Natural Frequency Of System Research Articles

Natural Frequency Perturbations Using a Scalar Expression with Reference Plots to Predict Associated Errors

PurposeEigenvalues are the natural frequencies of system squared. When designing a system it is important to know the natural frequencies, because if the system is forced near one of these natural frequencies the magnitude of vibration becomes very large. The eigenvalues are typically determined by solving an eigenvalue problem, which is an iterative produce that is expensive for larger systems. If multiple perturbations to the system are made or tested re-solving an eigenvalue problem every time becomes prohibitive. Perturbation methods exist to predict perturbed eigenvalues more quickly. However, these methods typically require matrix–vector products and do not quantify what is considered a small enough perturbation to use these methods.MethodsThis paper looks to address these issues using a scalar perturbed eigenvalue expression that avoids calculating matrix–vector products for every perturbation and developing reference plots that can be used to predict the associated error. The reference plots may be used to predict errors in the approximated natural frequencies from nominal modal parameters. The scalar perturbed eigenvalue expression and reference plots for errors were tested using numerical examples.ResultsIn every case tested the plots were able to accurately predict the expected errors, to be within a predicted range.ConclusionThe proposed method allows one to use the developed scalar expression to predict perturbed eigenvalues, and the developed reference plots may be used to predict the errors associated with using the proposed expression.

Application of point interpolation mesh-free method for magneto/electro rheological vibrations of sandwich conical panels

Magneto-rheological or electro-rheological materials, due to many possible control-based applications have acquired lots of attention. So, in the present work, as the first attempt, a point interpolation method is exploited to evaluate vibrations of the sandwich conical panel with magneto-rheological or electro-rheological core, and composite face sheets. Through the linear viscoelastic hypothesis, and in the pre/post-yield zones, the correlation of shear stress-strain for magneto-rheological fluid is applied. The equilibrium equations are governed by the principle of Hamilton in addition to the first-shear deformation (FSD) hypothesis. By satisfying the compatibility equations, the interfaces of the core along with composite face layers are simulated. Upon polynomial basic function, shape functions together with the approximate form of the displacement field are constructed. The correctness of the computational approach is certified through the analogical investigation. The results show that number of face layers, thickness, the pattern of composite layers, weight fraction, and electric/magnetic field, together with the geometry of the conical panel have an imperative impact on the system's natural frequency in addition to the loss factor index.

Coupled fluid-structure simulations and experiments of DBD plasma actuator for damping of plate vibrations

In the present work, an experimental and numerical study has been performed to evaluate the suitability of the dielectric barrier discharge plasma actuators to reduce free vibrations of plates. Two configurations composed of two different plexiglass plates and identical actuators were manufactured. Experimental campaigns, in terms of laser doppler anemometer velocity measurements, flow visualization and proper orthogonal decomposition were carried out to characterize the plasma actuator and to validate the numerical models. The plasma-induced virtual body force for the numerical analysis has been calibrated based on the experimental body force and the induced flow field. Then numerical studies of the plasma effect on the flow around an oscillating plate was performed. An experimental modal analysis has been also performed to understand the vibration characteristics of the plate and to support the numerical study. A reduced-order model of the system, based on the harmonic oscillator, was developed to properly design an open loop controller (system natural frequency, amplitude and timing of the attenuation force). Eventually, the work demonstrates that the control strategy based on the plasma actuation is effective in reducing the plate vibrations. However, a deeper study and a more sophisticated control strategy would be required to significantly mitigate the flutter oscillations of airfoils in real applications.

Resonant vibrations of a non-ideal gyroscopic rotary system with nonlinear damping and nonlinear stiffness of the elastic support

When motor performance characteristic is unknown, non-linear differential equations of motion of nonideal gyroscopic rigid rotary system with nonlinear cubic damping and nonlinear stiffness of the elastic support turn out to be numerically unsolvable.•In this case, the method uses the motor performance characteristic expression found from the frequency equation of forced stationary oscillations based on assumption that the angular acceleration is many times less than the square of the angular speed of rotation, replacing the stationary rotation angular speed with the shaft rotation angle derivative.•The method correctness is evidenced by a good consistency between the rotor motion equation numerical solution results and the analytical solution results, and by the nonlinear cubic damping of the shaft angular coordinate oscillograms obtained by direct simulation, as well as by comparison with the results of numerical simulation for a straight-line DC motor performance characteristic.•The method limitations are that it is used for the first approximation and weak nonlinear oscillations in the resonance region, where the shaft rotation speed is of the order of the oscillating system natural frequency.

