Nonlinear Vibration Research Articles

A Positioning and Tracking Performance–Enhanced Composite Control Algorithm for the Macro–Micro Precision Stage

In the macro–micro composite motion process, the macro stage achieves rapid motion in a large workspace, while the micro stage realizes precise positioning within a small displacement. The large-stroke and high-speed motion of the macro stage is influenced by nonlinear friction, overshooting, and vibrations, making it challenging to achieve rapid stabilization within the travel range of the micro stage, thereby impacting equipment operation efficiency. This paper proposes a composite controller structure designed to solve the fast point-to-point positioning of the macro stage driven by a linear motor and enhance the trajectory tracking performance of the stage. The proposed composite control algorithm (CCA) includes velocity feed-forward, gain-scheduled proportional integral differential (PID) control, and a plug-in repetitive control method. By employing a tracking differentiator, velocity feed-forward, and a gain-scheduled PID controller, the control algorithm can realize rapid stabilization and positioning for the macro stage. Through velocity feed-forward, gain-scheduled PID control, and a plug-in repetitive controller, the control algorithm can reduce the trajectory tracking error of the stage. Simulations and experimental studies of point response and sinusoidal trajectory tracking are carried out on a macro–micro stage to verify the effectiveness of the composite controller for the linear motor. The experimental results demonstrate that the proposed composite controller effectively reduces the macro stage’s settling time and overshoot, and improves the accuracy of sinusoidal trajectory tracking, which lays a foundation for submicron positioning accuracy in high-speed macro–micro motion stages.

Nonlinear free vibrations of sandwich plates with FG GPLR face sheets based on the full layerwise finite element method

Nonlinear free vibrations of sandwich plates with FG GPLR face sheets based on the full layerwise finite element method

Multiple nonlinear vibration model of drilling string and mechanism of drilling speed increase in ultra-HPHT oil & gas wells

Multiple nonlinear vibration model of drilling string and mechanism of drilling speed increase in ultra-HPHT oil & gas wells

Investigation of nonlinear dynamic behaviors of vertical rotor system supported by aerostatic bearings

Abstract A common issue associated with gas bearing-rotor systems is the tendency to generate self-excited vibrations, leading to instability. To address this problem, the fluid-structure coupling model of an aerostatic bearing-rotor system is established in this paper. Then a hybrid method combining the finite difference method and direct integration method, is employed to solve the bearing lubrication equation and rotor motion equation simultaneously. Furthermore, based on the orbit of rotor center,frequency spectrum diagram, Poincaré map, waterfall diagram and bifurcation diagram, the effects of rotational speed, rotor mass, orifice diameter and nominal clearance on the nonlinear dynamic behaviors of the bearing-rotor system are investigated. The results indicate that the system exhibits rich nonlinear behaviors with increasing rotational speed and rotor mass, including the occurrence of typical half-speed whirl. However, the nonlinear vibrations of the system can be restricted by selecting appropriate bearing structural parameters, providing theoretical guidance for the design of aerostatic bearing-rotor systems.

Research on nonlinear vibration of a dual-rotor system with combined misalignment of cylindrical roller bearing and coupling

Research on nonlinear vibration of a dual-rotor system with combined misalignment of cylindrical roller bearing and coupling

Nonlinear vibrations and static bifurcation of a hyperelastic pipe conveying fluid

This study investigates the free and forced nonlinear vibrations, along with the static bifurcation behavior, of a fluid transmission hyperelastic pipe supported by clamped ends. Both geometrical and material nonlinearities are taken into account. The mathematical model is grounded in Euler–Bernoulli beam theory, incorporating von Kármán nonlinearity to capture the complexities of the system. For the hyperelastic materials, the strain energy function proposed by Yeoh is employed. Considering that the lowest critical velocity of the fluid flow is associated with the first mode, after deriving the governing equation using Hamilton’s principle, Galerkin discretization is applied to obtain the reduced order model in the first mode. Analysis of static bifurcation and stability is performed and critical velocity values of fluid flow are determined. The analysis of free and forced vibrations of the hyperelastic pipe is conducted using the method of multiple time scales, specifically focusing on the sub-critical velocity range of fluid flow, which corresponds to the unbuckled state of the pipe. This approach allows for a comprehensive examination of how fluid flow velocity influences the frequency characteristics of free vibrations, as well as the frequency response in the vicinity of the main resonance. The results obtained from this study can significantly enhance our understanding of the design and application of hyperelastic pipes in various fluid transfer scenarios. By elucidating the dynamic behavior and critical parameters associated with these pipes, the findings can inform engineering practices and contribute to the optimization of systems that utilize hyperelastic materials for fluid conveyance.

