Many researchers have analyzed the non-conservative instability of beams resting on an elastic foundation. De Rosa and Franciosi [1], and Takahashi and Yoshioka [2] have studied the influence of an intermediate support on the stability behaviour of cantilever beams and double beams subjected to follower forces. Singh, et al. [3] discussed the implementation of follower and axial end forces in a beam-type MEMS resonator for the application of resonant frequency tuning. Furthermore, Takahashi and Yoshioka [4] have studied the stability behaviour and in-plane vibration of L-shaped cantilever beams subjected to follower forces. Takahashi [5] has studied the identification method for the critical force of a non-uniform L-shaped cracked shaft using the neural networks.

The problem of experimental design or design of experiments (DOE) is encountered in many fields. A common situation for using DOE is when the designer does not know the exact underlying relationship between response and design variables. The basic idea of response surface methodology is to create explicit approximation functions to the objective and constraints, and then to use these when performing the optimization. The approximation functions are typically in the form of low-order polynominals fit by least squares regression analysis.

In this paper the possibility of using a response surface methodology, which consists of a design of experiments, for estimating the critical flutter load of the L-shaped beam is studied. An analysis is presented for the vibration and stability of a tapered L-shaped beam subjected to a follower force using the transfer matrix approach. The method is applied to beams, and the natural frequencies and flutter loads are calculated numerically, to provide information about the effect on them when varying cross-section, span and stiffness of intermediate supports, opening angles and the slenderness ratio.

Some numerical examples were presented to demonstrate the possibility of the response surface approximation. From the results of the numerical examples we can draw the following conclusions. First, the critical flutter load can be predicted by using the response surface approximation with three-level orthogonal Latin squares. Second, the generalization capability of the response surface with three-level orthogonal Latin squares is sufficient for estimating the critical flutter loads.

1

M.A. De Rosa, C. Franciosi, "The influence of an intermediate support on the stability behavior of cantilever beams subjected to follower forces", J. Sound Vibr., 137, 107-115, 1990. doi:10.1016/0022-460X(90)90719-G

2

I. Takahashi, T. Yoshioka, "Vibration and stability of a non-uniform double-beam subjected to follower forces", Computers Struct., 59, 1033-1038, 1996. doi:10.1016/0045-7949(95)00346-0

3

A. Singh, et al., "MEMS implimentation of axial and follower end forces", J. Sound Vibr, 286, 637-644, 2005.

4

I. Takahashi, T. Yoshioka, "Vibration and stability of a non-uniform L-shaped beam subjected to follower forces (in-plane vibration)", J.Sound Vibr., 171, 255-265, 1994. doi:10.1006/jsvi.1994.1117

5

I. Takahashi, "Identification for Critical Flutter Load of a Non-uniform L-shaped Cracked Shaft subjected to a Follower Force (Out-of-Plane Vibration)", in B.H.V. Topping, (Editor), "Proceedings of the Seventh International Conference on the Application of Artificial Intelligence to Civil and Structural Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 58, 2003. doi:10.4203/ccp.78.58

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