A study of the eigenvalue sensitivity by homotopy and perturbation methods

Sliva, G.; Brézillon, Anthony; Cadou, Jean-Marc; Duigou, Laëtitia

doi:10.1016/j.cam.2009.08.086

Cited by 46 publications

(8 citation statements)

References 7 publications

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“…One of the effective methods for the solution of nonlinear eigenvalue problems is the perturbation theory technique based on an artificially introduced small parameter [Andrianov and Awrejcewicz 2013;Andrianov et al 2014;Nayfeh 2000;2011;Sliva et al 2010]. An analytical expression for the eigenvalues of the nonlinear eigenvalue problem (2-7)-(2-8) can be derived by applying the perturbation theory method.…”

Section: Governing Equations and Asymptotic Analysis Mixity Parametermentioning

confidence: 99%

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Степанова

Yakovleva

2015

J. Mech. Mater. Struct.

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The stress-strain state analysis near the crack tip in a power-law material under mixed-mode loading conditions is investigated. By the use of the eigenfunction expansion method the stress-strain state near the mixed-mode crack tip under plane stress conditions is found. The type of the mixed-mode loading is specified by the mixity parameter, which varies from 0 to 1. The value of the mixity parameter corresponding to a mode II crack loading is equal to 0, whereas the value corresponding to a mode I crack loading is equal to 1. It is shown that the eigenfunction expansion method results in a nonlinear eigenvalue problem. The numerical solutions of the nonlinear eigenvalue problems for all values of the mixity parameter and for all practically important values of the strain hardening (or creep) exponent are obtained. It is found that mixed-mode loading of the cracked plate gives rise to a change in the stress singularity in the vicinity of the crack tip. Mixed-mode loading of the cracked plate results in new asymptotics of the stress field, which is different from the classical Hutchinson-Rice-Rosengren (HRR) stress field. The approximate solution of the nonlinear eigenvalue problem is obtained by a perturbation theory technique (artificial small parameter method). In the framework of the perturbation theory approach, a small parameter representing the difference between the eigenvalue of the nonlinear problem and the undisturbed linear problem is introduced. The asymptotic analysis carried out shows clearly that the stress singularity in the vicinity of the crack tip changes under mixed-mode loading in the case of plane stress conditions. The angular distributions of the stress and strain components (eigenfunctions) in the full range of values of the mixity parameter are given. The authors would like to acknowledge the financial support of the Russian foundation of basic research, project 13-01-97009-a.

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Section: Governing Equations and Asymptotic Analysis Mixity Parametermentioning

confidence: 99%

Untitled

Степанова

Yakovleva

2015

J. Mech. Mater. Struct.

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“…Many approaches have been proposed to simplify the formulation, such as proper selection of reference frames [11], generalized coordinates [12][13][14][15], mechanics principles [16,17], and other methods [18,19]. On the other hand, despite sensitivity analysis of SR-MBS based on the conventional method is well documented [20][21][22][23][24], the formulation is quite complicated because the resulting equations are implicit functions of the design parameters. Actually, what people concern, for many kinds of mechanical systems under working conditions, are eigenvalue problems and the relationship between the modal parameters and the design parameters.…”

Section: Introductionmentioning

confidence: 99%

Vibration Calculation of Spatial Multibody Systems Based on Constraint-Topology Transformation

et al. 2011

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Many kinds of mechanical systems can be modeled as spatial rigid multibody systems (SR-MBS), which consist of a set of rigid bodies interconnected by joints, springs and dampers. Vibration calculation of SR-MBS is conventionally conducted by approximately linearizing the nonlinear equations of motion and constraint, which is very complicated and inconvenient for sensitivity analysis. A new algorithm based on constraint-topology transformation is presented to derive the oscillatory differential equations in three steps, that is, vibration equations for free SR-MBS are derived using Lagrangian method at first; then, an open-loop constraint matrix is derived to obtain the vibration equations for open-loop SR-MBS via quadric transformation; finally, a cut-joint constraint matrix is derived to obtain the vibration equations for closed-loop SR-MBS via quadric transformation. Through mentioned above, the vibration calculation can be significantly simplified and the sensitivity analysis can be conducted conveniently. The correctness of the proposed method has been verified by numerical experiments in comparison with the traditional approaches.

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“…According to the perturbation theory and interval algorithms, Qiu et al [10] presented the interval parameter perturbation method to solve the structural mechanical problems, where the interval matrix inverse is calculated by the first-order Neumann series. Resorting to its small computational cost and easily guaranteed convergence condition, the interval perturbation method has been widely applied in engineering problems [11,12]. However, due to the unpredictable effect of neglecting high-order terms in Neumann series, the traditional perturbation method will not be effective for the cases with large uncertainty level.…”

Section: Introductionmentioning

confidence: 99%