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1.
1:2 Internal Resonance of Coupled Dynamic System with Quadratic and Cubic Nonlinearities 总被引:1,自引:0,他引:1
The1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this
system in1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities
were remained. Based on a new convenient transformation technique, the4-dimension bifurcation equations were reduced to3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding
were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior
in strong internal resonance on4-dimension center manifolds.
Paper from Chen Yu-shu, Member of Editorial Commuttee, AMM
Foundation item: the National Natural Science Foundation of China (1990510); the National Key Basic Research Special Fund (G1998020316);
the Doctoral Point Fund of Education Committee of China (D09901)
Biography: Chen Yu-shu (1931-) 相似文献
2.
戴天民 《应用数学和力学(英文版)》1981,2(2):183-188
In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived. 相似文献
3.
João C. André 《Nonlinear dynamics》1996,11(3):275-293
In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude. 相似文献
4.
Self-Excited Oscillators with Asymmetric Nonlinearities and One-to-Two Internal Resonance 总被引:1,自引:0,他引:1
An analysis is presented on the dynamics of asymmetric self-excited oscillators with one-to-two internal resonance. The essential behavior of these oscillators is described by a two degree of freedom system, with equations of motion involving quadratic nonlinearities. In addition, the oscillators are under the action of constant external loads. When the nonlinearities are weak, the application of an appropriate perturbation approach leads to a set of slow-flow equations, governing the amplitudes and phases of approximate motions of the system. These equations are shown to possess two different solution types, generically, corresponding to static or periodic steady-state responses of the class of oscillators examined. After complementing the analytical part of the work with a method of determining the stability properties of these responses, numerical results are presented for an example mechanical system. Firstly, a series of characteristic response diagrams is obtained, illustrating the effect of the technical parameters on the steady-state response. Then results determined by the application of direct numerical integration techniques are presented. These results demonstrate the existence of other types of self-excited responses, including periodically-modulated, chaotic, and unbounded motions. 相似文献
5.
RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY 总被引:1,自引:1,他引:1
The theories of thermopiezoelectricity and magnetoelasticity for micropolar continua have been systematically developed by
W. Nowacki. In this paper, the theories are restudied. The reason why they were restricted to linear cases is analyzed. The
more general conservation principle of energy, energy balance equation and Hamilton principle of thermopiezoelectricity and
magnetoelasticity for micropolar continua are established. The corresponding complete equations of motion and boundary conditions
as well as balance equations of energy rate for local and nonlocal micropolar thermopiezoelectricity and magnetothermoelasticity
are naturally derived. By means of two new functionals and total variation the boundary conditions of displacement, microrotation,
electric potential and temperature are also given.
Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235)
and the DFG (51520001); the Research Foundation of the Liaoning Education Committee (990111001)
Biography: DAI Tian-min (1931-) 相似文献
6.
In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content. 相似文献
7.
8.
The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors. 相似文献
9.
In this paper the nonlinear response of a base-excited slender beam carrying an attached mass is investigated with 1:3:9 internal resonances for principal and combinationparametric resonances. Here the method of normal forms is used to reduce the second order nonlinear temporal differential equation of motion of the system to a set offirst order nonlinear differential equations which are used to find the fixed-point, periodic, quasi-periodic and chaotic responses of the system.Stability and bifurcation analysis of the responses are carried out and bifurcation sets are plotted. Many chaotic phenomena are reported in this paper. 相似文献
10.
Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case 总被引:1,自引:0,他引:1
A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each
other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The
Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and
the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce
the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance
is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point
of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of
codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion,
but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact
periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are
also discussed.
The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation
of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang. 相似文献
11.
12.
Bifurcation and universal unfolding for a rotating shaft with unsymmetrical stiffness 总被引:1,自引:0,他引:1
The 1/2 subharmonic resonance bifurcation and universal unfolding are studied for a rotating shaft with unsymmetrical stiffness.
The bifurcation behavior of the response amplitude with respect to the detuning parameter was studied for this class of problems
by Xiao et al. Obviously, it is highly important to research the bifurcation behavior of the response amplitude with respect
to the unsymmetry of stiffness for this problem. Here, by means of the singularity theory, the bifurcation and universal unfolding
of amplitude with respect to the unsymmetrical stiffness parameter are discussed. The results indicate that it is a high codimensional
bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, we study four
forms of two parameter unfoldings contained in the universal unfolding. The transition sets in the parameter plane and the
bifurcation diagrams are plotted. The results obtained in this paper show rich bifurcation phenomena and provide some guidance
for the analysis and design of dynamic buckling experiments of this class of system, especially, for the choice of system
parameters.
