Bifurcation analysis of an aeroelastic system with concentrated nonlinearities

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system??s instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed.     

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton??s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.     

Dynamic response of a single-mesh gear system with periodic mesh stiffness and backlash nonlinearity under uncertainty

Parametric uncertainties play a critical role in the response predictions of a gear system. However, accurately determining the effects of the uncertainty propagation in nonlinear time-varying models of gear systems is awkward and difficult. This paper improves the interval harmonic balance method (IHBM) to solve the dynamic problems of gear systems with backlash nonlinearity and time-varying mesh stiffness under uncertainties. To deal with the nonlinear problem including the fold points and uncertainties, the IHBM is improved by introducing the pseudo-arc length method in combination with the Chebyshev inclusion function. The proposed approach is demonstrated using a single-mesh gear system model, including the parametrically varying mesh stiffness and the gear backlash nonlinearity, excited by the transmission error. The results of the improved IHBM are compared with those obtained from the scanning method. Effects of parameter uncertainties on its dynamic behavior are also discussed in detail. From various numerical examples, it is shown that the results are consistent meanwhile the computational cost is significantly reduced. Furthermore, the proposed approach could be effectively applied for sensitivity analysis of the system response to parameter variations.     

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｄｙｎａｍｉｃｓｙｓｔｅｍｓｄｅｓｃｒｉｂｅｄｂｙｔｈｅｆｏｌｌｏｗｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ : ｘ=ｆ(ｘ) ,( 1 )ｗｈｅｒｅｘａｎｄｆａｒｅｎ＿ｄｉｍｅｎｓｉｏｎａｌｒｅａｌｖｅｃｔｏｒｓ,ａｎｄｔｈｅｄｏｔｉｎｄｉｃａｔｅｓｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｉｍｅ ,ｔ.Ｓｕｐｐｏｓｅｔｈａｔｘ=ｕ(ｔ)ｉｓａｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆＥｑ .( 1 )ｗｉｔｈｐｅｒｉｏｄｉｃ,ｓａｙ ,Ｔ .ＫｒｙｌｏｖａｎｄＢ…     

Nonlinear normal modes in pendulum systems

Dynamics of the spring pendulum and of the system containing a pendulum absorber is considered by using the nonlinear normal modes?? theory and the asymptotic-numeric procedures. This makes it possible to investigate the pendulum dynamics for both the small and large vibration amplitudes. The vibration modes stability is analyzed by different methods. Regions of the nonlinear normal modes?? stability/instability are obtained. The nonlinear normal modes?? approach and the modified Rauscher method are used to construct forced vibration modes in the system with a pendulum absorber.     

Adaptive control of a chaotic permanent magnet synchronous motor

This paper proposes a simple adaptive controller design method for a chaotic permanent magnet synchronous motor (PMSM) based on the sliding mode control theory which has given an effective means to design robust controllers for nonlinear systems with bounded uncertainties. The proposed sliding mode adaptive controller does not require any information on the PMSM parameter and load torque values, thus it is insensitive to model parameter and load torque variations. Simulation results are given to verify that the proposed method can be successfully used to control a chaotic PMSM under model parameter and load torque variations.     

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M ₃) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.     

Study of the inertial effect and the nonlinearities of?the?CRONE suspension based on the hydropneumatic technology

In this paper, we emphasize two main effects involved in the CRONE car suspension technology (CRONE: French acronym for Commande Robuste d??Ordre Non Entier). In a first time, we present the influence of the inductive or inertial effect of the pipes that links the different cells of the hydropneumatic car suspension. These components are mainly resistive and capacitive devices. Then, we analyze the nonlinear relations that link the hydraulic power variables (the flow and the pressure) of the hydraulic resistors and the hydropneumatic accumulators and we study the effect of the nonlinear terms on the car suspension response. Our study is based on the gamma RC arrangement developed in Altet et al. (In: Analysis and design of hybrid systems??proceedings of ADHS03, pp. 63?C68. Elsevier, Amsterdam, 2003) and Serrier et al. (In: Proceedings of IDETC/CIE 2005: ASME 2005 international design engineering technical conferences and computers and information in engineering conference, Long Beach, CA, USA, 24?C28 September 2005). In a second time, we focus only on the gamma RLC arrangement, introduced in Abi Zeid Daou et al. (Int. J. Electron. 96(12):1207?C1223, 2009). We show whether the parasite effect due to the pipes or the nonlinear RC components affect the system??s response. The simulation results show that neither the inertial effect caused by these parasite pipes of one meter length nor the use of the nonlinear resistors or the accumulators modifies the response of the gamma RC arrangement.     

Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

On the behavior of parametrically excited purely nonlinear oscillators

In this study, parametrically excited purely nonlinear oscillators are considered. Instabilities associated with 2:1, 3:1, and 4:1 subharmonics resonances are analyzed by assuming the solution for motion in the form of a Jacobi elliptic function, the elliptic parameter, and the frequency of which are calculated based on the energy conservation law of the corresponding conservative system. Chirikov??s overlap criterion is used to obtain the approximate critical value of the amplitude of the parametric excitation that causes the transition from local irregular behavior (seen as chaotic) to global chaos. The analytical results derived are compared with numerically results.     

Nonlinear vibrations of blade with varying rotating speed

This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton??s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin??s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.     

Experimental characterization of nonlinear systems: a real-time evaluation of the analogous Chua’s circuit behavior

This paper presents an experimental characterization of the behavior of an analogous version of the Chua’s circuit. The electronic circuit signals are captured using a data acquisition board (DAQ) and processed using LabVIEW environment. The following aspects of the time series analysis are analyzed: time waveforms, phase portraits, frequency spectra, Poincaré sections, and bifurcation diagram. The circuit behavior is experimentally mapped with the parameter variations, where are identified equilibrium points, periodic and chaotic attractors, and bifurcations. These analysis techniques are performed in real-time and can be applied to characterize, with precision, several nonlinear systems.     

三轴液压仿真转台非线性多输入多输出QFT控制器设计

三轴液压仿真转台本质上是非线性变参数多输入多输出系统.系统负载、油源压力、增益的变化及耦合作用随速度,加速度的变化造成很大的参数不确定性.针对液压转台提出一种基于SIMULINK模型线性化的多输入多输出非线性定量反馈设计方法.其本质是找到液压伺服系统一系列控制特性最差的运行点,直接用它们的频率响应集合来综合非线性定量反馈控制器.这种方法简化了目前的QFT多输入多输出控制系统设计方法,显示了很大的优越性.文中举例说明这种方法的效果.     

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Lyapunov＇s first method, extended by V. V. Kozlov to nonlinear mechani- cal systems, is applied to the study of the instability of the position of equilibrium of a mechanical system moving in the field of conservative and dissipative forces. The mo- tion of the system is limited by ideal nonlinear nonholonomic constraints. Five cases determined by the relationship between the degree of the first nontrivial polynomials in Maclaurin＇s series for the potential energy and the functions that can be generated from the equations of nonlinear nonholonomic constraints are analyzed. In the three eases, the theorem on the instability of the position of equilibrium of nonholonomic systems with linear homogeneous constraints （V. V. Kozlov （1986）） is generalized to the case of nonlin- ear nonhomogeneous constraints. In the other two cases, new theorems are set extending the result from V. V. Kozlov （1994） to nonholonomic systems with nonlinear constraints.     

The lateral stability of tractor and trailer combinations

A model has been developed to predict the lateral stability characteristics of tractor and unbalanced trailer combinations. For present combinations, stability deteriorates with speed culminating in instability at forward speeds in the region of 18 m/s. The effect of tractor and trailer size and other parameter variations on this speed dependent instability are examined.The effect of braking with and without axle locking are analysed. The stability of the combination is sensitive to the braking distribuion between the axles, which affect the hitch forces developed. Locking the tractor rear or trailer axle results in instabilities, commonly termed jack-knifing and trailer swing respectively. Jack-knifing is the more hazardous instability, whereas trailer swing although potentially dangerous has a divergence approximately an order of magnitude less.The potential of the model for predicting lateral dynamic behaviour of design concepts for future high speed farm transport which would operate at higher speeds than the current maximum of 9 m/s for tractor and trailer combinations is discussed. The scope for generalizing the model to examine other aspects of lateral behaviour, such as steering response is restricted by the limited amount of data available on the side force generated by tyres in agricultural conditions.     

