Small-amplitude Vibrations Research Articles

Dynamic Control of Piezoelectric Sandwich Beams with Multi-Frequency Excitations

In this work, we present a non-linear analysis of the dynamics of a sandwich beam with piezoelectric patches under multi-frequency excitations. The model takes into account geometrical non-linearities, piezoelectric non-linearities, and inertia. Hamilton’s principle is used to determine the electromechanical equation of motion, and Garlekin’s technique is used to determine the equivalent model of the piezoelectric generator. A mathematical methodology based on the multi-scale method for vibration control and stability is reviewed in this work to determine a system of amplitude and phase equations. The control of primary and secondary mode resonances is developed, and feedback effects are analyzed for small and large amplitudes of vibrations of sandwich beams. Frequency response curves are presented and discussed for various gain parameters.

Experimental and theoretical analysis of charge length on single-hole vibration amplitude from underground deep-hole blasting

Deep-hole blasting, characterized by large diameter and long charge length, is a prevalent practice in large-scale underground mining. However, the increasing charge length in each borehole, together with the growing charge weight, necessitates a heightened focus on the induced vibration effects. Understanding the impact of long charge length on induced vibration in comparison with traditional vibration characteristics is crucial for the safety control of deep-hole blasting. Therefore, experiments involving 6 charge lengths and 4 monitoring stations were conducted in an underground mine to clarify the variation of waveform and peak particle velocity (PPV). The entire borehole waveform superposed by the short charge elements was theoretically calculated and compared with the measured waveforms. The exact full-field model was further used to gain a comprehensive understanding of the effect of charge length and the related parameters of initiation positions and velocity of detonation (VoD). The experimental data and model analysis allow the following conclusions: (1) the effect of charge length on amplitude is highly dependent on the Mach wave, irrespective of varying initiation positions and VoDs; (2) an increase in the charge length does not necessarily lead to an increase of the PPV in the far-field, but it would result in a faster attenuation rate below a certain length; (3) given a constant charge weight, a greater length-to-diameter ratio of the explosive column results in a smaller vibration amplitude; (4) the appropriate duration of the explosive pressure should be considerably longer than the actual duration for investigating blast vibration.

Wire rope isolator identification and dynamic modeling for small amplitude vibrations

Wire rope isolators are used in variety of applications to protect sensitive equipment from vibration. The nonlinear hysteretic behavior of steel wires provides advantages when compared to linear vibration isolators. This study proposes an amplitude dependent stiffness and damping model for low amplitude vibrations under axial loading, where the parameters can be determined with an experimental procedure. Comprehensive experimental results of forced vibration tests with varying loading, frequency and preload were considered in the model identification. Amplitude dependent stiffness and loss energy models were determined from the test data, and the effect of the preload and loading frequency on the model parameters were studied. It is shown, that the effect of preload and frequency is not evidently clear, while the effect of vibration amplitude is more significant. The mathematical model was further verified against measurements from base excitation loading. The proposed model can be used to study the effectiveness of the selected wire rope isolator configurations in chosen application, and to effectively perform dynamic design studies.

Performance of surge-induced synchronized switch harvesting on inductor as an efficient technique under strong electromechanical coupling and low amplitudes of vibration

The process of harvesting energy from ambient sources is key for various applications. This study examined the performance of two representative techniques from the surge-induced synchronized switch harvesting on inductor (S3HI) family under strong electromechanical coupling and compared this performance with that of some established techniques. S3HI techniques exploit the surge voltage to overcome the voltage barrier of diodes for rectifiers and storage capacitor voltages. One of their features, revealed in a previous study on weakly coupled systems, is that they can harvest substantial energy from low-level vibrations. This feature is desirable for certain use cases. This study aimed to clarify whether this feature is applicable even when the coupling is strong. The performance of various established techniques and two representatives from the S3HI group was studied by formulating an approximate analytical solution and performing numerical simulations and experiments. The theoretical results were confirmed to be consistent, and the discrepancy between experimental and theoretical results was minor. These results clearly demonstrate that the techniques from the S3HI family can effectively harvest substantial energy from small-amplitude vibrations, even when the coupling is strong. Moreover, the performance advantage of S3HI methods is even greater when the coupling is strong.

