Nonlinear Curvature Effect Research Articles

Theory for highly stressed thick-walled elastic spheres and their vibration without small strain approximations

This paper considers how the radii and vibration of an elastic thick-walled sphere is affected by high stresses imposed by high external pressures or by high internal pressures. For simplicity, the sphere consists of idealized Hookean material allowing unlimited compression or expansion. The sphere is filled by material with an isotropic pressure that generally differs from the pressure on the outer surface of the sphere. The analysis includes large strain nonlinear curvature effects where, unlike linear elasticity theory, the difference in radial and tangential strains is not small compared to those strains. Despite the sphere’s robustness, destructive physical consequences are predicted for nonzero Poisson coefficients where, for instance, a high-pressure tank inner stresses make the inner radius approach the less stressed outer radius. This would be avoided if the Poisson coefficient changed with large strains and approached zero. To model the effect of high stresses and strains on breathing mode vibration, an effective spring constant is derived from quadratic radius deviations of the system potential energy from equilibrium. Insights from analysis avoiding linear elasticity approximations may be applicable to improved understanding of deep-sea marine creature survival, improved underwater vessel design for large depths, and safer containers of fluids at high pressures.

Nonlinear curvature effects in gravitational waves from inspiralling black hole binaries

Gravitational waves (GWs) from merging black holes allow for unprecedented probes of strong-field gravity. Testing gravity in this regime requires accurate predictions of gravitational waveform templates in viable extensions of general relativity. We concentrate on scalar Gauss-Bonnet gravity, one of the most compelling classes of theories appearing as the low-energy limit of quantum gravity paradigms, which introduces quadratic curvature corrections to gravity coupled to a scalar field and allows for black hole solutions with scalar charge. Focusing on inspiraling black hole binaries, we compute the leading-order corrections due to curvature nonlinearities in the GW and scalar waveforms, showing that the new contributions, beyond merely the effect of scalar field, appear at first post-Newtonian order in GWs. We provide ready-to-implement GW polarizations and phasing. Computing the GW phasing in the Fourier domain, we perform a parameter-space study to quantify the detectability of deviations from general relativity. Our results lay important foundations for future precision tests of gravity with both parametrized and theory-specific searches.

Nonlinear oscillations of viscoelastic microcantilever beam based on modified strain gradient theory

A viscoelastic microcantilever beam is analytically analyzed based on the modified strain gradient theory. The Kelvin Voigt scheme is used to model the beam viscoelasticity. Applying Bernoulli-Euler inextensibility of the centerline condition via Hamilton’s principle, the nonlinear equation of motion and related boundary conditions are derived based on shortening effect theory and discretized by Galerkin method. Inner damping, nonlinear curvature effect, and nonlinear inertia terms are applied. The generalized derived formulation in this article, allows modeling of any nonlinearity combinations such as nonlinear terms arises due to inertia, damping, and stiffness, as well as modeling the size effect via considering modified coupled stress or modified strain gradient theories. First mode nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated utilizing multiple time scale method and validated by numerical findings. Results indicate that the nonlinear terms have an appreciable effect on natural frequency and time response of a viscoelastic microcantilever. Furthermore, the investigation reveals that due to the size effects, natural frequency enhances drastically, especially when the thickness of the beam and the length scale parameter are comparable. Outcomes clarify the importance of size effects in analyzing of the mechanical behavior of small scale structures.

