Quasi-periodic Vibrations Research Articles

Quasi-Periodic and Periodic Vibration Responses of an Axially Moving Beam under Multiple-Frequency Excitation

Quasi-Periodic and Periodic Vibration Responses of an Axially Moving Beam under Multiple-Frequency Excitation

Delayed feedback active suspension control for chaos in quarter car model with jumping nonlinearity

Vehicles that operate on off-road terrain, such as rough and unpaved roads, sometimes suffer from severe vibrations. This vibration causes vehicle tires to lose contact with the supporting ground and subsequently collide with it. This jumping leads to nonlinear impact dynamics similar to those of a bouncing ball. In this study, a quarter car model with jumping nonlinearity is newly developed to represent the impact dynamics of an off-road vehicle under harsh terrain conditions. Subsequently, complicated vibrations, such as quasi-periodic and chaotic vibrations in the developed model, are identified using a bifurcation diagram and Lyapunov exponents. Additionally, delayed feedback (DF) control of the active suspension is designed for the developed model. The numerical results show that DF control is effective at eliminating chaos and reducing vibration levels. The noise robustness of the DF control is validated for sinus excitation and small random road noise. The results indicate that DF is effective even when road noise exists. Thus, this study demonstrates the feasibility of DF active suspension control for future suspension control design of the off-road vehicle.

Study on the vortex-induced vibration and flow control of ribbed circular cylinder

This work investigates the vortex suppression performance and mechanism of ribs on high-quality ratio cylinders. Through wind tunnel tests and numerical simulations, the surface wind pressure distribution characteristics and flow separation phenomena of different ribbed cylinders are explored, and the spanwise correlation and nonlinear vibration characteristics of vortex-induced vibrations of ribbed cylinder models are analyzed. The main conclusions are as follows: ribs change the position of the boundary layer separation point, and the difference in size of left and right separated vortices causes a pressure jump phenomenon, altering the wind pressure distribution of the segment model and reducing the wind pressure, resulting in an increase in the locked wind speed of the ribbed cylinder. Complex separated vortices form behind the ribs, affecting the size of the wake vortex and reducing the stability of the segment model at locked wind speeds. Cylinders with four ribs exhibit good vortex-induced vibration suppression performance at 0° and 45° positioning angles. In addition, the cylinder with four installed ribs cylinders exhibits two locked wind speed regions, each showing different motion states: at the primary locked wind speed, they mainly demonstrate quasi-periodic vibrations and degraded quasi-periodic vibrations, while at the secondary locked wind speed, primarily in a chaotic state dominated by high-frequency harmonic components. These research findings have significant implications for future studies and practical engineering applications.

A piezoelectric nonlinear energy sink shunt for vibration damping

The theoretical study and experimental validation of a nonlinear shunt circuit for piezoelectric vibration damping is investigated here. The circuit consists of a resistor, an inductor and a nonlinear cubic voltage source. The shunt acts as an electrical analog to the mechanical nonlinear energy sinks (NESs). These mechanical NESs are passive vibration absorbers that typically have a cubic nonlinear stiffness. They have attractive properties such as saturation of the host system’s vibration amplitude and strongly modulated response. This increases its operational frequency bandwidth and robustness against variations in the properties of host systems compared to linear vibration absorbers. However, the nonlinear nature may induce isolated responses in the host system that induce high vibration amplitudes. This paper investigates if these attractive properties also occur in the electric nonlinear energy sink shunt. An analytical expression for the frequency response is derived through the complexification-averaging method. Bifurcations in the frequency response reveal the occurrence of a quasi-periodic vibration energy exchange between the host system and the voltage over the electrodes of piezoelectric material. This is the main mechanism behind the amplitude saturation of the host system. Other bifurcations also reveal the existence of isolated responses. The nonlinear shunt is then realized with analog multipliers and a synthetic inductor and its performance in vibration damping is experimentally verified for a cantilever beam.