A vertical track nonlinear energy sink

Eliminating the effects of gravity and designing nonlinear energy sinks (NESs) that suppress vibration in the vertical direction is a challenging task with numerous damping requirements. In this paper, the dynamic design of a vertical track nonlinear energy sink (VTNES) with zero linear stiffness in the vertical direction is proposed and realized for the first time. The motion differential equations of the VTNES coupled with a linear oscillator (LO) are established. With the strong nonlinearity considered of the VTNES, the steady-state response of the system is analyzed with the harmonic balance method (HBM), and the accuracy of the HBM is verified numerically. On this basis, the VTNES prototype is manufactured, and its nonlinear stiffness is identified. The damping effect and dynamic characteristics of the VTNES are studied theoretically and experimentally. The results show that the VTNES has better damping effects when strong modulation responses (SMRs) occur. Moreover, even for small-amplitude vibration, the VTNES also has a good vibration suppression effect. To sum up, in order to suppress the vertical vibration, an NES is designed and developed, which can suppress the vertical vibration within certain ranges of the resonance frequency and the vibration intensity.

Effects of wind-induced static angle of attack on flutter performance of long-span bridges using 2D bimodal and 3D multimodal analysis

Flutter performance of long span bridges is directly affected by winds featured with large angles of attack (AoAs) in typhon climates and mountainous areas. Flutter critical wind speed and flutter derivatives (FDs) of a streamlined closed-box girder under various initial AoAs are obtained by free and forced vibration wind tunnel tests of a section model, based on which the two-dimensional(2D) and three-dimensional (3D) flutter performance and evolution mechanisms are further investigated. The results indicate that FDs identified at small vibration amplitudes through different forced vibration control methods, i.e., fixing wind speed or vibration frequency, in wind axis or body axis are almost identical. The flutter performance of the closed-box girder is susceptible to the AoA. The flutter critical wind speed changes significantly with the initial AoA due to the nonlinear variation of some typical FDs. The sign of A2* shifts from negative to positive and the reduced wind speed associated with the sign change becomes lower as the initial AoA increases, which worsens the flutter performance. The flutter critical wind speed would significantly decrease because of the consideration of the static additional AoA in the range of positive initial AoA. In addition, the participation of multi modes and different longitudinal distribution of static additional AoA are taken into consideration in 3D numerical analysis, which have little influence on the flutter critical wind speed of long span bridges between the 2D and 3D numerical analysis.

Procedure in developing a longitudinal-torsional vibration-assisted micro-milling system

Micromachining is an advanced microfabrication technique for micro-sized or micro-accuracy products through subtractive manufacturing such as micro-milling. Vibration-assisted machining (VAM) is a method in which small amplitude vibrations are applied to the tool or workpiece to enhance the fabrication process. Milling results could be improved by adding longitudinal and torsional vibration to the tool using piezoelectric components vibrating at the ultrasonic frequency with an amplitude of less than 1 μm is used. A slip ring is required to transmit electric power from a static structure to a rotating frame. After the power has been transmitted to the system, an ultrasonic horn, called an acoustic horn or sonotrode, amplifies the vibrations at the tool’s tip. Since the excitation vibration is only longitudinal, the magnification of the vibration at the tooltip is also only longitudinal. Grooves are added to the side of the ultrasonic horn to produce torsional vibrations. This vibration is then simulated through finite element analysis software, explicitly using explicit dynamics. A 3D simulation is run for a quarter cycle of micro-milling through a Ti-6Al-4V material. It could be concluded that the LTVAM application may improve machining quality, such as temperature reduction of up to 9%, cutting force reduction of up to 35%, and surface roughness improvement of up to 27%.

