Many researchers have analyzed the non-conservative instability of beams resting on an elastic foundation. Furthermore, there are some papers on the stability problems of cracked beams, for making a diagnostic system to detect the crack. The stability of columns with a single crack subjected to follower and vertical loads was studied by Anifantis and Dimarogonas [1]. Takahashi [2] studied the vibration and stability of a cracked Timoshenko beam subjected to a follower force. Takahashi [3] proposed the identification method for the axial force (or critical force) and boundary conditions of a beam 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 responses and design variables. The basic idea of response surface methodology is to create explicit approximation functions to the objective and constraints, and then use these when performing the optimization. The approximation functions are typically in the form of low-order polynominals fitted by least squares regression analysis. In order to construct the approximation function, it is necessary to have some results for a minimum number of points in the design space. The proper selection of points could drastically improve the quality of a response surface model. The response at the most suitable points, which are selected by the design of experiments (DOE) could have been obtained either by some analysis program or through physical experiments.

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 shaft is studied. An analysis is presented for the vibration and stability of a tapered shaft simultaneously subjected to a follower force with an axial force by the use of the transfer matrix approach. The method is applied to shafts with varying cross-sections, and the natural frequencies and flutter loads are calculated numerically, to provide information about the effect on them of varying cross-section, the slenderness, the axial force and the crack.

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 L₂₇(3¹³) is sufficient for estimating the critical flutter loads.

1

Anifantis N., Dimarogonas A.D., "Stability and columns with a single crack subjected to follower and vertical loads", Int. J. Solid Struct., 19, pp.281-291, 1983. doi:10.1016/0020-7683(83)90027-6

2

Takahashi I., "Vibration and stability of a cracked shaft simultaneously subjected to follower force with an axial force", Int. J. Solid Struct., 35, pp.3071-3080, 1998. doi:10.1016/S0020-7683(97)00364-8

3

Takahashi I., "Identification for critical force and boundary conditions of a beam using neural networks", J. Sound Vibr, 228, pp.857-870, 1999. doi:10.1006/jsvi.1999.2451

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