Amplitude-frequency Curves Research Articles

Synchronous detection of multiple micro-substances with a single output channel using a weakly coupled resonator array

Abstract In this paper, a multiple micro-substances sensing scheme with a single output channel using a weakly coupled resonator array is proposed. An analytical model of the resonator using the Euler-Bernoulli beam theory has been developed, as well as the dynamic behavior has been further explored using the Galerkin method. We have discovered numerically that, in the weakly coupled cantilever array, the A-f (amplitude-frequency) curve of any cantilever physically reflects the vibration features of other cantilevers. Hence, by measuring the output signal of a single cantilever, multiple substances applied on each cantilever respectively can be identified and detected synchronously. A single output channel via a weakly coupled micro-resonator array is constructed and validated numerically for picogram level mass detection. Equivalent experiments with a macro coupled five-cantilever array have been further conducted for verification. Under a common sweep-driven signal, the five analytes applied on each cantilever with masses at microgram level can be detected synchronously and independently by measuring the frequency shifts of the five resonant peaks of the center cantilever. Multiple substances sensing with a single output channel and a single driving signal is thus realized with low relative errors and high linearities. By enhancing the driving voltage, the mass resolution can also be improved via Duffing bifurcation. This work not only reveals a new coupled vibration behavior but also provides a new avenue for multiple analytes detection.

Fluid Interaction Analysis for Rotor-Stator Contact in Response to Fluid Motion and Viscosity Effect

Fluid–structure interaction introduces critical failure modes due to varying stiffness and changing contact states in rotor-stator systems. This is further aggravated by stress fluctuations due to shaft impact with a fixed stator when the shaft rotates. In this paper, the investigation of imbalance and rotor-stator contact on a rotating shaft was carried out in viscous fluid. The shaft was modelled as a vertical elastic rotor system based on a vertically oriented elastic rotor operating in an incompressible medium. Implicit representation of the rotating system including the rotor-stator contact and the hydrodynamic resistance was formulated for the coupled system using the energy principle and the Navier–Stokes equations. Additionally, the monolithic approach included an implicit strategy of the rotor-stator fluid interaction interface conditions in the solution methodology. Advanced time-frequency methods, such as Hilbert transform, continuous wavelet transform, and estimated instantaneous frequency maps, were applied to extract the vibration features of the dynamic response of the faulted rotor. Time-varying stiffness due to friction is thought to be the main reason for the frequency fluctuation, as indicated by historical records of the vibration displacement, whirling orbit patterns of the centre shaft, and the amplitude–frequency curve. It has also been demonstrated that the augmented mass associated with the rotor and stator decreases the natural frequencies, while the amplitude signal remains relatively constant. This behaviour indicates a quasi-steady-state oscillatory condition, which minimises the energy fluctuations caused by viscous effects.

Research on vertical vibration characteristics of rolling mill based on magnetorheological fluid damper absorber

The study of rolling mill vibration theory has always been a scientific frontier in the field of rolling forming, which is very important to the quality of sheet metal and the stable operation of equipment. A magnetorheological fluid damper absorber is designed to control the nonlinear vertical vibration of rolling mill. Considering the fractional order and delay factors in the control effect of magnetorheological fluid, the fractional order delay nonlinear vertical vibration equation with magnetorheological damping damper is established. The amplitude-frequency characteristic equations of the main resonance and sub-resonance of the system are solved by multi-scale method. The effects of stiffness coefficient, damping coefficient, time delay, fractional order, and exciting force on the vibration characteristics of roller system are analyzed by comparing the amplitude-frequency curve, time-domain curve and phase diagram. The transition set of the steady-state response of the system and the corresponding bifurcation topology structure are analyzed by using the singularity theory. A magnetorheological fluid damper absorber platform of rolling mill is built based on modern test technology, and the influence laws of the control threshold, control current, type of magnetorheological fluid and reduction rate are analyzed. The correctness and feasibility of the design of the magnetorheological fluid damper absorber are verified, which provided theoretical guidance and technical support for the nonlinear dynamic analysis and stability control of the rolling mill.

