2-DOF Vibration Research Articles

Comparison of the dynamic performance of nonlinear one and two degree-of-freedom vibration isolators with quasi-zero stiffness

Nonlinear stiffness isolation mounts, which offer a high static stiffness alongside a low dynamic stiffness or even quasi-zero stiffness (QZS) over a displacement range, have been proposed. These vibration isolators offer a higher isolation frequency band of low transmissibility than conventional linear devices. Here, three kinds of nonlinear two degree-of-freedom (DOF) vibration isolators with QZS characteristic are analyzed in order to further improve the isolation performance. The dynamic response is obtained using the harmonic balance method, and the peak dynamic displacement is obtained using backbone curve analysis and energy balancing method. The optimum isolation performance of the nonlinear 2DOF vibration isolators is evaluated for four performance indexes and compared with three baseline vibration isolators. These are a linear and a QZS 1DOF vibration isolator as well as a linear 2DOF vibration isolator. To ensure a fair comparison, the static displacement of each vibration isolator is kept constant. The comparison demonstrates that a nonlinear 2DOF vibration isolator can be tuned to achieve a better isolation performance in the higher isolation frequency band than the baseline vibration isolators, while retaining a moderate peak dynamic displacement and peak transmissibility. In addition, the best vibration isolator is identified for each of the four performance indexes.

Bifurcations and chaos of a vibration isolation system with magneto-rheological damper

Magneto-rheological (MR) damper possesses inherent hysteretic characteristics. We investigate the resulting nonlinear behaviors of a two degree-of-freedom (2-DoF) MR vibration isolation system under harmonic external excitation. A MR damper is identified by employing the modified Bouc-wen hysteresis model. By numerical simulation, we characterize the nonlinear dynamic evolution of period-doubling, saddle node bifurcating and inverse period-doubling using bifurcation diagrams of variations in frequency with a fixed amplitude of the harmonic excitation. The strength of chaos is determined by the Lyapunov exponent (LE) spectrum. Semi-physical experiment on the 2-DoF MR vibration isolation system is proposed. We trace the time history and phase trajectory under certain values of frequency of the harmonic excitation to verify the nonlinear dynamical evolution of period-doubling bifurcations to chaos. The largest LEs computed with the experimental data are also presented, confirming the chaotic motion in the experiment. We validate the chaotic motion caused by the hysteresis of the MR damper, and show the transitions between distinct regimes of stable motion and chaotic motion of the 2-DoF MR vibration isolation system for variations in frequency of external excitation.

Vibration energy harvesting with a nonlinear structure

In the paper, beneficial nonlinearities incurred by an X-shape structure are explored for advantageous vibration energy harvesting performance. To this aim, a nonlinear structure beneficial for vibration energy harvesting is proposed, which is composed by X-shape supporting structures and a rigid body. By designing structure nonlinearities, which are determined by several key structure parameters, the power output peak of the harvesting system can be much improved and the effective frequency bandwidth for energy harvesting can be obviously increased, especially at the low frequency range. A coupling effect can be created among nonlinear stiffness and damping characteristics by constructing a 2-DOF vibration system, which has great influence on the energy harvesting performance. The proposed nonlinear energy harvesting systems can obviously outperform the corresponding linear systems in the whole frequency range and also demonstrate advantages compared with some other existing nonlinear energy harvesting systems in the literature. The results in this study provide a novel and practical method for the design of effective and efficient energy harvesting systems (especially in the low frequency range).

Design of double dynamic vibration absorbers for reduction of two DOF vibration system

This research is aimed to design and analyze the performance of double dynamic vibration absorber (DVA) using a pendulum and a spring-mass type absorber for reducing vibration of two-DOF vibration system. The conventional fixed-points method and genetics algorithm (GA) optimization procedure are utilized in designing the optimal parameter of DVA. The frequency and damping ratio are optimized to determine the optimal absorber parameters. The simulation results show that GA optimization procedure is more effective in designing the double DVA in comparison to the fixed-points method. The experimental study is conducted to verify the numerical result.

Use of Bayesian Networks for Qualification Planning: Early Results of Factor Analysis

This paper discusses the factor analysis that provides the basis for development and use of Bayesian Network (BN) models to support qualification planning in order to predict the suitability of Six Degrees of Freedom (6DOF) vibration testing for qualification. Qualification includes environmental testing such as temperature, vibration and shock to support a stochastic argument about the suitability of a design. Qualification is becoming more complex because it involves significant human expert judgment and relies on new technologies that have often never been fully utilized to support design assessment. Technology has advanced to the state where 6DOF vibration tests are possible, but these tests are far more complex than traditional single degree of freedom tests. This challenges systems engineers as they strive to plan qualification in an environment where technical and environmental constraints are coupled with the traditional costs, risk and schedule constraints. BN models may provide a framework to aid Systems Engineers in planning qualification efforts with complex constraints. Previous work identified a method for building a BN model for the predictive framework. This paper discusses validation efforts of models derived from the factor analysis and summarizes some recommendations on the factor analyses from industry subject matter experts.