Chaotic snap-through vibrations of bistable asymmetric deployable composite laminated cantilever shell under foundation excitation and application to morphing wing

The realization of the dynamic snap-through behaviors provides the new design ideas in the fields of the morphing aircraft and piezoelectric energy harvesting. This paper studies 1:2 internal resonance, nonlinear vibrations and chaotic snap-through phenomena of the bistable asymmetric composite laminated (BSCL) cantilever shell with the lower-frequency and higher-frequency primary resonances for the first time. The transverse foundation excitation subjects to the fixed end of the bistable cantilever shell. The perturbation analysis of two-degrees-of-freedom nonlinear ordinary differential equations is carried out by using the first-order approximate multiple scale method. The analytical results of the frequency-amplitude and force-amplitude response curves are obtained under the small foundation excitation. The obtained results reveal that the BSCL cantilever shell exhibits the double-jumping characteristics when 1:2 internal and primary resonances occur. There is a continuous energy exchange back and forth between two modes of the bistable laminated cantilever shell. As the foundation excitation increases, the BSCL cantilever shell exhibits saturation phenomenon. Numerical simulations are finished to further investigate the effects of the large excitation on the chaotic, quasi-periodic and snap-through vibrations for the BSCL cantilever shell. The vibration experiment is carried out to investigate the internal resonance and dynamic snap-though motions of the BSCL cantilever shell. Using the snap-though behaviors of the BSCL cantilever shell, we obtain the morphing structure of the aircraft wing.

A study of controlling the transverse vibration of a beam-plate system by utilizing a nonlinear coupling oscillator

In the engineering field, prolonged exposure of complex structures to high-strength vibration environments can lead to various engineering challenges. This underscores the importance of effectively managing the vibration of complex structures. Given the extensive utilization of beams and plates in the construction of intricate structures, this study incorporates a nonlinear coupling oscillator into the beam-plate system and formulates a vibration analysis model for this system. The governing equations of the system are derived theoretically and solved numerically using the Galerkin truncation method. This study focuses on the operational modes of the nonlinear coupling oscillator based on accurate numerical results. Meanwhile, the influence of parameters belonging to the nonlinear coupling oscillator on the transverse vibration responses of the beam-plate system is systemically studied. Upon meticulous examination of the numerical results, the operational states of the nonlinear coupling oscillator encompass the normal vibration suppression mode and the quasi-periodic vibration suppression mode. The alteration in the linear elastic coupling stiffness of the beam-plate system impacts the effectiveness of vibration suppression in the nonlinear coupling oscillator. This influence stems from the fact that changes in the linear elastic coupling stiffness directly affect the nonlinear forces exerted on the beam-plate system within a nonlinear coupling oscillator. Within a feasible range, the nonlinear stiffness and viscous damping can be chosen as the adjustable parameters for the nonlinear coupling oscillator to regulate the vibration of the beam-plate system by adjusting the motion mass of the nonlinear coupling oscillator, which presents a challenge. The appropriate utilization of the nonlinear coupling oscillator demonstrates favorable control effectiveness in managing the transverse vibration of the beam-plate system. The study presented in this work offers a method to simultaneously control the vibration of each component within beam-plate coupling systems.

Study on vortex-induced multimodal coupled vibration of arch bridge suspenders

The objective of this paper was to investigate the 2-DOF suspender response under the condition of vortex-induced force and reveal the multimodal coupling mechanism. Firstly, an elastically mounted segmental model of suspender was established. Wind tunnel experiments were conducted to obtain the time history displacement signals in cross-flow and in-line directions, the empirical parameters were obtained by identify those signals. The nonlinear partial differential motion equation of suspender with two degrees of freedom was established based on Hamilton's principle, which considered the dynamic strain, bending stiffness and vortex-induced force of suspender. The Galerkin method was used to discretize the nonlinear partial differential equations into nonlinear motion governing equations, which considered the first three order modes. A numerical method was used to solve the motion governing equation, the time history displacement curve, phase diagram, amplitude spectrum and Poincare map of the multi-modal responses in the cross-flow and in-line directions were obtained. With the weak coupling effect of nonlinear stiffness and vortex-induced force, the cross-flow and in-line responses demonstrate period doubling bifurcation, quasi-periodic vibration, period doubling vibration, crises and transient chaotic phenomena. However, with the strong coupling effect, the crises and transient chaos in response amplitude disappear. More importantly, planar/non-planar multimodal internal resonance were found in this work. With the weak multimodal coupling effect in suspender response, the low-order modal response is dominant; as the multimodal coupling effect increases, the high-order modal response becomes predominant. In addition, the system energy not only transfers from high-order modal response to low-order modal response, but also the system energy transfer from low-order modal response to high-order modal response. These conclusions are useful for the design and practical engineering of suspenders, which have high theoretical research value in understanding the mechanism of multimodal coupling effects on suspender response.

