| In practical engineering problems, the theories of the design and analysis of structures are always established on the basis of the definite mathematics models. However, there are always uncertain factors in the structural engineering practices, such as the inaccuracy of the measurement, the complexity of the structures or errors in manufacture, etc. When the structures are large and complex, the combination of the uncertainty can have some effect on the systems. Therefore, it is necessary to design and analyze the structures with uncertain models directly.The uncertainty can be described as following three kinds: 1.physical uncertainty. It is correlated to load, material and geometrical size. Generally speaking, the change of working conditions and the errors in manufacturing or installation can bring this kind of uncertainty. 2.statistical uncertainty. Nowadays, probability statistical methods are used in solving uncertainty. But because there is no enough statistic information, samples cannot represent all the system information. 3.model uncertainty. During structural analysis and design, the models are constructed between the inputs, including load, geometric size and plastic modules, and the outputs, the displacement, stress and strain. Even beside the physical uncertainty, there are uncertainties in modeling such as the theoretical simplifying and unknown boundary conditions.There are several mathematical models in present researches or applications about uncertain structural analysis, such as stochastic model, fuzzy model, convex model and interval one etc. Due to the diversity and complexity of the uncertain parameters, it is impossible to apply one uncertain model to all kinds of problems. We have to select the better ones for each specific problem. For instance, although the stochastic one has been developed well, it is difficult to use in the following two cases: (1) no more data can be given to obtain the statistical character; (2) the parameters disagree the random mechanism. For the fuzzy and convex models, there also is certain subjective information limited.More attention has been given to the interval model since the interval mathematics appears, because it needs only the bounds of the structural parameters, which can be always obtained in the engineering.The vibration control problems are very important in engineering and the deterministic one has been well developed. The closed-loop control systems are defined as following: the real-time control is applied to the controlled objects according the state of the objects to obtain the requirement in advance. Because of the uncertainties of the structures, the research of the robustness has important positions in controlling engineering. The extensive engineering background of the robustness question is already fully approved.Using the interval analysis, the vibration control problem of structures with interval parameters is discussed,which is approximated by a deterministic one. The methods to solve the interval eigenvalues and dynamics responses of the closed-loop system are presented respectively. The expressions of the interval stiffness and interval mass matrix are developed directly from the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of the interval complex eigenvalue and response are developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The numerical results show that the present methods are valid.First, we briefly introduce the theory of interval mathematics and control question as the base of solving questions of closed-loop systems. Then we give the general form of the vibration control problem based on the 2nd systems. Using the pole allocation method, the modal gains are obtained to guarantee the asymmetric stability for the approximate deterministic system. The expressions o...
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