Predicting stochastic characteristics of generalized eigenvalues via a novel sensitivity-based probability density evolution method |
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| Institution: | 1. Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China;2. College of Mathematics, Sichuan University, Chengdu 610043, China;1. Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia;2. Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt;3. Department of Mathematics, College of Science and Humanities at Howtat Sudair, Majmaah University, Majmaah 11952, Saudi Arabia;4. Department of Mathematics, Faculty of Science, University of Tabuk, P.O.Box 741, Tabuk 71491, Saudi Arabia;5. Institute of Engineering, Polytechnic of Porto, Rua Dr. António Bernardino de Almeida, 431, Porto 4249-015, Portugal;1. College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 200016, China;2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China;3. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China |
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| Abstract: | This paper proposes a novel numerical method for predicting the probability density function of generalized eigenvalues in the mechanical vibration system with consideration of uncertainties in structural parameters. The eigenproblem of structural vibration is presented by first and the sensitivity of generalized eigenvalues with respect to structural parameters can be derived. The probability density evolution method is then developed to capture the probability density function of generalized eigenvalues considering uncertain material properties. Within the proposed method, the probability density evolution equation for the generalized eigenvalue problem is established accounting for the sensitivity of generalized eigenvalues with respect to structural parameters. A new variable which connects generalized eigenvalues to structural parameters is then introduced to simplify the original probability density evolution equation. Next, the simplified probability density evolution equation is solved by using the finite difference method with total variation diminishing schemes. Finally, the probability density function as well as the second-order statistical quantities of generalized eigenvalues can be predicted. Numerical examples demonstrate that the proposed method yields results consistent with Monte-Carlo simulation method within significantly less computation time and the coefficients of variation of uncertain parameters as well as the total number of them have remarkable effects on stochastic characteristics of generalized eigenvalues. |
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