Critical Amplitude Research Articles

External excitation effects on the flutter characteristics of a 2-DOF rigid rectangular panel

The flutter of a rigid panel is considered as a super/subcritical Hopf bifurcation. However, the initial conditions of such characteristics have not been extensively explored. In this study, a 2-DOF rigid rectangular panel was tested in a wind tunnel at zero angle of attack to investigate the flutter stability, with and without external excitations, in the Reynolds number range of 6.15–8.21×104. Coupled flutter occurs abruptly and evolves into four stages as the inflow wind velocity reaches a critical flutter velocity, Ucr. A hysteresis loop is observed when the wind velocity is decreased from Ucr to 0.75Ucr. External excitations are applied to the test model stepwise in the hysteresis region to examine the stability conditions. The critical excitation amplitude, Acr, is determined, which corresponded to the unstable limit cycle of the subcritical Hopf bifurcation. A fitted equation of the bifurcation is provided to correlate the flutter amplitude and the critical excitation amplitude with wind speed. PIV measurements show the flow field evolution around the panel during the entire flutter occurrence process. The size of the leading-edge vortex expands as the oscillation amplitude increases, then convects to the trailing edge and eventually sheds into the wake region. Wake patterns of each flutter stage were illustrated and analyzed using the Dynamic Mode Decomposition (DMD) method.

Effects of differential subgrade settlement on damage distribution and mechanical properties of CRTS II slab track

Differential subgrade settlement adversely affects the slab track. In this study, a three-dimensional model of a slab track based on damage mechanics theory was established for analysing the effects of differential subgrade settlement on the damage distribution and mechanical properties of the slab track. The prestressed steel bars and ordinary steel bars in the precast slab of the slab track were refined. The rail supporting forces were employed as the train loads. The damage distribution, deformation, and longitudinal stress of the track under differential subgrade settlement cases were calculated using the model. The results indicated that with the increasing of settlement amplitude, the concrete base is damaged first. If the settlement amplitude continues to increase, the precast slab is damaged. The critical subgrade settlement amplitude of the initial damage of the track structure increases with the increasing of settlement wavelength. Under differential subgrade settlement, the steel bars in the precast slab were subjected to additional loads. However, the precast slab may also crack. Damage to the precast slab and concrete base destroys the integrity of the longitudinal connection system of the CRTS II slab track. Subgrade with differential settlement should be repaired in time to avoid serious damage to the track structure.

Linear stability of transversely modulated finite‐amplitude capillary waves on deep water

AbstractWe investigate the three‐dimensional linear stability of the periodic motion of pure capillary waves progressing in permanent form on water of infinite depth for the whole range of wave amplitudes. After introducing a coordinate transformation based on a conformal map for two‐dimensional steady capillary waves, we perform linear stability analysis of finite‐amplitude capillary waves in the transformed space. To solve the linearized equations for small amplitude disturbances, it is assumed that the wavelengths of the disturbances in the transverse direction are much longer than those in the propagation direction and, therefore, the disturbances are weakly three‐dimensional. This assumption along with the periodicity of solutions allows us to write the linearized equations as an eigenvalue problem in matrix form. Following a perturbation theory for matrices, we expand the solutions of this eigenvalue problem in terms of a small parameter measuring the weak three‐dimensionality, and numerically obtain approximate eigenvalues. For weakly three‐dimensional superharmonic disturbances, the numerical results demonstrate that the pure capillary waves are two‐dimensionally stable, but three‐dimensionally unstable for almost all wave amplitudes. On the other hand, for subharmonic disturbances that are known to be two‐dimensionally unstable, it is found that the long‐wavelength disturbances in the transverse direction reduce the two‐dimensional growth rate near the critical amplitude beyond which the pure capillary waves are unstable.

Nonlinear Supratransmission in Quartic Hamiltonian Lattices With Globally Interacting Particles and On-Site Potentials

Abstract We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

Nonlinear responses and bifurcations of a rotor-bearing system supported by squeeze-film damper with retainer spring subjected to base excitations

