The Approximate Analysis Of Eigenvalues For Structures With Interval Parameters

In actual engineering situations, the structural parameters are often uncertain, such as the inaccuracy of the measurement, errors in the manufacturing process, invalidity of some components, etc. Although in most instances, errors or uncertainties are likely to be small. But these errors or uncertainties may combine to make structural response have big deviations or unpredictability, especially in the multi-components system. Therefore, this paper presents a new method to estimate the upper and lower bounds on eigenvalues of structures with interval parameters.In first chapter, the knowledge about dynamic finite element method is introduced. The differential equation of vibration can be derived the Hamilton theory. A brief review of interval mathematical backgrounds is given in Chapter 2. Then a quick and high efficient method for eigensolution reanalysis with Epsilon-algorithm is introduced. The details will be intensively discussed in Chapter 3. In the following chapter, the eigenvalues are considered as the functions of the structural parameters. Using Taylor expansion to expand the eigenvalues on each structural parameter, according to interval extension theory we get the interval eigenvalues affected by all parameters by adding up the all values affected by each parameter. Solving bounds on eigenvalues of the structures affected by every single parameter based on eigensolution reanalysis with Epsilon-algorithm, the bounds on eigenvalues of the structures affected by all parameters are evaluated finally. Epsilon-algorithm in computing process saves a large amount of time which consumes in the dynamic finite element calculations. At the same time, the reanalysis accuracy is guaranteed while the structural parameters have large changes. So this method shows us a satisfying result. In Chapter 5, three examples are given to testify the method in this paper.