Prediction of chatter stability for milling process using Runge-Kutta-based complete discretization method

On the basis of the classical Runge-Kutta method and the complete discretization method, a Runge-Kutta-based complete discretization method (RKCDM) is proposed in the paper to predict the chatter stability of milling process, in which the regenerative effect is taken into consideration. Firstly, the dynamics model of milling process is simplified as a 2-DOF vibration system in the two orthogonal directions, which can be expressed as coefficient-varying periodic differential equations with a single time delay. Then, all parts of the delay differential equation (DDE), including delay term, time-domain term, parameter matrices, and most of all the differential terms are discretized using the classical fourth-order Runge-Kutta iteration method to replace the direct integration scheme used in the classical semi-discretization method (C-SDM) and the classical complete discretization scheme with the Euler method (C-CDSEM), which can simplify the complexity of the discretization iteration formula greatly. Lastly, the Floquet theory is adopted to predict the stability of milling process by judging the eigenvalues of the state transition matrix corresponding to certain cutting conditions. Comparing RKCDM with C-SDM and C-CDSEM, the numerical simulation results show that RKCDM has the highest convergence rate, computation accuracy, and computation efficiency. As dichotomy search rather than sequential search is used in the algorithm, the calculation time for obtaining the stability lobe diagrams (SLDs) is greatly reduced. As a result, it is practical to determine the optimal chatter-free cutting conditions for milling operation in shop floor applications.

Flow-induced vibrations of square and rectangular cylinders at low Reynolds number

One-degree-of-freedom (1-dof) and two-degrees-of-freedom (2-dof) flow-induced vibrations (FIVs) of square and rectangular cylinders at a mass ratio of 10 and a low Reynolds number of 200 are studied numerically by solving the two-dimensional incompressible Navier–Stokes equations. The aim of the study is to identify the effects of the aspect ratio α, defined to be the ratio of the cylinder dimension in the cross-flow direction to that in the inline direction, on the vortex-induced vibration (VIV) and galloping responses. Simulations are conducted for aspect ratios of 0.3, 0.5, 0.7, 1 and 1.25 and reduced velocities ranging from 1 to 30. Distinct VIV lock-in and galloping regimes are found for all the aspect ratios except α = 0.3, for which only VIV lock-in is found. The VIV lock-in regime and the galloping regime are separated by a reduced velocity range, where the response amplitude is very small and the response frequency is a linear function of the reduced velocity. It is found that the maximum amplitude in the VIV lock-in regime decreases with increasing aspect ratio. Galloping does not start until the reduced velocity exceeds a critical value. The critical reduced velocity for galloping increases with increasing aspect ratio. For α = 0.5, galloping starts at Vr = 7 and 6 for 1-dof and 2-dof vibrations, respectively. The critical reduced velocity for galloping is increased to 17 at α = 1.25 for both 1-dof and 2-dof vibrations. Because the response amplitude in the inline direction is much smaller than that in the cross-flow direction, the response amplitude and frequency in 2-dof vibration are very similar to their counterparts in 1-dof vibration. However, the response amplitude in 2-dof galloping is greater than that in 1-dof galloping. A 2T vortex shedding mode is observed in the VIV lock-in regime for α = 0.3 and 0.5.

A dimensionless analysis of a 2DOF piezoelectric vibration energy harvester

In this study, a dimensionless analysis method is proposed to predict the output voltage and harvested power for a 2DOF vibration energy harvesting system. This method allows us to compare the harvesting power and efficiency of the 2DOF vibration energy harvesting system and to evaluate the harvesting system performance regardless the sizes or scales. The analysis method is a hybrid of time domain simulation and frequency response analysis approaches, which would be a useful tool for parametric study, design and optimisation of a 2DOF piezoelectric vibration energy harvester. In a case study, a quarter car suspension model with a piezoelectric material insert is chosen to be studied. The 2DOF vibration energy harvesting system could potentially be applied in a vehicle to convert waste or harmful ambient vibration energy into electrical energy for charging the battery. Especially for its application in a hybrid vehicle or an electrical vehicle, the 2DOF vibration energy harvesting system could improve charge mileage, comfort and reliability.