Investigation on the primary resonance of a cylindrical bubble in compressible liquid

In the present paper, the paramount characteristics of the resonance of cylindrical bubbles in a compressible liquid are theoretically investigated with multi-scale analysis. Considering the liquid's compressibility, a dimensionless equation of the cylindrical bubble wall motion is established for the primary resonance under a single-frequency acoustic field. Comparing with the numerical results, the present analytical solution is verified in terms of accuracy. The key parameters on the characteristics of primary resonance are further explored including the equilibrium bubble radius, dimensionless amplitude of the acoustic field, and other detuning parameters. The main conclusions are given as follows: (1) During primary resonance, three typical nonlinear phenomena are observed: multivalued solutions, jumps, and hysteresis phenomena. (2) The liquid's compressibility affects the intensity of acoustic waves emitted by the bubbles during primary resonance. The maximum pressure at the bubble interface in the incompressible liquid is higher than that in the compressible liquid. (3) In the amplitude–frequency curve, the unstable region of the compressible liquid is smaller and the peak is lower than in the incompressible liquid.

Nonlinear vibration analysis of multi-support oil-tank system in aero-engine based on asymptotic methods

This article is primarily focused on the complicated nonlinear vibration behavior of an aero-engine lubricating oil-tank system with multiple nonlinear pedestals. First, the rubber damping ring’s nonlinear factors are introduced into a cubic nonlinear dynamic model of lubricating oil-tank system. The nonlinear system parameters are identified by testing a simplified aero-engine lubricating oil-tank structure. Then, the nonlinear dynamic model is solved via asymptotic approach, and results are confirmed using Runge-Kutta numerical analysis approach. The system’s vibration responses and amplitude-frequency curves are also acquired, and influence of system parameters on nonlinearity is analyzed. Finally, an experimental investigation on the simplified oil-tank is performed. The nonlinear vibration characteristics of the rubber damping ring and its vibration reduction mechanism are verified using vibration transmissibility. The experimental results are consistent with numerical results. This paper’s recommended analysis approach has engineering significance for vibration and noise reduction design of aero-engine components.

A new semi-analytical approach for nonlinear dynamic responses of functionally graded porous graphene platelet-reinforced circular plates and spherical shells

This paper presents the semi-analytical approach for nonlinear dynamic responses of porous graphene platelet-reinforced circular plates (CiPs) and spherical shells (SpSs) resting on the Pasternak foundation in thermal environment. The graphene platelets (GPLs) are distributed in the functionally graded (FG) distribution patterns and uniform distribution pattern through the CiP and SpS thickness direction. The governing equations of the GPL-reinforced structures subjected to time-dependent external pressure respected to the harmonic and linear functions of time are established based on the geometrically nonlinear higher-order shear deformation theory. A simple and effective semi-analytical solution for the considered problem is established using the Euler-Lagrange equations, and the damping viscosity of the foundation can be counted using the Rayleigh dispassion function. The Runge-Kutta method is employed to determine the dynamic responses of the GPL-reinforced structures, while the fundamental frequencies of free and linear vibration, and frequency-amplitude curves are obtained in explicit form. The numerical results obtained reveal that the GPL and porous distributions, geometrical and material properties can significantly affect the frequency-amplitude curves, and dynamic responses of harmonically loaded and linearly loaded structures, specially, the dynamic buckling phenomenon is clearly observable with the SpSs but not with the CiPs.

Vibration control of nonlinear viscoelastic systems with displacement delay feedback

In this study, the nonlinear vibration characteristics of a nonlinear Zener model under harmonic excitation, equipped with linear and nonlinear displacement delay feedback control, were investigated. The amplitude–frequency curve of the system’s main resonance was computed using the multi-scale method. Stability conditions for the system were then established based on the Lyapunov stability theory. The influence of several parameters on the system’s dynamic behavior was also analyzed, including time delay, displacement feedback gain coefficient, nonlinear term coefficient, damping coefficient, and external excitation. The findings revealed that the nonlinear displacement feedback gain coefficient was observed to exert a more substantial effect on the system’s amplitude than other factors. Further, the nonlinear displacement delay feedback gain coefficient and time-delay value diminish the amplitude of the vibration, leading all solutions to achieve a stable state. In instances with time-delay control, the impact of the main system parameters on system vibration was significantly diminished. The aim of this study is to provide a theoretical basis for the implementation of displacement delay controllers in nonlinear viscoelastic systems.

