Transient mode localization in coupled strongly nonlinear exactly solvable oscillators

Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.     

Experiments on the dynamic behavior of fluid-coupled concentric cylinders

This paper describes an experimental vibration study of fluid-coupled coaxial cylinders that simulates the vibration of a reactor vessel with a thermal liner. The model cylinders are made of acrylic. Thickness and gap-size parameter studies are performed by a series of different compinations of three outside cylinders and nine inside cylinders that have variable thicknesses and diameters. Damping ratios are measured on a mode-by-mode basis for several combinations of cylinders. The vibrated cylinders are mounted to a rigid stand, with the cuter cylinder supported at both ends and the inner cylinder supported at either one end (pendulum mode) or both ends, as the case may be. The natural frequencies are obtained first in air and then with coaxial cylinders coupled by water. The mode shapes are obtained by circumferential (shell modes) and axial (shell/beam modes) mapping of the response with two diametrically opposite ‘roving’ Dymac eddy probes. In general, the natural vibration of the system has two distinct responses in-phase and/or out-of-phase modes, i.e., the radial displacement phase relationship between inner and outer cylinders. In the out-of-phase modes the frequency is shown to decrease to either zero or a very low limiting value as the gap size cecreases. The opposite occurs for in-phase modes. Damping ratios are found to be much higher for out-of-phase modes and for relatively rigid cylinders than for in-phase modes and flexible cylinders, respectively.     

Shock Hugoniots based on the self-consistent average atom (SCAA) model. Theory and experiments. (Second revision)

We use a “self-consistent average atom” (SCAA) model to compute shock Hugoniots for aluminum, iron, molybdenum, strontium, barium and thulium. The pressures and energies include relativistic effects. We make comparisons with the Thomas-Fermi-Dirac (TFD) model and with the available experimental data including pressures, shock and particle speeds and energy deposition. The connection between the usage of the “average atom” (AA) model and “detailed configuration accounting” (DCA) is discussed in the Appendix.     

Symmetry limit theory for cantilever beam-columns subjected to cyclic reversed bending

The behavior of a linear strain-hardening cantilever beam-column subjected to completely reversed plastic bending of a new idealized program under constant axial compression consists of three stages: a sequence of symmetric steady states, a subsequent sequence of asymmetric steady states and a divergent behavior involving unbounded growth of an anti-symmetric deflection mode. A new concept “symmetry limit” is introduced here as the smallest critical value of the tip-deflection amplitude at which transition from a symmetric steady state to an asymmetric steady state can occur in the response of a beam-column. A new theory is presented for predicting the symmetry limits. Although this transition phenomenon is phenomenologically and conceptually different from the branching phenomenon on an equilibrium path, it is shown that a symmetry limit may theoretically be regarded as a branching point on a “steady-state path” defined anew. The symmetry limit theory and the fundamental hypotheses are verified through numerical analysis of hysteretic responses of discretized beam-column models.     

The effects of out-of-phase biaxial-strain cycling on low-cycle fatigue

The effects of out-of-phase or nonsynchronous straining on low-cycle fatigue was investigated. Biaxial strains were imposed on thin-walled tubular 7075-T6 aluminum specimens by tension—compression and torsion. Phase angles of 0 deg, 30 deg, 45 deg, 60 deg and 90 deg were applied between two strains. It was found that out-of-phase cycling has an effect on the failure mode in the low-cycle-fatigue range. An analysis based on the maximum total strain in three-dimensional strain is proposed for treating “out-of-phase” straining conditions in low-cycle fatigue.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Stability and coupled dynamics of three-dimensional dual inverted flags

Fluid–structure interaction of an inverted flag, which has a free leading edge and a clamped trailing edge, has drawn attention recently because of its novel properties such as divergence stability, a low stability threshold, and large-amplitude flapping motion. In this study, the stability and flapping behaviors of dual inverted flags with finite height are investigated for a side-by-side arrangement, and their noticeable characteristics are compared to those of dual conventional flags. The critical velocity at which the inverted flags break the equilibrium of a straight configuration reduces monotonically when a gap distance between the two flags becomes smaller and an aspect ratio becomes larger, which is also predicted by our linear stability analysis using simple theoretical models of two-dimensional flags and slender flags. After bifurcation, in addition to the synchronized in-phase and out-of-phase modes commonly observed in dual conventional flags, a novel attached mode appears which is mainly observed for small gap distance and small aspect ratio. In this non-linear mode, the leading edges of the two inverted flags touch each other on a midline, and the deformed inverted flags maintain static equilibrium. In a non-linear flapping regime, a new mechanism of a mode transition from an out-of-phase mode to an in-phase mode is identified, which is allowed by the collision of the two flags flapping with large amplitude.     

