Multi-mode Vibration Research Articles

Multi-mode vibration attenuation of mistuned bladed disks by frictional tuned mass dampers array

An effective method is presented for multi-mode vibration attenuation of mistuned bladed disk (blisk) by employing ring-like array structure composed of multiple frictional tuned mass dampers (FTMDs). The multi-mode vibration attenuation is realized by arranging multiple series of FTMDs. A novel design scheme of the FTMD array is presented. 3D finite element method is employed for quantitative modelling of the blisk-FTMD array system, and the rotordynamic effects of Coriolis forces, spin softening and centrifugal stiffening are taken into account to reach better tuning in natural frequencies in the rotating conditions. The free-interface component mode synthesis method is used to reduce the order of the model. An efficient technique, which combines the harmonic balance method, the analytical formulation of Jacobian matrix and the parameterized Newton-Raphson method, is employed to solve the nonlinear equations. Finally, a blisk, with its first 4ND and 5ND modes being targeted, is employed as an example to illustrate the effectiveness of the presented method. Numerical results show that the method can obviously mitigate the multi-mode vibration of blisk with wide levels of mistuning and is robust against the small tuning deviations of the oscillators.

Response adjustable performance of a visco-elastomer sandwich plate with harmonic parameters and distributed supported masses under random loading

Vibration control of composite structures with distributed masses under random loadings is a significant issue. Adjustability of dynamic characteristics including response spectrum peaks and valleys is important for structural vibration control. The vibration control design in space contains structure and conformation designs which combination results in periodic composite structures. In the present paper, spatial periodicity control design is proposed. Stochastic response adjustable performance of a visco-elastomer sandwich plate with harmonic distribution of geometrical and physical parameters and distributed supported masses under random base motion loading is studied. Both facial layer thickness and core layer modulus of the sandwich plate are considered as harmonic distribution in length and width directions as well as periodically distributed masses. Partial differential equations of coupling motions of the sandwich plate system are derived and converted into ordinary differential equations for multi-mode coupling vibration. Generalized stiffness, damping, and mass coefficients are functions of the harmonic distribution parameters. An analysis solution with frequency response function and response spectral density expressions of the sandwich plate system is obtained. Numerical results are given to show the response adjustable performance through the harmonic geometrical and physical parameters and distributed masses. The results have a potential application to stochastic vibration control or dynamic optimization design of smart composite structure systems.

Effect of yaw angle on vibration mode transition and wake structure of a near-wall flexible cylinder

The multi-mode transition and vortex structures in the vortex-induced vibration (VIV) of a near-wall flexible cylinder under different yaw angles are investigated through three-dimensional direct numerical simulation. Yaw angles α = 0°–60°, gap ratio G/D = 0.8, and Re = 500 are adopted. With the increase in α, the dominated vibration mode decreases from the 6th to 1st mode in the in-line (IL) direction and the 3rd to 2nd mode in the cross-flow (CF) direction. For the IL vibration, no mode transition occurs at α = 0°, whereas frequently mode transition is observed at α > 0°, due to the intermittent participation and spanwise competition of different modes, thus showing an intensified traveling-wave characteristic. For the CF vibration, mode transition is not excited at any α case even with spanwise mode competitions, due to the significant weight of the dominated mode, thus showing a strong standing-wave characteristic. The asymmetrical distributions of vibration displacements and force coefficients are established because of irregular energy transfer along the span. The spanwise vortex tubes at α = 0°–30° are separated into several cells associated with the dominated vibration mode, showing a locally parallel vortex shedding. However, positively yawed and negatively yawed vortex shedding are observed at α = 45° and 60°, respectively. The vortex strengths vary along the cylinder, where large-scale and small-scale vortices are observed at the CF anti-node and node planes, respectively. The independence principle is only valid at α < 15° for predicting the multi-mode vibrations and hydrodynamics, significantly reduced from that of α < 45° in the wall-free case or the mono-mode VIV case.

교량 바닥판의 다중모드 진동 저감을 위한 2자유도 진동흡진기

Vibrations that impact bridges need to be addressed because they can cause problems in terms of serviceability and structural safety. Therefore, the serviceability standards for vibration are considered by performing wind resistance design and loading tests. This study proposes a two-degree-of-freedom (2DOF) dynamic absorber that reduces the multi-mode vibrations of a bridge deck affected by wind load. The bridge decks have multi-mode vibration characteristics because various forces such as drag, lift, and moment are generated under the influence of wind loads. Therefore, a dynamic model is constructed and the parameters of the reduced model are used. In addition, the design formula for a single rigid 2DOF dynamic absorber is proposed to reduce vibrations occurring at two resonant frequencies acting on the bridge deck. Finally, the performance is confirmed by numerical simulations. In addition, it was shown that the vibration damping performance of the proposed dynamic absorber was affected by the radius of gyration as well as the mass ratio.

