Primary Resonance Research Articles

Secular dynamics of stellar spin driven by planets inside Kozai–Lidov resonance

Abstract In many exoplanetary systems with ‘hot Jupiters’, it is observed that the spin axes of host stars are highly misaligned to planetary orbital axes. In this study, a possible channel is investigated for producing such a misalignment under a hierarchical three-body system where the evolution of stellar spin is subjected to the gravitational torque induced from the planet inside Kozai–Lidov (KL) resonance. In particular, two special configurations are explored in detail. The first one corresponds to the configuration with planets at KL fixed points, and the second one corresponds to the configurations with planets moving on KL librating cycles. When the planet is located at the KL fixed point, the corresponding Hamiltonian model is of one degree of freedom and there are three branches of libration centres for stellar spin. When the planet is moving on KL cycles, the technique of Poincaré section is taken to reveal global structures of stellar spin in phase space. To understand the complex structures, perturbative treatments are adopted to study rotational dynamics. It shows that analytical structures in phase portraits under the resonant model can agree well with numerical structures arising in Poincaré sections, showing that the complicated dynamics of stellar spin are governed by the primary resonance under the unperturbed Hamiltonian model in combination with the 2:1 (high-order and/or secondary) spin-orbit resonances.

Theoretical and Experimental Analysis on Vibration Absorber with Particle Damping

To better suppress low-frequency vibrations of flexible manipulators induced by the rotation of motors and eccentricity, a novel type of tuned particle damper (TPD) is designed by combining the advantages of classical dynamic vibration absorber (DVA) and particle dampers (PD). Compared to traditional DVAs, this TPD can reduce additional mass and effectively broaden the frequency band of the DVA. Firstly, an equivalent theoretical model is established to describe the frequency tuning principle of the designed TPD. Based on the theory of a single particle damper, the equivalent damping and stiffness of the particles are calculated through an approximate approach. Then, a three-degree-of-freedom vibration model of the manipulator system with the TPD is built, and the dynamical characteristic of the primary resonance for the coupled system are analyzed by the perturbation method. Finally, the experimental platform is set up to verify the theoretical results. A manipulator is applied to test the low-frequency vibration absorption of the designed TPD, and the vibration suppression effect is discussed both in theoretical analysis and experiments.

Three-dimensional resonance orbit design for binary asteroid system

The solar radiation pressure can affect significantly orbits around binary asteroid system (i.e., BAS). One novel three-dimensional resonance orbit family for BASs is constructed due to primary resonance and 1:1 internal resonance between the in-plane motion and the out-of-plane motion, by treating the solar radiation pressure as excitation. Since the periodic orbits are disturbed by the solar radiation, the multiple scales method can be used to derive the first-order approximated solution. Then, the stabilities of such orbit solutions are investigated. The energy transferring mechanism that “pumps” SRP energy from in-plane motion to out-of-plane motion is also illustrated. The accuracy of such approximated solutions is further verified by numerical iterations. The 7088 Ishtar BAS and1996 FG3 BAS are taken as examples to verify the proposed method.

Design of phononic crystal using open resonators as harmful gases sensor

This paper investigates the ability to use a finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect to detect the concentration of harmful gases such as CO2. This research investigates the impact of periodic open resonators, defect duct at the center of the structure, and geometrical parameters such as cross-sections and length of the primary waveguide and resonators on the model's performance. As far as we know, this research is unique in the sensing field. Furthermore, these simulations show that the investigated finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect is a promising sensor.

Investigation on Transmission Property of Warp-Knitted Spacer Fabric in Primary Resonance Using Nonlinear System Model

Investigation on Transmission Property of Warp-Knitted Spacer Fabric in Primary Resonance Using Nonlinear System Model

A Non-Linear Non-Planar Coupling Mechanism of Suspended Cables in Thermal Conditions

Slight variations induced by thermal effects may bring unexpected discrepancies in both the system’s linear and non-linear responses. The present study investigates the temperature effects on the non-linear coupled motions of suspended cables subject to one-to-one internal resonances between the in-plane and out-of-plane modes. The classical non-linear flexible system is excited by a uniform distributed harmonic excitation with the primary resonance. Introducing a two-mode expansion and applying the multiple scale method, the polar and Cartesian forms of modulation equations are obtained. Several parametric investigations have highlighted the qualitative and quantitative discrepancies induced by temperature through the curves of force/frequency-response amplitude, time history diagrams, phase portraits, frequency spectrum, and Poincaré sections. Based on the bifurcation and stability analyses, temperature effects on the multiple steady-state solutions, as well as static and dynamic bifurcations, it is observed that the periodic motions may be bifurcated into the chaotic motions in thermal environments. The saddle-node, pitch-fork, and Hopf bifurcations are sensitive to temperature changes. Finally, our perturbation solutions are confirmed by directly integrating the governing differential equations, which yield excellent agreement with our results and validate our approach.