Stability of an equilibrium position of a pendulum with step parameters

A mathematical pendulum affected by parametric disturbance with potential energy being periodic step function is considered. Non-linear equation of the pendulum depends on two parameters characterizing the mean value in time of the parametric disturbance and range of its “ripple”. Values of the parameters can be set arbitrarily. The non-linear problem of stability for two particular solutions of the equation corresponding to a hanging and inverse pendulum is solved.     

Buckling of spherical sandwich shells

An experimental investigation was carried out to determine the critical buckling loads of several shallow spherical sandwich shells. A cold-forming process simultaneously using pressure and vacuum was employed to manufacture the nearly perfect spherical facing layers from 5052 aluminum-alloy sheets of 0.006 and of 0.008-in. thicknesses. Eight shallow spherical-shell specimens of 20-in. base diameter and of 20 and 30-in. radii with 1/8 and 1/4-in. thickness of “Flexcore” have been tested in a 300-psi autoclave specifically designed for these experiments. The pressure on shells was developed by the differential pressure between the inner and the outer chambers separated by the shell being tested. When the inner chamber was maintained at atmospheric pressure and gas pressure was applied in the outer chamber, the testing procedure was termed “soft.” Alternatively, the inner chamber would be filled with fluid with the outer chamber remaining filled with gas. By initially pressurizing both chambers equally, a load on the shell could be developed by the differential pressure due to controlled bleeding of the fluid inside the inner chamber, while the gas in the outer chamber was maintained at the initial pressure. This is an accurate volume-control experiment and this testing procedure was termed “hard.” In the latter case, it was possible to monitor the displacements of the shell for each load increment with a nest of clip gages of an unique design. It was found that there is no substantial difference in the buckling loads between the hard and “soft” systems. All shells buckled in the plastic range. A reasonably good correlation is obtained with a linear theory using the double modulus for the sandwich segments.     

一种特殊瞬态动力学现象释疑

引入瞬态平衡点的概念 ,来解释一种简单的摆式减振器模型中的瞬态动力学行为中的一种特殊现象 :即振子从开始小幅振动突然增大最后达到大幅振动 ,而减振摆也从铅垂位置附近的小幅振动过渡到绕悬挂点的圆周运动。分析表明 ,产生此现象的原因是系统瞬时平衡点的剧烈变化     

An experimental investigation of one- and two-degree of freedom VIV of cylinders

In the paper,an experiment investigation was conducted for one-and two-degree of freedom vortex-induced vibration(VIV) of a horizontally-oriented cylinder with diameter of 11 cm and length of 120 cm.In the experiment,the spring constants in the cross-flow and in-line flow directions were regulated to change the natural vibration frequency of the model system.It was found that,in the one-degree of freedom VIV experiment,a "double peak" phenomenon was observed in its amplitude within the range of the reduced velocities tested,moreover,a "2T" wake appeared in the vicinity of the second peak.In the two-degree of freedom VIV experiment,the trajectory of cylinder exhibited a reverse "C" shape,i.e.,a "new moon" shape.Through analysis of these data,it appears that,besides the non-dimensional in-line and cross-flow natural vibration frequency ratios,the absolute value of the natural vibration frequency of cylinder is also one of the important parameters affecting its VIV behavior.     

