### Nonlinear Mathematics for Uncertainty and its Applications

Considering the governing equation of motion for the Duffing-harmonic oscillator:. Homotopy analysis is a general analytic method for solving the non-linear differential equations. The HAM transforms a non-linear problem into an infinite number of linear problems with embedding an auxiliary parameter q that typically ranges from zero to one. As q increases from 0 to 1, the solution varies from the initial guess to the exact solution.

By suitable choice of the auxiliary parameter q , we can obtain reasonable solutions for large modulus. This method is a strong and easy-to-use analytic tool for investigating nonlinear problems, which does not need small parameters. In , Liao employed the basic ideas of homotopy in topology to propose a general analytic method for nonlinear problems, namely homotopy analysis method HAM [].

The basic idea of HAM is introduced and then its application in nonlinear vibration is studied. To illustrate the basic ideas of the HAM, consider the following non-linear differential equation:. Where N is a nonlinear operator, t denotes the independent variable and u t is an unknown variable. The homotopy function is constructed as follows:. The parameter L denotes an auxiliary linear operator. The m-order deformation equation can be calculated by differentiating Eqs. Where the m -1 , m -1 and R m m -1 , m -1 are defined as follows:.

The zero-order deformation equation can be written as below:. The same result was obtained in the first example of section 2. Solving Eqs. It has reviewed new asymptotic methodologies throughout numerous examples. The analytical solutions yield a thoughtful and insightful understanding of the effect of system parameters and initial conditions. Also, Analytical solutions give a reference frame for the verification and validation of other numerical approaches.

The most significant feature of those methods is their excellent accuracy for the whole range of oscillation amplitude values. Also, it can be used to solve other conservative truly nonlinear oscillators with complex nonlinearities. The solutions are quickly convergent and its components can be simply calculated. Also, compared to other analytical methods, it can be observed that the results of those methods require smaller computational effort and only the one iteration leads to accurate solutions.

The successful implementations of the mentioned methods for the large amplitude nonlinear oscillation problem were considered in this review. All reviewed methods can be applied to various kinds of weak and strong nonlinear problems, and the examples studied in this review can be utilized as paradigms for oscillator problems.

Through nonlinear oscillators, all the reviewed methods yield high accurate approximate periods which indicated above. Abdollahzade, M. Bayat, and M. Application of two analytic approximate solutions to an oscillation of a mass attached to a stretched elastic wire. Acton and P. Solving Equations with Physical Understanding. Adam Hilger Ltd. Agrwal and H. Weighted linearization technique for period approximation in large amplitude non-linear oscillations.

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Application of he's variational approach method for periodic solution of strongly nonlinear oscillation problems. Amore and A. Andrianov, J. Awrejcewicz, and V. Analysis of natural in-plane vibration of rectangular plates using homotopy perturbation approach. Mathematical Problems in Engineering , , Awrejcewicz, and L.

Asymptotical mechanics of thin-walled structures. Springer Verlag, Aziz and V. Perturbation techniques in phase change heat transfer. Applied Mechanics Reviews , , Azrar, R. Benamar, and R. Semi-analytical approach to the non-linear dynamic response problem of ss and cc beams at large vibration amplitudes part i: General theory and application to the single mode approach to free and forced vibration analysis.

Babazadeh, G. Domairry, A. Barari, R. Azami, and A. Numerical analysis of strongly nonlinear oscillation systems using he's max-min method. Bayat and G. Analysis of the steel braced frames equipped with adas devices under the far field records. On the effect of the near field records on the steel braced frames equipped with energy dissipating devices. Bayat, G. Abdollahzadeh, and M. Analytical solutions for free vibrations of a mass grounded by linear and nonlinear springs in series using energy balance method and homotopy perturbation method.

Bayat, A. Barari, and M. Dynamic response of axially loaded euler-bernoulli beams. Bayat, M. An analytical approach on a mass grounded by linear and nonlinear springs in series. Bayat and I. Application of he's energy balance method for nonlinear vibration of thin circular sector cylinder. Bayat, I. Pakar, and M. Analytical study on the vibration frequencies of tapered beams. Analytical periodic solution for solving nonlinear vibration equations. Tehnicki Vjesnik , Application of he's energy balance method for pendulum attached to rolling wheels that are restrained by a spring.

Analysis of nonlinear vibration of coupled systems with cubic nonlinearity. Shahidi, A. Barari, and G. The approximate analysis of nonlinear behavior of structure under harmonic loading. Analytical evaluation of the nonlinear vibration of coupled oscillator systems. Shahidi, and M. Application of iteration perturbation method for nonlinear oscillators with discontinuities. Neipp, A. Hernndez, and M. Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method.

