February 22, 2017

Advances in Nonlinear Dynamics: Methods and Applications by D. Quinn, R. Rand, J. Bridge (auth.), Anil K. Bajaj, Steven

By D. Quinn, R. Rand, J. Bridge (auth.), Anil K. Bajaj, Steven W. Shaw (eds.)

This is the second one and ultimate factor of the gathering of papers that have been contributed by means of pals and co-workers of (Late) Professor P. R. "Pat" Sethna of the collage of Minnesota to commemorate his seventieth birthday on may well 26, 1993. the 1st set of contributions used to be released in Nonlinear Dynamics because the final factor (no. 6) of Vol. four in 1993. As situations might have it, Professor Sethna was once clinically determined with melanoma within the fall of 1992 and, after a longer conflict with the sickness, he kicked the bucket on November four, 1993, quite a few days ahead of the 1st set of contributed papers seemed in print. it's pleasurable to record that the organizers of those vi Foreword commemorative concerns in Nonlinear Dynamics have been in a position to current to Professor Sethna, at the social gathering of his seventieth birthday, entire info of the deliberate commemorative concerns. This moment set of contributions is devoted, in memoriam, to Professor P. R. Sethna. As a lot of you're good acutely aware, Professor Sethna was once an energetic researcher within the box of nonlinear vibrations and dynamics for almost 40 years, making many basic and critical contributions to either the theoretical and utilized points of this box. He used to be additionally well-known for his impressive management and administrative skills, amply established via his place because the Head of the dep. of Aerospace Engineering and Mechanics on the collage of Minnesota for twenty-six years (1966-1992).

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C. Sinha reported in the case of non-autonomous systems. Although the method for computing normal forms for periodic systems has been known for some years now, it requires the computation of a special periodic transformation known as the Liapunov-Ploquet (L-F) transformation [1]. It is well known that such a L-F transformation can be used to convert the quasi-linear periodic system to a vector field whose linear part is time-invariant and the nonlinear part consists of time-varying coefficients which are periodic.

Pandiyan and S. C. Sinha ~ k( + ~ Cos(rot) CD - (h ) o 't'l bl(~:- ~l) Fig. 4. A double inverted pendulum subjected to a periodic follower force. load. 2S(YI - Y2)2[k(2p - 7)YI + k(S + ph - 3))Y2 - (2BI + SB2)Y3 + SB2Y2]] where {YI, Y2, Y3, Y4} = {cPI, cP2, ¢I, ¢2}. In the following, the dynamics of a primary single Hopf and a single flip bifurcation of the above four dimensional system is discussed via the center manifold principle by reducing the problem to a two and a single dimension, respectively.

For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method.

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