Resonance Case Research Articles

Scalar Love numbers and Love symmetries of 5-dimensional Myers-Perry black holes

The near-zone “Love” symmetry resolves the naturalness issue of black hole Love number vanishing with SL (2, ℝ) representation theory. Here, we generalize this proposal to 5-dimensional asymptotically flat and doubly spinning (Myers-Perry) black holes. We consider the scalar response of Myers-Perry black holes and extract its static scalar Love numbers. In agreement with the naturalness arguments, these Love numbers are, in general, non-zero and exhibit logarithmic running unless certain resonant conditions are met; these conditions include new cases with no previously known analogs. We show that there exist two near-zone truncations of the equations of motion that exhibit enhanced SL (2, ℝ) Love symmetries that explain the vanishing of the static scalar Love numbers in the resonant cases. These Love symmetries can be interpreted as local SL (2, ℝ) SL (2, ℝ) near-zone symmetries spontaneously broken down to global SL (2, ℝ) × U (1) symmetries by the periodic identification of the azimuthal angles. We also discover an infinite-dimensional extension of the Love symmetry into SL (2, ℝ) ltimes hat{U}{(1)}_{mathcal{V}}^2 that contains both Love symmetries as particular subalgebras, along with a family of SL (2, ℝ) subalgebras that reduce to the exact near-horizon Myers-Perry black hole isometries in the extremal limit. Finally, we show that the Love symmetries acquire a geometric interpretation as isometries of subtracted (effective) black hole geometries that preserve the internal structure of the black hole and interpret these non-extremal SL (2, ℝ) structures as remnants of the enhanced isometry of the near-horizon extremal geometries.

Flexi Pack Dubai – a case of strategic resonance

Flexi Pack Dubai – a case of strategic resonance

Jump and Pull-in Instability of a MEMS Gyroscope Vibrating System.

Jump and pull-in instability are common nonlinear dynamic behaviors leading to the loss of the performance reliability and structural safety of electrostatic micro gyroscopes. To achieve a better understanding of these initial-sensitive phenomena, the dynamics of a micro gyroscope system considering the nonlinearities of the stiffness and electrostatic forces are explored from a global perspective. Static and dynamic analyses of the system are performed to estimate the threshold of the detecting voltage for static pull-in, and dynamic responses are analyzed in the driving and detecting modes for the case of primary resonance and 1:1 internal resonance. The results show that, when the driving voltage frequency is a bit higher than the natural frequency, a high amplitude of the driving AC voltage may induce the coexistence of bistable periodic responses due to saddle-node bifurcation of the periodic solution. Basins of attraction of bistable attractors provide evidence that disturbance of the initial conditions can trigger a jump between bistable attractors. Moreover, the Melnikov method is applied to discuss the condition for pull-in instability, which can be ascribed to heteroclinic bifurcation. The validity of the prediction is verified using the sequences of safe basins and unsafe zones for dynamic pull-in. It follows that pull-in instability can be caused and aggravated by the increase in the amplitude of the driving AC voltage.

Nonlinear vibration and resonance analysis of a rectangular hyperelastic membrane surrounded by nonlinear elastic foundation

In this article, the nonlinear vibration analysis of a rectangular hyperelastic membrane embedded within the nonlinear elastic foundation exposed to uniformly distributed hydrostatic pressure is investigated. The aim is to study the nonlinear vibration behaviors of a rectangular hyperelastic membrane when both primary and 1:1 internal resonances happen in the system. Also, by considering the system embedded within the nonlinear elastic foundation, the model of the structure has been developed close to the real situation. The structure is made of an isotropic, homogeneous, incompressible, and hyperelastic material. To model the system, the neo-Hookean hyperelastic constitutive law is used. The nonlinear elastic foundations include two linear terms Winkler and Pasternak, and a nonlinear cubic term. The governing equations are obtained with Hamilton’s principle of thin hyperelastic membrane and the assumption of finite deformations. Then, using Galerkin’s method, the equation of motion in the transverse direction, which is considered in two modes, is discretized. The resonant case includes the 1:1 internal resonance between two modes and primary resonance, in which the excitation frequency is close to the natural frequency of the first mode. In this analysis, the responses of nonlinear vibration behavior, including primary and 1:1 internal resonances, are obtained with analytical and numerical methods. The analytical and numerical methods are, respectively, the multiple scales method and the fourth-order Runge–Kutta method. Finally, the effects of the elastic foundation parameters and geometrical characteristics on the nonlinear vibration behavior are examined.

