TY - GEN
T1 - Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems
AU - Boivin, Nicolas
AU - Pierre, Christophe
AU - Shaw, Steven W.
PY - 1993
Y1 - 1993
N2 - Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by defining proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems - one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied - are simultaneously applied to the same study case - an Euler-Bernoulli beam constrained by a non-linear spring -, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.
AB - Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by defining proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems - one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied - are simultaneously applied to the same study case - an Euler-Bernoulli beam constrained by a non-linear spring -, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.
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M3 - Conference contribution
AN - SCOPUS:0027836229
SN - 0791811719
T3 - American Society of Mechanical Engineers, Design Engineering Division (Publication) DE
SP - 1
EP - 9
BT - Nonlinear Vibrations
A2 - Shahinpoor, Mo
A2 - Tzou, H.S.
T2 - 14th Biennial Conference on Mechanical Vibration and Noise
Y2 - 19 September 1993 through 22 September 1993
ER -