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Liao H.,Chinese Aeronautical Establishment
Journal of Computational Physics | Year: 2016

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method. © 2016 Elsevier Inc.. Source


Liao H.,Chinese Aeronautical Establishment
Mathematical Problems in Engineering | Year: 2013

In order to calculate the limit cycle oscillations and bifurcations of nonlinear aeroelastic system, the problem of finding periodic solutions with maximum vibration amplitude is transformed into a nonlinear optimization problem. An algebraic system of equations obtained by the harmonic balance method and the stability condition derived from the Floquet theory are used to construct the general nonlinear equality and inequality constraints. The resulting constrained maximization problem is then solved by using the MultiStart algorithm. Finally, the proposed approach is validated, and the effects of structural parameter uncertainty on the limit cycle oscillations and bifurcations of an airfoil with multiple nonlinearities are studied. Numerical examples show that the coexistence of multiple nonlinearities may lead to low amplitude limit cycle oscillation. © 2013 Haitao Liao. Source


Liao H.,Chinese Aeronautical Establishment
Nonlinear Dynamics | Year: 2016

A novel method is presented to calculate the sensitivity gradients of the largest Lyapunov exponent (LLE) in dynamical systems. After the elimination of the discontinuity of state perturbation vector, the augmented system of differentiation equations is constructed to govern the time evolution of the LLE. To overcome the ill-posed property of the sensitivity problem associated with the augmented differentiation system, the improved least squares shadowing approach is developed. The simple algebraic formula depending on the final state value of the Lagrange multipliers is deduced from the discretization representation for the first-order optimal conditions of the improved least squares shadowing formulation. The LU factorization technique is introduced to solve the set of discretized linear equations, resulting in a better performance of the convergence problem and computational expense. The correctness and effectiveness of the present approaches are validated. © 2016 Springer Science+Business Media Dordrecht Source


A method is proposed to calculate the periodic solutions of piecewise nonlinear systems. The method is based on analytical derivation of nonlinear multi-harmonic equations of motion. Since periodic variations of nonlinear forces are characterized by different states, the vibration cycle is broken into sequential transition intervals according to the instant sets of state transitions. Analytical formulations of the harmonic coefficients of the nonlinear forces and its derivatives with respect to the harmonic coefficients of displacements are developed. Sensitivities of the harmonic coefficients of periodic solutions are determined for constructing explicit expressions for vibration amplitude levels as a function of structural parameters. Numerical investigations of the limit cycle oscillations and its sensitivities of an airfoil with different piecewise nonlinearities have been performed. The results show that the developed method is capable of determining the periodic solutions and its sensitivities with respect to the structural parameters. In order to guarantee time continuity of the nonlinear force, for the hysteresis model it is not right to track the periodic solutions by using the preload or freeplay as the continuation parameters. © 2015 Elsevier Ltd. Source


Liao H.,Chinese Aeronautical Establishment
Mechanics Based Design of Structures and Machines | Year: 2016

The periodic motions of the fractional order and/or delayed nonlinear systems are investigated in the frequency domain using a harmonic balance method with the analytical gradients of the nonlinear quality constraints and the sensitivity information of the Fourier coefficients can also obtained. The properties of fractional order derivatives and trigonometric functions are utilized to construct the fractional order derivatives, delayed and product operational matrices. The operational matrices are used to derive the analytical formulae of nonlinear systems of algebraic equations. The stability of periodic solutions for the delayed nonlinear systems is identified by an eigenvalue analysis of quasi-polynomials characteristic equations. Sensitivity analysis is performed to study the influence of the structural parameters on the system responses. Finally, three numerical examples are presented to illustrate the validity and feasibility of the developed method. It is concluded that the proposed methodology has the potential to facilitate highly efficient optimization, as well as sensitivity and uncertainty analysis of nonlinear systems with fractional derivatives and/or time delayed. 2016 Copyright © Taylor & Francis Group, LLC Source

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