Nonlinear regimes of inertial wave attractors generated by a precessing lid: Zonal flows and Rossby waves

Fluid flow excited by inertial waves in a rotating annulus with a trapezoidal cross section is experimentally investigated. The forcing is introduced by the precession of a flat annular lid bounding the cavity. Such a setup allows to investigate interactions between inertial waves and zonal flows in the presence of equivalent topographic beta-effect, and its dynamic behavior under weak and strong forcing is experimentally studied for the first time. Due to the specific dispersion relation of inertial waves, a cavity shape supports wave attractors that appear only for retrograde lid precession. At a relatively low wave intensity, a zonal flow is generated in the vicinity of a “virtual” axial cylinder, the radius of which coincides with the radius of the focusing reflection from the conical bottom. A succession of non-linear regimes is observed as forcing increases, starting with a clearly identifiable case of triadic resonance. Further, the frequency spectrum is progressively enriched by emergence of additional discrete components, gradually reaching the state of “overheated” wave turbulence with a significant continuous background of the frequency spectrum. Finally, a strong low-frequency component develops in the frequency spectrum, and the continuous background of the spectrum decreases dramatically. The low-frequency component corresponds to azimuthally periodic structures identified as Rossby waves. In the case of strong forcing, a long-term variability of zonal-flow velocity at the timescale of tens of forcing periods is observed, which is likely a consequence of the interaction of azimuthal structures corresponding to a rich wave number set.

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He’s definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, “the non-perturbative approach”, that deals with the obtained equation has been introduced.

A well-balanced finite volume scheme based on planar Riemann solutions for 2D shallow water equations with bathymetry

We consider in this paper a finite volume scheme based on local planar Riemann solutions for the two-dimensional shallow water equations with bathymetry. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The scheme to be presented is a development of the preliminary works that will be cited below. Our foremost purpose is to extend those results to two-dimensional formalism while preserving the physical and mathematical properties, including the well-balancedness. The proposed scheme is applied to some specific families of solutions, especially lake at rest and partially well-balanced solution. The numerical results show that this approach can give a good accuracy, except for resonant cases. Furthermore, it is proved that our finite volume scheme can preserve the C-property in the sense that it can capture exactly the lake at rest solution.

Broadband multi-channel quantum noise suppression and phase-sensitive modulation based on entangled beam

We present a theoretical scheme for broadband multi-channel quantum noise suppression and phase-sensitive modulation of continuous variables in a coupled resonant system with quantum entanglement properties. The effects of different coupling strengths, pumping power in suppressing quantum noise and controlling the width of quantum interference channels are analyzed carefully. Furthermore, quantum noise suppression at quadrature amplitude is obtained with phase-sensitive modulation. It shows that the entanglement strength of the output field and the quantum noise suppression effect can be enhanced significantly by a strong pumping filed due to interaction of pumping light with the nonlinear crystal. The full width at half maxima (FWHM) of the noise curve at the resonant peak (Δ = 0 MHz) is broadened up to 2.17 times compared to the single cavity. In the strong coupling resonant system, the FWHM at Δ = 0 MHz (Δ = ±3.1 MHz) is also broadened up to 1.27 (3.53) times compared to the weak coupling resonant system case. The multi-channel quantum interference creates an electromagnetically induced transparent-like line shape, which can be used to improve the transmission efficiency and stability of wave packets in quantum information processing and quantum memory.

Vibration Study of Rotating Rigid Hub and a Flexible Composite Beam under Simultaneous Primary-internal Resonance

In this paper, the vibration of a composite system consisting of a rotating rigid hub and a flexible thin-walled beam is considered and studied. The equation of motion is derived in the previous work of Warminski and Latalski. The method of multiple scale technique has been applied to obtain frequency response equations near the simultaneous internal and primary resonance case in the absence of the acceleration of the hub. The vibration stability at this resonance case is investigated from the frequency response equations and studied using Liapunov’s methods. The effects of changes in selected structural parameters on the vibrating system behavior are investigated and studied numerically. Through the performed studies of the effects of changes in selected system a shift of the steady state amplitudes and the multi-valued of the bent curves are observed and the steady state amplitudes of the beam and hub have decreasing in the instability regions for natural frequencies. Finally, a comparison with the papers of previously published work is reported.

Investigation of Bragg resonant reflection of linear and nonlinear water waves by multiple fixed floating structures based on viscous meshfree method

The Bragg resonant reflection of linear and nonlinear water waves over a classical Bragg structures is numerically studied. The Smoothed Particle Hydrodynamics (SPH) method is a Lagrangian meshfree method, and SPH is adopted based on the code DualSPHysics, which takes the effect of viscosity into consideration. The sensitivity of particle spacing and a benchmark test to the transmission coefficient is analyzed. In the propagation of both linear and nonlinear water waves over Bragg floating structures, the Bragg resonant reflection occurs when the wavelength (λ) is about double of the spacing (L). The phase upshift can be observed in the cases of linear and nonlinear water waves, which means the minimum transmission coefficient appears when 2L/λ>1.0, and the phase upshift occurs with the increase of the width of floating in the nonlinear water waves. Besides, in the resonant case with three or more floating structures, the transmitted wave height is reduced by more than 70.9% and 73.5% relative to the incident wave height under linear and nonlinear water waves, respectively. The results give a reference for the design of wave attenuation in floating periodic structures in both linear and nonlinear water waves based on the Bragg rules.