The project supported by the National Natural Science Foundation of China (19990510), the National Key Basic Research Special
Foundation (G1998020316) and Liuhui Center for Applied Mathematics, Nankai University and Tianjin University 相似文献
13.
Sets of physical constants are tabulated for three structural models of fibrous composites with fibers of four types: Thornel-300
carbon microfibers, graphite whiskers, carbon zigzag nanotubes, and carbon chiral nanotubes. The matrix for all the types
of composites is always éPON-828 epoxy rosin (in some cases with polystyrene or pyrex additive). The values of the physical
constants are commented on and used to study the distinctions in the evolution of three types of waves (plane longitudinal,
plane transverse, and cylindrical) propagating in materials with soft and hard nonlinearities
__________
Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 47–60, December 2005. 相似文献
14.
This paper presents the global bifurcation and chaotic behavior for the coupling of longitudinal and transverse vibration of a thin elastic plate in large overall motion. First the parametric equations of the homoclinic orbits of such a system is obtained. Then, by using the Melnikov method and digital computer simulation. the behavior of bifurcation and chaos of this vibration system is investigated in the cases of different resonances. The obvious difference between the transverse vibration and the coupling of transverse and longitudinal vibration is also shown.The project supported by the National Natural Science Foundation of China. 相似文献
15.
拉压性能不同材料全量型本构关系及厚壁筒的应力分析 总被引:1,自引:0,他引:1
将经典全量理论作了推广,考虑了应力状态及塑性体积变形对拉压性能不同材料的塑性行为的影响。应用该本构模型分别计算了厚壁筒在内压和外压作用下的应力分布。给出了径向应力、环向应力和轴向应力沿壁厚的分布图。将本文的计算解与拉压性能相同(不考虑体积变形、强化曲线唯一)的幂函数强化材料的厚壁筒的理论解进行了比较。结果表明,材料的拉压性能不同对厚壁筒的环向应力和轴向应力影响较大。因此,对于拉压性能不同材料,考虑到其对应力状态及塑性体积变形敏感时,是不能将其简化成拉压性能相同、体积不可压缩、强化曲线唯一的理想材料。 相似文献
16.
17.
Patrizio Neff 《Continuum Mechanics and Thermodynamics》2003,15(2):161-195
This paper is concerned with the formulation of a phenomenological model of finite elasto-plasticity valid for small elastic
strains for initially isotropic polycrystalline material. As a basic we assume the multiplicative split of the deformation gradient into elastic and plastic part. A key feature of the model is the introduction of an independent
field of 'elastic' rotations which eliminate the remaining geometrical nonlinearities coming from finite elasticity in the presence of small elastic strains.
In contrast to micro-polar theories an evolution equation for is presented which relates to making use of a new device found by the author to perform the polar decomposition asymptotically. The model is shown to be
invariant under both change of frame and rotation of the so called intermediate configuration. The corresponding equilibrium
equations at frozen plastic and viscoelastic configuration constitute then a linear, elliptic system with nonconstant coefficients
which makes this model amenable to a rigorous mathematical analysis. The introduced hysteresis effects within the elastic
region are related to viscous elastic rotations of the grains of the polycrystal due to internal friction at the grain boundaries
and constitute as such a rate dependent transient texture effect. The inclusion of work hardening will be addressed in future
work.
Received March 07, 2002 / Published online February 17, 2003
RID="*"
ID="*"Communicated by Kolumban Hutter, Darmstadt 相似文献
18.
Fangqi Chen Jianshu Liang Yushu Chen Xijun Liu Hongcheng Ma 《Mechanics Research Communications》2007,34(3):213-221
The subharmonic bifurcation and universal unfolding problems are discussed for an arch structure with parametric and forced excitation in this paper. The amplitude–frequency curve and some dynamical behavior have been shown for this class of problems by Liu et al. Here, by means of singularity theory, in the case of strict 1:2 internal resonance, the bifurcation behavior of the amplitude with respect to a parameter (which is related to the amplitude of the live load imposed on the arch structures) is studied. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, 20 forms of two parameter unfoldings with some constraints are studied. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper present some new dynamic buckling patterns and abundant bifurcation phenomena. 相似文献