High-efficient internal resonance energy harvesting: Modelling and experimental study

This research investigates a U-shaped vibration-based energy harvester with broadband and bi-directional features via an internal resonance. The proposed U-shaped structure has been tailored in a way that displays a two-to-one internal resonance between the selected transverse modes. The resonant modes and mode shapes of the proposed device have been designed and validated through an analytical method, the finite-element simulations as well as experimental tests, showing good agreement. The experimental tests imply the benefits of energy transfer between two modes, as resonance phenomena in all segments have been successfully triggered to obtain large-amplitude oscillations for high-efficient energy conversion via piezoelectric effects. Under harmonic excitations, different from conventional multi-mode vibration energy harvesters, the proposed device presented broadened resonance regions for both modes, and exhibited equally effective bandwidths under forward/backward sweeps. Moreover, the device is capable of being excited in two orthogonal directions as a bi-directional energy harvester. Parametric studies on the acceleration levels are also presented to evaluate the system performance and reveal the critical excitation level for triggering the internal resonance. It has been found that even under a 0.1 g acceleration level, the internal resonance phenomenon can still be observed between the first and second modes, which implies the feasibility of the proposed device even for ambient vibration sources with low energies.

A novel soft encapsulated multi-directional and multi-modal piezoelectric vibration energy harvester

Advances in the design of various piezoelectric vibration-based energy harvesters (PVEHs), as a kind of power devices that can convert ambient energy to useable electrical energy, have become a hot topic in recent years due to their potential perspectives in wireless sensor networks, wearable electronics and low-power microelectronics. Unfortunately, conventional PVEHs modelled by beam- or plate-type planar structures are mainly restricted to harness kinetic energy in a single direction only. However, ambient vibration sources often work in multiple directions and broadband frequencies. To address this challenging problem, a mechanically-guided three-dimensional (3D) assembly structure is strategically designed to construct a soft cruciform-encapsulated PVEH in this work. Meanwhile, a reliable soft encapsulation technology is introduced to maintain the 3D configuration, which not only can avoid the collapse problem and also improve the dynamical performance. The entire system consists of a compressive buckling cruciform structure with a proof mass, PVDF thin film, and soft encapsulation. The dynamic characteristics of the system can be adjusted by changing the structural parameters, such as the pre-stressing force of the pre-stretched elastic substrate and the quality of the additional mass block. Finite element analysis and experimental studies are conducted to investigate the modal characteristics of system. A comparison of the vibration energy harvesting performance between the encapsulated and unencapsulated piezoelectric harvesters is presented. The results demonstrate that the encapsulated one can work well under multi-directional, multi-modal and low-frequency conditions. A maximum power output of 9.8 mW in the frequency range of 1–200 Hz can be achieved, which is almost 560 times more than that of the unencapsulated one. The proposed methodology also offers a new perspective for the fabrication design of soft-type PVEHs with higher working performance.

Modal vibration decomposition method and its application on multi-mode vibration control of high-speed railway car bodies

The vibration of a railway car body is a superposition of the vibrations of its various modes. It is typically easy to obtain the physical vibration of the car body using sensors in an in situ or a simulated test vehicle. However, it is difficult to determine the modal vibration of the body and its contribution. There are no effective multi-mode vibration control methods for the car bodies. This study proposes a modal vibration decomposition method (MVDM) based on singular value decomposition (SVD) and least squares fitting (LSF). Accordingly, the physical vibration of a railway car body is decomposed into modal vibrations. A method for calculating the modal contribution factor (MCF) is presented, and the dominant flexible modes of the car body are determined and considered the target for the vibration control method. Several pieces of equipment are considered as dynamic vibration absorbers (DVAs) to control the multi-mode vibration of the car body using the dynamic vibration absorption theory and determine the installation parameters of the individual equipment. Finally, the effectiveness of vibration control is verified through dynamic simulations. The results demonstrate the effective decomposition of the physical vibration of the car body into various modal vibrations using the MVDM. This provides accurate data for the MCF calculation and determination of the flexible modes of the car body. The proposed method reduces the vibration of the target modes and improves the ride quality of the railway vehicle. At the optimal damping ratio, the vibration of the DVA-based equipment itself is acceptable. This allows for multi-mode vibration control without requiring extensive modification to the car body structure or suspension system parameters of the vehicle.