Test and analysis of multi-cavity particle damper for horizontal vibration control of pipeline structures

Undesirable vibration in pipeline structures due to internal gas/liquid coupling, mechanical vibration, and water hammer action may cause machine shutdown, leakage, fatigue failure, and sometimes catastrophic explosion. It is necessary to design a device to mitigate pipeline vibration. This paper proposes a new type of particle damper, the multi-cavity particle damper (MPD). A mechanical MPD model is first proposed. Second, the analytical solutions of the MPD are given for the case of two symmetric impacts per cycle. A corresponding method for numerically simulating more general cases is also proposed, and the optimal movement distance is established using the analytical solution. Third, shaking table tests are carried out on pipeline structures controlled by an MPD subjected to harmonic excitation, and the control effects are verified. Finally, a parametric study is conducted, and the MPD design is discussed. The MPD is found to have a significant damping effect that can reduce the response by 98% in resonant cases. Shaking table tests verify the analytical solution, numerical simulation, and optimization method.

Fucik spectrum with weights and existence of solutions for nonlinear elliptic equations with nonlinear boundary conditions

We consider the boundary value problem $$\displaylines{ - \Delta u + c(x) u = \alpha m(x) u^+ - \beta m(x) u^- +f(x,u), \quad x \in \Omega, \cr \frac{\partial u}{\partial \eta} + \sigma (x) u =\alpha \rho (x) u^+- \beta \rho (x) u^- +g(x,u), \quad x \in \partial \Omega, }$$ where \((\alpha, \beta) \in \mathbb{R}^2\), \(c, m \in L^\infty (\Omega)\), \(\sigma, \rho \in L^\infty (\partial\Omega)\), and the nonlinearities f and g are bounded continuous functions. We study the asymmetric (Fucik) spectrum with weights, and prove existence theorems for nonlinear perturbations of this spectrum for both the resonance and non-resonance cases. For the resonance case, we provide a sufficient condition, the so-called generalized Landesman-Lazer condition, for the solvability. The proofs are based on variational methods and rely strongly on the variational characterization of the spectrum. See also https://ejde.math.txstate.edu/special/02/m2/abstr.html

Mitigation of high-speed train-induced environmental ground vibrations considering open trenches in the soft soil conditions by in-situ tests

This study focuses on the use of open trench wave barriers to prevent undesired high-speed train-induced ground vibrations. A series of full-scale passive isolation tests are performed in the most unfavorable geotechnical conditions at the highest speed passage of the actual moving load on the elevated embankment in resonance case. For this purpose, Sakarya-Pamukova has been considered the target test area of the Istanbul-Ankara HST line in North-Western Turkey. Site-specific acceleration time-histories, both response and frequency spectra in all directions were obtained from the ground motions recorded simultaneously in the far free-field conditions with and without passive isolation applications. In particular, the variation of the surface vibrations perpendicular to the railway line with the trench effect was observed according to the insertion loss in decibels. Results indicated that the peak values of measured velocity time-histories and spectral velocity response at ground surface decrease remarkable in the transversal vibration direction with passive isolation method, even in superficial depth of the open trench. The data measured at the most unfavorable geo-dynamic conditions will contribute to the development of more cost-effective, environmentally friendly and efficient vibration isolation barriers for researchers.

Nonlinear modelling of unimorph and bimorph magneto-electro-elastic energy harvesters

This paper nonlinearly models cantilever-based functionally graded magneto-electro-elastic energy harvesters (FGMEEEH) for the first time. The coupled magneto-electro-mechanical model is obtained on the basis of the Euler-Bernoulli beam theory. A hybrid procedure including Ritz's method is then utilized to generate reduced order models for both asymmetric unimorph and symmetric bimorph configurations. The resulting sets of initial value problems, whose convergence will be examined, are then analytically solved using the method of multiple time scales for both the free vibrations and primary resonance cases. The analytical time-histories of the system are compared by those obtained numerically and excellent agreements between them are observed. In addition, simplifying the frequency response function of the system in the primary resonance case, the present findings are validated by those available in the literature for linear unimorph systems. The influences of the harvester configurations, base acceleration amplitude, the value of the tip mass, the material gradation index as well as the resistances of the piezoelectric and magnetic circuits on the nonlinear response of the system are also studied in detail. It is observed that FGMEEEEHs enjoy much more efficiency in comparison to piezoelectric-based systems.

Analytical and numerical study of a vibrating magnetic inverted pendulum

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable.