Experimental characterization of a nonlinear mechanical oscillator with softening behaviour for large displacements

This paper presents an experimental insight into the performance of a mechanical oscillator consisting of an X-shaped-spring configuration. This configuration achieves an overall softening characteristic with quasi-zero stiffness behaviour far away from the static equilibrium point. Such a geometrical nonlinear configuration has attracted significant research attention in the last few years, particularly for its application as a vibration isolator with the possibility to extend the quasi-zero-stiffness region beyond that of the classical three-spring nonlinear isolator. However, previous experimental evidence has been limited to small amplitude vibration excitation only. Furthermore, it has been focused mainly on the isolation region, rather than on the large amplitude response, thus circumventing an insight on the damping effects and its modelling. To address this gap, in this paper, both frequency sweeps and random excitations are applied to a prototype device for experimental characterization. A nonlinear stiffness model is developed based on the geometry of the system and a nonlinear damping model is assumed based on experimental observation. The proposed model accurately describes the dynamic behaviour of the system as shown by comparison of theoretical and experimental data.

Molten Salt Pump Journal-Bearings Dynamic Characteristics Under Hydrodynamic Lubrication Conditions

Abstract A reliable high-temperature molten salt pump is critical for the development of Fluoride-salt-cooled, High-temperature Reactors (FHRs). By supporting the rotating journal, the appropriate journal bearing can ensure that the high-temperature molten salt pump runs smoothly and efficiently in the high-temperature fluoride salt over a long period of time. However, many bearing candidates served well for only a short period and experienced several issues. Moreover, the alignment of the molten salt pump journal bearings is a key factor for the molten salt pump's long-term steady running. In the long-term operation, a misalignment in the journal bearing can result in vibrations and excessive wear on the bearing surface of the molten salt pump. The journal bearing dynamic characteristics can be used to accurately assess the journal misalignment. Therefore, it is necessary to investigate the detailed journal-bearing dynamic behavior under the high-temperature hydrodynamic fluoride salt lubrication conditions for FHR applications. In this study, a small amplitude vibration is superimposed on a steady-running journal bearing to simulate the molten salt operating conditions. A Fortran 90 code has been written for the journal-bearing dynamic behavior analysis. The code was verified using the numerical data reported in the literature. The code is then employed to predict the dynamic coefficients of high-temperature fluoride salt hydrodynamic lubricated journal-bearing with various Sommerfeld numbers. These journal-bearing dynamic coefficients can be used to provide guidelines in the design of molten salt pumps.

Theory of nonlocal modal hydrodynamic functions for beam and plate vibrations in viscous fluids

In this paper, we introduce a new nonlocal modal hydrodynamic theory for fluid–structure interactions (FSI) of light, flexible cantilever beams and plates undergoing small amplitude vibrations in Newtonian, incompressible, viscous, heavy fluids otherwise at rest. For low aspect ratio flexible structures and high mode numbers, three dimensional (3D) and nonlocal fluid effects become prominent drivers of the coupled dynamics, to the point that existing local hydrodynamic theories based on two dimensional (2D) fluid approximations become inadequate to predict the system response. On the other hand, our approach is based on a rigorous, yet efficient, 3D treatment of the hydrodynamic loading on cantilevered thin structures. The off-line solution of the FSI problem results in the so-called nonlocal modal hydrodynamic function matrix, that is, the representation of the nonlocal hydrodynamic load operator on a basis formed by the structural modes. Our theory then integrates the nonlocal hydrodynamics within a fully coupled structural modal model in the frequency domain. We compare and discuss our theory predictions in terms of frequency response functions, mode shapes, hydrodynamic loads, quality factors, added mass ratios with the predictions of the classical local approaches, for different actuation scenarios, identifying the limitations of the hypotheses underlying existing treatments. Importantly, we also validate our new model with experiments conducted on flexible square plates. While computationally efficient, our fully coupled theory is exact up to numerical truncation and can bridge knowledge gaps in the design and analysis of FSI systems based on low aspect ratio flexible beams and plates.