The Influence of Speed Ratio on the Nonlinear Dynamics of a Magnetic Suspended Dual-Rotor System with a Fixed-Point Rubbing

Magnetic suspended dual-rotor systems (MSDS) provide the potential to significantly improve the performance of aero-engines by eliminating the wear and lubrication system, and solve vibration control issues effectively. However, the nonlinear dynamics of MSDS with rubbing is rarely investigated. In this work, the nonlinear support characteristics of active magnetic bearings (AMBs) are described by the equivalent magnetic circuit method, the impact force is characterized by the Lankarani–Nikravesh model, and the nonlinear dynamic model is established using the finite element method. On this basis, the influence of speed ratio on the nonlinear dynamics is investigated. Simulation results show that the fundamental sub-synchronous vibration of period n is the dominant motion of MSDS, where n is determined by the speed ratio. The frequency components of sub-synchronous vibrations of period k are integer multiples of the minimum dimensionless frequency component 1/k, where k is a positive integral multiple of n. Quasi-periodic and chaotic vibrations are more likely to occur near critical speeds, and their main frequency components can be expressed as a variety of combined frequency components of the rotating frequency difference and its fractional frequency. To reduce the severity of fluctuating stresses stemming from complicated non-synchronous vibrations, speed ratios, corresponding to smaller n and AMB control parameters attenuating vibration amplitude or avoiding critical speeds, are suggested.

Confinement of non-neutral proton plasma within a bent magnetic mirror field: 3D PIC simulation

We conducted a study on the confinement properties of non-neutral proton plasma in an asymmetric magnetic mirror using the three-dimensional (3D) particle-in-cell simulation method. Our findings indicate a close relationship between the ability of the magnetic mirror to confine protons and the incidence orientation of the proton beam. This relationship can only be accurately assessed through 3D simulation, as opposed to two-dimensional simulation. In addition, we observed that a higher velocity parallel to the magnetic field direction results in a shorter confinement time. The electric field at and near the central point of the mirror exhibits quasi-periodic vibrations, attributed to the electrostatic oscillation of the proton and the cyclotron motion. Furthermore, we discovered a drift motion of the confined proton plasma in the direction of the magnetic field gradient. These findings hold implications for the design of magnetic mirrors capable of effectively confining charged particle beams.

High-precision semi-analytical solution for the quasi-periodic nanobeam system based on the weight time-domain minimum residual method

Nanobeams have been widely used in various micro-electromechanical systems because of their small size and high sensitivity. At the nanoscale, quasi-periodic (QP) vibration often occurs in nanobeams due to the surface effects such as surface elasticity, residual surface tension and geometric nonlinearity. According to the QP motion properties of the nanobeam system, a weight time-domain minimum residual method (WTMRM) is presented in this paper by introducing the weight vector into the original time-domain minimum residual method (TMRM), and the high-precision semi-analytical QP (SAQP) solution is obtained by the WTMRM. This method mainly includes four aspects. Firstly, according to the properties of QP motion of the nanobeam system, a new SAQP solution constructed by the dominant frequency and frequency interval is presented. Then, the problem of solving the SAQP solution is converted into a nonlinear minimum optimization issue by deducing the presumptive solution and substituting the state variables back to the original control equations, and this optimization issue is resolved by the enhanced response sensitivity approach (ERSA) iteratively. Next, the weight vector is introduced to deal with the contradiction of the residual between the control equation and the initial condition constraints. Finally, the numerical results of the Runge–Kutta (RK) method are compared with those of the proposed method. The comparison results show that the precision of the SAQP solution achieved by the WTMRM is much higher than the existing results.

Quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass

In this work, quasi-periodic vibration of an axially moving beam under conveying harmonic varying mass are studied. Coupled modelling equations of nonlinear vibration of an axially moving beam are established with the aid of the generalized Hamilton principle. A nonlinear partial-integro-differential equation as the transverse vibration modelling equation is deduce by decoupled method. After obtaining the equilibrium buckled state of the axially moving beam, a transverse vibration governing equation around the buckled state is presented. The Galerkin method is applied to truncate the governing equation into a set of nonlinear ordinary differential equations. The harmonic balance method with three time variables is applied to analyze quasi-periodic vibration of the nonlinear dynamic system with combined parametric and forced external excitation. Time traces and phase-plane portraits of quasi-periodic vibration are obtained, which are in excellent agreement with those of the direct time integration method. Frequency amplitude responses of the vibration are determined by the harmonic balance method with three time variables. Effects of the velocity of the axially moving beam, amplitude of the varying mass, and the external excitation frequency on the vibration amplitude of axially moving beam and the ratio of the two fundamental frequencies are investigated.

A data driven method applied on nonlinear vibration of flexible rotor supported on centralized squeeze film damper

This paper discusses the application of Dynamic Mode Decomposition (DMD) on vibration response of flexible rotor supported by a Centralized Squeeze Film Damper (CSFD). The mathematics behind DMD and its applications to rotor vibration data such as linearization, reconstruction has been established. Vibration data from the flexible rotor system supported on CSFD exhibits both linear and nonlinear behavior. Further, DMD has been applied on periodic and quasi-periodic vibration data of CSFD for prediction and forecasting. DMD is observed to have successfully linearized a nonlinear system with the linear Henkel Matrix. The reconstruction and forecasting of time series data from rotor has been achieved with a maximum error of 0.2% even for nonlinear vibration data. Further, White Gaussian Noise has been intermixed with CSFD data to demonstrate the noise reduction capability of DMD method for nonlinear systems. Lastly, the application of DMD algorithm on experimental data from Jeffcott rotor based test setup has been shown.

Stability and bifurcations of complex vibrations in a nonlinear brush-seal rotor system.

A brush seal has the advantages of adapting to different vibration conditions and increasing the stability of the nonlinear rotor system. In this research, the stability and bifurcations of complex vibrations in a brush-seal rotor system are studied. An analytical seal force model is obtained through the beam theory and mutual coupling dynamics of the bristles and the rotor. The interaction between the bristles and the rotor is clearly depicted by a geometric map. Periodic and chaotic vibrations as well as the corresponding amplitude-frequency characteristics are first predicted by a numerical bifurcation diagram and 3D waterfalls. Discrete dynamic eigenvalue analysis is adopted for a detailed investigation of the stability and bifurcations of nonlinear vibrations. Jumping, quasi-periodic, and half-frequency vibrations are warned during the speeding up and down process. Four separate nonlinear vibration evolving routes are discovered. Two period-doubling bifurcation trees evolving to chaos are illustrated for the observation of global and independent periodic vibrations. Nonlinear vibration illustrations are presented through displacement orbits as well as harmonic amplitudes and phases. Chaotic vibration and unstable semi-analytical vibration solutions are compared. The obtained results and analysis methods provide new perspectives on nonlinear vibrations in the brush-seal rotor system.

Nonlinear Dynamics and Vibration Localization of Shrouded Blisk with Contact and Friction Effects

Shrouds have been widely used to reduce the level of excessive vibration of blisks. However, complicated nonlinear motion can be induced by the contact and friction between shrouds. Even worse, harmful localization of vibration can be encountered due to nonsmooth behaviour at the shroud contact interfaces in the tuned disk. In this paper, the nonlinear dynamics and localization of vibration of a shrouded blisk are studied considering the spin softening. The continuous parametric model of a shrouded blisk is established, and the transition boundaries between different status (i.e., stick, slip and separation) of shrouds are determined based on the Coulomb friction model. The steady-state responses of the blisk are analysed using variable rotation speed, and the primary resonance, beat and quasi-periodic vibration are presented in connection with the non-smooth behaviour of contact. A particular type of vibration localization initiated by the contact and friction effect of the shrouds in the form of asymmetric vibration in the tuned blisk is discovered. It is found that the blades-disk coupling has strong influences on the level of the localization. The effects of the contact stiffness and localization on the blade vibration are demonstrated through the change in the powers of the blade motions.

