Showing papers in "Journal of Applied Nonlinear Dynamics in 2012"
TL;DR: In this article, the analytical solutions of period-m motions are presented by the Fourier series, and the corresponding Hopf bifurcation of periodic motions leads to new periodic motions with period doubling.
Abstract: In this paper, analytical routes of period-1 motions to chaos in the Duffing oscillator with a twin-potential well are investigated through the generalized harmonic balance method. The analytical solutions of period-m motions are presented by the Fourier series, and the corresponding Hopf bifurcation of periodic motions leads to new periodic motions with period-doubling. Three analytical routes of asymmetric period-1 motions to chaos are presented comprehensively. To verify approximate, analytical periodic solutions, numerical simulations are carried out. In the analytical routes, the unstable periodic motions are presented, and such analytical routes with unstable periodic motions can help one find unstable chaos. Such unstable chaos cannot be obtained simply via the time going to infinity (i.e., t→ꝏ)
49 citations
TL;DR: In this article, free vibration and flutter analyses of two types of high aspect ratio aircraft wings are presented, where the wing is idealised as an assembly of bending-torsion coupled beams using the dynamic stiffness method leading to a nonlinear eigenvalue problem.
Abstract: Free vibration and flutter analyses of two types of high aspect ratio aircraft wings are presented. The wing is idealised as an assembly of bending-torsion coupled beams using the dynamic stiffness method leading to a nonlinear eigenvalue problem. This problem is solved using the Wattrick-Williams algorithm yielding natural frequencies and mode shapes. The flutter analysis is carried out using the normal mode method in conjunction with generalised coordinates and two-dimensional unsteady aerodynamic theory of Theodorsen. This is essentially a complex eigenvalue problem in terms of both air-speed and frequency. The flutter determinant is solved by an iterative procedure covering a wide range of air-speeds and frequencies. The computed natural frequencies, mode shapes, flutter speeds and flutter frequencies are compared and contrasted for the two type of aircraft wings and some conclusions are drawn.
20 citations
TL;DR: In this article, a probabilistic interpretation of fractional derivatives is proposed and the Shannon entropy is adopted to assess the truncation effect produced by the approximations. But the truncations are not considered.
Abstract: Fractional derivatives (FDs) need to be calculated using series or rational fractions. The effect of such approximations are usually analysed by means of the frequency or the time responses. This paper takes advantage of a probabilistic interpretation of FDs and adopts the Shannon entropy for assessing the truncation effect produced by the approximations.
20 citations
TL;DR: In this article, the description of shift-invariant systems in terms of Riemann- Liouville and Caputo derivatives is studied according to their initial conditions, and the situation of a past excitation of a linear system is considered and shown that the referred initial conditions may be either null or unavailable.
Abstract: The description of shift-invariant systems in terms of Riemann- Liouville and Caputo derivatives is studied according to their “initial conditions”. The situation of a past excitation of a linear system is considered and shown that the referred initial conditions may be either null or unavailable. This may lead to question the use of such derivatives.
17 citations
TL;DR: In this article, the generalized long short wave equation with power law nonlinearity was studied and several approaches that are used to solve this coupled system nonlinear evolution equations were presented. And numerical simulations were also given to this equation.
Abstract: This paper studies the generalized long-short wave equation with power law nonlinearity. There are several approaches that are used to solve this coupled system nonlinear evolution equations. The series solution approach yields the topological 1-soliton solution or shock wave solution. The ansatz method and the semiinverse variational principle leads to the non-topological 1-soliton of the equation. Additionally, the variational iteration method is used to study the equation. Finally, numerical simulations are also given to this equation.
17 citations
TL;DR: In this article, the Prony's spectroscopy is used to transform a wide class of multi-periodic and random signals (associated with a clearly expressed trend) to their amplitudefrequency response (AFR).
Abstract: The basic purpose of this paper is related to creation of the basis of new metrology based on quantitative analysis of the Prony’s spectra. The Prony’s spectroscopy gives a possibility to transform a wide class of multi-periodic and random signals (associated with a clearly expressed trend) to their amplitudefrequency response (AFR). For the strongly-correlated complex systems with memory the corresponding AFR is generated by a few initial frequencies that allow realizing the further compression of the initial random signal and reading it in terms of the limited number of quantitative parameters. This important peculiarity of the Prony’s spectroscopy allows in creating of a “universal” language for reading of different fluctuations and comparing them with each other. One interesting example of reading of the infinite sequences that are formed from the transcendental numbers (the Euler’s constant E = 2.71828... and number π = 3.14159...) is considered. The new elements that are added to calculation scheme/algorithm increase the stability and accuracy of the Prony’s spectra considered. In general, it opens new possibilities in creation of the basis of the fluctuation metrology that enables to read a wide class of random signals and compare them with other signals in terms of the quantitative parameters entering into the corresponding AFRs that, in turn, are governed by the corresponding distribution of frequencies.
