Stationary Vibration Research Articles

Multi-shaker time-varying root mean square non-stationary random vibration environmental control test

The current multi-shaker random control methods concentrate on the simulation of stationary vibrations. This work presents an innovative technique for multi-shaker non-stationary random vibration environmental control test. Firstly, an amplitude modulation method is introduced to generate multi-input non-stationary random signals with time-varying root mean square values, whose kurtoses and moving RMS values are specified by amplitude modulation functions. Then, the generated signals are applied as excitation signals for the non-stationary random vibration environmental control test. The response signals of a linear vibration system under non-stationary excitations are derived in the frequency domain. The derived formulas for kurtoses and moving RMS values are also applicable to non-stationary response signals. Meanwhile, the stationary power spectral densities of response signals are put forward. When the amplitude modulation functions are zero mean Gaussian distributions, the power spectral densities of the response signals can be directly calculated. Ultimately, the control of kurtoses, moving RMS values and stationary power spectral densities of response non-stationary random signals is realized. Simulation and experimental examples both adopt a two-input two-output beam vibration system and the results prove the correctness and practicability of the proposed method. The presented control procedure is concise and efficient in simulating multi-shaker non-stationary random vibration environments.

Ritz Solutions of Stationary Random Vibrations for GPL-Reinforced FG Rectangular Plate

This paper aims to establish a comprehensive model for evaluating the response characteristics of functionally graded graphene platelet-reinforced composite (FG-GPLRC) rectangular plates under stationary random acceleration excitation. Utilizing the first-order shear deformation theory (FSDT) and Halpin–Tsai assumptions, a dynamic model for the rectangular plate is derived, followed by obtaining the variational solution using the Rayleigh–Ritz method. To describe displacement components associated with dynamic equations and boundary conditions, a spectral geometry method (SGM) is employed. Additionally, a pseudo-excitation method (PEM) is utilized for the analysis of stationary random vibration. The proposed calculation method is validated through convergence analysis and comparative evaluation with existing literature and finite element models. Furthermore, the study investigates the influence of various factors, such as boundary conditions, the number and size of functionally graded rectangular plate layers, GPL distribution types, and material parameters, on the stationary random response attributes of FG rectangular plates. This research contributes to a deeper understanding of the dynamic response of functionally graded materials in structural configurations and offers a valuable analytical approach for researchers in this field.

Speckle interference for naked-eye detection of vibrations

An affordable optical project for students is explained in this work. The experiment only requires a handheld laser illuminating a structure with stationary vibrations induced, but the interpretation of this phenomenon demands a strong understanding of optical interference and modal behavior of a bar. The operating principle employs the speckle phenomenon, which is a type of interference between coherent light incident on a rough surface and its reflection. This interference occurs in a luminous plane outside of the vibrating object and shows a granular structure due to the brightness fluctuations. Speckle is static but if the surface illuminated oscillates, the plane of interference moves from one side to the other, causing a streak pattern. Mode shapes of the object under examination can be naked eye detected from streak patterns. What is more, this phenomenon can be photographed using a manual focus digital camera. Therefore all the components of the procedure can be available (or acquired) by almost any physics laboratory, at any level, which would be unthinkable for optical experiments just a few years ago.

Modified time domain randomization technique for multi-shaker non-stationary non-Gaussian random vibration control

Traditional random vibration environmental tests simulate stationary Gaussian random vibration environments. In this work, the generation and closed-loop control of multi-shaker non-stationary non-Gaussian random vibration signals with the modified time domain randomization technique are studied. The amplitude modulation function is introduced to control the non-stationary and non-Gaussian characteristics that the root mean square value of the generated signal changes with time. The kurtosis relationship between the one frame pseudo-random signal and consecutive true random signal is derived theoretically. It is found that the kurtosis of the consecutive non-stationary random signal is determined by the amplitude modulation function and kurtosis of the pseudo-random signal. The standard deviation of the time-varying root mean square reflects the non-stationary characteristics of the generated signal. A control strategy for multi-shaker non-stationary non-Gaussian random vibration environmental test is proposed. The non-stationary and non-Gaussian characteristics are extracted from the measured response signals so that the power spectral densities, kurtoses and non-stationary characteristics can be controlled simultaneously. The results from the simulation examples and multi-shaker vibration experiments prove the feasibility and practicability of the proposed non-stationary non-Gaussian random signal generation method and control strategy.

Different Boundary Problems Governed by the Dynamic and Stationary Operator Nonlinear Vibration of the Plates

Bouzeghaya Fouzia, Merouani Boubakeur In this paper, we propose to study some nonlinear boundary problems for the dynamically modified operator by adding a viscosity term -αΔu'' to the nonlinear vibrations of the plates. The field of application for vibrating plates is extensive. To meet user needs, we have considered the geometric shape, the density of the material constituting the plate, the plate thickness, and Poisson's ratio. Once the problems have been posed, our approach then consists of transforming them into nonlinear problems of the hyperbolic type. In this work, we study six boundary value problems and we prove for each problem an existence and uniqueness theorem. Finally, we demonstrate the existence of a solution to the stationary problem using a variant of Brouwer's fixed point theorem.

