Analysis of periodically excited non-linear systems by a parametric continuation technique

Stationary response determination of MDOF fractional nonlinear systems subjected to combined colored noise and periodic excitation

An analytical approximate technique for a class of strongly non-linear oscillators

Bifurcation analysis on a turning system with large and state-dependent time delay

Harmonic balancing method has more powerful superiority than other perturbation methods in the respect of solving steady-state response of strong nonlinear system. The amplitude-frequency equation of dry-friction vibration system considering cubic nonlinear displacement under harmonic excitation was derived by harmonic balancing method. Because the effect of even nonlinear term on vibration response of system was not evident, even nonlinear term of elastic restoring force was neglected. The first term coefficient and third term coefficient obtained by static experimental data for constitutive relationship of metallic rubber were substituted into frequency response equation. Amplitude-frequency characteristics curves of vibration response for cubic nonlinear displacement dry-friction system of metallic rubber were obtained. Natural frequencies obtained by computation were compared with natural frequencies by experiment, and the two Natural frequencies consisted with each other. It showed that the method that vibration response for cubic nonlinear displacement dry-friction system of metallic rubber was computed is reasonable.

The present chapter considers well-known approaches to mathematical simulation of subsonic turbulent jets under periodical harmonic excitation. Particular emphasis is placed on the description (in the framework of these approaches) of generation and interaction of large-scale coherent structures, as well as of their susceptibility to periodical excitation. The calculation results illustrating the turbulent mixing intensification (turbulence generation) in jets under low-frequency harmonic excitation and the mixing attenuation (turbulence suppression) under high-frequency excitation are presented. It is stressed that the mathematical simulation of turbulent jets under periodical excitation allows describing the laws of periodical (acoustical) excitation known from the experimental studies.

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-periodic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations.

The harmonic balance method is a very useful tool for characterizing and predicting the response of nonlinear dynamic systems undergoing periodic oscillations, either self-excited or due to harmonic excitation. The method and several of its variants were applied to nonlinear aeroelastic systems over the last two decades. This paper presents a detailed description of several harmonic balance methods and a continuation framework allowing the methods to follow the response of dynamic systems from the bifurcation point to any desired parameter value, while successfully negotiating further fold bifurcations. The continuation framework is described for systems undergoing subcritical and supercritical Hopf bifurcations as well as a particular type of explosive bifurcation. The methods investigated in this work are applied to a nonlinear aeroelastic model of a generic transport aircraft featuring polynomial or free-play stiffness nonlinearity in the control surface. It is shown that high-order harmonic balance solutions will accurately capture the complete bifurcation behavior of this system for both types of nonlinearity. Low-order solutions can become inaccurate in the presence of numerous folds in the limit-cycle oscillation branch but can still yield practical engineering information at a fraction of the cost of higher-order solutions. Time-domain harmonic balance schemes are shown to be more computationally expensive than the standard harmonic balance approach.

With rigorous dynamic performance of mechanical products, it is important to identify dynamic parameters exactly. In this paper, a response surface plotting method is proposed and it can be applied to identify the dynamic parameters of some nonlinear systems. The method is based on the principle of harmonic balance method (HBM). The nonlinear vibration system behaves linearly under the steady-state response amplitude, which presents the equivalent stiffness and damping coefficient. The response surface plot is over two-dimensional space, which utilizes excitation as the vertical axis and the frequency as the horizontal axis. It can be applied to observe the output vibration response data. The modal parameters are identified by the response surface plot as linearity for different excitation levels, and they are converted into equivalent stiffness and damping coefficient for each resonant response. Finally, the HBM with first-order expansion is utilized for identification of stiffness and damping coefficient of nonlinear systems. The classical nonlinear systems are applied in the numerical simulation as the example, which is used to verify its effectiveness and accuracy. An application of this technique for nonlinearity identification by experimental setup is also illustrated.

