Amplitude-frequency Curves Research Articles

Size-dependent linear and nonlinear vibration of functionally graded CNT reinforced imperfect microplates submerged in fluid medium

The linear and nonlinear vibrations of functionally graded carbon nanotubes reinforced microplate (FGCNTRM) submerged in fluid medium have been studied analytically. The primary novelty of this research is to take into consideration the effects of using carbon nanotubes (CNT) and geometric imperfection on the vibrational behavior of microplates submerged in the fluid. Equations of motion are obtained by first-order shear deformation theory of plates and considering the size effects and large deformations. Effective mechanical properties have been determined using the rule of mixtures for four various types of CNT distributions. The nonlinear equations of motion are discretized using the Galerkin method, and the equation response is acquired applying the method of multiple time scales. After verifying the results, the effect of microplate geometric characteristics, small size parameter, fluid height, geometrical imperfection amplitude, weight fraction, and distribution of CNT on linear and nonlinear frequencies as well as amplitude-frequency curves have been studied. The results demonstrate that an increase in fluid height leads to a decrease in the natural frequency. Also, geometric imperfection causes the stiffening behavior of the softening spring and substantially bends the response curve to the right. This bending can induce jump instabilities as the frequency slowly varies.

Study on the dynamics of two-degree-of-freedom fractional differential piecewise nonlinear systems under harmonic excitation

In this paper, a two-degree-of-freedom piecewise nonlinear system model with fractional differential term is constructed. The incremental harmonic balance method (IHB) is used to solve the two-degree-of-freedom piecewise nonlinear system with fractional differential term and the amplitude–frequency curve of the system is obtained. The correctness of the solution of the incremental harmonic balance method is verified by numerical methods. Then the influence of system parameters on the amplitude–frequency response characteristics of the system is studied. The bifurcation diagram of the system is analyzed by numerical method, and the dynamic characteristics of the system under typical parameters are obtained.

Bifurcation and stability analysis of fractional quintic oscillator system with power damping term

In this study, the resonance, bifurcation and stability of a class of fractional quintic systems are discussed. The implicit function expressions of the amplitude–frequency relationship of the steady-state response of the system under Coulomb dry friction, linear viscous damping, square damping, and cubic damping are derived by using the average method. Lyapunov’s first method is used to study the stability of the steady-state response of the system under the action of Coulomb dry friction, linear viscous damping, cubic damping, and analyze the influence of system parameters on the steady-state response. The amplitude–frequency curve of the system is drawn based on the implicit function expression and compared with the numerical solution. The primary resonance behavior of the system under four damping conditions is discussed when the fractional differential order, cubic stiffness coefficient, quintic stiffness coefficient, and fractional differential term coefficient take different values. The fork bifurcation and saddle–node bifurcation caused by the change in damping coefficient and fractional differential term coefficient under different parameters are studied.

An Efficient Analytical Method Based on Averaging and Memory-Free Principle for Variable Fractional Oscillators

Abstract It has been a difficult task to solve fractional oscillators analytically, especially when variable-order fractional derivatives (FDs) are included. The major difficulty consists in deriving analytical expressions for the variable FDs of trigonometric functions. To tackle this problem, a memory-free transformation for constant-order FDs is modified to transform the variable FDs equivalently into a nonlinear differential equation of integer order. Based on the equivalent equation, an analytical solution is obtained for the variable FD, showing nice agreement with numerical results. According to the approximate analytical solution in closed form, the frequency amplitude curve and the backbone line of variable fractional oscillators are determined accurately. In addition, it provides us with convenience in analyzing the primary resonance.

Softening-spring nonlinearity in large amplitude vibration of unsymmetric double-layer lattice truss core sandwich beams

In recent years, many novel sandwich structures with multilayer or graded lattice truss cores that exhibited superior structural performance have been proposed. However, only limited works have studied the nonlinear behavior of sandwich structures with unsymmetric lattice truss cores. This paper aims to provide an analytical study on the nonlinear vibration of the unsymmetric double-layer lattice truss core sandwich beams (LTCSBs). The double-layer LTCSB is designed to be unsymmetric and possesses varying material property and structural geometry in each layer. In this study, six unsymmetric cases of LTCSB classified as two categories according to the midplane locations are considered. Subsequently, an analytical model for the unsymmetric double-layer LTCSB is developed based on the Allen’s model and von Kármán nonlinear theory. The axial displacement of the midplane of LTCSB is considered in the analytical model, therefore the proposed model is more generalized compared with previous models for the symmetric double-layer LTCSB. The Ritz method with a direct iterative procedure is applied to solve the nonlinear governing equations and determine the nonlinear frequencies for the unsymmetric double-layer LTCSB. Finally, the effects of six unsymmetric cases of LTCSBs on the nonlinear frequency ratio versus amplitude curve under three different types of boundary conditions are discussed detailly. An interesting phenomenon of softening-spring nonlinearity is found for hinged–hinged and clamped–hinged sandwich beams with large bending–extension coupling.