14 citations
TL;DR: In this article, a generalization of the Prony's problem for strongly-correlated systems with memory is presented. But it is only for the case of strongly correlated systems with a memory.
Abstract: The new law that governs by the generation of frequencies ωk (k=1,2,...,K-1)for the strongly-correlated systems (having a memory) has been found. The generalization of the present idea is based on detailed analysis of the previous results obtained in paper [1] that were devoted to new solutions of the Prony’s problem. It was turned out that many complex systems with memory generate new set of frequencies based on frequencies that have been generated in the nearest past. For justification of this relationship we collected different data that confirm this statement. We created also a special mathematical program, which selects (based on some criteria) a desired hypothesis that is chosen from other six similar ones. For all available data considered there is an optimal hypothesis that describes the distribution of frequencies that follows from the recurrence relationship including in itself the neighboring frequencies. The found hypothesis provides the optimal fit of the random smoothed sequence with high accuracy (the relative error less that 10%) including also the fit of the remnant function.The physical interpretation of this law is given also. This “unexpected” discovery found for a wide class of the strongly-correlated systems with memory allows to replace the probability distribution function associated with some process by its Prony’s spectrum. From mathematical point of view it will help to obtain new solutions of the old Prony’s problem and replace also the Fourier spectrum containing usually the excess of artifact frequencies by the informative-significant band of frequencies obtained from new general law that, in turn, was found for the strongly-correlated systems.
13 citations
TL;DR: The necessity of having a dual behavior suspension and the design of the two practical designs are reviewed, which have some challenging and unsolved problems.
Abstract: Vibration isolators are needed to control the relative displacement, acceleration, or most importantly the transmitted force to the base or to the isolated device. As a general rule, a good vibration isolator should be as soft as possible to reduce the transmitted force, and it should be hard to limit the relative displacement. Soft suspension provides large relative displacement in resonance zone. To limit the relative displacement while having a soft suspension, a limiting displacement design is needed. There are two practical designs, Hydraulic Engine Mount (HEM) and Piecewise Linear Suspension (PLS) which are being used in industry since 80s. Hydraulic engine mounts were invented to passively produce a low damping at low amplitude and a high damping at high amplitude. Similarly, piecewise linear suspension were introduced to provide a soft suspension at low amplitude and a hard suspension at high amplitude. Having dual behavior puts both HEM and PLS in the domain of nonlinear system which in turn brings many new phenomena never appeared in linear analysis. This article will review the necessity of having a dual behavior suspension and the design of the two practical designs. However, both of these designs have some challenging and unsolved problems. Introducing the problems opens avenue for supervisors, researchers, and students to direct their research practice and hopefully solve them.
10 citations
8 citations
TL;DR: In this article, a discrete wavelet-based adaptive controller is explored to stabilize the nonlinear response of the milling tool in the time and frequency domains simultaneously, and an adaptive controller along with an adaptive filter effective for online system identification is implemented in the wavelet domain.
Abstract: A highly interrupted machining process, milling at high speed can be dynamically unstable and chattering with aberrational tool vibrations. While its associated response is still bounded in the time domain, however, milling could become unstably broadband and chaotic in the frequency domain, inadvertently causing poor tolerance, substandard surface finish and tool damage. Instantaneous frequency along with marginal spectrum is employed to investigate the route-to-chaos process of a nonlinear, time-delayed milling model. It is shown that marginal spectra are the tool of choice over Fourier spectra in identifying milling stability boundary. A novel discrete-wavelet-based adaptive controller is explored to stabilize the nonlinear response of the milling tool in the time and frequency domains simultaneously. As a powerful feature, an adaptive controller along with an adaptive filter effective for on-line system identification is implemented in the wavelet domain. By exerting proper mitigation schemes to both the time and frequency responses, the controller is demonstrated to effectively deny milling chatter and restore milling stability as a limit cycle of extremely low tool vibrations.
8 citations
TL;DR: In this paper, the New Iterative Method proposed by Daftardar-Gejji and Jafari with classical discretization technique was used to integrate some linear and nonlinear systems.
Abstract: In this article, we couple the New Iterative Method proposed by Daftardar-Gejji and Jafari [J. Math. Anal. Appl. 2006;316:753– 763] with classical discretization technique to integrate some linear and nonlinear systems. The numerical examples are presented to explain the method. Examples include some nonlinear dynamical systems such as Financial system, Duffing oscillator, Van der Pol oscillator and a non-autonomous system.
TL;DR: In this paper, a modified fuzzy sliding mode control (MFSMC) strategy was developed for actively controlling and stabilizing the non-linear vibration of a laminated composite cantilever beam.
