Frequency Response Diagrams Research Articles

Secondary Resonances of Asymmetric Gyroscopic Spinning Composite Box Beams

A comprehensive theoretical investigation on the occurrence of secondary resonances in parametrically excited unbalanced spinning composite beams under the stretching effects is conducted numerically and analytically. Based on an optimal stacking sequence and Rayleigh’s beam theory, the governing equations of the system are derived using extended Hamilton’s principle. The system’s partial differential equations are then discretized using the Galerkin method. Numerical (Runge–Kutta technique) and analytical (multiple scales method) approaches are exploited to solve the reduced-order equations, and their results are compared and verified accordingly. Comparison and convergence investigations are performed to guarantee the validity of the outcomes. Stability and bifurcation analyses are accomplished, and resonance effects are thoroughly studied utilizing frequency-response diagrams, phase portraits, Poincaré maps and time-history responses. It is observed that among the various types of secondary resonance, only a combination resonance can be observed in the system dynamics. The outputs reveal that, in this resonance, the gyroscopic coupling results in the steady-state time response consisting of three main frequencies. By examining the effects of damping, eccentricity, and beam length, it is exhibited that this resonance does not occur in the system’s dynamics for any combination of these parameters. Therefore, these parameters can be adjusted in the design of asymmetric beams to prevent this type of resonance.

VIBRATIONS OF A VERTICAL BEAM ROTATING WITH VARIABLE ANGULAR VELOCITY

An Euler-Bernoulli beam in vertical position rotating about its symmetry axis along its length is considered. The angular velocity is assumed to have small fluctuations about a constant mean velocity. The partial differential equation of motion is derived first. The equation is cast into a non-dimensional form. The natural frequencies are calculated for the pinned-pinned case. Principle parametric resonances such that the fluctuation frequency being close to two times one of the natural frequencies are considered. By employment of the Method of Multiple Scales, an approximate perturbation solution is found. The frequency response diagrams are drawn and the bifurcation points for transition from the trivial solution to the non-trivial solution are calculated. The conditions for which such resonances occur are exploited in the numerical results.

Exploring nonlinearity in quarter car models with an experimental approach to formulating fractional order form and its dynamic analysis.

This study explores the inherent nonlinearity of quarter car models by employing an experimental and numerical approach. The dynamics of vehicular suspension systems are pivotal for ensuring passenger comfort, vehicle stability, and overall ride quality. In this paper we assessed the impact of various parameters and components on suspension performance, enabled the optimization of ride comfort, stability, and handling characteristics. Firstly, experimental analysis allowed for the investigation of factors that are challenging to model theoretically, such as stiffness nonlinearity and damping characteristics, which may vary under different operating conditions. Time domain and frequency response diagram of the model has been obtained. Secondly, a quarter-car with single degree-of-freedom presented and investigated in fractional order form. Fractional order dynamics emphasize nonlinearities in quarter car models, capturing real-world dynamics effectively. The proposed fractional-order nonlinear quarter car model employed Caputo derivative. For numerical analysis of fractional order system, the Adam-Bashforth-Moulton method is used and the disturbance of road assumed to be stochastic. Results show that the dynamic response of the vehicle can be chaotic. Influence of road roughness amplitude and frequency on vehicle vibration is investigated.

Design and analysis of an exponentially tapered piezoelectric energy harvester under nonlinear rotational indirect magnetic excitation

Piezoelectric energy harvesting from rotational motions currently receives attention as a clean and permanent energy source. However, because of low output power, optimizing and improving the efficiency of this strategy through different methods is of great importance. This study investigates a new high-efficiency piezoelectric energy harvester under rotational indirect magnetic excitation using frequency up-conversion. The studied piezoelectric energy harvester is composed of an exponentially tapered cantilever beam with a piezoelectric layer, excited magnets, a tip mass, an octagonal disk attached to the rotating shaft, and exciting magnets on the disk. The potential energy and nonlinear magnetic force are determined and substituted into an electromechanically coupled governing equation after derivation. Numerical methods are applied to solve the governing equations due to their nonlinear nature and periodic oscillations. The tip mass varies to maintain a constant natural frequency and cancel out the effect of concurrent variations of the natural frequency and tapering parameter. Verification of the theoretical model is performed by the finite element and experiment methods. The effects of some parameters, such as the tapering parameter, total resistance of the circuit, number of magnets, and rotational speed, on the output voltage and power densities are evaluated. Results show a fourfold increase in output power density when comparing exponentially tapered width to constant width. Finally, frequency response curves are plotted and analyzed to examine the effect of shaft rotational speed, subharmonics, and bandwidth. The most important result extracted from the frequency response diagram is the increase in the number of subharmonics and the observation of 2 ω/5 and 2 ω/7 subharmonic components in the frequency response of the harvester with exponentially tapered width.