A quasi-zero-stiffness vibration isolator inspired by Kresling origami

Vibration isolators are widely used to mitigate undesirable vibrations in engineering systems and structures, but low-frequency vibration isolation remains a challenge. Origami-inspired structures, with their tunable stiffness and transformative capabilities, offer potential for vibration isolation. Inspired by conical Kresling origami, this study proposes a quasi-zero-stiffness vibration isolation system. Static analysis of the origami structure reveals that the system exhibits near-zero dynamic stiffness at the static equilibrium position, while maintaining static stiffness during the load-bearing process. The amplitude-frequency characteristic formula and vibration transmissibility formula are derived using the harmonic balance method. The effects of varying damping ratios and excitation amplitudes on the amplitude-frequency and transmissibility curves are examined. It is shown that the vibration isolation range increases with the number of layers. A comparison of the system's response to harmonic vibrations under different compressive deformations highlights the effectiveness of the origami quasi-zero-stiffness system as a vibration attenuator. Compared to existing quasi-zero-stiffness dampers, the origami-inspired system features a wider isolation frequency band, a lower initial isolation frequency, and reduced vibration transmissibility, demonstrating superior isolation performance, particularly at low frequencies.

Gyroscopic rotor dynamics simulation with anisotropy of elastic and damping characteristics of the support

In the work, the differential equations of motion of the gyroscopic rotor, built taking into account the anisotropy of stiffness and damping of the flexible support, are solved analytically, by the method of harmonic balance, convenient for obtaining separately amplitude-frequency and phase-frequency characteristics in the direction of oscillations. The equations of the non-stationary process are obtained by the method of changing amplitudes. It has been found that when the linear stiffness of the elastic support is different in two orthogonal directions, two critical velocities and the corresponding resonant regions arise. In the area of each critical speed, there are two amplitude-frequency curves of oscillations of the main direction and the direction perpendicular to it, respectively. The geometric nonlinearity of damping suppresses the elevations of these amplitude-frequency curves more significantly than linear damping. If only one of the two directions has a damping nonlinearity, then its effect is on the amplitude-frequency curves of the corresponding critical speed. It is more efficient to control resonant amplitudes for smooth resonant transitions by enhancing the linear damping with geometrically nonlinear damping. The results of the analytical solution of the equations of motion agree well with the results of direct modeling and experimental studies.

Resonance and attraction domain analysis of asymmetric duffing systems with fractional damping in two degrees of freedom

This article establishes a two degree of freedom asymmetric Duffing system model with fractional damping through the study of the Jeffcott rotor system with horizontal support, and analyzes the dynamic behavior of its resonance and attraction domain structures. The second-order approximate solution of the system equation is obtained using a multiscale method, and the slow flow modulation equations of both the amplitudes and phases are extracted. The comparison between approximate and numerical solutions has shown their high consistency and the Routh Hurwitz criterion is utilized to study the stability of the system. Finally, we explore the effect of fractional damping on the vibration behavior of the system using amplitude frequency curves, attractive watersheds, and time trajectory plots, including both negative and positive values of the nonlinear stiffness coefficient. The analysis shows that if the coefficient and order of the fractional damping term are reasonably selected, the horizontal and vertical displacements of the system vibration can be effectively controlled. The importance of this work lies in its application in rotor vibration, which has significant implications for the study of the system under investigation.

Vibration control of FG-pipe conveying fluid using a nonlinear absorber with nonlinear damping in longitudinal direction