Use of Bayesian Networks for Qualification Planning: A Predictive Analysis Framework for a Technically Complex Systems Engineering Problem

Abstract This paper discusses the viability of using Bayesian Network (BN) models to support qualification planning in order to predict the suitability of Six Degrees of Freedom (6DOF) vibration testing for qualification. Qualification includes environmental testing such as temperature, vibration, and shock to support a stochastic argument about the suitability of a design. Qualification is becoming more complex and restricted yet available new technologies are not fully utilized. Technology has advanced to the state where 6DOF vibration shakers and control systems capable of high frequency tests are possible, but the problem using these systems is far more complex than traditional single degree of freedom (SDOF) tests. This challenges systems engineers as they strive to plan qualification in an environment where technical, environmental, and political constraints are coupled with the traditional cost, risk and schedule constraints. New technologies are also available for systems engineers to combine technical understanding with cost, risk and schedule factors to aid in decision making for complex problems such as qualification planning. BN models may provide the framework to aid Systems Engineers in planning qualification efforts with complex constraints. This paper discusses related work, the current approach and results of this research.

A 2 DOF vibration harvester for broadband and multifrequency harvesting using a single electro-magnetic transducer

The narrow bandwidth of resonant vibration energy harvesters has long been seen as a drawback to exploitation. The narrow bandwidth is necessitated by the requirement to sufficiently amplify small source vibrations, but results in devices vulnerable to changes in excitation frequency, de-tuning due to ageing of components, and also makes efficient harvesting from sources with multiple frequency components difficult. In this paper a harvester based on a 2 degree of freedom oscillator is presented that not only enjoys the wider bandwidth of the higher order system, but configures the electromagnetic transducer in such a way that it requires no more components than the transducer of a typical single degree of freedom harvester. Theoretical models of the harvester system predict a range of possible frequency response functions dependent on easily-adjusted electrical parameters. These predictions are validated with experimental results.

Analysis and Optimization of a Vibration Isolation Platform Based on 6-DOF Parallel Mechanism

In this paper a 6-DOF(Degree of Freedom) vibration isolation platform with structural configuration of 3CPS (3 Cylindrical-Prismatic-Spherical)-1UPS (1 Universal-Prismatic-Spherical) parallel mechanism is proposed. Firstly, the dynamic model is developed through virtual work principle. The transfer function and the complex frequency response function were obtained from the dynamical equation. Then, 36 amplitude frequency diagrams are obtained from the frequency response function. By analyzing the 36 amplitude frequency diagrams, 10 vibration coupling terms were chosen. The range of the spring stiffness coefficient and the range of the damping coefficient of the magneto-rheological damper are determined by analyzing the acceleration transmissibility of the 10 vibration coupling terms. At last, an integrated optimization combining structural and controller optimization is developed. The optimal parameters of the parallel mechanism are obtained through Genetic Algorithm (GA). This work is significant for the prototype design and verification of the vibration isolator.

Using a high resolution motion capture system to determine 6-DOF whole-body vibration accelerations

Using a high resolution motion capture system to determine 6-DOF whole-body vibration accelerations

Design Methodology of a Micro-Scale 2-DOF Energy Harvesting Device for Low Frequency and Wide Bandwidth

A detailed design methodology of a micro-scale 2-DOF energy harvesting device that can harvest human motion energy of low frequency and wide bandwidth is developed. Based on the concept of the 2-DOF vibration absorber, device parameters are selected to harvest energy at low frequency of 1-10 Hz and wide bandwidth with ±20% of the mean frequency, which matches the human motion. The device dimensions are limited to 40 × 30 × 10 mm3 to fit with the human wrist size. Then, a finite element model is developed to investigate the system performance with the selected parameters. When subjected to harmonic excitation of 1 g, the proposed 2-DOF device is able to provide a power of at least 10 μW in between the two close resonant peaks of 4 Hz and 6 Hz, which is the target frequency range. The device shows very high power per square frequency compared with the reported harvesters.

Selecting seats for steel industry mobile machines based on seat effective amplitude transmissibility and comfort

