Analytical Solutions for Stable and Unstable Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Harmonic Balance ◽  
Closed Form Solution ◽  
Form Solution ◽  
Quadratic Nonlinearity ◽  
Periodic Motions ◽  
Harmonic Term ◽  
Forced Oscillator ◽  
Small Parameters ◽  
Constant Coefficients

In this note, a closed-form solution of periodic motions in a periodically forced oscillator with quadratic nonlinearity is presented without any small parameters. The perturbation method is based on one harmonic term plus perturbation modification, and the traditional harmonic balance is to arbitrarily select harmonic terms with constant coefficients. If harmonic terms are not enough included in the approximate solution, such a solution is not an appropriate, analytical solution for periodic motions, and some analytical solutions cannot be caught.

Analytical Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Series Expansions ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator ◽  
The Stability ◽  
Generalized Harmonic Balance Method

In this paper, period-1 motions in a quadratic nonlinear oscillator under excitation are investigated by the generalized harmonic balance method. The analytical solutions of period-1 motion for such an oscillator are presented by the Fourier series expansions. The stability and bifurcation analysis of period-1 motion is carried out via eigenvalue analysis. To verify the approximate analytical solutions, numerical simulations are performed for a better understanding of the parameter characteristics of the period-1 solutions, and the stable and unstable periodic motions are illustrated. The analytical period-1 solutions are different from the perturbation analysis.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

Analytical Solutions of Shock Fitting Equations for the Multishock Compaction of Die-Contained Powder Media

1992 ◽  
Vol 114 (1) ◽  
pp. 63-70
Author(s):  
Yukio Sano ◽  
Koji Tokushima ◽  
Yuji Inoue ◽  
Yoshihito Tomita
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Specific Volume ◽  
Linear Equations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Material Constants ◽  
The Difference ◽  
Volume Relation ◽  
Compaction Processes

In an earlier paper [4], two sets of equations which governed the processes of propagation of shock waves reflected from the punch and plug surfaces in a die-contained copper powder medium were presented. The pressure-specific volume relation included in the sets of equations was composed of three partial relations having different material constants. In the present paper the sets of equations are simplified by assuming that the pressure and specific volume at the front and back sides of the shock front are always related by the same material constants, and linear equations are obtained by introducing a further minor assumption into the simplified nonlinear equations included in the sets of equations. Two sorts of analytical solutions of the linear equations are obtained. One is a general-form solution, while the other is a closed-form solution. The general-form solution calculated is compared satisfactorily with the difference solution computed in the previous study, confirming that the assumption introduced into the simplified equations is minor. Furthermore, calculated characteristics of the general-form solution are revealed by the consideration of the simplified equations and the linear equations, giving greater insight into the compaction processes. The closed-form solution, which is obtained only for the propagation of the shock wave starting from the punch surface and returning from the plug surface, agrees well with the general-form solution.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

A Dynamic Model of the Deformation of a Diamond Mesh Cod-End of a Trawl Net

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
F. G. O’Neill ◽  
R. D. Neilson
Keyword(s):  
Dynamic Model ◽  
Harmonic Balance ◽  
Second Harmonic ◽  
Closed Form Solution ◽  
Harmonic Balance Method ◽  
Form Solution ◽  
Body Motion ◽  
Harmonic Series ◽  
Pressure Loading ◽  
Dependent Variables

A dynamic model of a diamond mesh cod-end subject to harmonic forcing is developed. The partial differential equations governing the displacements of the cod-end and the tension in the twine are first derived and then analyzed using the harmonic balance method by substituting a harmonic series for the dependent variables and the forcing term. A closed-form solution is derived for the case of rigid-body motion, where there is no deformation of the cod-end geometry, along with the conditions for the forcing under which this motion occurs. A pressure loading, which varies linearly over a portion of the cod-end and varies harmonically with time, is then introduced as a first representation of the loading on the cod-end that results from the pressure and acceleration forces on the catch due to surge motion of the towing vessel. The resulting sets of equations for the static and the first and second harmonic terms are solved numerically in a sequential manner, and the results presented for a number of cases. These results show that, due to the nonlinearity of the system, the oscillatory motion of the cod-end is asymmetric, and that the deformation of the net and the amplitude of oscillation increases as the region over which the forcing is applied increases. The model is the basis for a more complete coupled catch/cod-end model.

Analytical Solutions for Heat Transfer During Cyclic Melting and Freezing of a Phase Change Material Used in Electronic or Electrical Packaging

2003 ◽  
Vol 125 (1) ◽  
pp. 126-133 ◽  
Author(s):  
Suman Chakraborty ◽  
Pradip Dutta
Keyword(s):  
Heat Transfer ◽  
Phase Change ◽  
Phase Change Material ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Electronics Cooling ◽  
Form Solution ◽  
Transfer Model ◽  
Change Material

In this paper, we develop an analytical heat transfer model, which is capable of analyzing cyclic melting and solidification processes of a phase change material used in the context of electronics cooling systems. The model is essentially based on conduction heat transfer, with treatments for convection and radiation embedded inside. The whole solution domain is first divided into two main sub-domains, namely, the melting sub-domain and the solidification sub-domain. Each sub-domain is then analyzed for a number of temporal regimes. Accordingly, analytical solutions for temperature distribution within each sub-domain are formulated either using a semi-infinity consideration, or employing a method of quasi-steady state, depending on the applicability. The solution modules are subsequently united, leading to a closed-form solution for the entire problem. The analytical solutions are then compared with experimental and numerical solutions for a benchmark problem quoted in the literature, and excellent agreements can be observed.

Analytical Periodic Motions of a Parametric Oscillator With Quadratic Nonlinearity

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Parametric Oscillator ◽  
Quadratic Nonlinearity ◽  
Fourier Series Expansion ◽  
Periodic Motions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical solutions of periodic motions in a parametric oscillator are presented by the finite Fourier series expansion, and the stability and bifurcation analysis of periodic motions are performed. Numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum.

scholarly journals Fourier Transform of the Continuous Arithmetic Asian Options PDE

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Zieneb Ali Elshegmani ◽  
Rokiah Rozita Ahmad
Keyword(s):  
Fourier Transform ◽  
Direct Method ◽  
Option Price ◽  
Closed Form Solution ◽  
Form Solution ◽  
Three Dimensions ◽  
Asian Option ◽  
Asian Options ◽  
Partial Differential ◽  
Constant Coefficients

Price of the arithmetic Asian options is not known in a closed-form solution, since arithmetic Asian option PDE is a degenerate partial differential equation in three dimensions. In this work we provide a new method for computing the continuous arithmetic Asian option price by means of partial differential equations. Using Fourier transform and changing some variables of the PDE we get a new direct method for solving the governing PDE without reducing the dimensionality of the PDE as most authors have done. We transform the second-order PDE with nonconstant coefficients to the first order with constant coefficients, which can be solved analytically.

Sign in / Sign up

Export Citation Format

Share Document