A novel eddy current damper system for multi-mode high-order vibration control of ultra-long stay cables

Recently, the multi-mode high-order wind-induced vibration of the ultra-long stay cables has been observed on several cable-stayed bridges, which may result in a visual panic of the users and the fatigue of the stay cables at anchorage ends. In order to investigate the multi-mode high-order wind-induced vibration characteristics of ultra-long stay cables, a vibration monitoring system (VMS) of the stay cables is established on the Sutong Bridge (STB). Furthermore, to suppress the in-plane and out-of-plane vibration responses of the stay cables simultaneously, a novel eddy current damper system (ECDS) is proposed in this study. The damping coefficients of the ECDS for multi-mode high-order vibration of the stay cables are determined via a modified method. Moreover, laboratory tests are conducted to evaluate the performance of the developed dampers. Furthermore, the field tests of seven stay cables of the STB are carried out to identify the modal damping ratios of the cable-damper system via the free vibration method. The results show that it is feasible to provide sufficient additional modal damping ratios for multi-mode high-order vibration of ultra-long stay cables applying the ECDS proposed in this study. Finally, the one-year field measurements of the wind-induced vibration responses of the seven stay cables of the STB are carried out to validate the ECDS suppress effects on multi-mode high-order vibration control of ultra-long stay cables. The measurement results show that the multi-mode high-order vibration responses of the monitored stay cables of the STB are effectively suppressed.

Nonsynchronous Rotor Blade Vibrations in Last Stage of 380 MW LP Steam Turbine at Various Condenser Pressures

This paper presents an analysis of nonsynchronous rotor blade vibrations in the last stage of an LP steam turbine at various condenser pressures. The nonlinear least squares Levenberg–Marquardt method is used in a tip-timing analysis to determine nonsynchronous multimode rotor blade vibrations, which is a novelty. This is done with two sensors in the casing and a once-per-revolution sensor. The accuracy of the nonlinear least squares Levenberg–Marquardt multimode method is compared with the one-mode linear method. The algorithm is verified by comparing it with one-mode tip-timing methods for synchronous and nonsynchronous vibrations. The analysis shows that the rotor blades vibrate simultaneously with two modes in non-nominal conditions, which is also a novelty. The rotor frequencies are unchanged, although the blade vibration amplitudes vary, depending on the pressure in the condenser. Flutter does not appear in the last stage for the various condenser pressures and powers that were tested.

Effectiveness of damping and inertance of two dampers on mitigation of multimode vibrations of stay cables by using the finite difference method

Stay cables of several cable-stayed bridges with a span of nearly 1000 m are confronted with the simultaneous actions of low-mode rain-wind induced vibration and high-mode vortex-induced vibration. It is difficult to effectively reduce these two kinds of cable vibrations by installing one damper at the end of the stay cable. In this paper, the effectiveness of two dampers attached at the lower end of the stay cable, including viscous dampers and viscous inertial mass dampers, is investigated by the finite difference method. First, the stay cable was simplified as an inclined cable with sag, and equation of motion governing the stay cable attached with two dampers was derived. This equation was then numerically solved by the finite difference method. Second, taking Cable A30 of the Suzhou-Nantong Yangtze River Bridge as a reference, a series of numerical simulations were carefully conducted to obtain the damping ratios of the first fifty modes. The results show that it is impossible to enhance the damping ratios of all of the concerned modes up to the minimum target value only by installing a single viscous damper or a single viscous inertial mass damper at the end of the stay cable. However, two viscous dampers installed at the lower end of the cable (l1/l = 2%, l2/l = 5%) can overcome the shortcoming of a single viscous damper or viscous inertial mass damper to enhance the damping ratios of the first 34 modes up to the minimum target value (0.5%). In general, the optimized relative location of the two dampers (l2/l1) should be near 2. The optimized ratio of the damping coefficients (η1/η2) should be lower than l2/l1, and the lower limit of η1/η2 is determined by the minimum target values for modal damping ratios. Moreover, it appears that a lower installation position of two viscous dampers can realize the mitigation of higher modes even up to the 40th ∼ 50th modes.