The purpose of this work was to help a steel industry partner select the most appropriate of three high end heavy equipment seats to retrofit a number of their heavy mobile machines used in the steel making process. The participants included 8 males (22.3 ± 2.0 yrs.) and 8 females (23.5 ± 1.8 yrs.) with no experience operating heavy mobile equipment. Previously recorded 6-DOF chassis acceleration data from a Pot Hauler (a machine which picks up and transports pots of slag) were used to extract six, 20 second representative profiles for implementation on a lab-based heavy machine simulator (6-DOF Parallel Robotics System Corporation robot). Subjects sat on three heavy equipment seats (BeGe7150, Grammar MSG 95G1721, and a 6801 Isringhausen with the seat pan cushion retrofitted with a Skydex cushion) mounted on the simulator. Each subject completed three trials for each combination of seat (n=3) and vibration profile (n=6). Chassis and operator/seat interface vibration were measured by 2, 6-DOF vibration transducers. Variables included Seat Effective Amplitude Transmissibility (SEAT) (X,Y,Z,Roll,Pitch,Yaw,6DOF Vector Sum) to determine if the seat was attenuating or amplifying the vibration, 6-degree of freedom (DOF) vibration total value weighted predicted comfort (Avc) (according to ISO 2631-1) and operator reported comfort (ORC). Factorial ANOVAs revealed significant differences (p < or = 0.05) between seats for all SEAT variables but different seats performed better than others depending on the axis. Significant differences between males and females were observed for SEAT in X,Y, and Pitch as well as for Avs. As expected there were significant differences between vibration profiles for all assessed variables. A number of interaction effects were observed, the most frequently occurring of which was between seat and vibration profile. Based upon the number of seat and vibration profile interactions, results suggest that a single seat is not suited for all tested conditions. However, SEAT values for all of the seats tested were extremely low (e.g., 6-DOF SEAT < 30%) indicating that all of the seats were capable of providing good vibration attenuation.

Simulation on Roadheader Rear Supporting Leg Vibration Moulding

Using three-dimensional discrete element method (DEM) to simulate the vibration of the evaporated pattern casting molding process, accordingly summary the influence law of vibration direction effecting on dry sand filling. The results show that the 3DOF vibration filling effect is best, 2DOF takes second place, 1DOF is the worst. In the 2DOF vibration the YZ direction is best, XY direction worst. Meanwhile Y direction vibration works best in 1DOF vibration, second is X direction, and Z direction gets worst.

Bandwidth Optimization Design of a Multi Degree of Freedom MEMS Gyroscope

A new robust multi-degree of freedom (multi-DOF) MEMS gyroscope is presented in this paper. The designed gyroscope has its bandwidth and amplification factor of the sense mode adjusted more easily than the previous reported multi-DOF MEMS gyroscopes. Besides, a novel spring system with very small coupling stiffness is proposed, which helps achieve a narrow bandwidth and a high amplification factor for a 2-DOF vibration system. A multi-DOF gyroscope with the proposed weak spring system is designed, and simulations indicate that when the operating frequency is set at 12.59 kHz, the flat frequency response region of the sense mode can be designed as narrow as 80 Hz, and the amplification factor of the sense mode at the operating frequency is up to 91, which not only protects the amplification factor from instability against process and temperature variations, but also sacrifices less performance. An experiment is also carried out to demonstrate the validity of the design. The multi-DOF gyroscope with the proposed weak coupling spring system is capable of achieving a good tradeoff between robustness and the performance.

A Study on Vibration Characteristics and Stability of the Ambulance Nonlinear Damping System

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.

Studies on Dynamic Parameters of Intellectual Vibratory Roller

In order to anlyse the compaction process of vibrating roller more accurately, this paper built a new 6-DOF Vibration system dynamic model under the coupling operating condition of intellectual vibratory roller considering the qualities of soilmass and applied the new model on a type of intellectual vibratory roller’ dynamic analysis. On that basis, we optimized the design parameter of this vibratory, and found out the optimum working parameters. It would realize the effective control of intellectual vibratory roller in the process of compaction, advance the density and compaction efficiency in compacting process, and improve comfortabilityg of the complete machine. At the same time, it also provided an effective numerical basis for the optimization design of vibratory roller.

Validation Techniques for 6-DOF Vibration Data Acquisition

Multiple Degree of Freedom (MDOF) excitation systems and MDOF vibration control systems continue to improve, and are now standard equipment in many dynamic test laboratories. Determination of an input specification for such MDOF systems is critically dependent on properly acquired field data. Validation of field data will be discussed and demonstrated employing the same transformation tools used in both transformation-based 6-degree-of-freedom (6-DOF) vibration control and generalized MDOF vibration specification development (VSD).

Stability Analysis of Mode Coupling Vibration Systems with Coulomb Friction

In the article, the stability analysis of mode coupling systems with Coulomb friction is performed. A two-degree-of-freedom (2-DOF) vibration system with Coulomb friction is constructed. Taking the friction coefficient as a bifurcation parameter, a Hopf bifurcation graph is obtained using the Routh criterion. It is illustrated that Coulomb friction can induce self-excited oscillation in the mode coupling system. The analytic expressions of the limit cycle amplitudes of the oscillation system are achieved based on the extended harmonic balance method. Moreover, the effects of some parameters on the size of limit cycles are investigated. All the research method and results presented can provide a valuable reference for the structure optimal design of mechanical systems and vibration reduction.