Multi-mode Vibration Research Articles

Near unity broadband emissivity and ultralow thermal conductivity from yttrium niobate-based ceramics

Maximizing energy output via thermal radiation when blocking unfavored heat transfer through thermal conduction is the main challenge of radiative thermal management for various industrial and aerospace applications. Here we report a Ca-Cr co-doped Y3NbO7 ceramic that integrates a near-unity broadband (0.3–14 μm) emissivity (>0.9) with an ultralow thermal conductivity (<1.6 W·m−1·K−1). Experiments and theoretical calculations demonstrate the formation of impurity energy levels as gap states, enabling optical absorption at low photon energies and thus improving the emissivity in visible and near-infrared bands (0.3–2 μm). The lattice distortion and extra multi-mode vibrations induced by Ca-Cr doping improve the emissivity in mid-infrared range (2–14 μm). The intrinsic inhomogeneity of chemical bonds in Y3NbO7 enables strong scattering of heat-carrying vibrational modes and thus an ultra-low thermal conductivity. This doped Y3NbO7 exhibits a record high emissivity to thermal conductivity ratio of 0.56, showing great potential in radiative thermal management.

A novel multimodal piezoelectric energy harvester with rotating-DOF for low-frequency vibration

This paper proposes a novel piezoelectric energy harvester with rotating-DOF to efficiently achieve multimodal vibration energy harvesting in low-frequency environments. The designing of the harvester based on the beam whose ends were connected to rotating shafts was accomplished firstly, and then the dynamic properties of the structure were numerically calculated by finite element analysis. Finally, experimental prototypes and test systems were designed to verify the correctness of the theory. This paper analyzed how that different key structural parameters and acceleration affect its output characteristics. Especially, the effect of double-rotating shafts on the energy harvesting performance was analyzed innovatively. The results acquired by experiments were in accord with those of simulation. Its first-order resonant frequency is down to 12.65 Hz, at which excellent output characteristics were obtained. The harvesting performance at the third-order resonant frequency was also remarkable, indicating the application prospect in the multimodal vibration circumstances. Moreover, the harvesting performance can be enhanced by the adjustable counterweight and proof mass, which improve environmental adaptability. Its maximum output is 35.73 V (12.65 Hz) and 24.69 V (43.64 Hz), and the output power is 2.13 mW and 1.02 mW under an acceleration of 1.96 m/s2. This paper provides a novel idea for low-frequency multimodal energy harvesting techniques.

Multimode damping optimization of a long-span suspension bridge with damped outriggers for suppressing vortex-induced vibrations

Long-span suspension bridges are subjected to wind-induced vibrations due to their large flexibility and low damping. Damped outriggers have been proposed recently to supplement damping and suppress multimode vibrations of main girders of long-span bridges, e.g., vortex-induced vibrations. However, when installed at multiple positions including tower-girder junction points and girder ends to further improve multimode damping effect, damping effects of the damped outriggers are interdependent. Additionally, the frequency curves of adjacent modes with respect to varying parameters of the damping devices could approach and intersect with each other, making it difficult to track mode orders and modal damping variations in design optimization. This study therefore proposes an optimization method to design damped outrigger parameters for maximizing multimode damping of the bridge. First, the bridge with damped outriggers is modeled generally using finite element method for dynamic analysis. Subsequently, two critical issues in the design, i.e., the mode tracking and multi-objective optimization, are respectively addressed by using the modal assurance criterion and the genetic algorithm. The proposed method is then applied to the Xihoumen Bridge with a main span of 1650 m. The results show that through the proposed optimization, 6 out of 7 modes vulnerable to vortex-induced vibrations can reach a damping ratio over 1.0%, which cannot be achieved by using damped outriggers of a uniform size. Furthermore, the required damper coefficients are decreased by 49.4%. The flexibility of the outriggers has been considered in the design. The developed method is promising in the design of other types bridge damping devices, e.g., multiple tuned mass dampers.

