Steady-state Displacement Research Articles

A Keyphasor-Free Traveling Waves Analysis Method Based on Blade Tip Timing and Experimental Validation

Abstract Blade Tip Timing (BTT) emerges as a non-intrusive, promising method for assessing blade vibration. Despite its potential, BTT's inherent measurement limitations often result in the under-sampling of individual blade data. Assuming that the bladed-disk assembly vibrates in nodal diameter modes, traveling wave analysis (TWA) can overcome the limitations of under-sampling and identify the nodal diameter modes. Traditional TWA methods rely on a keyphasor or once-per-revolution (OPR) sensor to provide shaft reference signals. However, such a sensor could be damaged during operation or challenging to install due to limited space. In this paper, a new keyphasor-free TWA (KF-TWA) method is introduced, which is capable of measuring the order and nodal diameter of nodal diameter vibrations with as few as two sensors. Theoretical derivations reveal the method's resilience to the steady-state displacement of blades. The feasibility and effectiveness of the KF-TWA method are validated through numerical experiments and compressor tests. In the numerical experiments, blade detuning and nodal diameter vibration responses are simulated, leading to the development of a criterion for identifying nodal diameter vibrations. A compressor test rig, equipped with a 64-keyphasor disk and a sufficient number of circumferentially distributed sensors, is designed to provide blade vibration benchmark. In the experiments, the KF-TWA method successfully identified nodal diameter vibrations with amplitudes as low as 0.014 mm, consistently yielding identical results across multiple sensor pairs, which demonstrates a higher robustness compared to the traditional TWA method.

Free and Forced Vibration Analysis of Carbon/Glass Hybrid Composite Laminated Plates Under Arbitrary Boundary Conditions

This paper presents the theoretical investigations on the free and forced vibration behaviours of carbon/glass hybrid composite laminated plates with arbitrary boundary conditions. The unknown allowable displacement functions of the physical middle surface are expressed in terms of standard cosine Fourier series and sinusoidal auxiliary functions to ensure the continuity of the displacement functions and their derivatives at the structural boundaries. Arbitrary boundary conditions are achieved through the introduction of an artificial spring technique. The first shear deformation theory and Lagrange equations are utilized to derive the energy expression, and the eigenvalue equations associated with free and forced vibration are obtained by Rayleigh-Ritz variational operations. Subsequently, these equations are then solved to determine the natural frequency, mode of vibration, and the steady-state displacement response under forced excitation. The new results are compared with those from references and finite element methods to verify the convergence, accuracy and efficiency of the analytical method. The effects of hybrid ratios, stacking sequences, lamination schemes, fibre orientation, boundary conditions and excitation force on the free and forced vibration behaviours of the carbon/glass hybrid composite laminated plates are analyzed in detail.

Inertial active harmonic particle with memory induced spreading by viscoelastic suspension.

We investigate the self-propulsion of an inertial active particle confined in a two-dimensional harmonic trap. The particle is suspended in a non-Newtonian or viscoelastic suspension with a friction kernel that decays exponentially with a time constant characterizing the memory timescale or transient elasticity of the medium. By solving the associated non-Markovian dynamics, we identify two regimes in parameter space distinguishing the oscillatory and non-oscillatory behavior of the particle motion. By simulating the particle trajectories and exactly calculating the steady-state probability distribution functions and mean square displacement; interestingly, we observe that with an increase in the memory time scale, the effective temperature of the environment increases. As a consequence, the particle becomes energetic and spread away from the center, covering larger space inside the confinement. On the other hand, with an increase in the duration of the activity, the particle becomes trapped by the harmonic confinement.