A global model is necessary to understand and improve the dynamic behavior of electric power steering (EPS) systems. This paper presents the transient response of a worm gear under harmonic excitation. The gear model is formulated by extending the method used for spur gears and is based on the lumped parameter theory. Backlash clearances and error excitation are taken into account. It is also possible to couple axial, lateral and torsional motions. To avoid numerical problems encountered with direct numerical integration, meshing stiffness and damping, which are often considered as piecewise linear functions owing to backlash, are approached by a continuous function. The effects of several parameters such as mesh stiffness, damping values and backlash value are analyzed. Certain discontinuities are shown to exist in the accelerations of both wheel and worm centers due to impacts between the gear pair.Copyright © 2005 by ASME

Torus doublings and chaotic amplitude modulations in a two degree-of-freedom resonantly forced mechanical system

Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

Effective and efficient simulation and verification techniques are always highly demanded in the electronic design automation (EDA) community. However, existing modeling and simulation approaches could no longer fulfill this growing demand due to the presence of large-scale distributed components and strongly nonlinear devices. In this thesis, two novel techniques are presented for fast simulation and verification of analog and radio-frequency (RF) circuits. Emphasis is placed on developing an analytical approach for periodic steady-state (PSS) analysis and an effective model order reduction (MOR) technique for nonlinear, especially highly nonlinear, systems arising in analog/RF circuit applications.
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\nThe first approach, named autonomous Volterra (AV), achieves efficient PSS analysis of nonlinear circuits. With elegant analytic forms and availability of efficient solvers, AV constitutes a competitive steady-state algorithm besides the two mainstream PSS algorithms, namely, shooting Newton (SN) and harmonic balance (HB). Nonlinear systems are first captured in nonlinear differential algebraic equations (DAEs), followed by expansion into linear Volterra subsystems. A key step of steady-state analysis lies in modeling each Volterra subsystem with autonomous nonlinear inputs. The PSS solution of these subsystems then proceeds with a series of Sylvester equation solves, completely avoiding the guesses of initial condition and time stepping as in SN, as well as the uncertain length of Fourier series as in HB. Error control in AV is also straightforward by monitoring the norms of the Sylvester equation solutions. It is further demonstrated that AV is readily parallelizable with superior scalability towards large-scale problems.
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\nBesides PSS analysis, another important step in analog/RF design is transient as well as general time simulation. To this end, the second part of this thesis features a tensorbased nonlinear model order reduction (TNMOR) algorithm that allows efficient simulation of nonlinear circuits via the emerging techniques utilizing tensors, namely, a multidimensional generalization of matrices. Unlike existing nonlinear model order reduction (NMOR) methods, high-order nonlinearities are captured using tensors in TNMOR, followed by decomposition and reduction to a compact tensor-based reduced-order model (ROM). Consequently, TNMOR completely avoids the dense reduced-order system matrices, which in turn permits faster simulation and less memory requirement. Numerical experiments on transient and PSS analyses confirm the superior accuracy and efficiency of TNMOR, particularly in highly nonlinear scenarios.

The stability of steady state synchronous and nonsynchronous response of a nonlinear rotor system supported by squeeze-film dampers is investigated. The nonlinear differential equations which govern the motion of rotor bearing system are obtained by using the Generalized Polynomial Expansion Method. The steady state response of system is obtained by using the hybrid numerical method which combines the merits of the harmonic balance and collocation methods. The stability of system response is examined using Floquet-Liapunov theory. Using the theory, the performance may be evaluated with the calculation of derivatives of nonlinear hydrodynamic forces of the squeeze-film damper with respect to displacement and velocity of the journal center. In some cases, these derivatives can be expressed in closed form and the prediction of the dynamic characteristic of the nonlinear rotor system will be more effective. The stability results are compared to those using a direct numerical integration method and both are in good agreement.

In this paper, we consider a class of second-order Hamiltonian system in $\mathbb {R}^{N}$ with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., $x(t+T)=Qx(t)$ with $T>0$ and Q is an $N\times N$ orthogonal matrix. In the case $Q^{k}\neq I_{N}$ for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case $Q^{k}=I_{N}$ for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.

A method of calculating field functions of circular coils by direct numerical integration is explained, and formulae for the integrands required are given. The method is not only more straightforward mathematically than classical methods, but is also faster. A known principle has been extended, enabling a comprehensive treatment to be presented. The method can be used for calculating both magnetic fields of current-carrying coils, and induced EMFS of sense coils: a complete set of the formulae involved is given. Ways of extending the method to the cases of sets of coils and finite coil volumes and finite field volumes are explained.