On Modal Interaction,Stability and Nonlinear Dynamics of a Model Two D.O.F. Mechanical System Performing Snap-Through Motion

Dynamics of a simple two degrees of freedom (d.o.f.) mechanical system is considered, to illustrate the phenomena of modal interaction. The system has a natural symmetry of shape and is subjected to symmetric loading. Two stable equilibrium configurations are separated by an unstable one, so that the model system can perform cross-well oscillations. Nonlinear statics and dynamics are considered, with the emphasis on detecting conditions for instability of symmetric configurations and analysis of bi-modal non-symmetric motions. Nonlinear local dynamics is analyzed by multiple scales method. Direct numerical integration of original equations of motions is carried out to validate analysis of modulation equations. In global dynamics (analysis of cross-well oscillations) Lyapunov exponents are used to estimate qualitatively a type of motion exhibited by the mechanical system. Modal interactions are demonstrated both in the local dynamics and for snap-through oscillations, including chaotic motions. This mechanical system may be looked upon as a lumped parameters model of continuous elastic structures (spherical segments, cylindrical panels, buckled plates, etc.). Analyses performed in the paper qualitatively describe complicated phenomena in local and global dynamics of original structures.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part II: Experimental Analysis

In Part I, theoretical analysis of the dynamic behaviour of a rigid rotor with nonlinear elastic restoring forces was carried out. In this part (Part II), an experimental confirmation of the theoretical data from that analysis was sought. With this aim, an experimental model was set up consisting mainly of a practically rigid rotor clamped onto a small diameter piano wire symmetrical to the wire supports. These supports were rigid and equipped with roller bearings and a device that made it possible to adjust the initial tension in the wire so as to make the elastic restoring forces less or more linear. The rotor was dynamically unbalanced and was driven by an asynchronous motor regulated by means of an inverter in order to adjust the rotor speed. A series of tests was performed on this rig with different values of the initial tension in the wire, and the trajectories of two points on the rotor axis were recorded in the course of the tests. These trajectories were obtained, under the hypothesis of similarity, from the orbits covered by two given sections of the wire and detected with two pairs of capacitive transducers. The collected data was compared with the theoretical results from Part I of the present investigation. Comparison of the collected data with the corresponding theoretical results made it possible to infer that system nonlinearity in the presence of small damping can give rise to motions that are periodic, whether synchronous or not, or quasi-periodic, but never chaotic.     

Nonlinear dynamics and chaos of a finite-degree-of-freedom model of a buckled rod

The simple example of a mechanical system expressively exhibiting unpredictable and chaotic motions is a rod compressed by a supercritical force and subjected to a time-dependent transverse loading. Dynamics of this system can be analyzed either through modal analysis or through another lumped parameter modelling, for example, by discretization of the rod into an ensemble of segments. The paper is aimed to present the latter formulation of the problem and to discuss numerical results obtained in this framework.

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Sommario Un semplice esempio di sistema meccanico in grado di esibire in modo espressivo comportamenti dinamici non predicibili e caotici è rappresentato da una trave compressa in regime supercritico e soggetta ad un carico trasversale dipendente dal tempo. La dinamica di questo sistema può essere analizzata tramite approssimazioni modali, ovvero attraverso una modellazione a parametri concentrati, ad esempio discretizzando la trave in elementi rigidi con deformabilità localizzate. Il lavoro presenta quest'ultima formulazione del problema e ne discute i relativi risultati numerici.

     

Stability and Bifurcation Analysis of a Modified Geometrically Nonlinear Orthotropic Jeffcott Model with Internal Damping

In this paper, a modified Jeffcott model is proposed and studied in order to shed light into the dynamics of a complex system, the Short Electrodynamic Tether (SET), which is similar to an unbalanced rotor. Due to the internal damping, a geometrically linear SET model appears to be unstable as predicted by the linear rotordynamics theory. Some studies in the field of rotordynamics suggest that this instability caused by internal damping do not appear if geometric nonlinearities are taken into account in the system equations of motion. Stability and bifurcation analysis have been carried out on the modified Jeffcott model, which accounts for geometric nonlinearities, orthotropy in the shaft's cross section, and a viscous damping-based internal damping model. The stability results analytically obtained have been compared with a nonlinear multibody model by means of time simulations and good agreement has been found.     

