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 associates of (Late) Professor P. R. "Pat" Sethna of the collage of Minnesota to commemorate his seventieth birthday on may possibly 26, 1993. the 1st set of contributions was once released in Nonlinear Dynamics because the final factor (no. 6) of Vol. four in 1993. As conditions could have it, Professor Sethna used to be clinically determined with melanoma within the fall of 1992 and, after a longer conflict with the illness, he kicked the bucket on November four, 1993, quite a few days prior to the 1st set of contributed papers seemed in print. it truly is fulfilling to record that the organizers of those vi Foreword commemorative matters in Nonlinear Dynamics have been in a position to current to Professor Sethna, at the social gathering of his seventieth birthday, entire information of the deliberate commemorative matters. 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 primary and demanding contributions to either the theoretical and utilized facets of this box. He used to be additionally famous for his striking management and administrative skills, amply confirmed via his place because the Head of the dept of Aerospace Engineering and Mechanics on the college of Minnesota for twenty-six years (1966-1992).
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And Qij are the elements of Q (t). Here (:) represents the differentiation with respect to tf. Because of the transformation, the two Floquet multipliers of equation (29) on the unit circle transform to a purely imaginary pair in equation (30). Further, it is to be noted that the nonlinear part of equation (30) can be expressed in terms of Fourier expansions due to the periodic nature of the Liapunov-Floquet transformation. 21181i} are the eigenvalues of R, and the periodic coefficients fij(t, T); i, j = 1, ...
The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in time t which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one.
The model captures the interaction of a symmetric in-plane mode with two out-of-plane modes. The modes are coupled through quadratic and cubic nonlinearities arising from nonlinear cable stretching. For particular magnitudes of equilibrium curvature, the natural frequency of the in-plane mode is simultaneously commensurable with the natural frequencies of the two out-of-plane modes in 1: 1 and 2: 1 ratios. A second nonlinear order perturbation analysis is used to determine the existence and stability of four classes of periodic solutions.