Multimodal analysis of generic cantilever-based piezoelectric nanogenerators with offset-proof mass

The present work investigates analytically and experimentally the multimodal performance analysis of piezoelectric (PZT-5A) based generic nanogenerators. The present nanogenerators consider both uni-morph and bimorph with offset-proof mass subjected to sinusoidal base excitation. The finite element-based multi-mode electromechanical coupled dynamics governing equations are derived to formulate the analytical expressions of uni-morph, and bimorph nano-harvesters for frequency responses function of output voltage, current, and power, respectively. The impact of base excitation, load resistance, thickness, length, and mass of offset-proof mass on the output voltage and harvested energy is investigated by considering multi-mode vibration bodies. PZT-5A is evenly distributed over the substructure for the single voltage output. Interestingly, we noticed that a uni-morph cantilever with offset-proof mass is the best among all the investigated configurations, showing the maximum power at different load conditions. Analytical solutions are in good agreement with nearly no error for the first few vibration modes as compared to what was obtained using numerical simulation. The experimental realization for a single mode of vibration has been explored, and results showed that the fundamental mode swings very similar behaviors in voltage and power output, mostly in pre-and post-resonance conditions as compared to analytical results, and the difference between these two results is about nearly 5%. Thus, the present work depicts an effectual correlation between analytical development, numerical results using COMSOL 2D, and experimental realization for analyzing energy harvesting devices. These results may be helpful for the experimentalists to design efficient nanogenerators.

Multimodal MEMS vibration energy harvester with cascaded flexible and silicon beams for ultralow frequency response

Scavenged energy from ambient vibrations has become a promising energy supply for autonomous microsystems. However, restricted by device size, most MEMS vibration energy harvesters have much higher resonant frequencies than environmental vibrations, which reduces scavenged power and limits practical applicability. Herein, we propose a MEMS multimodal vibration energy harvester with specifically cascaded flexible PDMS and “zigzag” silicon beams to simultaneously lower the resonant frequency to the ultralow-frequency level and broaden the bandwidth. A two-stage architecture is designed, in which the primary subsystem consists of suspended PDMS beams characterized by a low Young’s modulus, and the secondary system consists of zigzag silicon beams. We also propose a PDMS lift-off process to fabricate the suspended flexible beams and the compatible microfabrication method shows high yield and good repeatability. The fabricated MEMS energy harvester can operate at ultralow resonant frequencies of 3 and 23 Hz, with an NPD index of 1.73 μW/cm3/g2 @ 3 Hz. The factors underlying output power degradation in the low-frequency range and potential enhancement strategies are discussed. This work offers new insights into achieving MEMS-scale energy harvesting with ultralow frequency response.

Theoretical analysis and experiment of multi-modal coupled vibration of piezo-driven Π-shaped resonator

Nonlinear modal interactions have recently become the focus of intense research in mechanical resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. This paper reports for the first time a piezoelectric driven Π-shaped beam with 1:1:2 internal resonance and systematically studies the multi-mode coupling dynamic behavior. Typically, the influence mechanism of driving voltage on the energy transfer path of the multi-modal vibration is experimentally observed. With the increase of the driving voltage, number of frequency intervals of modal coupled vibration increases from 1 to 3, and then decreases to 2, which is unique to multi-mode coupled resonators. To explain the various complex nonlinear phenomena observed experimentally, a theoretical framework for multi-modal coupled vibration analysis of Π-shaped resonator is proposed for the first time. Perturbation and bifurcation analysis are introduced to deduce the discriminant formula of multi-modal coupled vibration theoretically. Both experimental measurements and theoretical predictions show that the multi-modal coupling greatly complicates the dynamic behavior of the resonant system and induces multiple amplitude jumps. In particular, based on the multi-modal coupled vibration behavior of the Π-shaped resonator, a conceptual model of trigger resonance sensor with multiple mass warnings is creatively proposed. The experimental results show that the amplitude of the resonant system has six distinct changes with the increase of the added mass, accompanied by bifurcation phenomenon. The research results of this paper provide a systematic framework for the development of multi-mode coupled resonant devices.