A high-speed rotating rotor system mounted on a moving vehicle is inevitably subjected to parametric excitations and exciting forces induced by base motions. Dynamic characteristics of a rotor-bearing system supported by squeeze-film damper with retainer spring subjected to unbalance and support motions are investigated. Using Lagrange’s principle, equations of motion for rotor system relative to a moving support are derived. Under base excitations, steady-state and transient responses are analyzed by frequency–amplitude curve, waveform, orbit, frequency spectrum, and Poincare map. Changing with rotating speed or base harmonic frequency, journal motions are analyzed by bifurcation diagram. The results indicate that under base axial rotation, increasing base angular velocity, first two critical speeds decrease but resonant amplitudes increase slightly. The journal whirls around the static eccentricity with noncircular orbit. Under base lateral rotation, critical speeds, and resonant amplitudes remain essentially unchanged, but orbit’s deviation is related to base angular velocity. Excited by base harmonic translation, the integral multiples of fundamental frequency $$k{\varOmega }\left( {k = 1,2} \right)$$ , base harmonic frequency $${\varOmega}^{z}$$ , and combined frequencies $$k{\varOmega } \pm j{\varOmega }^{z} { }\left( {k,j = 1,2} \right)$$ are stimulated, changing the motions from periodic to quasiperiodic. Overall, it provides a flexible approach with good expandability to predict dynamic characteristics of squeeze-film damped rotor system under base motions.

Nonlinear Approximations to Critical and Relaxation Processes

We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration.

Analytical study of thermal upheaval buckling for free spanning pipelines

Subsea pipelines with a free suspended span segment may be subject to upheaval buckling under high temperature, which may lead to a failure mode such as local buckling, fracture and fatigue. Mathematical models are established to study the upheaval buckling behaviour of subsea pipelines with a free span segment. Then, the accurate locations of the maxima of displacement and stress are obtained. The buckled configuration and the typical post-buckling behaviour and their dependence on free span length are analysed. Furthermore, a detailed analysis is presented for critical temperature difference, displacement amplitude and maximum axial compressive stress. Finally, the influence of touchdown on post-buckling behaviour during the process of upheaval buckling is discussed. The results show that the pipeline is prone to upheaval buckling when the free span length is large enough and the critical temperature difference decreases with increasing free span length. The displacement amplitude and the maximum stress within the span increase with the increase of the free span length before touchdown. However, after the pipeline touches the bottom of the free span, the maximum stress within the span decreases significantly compared to the case without touchdown, which becomes further smaller for smaller depth of the free span.

Alternating Shear Orientation During Cyclic Loading Facilitates Yielding in Amorphous Materials

The influence of alternating shear orientation and strain amplitude of cyclic loading on yielding in amorphous solids is investigated using molecular dynamics simulations. The model glass is represented via a binary mixture that was rapidly cooled well below the glass transition temperature and then subjected to oscillatory shear deformation. It was shown that periodic loading at strain amplitudes above the critical value first induces structural relaxation via irreversible displacements of clusters of atoms during a number of transient cycles, followed by an increase in potential energy due to the formation of a system-spanning shear band. Upon approaching the critical strain amplitude from above, the number of transient cycles required to reach the yielding transition increases. Interestingly, it was found that when the shear orientation is periodically alternated in two or three dimensions, the number of transient cycles is reduced but the critical strain amplitude remains the same as in the case of periodic shear along a single plane. After the yielding transition, the material outside the shear band continues strain-induced relaxation, except when the shear orientation is alternated in three dimensions and the glass is deformed along the shear band with the imposed strain amplitude every third cycle.

Contribution of the positional and orientational ordering in anisotropic particle-based MR fluids: static and dynamic rheological study

Flake shape-based MR fluids were prepared in a carrier with variable weight fraction of magnetite nanoparticles (0%, 1%, 2%, 3%, and 4%). The spherical shape particle-based MR fluid was purchased from LORD Corporation, USA, for comparison. The magneto-rheology of these fluids under dynamic conditions indicates the orientational ordering of flakes which increases due to the presence of nanoparticles, as it changes the sliding friction between the flakes. The presence of nanoparticles significantly reduces the elastic stress derived from storage modulus compared to that of pure MR fluid. The particle-matrix interaction can be analyzed from the cohesive energy density, Ec (the energy required to break all internal structure) using the critical strain amplitude and the storage modulus in the linear regime. The behavior of Ec is found to be dependent on the shape of the particles. These results are supported by thermal conductivity, sedimentation, and redispersion study.

Flapping of heavy inverted flags: a fluid-elastic instability

Abstract

Hyperuniform Structures Formed by Shearing Colloidal Suspensions.

In periodically sheared suspensions there is a dynamical phase transition, characterized by a critical strain amplitude γ_{c}, between an absorbing state where particle trajectories are reversible and an active state where trajectories are chaotic and diffusive. Repulsive nonhydrodynamic interactions between "colliding" particles' surfaces have been proposed as a source of this broken time reversal symmetry. A simple toy model called random organization qualitatively reproduces the dynamical features of this transition. Random organization and other absorbing state models exhibit hyperuniformity, a strong suppression of density fluctuations on long length scales quantified by a structure factor S(q→0)∼q^{α} with α>0, at criticality. Here we show experimentally that the particles in periodically sheared suspensions organize into structures with anisotropic short-range order but isotropic, long-range hyperuniform order when oscillatory shear amplitudes approach γ_{c}.