Nonlinear Phase Angle and Amplitude Analysis of Porous Functionally Graded Material Rotating Shaft Under Electromagnetic Loads

In this research, the nonlinear response including amplitude and phase angle of a functionally graded inextensional rotating shaft is studied under the electromagnetic load. Three types of porosity distributions through the radial direction are considered in this work. For the rotating shaft with nonlinear curvature and inertia, the governing equations and corresponding boundary conditions are derived. The Galerkin and the multiple scales methods are used to obtain the modulation equations. The effect of the power law index for a functionally graded shaft fabricated from the mixture of ceramic and metal is presented in the frequency response diagrams for primary resonance under electromagnetic force. Numerical simulations include the effects of the power law index of functionally graded material (FGM), porosity distribution, air-gap length, electromagnetic load, and vibration modes of a rotor on the phase angle and amplitude of steady state responses.

Radial Gradient Seismic Metamaterials with Ultra-Low Frequency and Ultra-Wide Band Gap

In this paper, a radial gradient seismic metamaterial (RGSM) is proposed. The structural unit cell is composed of an external square soil embedded with a triangular-cross-sectioned steel ring, which is filled at different angles of multiple steel rings to form a supercell. The dispersion curve and attenuation spectrum of the unit cell are calculated by the finite element method, and the opening mechanism of the band gap is explained by analyzing the modes at the band gap boundary. The influence of geometric parameters and material parameters on the band gap is further studied, and the optimized supercell radial gradient seismic metamaterial (OS-RGSM) structure is designed through structure and parameter optimization. The ultra-low broadband excellent band gap in the range of 2.35–20 Hz for seismic Lamb waves is realized, and its three-dimensional frequency response and displacement field diagram are calculated. In addition, the attenuation characteristics of the optimized supercell seismic metamaterial on the seismic surface wave are calculated and analyzed. It is found that the attenuation can reach more than 50% in the ultra-low frequency range of 3.5–9 Hz. The seismic wave barrier is verified by the vibration transmission characteristics of RGSM under finite period and dynamic time history analysis. The results show that RGSM can effectively shield from seismic Lamb waves in the ultra-wideband with the starting frequency of 2.35 Hz and can also effectively attenuate the seismic surface wave in semi-infinite space.

Numerical investigation on the coupled vibrations of piezoelectric energy harvester with a liquid-filled proof mass

Replacing the solid tip mass of a piezoelectric cantilever beam with a liquid-filled mass can increase its frequency bandwidth due to the effect of nonlinear liquid sloshing. To investigate the coupled vibrations of the piezoelectric beam and the sloshing liquid, as well as their contributions to the output power, a coupled two-dimensional finite element method-smoothed particle hydrodynamics model has been developed in this study. Using this model, the dynamic behavior of a piezoelectric beam with a liquid-filled rectangular container as the tip mass, subjected to vertical harmonic excitation, has been investigated. The effects of parametric sloshing, excitation level, and geometric nonlinearity on the output voltages have been studied in detail. The simulation results indicate that: (a) the parametric sloshing in the liquid container exhibits subharmonic characteristics, which can be triggered by matching the excitation frequency to twice the natural frequency of the sloshing mode; (b) the piezoelectric beam exhibits subharmonic or harmonic oscillations at parametric resonance; (c) due to the effect of coupled vibrations, the energy harvester with a liquid-filled proof mass has a broader bandwidth compared to the traditional harvester; (d) the frequency response diagram of the output voltage shows multiple peaks at high excitation amplitudes, and the bifurcations are caused by parametric sloshing.

Linking and Comparison of the Damping of Fluid Transients and Frequency Response Diagram Methods for Pipe Leak and Burst Detection and Localization

Linking and Comparison of the Damping of Fluid Transients and Frequency Response Diagram Methods for Pipe Leak and Burst Detection and Localization

Simulation analysis of electromagnetic noise of shaftless rimless thruster

In this paper, the shaftless rimless propulsion motor is taken as the research object, and the modal analysis of the stator in free state and constrained state is carried out by using the finite element method. The electromagnetic force is solved by numerical method, and then it is loaded on the stator. The harmonic response analysis of the stator is carried out, and the acceleration and displacement frequency response diagrams of the stator surface are obtained, and the vibration nephogram of the stator is analyzed. The results show that the acceleration and displacement in the radial direction are greater than those in the axial direction, which indicates that the vibration frequency in the radial direction is greater than that in the axial direction under the excitation of the radial electromagnetic force. When the frequency of the radial electromagnetic force is close to the natural frequency of the motor, the motor resonates and the vibration frequency increases.