Medium-high frequency vibration control of an industrial pipeline using squeeze-mode magnetorheological dampers

Due to fluid excitation and the vibration of connected equipment, industrial pipelines are prone to long-term medium-high frequency and small amplitude vibration. Magnetorheological (MR) dampers with a multi-stage squeeze plate structure were proposed to solve the problem of medium-high frequency vibration in the industrial pipeline. The dynamic characteristics of the industrial pipeline were analyzed to determine the installation position of the MR dampers. The mechanical properties of the proposed squeeze-mode MR dampers were tested, and the hyperbolic tangent mechanical model of the MR damper was established. The vibration control model of the industrial pipeline was built for simulation analysis. In addition, the Acceleration-Driven-Damper (ADD) control and the Proportional-Integral-Derivative (PID) control were used for simulation to predict the vibration reduction effect. Constant current experiments and control algorithm experiments were carried out on the vibration control platform of the industrial pipeline. The experimental results show that the maximum vibration attenuation rate of the MR dampers with constant current reaches 61.95%, while those with ADD control and PID control are 65.43% and 67.46%, respectively. Therefore, the squeeze-mode MR dampers can effectively reduce the medium-high frequency vibration of the industrial pipeline, and the control algorithms can further improve the vibration reduction performance.

Voltage-controlled non-axisymmetric vibrations of soft electro-active tubes with strain-stiffening effect

Material properties of soft electro-active (SEA) structures are significantly sensitive to external electro-mechanical biasing fields (such as pre-stretch and electric stimuli), which generate remarkable knock-on effects on their dynamic characteristics. In this work, we analyze the electrostatically tunable non-axisymmetric vibrations of an incompressible SEA cylindrical tube under the combination of a radially applied electric voltage and an axial pre-stretch. Following the theory of nonlinear electro-elasticity and the associated linearized theory for superimposed perturbations, we derive the nonlinear static response of the SEA tube to the inhomogeneous biasing fields for the Gent ideal dielectric model. Using the State Space Method, we efficiently obtain the frequency equations for voltage-controlled small-amplitude three-dimensional non-axisymmetric vibrations, covering a wide range of behaviors, from the purely radial breathing mode to torsional modes, axisymmetric longitudinal modes, and prismatic diffuse modes. We also perform an exhaustive numerical analysis to validate the proposed approach compared with the conventional displacement method, as well as to elucidate the influences of the applied voltage, axial pre-stretch, and strain-stiffening effect on the nonlinear static response and vibration behaviors of the SEA tube. The present study clearly indicates that manipulating electro-mechanical biasing fields is a feasible way to tune the small-amplitude vibration characteristics of an SEA tube. The results should benefit experimental work on, and design of, voltage-controlled resonant devices made of SEA tubes.

Nonlinear electromagnetic vibration energy harvester comprising dual helical-plane springs and multiple Halbach arrays for low-frequency and small-amplitude vibrations

Electromagnetic vibration energy harvesters are widely investigated for self-powered wireless sensors and nonlinearity has been introduced for low-frequency and broadband vibration energy harvesting. But how to realize the nonlinearity with small amplitudes, high reliability and few complexities is still a challenge in real-world applications. In this paper, a novel electromagnetic vibration energy harvester comprising dual helical-plane springs and multiple Halbach arrays is proposed. The structural characteristics are analyzed, which indicate that the helical-plane springs can have nonlinear stiffness under small-amplitude vibrations and multiple Halbach arrays can greatly enhance the magnetic field. Then a magnet-electro-mechanical model is built by combining mechanical dynamics &electrodynamics, which is numerically solved by using the Runge-Kutta algorithm. Finally, the feasibility of the nonlinear electromagnetic vibration energy harvester is validated both numerically and experimentally. The results show that it has a nonlinear stiffness, a resonance bandwidth of 3 Hz and a peak power of 14 mW when the excitation amplitude is only 0.5 g. In particular, the resonance frequency range depends on the excitation amplitude. Furthermore, the prototype of a self-powered wireless temperature sensor is constructed and testified. The results of this study indicate that the proposed structure can be utilized and extended to build compact, reliable and nonlinear electromagnetic vibration energy harvesters for low-frequency, small-amplitude and broadband vibrations in real-world applications.