Hernandez, T. Neipp, and A. Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by he's homotopy perturbation method. Pascual, T. Solution for an anti-symmetric quadratic nonlinear oscillator by a modified he's homotopy perturbation method. Pascual, E. Fernndez, C. Neipp, and T. Higher-order approximate solutions to the relativistic and duffing-harmonic oscillators by modified he's homotopy methods. Physica Scripta , 77 , Pascual, S. Gallego, M. Ortuno, and C. Pascual, M. Ortuno, T. Application of a modified he's homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities.

Bender, K. Milton, S. Pinsky, and Jr. A new perturbative approach to nonlinear problems. Journal of Mathematical Physics , , Biazar and H. Variational iteration method-a kind of non-linear analytical technique: some examples. Cai and J. Application of the modified frequency formulation to a nonlinear oscillator. Cai, W. Wu, and M. Approximate period solution for a kind of nonlinear oscillator by he's perturbation method.

Cai and W. He's frequency formulation for the relativistic harmonic oscillator. Chan, K. Chung, and Z. A perturbation-incremental method for strongly non-linear oscillators. Chen and J. Homotopy analysis method for limit cycle flutter of airfoils. Homotopy analysis method for limit cycle oscillations of an airfoil with cubic nonlinearities.

Cheung, S. Chen, and S. Vibrations of a coupled two-degree-of-freedom system. The motion of a two-mass system with non-linear connection. Cveticanin and I. Parametrically excited vibrations of an oscillator with strong cubic negative nonlin earity. Darvishi, A. Karami, and B. Application of he's parameter-expansion method for oscillators with smooth odd nonlinearities.

Das and P. Application of homotopy perturbation method and homotopy analysis method to fractional vibration equation. Dautray, J. Lions, M. Artola, and M. Mathematical analysis and numerical methods for science and technology: Spectral theory and applications , volume 3. Dehghan, J. Manafian, and A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Ding and L. Applying he's parameterized perturbation method for solving differential-difference equation.

Domairry and H. An approximation of the analytic solution of some nonlinear heat transfer equations: A survey by using homotopy analysis method. He's frequency-amplitude formulation for the duffing harmonic oscillator. Feng and L. Homotopy analysis approach to duffing-harmonic oscillator. Francis, I. Morse, and R. Concrete damage evolution analysis by backscattered ultrasonic waves. Prentice-Hall of Japan, Fu, J. Zhang, and L. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system mems.

Afrouzi, D. Ganji, and R. He's energy balance method for nonlinear oscillators with discontinuities. Ganji, G. Afrouzi, and R. Application of he's variational iteration method for solving the reaction-diffusion equation with ecological parameters. Ganji, M. Gorji, S.

Soleimani, and M. Solution of nonlinear cubic-quintic duffing oscillators using he's energy balance method. Ganji, N. Jamshidi, and Z. Hpm and vim methods for finding the exact solutions of the nonlin ear dispersive equations and seventh-order sawada-kotera equation. Ganji, S. Karimpour, and S. Approximate analytical solutions to nonlinear oscillations of non-natural systems using he's energy balance method.

He's iteration perturbation method to nonlinear oscillations of me chanical systems with single-degree-of freedom. Nourollahi, and M. A comparison of variational iteration method with adomian's decomposition method in some highly nonlinear equations. Ganji and A. Application of he's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations.

Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. Ganji, D. Ganji, Z. Ganji, and S. Periodic solution for strongly nonlinear vibration systems by he's energy balance method. Determination of the frequency-amplitude relation for nonlinear oscillators with fractional potential using he's energy balance method. He's energy balance and he's variational methods for nonlinear oscillations in engineering.

## Sensitivity analysis - Wikipedia

Karimpour, and H. Applications of he's homotopy perturbation method to obtain second-order approximations of the coupled two-degree-of-freedom systems. Sfahani, S. Tonekaboni, A. Moosavi, and D. Higher-order solutions of coupled systems using the parameter expansion method. Mathematical Problems in Engineering , 20, Barari, and D.

Approximate analysis of two-mass-spring systems and buckling of a column. Ganji, H. Babazadeh, and S. Variational approach method for nonlinear oscillations of the motion of a rigid rod rocking back and cubic. Babazadeh, and N. Application of amplitude-frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back. Davodi, and S. Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach. Geng and X. He's frequency formulation for nonlinear oscillators.

European Journal of Physics , , Visco-elastic mhd flow of walters liquid b fluid and heat transfer over a non-isothermal stretching sheet.