Vortex induced vibration energy harvesting using magnetically coupled broadband circular-array piezoelectric patch: Modelling, parametric study, and experiments

Piezoelectric composites have become increasingly important in energy harvesting from vibration and, more recently, flow-induced vibration. The compatibility of piezoelectric devices has allowed tuning harvesters to the system's natural frequency to maximize the energy harvesting performance. However, the narrowband characteristic in piezoelectric composites is susceptible to efficiency losses when there are changes in the ambient surrounding. Thus, magnetic coupling introduces nonlinearity and can increase the broadband energy harvesting performance to account for slight variations. This paper proposes a piezoelectric energy harvester used in a pipe array to harvest oscillations from vortex-induced vibration. Adding a magnetic coupler introduces bandwidth enhancement characteristics that can accommodate slight changes in the freestream velocity. A computational simulation was used to compare different coupler shapes that can direct water flow toward the harvester array. The results show that an elliptical coupler works best and that a solid coupler has up to 50 % better performance than a hollow coupler. The experimental results showed that narrowband voltage output could reach up to 9 V in a pipe array. Furthermore, the broadband performance of magnetic coupling increased the bandwidth by up to 33 % in different orientations and distances.

Dynamic properties analysis of ball screw feed system based on improved hybrid model

Ball screws are widely used as feed systems for machine tools due to their high transmission efficiency and long service life. However, the large inertia force generated by the ball screw in high-speed motion will excite its flexible mode and cause system vibration. In the actual working condition, the dynamics of the system will also change with the nut position and workpiece mass. In order to further study the modeling method and dynamic properties of the ball screw feed system, this paper firstly improves the traditional hybrid model by introducing an energy loss factor to modify the hysteresis damping of the model. Further, the rationality of the improved model is verified by hammer test. Finally, the model was used to analyze the dynamic properties of the system and to obtain the influence law of workpiece mass and nut position on the system natural frequency.

Effect of scour erosion on mode shapes of a 5 MW monopile-supported offshore wind turbine

This paper investigates the influence of scour erosion on the mode shapes of the National Renewable Energy Laboratory (NREL) 5 MW offshore wind turbine (OWT) in varying soil conditions, where modes are derived from wind and wave-induced accelerations. A numerical model, which considers aerodynamic and hydrodynamic loading with soil-structure interaction (SSI) and servo-dynamics, is developed. The superstructure is modelled using OpenFAST and the foundation is modelled with SESAM, where SSI is incorporated using Winkler (p-y) springs. The foundation stiffness is subsequently integrated in OpenFAST using mudline interface springs. The influence of soil density, scour hole depth, and loading on the derived mode shapes is studied. Accelerations are extracted at nine locations along the tower. The first two system natural frequencies and mode shapes are estimated using frequency domain decomposition (FDD) applied to the acceleration time-histories for each scour depth and soil condition. Modal Assurance Criterion (MAC) values are calculated to evaluate the changes in mode shapes that occur due to scour. It is shown that the mode shapes present consistent changes as scour depth increases, and the second mode shape exhibits more sensitivity to scour. The sensitivity of mode shape changes to scour under variations in wind speed and measurement noise are also investigated.

Stiffness contributions of ballast in the context of dynamic analysis of short span railway bridges

Railway bridges are subjected to intensive dynamic loading. In structural calculations, the resulting response is estimated by dynamic calculations. As the basis of these calculations, the system's natural frequencies are to be numerically determined. Various studies have shown that a discrepancy between these measured natural frequencies and the numerical determination often occurs for short-frame bridges. In common practice, the ballast is assumed to be non-contributing to the global structural stiffness. In some cases, rheological models are developed to consider possible ballast stiffnesses based on small-scale model tests. The resulting spring-damper models are based on empirical values and are not directly related to the material behaviour of ballast.This paper discusses the ballast stiffness at the material level using relationships known from soil dynamics. The research shows that the material properties of the ballast can be determined in terms of stress and shear strain dependence. An equivalent ballast stiffness can be determined and considered in dynamic calculations for a known stress state and corresponding superstructure accelerations. The paper presents an approach for determining vertical and horizontal ballast stiffness according to shear strains and superstructure accelerations.This developed approach is validated by experimental tests using a small-scale bending beam and accompanying numerical calculations.Furthermore, a parameter study shows that the stiffness of the ballast does not contribute significantly to the global bending stiffness. Thus, the first bending natural frequency is hardly influenced by the ballast track.