Nonlinear dynamic characteristics analysis and adaptive avoid vortex-coordinated optimal control of hydropower units under grid connection

Hydropower units operate as the first option for renewable energy supply. The coupling characteristics of hydro-turbine governing system (HTGS), misaligned shafting (MSH), and power grid (PG) (HTGS-MSH-PG) have always been omitted. Meanwhile, the safety hazard from the draft tube vortex zone (DTVZ) and the difficulty of coordinating the ultra-low frequency oscillation (ULFO) and primary frequency regulation (PFR) performance of hydropower unit grid-connected operation have not been resolved. In this study, we aim to investigate the interaction mechanism and nonlinear dynamic characteristics of the HTGS-MSH-PG coupling system and determine the optimal control strategy for considering the DTVZ, ULFO, and PFR of hydropower units. First, a novel coupling model of HTGS-MSH-PG is established. Thereafter, the stability of the HTGS-PG coupling system is examined by employing the Hopf bifurcation theory, and the damping characteristics are obtained using the hydraulic damping model. The nonlinear dynamic behavior of misaligned distance, nonlinear vibration characteristics of MSH, and influence of MSH vibrations on HTGS and PG are revealed based on this model. Finally, an adaptive avoid vortex-coordinated optimize control strategy which considers the means to prevent a strong DTVZ and coordinate the ULFO and PFR performance of HTGS, MSH, and PG is proposed. NSGA-II is introduced to obtain the Pareto optimal solution set, and the optimal scheme is determined by the fuzzy satisfaction method. Moreover, the nonlinear dynamic characteristics of hydropower units under optimal control are revealed. The results demonstrate that the nonlinear dynamic characteristics of the HTGS-MSH-PG coupling system mainly include the quasi-periodic multi-frequency vibration of MSH and multi-frequency dynamic performance of HTGS and PG. The adaptive avoid vortex-coordinated optimal control strategy can effectively prevent a strong DTVZ, coordinate the ULFO and PFR performance of HTGS, MSH, and PG, and increase the satisfaction rate by 72.2%. These results lay a theoretical and technical foundation for the safe, stable, and optimal operation of hydropower units.

Optimal design for vibration energy harvesters based on quasi-periodic structures

In this paper, the design of large-scale quasi-periodic Vibration Energy Harvesters (VEH) is optimized to enhance the harvested power of an electromagnetic mode localized structure. This work aims to optimize the output power by employing the energy localization phenomenon in a large-scale periodic configuration by introducing the minimum number of perturbations. The harvested power, number and location of perturbations are among the objectives that need to be optimized. A genetic-based mixed-integer optimization algorithm is used to meet the objective functions within a constraint on the system kinetic energy. Numerical simulations for quasi-periodic systems with 20 and 100 Degrees of Freedom (DOF) are performed. It is shown that the ratio of harvested power increases as the number of perturbations rises and it exceeds 80% of the total output power by perturbing almost one-third of the total DOFs. The proposed methodology is a decision-making aid to provide an optimal design in a generalized quasi-periodic VEH in order to reduce the number of harvesting transducers while providing a significantly high amount of harvested power.

Delayed oscillator model of pressure relief valves with outlet piping

The system of a vessel and a direct spring operated pressure relief valve is extended by a downstream pipe. The presented mathematical model consisting of ordinary and partial differential equations can be transformed to a system of differential difference equations by means of the travelling wave solutions of the partial differential equations. This model contains a time delay originated in the pipe length and the speed of sound in the pipe. The linear stability analysis presents stability charts in a parameter plane characterising the environment where the valve is built in. The boundary lines and the corresponding vibration frequencies of the self-excited vibrations are determined as compact closed form expressions. Several analogies are identified with other physical systems modelled by delayed oscillators. Both the stability charts and the vibration frequencies are checked with numerical simulations of the original nonlinear system consisting of coupled ordinary and partial differential equations. The connection between the travelling wave solution and the standard quarter standing wave approximation is explained. A qualitatively relevant phenomenon is identified, namely the appearance of the quasi-periodic vibration at certain parameter combinations.