Application of Special Nonsmooth Temporal Transformations to Linear and Nonlinear Systems under Discontinuous and Impulsive Excitation

Linear and nonlinear mechanical systems under periodic impulsive excitation are considered. Solutions of the differential equations of motion are represented in a special form which contains a standard pair of nonsmooth periodic functions and possesses a convenient structure. This form is also suitable in the case of excitation with a periodic series of discontinuities of the first kind (a stepwise excitation). The transformations are illustrated in a series of examples. An explicit form of analytical solutions has been obtained for periodic regimes. In the case of parametric impulsive excitation, it is shown that a nonequidistant distribution of the impulses with dipole-like temporal shifts may significantly effect the qualitative characteristics of the response. For example, the sequence of instability zones loses its different subsequences depending on the parameter of the shifts. It is shown that the method's applicability can be extended for nonperiodic regimes by involving the idea of averaging.     

Dynamic analysis of a simply supported beam resting on a nonlinear elastic foundation under compressive axial load using nonlinear normal modes techniques under three-to-one internal resonance condition

In this paper, the Nonlinear Normal Modes (NNMs) analysis for the case of three-to-one (3:1) internal resonance of a slender simply supported beam in presence of compressive axial load resting on a nonlinear elastic foundation is studied. Using the Euler?CBernoulli beam model, the governing nonlinear PDE of the beam??s transverse vibration and also its associated boundary conditions are extracted. These nonlinear motion equation and boundary condition relations are solved simultaneously using four different approximate-analytical solution techniques, namely the method of Multiple Time Scales, the method of Normal Forms, the method of Shaw and Pierre, and the method of King and Vakakis. The obtained results at this stage using four different methods which are all in time?Cspace domain are compared and it is concluded that all the methods result in a similar answer for the amplitude part of the transverse vibration. At the next step, the nonlinear normal modes are obtained. Furthermore, the effect of axial compressive force in the dynamic analysis of such a beam is studied. Finally, under three-to-one-internal resonance condition the NNMs of the beam and the steady-state stability analysis are performed. Then the effect of changing the values of different parameters on the beam??s dynamic response is also considered. Moreover, 3-D plots of stability analysis in the steady-state condition and the beam??s amplitude frequency response curves are presented.     

Dynamic buckling of discrete structural systems under combined step and static loads

The dynamic instability of discrete, elastic, multiple degree of freedom (d.o.f.) systems under a combination of static and step loads is studied. Conservative, autonomous and holonomic systems are considered, in which the associated static response is a bifurcation under one load parameter, and a limit point under the second parameter. A review of different criteria and algorithms obtained from them for the computation of dynamic buckling loads is first presented, followed by a procedure derived from previous investigations on one d.o.f. systems. The different procedures are applied to a two d.o.f. problem under axial and lateral load, with quadratic and cubic non-linearities. The response in time shows that the system oscillates about the static equilibrium position before dynamic buckling is reached, with the kinetic energy tending to zero as assumed in the static (energy) procedures of stability.     

Maximum of action for Hamiltonian systems with unilateral constraints

Hamilton??s variational principle for mechanical systems with unilateral constraints is considered. It is shown that the action functional attains its local maximum on the class of variations belonging to the interior of the domain admissible for motion. An example is given.     

On the purposeful coarsening of smooth vector fields

This paper argues for the possibility of purposely approximating smooth vector fields with highly localized variability in terms of piecewise smooth vector fields for the purpose of analyzing the bifurcation characteristics of the corresponding dynamical systems. Here, emphasis is placed on the changes in system response that result as a periodic trajectory begins to incorporate a brief flow segment in the region of high variability under variations in some system parameter. In particular, it is shown that tools from the theory of grazing bifurcations in piecewise-smooth systems may be employed to qualitatively predict the bifurcation scenario associated with such a transition both in terms of the shape of the branch of periodic trajectories and in terms of the persistence of a local attractor in the vicinity of the original periodic trajectory.