Resonant enhancement of an X-ray laser electric field inside an atom

In the frame of a time-dependent oscillator model, the field induced by the coupling between the atomic electric dipole and an external X-ray laser field is investigated. The aim of the paper is to asses if in the resonant case (the frequency of the incoming field matches the frequency of the atom’s collective dipole oscillation) the damped collective oscillation induced by the external time-varying electric field can produce a significant amplification of the total electric field at the nucleus position.

Initial release condition dependent decoupling motion of a Wilberforce pendulum in the case of near resonance

The vertical and angular oscillation of a Wilberforce pendulum was found to be coupled in the case of resonance. Many studies have discussed the effects of the physical properties of a Wilberforce pendulum on coupled motion. In this article, we mainly study the influence of initial release conditions on the coupled motion of a Wilberforce pendulum in the near resonance case. The experimental results clearly exhibit that the coupled motion phenomenon depends on the initial release condition. Theoretical calculation further demonstrates that the coupled strength between two normal modes is determined by the ratio of initial angular to vertical displacement in the case of near resonance. The experimental results also verify the calculated variation of decoupling coefficient with this displacement ratio. Our study indicates that even in the case of resonance, the motion of a Wilberforce pendulum is probably decoupled at a given initial release condition. Complete decoupled motion is found to appear only when the damping coefficients in vertical and angular motion generate equal energy dissipation rates. The corresponding decoupling phenomenon in the case of resonance, occurs when the initial release energy of vertical and angular motion is equal.

Methodology of investigation the dynamics of longitudinally moving systems under the action of impulse perturbations

A methodology for researching oscillations of one-dimensional elastic systems under the influence of impulse disturbances has been developed. The mathematical model of the considered process is boundary value problems for hyperbolic equations of the second order with an irregular right-hand side and a mixed derivative with respect to linear and time variables. As for the irregularity of the right part, it takes into account the impulse effect on the research object and has the most important theoretical and practical periodic character; and with the help of the mixed derivative in the Euler variables, the longitudinal movement of the studied systems is partially taken into account. Using the basic ideas of asymptotic integration of equations with partial derivatives, the basic propositions of the wave theory of motion adapted to the considered class of problems, the principle of single-frequency oscillations in nonlinear systems with concentrated masses and distributed parameters, the method of regularization of impulse disturbances, equations in the standard form were obtained that describe the determining parameters of nonlinear oscillations for resonant and non-resonant cases. It is shown: that the longitudinal movement of the systems has a significant effect on the natural frequency of oscillations, and therefore on the period of impulse disturbances during which the resonance process takes place; the amplitude of the passage of resonance depends significantly on the speed of movement. The results obtained in the work can serve as a basis for choosing the operating modes of some classes of systems characterized by longitudinal movement in order to avoid resonant processes in them.

Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system

This paper focuses on the dynamical analysis of the motion of a new three-degree-of-freedom (DOF) system consisting of two segments that are attached together. External harmonic forces energize this system. The equations of motion (EOM) are derived utilizing Lagrangian equations, and the approximate solutions up to the third order are investigated using the methodology of multiple scales. A comparison between these solutions and numerical ones is constructed to confirm the validity of the analytic solutions. The modulation equations (ME) are acquired from the investigation of the resonance cases and the solvability conditions. The bifurcation diagrams and spectrums of Lyapunov exponent are presented to reveal the different types of the system’s motion and to represent Poincaré maps. The piezoelectric transducer is connected to the dynamical system to convert the vibrational motion into electricity; it is one of the energy harvesting devices which have various applications in our practical life like environmental and structural monitoring, medical remote sensing, military applications, and aerospace. The influences of excitation amplitude, natural frequency, coupling coefficient, damping coefficient, capacitance, and load resistance on the output voltage and power are performed graphically. The steady-state solutions and stability analysis are discussed through the resonance curves.

Dynamical analysis and numerical verification of a non-smooth nonlinear energy sink

The performance of the vibration reduction of a lightweight non-smooth nonlinear energy sink (NES) attached to a two-story linear damped primary structure (PS) is studied through dynamical analysis and numerical verification. First, a three-degree-of-freedom non-smooth nonlinear system is formulated to describe the coupled dynamics involving the PS and non-smooth NES, in which nonlinearity is induced by the piecewise linear restoring force of the NES and described by a non-smooth function. Second, when a 1:1:1 internal resonance case of the coupled system is considered, a modified complex-average technique and the method of multiple scales are employed to obtain a set of algebraic equations between the vibration amplitudes of the PS and attached non-smooth NES. The stability conditions and initial critical energy for activating targeted energy transfer (TET) can be analytically obtained. Finally, the linear stiffness, equivalent viscous damping coefficients and impact clearance of the non-smooth NES are numerically optimized by the peak estimates. The initial critical energy for the TET of NES is numerically verified to show the excellent performance of vibration reduction for the non-smooth NES by comparing with other attached dissipation devices under a wide range of impulsive loads.