Highly Sensitive Mass Sensing Scheme via Energy Relocalization With a Coupled Three-Beam Array

This article clarifies a high-sensitivity mass sensing mechanism, intrinsically correlated with energy relocalization. A sensing theory is derived to predict the amplitude–frequency curve with respect to the effect of mass perturbation, where high linearity is observed for both amplitude changes and frequency shifts. A localization factor is defined to characterize the degree of the energy localization, and its physical rationality is demonstrated with a coupled three-beam array as an example. A higher detection sensitivity via energy relocalization, increased by about 123.8% based on the amplitude change and by 250% based on the localization factor, is experimentally obtained with a high vibration amplitude compared to a low one. A mass sensing-warning scheme is further proposed for low-concentration monitoring and high-concentration warning, applicable to the detection and warning of a gas leak, dust pollution, harmful chemicals, and so on.

Inclination Effect on the Periodic Response of a Symmetrical MEMS Gyroscope.

The inclination effect caused by fabrication errors on the periodic response of a symmetric MEMS gyroscope is investigated. The dynamic equation is established considering the inclination effect on support stiffness and electrostatic forces. The periodic response is obtained by the averaging method. The two-variable singularity theory is employed to study the bifurcation characteristics and give transition sets on the DC-AC voltage plane, which divide the plane into four persistent regions. The amplitude-frequency curves demonstrate that only the two persistent regions with low voltages are feasible for the gyroscope. Both over-etching and under-etching reduce the feasible region. The effect of parameters on the performance is present. The mechanical sensitivity and nonlinearity increase with the voltages. With the increase in the inclination angle, the mechanical sensitivity and nonlinearity decrease first and then increase. The full temperature stability of the mechanical sensitivity is also considered. The variation in mechanical sensitivity with temperature is small at a large voltage and negative inclination angle. Under-etching, which leads to small nonlinearity and good temperature stability, is more beneficial to the performance of the gyroscope than over-etching.

Characterization of surface properties of a circular tube using nonlinear circumferential guided wave

The feasibility of using the nonlinear effect of primary circumferential guided wave (CGW) propagation for characterizing the surface properties in circular tubes has been investigated. This nonlinear approach will be suitable for some practical applications, where ultrasonic transducers cannot touch the surfaces to be inspected. An appropriate mode pair of primary CGW and double-frequency CGW (DFCGW) has been chosen to enable the second harmonic generation (SHG) of primary CGW propagation in the given circular tube to have a strong nonlinearity, i.e., the second-harmonic amplitude can accumulate along with the propagation circumferential angle. Here, the theoretical predictions, finite element simulations, and experimental measurements were conducted to analyze the influence of changes in the surface properties of the circular tube on the effect of SHG of primary CGW propagation, where a surface coating with the different properties was used to simulate changes in the surface properties of a circular tube. When the changes in the areas and thicknesses of surface coatings took place, the amplitude-frequency curves of both the fundamental waves and the second harmonics generated by CGW propagation were measured. The Stress Wave Factors (SWFs) of CGW propagation were introduced to characterize the changes in the surface properties. It was found that changes in the surface properties might clearly affect the efficiency of SHG of CGW propagation. There is a direct correlation between changes in the surface properties and the SWFs of CGW propagation. The consistency of theoretical predictions, finite element simulations, and experimental results validate the feasibility of characterizing changes in the surface properties of the given circular tube using the second-harmonic SWF of CGW propagation.

