Multi-degree-of-freedom Research Articles

Development of a Novel Seat Suspension Based on the Cubic Stewart Parallel Mechanism and Magnetorheological Fluid Damper

To alleviate the impact and vibrations to a driver in multiple directions during the driving of non-road vehicles, the authors of this paper proposed a multi-degree-of-freedom (MDOF) seat damping suspension that was based on the cubic Stewart mechanism and magnetorheological fluid (MRF) damper. A kinematics analysis of the cubic Stewart mechanism was carried out. The relative motion velocity of each leg of the Stewart mechanism was calculated from the center velocity of the upper and lower platforms, according to a reverse kinematics equation. Furthermore, forward and inverse dynamic models of the MRF damper were established, which laid the foundation for semi-active control of the seat suspension. Finally, a semi-active control method for multidimensional damping based on the optimized fuzzy skyhook control method was proposed. The research results showed that using this method could simultaneously improve the vibration damping performance of a seat suspension in the vertical, horizontal, and roll directions.

Vision-Based Dynamic Response Measurements of Structures by Using Smartphone Videos and Alternative Techniques

This study aimed to measure dynamic responses of structural systems using smartphone videos and vision-based sensing processes. Two algorithms, based on template matching and feature extraction, are employed for this purpose. They were verified on the shake table experiments with single and multi-degree freedom steel specimens where their videos were captured by an ordinary smartphone during excitation. Dynamic responses such as strains at discrete sections as well as displacements, velocities, and accelerations at the floor levels of the specimens were obtained by following the signs of physical or virtual markers during the video recording. Through comprehensive experiments and one available full-scale multi-story shake table experiment in the literature, the developed vision-based algorithms were validated to be used with smartphone videos. It was also shown that regardless of the quality of the video record, substantial characteristics of a specimen or a structure could be determined reasonably by smartphone videos since the absolute mean relative differences varied between 10% to 20%.

Performance-aware design for piezoelectric energy harvesting optimisation via finite element analysis

Most of the optimisation studies of Vibration Energy Harvesters (VEHs) account for a single output quantity, e.g. frequency bandwidth or maximum power output, but this approach does not necessarily maximise the system efficiency. In those applications where VEHs are suitable sources of energy, to achieve optimal design it is important to consider all these performance indexes simultaneously. This paper proposes a robust and straightforward multi-objective optimisation framework for Vibration Piezoelectric Energy Harvesters (VPEHs), considering simultaneously the most crucial performance indexes, i.e., the maximum power output, efficiency, and frequency bandwidth. For the first time, a rigorous formulation of efficiency for Multi-Degree of Freedom (MDOF) VPEHs is here proposed, representing an extension of previous definitions. This formulation lends itself to the optimisation of FE and MDOF harvesters models. The optimisation procedure is demonstrated using a planar-shape harvester and validated against numerical results. The effects of changing some structural parameters on the harvester performance are investigated via sensitivity analysis. The results show that the proposed methodology can effectively optimise the global performance of the harvester, although this does not correspond to an improvement of every single index. Furthermore, the optimisation of each performance index individually results in a variety of design configurations that greatly differs from one another. It is here demonstrated that the design obtained with the multi-objective function here proposed is similar to the design obtained when optimising the efficiency.

Active/passive vibration isolation with multi-axis transmission control: Analysis and experiment

It is necessary to cut off all vibration transmission paths for an active vibration isolation system. A greater suppression of local responses will result in a better overall attenuation. In order to realize impediment at possible transmission paths, a new scenario of multi-axis transmission control is proposed, in which the active/passive isolator attenuates vibration transmission at multi-degrees of freedom by vertical and horizontal actuators mounted orthogonally on the intermediate inertial mass. This scenario is able to weaken the cross-coupling of control action between isolators by the passive parts. In order to investigate the characteristics of multi-axis transmission control, the dynamic model of a partitioned isolation system with one active/passive isolator is established and the involved transmission control is analyzed. The numerical results show that the foundation vibration can be attenuated along with the decrease of vibrations of the intermediate inertial mass. It is further revealed by experimental results that the multi-axis transmission control at the intermediate inertial mass is necessary to achieve sufficient vibration attenuation in the foundation, whereas the control in a single axis is ineffective. These verified results imply that the active/passive isolation with local transmission path control makes it possible to apply decentralized control in a large active vibration isolation system.