Nonlinear damping in micromachined bridge resonators

This study presents a thorough theoretical and experimental investigation on the nonlinear damping of in-plane micromachined electromechanical resonators. More specifically, experiments are conducted on an electrically actuated bridge resonator, and the primary resonance response of the system is obtained at various AC and DC voltages. A nonlinear theoretical model is developed using the Euler–Bernoulli beam theory while accounting for the geometric, electrostatic (including fringing field effect), and damping nonlinearities. Two damping models are considered in the theoretical model: the Kelvin–Voigt model, which for this system is a nonlinear damping model due to the presence of geometric nonlinearities. The second damping model consists of linear, quadratic, and cubic damping terms. A high-dimensional discretisation is performed, and the nonlinear dynamics of the resonator are examined in detail in the primary resonance regime by constructing the frequency response diagrams at various AC and DC voltages. Thorough comparisons are conducted between the experimental data and the theoretical results for different damping conditions. It is shown that the microresonator displays strong nonlinear damping. Detailed calibration procedures for the nonlinear damping models are proposed, and the advantages and disadvantages of each nonlinear damping model are discussed.

A modified power decoupling control strategy for a grid-connected inverter with a low switching frequency under unbalanced grid voltages

In the photovoltaic grid-connected power generation system, when proportional resonant (PR) control is adopted for the grid-side inverter in the two-phase stationary coordinate (αβ), there is a coupling problem between active power (P) and reactive power (Q). Under unbalanced grid conditions, especially in high-power applications where the switching frequency of the grid inverter needs to be reduced, the power coupling problem will be exacerbated. Based on the instantaneous power response characteristics of the grid inverter, expressions of the power coupling coefficient under unbalanced power grids are derived in this paper, and the influence of the unbalanced grid voltage on the power coupling is revealed. Moreover, a decoupled grid-side model of the grid inverter is established, and a modified controller based on positive and negative sequence (P–N sequence) decoupling is proposed to solve the power coupling problem. The closed-loop transfer function of the system with the modified current controller is studied, and the stability of the modified controller is analyzed according to the bilateral frequency response diagram. Based on analysis of the coupling coefficient, the degree of coupling between P and Q can be reduced significantly by applying the proposed control method. Furthermore, in order to solve the influence of the delay on the power coupling caused by the low switching frequency, the decoupling compensation link for the delay is introduced, which can eliminate the influence of delay on power coupling. Finally, experiments are performed on a grid-connected inverter prototype. The experimental results show that when the grid voltage is unbalanced, the proposed control method can reduce the coupling degree of the P and Q, which also improves the dynamic performance of the grid-connected inverter with a low switching frequency.

BVD and Mason’s modelling of piezoelectric bulk acoustic resonators for high frequency applications

This paper deals with the electrical features of BAW devices designed specifically for 3.83 GHz applications which are highly in demand as per the keyless controls or wireless telecommunication technologies such as mobile phone applications. Results of the BVD analytical modelling are shown in form of 1D BVD circuit parametric analysis for one vibration mode of piezoelectric plate of AlN of 250 nm. The relation between the two resonant frequencies with the physical parameters of a resonator is important for designing circuits and for analysis of electrical behaviour of piezoelectric materials. Resonators are used where narrow filtering characteristics are required such as entry remote controls, cellphone filters or GPS receivers. Surface acoustic wave resonators are not very suitable for most demanding applications. Compared to it and dielectric ceramic filters, the BAW technology provides advantages in insertion loss, Q values, operating frequency, power handling capacity, temperature stability with also having added benefits of small size & low cost. We used MATLAB Simulink, 1D modelling to get an impedance frequency response of 3.83 GHz BAW resonator. Calculations were performed to get the desired resonant frequency & in accordance with the major parameters of interest, we got the required magnitude plot in Bode frequency response diagram corresponding to motional leg of a modified BVD Circuit. It was found that the maximum value of Impedance was 149 dB at Anti resonant frequency of 3.909 GHz and minimum value of Impedance as −75.4 dB at resonant frequency of 3.83 GHz with higher gain & phase margins. After that we performed similar analysis for Mason’s model implementation of the Impedance equation using MATLAB software.