Dynamic characteristics of a relief valve captured by a combination of fluid–solid coupling and feedforward neural network (FNN)

Considering multidisciplinary characteristics of thin plate vibration, fluid–solid coupling, and other aspects of a relief valve controlled by annular thin plates, a dynamic finite element (FE) model in view of fluid–solid coupling is firstly established for capturing relationships between dynamic characteristics of crucial indexes and partial working conditions. Secondly, the partial dataset of FE model under different conditions is statistically analyzed, and it will be utilized to train the feedforward neural network (FNN) model. The training process of FNN could be completed if results drawn from the FNN model are highly consistent with those of the FE model. Thirdly, dynamic characteristics under more conditions will be predicted through such a trained model, and dynamic behaviors from the FE model for same conditions of the FNN model are also obtained. Finally, comparing with results from the FE model, the maximum absolute error of steady-state displacement from the FNN is 0.0052 mm in an instance, thus verifying the rationality of this combined method. Consequently, such a combination of the FE model and the FNN model presents high accuracy and avoids repeated calculations of FE model with long times.

Proportional-retarded control of a quasi-zero-stiffness vibration isolator

This paper explores the control design and performance analysis of an active quasi-zero-stiffness vibration isolator utilizing the proportional-retarded (PR) control architecture. The PR architecture replaces the conventional “derivative” term in the proportional-derivative (or displacement-velocity) controller with the novel “retarded (time-delayed)” term, which is a new and effective way of active vibration suppression. Comprehensive gain-delay conjoint regulation strategies are developed for the strengthened isolation performance. For impact excitations, a complete stability analysis is conducted near the equilibrium position. Furthermore, a parameterized dominant pole placement algorithm is proposed to optimize the transient response of the PR-actuated isolator. Focusing on the rectangular pulse and long-duration step impacts, the algorithm adjusts the dominant poles to optimize the transient performance metrics, including the maximum displacement overshoot, steady-state displacement overshoot, and settling time. Regarding harmonic excitations, the delay-coupled effect is scrutinized for better low-frequency isolation performance, including the resonance frequency, nonlinear steady-state amplitude, multiple steady-state bands and nonlinearity elimination, equivalent active damping and stiffness, and absolute displacement transmissibility. In the case of random excitations, we demonstrate the superiority of the PR structure over the passive system.

A novel interval dynamic topology optimization methodology of piezoelectric structures under reliable active control

In this paper, considering the influence of uncertainties in piezoelectric intelligent structures with active vibration control law, a robust topology optimization method to reduce the sensitivity of structural dynamic flexibility to uncertainty factors for piezoelectric intelligent structures is proposed. The effect of active control can be equivalent to the damping. Considering the influence of uncertainty, the structural dynamic response of the system will be within the interval range. Due to the monotonicity of the steady-state displacement response on uncertain parameters and the monotonicity of dynamic flexibility on uncertain parameters are both difficult to determine, a more accurate upper and lower limit is obtained using the collocation method. With the constraints of reliability and volume, the robustness index is defined based on the dynamic flexibility of the structure and defined as the objective function to establish the robust topology optimization model. In terms of optimization solution, the reliability index is transformed into the constraint translation distance. The sensitivity of constraint functions and objective functions to design variables is derived by employing the chain derivative method and the adjoint vector method. Finally, two examples are used to verify the effectiveness of the proposed method.

Initial Vacancy-Dependent High-Temperature Creep Behavior of Nanocrystalline Ni by Molecular Dynamics Simulation

Nanocrystalline metals possessing excellent mechanical strength have great potential to replace traditional metal materials as structural materials, but their poor resistance to creep deformation seriously restricts their engineering applications at high temperatures. The high-temperature creep behavior of nanocrystalline Ni with different volume fractions of initial vacancies ranging from 0% to 10% was studied systematically by molecular dynamics simulation in this study. The results showed that the steady-state creep displacement first increased and then decreased with increasing initial vacancy concentration, reaching the maximum when the initial vacancy concentration was 6%. The microstructural characteristics, such as quantity increment and distribution of the vacancies, the number and types of dislocations, and shear strain distribution during creeping, were analyzed in detail. The deformation-induced vacancies formed at the grain boundary (GB) in the initial creep stage, and their variation trend with the initial vacancy concentration was consistent with that of the creep displacement, indicating that the initial vacancy-dependent high-temperature creep behavior of nanocrystalline Ni was mainly determined by the rapidly increasing number of vacancies at the GB in the initial creep stage. Afterwards, the deformation-induced, vacancy-assisted 1/6{112} Shockley partial dislocation activities dominated the creep deformation of nanocrystalline Ni in the steady-state creep stage. The results can provide theoretical support for expanding the application of nanocrystalline metals from the perspective of crystal defect engineering.