Evaluation of interfacial friction characteristics in multi-mode ultrasonic vibration assisted micro-extrusion process

Ultrasonic vibration assisted micro-forming process has been widely used for its advantages of increasing forming performance, reducing forming stress and interface friction. Existed studies were concentrated on the effect of ultrasonic vibration on the tool-sample interface friction, but few studies concerned the mold cavity-sample interface friction. In this paper, the variation characteristics of the mold cavity-sample interface friction during ultrasonic-assisted micro-extrusion were studied. The multi-mode ultrasonic vibration assisted copper T2 micro-extrusion experiments were conducted, and a theoretical model was developed to predict the changing rule of the interface friction coefficient and quantify the ultrasonic friction reduction effect during the ultrasonic vibration assisted micro-extrusion forming process. The results show that the mold cavity-sample interface friction coefficient was decreased with the ultrasonic amplitude increasing, and decreased sequentially with tool ultrasonic vibration (TUV), mold ultrasonic vibration (MUV) and compound ultrasonic vibration (CUV). The extrusion stress of the theoretical model agrees well with that of the experiments and the maximum relative errors of TUV, MUV and CUV are 6.9 %, 7.6 % and 11.6 %, respectively. This research result will be beneficial for further investigation on the friction mechanism in the ultrasonic vibration assisted micro-forming processes.

Multimode vibration control of stay cables using pseudo negative stiffness MR damping system

At present the vibration control of stay cable suffers from low damping effectiveness in single mode vibration caused by the limitation of damper installation position as well as the poor modal compatibility, that is, a rapid degradation performance when the actual mode deviates from the designed optimal mode. In this paper, one practical semi-active magnetorheological (MR) damper based control solution is proposed to address the problems faced in multi-modal vibration control of stay cables. The semi-active pseudo negative stiffness (PNS) control strategy takes full account of the MR damper’s mechanical characteristics and the demands of the stay cable vibration control. Compared with the linear equivalent model, the proposed time-varying model exhibits more details of the damping force and the nonlinear response of the damped stay cable, which shows its essential role in the optimal design of MR-PNS scheme. Then the optimal design method of MR-PNS multi-modal vibration control for stay cable is summarized by taking the first three modes vibration control of the J20 cable in Nanjing Baguazhou Yangtze River Bridge as simulation examples. The simulations of cross-modal, multi-modal and wind-induced vibration cases are conducted respectively, while the results show that the optimal designed multi-modal MR-PNS scheme can simultaneously exceed the passive maximum modal damping ratio within first three modes. The advantages of the proposed MR-PNS method in high damping efficiency and modal compatibility could be verified by comparing with the passive multi-modal damping solution.

Free vibration of a taut cable with internal high damping rubber dampers and an external negative stiffness damper

Frequent vibration events on the long stay cables highlight the need to control their multi-mode vibrations. Compared to commonly used viscous dampers for cable vibration control, the negative stiffness damper (NSD) demonstrates its superior performance. However, a single NSD is incapable of providing sufficient supplemental damping for a super-long cable, especially for the multi-mode vibration mitigation. The combined internal high damping rubber (HDR) dampers and the external NSD for cable vibration control is proposed to achieve better multi-mode control performance in this study. First, the complex modal analysis of a taut cable coupled with an external NSD and two internal HDR dampers is conducted to reveal the coupling performance of the damping system. The asymptotic and iterative solutions of transcendental equation obtained in complex modal analysis are derived, respectively. Subsequently, the comparison of asymptotic and iterative solutions verifies the applicability of asymptotic solutions. Finally, parametric studies are performed to investigate the effect of the parameters of NSD and the coupling configuration on their combined control performance. Results show that the coupling schemes with different configurations can possess a wider effective modal range or a better damping effect in the narrower modal ranges according to different damping targets. It provides a diversified solution for multi-mode damping enhancement of stay cables.

A magnetic negative stiffness eddy-current inertial mass damper for cable vibration mitigation

Previous studies have shown superior control performance in mitigating vibrations of stay cables by using passive control devices such as negative stiffness dampers (NSDs) and inertial mass dampers (IMDs). Integrating the advantages of the two dampers, we propose a novel passive control device termed magnetic negative stiffness eddy-current inertial mass damper (MNS-ECIMD) for cable vibration mitigation. A mechanical model of the MNS-ECIMD is proposed and validated against dynamic test data of a small-scale prototype. A parametric analysis is conducted based on the finite difference model of the cable-damper system to investigate the effects of the magnetic negative stiffness coefficient, inertance coefficient, and eddy-current damping coefficient of the MNS-ECIMD on cable vibration mitigation performance. The MNS-ECIMD performance is further validated using a small-scale cable test, of which the results illustrate its superior performance for the first two modes compared with the ECIMD. A simple parameter optimization procedure is then presented for cable multi-mode vibration control, which is then demonstrated via numerical simulation of a 306 m-long stay cable of Stonecutter Bridge in Hong Kong. Numerical results illustrate that the MNS-ECIMD significantly outperforms conventional viscous dampers and the ECIMD for cable multi-mode vibration control, owing to the additional negative stiffness effect.