Design and Experiment of Mass Warning Resonant Sensor Induced by Modal Coupling

A new mass warning scheme via integrating modal coupled vibration of T-type resonant sensor is proposed in this paper, which can utilize the amplitude jump induced by mass perturbation to achieve the first and second mass warnings. A T-type resonant sensing structure with 1:2 internal modal coupling is designed and its coupling dynamic behavior is experimentally measured. Then, the theoretical expression is established and derived by Hamilton’s principle and multi-scale method to predict the amplitude-frequency curve with respect to the effect of mass perturbation. Besides, the physical conditions of modal coupled vibration under different perturbation masses are obtained. The working range of the resonant mass sensor is investigated by bifurcation analysis. Both theoretical and experimental results show that when the driving frequency is slightly greater than the upper critical frequency of the modal coupled vibration of the resonator, the resonant sensor has the first mass warning and the second mass warning with the increase of the perturbation mass.

Vibration and nonlinear dynamic response of nanocomposite multi-layer solar panel resting on elastic foundations

This paper dealt with the nonlinear dynamic response and vibration of multilayer organic solar panels subjected to mechanical and thermal loadings. The nanocomposite multilayer organic solar panel contains five layers which are manufactured by Al, P3HT: PCBM, PEDOT: PSS, Glass and Graphene which is a flexible replacement for indium tin oxide as a transparent and conducting electrode in the construction of solar panel. Based on classical theory, the basic equations are established by taking into account the effect of elastic foundations, von Kármán nonlinearity terms, and initial imperfection. The closed form expressions of the frequency–amplitude curves and natural frequency are obtained by using the Galerkin and Runger–Kutta methods. The numerical results illustrate positive effect of elastic foundations, negative effect of temperature increment and initial imperfection, considerable influence of geometrical parameters as well as excitation force on the free vibration and dynamic response of multilayer organic solar panel.

Dynamics analysis on the MDOF model of ball screw feed system considering the assembly error of guide rails

The dynamics of the ball screw feed system is very important in high-precision machining of CNC machine tools. However, the existing literatures rarely study the dynamic behaviors of the coupled feed system and didn’t take the influence of the assembly error of the guide rails into consideration. Because the stiffness of the feed system is mainly affected by the ball screw, linear guide and ball bearings, this paper presents the fourteen degree-of-freedom dynamic model of the feed system by deriving the nonlinear restoring force functions of the screw nut, linear guide and ball bearings and the screw shaft’s stiffness. An algorithm of combining the Newmark Method and the Monte Carlo simulation is proposed to numerically solve for the vibration responses of the feed system, and the proposed dynamic model is experimentally verified. Based on the established dynamic model, the effects of the excitation amplitude, system parameters and assembly error on the dynamic characteristics of the feed system are discussed by means of the amplitude-frequency curve and 3-D frequency spectrum, which demonstrate that the vibration responses of the feed system for the fourteen degrees of freedom influence each other, and the assembly error affects the nonlinear dynamic characteristics of the feed system.

Surface acoustic wave immunosensor based on Au-nanoparticles-decorated graphene fluidic channel for CA125 detection

A delay line configuration Rayleigh SAW immunosensor with an Au-nanoparticles-decorated graphene (AuNPs-Gr) fluidic channel is proposed for the detection of CA125. The tube composed of Gr sheets is prepared by chemical vapor deposition (CVD) and Au-nanoparticles are decorated on the inner wall of the Gr tube by electrodeposition method. The AuNPs-Gr tube with a volume of 4.8 μL is placed on the bus bar of the SAW device and acts as sensing element as well as a micro-fluidic chamber. To estimate the sensing performances of the immunosensor comprehensively, besides the amplitude-frequency characteristics of the insert loss (S21), the phase-frequency characteristics of S21, the equivalent conductance-frequency and susceptance-frequency characteristics are also analyzed. The deposition time of the Gr tubes, antibody concentration, antibody incubation time, and antigen incubation time are optimized to enhance the response of the SAW biosensor. Although the response sensitivity extracted from the amplitude-frequency curves (8.57 Hz/log(mU/mL)) is a bit lower than that from phase-frequency data (11.88 Hz/log(mU/mL)) and that from susceptance-frequency data (12.14 Hz/log(mU/mL)), a wider linear response range of 0.01–300 mU/mL is obtained from the amplitude-frequency curves and the limit of detection is calculated to be 0.00371 mU/mL. In addition, the proposed AuNPs-Gr fluidic channel SAW immunosensor exhibits good specificity and long-time stability and can be used to detect the CA125 in human serum.