Shaking‐table substructure test using a MDOF boundary‐coordinating device

AbstractBoundary coordination is necessary for a shaking‐table substructure test (STST), which includes transferring forces and displacements between the tested and numerical substructures. Unlike other real‐time hybrid tests (RTHTs), the physical substructure is mounted on a shaking table and excited by the inertial force, which naturally requires a force control for boundary coordination. However, the natural velocity feedback in hydraulic actuators and high‐frequency noise in the force feedback signals makes it difficult to achieve force control. To solve these difficulties, in this study, a multi‐degree‐of‐freedom (MDOF) boundary‐coordinating device (BCD) is developed to simultaneously implement the boundary shear force and bending moment by the inertial force generated from shaking masses that are uniaxially driven by electromagnetic actuators. An STST framework was developed based on the BCD. The bottom two stories of a four‐story steel moment frame were selected as the physical substructure. A hydraulic shaking table was employed for ground excitation, and the BCD was used as the boundary coordinator for the numerical substructure, that is, the top two stories of the frame. To trace high‐frequency force signals, an enhanced three‐variable control (ETVC) method is proposed. Furthermore, an error‐response negative‐feedback compensation (ENC) method was developed to reduce the adverse effects of loading errors. Several rounds of STST tests demonstrated that the boundary commands were effectively traced by the BCD, so that the STST framework could consistently reproduce the seismic behavior of engineering structures.

Chaotic analysis and entropy estimation of the entropy source based on semiconductor superlattice chaos

A true random number generator (TRNG) is a fundamental security primitive. Recently, the semiconductor superlattice (SSL) has been investigated as an ideal physical entropy source with high speed. It shows multi-degree freedom nonlinearity based on the effects of quantum mechanics. However, there is still a lack of systematic evaluation schemes for SSL. In this study, a chaotic assessment model for SSL is proposed. Chaotic characteristic assessment and nonlinear dynamics estimation are carried out using the model. Furthermore, a set of entropy estimation procedures are introduced to evaluate the uncertainty of the SSL. The estimation results demonstrate that SSL is chaotic, highly complex, and nonlinear. SSL shows stable performance and has an entropy rate greater than 50% with 8 sample bits. Moreover, the throughput of the assessment prototype reaches 300 Mbit/s. In brief, SSL is an excellent physical entropy source for high-quality and high-speed TRNG.

Closed-form critical response of elastic-plastic shear building with viscous damper under pseudo-double impulse for simulating resonant response under near-fault fling-step motion

A set of approximate closed-form solutions of the maximum interstory drifts under the critical pseudo-double impulse (PDI) is derived for non-proportionally damped multi-story shear building models with bilinear hysteresis. The use of PDI and the closed-form solutions efficiently and accurately enable the capture of the critical responses of elastic-plastic multi-degree-of-freedom (MDOF) systems under the one-cycle sine wave, which substitutes for the main part of near-fault fling-step motions. The formulation of the closed-form maximum interstory drifts is based on the energy balance law. In the formulation, a quadratic function approximation of the damping force-interstory drift relation is introduced together with an updated mode-controlled energy-based approach (UMEA). While UMEA was proposed in the previous paper to derive the approximate maximum interstory drifts of undamped elastic-plastic MDOF systems under the critical PDI, it is extended so that it can be applied to non-proportionally damped MDOF systems. It is demonstrated that the proposed method can estimate the maximum interstory drifts of elastic-plastic non-proportionally damped MDOF systems under the critical PDI and the corresponding one-cycle sine wave with high accuracy. The estimation by the proposed method can be conducted much more efficiently and stably than the time-history response analysis (THRA). It is also shown that the hysteretic energy dissipation is not large enough to reduce the maximum interstory drifts. Finally, it is demonstrated that the proposed method can accurately estimate the maximum interstory drifts of elastic-plastic moment-resisting frames with viscous dampers under the resonant one-cycle sine wave.

