Hello and thank you for your question.
In providing the following one-page summary, I have indicated in
brackets, e.g. [14], where the mathematical formula given in the paper
needs to be set forth in full.
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Richard-ga
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Rao, G.K. and K.K. Raju, Large-Amplitude Vibrations of Spring-Hinged
Beams, uses a direct numerical integration method to evaluate the
nonlinear vibration behavior of beams with symmetrically placed
spring-hinges. The numerical results presented indicate the efficacy
of the method used, whereby the admissible lateral displacement
functions are used to represent the spring-hinged boundary functions.
The authors report that to the best of the their knowledge, this is
the first time that the reported method has been used to determine the
nonlinear vibrations of spring-hinged beams.
Components incorporating non-uniform beams are encountered in many
engineering structures. Because these structural elements are
slender, they are prone to large-amplitude vibrations. Previously
applied analytical methods, including those of Rich-Galerkin and
Singh, apply classical boundary conditions to the study. In actual
practice, depending on the degree of end-support flexibility, the ends
of the beams can be treated as elastically restrained against
rotation. The authors’ Note presents a simple analytical method to
obtain the large-amplitude behavior of spring-hinged beams that are
symmetric with respect to the midpoint of the span of the beam.
The authors adapt the differential equation governing the
spring-hinged beam
[1]
to this purpose by taking into account the appropriate assumptions
regarding the boundary conditions of the function, which they combine
and solve for w(x), the spatial function of lateral displacement
[6]
Combining these two formulae, they derive a final differential
equation in terms of time
[8]
This equation is Duffing’s function. The authors apply a direct
numerical integration method to Duffing’s function, which yields the
desired degree of accuracy for a given [gamma] and [alpha] which they
present as
[14]
The accuracy of their formula is confirmed by considering the two
extreme cases of the rotational spring stiffness parameter, namely
when [gamma] = 0 and when [gamma] = 10^5, and is shown to give very
accurate results even for very high values of lateral displacement.
The results are shown to agree well with the finite element solution
of Rao. The authors rightly claim credit for having derived their
lateral displacement function in an elegant way, and for demonstrating
its accuracy.
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