DEVELOPMENT OF FINITE ELEMENT ANALYSIS PROGRAM FOR ROTORDYNAMIC PREDICTION OF SINGLE AND MULTI DISK ROTOR MODEL

The rotating equipment is a crucial part of system performance reliability. Vibration, as a characteristic of an oscillating body, is a fundamental parameter to determine the dynamic behaviour of the rotor system. The dynamic characteristics include natural frequency, critical speed, mode shape, and vibration response. This paper discussed the application of rotordynamics analysis as a tool to numerically predict the dynamic behaviour of a rotor system. The aim is to determine a rotor system's natural frequency and critical speed by a finite element analysis (FEA) program. The selected method is validated using various analytical methods from other references with small discrepancies in the result. The calculation of FEA using the computational program; they applied the developed program to describe the dynamic characteristic of single and multi-disk rotors. The developed works provided comprehensive results about a rotor system's natural frequency, mode shape, and critical speed and predicted the vibration response due to unbalance. However, other types of rotors, such as coaxial dual rotors, can be further investigated to make the program capable of the general purpose of the rotor. Validation with the real case problem could be an interesting investigation to satisfy the result of developed works.

Modeling Flexible Multi-Body Systems Within the Udwadia–Kalaba Framework, a Lumped Parameter Approach

Abstract It is common practice within multi-body dynamic (MBD) modeling to assume that individual bodies are rigid however this can be an oversimplification especially when slender bodies are present resulting in inaccurate estimations of the system's natural frequencies and overall behavior. To address this shortfall, we extend the Udwadia–Kalaba (U–K) MBD formulation in this paper to model flexible multibody systems for purposes of exploring system natural frequencies. To model the flexibility, a lumped parameter approach is proposed, which in this work idealizes a flexible beam as a series of discrete rigid elements connected by torsional springs. In the U–K formulation, a mechanical system can also be discretized into rigid elements and adapted. This is viewed as a benefit for incorporating a lumped parameter approach within the U–K formulation to model flexible multibody systems. A flexible crank-slider mechanism is introduced and modeled within the Udwadia–Kalaba formulation to capture the dynamics of flexibility through linkage compliance. The model is validated against an alternatively formulated MBD model and system natural frequencies and mode shapes numerically predicted. Results of the study show the effectiveness and potential of extending the application of the Udwadia–Kalaba formulation by using a lumped parameter approach to dynamically model flexible multibody systems.

Cross-flow oscillations of a circular cylinder with mechanically coupled rotation

Flow-induced vibrations (FIV) of an elastically mounted circular cylinder are investigated by means of two-dimensional simulations. A mechanical coupling between cross-flow translation and rotation provides a single degree-of-freedom system in which the coupled rotational oscillations affect the fluid–structure dynamics. The structural response of this system is investigated exploring the design space spanned by reduced velocity, coupling radius and phase density ratio. The kinematic coupling introduces the rotation-induced shear layer modifications, as well as an equivalent inertia effect connected to the coupling force. Such a computational campaign is carried out by means of direct numerical simulations with immersed boundary forcing at a Reynolds number equal to 100. The investigated system exhibits the wake-body synchronisation features typical of lock-in for non-rotating cylinders. However, the kinematic coupling provides a novel FIV scenario, in which the oscillation amplitude is magnified in the locked configurations with respect to the forced rotation case. Furthermore, it is found that there a significant widening of the reduced velocity domain where the lock-in condition takes place. In view of the proposed analyses, it is determined that the coupled rotation guarantees the phase alignment between lift and displacement necessary to sustain the lock-in condition, making the oscillation amplitude grow indefinitely with the reduced velocity. This is inherently achieved due to the rotational shear layer and the added mass contribution, which prevent the exact match between oscillation frequency and system natural frequency in vacuum. The outcomes of this study might potentially lead to an innovative water energy harvester offering larger power outputs and extended optimal operating regions.

A data-driven method for real-time compaction quality evaluation of a cement-stabilized base layer

Adequate and uniform compaction is essential for the safety and durability of a pavement structure. The current compaction quality control, which relies on limited spot tests data, cannot provide timely feedback about compaction quality. The main objective of this paper is to develop a novel data-driven method for compaction quality assessment of cement-stabilized base in a real-time manner. The basic idea is to consider the drum of the vibratory roller together with the underlying cement-stabilized base layer as a spring-mass-dashpot system. The compaction level of the base is connected to the stiffness of the system and to the corresponding natural frequencies. In this study, the vibration of the drum was monitored during operation. The fundamental mode natural frequency of the system at different compaction levels of the base was identified by the fast Bayesian fast Fourier transform method. The results indicate that the natural frequency of the vibratory system has a positive correlation with the degree of compaction (DOC) of the underlying cement-stabilized base layer. Therefore, a novel and practical method for on-site compaction quality assessment utilizing measured natural frequency is proposed in this paper. A data-driven classification method based on support vector machine (SVM) was developed to realize the compaction quality evaluation. The classification results show that the proposed method can recognize signal patterns corresponding to different DOCs. The proposed method combines vibration theory and machine learning to provide a technical reference for the research of continuous compaction control with vibratory rollers.