A broadband hybrid energy harvester with displacement amplification decoupling structure for ultra-low vibration energy harvesting

Abnormal vibration of electrical equipment requires a large number of distributed sensors to monitor. However, there are issues that require maintenance and power supply. This work proposes a working mode for harvesting the low-grade vibration energy of electrical equipment, which can replace vibration sensors and achieve self-powered monitoring of equipment vibration. Additionally, we present a novel energy harvesting module that solves the issue of converting random vibration into linear vibration. The introduction of laminated hinges achieves large displacement output under small amplitude vibration and energy harvesting from random low amplitude vibration within a bandwidth of 2–23 Hz. The impact of geometric parameters on the proposed system is evaluated through parameter analysis. Experimental findings reveal that the device can achieve a maximum power output of 17.84 mW at an ultra-low 0.05 g acceleration., resulting in a peak power density of approximately 44.87 W/m3. Compared to traditional methods, peak power has increased by 266.2 %. Wireless vibration online monitoring systems for visualizing vibration status are created by combining the equipment with repeaters and phones. This research introduces a unique strategy for remote online vibration monitoring that may be used in distributed sensor systems to identify malfunctioning machinery.

Frequency response of a mass grounded by linear and nonlinear springs in series: An exact analysis

An exact analytical solution for the frequency response of a system consisting of a mass grounded by linear and nonlinear springs in series was derived. The system is applicable in the study of beams carrying intermediate rigid mass. The exact solution was derived by naturally transforming the integral of the governing nonlinear ODE into a form that can be expressed in terms of elliptic integrals. Hence, the exact frequency–amplitude solution was derived in terms of the complete elliptic integral of the first and third kinds. Periodic solutions were found to exist for all real values of [Formula: see text] and for [Formula: see text]. On the other hand, linear or weakly nonlinear frequency–amplitude responses were found to occur during small-amplitude vibrations ([Formula: see text]), very large-amplitude vibrations with strong hardening nonlinearity ([Formula: see text] and [Formula: see text]), and for all amplitudes when [Formula: see text]. Simulations showed that the system’s periodic response is significantly influenced by the nonlinearity parameter ([Formula: see text]), linearity parameter ([Formula: see text]), and amplitude of vibration ([Formula: see text]). Besides, the existence of bifurcation points at [Formula: see text] and different values of [Formula: see text] was confirmed. Lastly, an approximate frequency solution obtained using the He’s frequency–amplitude formulation was found to produce errors less than 1.0% for a wide range of input parameters. In conclusion, the present study provides a benchmark solution for verification of other approximate solutions, and the He’s frequency–amplitude formulation can be used to obtain fast and accurate solutions for complex nonlinear systems.

Numerical study on flow-induced vibration of an oscillating cylinder with an unattached downstream square plate at a low Reynolds number

Based on the Lattice Boltzmann Method, we numerically studied the flow around a transversely oscillating cylinder with a downstream stationary square plate at a Reynolds number Re = 150 and a mass ratio m* = 2.0. This paper initially considered a zero-damping ratio and a wide range of reduced velocities (Ur = 3–30) to investigate the effects of the gap ratio (L/D = 0.3–3.0) and the dimensionless width of the square plate (W/D = 0.5–2.0) on the response characteristics, force coefficients, wake patterns, and pressure distribution. The energy conversion performance is further tested by applying different non-zero damping ratios (ζ = 0.01–0.3). The main conclusions are as follows: (1) Three vibration behaviors are identified: vortex-induced vibration (VIV), wake interference galloping, and a newly reported “bifurcation” phenomenon of oscillation. For L/D = 0.3, at high reduced velocities (Ur ≥ 15), the cylinder experiences the “bifurcation” phenomenon, where it oscillates on one side of the square plate with a small vibration amplitude. Similar behavior is achieved for the system with L/D = 0.5 and 0.7 but requires a wider square plate than one diameter and higher reduced velocities of Ur ≥ 17.5. (2) Considering both the response amplitude and drag coefficient exerted on the square plate, the cylinder-square plate system with L/D = 0.7 and W/D = 1.0 exhibits a wide wake interference galloping response and an expected vibration amplitude with a lower drag coefficient. (3) For L/D = 0.7 and W/D = 1.0, as Ur increases from 10 to 15, the inclusion of the square plate leads to an increased difference in pressure coefficient between the front of the square plate and the rear of the cylinder. The amplified pressure difference enhances the response of the cylinder, potentially explaining the transition from VIV to wake interference-induced galloping. (4) With a non-damping ratio (ζ = 0.01–0.3), the cylinder-square plate system with L/D = 1.0 and W/D = 1.0 achieves the highest maximum energy harvesting efficiency ηM = 12.01% at the optimum damping ratio ζO = 0.2.