This study investigates the use of a nonlinear absorber with nonlinear damping to mitigate the transverse vibrations of a fluid-conveying pipe made of functionally graded materials (FGM). The nonlinear effects arising from the pipe’s curvature and inertia are precisely accounted for in the equation of motion. The governing equations of the FGM pipe-nonlinear absorber system are derived using extended Hamilton’s principle and solved using the method of multiple scales. 1:1 internal resonance condition between the first natural frequency of the pipe and absorber is explored for transferring energy from the pipe to the absorber based on the modal interaction. The frequency-amplitude curves reveal that as the power-law index (n) increases, the nonlinear hardening behavior of the pipe decreases, while that of the absorber increases. Incorporating nonlinear damping in the absorber is found to further improve vibration suppression, leading to a strongly modulated response that reduces the pipe vibrations by over 84.57% and the absorber vibrations by over 94.57%. The impact of the absorber’s nonlinear stiffness on the strongly modulated response is also investigated, indicating that excessively high stiffness can cause the disappearance of this beneficial behavior. The findings of this study provide valuable insights into the optimal design of FGM pipes conveying fluid and the integration of nonlinear absorbers with nonlinear damping for enhanced vibration mitigation.

Problems and corrections of classical mathematical model for piecewise linear system

Due to the existence of gaps or backlash, many mechanical systems can be simplified into piecewise linear models. The dynamic study on mechanical systems should be based on reliable mathematical models. So that it is very important to determine the contact point and separation point between the primary system and the auxiliary spring system (ASS) in a piecewise linear system. In most existing literature, the contact point and separation point of the mathematical model are fixed at the gap. But in this paper, it is found that the contact point and separation point actually change with the system parameters when the ASS contains a damper, which implies the most existing mathematical models are incorrect. It is firstly demonstrated through numerical solution that the primary system will prematurely separate from the ASS before reaching the gap under harmonic excitation, which shows the incorrectness of the classical mathematical models. Then, based on the mechanical model and engineering practice, two corrected mathematical models are proposed. And the motions of the primary system and ASS after premature separation in the corrected models are studied. Finally, through comparisons of the contact points, separation points, amplitude-frequency curves and motion states between the corrected models and the classical mathematical model, it can be concluded that the corrected models are more reasonable. And comparisons with the experimental data imply that the corrected models can better reflect the engineering practice. These results will be helpful to the study and design of the piecewise linear system.

A methodology for evaluating damage in thin-walled cylindrical shells using bounded ultrasonic beam scattering

This paper introduces an ultrasonic detection methodology specifically designed to assess damage in thin-walled cylindrical shells, which crucially relies on the interaction between a bounded ultrasonic beam and a thin-walled cylindrical shell immersed in fluid, considering both water-filled and air-filled cavities. Initially, we derive the scattered sound field expression for a bounded ultrasonic beam obliquely striking the shell. Subsequently, through finite element simulations and experimental verification, we observe that when the incident bounded beam is emitted at the critical angles defined in this study, early damage significantly enhances the received sound pressure amplitude detected by a symmetrically positioned receiver. Notably, a mere 5 % decrease in the shell's elastic modulus leads to a notable 233.69 % increase in the area under the amplitude-frequency curve of the received sound pressure for water-filled cavities and a remarkable 642.85 % surge in area for air-filled cavities. Experimental data further validates the sensitivity of this method towards varying corrosion damage states. This approach combines the reliability of linear ultrasonic methods with the enhanced sensitivity of nonlinear ultrasonic techniques, offering a significant advancement over traditional ultrasonic detection methods for inspecting cylindrical shells. The proposed assessment method holds immense potential in providing novel insights for inspecting cylindrical shells in practical applications.

Nonlinear resonant analyses of graphene oxide powder reinforced hyperelastic cylindrical shells containing flowing-fluid

The resonant responses are investigated for the graphene oxide powder reinforced hyperelastic cylindrical (GOPRHC) shells containing flowing-fluid, and the shells are subjected to the radial harmonic excitations. Employing the improved Donnell's nonlinear shell theory, Halpin-Tsai model, hyperelastic constitution relation, velocity potential theory and Lagrange equation, the governing equation of motions are obtained for the GOPRHC shells containing flowing fluid. The amplitude frequency and force amplitude curves are presented by using the harmonic balance method and the pseudo-arc length continuation method. The effects of three distributions of the GOP, weight fraction of the GOP and fluid velocity on the natural frequency for the GOPRHC shells are discussed. The influences of different parameters on the dynamical responses for the GOPRHC shells containing flowing-fluid are conducted, including the external excitation, weight fraction of the GOP, fluid velocity and structural parameters. The results show that the GOPRHC shells with Hypere-X distribution containing flowing-fluid have the largest frequency. The super-harmonic resonance responses appear and present the synchronization effects with the primary resonant responses in the GOPRHC shells containing flowing-fluid. The increases of external excitations, fluid velocity, weight fraction of the GOP and structural parameters enrich the resonant responses for the GOPRHC shells. Compared to the fluid-filled hyperelastic cylindrical shells, the existence of the flowing-fluid makes the GOPRHC shells more prone to the chaotic vibrations. The hysteresis phenomena of chaotic vibrations occur in the GOPRHC shells containing flowing-fluid.