Influence of soil properties on the shift in natural frequencies of a monopile-supported 5MW offshore wind turbine under scour

This paper studies the change in the natural frequencies of the NREL 5MW monopile-supported offshore wind turbine (OWT) due to scour. The natural frequencies are derived from accelerations arising under wind and wave loads, for a range of different soil properties. A model of the wind turbine is developed using OpenFAST and SESAM software which considers aerodynamic and hydrodynamic loading with soil-structure interaction. The superstructure behaviour is modelled using OpenFAST and the foundation behaviour is modelled with SESAM, where soil-structure interaction is incorporated using p-y springs. The influence of soil density and scour hole depth on the obtained natural frequencies are studied. Accelerations are extracted at nine locations along the tower resulting from applied wind and wave loads. The first two system natural frequencies are estimated using frequency domain decomposition (FDD) applied to the acceleration time-histories for each scour depth and soil type. It is shown that the increase in scour depth around the foundation of the OWT leads to a decrease in natural frequencies when compared to the case where there is no scour.

Crack propagation mapping at component‐level fatigue testing by means of natural frequency tracking control

AbstractAdvancements seeking structural components' strength and weight reduction necessarily pass through fatigue testing. Understanding failure phenomena and crack evolution is a way of methodologically achieving such a goal. An own‐designed resonant fatigue test rig was used to study the behavior of three distinctly manufactured crankshaft batches. A control system and a resonant frequency estimator were developed to follow the overall stiffness reduction with crack advancement. The control algorithm followed the frequency decay to compensate for the input load. The control strategies were validated by finding the system's natural frequency at the beginning of the test and maintaining a constant nominal load. Fractographies made with failed and non‐failed specimens revealed the connection between the expected physical results and the natural frequency shift evolution. The implemented logic enables crack advancement tracking and failure determination. The developments here can be overflown to different resonance fatigue systems to characterize their endurance performance.

A dynamic substructuring technique for coupling and decoupling structure-cavity systems

Flexible structures adjacent to acoustic cavities are commonly encountered. In some instances, coupling between the structure and cavity causes non-negligible changes to the system frequency response. Here, we describe the use of modern dynamic substructuring techniques in coupling and decoupling structure-cavity systems. As input, free-interface in vacuo natural frequencies and modes of the structure are used along with an uncoupled modal representation of the cavity. In the acoustic model, the fluid-structure interface is typically modeled as pressure release, though faster frequency convergence is possible by enriching the representation with rigid-wall modes. The subsystems are assembled in modal coordinates with continuity at the interface enforced via Lagrange multipliers. This approach circumvents the calculation of coupling coefficients, and results in real-valued system natural frequencies and modes; however, the approach presents challenges when large numbers of interface degrees of freedom are present. Attempts to overcome these challenges with interface reduction techniques are described. Finally, this research led to the derivation of a formula approximating the in vacuo natural frequencies of test structures that are well coupled to adjacent acoustic cavities. The formula was successfully applied to an air-filled cylindrical test structure for which the frequency response was confounded by structure-cavity coupling.

Effect of Preload on the Vibrations of EHL Angular Contact Ball Bearings: Theoretical and Experimental Results

The vibrations of a shaft in rotary mechanical systems supported by angular contact ball bearings are investigated theoretically and experimentally for various preloads in this paper. In the theoretical part of the study, a dynamic bearing model is presented, a rigid shaft supported by EHL angular contact bearing has been modelled as 5 DoF. Non-linear equations of motion are solved numerically by the Runge–Kutta method. In the second part of the study, an experimental setup that enables performing different operating cases has been designed to validate the theoretical results. Theoretical and experimental data are investigated and compared in both time and frequency domains and the results are compared. It is observed from both the theoretical and experimental studies that preload has a significant effect on the vibration behaviour. Results show that the increase in preload reduces the amplitude of the variable compliance frequencies of bearing, the natural frequency of system is shifted to a higher value, and using signal processing with an envelope spectrum gives better results in spectral analysis; small deviations occur between the theoretical and the experimental data due to modulation and noise from machine elements such as gear, motor, misalignment, waviness, etc. Therefore, the presented dynamic bearing model can be used with a reasonable accuracy to examine effect of preload on the vibration of shaft-bearing system.