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

Nonlinear resonance of fractional order viscoelastic PET films under temperature loading

The effects of oven temperature during printing on nonlinear vibration for fractional-order PET films are considered in this paper. The effect of temperature, fractional order modelling and some other parameters are analysed with respect to the response of the resonance. Fractional order kelvin-Voigt ontological relationship is used to describe the characteristics of viscoelastic materials. The differential equations for nonlinear vibrations are inferred according to the second law of Newton and the theory of von Karman. Discretization for nonlinear equations on locomotion using the Bubnov–Galerkin method. Forced co-oscillatory amplitude-frequency response equations for thin-films systems under temperature loading were calculated using the multiple scales method. Results of numeral results show that temperature, and fractional-order visco-elastic modelling influence the membrane's response to resonance. These results provide a basis for studying fractional-order visco-elastic films vibrations and identifying regions of stable operation in moving systems to prevent divergent instabilities for flexible electronic device manufacturing.

Nonlinear bending and vibration analysis of a variable-width piezoelectric nanoplate with flexoelectric effects

Nonlinear bending and vibration analysis of a variable-width piezoelectric nanoplate with flexoelectric effects

Vibration control of a longitudinally vibrating rod connected through an elastic coupler by installing a nonlinear vibration reduction coupler

To study the potential application of the longitudinal vibration control of a rod coupling system by employing NESs, this study proposes a nonlinear vibration reduction coupler (NVRC), where the NVRC can be regarded as the NES which is installed between the two elastic structures. The introduction of NVRC is aimed at controlling the longitudinal vibration of the rod system. Based on the Lagrange method (LM), the longitudinal vibration analysis model of a rod system connected through a linear elastic coupler and an NVRC is established, in which the longitudinal vibration responses of the rod system with an NVRC are predicted. In the numerical analysis, the correctness and stability of LM in calculating longitudinal vibration responses of the rod system with an NVRC are verified, where the chosen reasons for observation points are explained. It can be concluded that the introduction of NVRC can reduce the vibration of each resonance area of the rod system at the same time. Under certain parameters, the vibration characteristics of the rod system are changed by employing NVRC. Furthermore, nonlinear stiffness and viscous damping of NVRC can be chosen as the adjustable parameters that strengthen the vibration reduction ratio of the rod system. Overall, the longitudinal vibration of the rod system can be effectively controlled by choosing reasonable parameters of NVRC. The introduction of NVRC provides an effective way to control the longitudinal vibration of the rod system.

Nonlinear forced vibration of the FGM piezoelectric microbeam with flexoelectric effect

In this paper, based on the extended dielectric theory, Euler beams theory and von Karman’s geometric nonlinearity, a nonlinear FGM piezoelectric microbeam model is established with flexoelectric effect. The governing equations, initial conditions and boundary conditions are obtained by applying Hamilton’s principle and then solved by combining the differential quadrature method (DQM) and iteration method. The innovation of this paper is to construct a nonlinear forced vibration model of piezoelectric microbeams. The coupling response between the inverse flexoelectric effect and the inverse piezoelectric effect is investigated. Various effects are examined, including the functional gradient index m and transverse distributed load q affecting the distribution of electric potential. Results indicated that the functional gradient index m, beam thickness h, and span-length ratio L/h have a significant impact on the dimensionless deflection of the FGM microbeam. The influence of the flexoelectric effect on dimensionless deflection increases with the decrease of scale. In addition, transverse load q and the functional gradient index m also have a significant impact on the distribution of electric potential. This paper will provide useful theoretical guidance for the design of micro-sensors and micro-actuators.