Geometric nonlinear stiffness and frequency of the traditional spring-suspended free-vibration testing device

Free vibration tests on bridge deck sectional models in wind tunnels have been widely adopted to study the wind-resistant performances of bridges. The traditional spring-suspended free-vibration testing device is suitable for small torsional amplitude vibration tests. When large torsional displacement occurs, a geometry nonlinear will be introduced, and the stiffness and frequency of the traditional device will change. However, quantitative research on this subject is inadequate at present. Based on proper simplifications and assumptions, new analytical formulas are derived that describe how the torsional and vertical stiffnesses of the traditional device vary as the torsional and vertical displacement change for different spring lengths, initial strains, joint spacings, and stiffnesses. The amplitude-dependent torsional frequency is also quantified based on the potential energy function. It is found that the torsional stiffness reduction rate Rα and frequency reduction rate RfAα nonlinearly increase with the torsional displacement α and amplitude Aα, respectively. In the parameter ranges considered, R10° is basically within 4%–6%, R20° is basically within 15%–28%; Rf10° is basically within 1%–2%, Rf20° is basically within 6%–11%. From the view of geometry nonlinear, the maximum torsional amplitude of the traditional device is suggested to be less than 4°. Although Rα can be effectively reduced by increasing the initial strain and length of the upper and lower springs, it cannot be completely eliminated whatever increasing the above parameters. The vertical stiffness reduction rate can be ignored.

Nonlocal Theory for Submerged Cantilever Beams Undergoing Torsional Vibrations

Abstract We propose a new theory for fluid–structure interactions of cantilever microbeams undergoing small amplitude vibrations in viscous fluids. The method is based on the concept of nonlocal modal hydrodynamic functions that accurately capture three-dimensional (3D) fluid loading on the structure. For short beams for which 3D effects become prominent, existing local theories based on two-dimensional (2D) fluid approximations are inadequate to predict the dynamic response. We discuss and compare model predictions in terms of frequency response functions, modal shapes, quality factors, and added mass ratios with the predictions of the local theory, and we validate our new model with experimental results.

A triboelectric nanogenerator based on bionic design for harvesting energy from low-frequency vibration

With the development of Internet of Things and the increase of wearable devices, the infinite self-power supply of small devices such as wireless sensors and portable devices has been widely concerned. Considering that the vibration energy in the environment is not affected by day and night, season and climate, and widely distributed, its mechanical energy can be collected. If the low-frequency and small amplitude vibration energy in the environment can be collected and converted into electric energy, it will provide a reliable solution for the self-power supply of these small devices. In this study, we designed a miniaturized energy harvester inspired by bionics. The resonance mechanism improved by nonlinearity is used to realize the amplification of vibration in low width frequency band. And triboelectric nanogenerator (TENG) is used to convert mechanical energy into electrical energy. Through the modeling and dynamic analysis of the system, we find a low resonance frequency band as the working range of the system. Subsequently, a prototype of the bird legs-type TENG was fabricated and tested to verify the design concept. We verify the ability of the prototype to amplify low-frequency vibration and optimized the structural parameters. Moreover, we demonstrate the excellent electrical output performance of the prototype in low-frequency and small external excitation environment and give two application prospects of wearable prototype and vehicle-mounted prototype.

ВЛИЯНИЕ УЛЬТРАЗВУКОВОГО СВЕРЛЕНИЯ НА КАЧЕСТВО ПОВЕРХНОСТИ ОТВЕРСТИЙ В КОНСТРУКЦИОННОМ УГЛЕПЛАСТИКЕ

The article describes the results of a study of the surface morphology of small-diameter through and blind holes obtained by ultrasonic drilling in structural carbon fiber. It is established that the message to the instrument of ultrasonic vibrations of small amplitude contributes to the formation of a uniform morphology of the surface of the holes without breaks, delaminations and shells. On the side surface of the holes treated with ultrasound, there are practically no pulled out and disoriented reinforcing fibers, which indicates an increase in the quality indicators of the dimensional processing of carbon fiber plastics.