Pendulum motions of extended lunar space elevator

In the usual everyday life, it is well known that the inverted pendulum is unstable and is ready to fall to “all four sides,” to the left and to the right, forward and backward. The theoretical studies and the lunar experience of moon robots and astronauts also confirms this property. The question arises: Is this property preserved if the pendulum is “very, very long”? It turns out that the answer is negative; namely, if the pendulum length significantly exceeds the Moon radius, then the radial equilibria at which the pendulum is located along the straight line connecting the Earth and Moon centers are Lyapunov stable and the pendulum does not fall in any direction at all. Moreover, if the pendulum goes beyond the collinear libration points, then it can be extended and manufactured from cables. This property was noted by F. A. Tsander and underlies the so-called lunar space elevator (e.g., see [1]). In the plane of the Earth and Moon orbits, there are some other equilibria which turn out to be unstable. The question is, Are there equilibria at which the pendulum is located outside the orbital plane? In this paper, we show that the answer is positive, but such equilibria are unstable in the secular sense. We also study necessary conditions for the stability of lunar pendulum oscillations in the plane of the lunar orbit. It was numerically discovered that stable and unstable equilibria alternate depending on the oscillation amplitude and the angular velocity of rotation. The study of the lunar elevator dynamics originates in [2]. The concept of lunar elevator was developed in detail in [3, 4]. Several classes of equilibria with the finiteness of the Moon size taken into account were studied in [5]. The possibility of location of an orbital station fixed to the Moon surface by a pair of tethers was investigated in [6]. The problem of orientation of the terminal station of the lunar space elevator was studied in [7]. The influence of the tether length variations on the motion of the lunar tether system was considered in [8]. The alternation of stable and unstable flat oscillations is well known in the problem of satellite oscillations in a circular orbit [9, 10].     

泡沫铝夹芯双圆管结构的准静态轴向压缩性能研究

对泡沫铝夹芯双圆管结构的准静态轴压性能进行了实验研究,发现该新型结构的比质量吸能效率远远高于传统的泡沫铝夹芯单管,并接近甚至超过相应的空管结构;其内外管变形模式均与空管不同且受内外管组合的影响.本文讨论了它的变形机理,分析了外管壁厚对其压缩行为的影响,发现增大外管壁厚有利于增大结构的行程利用率,提高结构的比质量能量吸收...     

低频振动隔离和能量采集双功能超材料

振动隔离和能量采集一体化是一种能够将有害振动隔离并转化为电能收集利用的动力学机制. 本文从局域共振超材料存在低频带隙特性出发, 研究了振动隔离和能量采集双功能超材料的动力学行为. 通过在球型磁腔内放置固接了感应线圈的球摆构成具有能量采集功能的球摆型谐振器, 并将其周期性的放置在基体梁中, 可以将带隙频率范围内的振动聚集在谐振器内, 以实现振动隔离和能量采集双功能. 建立了横向激励下双功能超材料梁的动力学方程, 应用Bloch's定理得到超材料的能带结构, 通过有限元仿真验证了理论模型和研究方法. 研究了不同参数下超材料梁的带隙特性. 进一步将一维拓展到二维, 研究了二维双功能超材料板的振动隔离和能量采集性能. 最后, 设计并建造了振动隔离和能量采集一体化双功能超材料动力学实验平台, 解析、数值和实验结果表明, 在局域共振带隙的频率范围内, 超材料梁主体的振动明显被抑制, 与此同时, 振动被局限在谐振器中, 使采集到的电压达到了最大值. 通过对附加谐振器和没有附加谐振器的能带结构和幅频响应的对比, 发现球摆型谐振器的加入可以在低频范围内形成了一个局域共振带隙, 有效提高了超材料梁在低频处的振动隔离和能量采集性能.      

Global Dynamics of a Coupled Chemotaxis-Fluid Model on Bounded Domains

This paper is concerned with the long-time behavior of large amplitude classical solutions to an initial-boundary value problem of a coupled chemotaxis-fluid model which describes the so-called “chemotactic Boycott effect” arising from the interplay of chemotaxis and diffusion of nutrients or signaling chemicals in bacterial suspensions. The result is proved via energy method.     

均匀来流条件下并行排列旗帜耦合运动模式的实验

利用高速摄影技术在低速风洞中记录了不同间距并行排列的两个旗帜在不同来流速度中的耦合运动。利用自编的时间-空间演化图像处理软件分析总结了旗帜的耦合运动模式以及旗帜摆动振幅、频率和St数的变化规律。实验结果显示,随着排列间距和来流速度的变化,两旗帜可能以静止、同向摆动、反向摆动和过渡状态这四种不同的模态耦合运动。两旗帜同向摆动时摆动频率明显低于单个旗帜在相同来流中的值,反向摆动时情况相反。在过渡状态中两旗帜摆动的振幅交替增减并且运动中同时包含有两个频率,而同向和反向摆动都是单频率的运动。     