Dynamical crossovers and correlations in a harmonic chain of active particles.

We explore the dynamics of a tracer in an active particle harmonic chain, investigating the influence of interactions. Our analysis involves calculating mean-squared displacements (MSDs) and space-time correlations through Green's function techniques and numerical simulations. Depending on chain characteristics, i.e., different time scales determined by interaction stiffness and persistence of activity, tagged-particle MSDs exhibit ballistic, diffusive, and single-file diffusion (SFD) scaling over time, with crossovers explained by our analytic expressions. Our results reveal transitions in bulk particle displacement distributions from an early-time bimodal to late-time Gaussian, passing through regimes of unimodal distributions with finite support and negative excess kurtosis and longer-tailed distributions with positive excess kurtosis. The distributions exhibit data collapse, aligning with ballistic, diffusive, and SFD scaling in the appropriate time regimes. However, at much longer times, the distributions become Gaussian. Finally, we derive analytic expressions for steady-state static and dynamic two-point displacement correlations. We verify these from simulations and highlight the differences from the equilibrium results.

A New Approach for Measuring Viscoelastic Properties of Soft Materials Using the Dynamic Response of a Spherical Object Placed at the Sample Interface

BackgroundThere are several techniques to characterize the mechanical properties of soft materials, such as the indentation method and the method based on the application of a spherical object placed inside the sample. The indentation systems usually yield the elastic properties of materials and their mathematical models do not consider the inertia of the sample involved in motion and radiation damping, while placing an object inside the sample is not practical and this procedure can alter the mechanical properties of the sample for the method based on the application of a bubble/sphere placed inside the sample.ObjectiveA new approach for the identification of the viscoelastic properties of soft materials using the dynamic response of a spherical object placed at the sample interface was proposed.MethodsThe spherical object placed at the sample interface was pressed using an electromagnet and the dynamic response of the spherical object was tracked using a high-speed camera, while the dynamic response of the spherical object placed at the sample interface was estimated using a comprehensive analytical model. The effects of the shear modulus, viscosity, Poisson’s ratio and density of the soft sample, the radius and density of the spherical object and the damping due to radiation were considered in this mathematical model. The shear modulus and viscosity of the soft sample were determined by matching the experimentally identified and theoretically estimated responses of the spherical object.ResultsThe shear moduli and viscosities of the three phantoms with the gelatin mass ratios of 0.20, 0.25 and 0.29 were measured to be 3450, 4300 and 4950 Pa and 12.5, 14.0 and 15.0 Pa⋅s, respectively. The shear modulus and viscosity of the phantom increases as the gelatin mass ratio increases. The frequency of oscillations of the hemisphere placed at the phantom interface increases as the gelatin mass ratio increases due to stiffness increase.ConclusionsAfter matching the experimental and theoretical steady-state displacements and amplitudes of oscillations of the hemisphere at the sample interface, the comparison of the experimentally identified and theoretically predicted frequency of oscillations further confirmed the identified material properties of the samples. The approach presented here is expected to provide valuable information on material properties in biomedical and industrial applications.

Broadband vibration attenuation in nonlinear meta-structures with magnet coupling mechanism: Theory and experiments

Nonlinear metamaterial structures may provide extensive applications in low-frequency and broadband vibration attenuation. Recently, there has been growing interest in nonlinear resonator design towards vibration attenuation of metamaterial structures. This work is focused on investigating the working mechanism of meta-beam by periodically attaching nonlinear coupling multi-frequency resonators for vibration suppression. The Harmonic Balance Method (HBM) and the energy method are first used to obtain theoretical expressions for the edge frequencies of the coupling multi-frequency band gaps. Additionally, the nonlinear coupling 2-DOF (degree of freedom) resonator attached to the primary beam is experimentally designed as a double cantilevered beam with different masses attached to both ends and whose coupling force is achieved by magnet masses on the same plane. The nonlinear coupling behaviors of the frequency response obtained by experimental test agree well with the theoretical analysis of steady-state displacements at different excitation levels. Finally, the dispersion properties of coupling band gaps and the nonlinear dynamic behavior of a finite-sized meta-beam are also investigated. Results show that compared to its rival linear metamaterial structures, the proposed nonlinear meta-beam can obtain a lower-frequency band gap. Furthermore, the nonlinearity of the proposed meta-beam can form broadband vibration attenuation by large excitation level.