Multimodal vibration control with piezoelectric patches connected to self-tuning RLC shunts

This paper presents experimental results on the control of stochastic flexural vibrations in a plate equipped with five piezoelectric patches connected to self-tuning multi-resonant shunts composed by N-parallel resistive-inductive-capacitive (RLC) branches. The study contrasts the tuning of the branches based on the minimization of the time-averaged total flexural kinetic energy and the maximization of the time-averaged electric power absorbed by each RLC branch relative to the resonant responses of the target modes. Next, it investigates the online tuning of the branches RL components with an extremum seeking algorithm set to maximize the time-averaged electric power absorbed by each RLC branch.

Analytical study on the effects of flexural rigidity and negative stiffness in the optimal tuning of inerter-based damper for cable vibration mitigation

The control performances of inerter-based dampers on stay cables, usually governed by relevant damper parameters (such as inertance, stiffness, and damping coefficients), are sensitive to parameter variation around the optimal range. Further given these inerter-based dampers amplify the vibration amplitude at the damper location, the effects of cable’s flexural rigidity, which is often ignored in previous studies, are examined in this study. The results suggest an approximate 10% increase in all three design parameters (i.e., inertance, stiffness, and damping coefficients) is required to achieve optimal control compared with the case ignoring the flexural rigidity. In addition, the potential combination of inerter-based dampers with negative stiffness elements is also discussed in this study, which offers a more flexible layout and enhances multi-mode cable vibration control performance. Consequently, the tuning procedures are updated, and the revised optimal tuning formulas taking account of both the cable’s flexural rigidity and the introduction of negative stiffness are presented in this paper.

Multi-mode vortex-induced vibration of a long-span bridge under non-uniform flows

An increasing number of modern long-span bridges have been built in complex valley terrain, and the influence of non-uniform flows on the vortex-induced vibrations (VIVs) of bridges should be considered. Therefore, this study investigated the multi-mode VIVs of long-span bridges in both the completed bridge stage and erection stage using a taut-strip model under a uniform flow and two types of non-uniform flow. The non-uniform flows were generated by adjusting spires (both their separation distance and shapes) in wind tunnel. The two non-uniform flows were represented by the linear and parabolic wind velocity flow distributions. The characteristics of the vibration responses and wake flows, mechanism of multi-mode VIVs, and aerodynamic differences between different stages and under different incoming flows were discussed. The results indicated that the occurrence of multi-mode VIVs arose from a wake frequency difference covering two or more modal frequencies owing to variations in the wind profile. The non-uniform flows disturb the regular vortices in the wake due to the variable wind velocity and higher turbulence intensity, leading to suppression of the VIV response.

Non-contact full-field dynamic strain reconstruction of rotating blades under multi-mode vibration

Dynamic strain measurements of rotating blades have attracted significant attention in the turbomachinery design, testing, and service stages. Measuring the blade full-field strain using traditional strain gauges (SGs) is difficult. Too many SGs may add damping to the measured blades, thereby lowering the peak strain that should have been experienced. The strain level was then underestimated for the blades without SGs. A non-contact displacement measurement technique, called blade tip timing (BTT), is promising for replacing SGs but requires effective displacement-to-strain transformation. The transformation under blade single-mode vibration has been studied in recent years. Although the multi-mode vibration of rotating blades is increasingly common, it is usually ignored. In contrast to single-mode vibration, when the blade is subjected to multi-mode vibration, the peak stress changes in both the temporal and spatial domains. This causes difficulties in predicting the blade stress limit and fatigue life. This study proposes a full-field dynamic strain reconstruction method for rotating blades under multi-mode vibration. Based on the BTT-measured displacement, the proposed method enables the full-field dynamic strain reconstruction of the blade in the time domain. First, the multi-mode displacement response of the rotating blades was measured using the BTT system. Second, a least-squares decoupling model for the multi-mode response was established. The blade tip displacement response was decoupled, and the multi-mode vibration parameters were identified, including the frequency, amplitude, and phase. Third, an analytical transform relationship between the tip displacement and full-field strain was derived based on the blade mode shapes. The blade full-field dynamic strain was reconstructed in the time domain using the decoupled displacement response and a displacement-to-strain transmissibility matrix. Finally, the full-field strain at different instants was visualised using a blade finite element model. The proposed method is validated on a rotor with five blades through both numerical simulations and spinning experiments. The reconstructed strain was compared with the result measured by the SG. The average strain relative errors of the five blades were within 7.1%. The peak stress is predicted, and the critical positions on the blade under multi-mode vibration are marked, which can provide valuable knowledge for blade fatigue life prediction and health monitoring.

