Finding Chebyshev series periodic solutions of nonlinear vibration systems via optimization method

Since shifted Chebyshev series can accurately approximate trigonometric function and Floquet transition matrix, a new method is presented for solving shifted Chebyshev series periodic solution of nonlinear vibration systems via optimization method. In the suggested method system state variables are expanded into the shifted Chebyshev series of the first kind with unknown coefficients. Then solving the unknown coefficients equals to an optimization issue on calculating the residual-force minimum value. It can be used to solve high dimension strongly nonlinear time-varying systems and parametrically excited systems. The accuracy of solutions can be controlled by adjusting optimization initial value, and Floquet transition matrix can be calculated effectively. As illustration examples the Chebyshev series periodic solutions and stability analysis of Duffing system and helicopter rotor coupling motion equation are studied. Compared with the harmonic balance method or time finite element method, the suggested method has a higher accuracy. It indicates that this method is accurate and effective.

[1]  S. Sinha,et al.  An efficient computational scheme for the analysis of periodic systems , 1991 .

[2]  K. Y. Sze,et al.  Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems , 2012 .

[3]  J.-X Xu,et al.  RESEARCH ON THE PERIODIC ORBIT OF NON-LINEAR DYNAMIC SYSTEMS USING CHEBYSHEV POLYNOMIALS , 2001 .

[4]  Ibrahim Çelik,et al.  Collocation method and residual correction using Chebyshev series , 2006, Appl. Math. Comput..

[5]  M. Dehghan,et al.  Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials , 2012 .

[6]  Chung-Jen Lu,et al.  A modified incremental harmonic balance method for rotary periodic motions , 2011 .

[7]  Fabrice Thouverez,et al.  On a new harmonic selection technique for harmonic balance method , 2012 .

[8]  J. H. Tang,et al.  A convolution integral method for certain strongly nonlinear oscillators , 2005 .

[9]  Ibrahim Çelik,et al.  Approximate solution of periodic Sturm-Liouville problems with Chebyshev collocation method , 2005, Appl. Math. Comput..

[10]  S. Sinha,et al.  Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds , 2011 .

[11]  Gareth D. Padfield.  Helicopter Flight Dynamics – an Introductory Tour , 2008 .

[12]  Haitao Ma,et al.  Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials , 2004 .

[13]  X. Xu,et al.  Finding the Periodic Solution of Differential Equation via Solving Optimization Problem , 2009 .

[14]  C. W. Clenshaw.  The numerical solution of linear differential equations in Chebyshev series , 1957, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Eric A. Butcher,et al.  On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations , 2011 .

[16]  S. C. Sinha,et al.  Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations , 1995, Nonlinear Dynamics.

[17]  S. Redkar,et al.  Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations , 2011 .

[18]  Lloyd N. Trefethen,et al.  Barycentric Lagrange Interpolation , 2004, SIAM Rev..

[19]  Firas A. Khasawneh,et al.  A multi-interval Chebyshev collocation approach for the stability of periodic delay systems with discontinuities , 2011 .