTRUCTIONS TO INTEGRABILITY OF HAMILTONIAN SYSTEMS II ∗ , 2001 .

[6]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[7]  CORRIGENDUM: On a simple formulation of the golf ball paradox On a simple formulation of the golf ball paradox , 2009 .

[8]  M. A. Almeida,et al.  On the Ziglin-Yoshida analysis for some classes of homogeneous hamiltonian systems , 1998 .

[9]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[10]  H. Moritz,et al.  Geodetic reference system 1980 , 1988 .

[12]  George Birkhoff.  Physical aspects of dynamical systems , 1927 .

[13]  Juan José Morales Ruiz,et al.  Differential Galois Theory and Non-Integrability of Hamiltonian Systems , 1999 .

[14]  N. Tufillaro,et al.  Swinging Atwood’s Machine , 1984 .

[15]  J. Humphreys,et al.  Linear Algebraic Groups , 1975 .

[16]  Nicholas B. Tufillaro,et al.  Unbounded orbits of a swinging Atwood's machine , 1988 .

[17]  Michael F. Singer,et al.  Necessary conditions for liouvillian solutions of (third order) linear differential equations , 2005, Applicable Algebra in Engineering, Communication and Computing.

[18]  Hyman Bass,et al.  Selected works of Ellis Kolchin with commentary , 1999 .

[19]  C. Simó,et al.  Non-integrability of the degenerate cases of the Swinging Atwood'sMachine using higher order variational equations , 2010 .

[20]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[21]  N. Tufillaro.  Integrable motion of a swinging Atwood’s machine , 1986 .

[22]  Juan José Morales Ruiz,et al.  Galosian Obstructions to Integrability of Hamiltonian Systems II , 2001 .

[23]  C. Simó,et al.  Integrability of hamiltonian systems and differential Galois groups of higher variational equations , 2007 .

[24]  Andy R. Magid,et al.  Lectures on differential Galois theory , 1994 .

[25]  C. Simó,et al.  Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples , 2009 .

[26]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[27]  H. Yehia.  On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood machine , 2006 .

[28]  Nicholas B. Tufillaro,et al.  Motions of a swinging Atwood's machine , 1985 .

[29]  G. A. Watson.  A treatise on the theory of Bessel functions , 1944 .

[30]  N. Tufillaro.  Periodic orbits of the integrable Swinging Atwood's Machine , 1994, chao-dyn/9411016.

[31]  Marta Net,et al.  Computation of invariant tori by Newton-Krylov methods in large-scale dissipative systems , 2010 .