Modal-based synthesis of passive electrical networks for multimodal piezoelectric damping

This work presents a new approach to design an electrical network which, when coupled to a structure through an array of piezoelectric transducers, provides multimodal vibration mitigation. The characteristics of the network are specified in terms of modal properties. On the one hand, the electrical resonance frequencies are chosen to be close to those of the targeted set of structural modes. On the other hand, the electrical mode shapes are selected to maximize the electromechanical coupling between the mechanical and electrical modes while guaranteeing the passivity of the network. The effectiveness of this modal-based synthesis is demonstrated using a free–free beam and a fully clamped plate.

Influence of Internal Rubber Damper on Cable External Viscous Damper Effectiveness

In practice, the internal rubber dampers are widely installed inside the cable guide pipe between the external viscous damper and the cable anchorage. In order to study the effect of the internal damper on the performance of the external viscous damper, the theoretical analysis model of a taut cable with an internal rubber damper and an external viscous damper is established in this paper, where the internal damper is assumed to be a high damping rubber damper with a flexible support and the external damper is considered to be a linear viscous damper. Then, the numerical iterative formula and approximate solution expression for the cable modal damping ratio are derived. Based on the approximate solution, the parameters of the internal rubber damper are analyzed comprehensively, and its influence on the cable modal damping in cable multimode is studied as well. Results show that, except for the support flexibility of the internal rubber damper, other parameters will have a negative effect on external viscous damper effectiveness. From the perspective of cable multimode vibration control, the installation of an internal rubber damper significantly weakens the modal damping, and this effect is more pronounced in lower frequency modes.

Multi-mode coupled vibration performance analysis of a radial-longitudinal (R-L) ultrasonic transducer.

A radial-longitudinal (R-L) ultrasonic transducer is designed by compounding a piezoelectric ceramic and an outer metal ring on the central coupling cylinder of a longitudinal cascade transducer. This design is used to realize multi-mode vibration and increase the radiation range. By applying longitudinal and radial double excitation, three coupled vibration modes of the transducer are generated in the frequency range of 15-65 kHz. The coupled vibration dominated by radial vibration is regarded as the best vibration mode of this transducer. The electromechanical equivalent circuit and the resonance frequency equation of the transducer's coupled vibration are derived by using the equivalent elastic method and one-dimensional vibration theory and verified by the finite element method and experimental method. The results show that the electrical impedance frequency curves of the transducer from the three methods are consistent. The transducer is expected to be used in ultrasonic cleaning and liquid processing applications.

Multi-mode vortex-induced vibration control of long-span bridges by using distributed tuned mass damper inerters (DTMDIs)

Tuned mass damper inerter (TMDI) has been proved to be a feasible device for vibration control of long-span bridges due to its excellent space performance. However, existing design method for TMDI is conducted mode-by-mode. When applied to multi-mode control of the bridge through multiple TMDIs, it cannot consider the synergistic effect among the TMDIs and thus leading to a suboptimum design result. This paper presents an efficient method for multi-mode vortex-induced vibration (VIV) control of long-span bridges by using spatially distributed TMDIs (DTMDIs). The governing equations of the bridge-DTMDIs system are established in modal coordinate. A global optimization method considering the synergistic effect among the DTMDIs is proposed and compared with conventional mode-by-mode approach through a numerical case study. The robustness of the control schemes against variations in structural and aerodynamic parameters is investigated. The influence of inerter span length on the control efficiency is also discussed. The results show that the proposed global design method can achieve a better control efficiency with even less TMDIs as compared with conventional mode-by-mode approach. The proposed method can also be extended to VIV control of some other line-like structures.

Multimodal Multidirectional Piezoelectric Vibration Energy Harvester by U-Shaped Structure with Cross-Connected Beams

This paper proposes a multidirectional piezoelectric vibration energy harvester based on an improved U-shaped structure with cross-connected beams. Benefitting from the bi-directional capacity of U-shaped beam and additional bending mode induced by cross-connected configuration, the proposed structure can well capture the vibrations in 3D space at the frequencies lower than 15 Hz. These features are further validated by finite element analyses and theorical formulas. The prototype is fabricated and the experiments under different conditions are carried out. The results show that the proposed harvester can generate favorable voltage and power under multidirectional vibrations at a low operating frequency. Practical applications of charging capacitors and powering a wireless sensor node demonstrate the feasibility of this energy harvester in supplying power for engineering devices.

