Non-Stationary Oscillations at Slow Transitions Across Instabilities

2007 ◽  
Author(s):  
Rudolf R. Pusˇenjak ◽  
Maks M. Oblak ◽  
Jurij Avsec
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Time Scales ◽  
Primary Resonance ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Excitation Parameter ◽  
Linear Dynamical Systems ◽  
Weakly Nonlinear ◽  
Non Linear

The paper presents the study of non-stationary oscillations, which is based on extension of Lindstedt-Poincare (EL-P) method with multiple time scales for non-linear dynamical systems with cubic non-linearities. The generalization of the method is presented to discover the passage of weakly nonlinear systems through the resonance as a control or excitation parameter varies slowly across points of instabilities corresponding to the appearance of bifurcations. The method is applied to obtain non-stationary resonance curves of transition across points of instabilities during the passage through primary resonance of harmonically excited oscillators of Duffing type.

On the Equivalence of Normal Form Theory and Multiple Time Scale Method

2007 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Perturbation Scheme ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales are introduced that serve as the independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear systems as simple as possible. The “simplest” differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the equivalence of these two methods for constructing periodic solutions is proven, and it is explained why some studies have found the results obtained by the two techniques to be inconsistent.

On the Formal Equivalence of Normal Form Theory and the Method of Multiple Time Scales

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Differential Equations ◽  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Normal Form Theory ◽  
Weakly Nonlinear ◽  
Formal Equivalence ◽  
Form Theory

Multiple time scales technique has long been an important method for the analysis of weakly nonlinear systems. In this technique, a set of multiple time scales is introduced that serve as independent variables. The evolution of state variables at slower time scales is then determined so as to make the expansions for solutions in a perturbation scheme uniform in natural and slower times. Normal form theory has also recently been used to approximate the dynamics of weakly nonlinear systems. This theory provides a way of finding a coordinate system in which the dynamical system takes the “simplest” form. This is achieved by constructing a series of near-identity nonlinear transformations that make the nonlinear terms as simple as possible. The simplest differential equations obtained by the normal form theory are topologically equivalent to the original systems. Both methods can be interpreted as nonlinear perturbations of linear differential equations. In this work, the formal equivalence of these two methods for constructing periodic solutions and amplitude evolution equations is proven for autonomous as well as harmonically excited nonlinear vibratory dynamical systems. The reasons as to why some studies have found the results obtained by the two techniques to be inconsistent are also pointed out.

Extended Lindstedt-Poincare Method With Multiple Time Scales for Nonstationary Oscillations

2006 ◽  
Author(s):  
Rudi R. Pusenjak ◽  
Jurij Avsec ◽  
Maks M. Oblak
Keyword(s):  
Dynamical Systems ◽  
Time Scales ◽  
Multiple Scales ◽  
Duffing Oscillator ◽  
Excitation Frequency ◽  
Van Der Pol Oscillator ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Response Curves ◽  
Nonstationary Oscillations

The paper presents the extended Lindstedt-Poincare (ELP) method, which applies multiple time scales to treat nonstationary oscillations arising in dynamical systems with cubic non-linearities. The passage through the resonance is conducted to study deviations from the stationary response. The method is applied to the dynamical systems such as Duffing oscillator and van der Pol oscillator, whereat effects of varying the excitation frequency and varying the excitation amplitudes, respectively are studied. It is shown that application of multiple scales benefits to find more accurate expressions of stationary responses in comparison to the conventional Lindstedt-Poincare method and consequently contributes to the versatile and effective calculation of the nonstationary frequency response curves.

Low- and High-Temperature Primary Resonance in Shape Memory Oscillator Observed by Multiple Time Scales and Harmonic Balance Method

2019 ◽  
Vol 14 (11) ◽  
Author(s):  
Andrzej Weremczuk ◽  
Joanna Rekas ◽  
Rafal Rusinek
Keyword(s):  
Shape Memory ◽  
Time Scales ◽  
Harmonic Balance ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Multiple Time Scales ◽  
Practical Implementation ◽  
Multiple Time ◽  
Fifth Order

Abstract This paper focuses on the primary resonance of a one degree-of-freedom (1DOF) oscillator with a spring made of shape memory alloy (SMA). The primary resonance is analyzed using the multiple time scales method (MTSM) and the harmonic balance method (HBM). The shape memory spring is described by a fifth-order polynomial function. The solutions are analyzed along with the results reported by another authors, and compared with numerical simulations. Three ranges of temperature are analyzed. Finally, the practical implementation aspect of the harmonic balance and MTSMs are discussed.

On the Vibrations of a Linear, and a Weakly Nonlinear Damped System

2001 ◽  
Author(s):  
Darmawijo Yò ◽  
W. T. van Horssen
Keyword(s):  
Time Scales ◽  
Transmission Lines ◽  
Power Transmission ◽  
Flexible Structures ◽  
Initial Boundary ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Power Transmission Lines ◽  
Weakly Nonlinear ◽  
Damped System

Abstract In this paper initial-boundary value problems for a linear, and a weakly nonlinear string (or wave) equation are considered. One end of the string is assumed to be fixed and the other end of the string is attached to a spring-mass-dashpot system, where the damping generated by the dashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead power transmission lines. For a linear problem a semigroup approach will be used to show the well-posedness of the problem as well as the asymptotic validity of formal approximations of the solution on long time-scales. It is also shown hov a multiple time-scales perturbation method as described in Kevorkian and Cole (Kevorkian and Cole, 1981) can be used effectively to construct asymptotic approximations of the solution on long timescales.

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