Impedance modelling and stability assessment for grid‐connected dual‐PWM adjustable speed drives with motor torque oscillations

The grid-connected dual-PWM adjustable speed drives (ASDs) modulate the energy of the motor torque oscillation and inject it into the power system, potentially causing low-frequency oscillation (LFO) in the grid. In this study, considering the motor torque oscillation and harmonic common-modulation of the both-side converters, the precise sequence impedance models are derived for dual-PWM ASDs using the harmonic linearisation method. Based on the derived impedance of dual-PWM ASD, the torque oscillation, which is equivalent to the time-varying impedance, causes a series of discontinuities at the torque-oscillation-related frequencies on its amplitude-frequency curve. Moreover, these discontinuous drops make it more likely to intersect with the amplitude-frequency curve of the grid impedance, which may lead to LFO of the grid. Then, by applying the proposed impedance model and Bode plot, the stability of the grid-ASD cascade system with motor torque oscillation is assessed. The conclusions of stability analysis demonstrate the exact mechanism that the mechanical system oscillations lead to the grid-converter cascade system instability through the common-modulation and coupling of the dual-PWM ASD. Finally, experimental results on a 3 kW ASD system in the laboratory and measured results on the distribution grid in the Shanghai Yangshan Deepwater Port area verify the correctness and effectiveness of the proposed model and stability assessment.

Nonlinear dynamics of flexible diaphragm coupling’s rotor system during maneuvering flight

In this paper, the flexible multi-diaphragm coupling which is used as flexible power transmission shaft of the aeroengine accessory is taken as the research object, and the coupling stiffness matrix and axial nonlinear stiffness of the diaphragms are considered in the coupling rotor system. On this basis, in order to consider the influence of aircraft maneuvering load, in non-inertial system the bending-pendular-axial coupled differential equations of flexible diaphragm coupling were established by Lagrange method. The modal characteristics of the flexible diaphragm coupling were analyzed and compared with the finite element solutions, and the correctness of model is verified. Runge-Kutta method is used to solve and analyze the influence of different maneuvering flight conditions on the vibration characteristics of the flexible diaphragm coupling. The research indicates that the coupling between diaphragm’s axial and radial stiffness leads to the right shift of resonant region, the increase of resonance peak value, and the nonlinear characteristics of amplitude-frequency curve such as jump and multi-value. In the non-inertial system, only the installation distance a of the flexible diaphragm coupling along the wingspan leads to the increase of the axial deformation offset of the flexible diaphragm coupling in the rolling flight state. The increase of climbing or diving angular velocity makes the flexible diaphragm coupling’s vibration changes from single period to multi-period, bifurcation or chaos state; With the increase of diving angular velocity and rolling angular velocity, the axial critical speed gradually increases; Each flight condition not only affects the vibration characteristics, but also causes the axial, radial and angular deformation of the flexible diaphragm coupling to a certain extent. This study provided a theoretical basis and method for the design and analysis of diaphragm coupling.

Dynamic Responses of a Pile with a Cap under the Freezing and Thawing Processes of a Saturated Porous Media Considering Slippage between Pile and Soil

The freezing/thawing stratification effect of seasonal factors or artificial disturbances in frozen soil regions has an important influence on the vertical vibration of the pile–soil–cap system. Taking into account the slippage between the pile and soil, a simplified layered analytical model of the vertical vibration of the pile–soil–cap system in a double-layered stratum under the freezing and thawing processes of a saturated porous medium was established, and the analytical solution of the dynamic response on the top of the pile cap was obtained. In this model, frozen saturated porous media and Biot’s porous media theory were used to simulate frozen soil and unfrozen soil, respectively. The validation of the slippage model was first verified by comparison with the results of the existing model tests. This was followed by a dynamic model test of the pile–soil–cap system in a self-made, ground-freezing system. In comparison with the analytical results and the experimental results of model tests under the freezing/thawing processes, the validation of the present model is further verified. A comprehensive parametric study reveals that the parameters of the frozen or thawed soil layer have significant effects on the amplitude–frequency curve of the vertical vibration of the pile foundation.

Calculating Parametric Oscillation of Third-Order Nonlinear System With Dynamic Friction and Fractional Damping

AbstractIn this paper, the parametric resonance of third-order parametric nonlinear system with dynamic friction and fractional damping is investigated using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the resonance solution is then obtained by means of Lyapunov theory. Additionally, the effect of the fractional derivative on the system dynamics is analyzed. The effects of the two parameters of the fractional-order derivative, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated.

