Keywords: nonlinear buckling, damped Mathieu/Duffing equation, modulated amplitude, noise, stability.

The lateral vibrations and stability of pin-ended columns subjected to time dependent end-loads can be analysed approximately by using a Mathieu/Duffing type equation with parametric excitation. The nonlinear term describes the effect of stretching of the neutral axis and yields a finite post-buckling response. Several limits are considered: very slow variation corresponds to quasi-static variation through the Euler buckling load, a more rapid periodic variation yields the characteristic bifurcations of the damped Mathieu equation. In each case, the response near the bifurcation due to slow variation through the bifurcation is analysed by using a combination of multiple time scales and matched asymptotic expansions. The analysis shows the effect of perturbations or additive noise on the growth of the response.

Small transverse motions of a long slender pinned-pinned column loaded by a time-varying axial force are considered. The effects of stretching of the neutral axis are included. A standard modal analysis of the equation of motion of the beam is reviewed briefly. A first mode approximation to the response yields the equation:

where is the transverse displacement, and the prime indicates differentiation with respect to time. The axial load is given by . The stationary Euler buckling load is at ; the post-buckling response amplitude in this case is then limited in growth by the nonlinear term, and is given by .

Equation (29) forms the basis for the analysis in the paper. A variety of nonstationary solutions are examined for various forms of the time-varying component of the axial load, , with particular attention, of course, to the response in the neighbourhood of some of the bifurcations exhibited by Equation (29). The analysis uses a combination of multiple time scale analysis, as in the text by Nayfeh and Mook [1], and matched asymptotic expansions, as in the analysis of similar types of bifurcations by Davies and Rangavajhula [2,3] and Davies [4].

A very slow linear or sinusoidal variation of can be analysed directly by using matched asymptotic expansions. The variation, in the form or where is a small parameter, is assumed to be sufficiently slow that the basic forms of the stationary Euler type buckling response can still be identified. Numerical simulations indicate that noise has an appreciable effect on the response, so a term , where is unit amplitude Gaussian white noise and is the noise amplitude, is added to Equation (29). The outer expansion, a power series in , is close to the stationary solutions away from the bifurcation, but is singular at the bifurcation. Stretching of the variables in terms of fractional powers of e is used to describe the response near the bifurcation. An inner-inner expansion describes how noise or an initial perturbation generates exponential growth of the response away from the initial trivial value. This matches in an overlap region to an inner expansion, which describes the jump, and this in turn matches in the usual way to the outer solution. For linear variation, the apparent delay in the value of the axial load at which buckling occurs is shown to be , a characteristic combination for bifurcations of this sort.

A second case of interest is at a more rapid sinusoidal variation of . The term in Equation (29) is written . The linear version of Equation (29) now exhibits the characteristic Mathieu equation transition curves between stable and unstable responses; these transition curves emanate from integer values of in the plane. For example, the transition curve near has the form . Again the inclusion of the nonlinear term limits the response in what were unstable regions. Previous analyses are generalized by allowing the amplitude to vary slowly through the bifurcations associated with these transition curves, the interest again being in determining how the response changes from its original trivial form to a finite value.

A multiple time scale analysis leads to another singular perturbation problem of similar form to that treated earlier. The outer solution is again close to the stationary solution, but is singular at the bifurcation. Examples of numerical simulations for either slow linear or slow sinusoidal variation through the bifurcations again demonstrate the critical effect of noise on the change in form of the response. The need to recognise the importance of low-level noise in cases of this sort is the most important conclusion of this work. Further analysis of this case using matched asymptotic expansions is in progress.

This research was supported by the Natural Sciences and Engineering Research Council of Canada.

1

A.H. Nayfeh, D.T. Mook, "Nonlinear Oscillations" Wiley, New York,1979.

2

H.G. Davies, K. Rangavajhula, "Dynamic Period-doubling Bifurcations of a Unimodal Map", Proceedings of the Royal Society, A453, 2043-2061, London,1997.

3

H.G. Davies, K. Rangavajhula, "A Period-doubling Bifurcation With Slow Parametric Variation", Proceedings of the Royal Society, A457, 2965-2982, London, 2001.

4

H.G. Davies, "Slow Sinusoidal Modulation through Bifurcations: the Effect of Additive Noise" Nonlinear Dynamics, to be published, 2004. doi:10.1023/B:NODY.0000045509.36253.31

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)