Deformation‐based design of seismically isolated bridges

AbstractA performance‐based design procedure is proposed for seismically isolated bridges with/without supplemental energy dissipation devices. Considering multiple performance levels (PLs), the proposed method initially identifies the critical hazard level and “near‐optimal” alternatives of the isolation system. Both performance and economy are evaluated based on the inelastic response of reduced‐degree‐of‐freedom (RDOF) isolation and energy dissipation systems. Nonlinear response history analysis (NLRHA) of the multi‐degree‐of‐freedom (MDOF) system is employed in successive design steps, corresponding to different PLs, to refine the design solution through the control of a broad range of material strains and deformations. The above characteristics aim to deliver a rigorous and broadly applicable design method that reduces the high computational effort characterising complex design frameworks, while addressing identified deficiencies of alternative displacement‐based methods. The efficiency of the proposed design methodology is demonstrated by applying it to an actual bridge that was previously designed for ductile behaviour. Alternative isolation and energy dissipation schemes are investigated. Assessment of the design through NLRHA reveals in all cases enhanced seismic performance and cost reduction in the substructure design, thus rendering base‐isolation an appealing design alternative.

Wide-Range Displacement Sensor for Vibration Measurement of Magnetically Suspended Air-Blower

The displacement measurement is critical to the stable suspension and position control of magnetically suspended air-blower, and the eddy current sensor is a typical sensor to measure the dynamic displacements between the rotor shaft and the stator part. The displacement coordinate of magnetically suspended air-blower on multi degrees of freedom is firstly developed, and then the detection principle of eddy current displacement sensor is researched. Furthermore, the design processes of displacement sensor including dimensional parameters and quality factor are presented, and then the simulation had been conducted. Finally, the measurement experiments of magnetically suspended air-blower using the displacement sensor system are conducted, the results indicate that the linear range of displacement measurement approaches to 7.5 mm, the relative error is smaller than 2%, and the temperature stability is 0.022%/°C. Therefore, the displacement measurement and vibration analysis of magnetically suspended air-blower could be realized by the designed displacement sensor system.

Globally-evolving-based generalized density evolution equation for nonlinear systems involving randomness from both system parameters and excitations

It has long been one of the main challenges in science and engineering to capture the probabilistic response of high-dimensional nonlinear stochastic dynamic systems involving double randomness, i.e. randomness in both system parameters and excitations. For this purpose, a globally-evolving-based generalized density evolution equation (GE-GDEE) is established. Generally, for a multi-dimensional nonlinear system involving double randomness, if one single physical quantity as a response of the system is of interest, a GE-GDEE, as a two-dimensional partial differential equation (PDE) governing the probability density function (PDF), can be derived. The effective drift coefficients, which represent the physically driving force function in the GE-GDEE, can be determined based on the data from some representative deterministic dynamic analyses of the underlying physical system. A new estimator for effective drift coefficients is developed based on the vine copulas. Once the effective drift coefficients are determined, the GE-GDEE can be solved to capture the probability distributions of the quantities of interest. Several numerical examples, including linear and nonlinear multi-degree-of-freedom (MDOF) systems subjected to white noise or non-stationary earthquake ground motions, are presented to verify the effectiveness of the proposed method. Finally, problems for future investigations are discussed.

