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In this talk, I will present a study of four problems in the area of nonlinear vibrations, using the asymp- totic method of multiple scales (MMS).
In this talk, I will present a study of four problems in the area of nonlinear vibrations, using the asymp- totic method of multiple scales (MMS).
The first problem, motivated by the study of a controlled, weakly damped, weakly forced oscillator, leads to the development of a new phase space based algorithm for limit cycle continuation. The algorithm approximates the limit cycle with an ordered set of points along with spline interpolation. The algorithm shows favourables results in several cases of limit cycles approaching homoclinic orbits.
The second problem deals with an analytical study of the classical van der Pol oscillator, but with an added fractional damping term. The fractional term is approximated here by a recently proposed Galerkin-based discretization scheme resulting in a set of ODE's on which MMS is applied. Excellent approximations for near-Hopf dynamics are obtained.
In the third problem, a well known tool vibration model in the large delay regime using the MMS up to second order is studied. In the present analysis, infinite dimensional dynamics caused by the large delay is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. The MMS slow flow shows excellent agreement with numerical computations involving infinite dimensional slow modulations.Finally, the last problem deals with the study of the weakly nonlinear whirl of an asymmet-ric, overhung rotor near its gravity critical speed using a well known two-degree of freedom modelusing MMS up to second order. The MMS slow flow reveals the weak role of inertia asymmetry, weak gravity resonances driven by nonlinearity, and an unanticipated periodic solution branch. The bifurcation structure for the zero damping case is deduced.
- Speaker
- Nandakumar Krishnan
- Venue
- Rm 111