Identification of Composite-Metal Bolted Structures with Nonlinear Contact Effect

The middle layer model has been used in recent years to better describe the connection behavior in composite structures. The influencing parameters including low pre-screw and high preload have the main effects on nonlinear behavior of the connection as well as the amplitude of the excitation force applied to the structure. Therefore, in this study, the effects of connection behavior on the general structure in two sections of increasing damping and reducing the stiffness of the structures that lead to non-linear phenomena have been investigated. Due to the fact that in composite structure we are faced to the limitation of increasing screw preload which tend to structural damage, so the investigation on the hybrid connection (metal-composite) behavior is conducted. In this research, using the two-dimensional middle layer theory, the stiffness properties of the connection are modeled by normal stiffness and the connection damping is modeled using the structural damping in the shear direction. Nonlinear frequency response diagrams have been extracted twice for two different excitation forces and then proposed by a high-order multitasking approximation according to the response range of the nonlinear finite element model for stiffness and damping of the connection. The effect of increasing the amplitude of the excitation force and decreasing the preload of the screw on the nonlinear behavior of the component has been extracted. The results show that the limited presented novel component model has been accurately verified on the model obtained from the vibration experimental test and the reduction of nonlinear model updating based on that is represented. The comparison results show good agreement with a maximum of 1.33% error.

Experimentally validated geometrically exact model for extreme nonlinear motions of cantilevers

A unique feature of flexible cantilevered beams, which is used in a range of applications from energy harvesting to bio-inspired actuation, is their capability to undergo motions of extremely large amplitudes. The well-known third-order nonlinear cantilever model is not capable of capturing such a behaviour, hence requiring the application of geometrically exact models. This study, for the first time, presents a thorough experimental investigation on nonlinear dynamics of a cantilever under base excitation in order to capture extremely large oscillations to validate a geometrically exact model based on the centreline rotation. To this end, a state-of-the-art in vacuo base excitation experimental set-up is utilised to excite the cantilever in the primary resonance region and drive it to extremely large amplitudes, and a high-speed camera is used to capture the motion. A robust image processing code is developed to extract the deformed state of the cantilever at each frame as well as the tip displacements and rotation. For the theoretical part, a geometrically exact model is developed based on the Euler–Bernoulli beam theory and inextensibility condition, while using Kelvin–Voigt material damping. To ensure accurate predictions, the equation of motion is derived for the centreline rotation and all terms are kept geometrically exact throughout the derivation and discretisation procedures. Thorough comparisons are conducted between experimental and theoretical results in the form of frequency response diagrams, time histories, motion snapshots, and motion videos. It is shown that the predictions of the geometrically exact model are in excellent agreement with the experimental results at both relatively large and extremely large oscillation amplitudes.

Oscillations of a soft viscoelastic drop

A soft viscoelastic drop has dynamics governed by the balance between surface tension, viscosity, and elasticity, with the material rheology often being frequency dependent, which are utilized in bioprinting technologies for tissue engineering and drop-deposition processes for splash suppression. We study the free and forced oscillations of a soft viscoelastic drop deriving (1) the dispersion relationship for free oscillations, and (2) the frequency response for forced oscillations, of a soft material with arbitrary rheology. We then restrict our analysis to the classical cases of a Kelvin–Voigt and Maxwell model, which are relevant to soft gels and polymer fluids, respectively. We compute the complex frequencies, which are characterized by an oscillation frequency and decay rate, as they depend upon the dimensionless elastocapillary and Deborah numbers and map the boundary between regions of underdamped and overdamped motions. We conclude by illustrating how our theoretical predictions for the frequency-response diagram could be used in conjunction with drop-oscillation experiments as a “drop vibration rheometer”, suggesting future experiments using either ultrasonic levitation or a microgravity environment.

Damage severity for cracked simply supported beams

This paper investigated the static and dynamic behaviors of isotropic cracked simply supported beam using finite element analysis (FEA), ANSYS software. Modal and harmonic vibration analysis of intact and damaged beam were performed in order to extract mode shapes of bending vibration, natural frequencies and obtain frequency response diagram. Static finite element analysis of undamaged and damaged simply supported beam was carried out to determine zero frequency deflection, then stiffness of intact and cracked beam was computed using conventional formula. Crack damage severity of damaged beam was calculated and it is noticed that as crack position is increased from left hand support of beam up to central point and crack depth is increased, then crack damage severity increases. The effect of mode shape pattern is investigated and it is found that the amount of decreasing of natural frequency is proportional to the normalized mode shape at position of crack. The exhibited correlation between results for damaged beam revealed that crack damage severity is proportional to zero frequency deflection and inversely proportional to first mode frequency.