流体动压滑动轴承-转子系统非线性动力特性及稳定性

根据油膜的物理特性,在动力积分、迭代过程中实时修正具有下游Reynolds边界条件的轴承流体润滑椭圆型变分方程,使其等价为变分不等式.运用八节点等参有限元方法,同时完成非线性油膜力及其Jacobian矩阵的计算.运用Newton-Raphson方法求得转子平衡点时,同时求得了作为副产品的轴承的刚度和阻尼系数.将预估-校正机理和Newton-Raphson方法相结合,提出了计算轴承-转子系统Hopf分岔点(对应于线性失稳转速)的方法.将预估-校正机理与Poincaré-Newton-Floquet方法相结合,分析了T周期运动的局部稳定性和分岔现象.结果表明,采用八节点等参有限元方法同时完成非线性油膜力及其Jacobian矩阵的计算时,同传统方法相比计算量减少,且精度协调一致;将预估-校正机理和Newton-Raphson方法相结合,可以方便地计算轴承-转子系统Hopf分岔点;将预估-校正机理与Poincaré-Newton-Floquet方法相结合,可以避免初值选取困难,快速求得系统周期解及其分岔点.所建立的计算方法具有省时、精度高等优点,可用于指导滑动轴承-转子系统设计.     

Periodic and Non-Periodic Oscillations of a Class of Hysteretic Two Degree of Freedom Systems

The equations governing the response of hysteretic systems to sinusoidal forces, which are memory dependent in the classical phase space, can be given as a vector field over a suitable phase space with increased dimension. Hence, the stationary response can be studied with the aids of classical tools of nonlinear dynamics, as for example the Poincaré map. The particular system studied in the paper, based on hysteretic Masing rules, allows the reduction of the dimension of the phase space and the implementation of efficient algorithms. The paper summarises results on one degree of freedom systems and concentrates on a two degree of freedom system as the prototype of many degree of freedom systems. This system has been chosen to be in 1:3 internal resonance situation. Depending on the energy dissipation of the elements restoring force, the response may be more or less complex. The periodic response, described by frequency response curves for various levels of excitation intensity, is highly complex. The coupling produces a strong modification of the response around the first mode resonance, whereas it is negligible around the second mode. Quasi-periodic motion starts bifurcating for sufficiently high values of the excitation intensity; windows of periodic motions are embedded in the dominion of the quasi-periodic motion, as consequence of a locking frequency phenomenon.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

Vibration of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation

Vibration analysis of a non-linear parametrically self-excited system with two degrees of freedom under harmonic external excitation was carried out in the present paper. External excitation in the main parametric resonance area was assumed in the form of standard force excitation or inertial excitation. Close to the first and second free vibrations frequency, the amplitudes of the system vibrations and the width of synchronization areas were determined. Stability of obtained periodic solutions was investigated. The analytical results were verified and supplemented with the effects of digital and analog simulations.     

Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaireDetermination of the macroscopic strength criterion of a porous medium by nonlinear homogenization

A porous medium, which matrix is a perfectly plastic solid, is considered. This paper proposes a method to determine the macroscopic admissible stress states. The method is based on a homogenization technique which takes advantage of the equivalence, under certain conditions, between a problem of limit analysis and a ficticious nonlinear elastic problem. The particular case of a Drucker–Prager solid matrix is considered. The method provides an analytical expression for the complete macroscopic strength criterion. To cite this article: J.-F. Barthélémy, L. Dormieux, C. R. Mecanique 331 (2003).     

2-D Simulation of Non-isothermal Fate and Transport of a Drip-applied Fumigant in Plastic-mulched Soil Beds. I. Model Development and Performance Investigation

Pre-plant application of toxic fumigants to soil beds covered by plastic film is commonly used in agriculture to control soil-borne pathogens. Plastic mulch covers tend to physically suppress the emissive loss of gaseous fumigant to the atmosphere. When liquid fumigant metham sodium (MS) is applied in irrigation water to field soil, it is rapidly transformed to the gaseous methyl isothiocyanate (MITC). The gaseous MITC is a potential atmospheric contaminant, and any untransformed MS is a potential contaminant of underlying groundwater due to the high water solubility of MS. A finite element numerical model was developed to investigate two-dimensional MITC fate/transport under non-isothermal soil conditions. Directional solar heating on soil beds, coupled heat and water flow in the soil, and non-isothermal chemical transport were included in the model. Field soil data for MITC distribution, soil water content, meteorological data, and laboratory data were used to verify the model for soil beds covered with plastic mulch. Four possible scenarios were considered: low and high drip-irrigation rates and low and high water contents. The movement of the center of MITC mass in the soil profile was effectively simulated. The lower drip-irrigation rate of MS yielded more extensive coverage of MITC in the plastic-covered soil bed. The lower soil air contents due to higher soil water contents for the higher irrigation rate resulted in high concentrations of soil MITC. NRMSE (normalized root mean square error) calculations further verified that the model predicted fumigant fate/transport well under these non-isothermal field conditions.