Optimized design and experimental study of a macroscopic mirror to achieve linear amplification of optical force-induced displacement.

A new thin plane mirror with an Archimedes spiral structure (Archimedes-structure thin plane mirror - ATPM) that implements an elastic support boundary is proposed in this study. An optimal structure of ATPM is developed to achieve a linear displacement response with respect to optical forces. The displacement response of the optimized ATPM is analyzed by considering the combined effects of optical force and gravity. The distribution of the optical force density is calculated based on a tilted Gaussian laser beam. Experimental results demonstrate that the optimized ATPM can produce a steady-state displacement of 24.18 nm on average in a normal-gravity environment when subjected to an average optical force of 132.17 nN. When the optical force exceeds 133 nN, the nonlinearity of the displacement response of the optimized ATPM is less than 6.28%. An amplification of the optical force-induced displacement is achieved by more than 15 times compared with that for an unstructured mirror of the same size. The results of this study can assist the development of a miniaturized macroscale optical force platform based on an ATPM for practical applications including the in-situ laser power measurement and nN level force source in the atomic and close-to-atomic scale manufacturing.

Numerical assessment of thermal performance of W/Cu monoblock during steady-state operation and plasma vertical displacement events under ITER-like conditions

This paper employs a two-dimensional conduction model to assess the spatiotemporally thermal performance of ITER W/Cu monoblock under ITER-like conditions. The model considers the heat conduction processes including the evaporation, radiation and melting of both W armor and Cu heat sink materials under a heat load either during steady-state discharge or during plasma vertical displacement event (VDE). Moreover, the model employs coupling cooling water condition instead of the fixed cooling temperature at bottom in authors’ previous work. For the steady-state case, the heat flux distribution on the target tile, the input data to the model, was specifically evaluated by a particle-in-cell code to take into account its geometrical effect. For the VDE case, the heat flux distribution was simply assumed to be uniform on the tile surface. The simulated results obtained from the heat conduction model indicate that cooling water conditions have important influence on the thermal performance of W/Cu monoblock, especially for Cu heat sink materials during steady state. In the case of VDE, the baffle tile surface would undergo intense vaporization and melting, while whether Cu heat sink materials melted was determined by the combination of the exposed heat load and its duration. Based on typical VDE heat loads, detailed analysis is presented herein.PACS: 52.55.Fa, 52.55.Rk, 52.40.Hf, 52.65.-y

Parameters optimization of three-element dynamic vibration absorber with inerter and grounded stiffness

Improving the control performance of dynamic vibration absorbers has recently been effective by introducing a grounded negative stiffness device. However, the negative stiffness structure is unstable and difficult to achieve in engineering practice , and its major drawback is that it amplif ies the vibration response of the primary system at low frequency region. Meanwhile, some mechanical devices can be combined to make the DVA work even better with a grounded positive stiffness. For this purpose, this paper combines for the first time the control effect of the inerter device and grounded positive stiffness into a three-element DVA model in order to better improve vibration reduction of an undamped primary system under excitation. First, the dynamic equation of motion of the system is written according to Newton ’s second law. Then, the steady-state displacement response of the primary system under harmonic excitation is calculated. In order to minimize the resonant response of the primary system around its natural frequency, the extended fixed point theory is applied. Thus, the optimized parameters such as the tuning frequency ratio, the stiffness ratio , and the approximate damping ratio are determined as a function of mass ratio and inerter – mass ratio. From the results analysis, it was found that the inerter – mass ratio has a better working range to guarantee the stability of the coupled system. Then , study on the effect of inerter – mass ratio on the primary system response is carried out. It can be seen that increasing the inerter – mass ratio in the optimal working range can reduce the response of the primary system beyond its uncontrolled static response. However, it is necessary to avoid the situation where the inerter – mass ratio is very large because it can lead to unrealistic optimal parameters. Finally, comparison with other DVA models is show n under harmonic and random excitation of the primary system. It is found that the proposed DVA model in this paper has high control performance and can be used in many engineering practice s .