Abstract: A modified fuzzy sliding mode control (MFSMC) strategy is devel-oped in this research for actively controlling and stabilizing the non-linear vibration of a laminated composite cantilever beam. The cantilever beam model, which is wildly seen in engineering applications, is established based on Hamilton’s principle through the application of Reddy’s third-order theory together with von Karman-type equations. Geometric nonlinearity of the beam is considered.The governing equations for the beam are derived corresponding to the higher order discretization of Galerkin method. By the model developed with n degrees of freedom, a vibration control strategy is developed on the basis of the modification of the fuzzy sliding mode control (FSMC). The vibration control strategy of the present research provides the availability for controlling the vibrations of such beam in n degrees of freedom. The approach is also proven to be effective in stabilizing the vibration of the nonlinear beam in a desired manner.
TL;DR: In this paper, the problem of relative controllability and Rn1 -observability for a linear stationary fractional differential-algebraic system with delay (FDAD) was studied.
Abstract: The paper deals with problems of relative controllability and Rn1 - observability for linear stationary fractional differential-algebraic system with delay (FDAD). FDAD system consists of fractional differential in the Caputo sense and difference equations. We present control systems and observation systems. We introduce the determining equation systems and their properties. By solution representations into series of their determining equation solutions we obtain effective parametric rank criteria for relative controllability and Rn1 -observability. A dual controllability result is also formulated.
TL;DR: In this paper, the authors studied the (3+1)-dimensional YTSF equation and a (3 + 1)-YTSF-type equation and derived two kinds of multiple wave solutions for each equation.
Abstract: n this work, we study the (3+1)-dimensional YTSF equation and a YTSF-type equation. We derive two kinds of multiple wave solutions for each equation. The simplified form of the direct method will be used to conduct the analysis.
TL;DR: In this article, a simple iterative adaptive emission control grasping only the main factors is developed, and it is shown that small modification of the parameters of the dynamic model have drastic influence on the emission so the use of adaptive techniques is essential.
Abstract: Precise emission models are so complex that their details practically are not available. The “physics” of hydrodynamic traffic models provides only ambiguousmathematical formulae. Therefore adaptive controllers iteratively improving the forecasts of a rough initial model are required that work without tuning any particular model parameters. For this purpose a simple numerical technique is reported. The stable stationary solutions of a particular model are determined. Then a simple iterative adaptive emission control grasping only the main factors is developed. It is shown that small modification of the parameters of the dynamic model have drastic influence on the emission so the use of adaptive techniques is essential.
TL;DR: In this paper, a delay differential equation (DDE) which exhibits a double Hopf or Hopf-Hopf bifurcation is studied using both Lindstedt's method and center manifold reduction.
Abstract: A delay differential equation (DDE) which exhibits a double Hopf or Hopf-Hopf bifurcation [1] is studied using both Lindstedt’s method and center manifold reduction. Results are checked by comparison with a numerical continuation program (DDEBIFTOOL). This work has application to the dynamics of two interacting microbubbles.
TL;DR: In this article, a method is presented for the numerical analysis of instabilities and bifurcations of the complex flows around body, where the formations, developments and evolvements of the complicated flow pattern in flow around a cylinder are numerically simulated using finite element method combined with explicit characteristic based split scheme (CBS) method, and the flow patterns are classified tentatively by streamline topology.
Abstract: A method is presented for the numerical analysis of instabilities and bifurcations of the complex flows around body. First, the formations, developments and evolvements of the complicated flow pattern in flow around a cylinder are numerically simulated using finite element method combined with explicit characteristic based split scheme (CBS) method, and the flow patterns are classified tentatively by streamline topology, in comparison with the existing results. The results show streamline topology method is powerful to investigate the instability and nature of the streamline patterns, especially, the vortex shedding. Further, the model reduction of fluid dynamics based on proper orthogonal decomposition (POD) and the stability analysis are given. Following POD, a low-dimensional dynamic system is derived to study numerically the stability and bifurcation of pattern. As an example, the stability analysis of flow around a circular cylinder is carried out, and is verified by the existing results. As a conclusion, the results show that the numerical method presented is powerful for the stability and bifurcation analysis of the complicated flows around body, and some nonlinear phenomena can be captured by the method.
TL;DR: In this paper, a singular perturbation technique (SPT) was proposed for the reconstruction of the shape of a scatterer from far field measurements, which is based on the linear sampling method.
Abstract: In this paper, we propose a new and efficient regularization technique (Singular Perturbation Technique-SPT) for the reconstruction of the shape of a scatterer from far field measurements. The approach used is based on the linear sampling method and uses singular perturbations of the identity along with Neumann series approximations to recover the shape of the unknown object. In comparison with the well established Tikhonov-Morozov regularization techniques our new algorithm appears to be more computationally efficient as it doesn’t require computation of the regularization parameter for each point in the grid. Moreover reconstructions obtained using (SPT) compare well with those obtained using other existing regularization methods.