Dynamic analysis of rolling ball bearing-rotor based on a new improved model

Dynamic analysis for a bearing-rotor system with the imbalance and the asymmetric gap is conducted in this paper. A new improved analytical model overall considering the gap, the varying compliance vibration and the time-dependent unbalanced force is established, especially the new model is more accurate and closer to reality by abandoning the assumption of the traditional model that the three center points of the inner ring, the outer ring and the rolling ball are collinear. More general vibration characteristics are described and the calculation results based on the new model are more universal than those based on the traditional model. The comparison of the calculation result between the improved model and the traditional model shows that the phase difference for the two results is obviously different from each other, the dominant frequency has no obvious difference between the two models and the amplitudes have somewhat difference. The parametric excitation vibration induced by the varying compliance force of the rolling ball on the inner ring-rotor is analyzed and then the influences of the rotating speed, the gap, the eccentricity and the mass of the rotor on the nonlinear responses are studied and some important conclusions are drawn. As the speed increases, the VC frequency gradually loses its domination of the frequency spectrum, and the rotational speed frequency and its combined frequency with the VC frequency dominate the vibration. The bearing-rotor system is susceptible to the variations of the rotational speed, the gap, the eccentricity and the mass of the rotor in certain ranges; the parameters can make the system in a relatively stable, stable and unstable state; the system shows the complex dynamic behaviors such as the periodical vibration, the quasi-periodic vibration, the chaotic motion and the jumping phenomenon, the bifurcation, sudden change. The research is significant for the quantitative calculation of the dynamic response for parameter designation and the fault diagnosis of the system.

Analysis of subharmonic and quasi-periodic vibrations of a Jeffcott rotor supported on a squeeze-film damper by the IHB method

In rotors supported on a squeeze-film damper (SFD) with static eccentricity, subharmonic and quasi-periodic vibrations are generated by rotor unbalance. In this paper, subharmonic and quasi-periodic vibrations in a Jeffcott rotor supported on an SFD are analyzed using the modified incremental harmonic balance (IHB) method. Oil film forces in the damper are calculated using π film theory and short bearing approximation. These forces are first calculated in the time domain and then transformed into the frequency domain by Fourier transformation. The oil film forces in the frequency domain are multiplied by a transformation matrix to match the matrices used in the classical IHB method. To apply the continuation technique, the Jacobian matrix of residuals is calculated analytically. The stability analysis of the calculated solutions is analyzed using Floquet theory. Using this theory, saddle-node, symmetry breaking, period-doubling, and secondary Hopf bifurcations are detected. The phase plane, Poincaré sections, and time history at these bifurcation points are calculated using the Runge–Kutta method.

Effects of upstream wake on vortex-induced vibrations and wake patterns of side-by-side circular cylinders

In this paper, vortex-induced vibrations and wake patterns of two side-by-side circular cylinders in the wake of a stationary upstream cylinder are numerically investigated for a uniform free stream flow at the Reynolds number of 100. The three cylinders are in equilateral triangle arrangements with a center-to-center spacing ratio of 2.0–5.0. The considered mass ratio is 2.0 and the reduced velocity range is 0.5–15.0. The side-by-side cylinders are allowed to vibrate only in the transverse direction. Variations of response amplitudes, frequencies, mean position shifts and wake patterns with the reduced velocity and spacing ratio are investigated using direct numerical simulations. In comparison with the case without an upstream cylinder, amplitudes, frequencies, mean position shifts and hydrodynamic forces of two side-by-side cylinders are significantly impacted by the upstream cylinder wake and the associated wake-vortex-structure interactions. Key observations include (a) vibrations initiated at higher reduced velocities for a small to medium spacing ratio, (b) a wider range of large-amplitude vibrations, (c) a lower-frequency component, (d) appearances of periodic, quasi-periodic, chaotic and asymmetric vibrations, and (e) a hysteresis phenomenon. The asymmetric vibration is characterized by dissimilar vibration amplitudes of the two cylinders whereas the hysteresis is related to a transition of wake patterns when increasing versus decreasing the reduced velocities. Behind two side-by-side circular cylinders, the flip-flopping, anti-phase synchronized, in-phase synchronized, and chaotic patterns are identified.