Structural control of high-rise buildings subjected to multi-hazard excitations using inerter-based vibration absorbers

High-rise buildings located in strong wind and earthquake-prone regions are exposed to multiple hazards that must be properly considered to prevent considerable socio-economic losses. In the present study, the Tuned Mass Damper-Inerter (TMDI) is adopted for the structural control of high-rise buildings under wind and earthquake excitations by exploiting its inherent multiple-mode control effects. A generalized multi-degree-of-freedom (MDOF) model of TMDI-controlled structure is established first and compared with the commonly-adopted generalized single-degree-of-freedom (SDOF) model. Analytical investigation on the multiple-mode control effects of TMDI demonstrates that the damping effects weighted by mode coordinate differences at two terminals of TMDI are mainly responsible for its multiple-mode control effects. Based on a real high-rise building project, analysis errors induced by the generalized SDOF model against the more accurate generalized MDOF model are quantified by a comprehensive Monte Carlo simulation, which suggests the necessity of adopting numerical search for parametric design of TMDI under most circumstances. Therefore, the parameters of TMDI and its two competitors, i.e., Multi-Tuned Mass Damper (MTMD) and Multi-Tuned Mass Damper-Inerter (MTMDI), are optimized using a genetic algorithm, assuming both wind- and earthquake-induced responses as performance objectives. Time and frequency domain analyses demonstrate the efficiency of utilizing the multiple-mode control effects of a single TMDI for multi-hazard design. The trend of the equivalent damping ratio of TMDI-controlled structure on an arbitrary mode is analyzed to identify the design parameters affecting most the TMDI multiple-mode control effect at last.

An Innovative Structural Dynamic Identification Procedure Combining Time Domain OMA Technique and GA

In this paper an innovative and simple Operational Modal Analysis (OMA) method for structural dynamic identification is proposed. It combines the recently introduced Time Domain–Analytical Signal Method (TD–ASM) with the Genetic Algorithm (GA). Specifically, TD–ASM is firstly employed to estimate a subspace of candidate modal parameters, and then the GA is used to identify the structural parameters minimizing the fitness value returned by an appropriately introduced objective function. Notably, this method can be used to estimate structural parameters even for high damping ratios, and it also allows one to identify the Power Spectral Density (PSD) of the structural excitation. The reliability of the proposed method is proved through several numerical applications on two different Multi Degree of Freedom (MDoF) systems, also considering comparisons with other OMA methods. The results obtained in terms of modal parameters identification, Frequency Response Functions (FRFs) matrix estimation, and structural response prediction show the reliability of the proposed procedure.

Experimental study on dynamic responses of a spar-type floating offshore wind turbine

In this paper, a scaled model experiment of a spar-type floating offshore wind turbine (FOWT) is carried out. The scaled wind turbine model is designed based on the OC3-Hywind FOWT, and the pitch control mechanism is considered. An optimal arrangement scheme of anemometers deployment on a disc plane is proposed to improve the efficiency of wind field measurement. Three categories of load case tests corresponding to the wind-only, wave-only as well as wind-wave concurrent loads are carried out. The rigid-body motion of the FOWT model and the deformation of the tower as well as the mooring tension are captured. It is revealed that the yaw is greatly affected by the rotor, and the trend and amplitude of the structural rigid-body motion are dominated by aerodynamic loads and hydrodynamic loads, respectively. Moreover, the coupling effects of multi-degree-of-freedom (MDOF) rigid-body motions of the structure are quite obvious. As for the flexible tower, although its deformation is quite small, the resonant response caused by doubling frequency coupling effects is significant. Besides, large amplitude of stress can be induced in the flexible tower under wave-contained conditions. The present experiment provides a benchmark for the validation of numerical simulation tools of FOWTs.