Nonlinear dynamics modeling and analysis of aviation drilling actuator feed system considering assembly error

The superior dynamic performance and machining accuracy of the aviation drilling actuator are crucial for high-quality drilling of aviation structural components. It is essential to construct a dynamic model that accurately predicts the dynamic characteristics of the actuator. However, the modeling rarely considers the influence of assembly errors, which makes it difficult to ensure the model precision. In this work, the nonlinear force of the joint surface between the ball screw (BS), rolling bearings and ball linear guide (LG) was derived, taking into account the moving and rotation errors of the guide rail (GR) and the misalignment errors of the bearings. A dynamic model with six degree-of-freedom (DOF) was established for the aviation drilling actuator. The Runge-Kutta method was selected to solve the model, and the model accuracy was verified by experiments. The influence of various assembly errors and worktable feed position on the vibration response of the aviation drilling actuator in three directions was analyzed by the time history, frequency spectrum, phase diagram, and amplitude-frequency curve. The research results indicate that variations in assembly errors and feed position affect the dynamic properties of the system. The moving errors along the horizontal and vertical directions and the rotation errors around the horizontal and vertical directions of the GR notably affect the amplitude-frequency characteristics and vibration displacement of the two directions. The bearing misalignment error in the horizontal plane significantly affects the axial vibration of the drilling actuator. The maximum vibration response in the three directions is achieved when the feed position is in the middle of the BS. Therefore, dynamic model analysis can provide theoretical guidance for adjusting assembly errors and selecting reasonable machining positions to improve the stability and machining accuracy of aviation drilling actuator.

Nonlinear poro-thermo-forced vibration in curved sandwich magneto-electro-elastic shells under hygrothermal environment

This research employs a multiple scales perturbation approach to evaluate the nonlinear wave propagation behaviors of a doubly curved sandwich composite piezoelectric shell with a flexible core in a hygrothermal environment. Stress and strain calculations for the flexible core and face sheets are carried out using Reddy's third-order shear deformation theory (TSDT) and third-order polynomial theory, respectively. The study explores the synergistic effects of a multilayered shell, flexible core, and magneto-rheological layer (MR) in revealing the nonlinearity of both in-plane and vertical moment within the core. The Halpin–Tsai model is employed to derive the properties of polymer/carbon nanotube/fiber (PCF) and polymer/graphene platelet/fiber (PGF) three-phase composite shells. The governing equations for the multiscale shell are derived using Hamilton's formulation. The research investigates temperature variations, diverse distribution patterns, curvature ratios, and magnetic fields through numerical analysis, presenting the results graphically and prior research has demonstrated the accuracy of these methods. Notably, these factors exert significant influence on the frequency-amplitude curves of the smart structure.

Comprehensive design strategy of the NSD–TMD controlled system with closed-form frequency-amplitude solutions

A negative stiffness device (NSD) utilizing pre-compressed helical springs has been integrated with diverse vibration control techniques. Recently, the incorporation of NSD has been proposed to enhance the efficacy of a tuned mass damper (TMD) controlled system. This study introduces two design strategies for the NSD–TMD system, which functions either as a TMD with an adjustable operating bandwidth or as a pure non-linear energy sink. First, the non-linear force of NSD was expressed in a Taylor series approximation form incorporating a negative linear stiffness and a positive cubic stiffness term. Second, the equations of motion were formulated, and closed-form solutions were derived using the complexification–averaging method. The accuracy of the Taylor series approximation was substantiated within a specified range through a comparison of force–displacement curves. Third, the accuracy of the dynamic closed-form solutions was affirmed by comparing the amplitude–frequency curves with the numerical simulations. For a case study, an actual timber pedestrian bridge with decaying natural frequency was selected, and the two design strategies were examined individually. The vibration control performance of the TMD and NSD–TMD was also comparatively analyzed. The assessment focused on the peak force transmissibility value and the effective operating frequency bandwidth. Results highlighted a notable enhancement in operational performance, particularly for primary structures exhibiting diminishing modal frequencies. In a TMD-controlled system, the efficacy of vibration control deteriorates and may even amplify the dynamic response when a modal frequency shift occurs in the primary structure. Conversely, both design strategies of NSD–TMD can achieve lower transmissibility and broaden the operating bandwidth.