Nonlinear primary resonance behavior of graphene platelet-reinforced metal foams conical shells under axial motion

Nonlinear primary resonance behavior of graphene platelet-reinforced metal foams conical shells under axial motion

Nonlinear vibrations of four-degrees of freedom for piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular plate with PPF control strategy

The nonlinear vibration suppression of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever (PFG-GRLCC) rectangular plate with positive position feedback (PPF) control strategy is investigated firstly. The material properties of the graphene-reinforced structure are calculated through the Halpin–Tsai micromechanical model. Considering the transverse external excitation and the converse effect of piezoelectricity, the governing equations of motion are formulated through von Karman large deformation theory, the classical laminated plate theory, and Hamilton principle. After adding the PPF controllers, a four-degrees of freedom model for the close-loop vibration control system is achieved via Galerkin truncation technique. The average equations in the case of the primary resonance and 1:1:3:3 internal resonance can be obtained using the multiple scale perturbation (MSP) method. The amplitude–frequency response curves are studied by the numerical continuation algorithm. The detailed parametric analyses show that the PPF controller can effectively reduce the nonlinear vibration response amplitudes of the PFG-GRLCC rectangular plate. In addition, the results reveal the energy transform between the host system and the PPF controller. This work is expected to provide theoretical guidance for nonlinear large amplitude vibration reduction of graphene-reinforced structure.

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.

Research on One-to-Two Internal Resonance of Sling and Beam of Suspension Sling–Beam System

An approach is presented to investigate the 1:2 internal resonance of the sling and beam of a suspension sling–beam system. The beam was taken as the geometrically linear Euler beam, and the sling was considered to be geometrically nonlinear. The dynamic equilibrium equation of the structures was derived using the modal superposition method, the D’Alembert principle and the Hamilton principle. The nonlinear dynamic equilibrium equations of free vibration and forced oscillation were solved by the multiple-scales method. We derived the first approximation solutions for the single-modal motion of the system. Numerical examples are provided to verify the correctness of formula derivation and obtain the amplitude–time response of free vibration, the primary resonance response, the amplitude–frequency response, and the amplitude–force response of forced oscillation. According to the analysis, it is evident that the combination system exhibits robust nonlinear coupling properties due to the presence of internal resonance, which are useful for engineering design.

Analysis of the Vibro-Impact Nonlinear Damped and Forced Oscillator in the Dynamics of the Electromagnetic Actuation

In this work, the effect of vibro-impact nonlinear, forced, and damped oscillator on the dynamics of the electromagnetic actuation (EA) near primary resonance is studied. The vibro-impact regime is given by the presence of the Hertzian contact. The EA is supplied by a constant current generating a static force and by an actuation generating a fast alternative force. The deformations between the solids in contact are supposed to be elastic and the contact is maintained. In this study, a single degree of freedom nonlinear damped oscillator under a static normal load is considered. An analytical approximate solution of this problem is obtained using the Optimal Auxiliary Functions Method (OAFM). By means of some auxiliary functions and introducing so-called convergence-control parameters, a very accurate approximate solution of the governing equation can be obtained. We need only the first iteration for this technique, applying a rigorous mathematical procedure in finding the optimal values of the convergence-control parameters. Local stability by means of the Routh-Hurwitz criteria and global stability using the Lyapunov function are also studied. It should be emphasized that the amplitude of AC excitation voltage is not considered much lower than bias voltage (in contrast to other studies). Also, the Hertzian contact coupled with EA is analytically studied for the first time in the present work. The approximate analytical solution is determined with a high accuracy on two domains. Local stability is established in five cases with some cases depending on the trace of the Jacobian matrix and of the discriminant of the characteristic equation. In the study of global stability, the estimate parameters which are components of the Lyapunov function are given in a closed form and a graphical form and therefore the Lyapunov function is well-determined.