Bubble pressure requirements to control the bubbling process in forced co-axial air-water jets

We analyse the controlled generation of bubbles of a given size at a determined bubbling rate in a co-flowing water stream forcing the gas flow. The temporal evolution of the bubble size, R(t), the air flow rate, Qa(t), and the pressure evolution inside the bubble, pb(t), during the bubbling process are reported. To that aim, the temporal evolution of the bubble shape and the pressure inside the air feeding chamber, pc(t), where a harmonic perturbation is induced using a loudspeaker, are obtained from high-speed images synchronized with pressure measurements. A model is developed to describe the unsteady motion of the gas stream along the injection needle, coupled with the Rayleigh-Plesset equation for the growing bubble, allowing us to obtain pb(t). Thus, the minimum pressure amplitudes required inside the forming bubble to control their size and bubbling frequency are provided as a function of the gas flow rate, the liquid velocity, uw, and the forcing frequency, ff. Two different behaviors have been observed, depending on the liquid-to-gas velocity ratio, Λ=uw/ua. For small enough values of Λ, the critical pressure amplitude is given by ps∼ρacuaStf3, associated to a rapid pressure increase taking place during an interval of time of the order of the acoustic time. However, for larger values of Λ, ps∼ρuw2Stf3Λ−1/5We−1/4. Here ρ and ρa are the liquid and gas densities respectively, c the speed of sound in air and Stf=ffro/uw and We=ρuw2ro/σ the Strouhal and Weber numbers, where ro denotes the outer radius of the injector.

Thermal cycling effects on the structure and physical properties of granular materials

Over the past two decades, various investigations have shown that the application of thermal cycles of heating and cooling to granular materials causes permanent densification. However, prior to this study, significant associated effects of thermal cycles on the structural reorganization and physical properties of granular materials remained mostly unknown. To address these elusive aspects, this paper investigates the relationship between the microstructural changes induced by thermal cycling and their influence on the physical properties of granular materials using a multiphysical and multiscale computational approach. Microstructural changes are analyzed using coupled thermo-mechanical discrete element simulations and macroscopic physical properties are upscaled using homogenization. Multiple initial porosities and particle size distributions are investigated for a large number of thermal cycles of varying amplitude. Results demonstrate significant variations in the structure and properties of granular materials. Volumetric densification and fabric anisotropy increase monotonically with the amplitude and number of cycles. Coordination number exhibits a maximum for a critical temperature amplitude established theoretically to correspond to an optimal particle rearrangement. Mechanical stiffness and thermal conductivity increase in anisotropic fashion due to stress relaxation and fabric anisotropy and typically exhibit the most variation at the critical temperature amplitude. Intrinsic permeability to fluid flow decreases isotropically and monotonically with the amplitude and number of thermal cycles due to pore volume reduction. The critical temperature amplitude provides a limit to the thermal energy that can seemingly induce optimal and permanent structural reorganizations, as well as maximum variations of specific physical properties of granular materials.

Effect of temperature fluctuation on the localized pattern of action potential in cardiac tissue

Based on the improved FitzHugh–Nagumo myocardial model driven by a constant external current, the effect of temperature fluctuation in a network of electrically coupled myocardial cells are investigated through analytical and numerical computations. Through the technique of multiple scale expansion, we successfully reduced the complex nonlinear system of equations to a more tractable and solvable nonlinear amplitude equation on which the analysis of linear stability is performed. Interestingly from this analysis, a plot of critical amplitude of action potential versus wave number revealed the growth rate of modulational instability (MI) is an increasing function of the thermoelectric couplings; T^{(1)} and T^{(2)}, under fixed conditions of nonlinear electrical couplings. In order to verify our analytical predictions through the study the long-time evolution of the modulated cardiac impulses, numerical computation is finally carried out. Numerical experiment revealed the existence of localized coherent structures with some recognized features of synchronization. Through the mechanism of MI, changes in thermoelectrical couplings promote wave localization and mode transition in electrical activities in the cell lattice. Results could provide new insights in understanding the underlying mechanism of the manifestation of sudden heart disorder subjected to heavily temperature fluctuation.

Floquet analysis on a viscous cylindrical fluid surface subject to a time-periodic radial acceleration

Parametrically excited standing waves are observed on a cylindrical fluid filament. These are the cylindrical analog of Faraday instability in a flat surface or spherical droplet. Using the Floquet technique, linear stability analysis has been investigated on a viscous cylindrical fluid surface, which is subjected to a time-periodic radial acceleration. Viscosity has a significant impact on the critical forcing amplitude as well as the dispersion relation of the non-axis symmetric patterns. The effect of viscosity on onset parameters of the pattern with azimuthal wavenumber, $m=1$, has shown different dependency from $m>1$. The effect of viscosity increases with an increasing $m$ is also observed.