Transition phenomena in the wake of a square cylinder

The transition phenomena in the wake of a square cylinder were investigated. The existence of mode A and mode B instabilities in the wake of a square cylinder was demonstrated. The critical Reynolds numbers for the inception of these instability modes were identified through the determination of discontinuities in the St–Re curves, and were found to have mean values of 160 and 204 for the onset of mode A and B instabilities, respectively. The spectra and time traces of the wake streamwise velocity component were found to display three distinct patterns in laminar, mode A and mode B flow regimes. Streamwise vortices with different wavelength at various Reynolds numbers were observed through different measures. The symmetries and evolution of the secondary vortices were observed using laser-induced-fluorescent dye. It was found that, just like the case of a circular cylinder, the secondary vortices from the top and bottom rows were out-of-phase with each other in the mode A regime, but in-phase with each other in the mode B regime. From the flow visualization, it was qualitatively proven that there is stronger interaction between braid regions in the mode B regime. At the same time, analysis of PIV measurements quantitatively demonstrated the presence of the stronger cross flow in mode B regime when compared to the mode A regime. It suggests that the in-phase symmetry of the mode B instability is the result of strong interaction between the top and bottom vortex rows. It was also observed that although the vorticity of the secondary vortices in the mode A regime was smaller, its circulation was more than twice that of mode B instability. Compared to primary vortices, the circulations of both mode A and mode B vortices were much smaller, which indicates that the secondary vortices most likely originate from the primary vortices. The wavelengths of the streamwise vortices in the mode A and B regimes were measured using the auto-correlation method, and were found to be 5.1 (±0.1)D, 1.3 (±0.1)D, and 1.1 (±0.1)D at Re=183 (mode A), 228 and 377 (both mode B), respectively. From the present investigation, mode A instability was likely to be due to the joint-effects of the deformation of primary vortex cores and the stretching of vortex sheets in the braid region. On the other hand, mode B instability was thought to originate from the “imprinting” process.     

In-plane perturbation of the tunnel-crack under shear loading I: bifurcation and stability of the straight configuration of the front

One considers a planar tunnel-crack embedded in an infinite isotropic brittle solid and loaded in mode 2+3 through some uniform shear remote loading. The crack front is slightly perturbed within the crack plane, from its rectilinear configuration. Part I of this work investigates the two following questions: Is there a wavy “bifurcated” configuration of the front for which the energy release rate is uniform along it? Will any given perturbation decay or grow during propagation? To address these problems, the distribution of the stress intensity factors (SIF) and the energy release rate along the perturbed front is derived using Bueckner–Rice's weight function theory. A “critical” sinusoidal bifurcated configuration of the front is found; both its wavelength and the “phase difference” between the fore and rear parts of the front depend upon the ratio of the initial (prior to perturbation of the front) mode 2 and 3 SIF. Also, it is shown that the straight configuration of the front is stable versus perturbations with wavelength smaller than the critical one but unstable versus perturbations with wavelength larger than it. This conclusion is similar to those derived by Gao and Rice and the authors for analogous problems.     

Natural frequencies of thin,rectangular plates with holes or orthotropic “patches” carrying an elastically mounted mass

The approximate solution for the title problem is obtained in the case of simply supported and clamped rectangular plates made of isotropic or orthotropic materials. A variational approach (the well known Rayleigh–Ritz method) is used, where the displacement amplitude is expressed in terms of beam functions. This means that each coordinate function satisfies identically all the boundary conditions at the outer edge of the plate. Free vibration analysis has been performed on various different cases; solid isotropic and orthotropic plates, orthotropic plates with a hole and isotropic plates with an orthotropic inclusion or “patch”, carrying an elastically mounted concentrated mass. It is important to point out that the case of an orthotropic patch is interesting from a technological viewpoint since it constitutes a model of a repair implemented on the virgin structural element when it has suffered damage. This approach has been implemented by the aeronautical industry in some instances. The obtained results are in very good agreement with those of particular cases of simply supported plates available in the literature.     

Normal modes of a double pendulum at low energy levels

Nonlinear Dynamics - This study is concerned with a double pendulum and its regular behaviour associated with low energy levels and the influence of the associated initial conditions on the...