The dynamic regimes and the energy harvesting of the energy harvester based on the bistable piezoelectric composite laminate

The dynamic regimes and the energy harvesting of the bistable energy harvester based on the bistable piezoelectric composite laminate are elaborated here. To identify the all-around dynamic regimes and assess the bistable energy harvesting performance, the forward frequency sweep, the reverse frequency sweep and the resistance sweep are conducted. The peak-to-peak frequency–amplitude response curves on the displacement and the voltage output are graphically presented. The single-well vibration, the limit cycle oscillations, the intermittency between the limit cycle oscillations and the chaotic snapthrough, the intermittency between the multiple periods snapthrough and the chaotic snapthrough, the multiple periods snapthrough and the chaotic snapthrough are elaborated by the Time history diagram, the Phase diagram and the Poincare map. The dynamic regimes are graphically presented delivering the stability zone for the dynamic responses and the optimal region for the energy harvesting. The RMS power output and the Max voltage output are checked to be the value judgements. Due to the softening nonlinear effect and the hysteresis, the conspicuous biases between the forward and reverse frequency sweeps are found to exist illustrating that the limit cycle oscillations expand the bandwidth of the double-well regimes during the reverse frequency sweep. From the perspective of the energy harvesting capacity, the limit cycle oscillations are demonstrated to be optimal, the double-well responses generally outperform the single-well responses concerning the off-resonance and the single-well responses concerning the primary resonance around the reduced modal frequency may outperform the double-well responses except the limit cycle oscillations. The load resistance is crucial to the power generation performance and needs optimizing with the change of the external excitation. The load resistance, the external excitation amplitude and frequency are interrelated and can be optimized simultaneously for the maximum power generation efficiency.

Restoring the reciprocity invariance in nonlinear systems with broken mirror symmetry

Circumventing the reciprocity invariance has posed an interesting challenge in the design of modern devices for wave engineering. In passive devices, operating the device in the nonlinear response regime is a common means for realizing nonreciprocity. Because mirror-symmetric systems are trivially reciprocal, breaking the mirror symmetry is a necessary requirement for nonreciprocal dynamics to exist in nonlinear systems. However, the response of an asymmetric nonlinear system is not necessarily nonreciprocal. In this work, we report on the existence of stable, steady-state nonlinear reciprocal dynamics in coupled asymmetric systems subject to external harmonic excitation. We restore reciprocity in the asymmetric system by tuning two symmetry-breaking parameters simultaneously under certain operating conditions. Specifically, we identify response regimes in the vicinity of the primary resonances of the system where the steady-state left-to-right transmission characteristics are identical to the right-to-left characteristics in terms of frequency, amplitude and phase. We interpret these regimes of reciprocal dynamics in the context of phase nonreciprocity, wherein incident waves undergo a nonreciprocal phase shift depending on their direction of travel. We hope these findings help design devices with new functionalities for controlling and steering of elastic waves.

Nonlinear Dynamical Analysis of Composite Boring Bar with Nano-carbon Materials

Preventing resonance in boring bar vibration is the key to achieve high quality and efficiency. In this paper, using the three-dimensional nonlinear dynamic model of boring process, the cutter bar is simplified as a rotating non planar bending axis, starting from the constitutive relationship and stress displacement relationship of composite materials. Based on Hamilton principle, the nonlinear motion differential equation of cutting process is established. The nonlinear partial differential equations of bending vibration are discretized by Galerkin method. Using the multi-scale method, the frequency response functions of the system in the primary resonance and super resonance are obtained. Using this model, the effects of the length diameter ratio of nano-carbon materials and the content of nano-carbon materials on the vibration frequency response of the boring bar are studied. The results show that the steady vibration amplitude of the cutter bar can be reduced by adding nano-carbon materials to the fiber reinforced polymer composite (FRP) beam.

Constrained Parameter-Splitting Multiple-Scales Method for the Primary/Sub-Harmonic Resonance of a Cantilever-Type Vibration Energy Harvester

In this paper, the approximate analytical solutions obtained by using the constrained parameter-splitting-multiple-scales (C-PSMS) method to the primary and [Formula: see text] sub-harmonic resonances responses of a cantilever-type energy harvester are presented. The C-PSMS method combines the multiple-scales (MS) method with the harmonic balance (HB) method. Different from the erroneous stability results obtained by using the Floquet theory and the classical HB method, accurate stability results are obtained by using the C-PSMS method. It is found that the correction to the erroneous solution when the HB method and Floquet theory are adopted in the stability analysis of the primary and [Formula: see text] sub-harmonic resonances of a largely deflected cantilever-type energy harvester is necessary. On the contrary, the C-PSMS method gives much improved results compared to those obtained by using Floquet theory and HB method when the numbers of terms in each response expression are the same. The frequency response curves of the primary resonance and the [Formula: see text] sub-harmonic resonance of the harvester obtained by the C-PSMS method are compared to those obtained by the HB method and verified by those obtained by the fourth-order Runge–Kutta method. Moreover, the basin of attraction based on the fourth-order Runge–Kutta method is presented to confirm the inaccurate stability results obtained by using the HB method and Floquet theory. The convergence examinations on the stability analysis carried out by the HB method and Floquet theory show that enough terms in the response assumption are needed to achieve relatively accurate stability results when studying the stability of the primary and sub-harmonic resonances of a cantilever by using the HB method and the Floquet theory. However, the low-order C-PSMS method is able to give an accurate frequency-amplitude response and accurate stability results of the primary and sub-harmonic resonances of a largely deflected cantilever-type energy harvester.