Wideband PZT energy harvesting from the wake of a bluff body in varying flow speeds

Wake-galloping flow energy harvesters vibrate linearly under fluid force which results in a narrow lock-in region. Hence, their efficiency drops significantly in variable flow speeds leading to inevitable discrepancies between the natural frequency of the energy harvester and vortex shedding frequency. This sensitive dependence on the flow speed hinders the performance of this type of energy harvesters. This paper present a novel design of a PZT energy harvester for capturing energy from a broad range of vortex shedding frequencies associated with varying wind speeds. The proposed system composed of an array of bimorphs placed in the downstream of a bluff body. First, the nonlinear partial differential equations (PDEs) governing the dynamics of aero-electromechanical system are obtained based on Euler-Bernoulli assumption and nonlinear curvature effects. Next, the ordinary forms (ODEs) of the PDEs are attained using a Galerkin's Schemes. Moreover, they are non-dimensionalized to generalize the results and analyses; a convergence analysis has been carried out to find the minimum number of modes required to the precise prediction of the response. Exploiting harmonic balance method (HBM), the approximate-analytical solutions of the ODEs are provided. Afterward, the effects of various properties of the proposed ensemble on the lock-in region are investigated. In addition, the efficiency of the energy harvester is assessed toward examining its optimal power and conversion factor. The results disclose that the proposed system has a wider lock-in region and larger conversion factor, compared to the typical wake-galloping energy harvesters. Hence, the superior performance of the proposed design is confirmed.

Nonlinear dynamic analysis of a Stockbridge damper

The purpose of this work is to validate a nonlinear mathematical model (finite element method) for dynamic simulation of Stockbridge dampers of electric transmission line cables. To obtain the mathematical model, a nonlinear cantilever beam with a tip mass was used. The mathematical model incorporates a nonlinear stiffness matrix of the element due to the nonlinear curvature effect of the beam. To validate the mathematical model, the numerical results were compared with experimental data obtained on a machine adapted from cam test. Five different circular cam profiles with eccentricities of 0.25, 0.5, 0.75, 1.25, and 1.5 mm were used. Vibration data were collected through three accelerometers arranged along the sample. A good concordance was found between the numerical and experimental data. The same behavior was observed in tests of another Stockbridge damper excited by a shaker. The nonlinear behavior of the system was evidenced.

Domain wall dynamics: Growth laws, localized structures and stable droplets

We discuss the relation between the dynamics of walls separating two equivalent domains and the existence of different kinds of localized structures in systems far from thermodynamic equilibrium. In particular we focus in systems displaying a modulational instability of a flat front where an amplitude equation for the dynamics of the curvature allows to characterize different growth regimes and to predict the existence of stable droplets, localized structures whose stability comes from nonlinear curvature effects.

A GEOMETRICALLY NON-LINEAR MODEL OF ROTATING SHAFTS WITH INTERNAL RESONANCE AND SELF-EXCITED VIBRATION

A geometrically non-linear model of the rotating shaft is introduced, which includes Kárman non-linearity, non-linear curvature effects, large displacements and rotations as well as gyroscopic effects. Through applying Timoshenko-type assumptions, the shear effects are also included in the model. Convenient matrix descriptions are used in order to facilitate the analysis based on Galerkin and continuation methods. The model is used to analyze the phenomenon of internal resonance. The influence of some of the system's parameters on the amplitude and frequency of self-excited vibration is investigated.

Theoretical determination of the critical conditions for the direct initiation of detonations in hydrogen-oxygen mixtures

The problem of the direct initiation of detonations by an energy source has been studied using both a one-step reaction model and a detailed chemistry for hydrogen-oxygen mixtures. The nonlinear curvature effect defines a minimum radius R c below which there is no generalised Chapman-Jourguet (CJ) detonation. Consequently, the detonation front with a radius smaller than R c cannot propagate in a self-sustained manner. For hydrogen-oxygen spherical detonations, this critical radius is about 500 times larger than the thickness of the planar CJ detonation front, in good agreement with our theoretical results based on a one-step reaction model. The so-determined critical conditions are applied on the direct initiation of a curved detonation. It has been found that the criterion based on the nonlinear curvature effect works well for the initiation of spherical or cylindrical detonations. The predicted critical radius and the critical energy are in good agreement with both numerical and experimental results. The influence of the one-dimensional intrinsic instability of the detonation front upon the direct initiation has also been examined via direct numerical simulations. For a sufficiently large activation energy, the strong instability appears in the propagation of the plane detonation front, a quasi-steady regime may be observed during the direct initiation for the initiation energy near the critical value. For very large activation energy, the planar detonation cannot propagate via auto-ignition mechanism, the detonation wave cannot be initiated even in the planar case.

