Identification technique for nonlinear boundary conditions of a circular plate
Introduction
To analyse the dynamic behaviour of machines and structures, numerical methods such as finite element method are often used. Such numerical methods, however, do not always yield accurate results. One of the reasons for this is its difficulty to specify the actual boundary conditions accurately.
As a means to grasp the characteristics of boundaries, experimental identification techniques have attracted interests of engineers, and many techniques have been proposed. Among them are techniques by Zhao et al. [1], Ren and Beards [2], [3], Ahmadian et al. [4], Pabst and Hagedorn [5], Xiang et al. [6], Takahashi [7] and Sanayei et al. [8], and Zhu and Huang [9]. All these techniques, however, have a restriction that they can be applied only to the case in which boundary conditions are linear. In practical cases, the boundary conditions often become nonlinear due to clearance, friction, material properties and so on. Thus, techniques applicable to nonlinear boundary conditions are desired. Sato et al. [10] proposed an identification technique for nonlinear boundary conditions. This technique, however, requires data measured on the boundaries. There are many cases in which measurement of data on the boundaries is difficult.
In previous papers [11], [12], [13], [14], [15], with the aim of developing an identification technique which is applicable to nonlinear boundary conditions and does not require data measured on the boundaries, the authors proposed techniques for one-dimensional structures. In this paper, as a continuation to the previous study, a technique applicable to two-dimensional structure is developed. Here a circular plate is considered.
In the first part of this paper, as a preparation, analytical solution for a circular plate with nonlinear boundary conditions is derived. Then an identification technique is proposed. In the technique, the boundary is modelled by springs and dampers. Depending on the case, effective mass and moment of inertia of the boundary are included in the model. Then their characteristics are determined by using the analytical solution together with the experimental data. Since the technique is based on the analytical solution, it is applicable to any structure, provided that its analytical solution can be derived. Finally, numerical simulation is conducted to discuss the applicability of the technique. In the simulation, responses of a circular plate are obtained from the equations of motion numerically, and random numbers are added to them as noise. With these data being regarded as experimental, identification of the boundary conditions is performed. In this way, it is shown that the proposed technique yields accurate results.
Section snippets
Formulation of the problem
A problem of identification of nonlinear boundary conditions for a linear circular plate is considered. Dynamic properties of the plate, except the boundary conditions are assumed to be known. In practice, it is often difficult to apply excitation to, or to measure responses on the boundaries. So, in developing a technique, it is imposed that the excitation to and measurement on the boundary are not required.
To formulate the problem, the boundary is modelled, as shown in Fig. 1, by springs and
Numerical simulation
To show applicability of the proposed technique, numerical simulation is conducted. First, dynamic response is calculated by solving the equations of motion of a plate whose parameters are given appropriately. Then using the data thus obtained, identification is performed. Finally, the obtained results are compared to the original values of the parameters used for calculation of the response.
Conclusions
As a basic study for developing an experimental identification technique applicable to nonlinear boundary conditions of a two-dimensional structure, a circular plate has been considered. A technique for it has been proposed. In the proposed technique, the boundary is modelled by springs, dampers, effective masses and moments of inertia. Their characteristics are determined by use of the data of steady-state deflection. By numerical simulation, it has been confirmed that the proposed technique
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