Vibration mitigation of long-span bridges with damped outriggers

Long-span bridges have low vibration frequencies and small intrinsic damping, and hence are subjected to large-amplitude vibrations, e.g., vortex-induced vibrations (VIVs), which jeopardize the serviceability and safety of the bridge structures. This study proposes a novel concept to supplement damping to such bridges for multimode vibration mitigation, by damping rotations of the bridge girder during vertical and torsional vibrations instead of translational displacements that have been previously the focus of relevant studies. A salient advantage of the concept is that bridge girder rotations near its ends are relatively large during vibrations enabling adequate supplemental damping and also convenient implementation. The concept is realized by a damped outrigger, consisting of a stiff outrigger rigidly connected to the bridge girder and horizontal dampers connecting the end of the outrigger to a bridge tower or a pier. A tensioned beam with rotational constraints is first used to demonstrate damping effects of damped outriggers on suspension bridges. Considering practical parameter ranges, the maximal damping ratio provided by one damped outrigger to a specific mode is about 1.0%. Influences of the relative importance of axial tension and bending stiffness and outrigger stiffness are studied. Furthermore, finite element analyses of an existing suspension bridge with damped outriggers are conducted to verify the damping performance. Numerical results show that by installing one damped outrigger, 6 out of 7 modes that have been involved in VIV events of the bridge can achieve a damping ratio of approximately 1.0% or above. By installing damped outriggers at the two towers, more modes can achieve a relatively large damping ratio, suggesting excellent performance of damped outriggers in suppressing multimode vibrations of long-span bridges.

Direct FE numerical simulation for dynamic instability of frame structures

Dynamic instability analysis of structures has been a critical issue in the application of engineering and mechanical designs. Can we directly simulate the dynamic stability of structures like buckling analysis in finite element software? To address this question, in this study, we explore the dynamic instability of frame structures, discretized into finite element structures, through developing a direct FE numerical simulation procedure. Benefiting from the advances of eigenvalue analysis of partial differential equations and finite element technology, a direct numerical simulation procedure is developed by using Floquet theorem, harmonic balance methods and finite element method. Dynamic instability characteristics of frame structures, concerning fundamental vibration, multi-mode coupling vibration and flexural-torsional vibration deformation due to their spatial characteristics, are thoroughly investigated. The results reveal that, the developed procedure can achieve dynamic stability analysis of damped and non-damped structures. Frame's flexural-torsional vibration deformation can significantly enlarge the size and instability intensity of unstable regions, threatening dynamic stability of frame structures. Moreover, damping ratio has a slight impact on instability mitigation of the frame induced by flexural-torsional vibration. Unstable regions of mode-coupling vibration of frame have a hidden attribute. No existing approximate formula can be available to estimate instability regions of frame's flexural-torsional vibration and mode-coupling vibration. The dynamic instability index presented here can characterize unstable regions intuitively. The proposed insights can provide design guidelines for frame structures, the developed direct FE simulation procedure may be insightful for extending to explore instability mechanisms of generalized structures with complex forms owing to parametric resonance.