## Nonlinear Mathematics for Uncertainty and its Applications

Ghorbani and J. An effective modification of he's variational iteration method. Mechanical and structural vibrations: theory and applications. Free vibrations of beams and shafts. Hamdan and N. On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass. Some new approaches to duffing equation with strongly and high order nonlinearity ii parametrized perturbation technique.

Homotopy perturbation technique. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. A new perturbation technique which is also valid for large parameters. Bookkeeping parameter in perturbation methods. Iteration perturbation method for strongly nonlinear oscillations.

## Nonlinear Mathematics for Uncertainty and Its Applications

Journal of Vibration and Control , 7 5 , Modified lindsted-poincare methods for some strongly nonlinear oscillations part iii: double series expansion. Modified lindstedt-poincare methods for some strongly non-linear oscillations: Part i: expansion of a constant. Preliminary report on the energy balance for nonlinear oscillations.

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Rebuild of king fang 40 bc musical scales by he's inequality. Introduction to perturbation methods. Springer, Hoseini, T. Pirbodaghi, M. Asghari, G. Farrahi, and M. Nonlinear free vibration of conservative oscillators with inertia and static type cubic nonlinearities using homotopy analysis method. Jalaal, E.

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Ghasemi, D. Bararnia, S. Soleimani, G. Nejad, and M. Effect of temperature-dependency of surface emissivity on heat transfer using the parameterized perturbation method. Thermal Science Suppl. Jamshidi and D. Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire. Kachapi, D. Periodic solution for strongly nonlinear vibration systems by he's variational iteration method.

Kaya and S. Altay Demirbag. Application of parameter expansion method to the generalized nonlinear discontinuity equation. Higher-order approximate periodic solutions of a nonlinear oscillator with discontinuity by variational approach. Mathematical Problems in Engineering , Khah and D.

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A study on the motion of a rigid rod rocking back and cubic-quintic duffing oscillators by using he's energy balance method. Application of he's variational approach method for strongly nonlinear oscillators. VAM , 2 2 :1, Khaleghi, D. Ganji, and A. Application of variational iteration and homotopy-perturbation methods to nonlinear heat transfer equations with variable coefficients.

Khan and Q. Homotopy perturbation transform method for nonlinear equations using he's polynomials. Khan, Q. Wu, H. Askari, Z. Saadatnia, and M. Nonlinear vibration analysis of a rigid rod on a circular surface via hamiltonian approach. Korenev and L. Dynamic vibration absorbers: theory and technical applications. Wiley London, Kutanaei, E. Ghasemi, and M. Mesh-free modeling of two-dimensional heat conduction between eccentric circular cylinders. Lai and C. Nonlinear vibration of a two-mass system with nonlinear stiffnesses. Homotopy analysis method and its application. Beyond perturbation: introduction to the homotopy analysis method , volume 2.

CRC Press, On the homotopy analysis method for nonlinear problems. Liao and A. Application of homotopy analysis method in nonlinear oscillations. Journal of applied mechanics , , Lim, B. Wu, and W. Higher accuracy analytical approximations to the duffing-harmonic oscillator. Lin and G. Probabilistic structural dynamics: advanced theory and applications.

McGraw-Hill Professional, Approximate period of nonlinear oscillators with discontinuities by modified lindstedt-poincare method. He's variational approach for nonlinear oscillators with high nonlinearity. He's variational iteration method for the modified equal width equation. Statistical energy analysis of dynamical systems: theory and applications. Marinca and N. A modified iteration perturbation method for some nonlinear oscillation problems. Periodic solutions for some strongly nonlinear oscillations by he's variational iteration method.

Nonlinear free vibration analysis of tapered beams by hamiltonian approach. Mehdipour, D. Ganji, and M. Application of the energy balance method to nonlinear vibrating equations. Momeni, N. Jamshidi, A. Application of he's energy balance method to duffing-harmonic oscillators. Naghipour, D. Hashemi, and H. Analysis of nonlinear oscillation systems using He's variational approach. IOP Publishing, Perturbation methods , volume 6. Wiley Online Library, Nayfeh and D. Nonlinear oscillations , volume Introduction to singular perturbations. Applied Mathematics and Mechanics , 14, DTIC Document.

Ozis and A. A note on he's homotopy perturbation method for van der pol oscillator with very strong nonlinearity. Generating the periodic solutions for forcing van der pol oscillators by the iteration perturbation method.

Pakar and M. Analytical solution for strongly nonlinear oscillation systems using energy balance method. Analytical study on the non-linear vibration of euler-bernoulli beams. Pakar, M. Analytical evaluation of the nonlinear vibration of a solid circular sector object. On the approximate analytical solution for parametrically nonlinear excited oscillators. Shahidi, D. Approximate analytical solutions for nonnatural and nonlinear vibration systems using he's variational approach method.

