16-05-2012, 12:09 PM
DESIGN AND NON-LINEAR ANALYSIS OF A PARABOLIC LEAF SPRING
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INTRODUCTION
As pointed out by Rahman et al. (2006), structural
problems, coupled with geometric non-linearity are always
challenges for engineers. This is because of the fact that
large deflection analysis of structures, inherently involves
non-linear differential equations having no closed form
solutions as pointed out by Bele’ndez et al. (2005). The
problem would become even more interesting if the shape
of the structure itself varies from point to point. That the
topic of large deflection analysis of cantilever beams is
ever interesting can be seen from a huge number of studies
reported in the literature; out of those researches only a few
relevant to the present study are discussed below.
Very recently Bele’ndez et al. (2005) carried out
numerical simulation using Runge-Kutta-Felhberg method
to find the tip deflection of a very slender beam under a
combined load. The authors studied the large deflections of
a uniform cantilever beam under the action of a combined
load consisting of an external vertical concentrated load at
the free end and a uniformly distributed load and compared
the numerical results with the experimental ones.
MATHEMATICAL ANALYSIS
Since the beams are quite slender for the present case,
only pure bending is considered for this study ignoring the
effect of shearing stresses. When deflection is large with
respect to the span of the beam the governing equations of
the elastic curve for a cantilever beam with a point load P
(Fig. 1), in terms of large deflection formulation
RESULTS AND DISCUSSION
We start discussion by proving soundness of our
numerical scheme by comparing few results, taking into
account only geometric non-linearity, for a highly flexible
cantilever beam of constant cross-section under a
combined load, as treated by Bele’ndez et al. (2005). Table
2 shows the comparison. The Young’s modulus for a
particular load was not explicitly given by Bele’ndez et al.
(2005). It was stated to be within 180-210 GPa. We used
its value as 200 GPa. As seen, the numerical non-linear
solution matches within an error of only 3.5% at the
highest experimental load found by Bele’ndez et al. (2005).
A better match would be possible with the known correct
value of E. For example, E = 194.5 GPa was found to give
least error as shown by Bele’ndez et al. (2005). Therefore,
our numerical predictions would match even better with
the experimental results with E = 194.5 GPa.
CONCLUSIONS
An innovative parabolic leaf spring has been designed
and analyzed solving highly non-linear differential
equations. The effects of two vitally important factors,
namely, the end-shortening and geometric nonlinearity, on
the response of parabolic shaped variable cross section,
have been demonstrated by numerical analysis. Nonlinear
solution plays vital role in determining the true stresses in
highly flexible structures.