Reliability evaluation method of vibration isolation performance of nonlinear isolator

Nonlinear isolators are widely used to reduce harmful vibrations due to their broad isolation bandwidth. The vibration isolation performance of nonlinear isolators can be quantitatively evaluated by the transmitted force or displacement. However, uncertainties exist in the excitation and structural factors of nonlinear isolators, which cause the dispersion of displacement and force transmissibility, and affect the vibration isolation performance of nonlinear isolators. In this paper, a reliability evaluation method of vibration isolation performance of nonlinear isolator is proposed by considering the influence of the uncertainties. First, quasi-zero stiffness characteristics of the nonlinear isolator is analyzed and a dynamic model is established to study the isolation performance of the nonlinear isolator. Then, the types, sources, and influence degrees of the uncertainty of dimensional, structural, and excitation parameters on the nonlinear isolator are studied, and sensitive parameters are extracted. Furthermore, the reliability model of the nonlinear isolator under the action of sensitive parameters is proposed by considering the amplitude of the steady-state displacement response of the nonlinear isolator as the evaluation index of vibration isolation performance. Finally, the reliability of the nonlinear isolator is evaluated using an advanced surrogate model method based on stochastic configuration network. An experimental platform is built to verify the validity of the reliability evaluation method. The proposed method can provide potential guidance for the reliability evaluation of the isolation performance of complex nonlinear isolators.

Development of a Novel Flexure-Based XY Platform Using Single Bending Hybrid Piezoelectric Actuator

In this article, a novel flexure-based XY platform using a bending hybrid piezoelectric actuator was proposed, which was inspired by a classical piezoelectric stack-actuated platform. First, the stack-actuated platform was designed, the configuration and working principle of the bending hybrid piezo-actuated platform were illustrated. Then, theoretical analyses were conducted to design the piezo-actuated platform, the finite element method was employed for the static and dynamic analyses, and the prototypes of the two platforms were fabricated. Eventually, a series of tests were carried out to verify the theoretical and simulated results, and to investigate and compare their output characteristics. The experimental results revealed that the proposed piezo-actuated platform could achieve a motion range of 16.48 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i> m × 16.84 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</i> m and the coupling error of less than 0.59%. Furthermore, the hysteresis ratios were about 8.3% and 8.2%, the first-order bending frequencies were 2162 and 2303 Hz in two axes, the displacement resolution was about 4.31 nm, and the steady-state displacement deviations of the point-to-point positioning control both are within ±30 nm in two axes. Compared with the traditional stack-actuated platform, the piezo-actuated platform not only had positive and negative output displacement, but also possessed superior performance, including low cross-coupling, low hysteresis, and high resolution, which was expected to be used in nanopositioning and micro/nanomanipulation.

Theory of sphere motions in viscous fluids including elasticity and compressibility

We study the motions of an elastic sphere and a compressible fluid sphere suspended in a compressible fluid. To this end, we use a scheme of a vector representation for the velocity in hydrodynamics and for the displacement in elasticity. First, we calculate the steady-state elastic displacement of a solid sphere under a gravity and a surface-tension gradient. Second, we examine the finite-size effects in a spherical container and find bulk acoustic resonance induced by an oscillating solid sphere. Third, applying periodic forces, we calculate the displacement, the velocity field, and the frequency-dependent friction constant for an elastic sphere and a compressible fluid sphere. We find complex acoustic effects sensitively depending on the frequency.