Efficient broadband vibration energy harvesting based on tuned non-linearity and energy localization

In this letter, non-linearity and energy localization are experimentally tuned in an electromagnetic vibration energy harvester in order to enhance its output performances. The non-linear device consists of two moving magnets guided by elastic beams and coupled by a repulsive magnetic force. The mechanical non-linearity is introduced by considering large displacements of the beams and the energy localization is achieved by mistuning the mass of one of the moving magnets. The critical load resistance is determined experimentally in order to tune the non-linearity level and to drive the harvester beyond its critical amplitude. Consequently, the performances of the device, in terms of harvested power only from the perturbed dof oscillator and frequency bandwidth, are enhanced respectively up to 19.4% and 116% compared to the performances of the non-linear periodic system.

Atomistic investigation of aging and rejuvenation in CuZr metallic glass under cyclic loading

The structural evolution and mechanical response of CuZr metallic glass (MG) under cyclic loading are investigated by molecular dynamics simulations. Either aging or rejuvenation was observed in the MG under cyclic loading with different cyclic strain amplitudes. Aging is more remarkable in the surface, while rejuvenation is correlated with the nucleation and development of shear transformation zone in the bulk of MG. The MG models with more structural heterogeneity were prepared by increasing the cooling rate during cooling process to exhibit larger degrees of structural evolution and aging during cyclic loading. While the fraction of ordered coordination polyhedron increases monotonically with increasing Cu content, the degree of aging during cyclic loading is not significantly impacted by Cu content. The glass-forming ability is the intrinsic character that governs the change in energy state and structural evolution of MG under cyclic loading. The critical cyclic strain amplitude for the transition from aging to rejuvenation is closely related to the initial structure of MG.

A delayed yielding transition in mechanically annealed binary glasses at finite temperature

The influence of strain amplitude, glass stability and thermal fluctuations on shear band formation and yielding transition is studied using molecular dynamics simulations. The model binary mixture is first gradually cooled below the glass transition temperature and then periodically deformed to access a broad range of potential energy states. We find that the critical strain amplitude becomes larger for highly annealed glasses within about one thousand shear cycles. Moreover, upon continued loading at a fixed strain amplitude, the yielding transition is delayed in glasses mechanically annealed to lower energy states. It is also demonstrated that nucleation of a small cluster of atoms with large nonaffine displacements precedes a sharp energy change associated with the yielding transition. These results are important for thermal and mechanical processing of amorphous alloys with tunable mechanical and physical properties.

Droplet Generation in a Flow-Focusing Microfluidic Device with External Mechanical Vibration.

The demand for highly controllable droplet generation methods is very urgent in the medical, materials, and food industries. The droplet generation in a flow-focusing microfluidic device with external mechanical vibration, as a controllable droplet generation method, is experimentally studied. The effects of vibration frequency and acceleration amplitude on the droplet generation are characterized. The linear correlation between the droplet generation frequency and the external vibration frequency and the critical vibration amplitude corresponding to the imposing vibration frequency are observed. The droplet generation frequency with external mechanical vibration is affected by the natural generation frequency, vibration frequency, and vibration amplitude. The droplet generation frequency in a certain microfluidic device with external vibration is able to vary from the natural generation frequency to the imposed vibration frequency at different vibration conditions. The evolution of dispersed phase thread with vibration is remarkably different with the process without vibration. Distinct stages of expansion, shrinkage, and collapse are observed in the droplet formation with vibration, and the occurrence number of expansion–shrinkage process is relevant with the linear correlation coefficient.

Passive vibration suppression in a coupled linear–bistable continuous module excited near resonance

Vibration suppression for marine structures has been an important concern for a long time. In this paper, a specifically designed continuous module that consists of an Euler beam, a preloaded bimorph beam and a coupled spring unit produces self-increasing damping and strongly suppresses the steady-state response near resonance. Based on Hamilton’s principle, we derive a two-degree-of-freedom coupled linear–bistable (CLB) model of this module. Utilizing the harmonic balance method and a numerical simulation, we analyze the proposed CLB model under a harmonic excitation. When the preloaded beam moves toward the critical amplitude of the saddle point of the system potential, the unstable equilibrium greatly increases the velocity phase difference between the Euler and preloaded beams and spontaneously enhances the dissipation power of the system. The self-increasing damping decreases the amplitude of the steady-state vibration of the coupled module excited near resonance by an order of magnitude. Numerical results demonstrate the essential roles of the damping ratio and the shape of the potential function in determining how this module can better suppress vibrations. This investigation provides opportunities to suppress dynamical responses using a continuous CLB module.