Evolution of magnetic phase in two-dimensional van der Waals Mn1−x Ni x PS3 single crystals

Metal thio(seleno)phosphates MPX3 have attracted considerable attentions with wide spanned band gaps and rich magnetic properties. In this series, two neighboring members MnPS3 and NiPS3 differ in magnetic atoms, magnetic easy axes, spin anisotropy, as well as nearest-neighbor magnetic interactions. The competition between these components may cause intriguing physical phenomena. In this article, the evolution of magnetism of Mn1−x Ni x PS3 series is reported. Despite the incompatible antiferromagnetic orders of two end members, the antiferromagnetism persists as the ground state in the whole substitution region. The magnetic ordering temperature T N show nonmonotonic V-shape behavior, and the reentrant spin glass phase at x= 0.5 is observed. In addition, abnormal bifurcation of T N occurs at x = 0.75, which may be due to the temperature-dependent spin reorientation or phase separation. The evolution of magnetism is further confirmed semi-quantitatively by our density functional theory calculations. Our study indicates that exotic magnetism can be intrigued when multi-degrees of freedom are involved in these low-dimensional systems, which call for more in-depth microscopic studies in future.

Choices of the Critical Frequency for φ in TL-φ Algorithms When Applied to Multi-Degree of Freedom Systems

TL-φ algorithms are newly developed explicit structure-dependent integration algorithms utilized for solving the temporally discretized equations of motion. In contrast to the existing algorithms, the most significant improvement of TL-φ algorithms is in diminishing the amount of period errors by introducing a precorrection coefficient φ into the integration parameters of TL-φ algorithms, which is related to the critical frequency of a system. In the previous work, the fundamental frequency of the system is deemed to be the critical frequency, so that φ is a constant scaling corresponding to the fundamental frequency for both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems. However, for a MDOF system, the first mode may not contribute to the total response more than other ones under a given external excitation, calculating φ only by the fundamental frequency will underestimate the contribution of the higher-frequency modes to structural dynamics. In this paper, choices of the critical frequency for φ when applying TL-φ algorithms to MDOF systems are investigated thoroughly. By considering the initial structural properties of the system and the frequency characteristics of the external excitation simultaneously, a calculation criterion of φ for MDOF systems under specific external excitations is proposed. Four numerical examples with different initial structure properties and loading conditions are designed, and the results demonstrate that the proposed criterion can be potentially used to solve structural dynamic problems of MDOF systems with a more desirable numerical dispersion performance.

A quadratic Wiener path integral approximation for stochastic response determination of multi-degree-of-freedom nonlinear systems

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.

Pseudo‐multi impulse for simulating critical response of elastic–plastic high‐rise buildings under long‐duration, long‐period ground motion

SummaryA pseudo‐multi impulse (PMI) is proposed as an extension of the ordinary base‐input multi impulse substituting a long‐duration, long‐period ground motion which can often be simulated by a multi‐cycle sine wave (MSW). PMI is treated as a multitude of impulsive lateral forces and hardly excites the higher‐mode responses of elastic multi‐degree‐of‐freedom (MDOF) models because the undamped fundamental participation vector is adopted as the influence coefficient vector. It is shown that the critical time interval of PMI can be obtained without any repetition. This enables a smart derivation of the critical response. The displacement transfer functions are derived for elastic MDOF models, where two kinds of the influence coefficient vectors are adopted: (1) one at every component (conventional one) and (2) the undamped fundamental‐mode participation vector. It is demonstrated that the critical PMI can efficiently and accurately evaluate the maximum responses under MSW and the critical input period of MSW for elastic–plastic MDOF models. Furthermore, PMI is extended to treat the critical case for the second eigenmode. Finally, the critical PMI is applied to high‐rise buildings for the investigation into the input energy and the cumulative plastic deformation ductility demand.

An experimental study of tuned liquid column damper controlled multi-degree of freedom structure subject to harmonic and seismic excitations.