Chaotic band-gap modulation mechanism for nonlinear vibration isolation systems based on time-delay feedback control

Abstract Systems designed for nonlinear vibration isolation that incorporate chaotic states demonstrate superior capabilities in vibration attenuation, adeptly modulating the spectral constituents of vibrational noise. Yet, the challenge of eliciting low-amplitude chaotic dynamics and perpetuating these states across a diverse array of parameters remains formidable. This study proposes a pioneering strategy and technique for modulating the chaos band by incorporating a time-delayed feedback control mechanism within the framework of nonlinear vibration isolation systems.The investigation commences with an exhaustive analysis of the nonlinear dynamics, shedding light on the principles dictating the evolution of chaos. The study then advances to scrutinize the dynamics of systems with delays to elucidate the chaos-inducing processes engendered by feedback with temporal lags. Building upon the system’s responses, the chaotic performance and the effectiveness of the vibration isolation are crafted. Consequently, the time-delayed feedback control parameters are identified as pivotal design variables, which are then employed to dissect the control mechanisms influenced by the time-delayed feedback on the chaos band. Utilizing the delineated control mechanism, the nonlinear vibration isolation system is precipitously transitioned from a state of stable periodicity to one of chaos, fostering low-amplitude chaotic dynamics across an expansive parameter space, and in turn, resolving the previously stated challenge. Perhaps most significantly, the mechanism for attaining low-amplitude chaos introduced here paves the way for innovative methodologies in the active vibration isolation design of similar systems. Furthermore, it is anticipated to yield theoretical guidance for the manipulation of chaos bands and the formulation of active vibration isolation strategies within the domain of nonlinear vibration isolation systems.

Pull-in instability and nonlinear vibration of microbeam with piezoelectric patch driven by electrostatic force using modified couple stress theory

The static and dynamic pull-in characteristics of electrostatically and piezo-electrically actuated microbeam are investigated. The static pull-in voltage is evaluated using analytical methods based on MCST, CCMT and validated through the use of FE based COMSOL multiphysics tools. The dynamic pull-in voltage obtained from numerical technique and verified with existing literature, lower than the static pull-in voltage. The piezoelectric patch has a significant influence on the pull-in voltage and an effect on the aspect ratio. Moreover, the pull-in band and instability zone of the microbeam with weak and hard forcing function have been investigated using multiple scale method.

Nonlinear dynamic modeling and vibration analysis for early fault evolution of rolling bearings

In rotating machinery, the condition of rolling bearings is paramount, directly influencing operational integrity. However, the literature on the fault evolution of rolling bearings in their nascent stages is notably limited. Addressing this gap, our study establishes an innovative nonlinear dynamic model for early fault evolution of rolling bearings based on collision impact. Firstly, considering the fault evolution characteristics, the influence of the rolling element and fault structure, the dynamic model of early fault evolution between the rolling element and the local fault is established. Secondly, according to the Hertzian contact deformation theory, a nonlinear dynamic model of rolling bearings expressed as mass-spring is established. Thirdly, the energy contribution method is used to integrate the fault evolution model and the nonlinear dynamic model of the rolling bearing. A nonlinear dynamic model of early fault evolution of the rolling bearing is proposed by using the Lagrangian equation. Comparing the simulation results of the nonlinear dynamic with the experimental results, it can be seen that the numerical model can effectively predict the evolution process and vibration characteristics of the fault evolution of rolling bearings in the early stage.

Effects of extreme temperature environments on nonlinear vibrations of the cable-stayed functionally graded beam

A cable-stayed functionally graded beam (CSFGB) model is proposed for the first time to study nonlinear behaviors of the system exposed to extreme temperatures. To this end, the in-plane one-to-one internal resonance between the global mode (FGB's mode) and the local mode (cable's mode) is explored under the condition of external primary resonance. First, governing differential equations of the system considering temperature effects are derived by using Hamilton's principle. To obtain the modal function of the FGB, the in-plane eigenvalue problem is solved through the separation-of-variables method. Subsequently, ordinary differential equations (ODEs) are yielded according to Galerkin discretization. The method of multiple time scales is then applied to deal with the ODEs and derive the modulation equations. Based on the above theoretical solutions, the frequency-/force-response curves at three different temperatures are delineated via Newton-Raphson method and the continuation of fixed points. Meanwhile, the time histories and phase portraits are also provided by directly integrating the ODEs. The results show that temperature changes have a significant influence on nonlinear characteristics of the system.