Sensitivity of tidal hydrodynamics to varying bathymetric configurations in a multi‐inlet rapidly eroding salt marsh system: A numerical study

AbstractWe describe the development of a high‐resolution, two‐dimensional hydrodynamic model for a multi‐inlet rapidly eroding tidal wetland on the western shore of Delaware Bay, using the finite‐volume, primitive equation community ocean model (FVCOM). Topo‐bathymetric surveys, together with water surface and current velocity measurements during calm and stormy conditions, have been conducted to support model validation. The tested model is then used to quantify the tide‐induced residual transport and asymmetry at major inlet entrances to determine the governing hydrodynamics. We chose a skewness method to calculate the tidal asymmetry and serve as a proxy for sediment transport estimates. The effects of the dredging of an artificial entrance channel and progressive channel deepening in shifting wetland hydrodynamics are shown by developing a scenario analysis. Model results show that the artificially dredged channel has altered the volume exchange at other inlet entrances and increased the net seaward export. The changes in the characteristic frequency of the frictional dissipation in the channel and the system's natural frequency are investigated using a simple ocean–inlet–bay analytical model. Subsequently, we have compared the channel friction scale to the inertia scale and observed that the new connection and gradual channel deepening reduce the overall frictional dominance. Ultimately, the study has shown how the short‐ and long‐term channel bathymetry changes, mainly the artificially dredged channel and progressive channel deepening, can affect the connected system's net circulation and trigger internal marsh erosion.

Analysis of design parameters of round-window stimulating type electromagnetic transducer by a nonlinear lumped parameter model of implanted human ear.

Round-window stimulating transducer is a new solution to treat mixed hearing loss. To uncover the factors affecting the round-window stimulation's performance, we investigated the influence of four main design parameters of round-window stimulating type electromagnetic transducer. Firstly, we constructed a human ear nonlinear lumped parameter model and confirmed its validity by comparing the stapes responses predicted by the model with the experimental data. Following this, an electromagnetic transducer's mechanical model, which simulates the floating mass transducer, was built and coupled to the human ear model; thereby, we established a nonlinear lumped parameter model of implanted human ear under round-window stimulation and verified its reliability. Finally, based on this model, the influences of the four main design parameters, i.e., the excitation voltage, the electromechanical coupling coefficient, the support stiffness, and the preload force, were analyzed. The results show that the change of excitation voltage does not alter the system's natural frequency. Chaotic motion occurs when the electromechanical coupling coefficient is small. Meanwhile, the stapes displacement appears to increase firstly and then decrease with the increase of the electromechanical coupling coefficient. The increase of the support stiffness enlarges the resonance frequency of the stapes displacement and reduces the stapes displacement near the resonance frequency, deteriorating the transducer's hearing compensation at low frequency. The preload force can improve the transducer's hearing compensation performance in mid-high frequency region.

Optimization based Feedforward mitigation of Elastic Flexure of Distributed Parameter System due to Reference Tracking

If a large scale structure performs a rigid body motion, this motion is superposed by an elastic flexure due to its inherent elasticity. In order to mitigate the elastic motion feedforward and feedback laws can be applied. By minimizing the L2-norm of the elastic flexure subject to a reference trajectory and the elastic system description we derive a feedforward signal that attenuates elastic motions of the structure. A comparison to simplified solution approaches shows a major improvement for reference signals in the range of the system's natural frequencies.

Vibration control of submerged floating tunnel in waves and earthquakes through tuned mass damper

In this study, a passive tuned mass damper (TMD) is investigated to dampen resonant motions of a submerged floating tunnel (SFT) in waves and earthquakes. TMD is adopted to control resonant lateral motion. A time-domain dynamics simulation model that considers the elasticity of the tunnel and mooring lines is first established, which is based on the lumped mass method and Morison equation, and the SFT-TMD interaction is considered through springs and dampers. Next, by using the harmony search (HS) algorithm, TMD's spring and damping coefficients are optimized; the dynamics simulations under white noise seismic excitations are continuously carried out with updated spring and damping coefficients produced by HS rules until the designed maximum iteration number. The influence of TMD's mass is individually investigated through a parametric study. After the optimization process is completed, the global performances of SFT with and without TMD are systematically evaluated in both waves and earthquakes. It is found that TMD plays a crucial role in controlling SFT vibrations when environmental loads are close to the system's fundamental lateral natural frequency under both wave and seismic excitations. TMD also enhances the comfort of passengers and reduces static and dynamic mooring tensions. • A time-domain dynamics simulation model is built for the interaction of a submerged floating tunnel and a tuned mass damper. • The harmony search algorithm shows good optimization performance for the tuning mass damper parameters. • A tuned mass damper effectively reduces resonant motions and mooring tensions under wave and seismic excitations.