A tuned liquid column damper (TLCD) is a passive vibration control device that not only mitigates unwanted structural vibrations but also acts as a water storage facility in a building. These aspects of TLCD make its application specifically suited for building structures. Previously, many experimental works on TLCDs have been conducted considering a single degree of freedom (SDOF) structure. However, the performance of TLCDs to control the response of multi-degree of freedom (MDOF) structure has rarely been studied experimentally. Therefore, this study has investigated the performance of a tuned liquid column damper (TLCD) on a multi-degree of freedom (MDOF) structure using shake table testing. A four-storey steel frame structure equipped with TLCD at the top of the fourth storey has been studied. Experimental normalized frequency response curves for MDOF structure equipped with TLCD have been determined. For this purpose, a series of harmonic loadings including frequencies 0.65 Hz, 1.17 Hz, 1.30 Hz, 1.43 Hz and 1.95 Hz have been applied in addition to historic earthquake loading. Peak and root-mean-square (RMS) accelerations have been discussed in detail for all the applied loadings at each storey level of the structure. For comparison purposes, the percentage reductions in peak and RMS accelerations have been calculated and compared. Also, RMS displacements and inter-storey drifts have been presented for resonant and seismic excitations. Both in time and frequency domains, responses of controlled MDOF structure have been analyzed and compared with uncontrolled structure. Results confirmed that TLCD has improved the MDOF structure responses at harmonic loadings frequencies near resonance and historic earthquake excitations. Furthermore, the improvement in the responses of MDOF structure with TLCD is more prominent at harmonic loadings compared to historic earthquake loading.

Semi-analytical solution to the steady-state periodic dynamic response of an infinite beam carrying a moving vehicle

Vehicle-induced vibration is among the most important aspects for riding comfort and structural durability. It is well known, however, that the span of structure is much larger than the size of vehicle, resulting in a huge computational cost for dynamic analysis, and meanwhile, the multi-degree-of-freedom (MDOF) vehicle and the nonlinear interaction between beam and vehicle are difficult to be involved in an explicit solution. In the present work, a joint harmonic balance and Fourier transform (HBFT) method is proposed for the steady-state periodic dynamic response of an infinite beam carrying a vehicle taking into account the surface roughness and the nonlinear interaction between beam and vehicle, and the semi-analytical steady-state periodic (SASP) solution is then obtained. First, the vehicle and beam are separated as two subsystems, where the infinite beam is supported by a viscoelastic foundation and the vehicle is described by a MDOF discrete model. The response of the beam is analytically solved from the Fourier transform approach and the response of the vehicle is explicitly formulated considering the Fourier expansion of the nonlinear interaction forces. Next, the Fourier coefficients for such interaction forces are numerically solved from the harmonic balance method based on the compatibility between the subsystems. By substituting the solved interaction forces back into the closed-form analytical solutions, the responses of both the subsystems are evaluated explicitly. This allows for accurate and efficient analysis for the steady-state periodic response of vehicle–beam system considering the infinite beam, the discrete vehicle and the nonlinear interaction between them. After validating the SASP solution by comparing with the assumed modal method, the effect from the vehicle speed on the response of the vehicle–beam system is discussed, indicating the existence of a few resonant speeds, having potential value for the optimal design of vehicles and road pavements. Apart from the present example, the SASP solution can be applied in broader cases and the proposed HBFT method is generally feasible to a variety of dynamic coupling problems consisting of a continuum and a discrete model.

Numerical Experiments on the Optimum Design and Location of a Passive Vibration Absorber Applied to Multi Degrees of Freedom System Under Base Excitation

Numerical experiments are used to investigate the optimal design parameters and the location of a passive vibration absorber and its application to reduce resonant vibration amplitudes of a base excited, discrete, Multi Degrees of Freedom (MDOF) vertical system. Attaching a passive vibration absorber (simple undamped resonator) to the system’s end mass (top mass) is preferable when dealing with the vibrations of multi degrees freedom systems. As the absorber is moved closer to the primary system's farthest end, the amplitude reduction results get better. The numerical analysis is less sensitive to slight variations in mass ratio than minor variations in the absorber’s natural frequency, according to perturbation analysis on a two degree of freedom system. Introducing damping to the used passive vibration absorber has no discernible effect on the absorber’s effectiveness.