Approximate analytical solutions of nonlinear vibration and bifurcation of dielectric elastomer balloon

Dielectric elastomers (DE), a novel class of actuators, are increasingly employed in fields such as soft robotics and vibration control. Most existing studies investigating the dynamic characteristics of DE structures neglect the viscoelasticity and strain-stiffening effects of materials, with only a few offering approximate analytical solutions. This paper establishes the governing equation for the dynamics of a DE balloon, utilizing the principle of virtual work and considering material viscoelasticity and strain-stiffening effects through linear damping and the Gent hyperelastic model, respectively. The nonlinear governing equations are simplified using the Chebyshev polynomial approximation. The multiple scales method is applied to derive approximate solutions for the structure's free vibration under static voltage excitation and parametrically excited vibration under harmonic voltage excitation. The accuracy of these approximate analytical solutions is validated by comparison with numerical solutions. Additionally, the paper analyzes stability and bifurcation phenomena through time-domain response curves, amplitude-frequency curves, and phase trajectories. Furthermore, the influence of the stretching limit, voltage, and damping on the nonlinear vibration of the balloon structure is studied, providing a valuable theoretical foundation for the future design and utilization of dielectric elastomers.

Achieving transition between internal resonance and mode localization through coupling strength modulation

Nonlinear mode coupling phenomena are prevalent in a wide range of dynamic systems, offering intriguing insights into the rich complexities of vibrational behavior. Internal resonance (IR) and mode localization (ML) are two typical mode coupling phenomena that have been widely studied and applied for sensing and energy harvesting. In this work, the transition between IR and ML has been studied theoretically and experimentally utilizing a mechanically coupled resonator, as well as its potential application has been explored. Numerical results of amplitude–frequency curves, phase–frequency curves, and the frequency veering curves under different internal and external conditions all indicate that the vibration behavior of the coupled resonator will transition from 1:1 IR region to ML region as the coupling stiffness increases. An equivalent experiment has also been implemented for verification, and its results are consistent with the theoretical and simulation ones. Furthermore, a self-powered sensor with a tunable coupling has been proposed, which can combine the advantage of IR for broadband energy harvesting and that of ML for ultra-high sensitivity mass detection.

Influence of contact characteristics of joint surface on vibration performance of the giant magnetostrictive transducer

The giant magnetostrictive transducer is a transducer device that realizes magneto-mechanical energy conversion. Each mechanical component is connected by the joint surface to realize the transmission of ultrasonic vibration energy. To ensure the efficient transmission of energy, the influence of the contact characteristics of the joint surface on the output stability of the transducer is studied. According to the contact conditions between the mechanical joint surfaces of the giant magnetostrictive ultrasonic transducer (GMUT), the key factors affecting the mechanical vibration performance of the ultrasonic system are obtained by analyzing the vibration transmission characteristics of the joint surfaces. The influence of different joint surface contact characteristics (JSCT) on the amplitude–frequency curve of the transducer is analyzed by the ANSYS software. Simulations and experiments have been conducted to verify the predicted results. As the preload force increases, the natural frequency of the transducer increases, but the growth rate decreases. The natural frequency of the double-rod GMUT with a small joint surface area is slightly higher in the optimal output mode shape, while its amplitude is significantly larger. The optimal prestress for the GMUT with different-sized joint surfaces is 3–5 MPa. This paper acquires the JSCT under a high-frequency vibration condition, reveals the influence of the joint surface on the mechanical output performance of the GMUT, and can provide a theoretical basis for the design and analysis of the rotary ultrasonic vibration processing system.