Response Adjustability Analysis of Partial and Ordinary Differential Coupling System for Visco-Elastomer Sandwich Plate Coupled with Distributed Masses under Random Excitation via Spatial Periodicity Strategy

Vibration control of composite structures coupled with distributed masses under random excitations is a significant issue. In this paper, partial and ordinary differential coupling equations are obtained from a periodic sandwich plate coupled with supported masses under random excitation. An analytical solution to the coupling equations is proposed, and the stochastic response adjustability of the system with various periodic distributions of geometrical and physical parameters is studied. Spatial periodic layer thickness and core modulus of the sandwich plate are considered based on the active‐passive periodicity strategy. The periodically distributed masses are supported on the plate by coupling springs and dampers. Partial and ordinary differential coupling equations for the system including the periodic sandwich plate and supported masses are derived and then converted into unified ordinary differential equations for multi-mode coupling vibration. Generalized system stiffness, damping and mass are functions of the periodic parameters. Expressions of frequency response function and response spectral density of the system are obtained. Numerical results show the response adjustability via the spatially periodic geometrical and physical parameters. The results have the potential for application to dynamic control or optimization of sandwich structure systems.

Dynamic behavior and damping enhancement of cable with negative stiffness inerter damper

To improve the control performance of stay cables, a novel negative stiffness inerter damper (NSID) that consists of a negative stiffness device in parallel with an eddy current inerter damper is proposed in this study. The novelty of the NSID is that the damper can utilize the damping enhancement of the negative stiffness and inerter for cable vibration control, simultaneously. A further novelty is to carefully explore the modal behavior of the cable based on optimum parameters of the NSID. On this basis, the damping enhancement principle and conditions of the NSID for cable vibration control are demonstrated. In addition, the closed-form equation of the cable modal damping ratio and the design equation of the NSID for multi-mode cable vibration control are derived. And the control efficacy of the NSID is evaluated through the damping enhancement of the design cable modes. The results show that the modal behavior in the nth cable mode can be divided into three distinct regimes according to the negative stiffness and the corresponding optimum inertance, which are underdamped mode, subcritical damped mode, and the (n − 1)th underdamped mode. In the first regime, the inerter and negative stiffness have similar effects on enhancing damping ratio of the cable and reducing the optimum damping coefficient of the NSID. In the second and third regimes, the mode crossover phenomenon is observed, and the actual damping ratio of the nth cable mode decreases as the negative stiffness or inertance increases. The NSID performs better than the VD, VID, and NSD in mitigating multi-mode cable vibrations. Increasing negative stiffness can further improve the control performance and reduce the optimum inertance and damping coefficient of the NSID.

Design of cylindrical metashells with piezoelectric materials and digital circuits for multi-modal vibration control

Thin-walled cylindrical shells are widely used in industries, such as the main parts of aircrafts, rockets, and submarines. Except for meeting the load-bearing capacities, such structures must also have good vibration and acoustic performances. However, it is still a challenge to control the multi-modal vibration of cylindrical shells at low frequencies. This study explores the cutting-edge local resonant piezoelectric metamaterials to control the low-frequency vibration of cylindrical shells. A novel cylindrical meta-shell with piezoelectric materials and digital circuits was proposed, and a multi-resonance transfer function is implemented in each digital circuit. A method to optimizing the parameters in the transfer function for the purpose of vibration reduction is developed. The vibrational characteristics of the meta-shell are numerically analyzed using the finite element method. Numerical results clearly demonstrate that by delicately designing the parameters in the transfer function, the meta-shell can reduce the peak amplitudes of the first five modes by 30 dB or more. Therefore, the proposed piezoelectric cylindrical meta-shell may open new opportunities in vibration mitigation of transport vehicles and underwater equipment.

A semi-active shunted piezoelectric tuned-mass-damper for multi-modal vibration control of large flexible structures

A semi-active electromechanical Tuned Mass Damper (SATMD), which consists of a combined spring-piezoelectric device connected to an external resistive-inductive electric circuit is presented, aiming to provide robust multi-modal vibration suppression capabilities in large flexible structures. The combined spring-piezoelectric device provides the stiffness for the resonant mass, and also electromechanical energy conversion and coupling. A reduced-ordered mode superposition structural dynamics model coupled with the SATMD device and the RL shunt electric circuit is formulated, to simulate the response of the integrated structural system. The broadband vibration reduction capabilities of the proposed SATMD are numerically and experimentally evaluated and quantified on a down-scaled simplified airframe model. Very good correlation between numerical and measured results is obtained. The results illustrate that a SATMD configuration with an auxiliary mass equal to 1% of the structural mass and combined resistive-inductive impedance, yields substantial simultaneous vibration suppression over the range of 3 low-frequency structural modes, and can be easily retuned to accommodate in-flight changes in structural and loading parameters.