Outlining the impact of structural properties of a multilayered porous channel with oscillating velocities and fields

In this study, we analyze the multilayered viscous potential flow, in which the effects of primary and combined resonances on the interfacial stability due to oscillating velocities and a periodic field are carried out. Fluids are assumed to be immiscible and incompressible with different electrical and dynamic properties and move through a porous nature. Viscosity occurs at the surface through the surface dynamic condition, where it is assumed to be weak in the fluid bulk and can be ignored. Given the boundary and surface conditions, the linear solutions of the motion and field equations led to simultaneous differential equations with complex periodic coefficients, which are a generalization of Mathieu’s equations. The special case of uniform velocity and the constant field is discussed where a quadruple algebraic equation has complex coefficients is obtained. The method of multiple time scales is applied to obtain analytical approximate solutions and perform the stability criteria for both the non-resonant and resonant states. The impacts of flow velocities and associated electric field properties, as well as the permeability of porous media and fluid sheet geometry, were taken into consideration. Through numerical discretization by varying the values of the quantities of the model, essential conclusions can be made that support an understanding of the stabilization process. For the primary resonance, it is found that there is an important role in increasing the thickness of the layers to the stability of the fluid sheet, while the porosity of the layers has a dual role phenomenon depending on the thickness of the upper layer. In the case of combined resonance, it was mentioned that increasing the dielectric constant between the lower layer and middle one tends to assist the system to instability, but it is worth noting that it supports it slowly, on the other hand, the electric horizontal field helps the system to be more stable.

Conceptual modeling and parametric study of the nonlinear dynamics of the floating wind turbine in the presence of primary and internal resonance

Conceptual modeling and parametric study of the nonlinear dynamics of the floating wind turbine in the presence of primary and internal resonance

Modelling and nonlinear analysis of the wave-induced vibrations of a single hybrid riser conveying two-phase flow

This paper analyses the nonlinear dynamic response of a single hybrid riser pipe to wave excitations. Detailed nonlinear equations of motion are derived for the riser pipe as it conveys two-phase flow. Hence its dynamics are described by two coupled nonlinear equations, relating to both longitudinal and transverse displacements. Wave excitations forcing the system are imposed as oscillating inline flow. In addition, considering that the riser is a structure that is vibrating in a moving fluid, the inline force is modelled with the extended Morison equation. These equations are solved by the method of multiple scale perturbation. Theoretical results show that longitudinal and transverse responses are coupled at some specified natural frequencies. Similarly, the riser pipe is also prone to primary and super-harmonic resonance excitations at some wave frequencies. This presents a complex nonlinear multi-frequency excitation problem identical to the numerical studies conducted on a typical single hybrid riser pipe based on the frequency response curves. The uncoupled primary resonance behaviour of the transverse displacements for all the void fractions studied exhibited a hardening nonlinear behaviour of non-unique responses and jump phenomenon. However, a distinctive feature of the harmonics was observed where the curves separate into two to form an isolated branch of periodic solutions. These are known as Isola. Also, the presence of internal resonance results in the transfer of energy between the planar axis resulting in undesirable axial resonance peaks which are destabilized by the presence of the nonlinear anti-resonance effect.

Quasi-3D nonlinear primary resonance of randomly oriented CNT-reinforced micro/nano-beams incorporating nonlocal and couple stress tensors

Quasi-3D nonlinear primary resonance of randomly oriented CNT-reinforced micro/nano-beams incorporating nonlocal and couple stress tensors

Wide temperature operation of piezoelectric sensors for detecting precursor levels in a canister

Liquid level detection using piezoelectric actuators and sensors is superior to other technologies in accuracy, stability, and durability. In the semiconducting industry, the accurate detection of precursor levels in a canister is directly connected to the quality of the atomic layer growth through chemical vapor deposition and atomic layer deposition processes. However, the sensitivity of the level detection using piezoelectric devices often decreases at a specific temperature range, limiting the wide temperature operation of the canister. We demonstrate reduced sensitivity of the piezoelectric sensors due to a change in detuning frequency by temperature. A model with a simple harmonic oscillator exhibits the fundamental behavior of the actuator amplitude after a finite number of driving pulses. The impedance measurement of a sensor assembly demonstrated a significant shift in the primary resonance frequency due to a change in environmental temperature. By analyzing the simulation data, we established a temperature-dependent number of driving pulses that could extend the operating temperature of the piezoelectric actuators, which can easily be applied to a wide temperature operation for a canister.