A hysteretic model of localized frictional contacts with instrumental stiffness

The steady-state dynamic response of a single-degree-of-freedom system comprising both a hysteretic element and a spring is considered. The Hertz–Cattaneo–Mindlin theoretical framework for modeling of local tangential contact with friction is applied in conjunction with the Masing model of hysteresis to describe the hysteretic behavior of the multiple localized frictional contact interface. The steady-state tangential displacement amplitude of a rigid body under harmonic tangential force excitation is approximately determined by means of the equivalent linearization technique, based on the harmonic balance principle. A special attention is paid to the evaluation of the frictional damping and the determination of the backbone curve of the Masing model from the dissipation-amplitude relation.

Nonlinear Dynamic Analysis of Bistable Piezoelectric Energy Harvester with a New-Type Dynamic Amplifier.

A distributed parametric mathematical model of a new-type dynamic magnifier for a bistable cantilever piezoelectric energy harvester is proposed by using the generalized Hamilton principle. The new-type dynamic magnifier consists of a two-spring-mass system, one is placed between the fixed end of the piezoelectric beam and the L-shaped frame, and the other is placed between the L-shaped frame and the base structure. We used the harmonic balance method to obtain the analytical expressions for the steady-state displacement, steady-state output voltage, and power amplitude of the system. The effect of the distance between the magnets, the spring stiffness ratio and mass ratio of the two dynamic magnifiers, and the load resistance on the performance of the harvester is investigated. Analytical results show that compared with the bistable piezoelectric energy harvester with a typical spring-mass dynamic magnifier, the proposed new-type energy harvester system with a two-spring-mass dynamic magnifier can provide higher output power over a broader frequency band, and increasing the mass ratio of the magnifier tip mass to the tip magnet can significantly increase the output power of the BPEH + TDM system. Properly choosing the stiffness ratio of the two dynamic amplifiers can obviously improve the harvested power of the piezoelectric energy harvester at a low excitation level.

Lateral contact yields longitudinal cohesion in active undulatory systems.

Many animals and robots move using undulatory motion of their bodies. When the bodies are in close proximity undulatory motion can lead to novel collective behavior such as gait synchronization, spatial reconfiguration, and clustering. Here we study the role of contact interactions between model undulatory swimmers: three-link robots in experiment and multilink swimmers in simulation. The undulatory gait of each swimmer is generated through a time-dependent sinusoidal-like waveform which has a fixed phase offset, ϕ. By varying the phase relationship between neighboring swimmers we seek to study how contact forces and planar configurations are governed by the phase difference between neighboring swimmers. We find that undulatory actuation in close proximity drives neighboring swimmers into planar equilibrium configurations that depend on the actuation phase difference. We propose a model for stable planar configurations of nearest-neighbor undulatory swimmers which we call the gait compatibility condition, which is the set of planar and phase configurations in which no collisions occur. Robotic experiments with two, three, and four swimmers exhibit good agreement with the compatibility model. To study the contact forces and the time-averaged equilibrium between undulatory systems we perform simulations. To probe the interaction potential between undulatory swimmers we apply a small force to each swimmer longitudinally to separate them from the compatible configuration and we measure their steady-state displacement. These studies reveal that undulatory swimmers in close proximity exhibit attractive longitudinal interaction forces that drive the swimmers from incompatible to compatible configurations. This system of undulatory swimmers provides new insight into active-matter systems which move through body undulation. In addition to the importance of velocity and orientation coherence in active-matter swarms, we demonstrate that undulatory phase coherence is also important for generating stable, cohesive group configurations.

Vibration attenuation of beam structure with intermediate support under harmonic excitation

The optimal position design of viscoelastic supports is studied for vibration attenuation of a beam under a harmonic excitation. By attaching the supports properly to the structure, the dynamic deflection of the steady-state response can be significantly suppressed. In this work, the derivative analysis of the dynamic deflection is first investigated with respect to the attachment location of a viscoelastic support. A closed-form formulation for the dynamic response sensitivity is gained readily by use of the discrete methodology. Then, a heuristic optimization algorithm is applied on the design sensitivity analysis to achieve the optimal solution of the supporting points to minimize the maximal steady-state displacement vibration of a beam structure. Finally, two typical examples are presented to demonstrate the validity and accuracy of the design sensitivity formulation developed. Numerical results show that the structural vibration deflections can be substantially reduced with optimization of the intermediate support positions.