On the dynamics of vibro-impact systems with ideal and non-ideal excitation

This study is concerned with modelling and analyses of a vibro-impact system consisting of a crank-slider mechanism and one oscillator attached to it, where the system is exposed to a non-ideal excitation. The impact occurs during the motion of the oscillator when it fits a base, and the excitation of the driving source is affected by this behaviour. The aim is to determine the interaction between a driving torque and the motion of the oscillator. To achieve this aim in a methodologically sound manner, both vibrating and vibro-impact systems with an ideal and non-ideal excitation are analysed. Analytical and numerical solutions are obtained for the vibrating system with the ideal excitation. Numerical analyses of the vibrating system with the non-ideal excitation is then conducted, where the characteristic curves for this system are found analytically. Numerical simulations are also carried out for other two systems and the results obtained are shown in terms of frequency–response diagrams, time-displacement diagrams and basins of attraction. The results found for different systems are compared mutually, and the differences between them are pointed out. Impact solutions for different regions of the excitation frequency are shown. For the vibro-impact system with the non-ideal excitation, the average value of its frequency is used.

Numerical analysis of a vibro-impact system with ideal and non-ideal excitation

This study is concerned with modelling and analyses of a vibro-impact system consisting of a crank-slider mechanism and one oscillator attached to it, where the system can be exposed to ideal or non-ideal excitation. The impact occurs during the motion of the oscillator when it hits a base, and the excitation of the driving source is affected by this behaviour. The aim is to determine the interaction between a driving torque and the motion of the oscillator. To achieve this aim in a methodologically sound manner, both vibro-impact systems with ideal and non-ideal excitation are analysed. For these system differential equations are formed and the impact model is provided in the paper. The impact causes a strong nonlinearity in the system. The mathematical model of the vibro-impact system with ideal excitation is presented as a second order differential equation where the vibro-impact system with non-ideal excitation is given as a coupled system of nonlinear second order differential equations. Numerical simulations are carried out for the two systems and the results obtained are shown in terms of frequency response diagrams as well as in terms of time-displacement diagrams. The results found for different systems are compared mutually, and the differences between them are pointed out. Impact solutions for different regions of the excitation frequency are shown. For a specific value of the excitation frequency in the frequency response diagram where multiple solutions are found, basin of attractor diagrams are formed. Average value of the excitation frequency is used for the vibro-impact system with non-ideal excitation.

Periodic response of a nonlinear axially moving beam with a nonlinear energy sink and piezoelectric attachment

An efficient semi-numerical framework is used in this paper to analyze the dynamic model of an axially moving beam with a nonlinear attachment composed of a nonlinear energy sink and a piezoelectric device. The governing equations of motion of the system are derived by using the Hamilton’s principle with von Karman strain-displacement relation and Euler - Bernoulli beam theory. The nonlinear energy sink is modeled as a lumped - mass system composed of a point mass, a spring with nonlinear cubic stiffness and a linear viscous damping element. The piezoelectric device is placed in the ground configuration. Frequency response curves of the presented nonlinear system are determined by introducing the incremental harmonic balance and continuation method for different values of material parameters. Based on the Floquet theory, the stability of periodic solutions was determined. Moreover, the presented results are validated with the results obtained by a numerical method as well as the results from the literature. Numerical examples show a significant effect of the nonlinear attachment on frequency response diagrams and vibration amplitude reduction of the primary beam structure.

Development of a Novel Tuning Approach of the Notch Filter of the Servo Feed Drive System

To alleviate the vibration effect of the feed drive system, a traditional approach is to apply notch filters to suppress vibrations in the velocity loop. The current approach is to adjust the notch filter parameters such as the center frequency, damping and depth based on observing the frequency response diagram of servo velocity closed loop. In addition, the notch filters are generally provided in the velocity loop control for the commercialized controllers such as FANUC, Siemens, etc. However, the resonance of the transmission system also appears in the position loop when the linear scale is used as the position feedback. The notch filter design without consideration of the resonance behavior of the position loop might cause degradation of performance. To overcome this problem, the paper proposes an innovative method which could automatically determine the optimal parameters of the notch filter under the consideration of resonance behavior of position loop. With the optimal parameters, it is found that both the gains of position and velocity loop controller could be increased such that the bandwidths of the position/velocity loops are higher. Based on the simulation results, the rising time is improved by 33% and the time for reaching the steady state is improved by 72% as comparing the cases of using the optimal approach and traditional approach.