A robust optimised shunted electrodynamic metamaterial for multi-mode vibration control

This paper presents a design approach for a shunted electrodynamic metamaterial (EDMM) for broadband robust vibration control. A unit cell of 12 inertial electrodynamic transducers is proposed, where the response of each transducer is tuneable via a connected resistive and inductive shunt circuit. The variations in the parameters of an off-the-shelf transducer are characterised experimentally, before the effect of this variation on the shunted response is investigated. It is shown that instability of the system is a limiting design factor. A problem is proposed whereby the resistive and inductive shunt values of an EDMM attached to a parametrically uncertain structure are to be found, and given the complexity of the design problem, a Particle Swarm Optimisation (PSO) is utilised to find a solution using an analytical model of the system. The results of the optimisation show that the effects of uncertainty in the actuators must be included, otherwise, the solution can be unstable. However, it is also shown that it is sufficient to ignore the uncertainty in the structure and optimise the EDMM considering actuator uncertainty alone, since the EDMM motion is then highly damped and, therefore, inherently robust to structural uncertainties.

Low-Frequency Multimode Vibration Suppression of an Acoustic Black Hole Beam by Shunt Damping

Abstract The ideal acoustic black hole (ABH) can achieve wave gathering and zero reflection of elastic waves. In practice, ABHs have to be truncated, limiting their application in lower frequency range. Aiming at improving the ABH beam's vibration suppression ability at low frequencies, this study proposes a shunt damping-ABH composite beam by pasting shunt damping instead of ordinary damping on the ABH tip. The energy method is employed to solve the vibration equation of the ABH beam. The admissible function is the Mexican hat wavelet. The proposed method is verified by the finite element method. Compared with the uniform beam, the numerical results show the ABH beam has a noticeable attenuating effect in high-frequency range due to the ABH effect, but almost has no attenuating effect in the low-frequency range. Therefore, we introduce shunt damping to enhance the low-frequency vibration control. The shunt damping is composed of circuits connected to a piezoelectric patch. The effects of different circuits connected to the piezoelectric patch are discussed. The R–L shunt circuit and L–C parallel blocking circuit can simultaneously suppress the multimode vibration peak of the ABH beam at the low frequency successfully. Finally, a vibration experiment of ABH beam combined with shunt damping is implemented to verify the present method's feasibility and the shunt damping effect. The proposed shunt damping-ABH composite beam could improve the suppressing ability in both the low and high-frequency ranges.

Multi-mode Vibration Averagely Control for Cylindrical Shells by Optimally Distributed Piezoelectric Sensors and Actuators

Multi-mode Vibration Averagely Control for Cylindrical Shells by Optimally Distributed Piezoelectric Sensors and Actuators

An improved multi-mode seismic vibration control method using multiple tuned mass dampers

Multiple vibration modes of an engineering structure might be excited by earthquake ground motions. Multiple tuned mass dampers (MTMDs) are widely used to control these multi-mode vibrations. However, in the commonly used MTMD system, the mass element in each tuned mass damper (TMD) is normally assumed to be the same. To improve the performance of MTMDs for seismic-induced vibration control, non-uniform MTMD masses are adopted in the present study to improve the mass utilization of TMD, and a location factor is proposed to determine the best location of each TMD in the MTMD system. The effectiveness of the proposed method is validated through numerical study. The results show that the proposed method effectively reduces the seismic responses of the structure induced by multiple vibration modes.

A comprehensive study on the coupled multi-mode vibrations of cylindrical shells

As is discussed in the previous work (Dong et al. 2021), the functions of beam-like modes are not suitable for simulating the cylindrical shells with the clamped and free ends. Also, ignoring the influences of the in-plane inertias can lead to large relative differences of the solutions of vibration characteristics. This work aims to accurately calculate the vibration characteristics of the cylindrical shells under various classical boundary conditions comprised of movable simply-supported, immovable simply-supported, sliding, clamped and free ends. Meanwhile, this work deals with the nonlinear coupled multi-mode vibrations of the cylindrical shell under different boundary conditions with the in-plane inertias taken into consideration. Firstly, several modified Chebyshev polynomials in terms of the axial coordinate and new harmonic functions in terms of the circumferential coordinate are employed to establish the equations of natural frequency in the frame of Lagrange equations for all the single-modes, where the polynomials are independent of geometries and material properties of the shells. And then, the circumferential-wave-dependent mode functions of the cylindrical shells are proposed, based on these mode functions coming from the linear vibration, a unified nonlinear differential equation of motion of the cylindrical shells under different boundary conditions is established for the first endeavor, in which the coupled multi-mode vibrations containing various vibration waves are considered. Finally, the high dimensional differential equation with the square and cubic nonlinearities is investigated by employing the incremental harmonic balance method. Results shown that polynomials accurately satisfy various boundary conditions, and the contribution of the single-mode to the coupled multi-mode vibration is related to the loading patterns.