Multi-frequency harmonic balance method for nonlinear vibration of pipe conveying fluid under arbitrary dual-frequency excitation

Multi-frequency harmonic balance method for nonlinear vibration of pipe conveying fluid under arbitrary dual-frequency excitation

Influence of Spiral Stiffeners’ Symmetric and Asymmetric Angles on Nonlinear Vibration Responses of Multilayer FG Cylindrical Shells

This study utilizes a semi-analytical approach to examine the nonlinear dynamic responses of multilayer functionally graded (MFG) cylindrical shells reinforced by FG spiral stiffeners (FGSSs), which may have symmetric or asymmetric angles, under a thermal condition. It is presumed that the temperature is distributed across the thickness direction. The shell includes three layers: an outer ceramic-rich layer, a middle FG layer, and an inner metal-rich layer. By applying classical shell theory (CST), the smeared stiffeners technique, von Kármán equations, and the Galerkin method, the problem of nonlinear vibrations (NVs) has been addressed. Furthermore, the method of multiple scales (MMSs) is applied to investigate the nonlinear vibrational characteristics of the MFG cylindrical shells reinforced by FGSS, particularly focusing on the 1:2:4 internal resonance and the subharmonic resonance of order 1/2. This study finds that FG spiral stiffeners with symmetric or asymmetric angles and ambient temperature significantly affect the vibrational properties of the MFG cylindrical shells reinforced by spiral stiffeners.

Time-delay feedback control on vertical vibration of maglev train under unsteady aerodynamic force

The stability and safety of maglev trains are significantly affected by unsteady aerodynamic forces during high-speed operation. In this paper, we employ a time-delay feedback control strategy to suppress the nonlinear vertical vibrations of maglev train under periodic unsteady aerodynamic force. Firstly, the amplitude-frequency response equation of the maglev vehicle is derived using the multiple scales method, and the stability of the steady-state solutions is determined. Then, the influences of velocity feedback gain coefficient and velocity time delay on the nonlinear vibrations of the maglev vehicle are investigated. The optimal value range of the time delay is identified. The results demonstrate that the time delay feedback control can effectively suppress the nonlinear vibration of maglev vehicles and enhance their operational stability under the premise of reasonable selection of time delay parameters. The theoretical research presented in this paper can serve as a theoretical reference for the design, improvement, and optimization of control systems for high-speed maglev trains.

Geometrically nonlinear vibrations of plate embedding a large circular acoustic black hole

The concept of acoustic black hole (ABH) designates a decrease in bending stiffness and a localized added damping within a thin structure aiming to localize and efficiently dissipate vibrations. The small thickness results in large amplitude vibrations, which suggests to take into account geometrical nonlinearities. The subject has been previously studied on a beam with a one dimensional ABH. Previous research studied a geometrically nonlinear plate embedding a one dimensional ABH. This study aims at solving the Von Karman equations for a variable thickness plate, via a modal projection using the eigenmodes obtained with a finite element model. The use of a finite element model in this context allows to take into consideration more complex geometries. The methodology is then applied to a rectangular plate embedding a large circular ABH with added damping. The convergence of the nonlinear coefficients is studied and time domain computations show the relative importance of the geometrical nonlinearity.

Nonlinear vibration and parametric excitation of magneto‐thermo elastic embedded nanobeam using homotopy perturbation technique

AbstractThe present study focuses on the modeling and analyzing the nonlinear vibration patterns and parametric excitation of embedded Euler–Bernoulli nanobeams subjected to thermo‐magneto‐mechanical loads. The Euler–Bernoulli nanobeam is developed with external parametric excitation. The governing equation of motion is derived by utilizing nonlocal continuum theory and nonlinear von Karman beam theory. Subsequently, the homotopy perturbation technique is employed to determine the vibration frequencies. Finally, the modulation equation of Euler–Bernoulli nanobeams is derived for simply supported boundary condition. In order to validate our findings, we conduct a comparative analysis against existing literature, thereby underscoring the effectiveness and robustness of our proposed methodology. The influence of stress, magnetic potential, temperature, damping coefficient, Winkler coefficient, and nonlocal parameters are tested numerically on nonlinear frequency‐amplitude and parametric excitation–amplitude responses. The numerical examples indicate the significant impact of physical variables on the nonlinear frequency and parametric excitation. The primary objective of this study is to investigate the effects of external physical variables on the dynamic behavior of nanobeams, particularly in nonlinear regimes. This study provides insights into the design and control of nanostructures under complex load conditions, contributing to developing advanced materials and nanosystems.