Direct Initiation of Gascous Detonations

A theoretical analysis of the direct initiation of gaseous detonations by an energy source has been published recently, the results are recalled here. Nonlinear curvature effects are proved to be essential mechanisms controlling the critical condition. These effects are first studied in a quasi-steady (q-s) state approximation valid for a characteristic time scale much larger than the reaction time scale. Two branches of q-s solutions are exhibited with a C-shaped curve and a critical radius below which generalised Chapman-Jouguet (CJ) solutions cannot exist. At ordinary conditions this critical radius is much larger than the thickness of the plane CJ detonation front (typically 300 to 500 times larger). Direct numerical simulations show that the upper branch of q-s solutions, acts as an attractor of the unsteady blast waves originating from a sufficiently strong energy source. The criterion of initiation derived here works to a good approximation and exhibits the huge numerical factor (10 6 -10 8 ) which is observed in the experimental data of the critical energy source and which was not explained up to now. Transient may induce additional failure mechanisms close to the critical condition.

On the direct initiation of gaseous detonations by an energy source

A new criterion for the direct initiation of cylindrical or spherical detonations by a localized energy source is presented. The analysis is based on nonlinear curvature effects on the detonation structure. These effects are first studied in a quasi-steady-state approximation valid for a characteristic timescale of evolution much larger than the reaction timescale. Analytical results for the square-wave model and numerical results for an Arrhenius law of the quasi-steady equations exhibit two branches of solutions with a C-shaped curve and a critical radius below which generalized Chapman–Jouguet (CJ) solutions cannot exist. For a sufficiently large activation energy this critical radius is much larger than the thickness of the planar CJ detonation front (typically 300 times larger at ordinary conditions) which is the only intrinsic lengthscale in the problem. Then, the initiation of gaseous detonations by an ideal point energy source is investigated in cylindrical and spherical geometries for a one-step irreversible reaction. Direct numerical simulations show that the upper branch of quasi-steady solutions acts as an attractor of the unsteady blast waves originating from the energy source. The critical source energy, which is associated with the critical point of the quasi-steady solutions, corresponds approximately to the boundary of the basin of attraction. For initiation energy smaller than the critical value, the detonation initiation fails, the strong detonation which is initially formed decays to a weak shock wave. A successful initiation of the detonation requires a larger energy source. Transient phenomena which are associated with the intrinsic instability of the quasi-steady detonations branch develop in the induction timescale and may induce additional mechanisms close to the critical condition. In conditions of stable or weakly unstable planar detonations, these unsteady phenomena are important only in the vicinity of the critical conditions. The criterion of initiation derived in this paper works to a good approximation and exhibits the huge numerical factor, 106–108, which has been experimentally observed in the critical value of the initiation energy.

Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. II. Forced Motions

Abstract The nonplanar, nonlinear, resonant forced oscillations of a fixed-free beam are analyzed by a perturbation technique with the objective of determining quantitative and qualitative information about the response. The analysis is based on the differential equations of motion developed in Part I of this paper which retain not only the nonlinear inertia but also nonlinear curvature effects. It is shown that the latter play a significant role in the nonlinear flexural response of the beam.

Out-of-plane vibrations of a beam including non-linear inertia and non-linear curvature effects

The harmonic, non-linear response and its stability, for a clamped-clamped/sliding beam subject to a planar excitation is investigated by a perturbation method taking into account the non-linear inertia and the non-linear curvature terms in the differential equations of motion. The influence of excitation and beam parameters in the planar and in the non-planar response of the beam is determined in detail and illustrated by several ‘response charts’.