Ahmadian, and M. On the homotopy analysis method for non-linear vibration of beams. Mechanics Research Communications , Pirbodaghi and S. Nonlinear free vibration of a symmetrically conservative two-mass system with cubic nonlinearity. Journal of Computational and Nonlinear Dynamics , Qian and S. Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der pol duffing oscillators by homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation , Qian, S. Lai, W. Zhang, and Y. Study on asymptotic analytical solutions using ham for strongly nonlinear vibrations of a restrained cantilever beam with an intermediate lumped mass.

Qian, W. Zhang, B. Lin, and S. Analytical approximate periodic solutions for two-degree-of-freedom coupled van der pol-duffing oscillators by extended homotopy analysis method. Qiu and X. Parameter perturbation method for dynamic responses of structures with uncertain-but bounded parameters based on interval analysis. Mechanical Vibrations 3rd edition ed. Addison Wesley, Ren and W. He's frequency formulation for nonlinear oscillators using a golden mean location. Ren, G. Liu, Y.

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Kang, H. Fan, H. Li, X. Ren, and W. Application of he's amplitude frequency formulation to nonlinear oscillators with discontinuities. Physica Scripta , , Younesian H. Askari Z. Saadatnia and M. Frequency analysis of strongly nonlinear generalized duffing oscillators using he's frequency-amplitude formulation and he's energy balance method. Sadighi and D. Solution of the generalized nonlinear boussinesq equation using homotopy perturbation and variational iteration methods.

Exact solutions of nonlinear diffusion equations by variational iteration method. Shaban, D. Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems. Shahidi, M. Pakar, and G. On the solution of free non-linear vibration of beams. Shakeri and M. Numerical solution of a biological population model using he's variational iteration method. Shen and L. The max-min approach to a relativistic equation.

Homotopy analysis method: A new analytic method for nonlinear problems. Applied Mathematics and Mechanics , Variational approach to the nonlinear oscillator of a mass attached to a stretched wire. The homotopy perturbation method for nonlinear oscillators. Variational approach for nonlinear oscillators with discontinuities. Siddiqui, T.

Haroon, S. Bhatti, and A. A comparison of the adomian and homotopy perturbation methods in solving the problem of squeezing flow between two circular plates. Slota and A. A new application of he's variational iteration method for the solution of the one-phase stefan problem. Soleimani, A. Ebrahimnejad, M. Esmaeilpour, D. Energy balance method to sub harmonic resonances of nonlinear oscillations with parametric excitation. Soltanian, S. Karbassi, and M. Application of he's variational iteration method for solution of differential-algebraic equations.

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Mechanical vibrations. Instead of only one surrogate model, multiple models are constructed with the clustered collocation points and combined based on the Dirichlet process mixture model, in order to improve the accuracy of polynomial chaos expansion. The new method is applied to two numerical examples, and the accuracy is comparable to the Monte Carlo sampling. The application of new quadrature rules to compute the generalized polynomial chaos expansion coefficients accurately reconstructs the output statistics by propagating the input uncertainty through a computational physics model while using far fewer quadrature nodes than with traditional methods.

The evolution of probability densities for state variables can be estimated and is demonstrated with examples of harmonic oscillators and two-degree-of-freedom airfoil oscillation. Compared to traditional Monte Carlo sampling, the described pseudospectral stochastic collocation framework can significantly reduce the number of samples and save simulation time. Based on the constructed dynamic Kriging surrogate models, the inverse reliability analysis is carried out using Monte Carlo simulations.

The proposed procedure is demonstrated in the generation of stochastic speed-made-good maps for a prototype wheeled vehicle platform over a significantly large geographical area. Using a method for approximating a field of scattered points in space, based on radial basis functions, this automated approach minimizes expert user input and facilitates the inclusion of uncertainties and variable geometries in CAE-based analysis tools, as demonstrated in the problem of reconstruction of the wet surface of a manufactured wind tunnel model.

This study quantifies the second order statistics for the output measures of a complex microswitch model that includes flexibility and an asperity-based contact model with parameters of material and geometric properties, actuation voltage, and tip asperity contact parameters. Sign In or Create an Account. Sign In. Advanced Search. Article Navigation. Guest Editorial. This Site. Google Scholar. Yan Wang Yan Wang. Kyung K. Choi Kyung K. Paramsothy Jayakumar Paramsothy Jayakumar. Author and Article Information. Radu Serban. Feb , 14 2 : 2 pages.

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