Parameter optimization of a grounded dynamic vibration absorber with lever and inerter

Mechanical vibration is mostly harmful, and it may not only generate noise but also affect the working life of the equipment. Lever, inerter, and grounded stiffness have good performance in the field of vibration control, but the dynamic vibration absorber simultaneously containing lever, inerter, and grounded stiffness is rarely studied. Based on the grounded dampertype dynamic vibration absorber, a dynamic vibration absorber with lever, inerter, and grounded stiffness is presented. And the optimal system parameters are analytically researched in detail. Firstly, the differential equation of motion is established according to Newton’s second law, and the analytical solution of the system is obtained. According to the amplitudefrequency curve of the system, it is obvious that there are two fixed points unrelated to the damping ratio. Meanwhile, the optimal frequency ratio of the dynamic vibration absorber is obtained based on the fixed-point theory. Under the premise of ensuring the system stability, the optimal grounded stiffness ratio is screened out, and the working range of inerter is further calculated. It is found that the inerter ratio has two working ranges when the coupling term values of magnification ratio and mass ratio are different. Furthermore, the approximate optimal damping ratio is derived by minimizing the maximum value of the amplitudefrequency curve. Using MATLAB, the numerical result is analyzed, and the correctness of analytical results is verified. Compared with other dynamic vibration absorbers under harmonic and random excitations, it is known that the model in this paper can evidently reduce the resonance amplitude and broaden the vibration band of the primary system. These results may provide a theoretical basis for the optimal design of similar dynamic vibration absorbers.

An Amplitude-Frequency Curve-Based Parameter Sensitivity Analysis for the Conveying Fluid Pipe

Sensitivity analysis is a common way to research the parameters’ impact on complex systems, for instance national economy, environment, ecosystem, etc. The parametric analysis of a conveying fluid pipe is a challenge because it is a complex high-order partial differential system. In this brief, a mechanism-based method is proposed to analyze the parameter sensitivity of the conveying fluid pipe. Firstly, the mathematical model in a high-order partial differential form of the conveying fluid pipe under a harmonic load is deduced and transformed into a nonlinear ordinary differential equation form through the Galerkin truncation. The amplitude-frequency curve of the target system is established by harmonic balance method and matrix analysis technique. Then, the parameter sensitivity is discussed by the analytical results of the amplitude-frequency curve through Newton’s method. Finally, some examples are applied to intuitively illustrate the conveying fluid pipe’s parameter sensitivity.

Parameter Optimization for a Novel Inerter-based Dynamic Vibration Absorber with Negative Stiffness

A novel dynamic vibration absorber(DVA) model with negative stiffness and inerter-mass is presented and analytically studied in this paper. The research shows there are still two fixed points independent of the absorber damping in the amplitude frequency curve of the primary system when the system contains negative stiffness and inerter-mass. The optimum frequency ratio is obtained based on the fixed-point theory. In order to ensure the stability of the system, it is found that inappropriate inerter coefficient will cause the system instable when screening optimal negative stiffness ratio. Accordingly, the best working range of inerter is determined and optimal negative stiffness ratio and approximate optimal damping ratio are also obtained. At last the control performance of the presented DVA is compared with three existing typical DVAs. The comparison results in harmonic and random excitation show that the presented DVA could not only reduce the peak value of the amplitude-frequency curve of the primary system significantly, but also broaden the efficient frequency range of vibration mitigation.

Vibration isolation performance of simply supported beam installed with a negative stiffness device

A negative stiffness device composed of two horizontal pre-compression springs is proposed to isolate the vibration of the simply supported beam. The nonlinear vibration equation of simply supported beam installed with a negative stiffness device is established by using Hamilton principle first. The lower degree of freedom dynamic equation of the system is derived by using Galerkin truncation approach. The influence of support position and negative stiffness parameters on the natural frequency of the beam are obtained from the modal analysis of the free vibration system. According to the dynamic internal force of the beam, the force transmissibility at the support end is defined and the displacement transmissibility of any point on the beam is also defined to facilitate the analysis of the vibration isolation effect. The influence of negative stiffness support parameters on the amplitude–frequency curve and the transmissibility curve of the simply supported beam is discussed. The results show that the fundamental frequency of the system can be significantly reduced by the negative stiffness device, while the frequencies of high-order modes are changed a little bit only. Increasing the horizontal spring stiffness will reduce the first-order resonance frequency and the initial frequency of vibration isolation. Consequently, the aim of lower frequency vibration isolation for the simply supported beam can be achieved. This paper provides an effective approach to the vibration isolating design of flexible structures.