The high/low frequency balance drives tactile perception of noisy vibrations.

Noisy vibrotactile signals transmitted during tactile explorations of an object provide precious information on the nature of its surface. Understanding the link between signal properties and how they are interpreted by the tactile sensory system remains challenging. In this paper, we investigated human perception of broadband, stationary vibrations recorded during exploration of textures and reproduced using a vibrotactile actuator. Since intensity is a well-established perceptual attribute, we here focused on the relevance of the spectral content. The stimuli were first equalized in perceived intensity and subsequently used to identify the most salient spectral features using dissimilarity estimations between pairs of successive vibration. Based on dimensionally reduced spectral representations, models of dissimilarity ratings showed that the balance between low and high frequencies was the most important cue. Formal validation of this result was achieved through a Mushra experiment, in which participants assessed the fidelity of resynthesized vibrations with various distorted frequency balances. These findings offer valuable insights into human vibrotactile perception and establish a computational framework for analyzing vibrations as humans do. Moreover, they pave the way for signal synthesis and compression based on sparse representations, holding significance for applications involving complex vibratory feedback.

Influence of pile length on dynamic performance of monopile wind turbine

The influence of pile configuration on safety of wind turbines has been attracting large amounts of attention recently. On the one side, the cost for the foundation accounts for 35 % of the total cost of monopile supported wind turbines (MWTs). On the other side, the working environment of wind turbines brings complex loads to monopiles. In this paper, a series of wind tunnel tests are performed with scaled model under different ground and operation conditions. The test results show that pile length factor (PLF) affect nacelle displacement and maximum pile moment negatively. Top acceleration confirmed a stationary random vibration. The natural frequency of model MWTs increases 11.6 % (dense ground) and 10.1 % (loose ground) when PLF increase from 0.264 to 0.362. While damping ratio decreases 1.447 % (dense ground) and 1.493 % (loose ground). These test results can be up scaled to estimate prototype's dynamic properties. According to test results, some numerical fitting surfaces including d-Ω-PLF, f-Ω-PLF and ξ-Ω-PLF have been established to check unfavorable working conditions, which can be used to guide the design of piles to support MWTs.

Stationary Stochastic Response Analysis of the FG-GPLRC Irregular Quadrilateral Plate Based on the Chebyshev–Ritz Method

The stationary random response characteristics of irregular quadrilateral plates under the base acceleration excitation are investigated in this paper. The computational model chosen in this paper is functionally graded graphene platelet reinforced composites (FG-GPLRC). Simultaneously, the material parameters are calculated employing the modified Halpin–Tsai model. A unified dynamic model based on the first-order shear deformation theory (FSDT) is proposed for FG-GPLRC irregular plates, where the displacement variables are considered as the first kind Chebyshev polynomials combined with coordinate transformation. The Chebyshev–Ritz method is utilized to derive the kinetic equations of irregular quadrilateral plates, and then the pseudo-excitation method (PEM) is employed to solve the stationary stochastic response of the system under the base acceleration excitation. The free vibration and stationary stochastic response of the plates under various boundaries are validated by a sufficient number of numerical studies, and the computational results display a favorable match with the published literature and finite element method (FEM), which verifies the correctness and reliability of the present method. Finally, a detailed parametric analysis of the stationary stochastic response characteristics of FG-GPLRC irregular quadrilateral plates is presented. The effects of various mass fractions, number of layers, geometric dimensions of GPL, and boundary constraints on the stationary stochastic vibration behaviors of FG-GPLRC irregular quadrilateral plates are considered.

Non-transient optimum design of nonlinear electromagnetic vibration-based energy harvester using homotopy perturbation method

Abstract In this paper the coupled differential equations governing the vibration of nonlinear electromagnetic energy harvesters are solved by the homotopy perturbation method. The amplitudes of odd harmonics of displacement of the magnet, coil current, and load voltage are derived up to the 5th harmonic. The frequency response of output power is plotted and it peaks at the linear mechanical resonance frequency. It should be noted that the optimum design of coil and load parameters, optimum electromagnetic coupling coefficient, and optimum vibration frequency of the magnet attached to a non-linear spring resulted in a stationary or non-transient vibration. Paying insufficient attention to this point and using typical parameters instead of optimum ones will result in transient vibration. The research aims at a rigorous semi-analytical method on a nonlinear problem which has previously solely investigated by numerical or experimental method.

Towards high-precision data modeling of SHM measurements using an improved sparse Bayesian learning scheme with strong generalization ability

Central to structural health monitoring (SHM) is data modeling, manipulation, and interpretation on the basis of a sophisticated SHM system. Despite continuous evolution of SHM technology, the precise modeling and forecasting of SHM measurements under various uncertainties to extract structural condition-relevant knowledge remains a challenge. Aiming to resolve this problem, a novel application of a fully probabilistic and high-precision data modeling method was proposed in the context of an improved Sparse Bayesian Learning (iSBL) scheme. The proposed iSBL data modeling framework features the following merits. It can remove the need to specify the number of terms in the data-fitting function, and automatize sparsity of the Bayesian model based on the features of SHM monitoring data, which will enhance the generalization ability and then improve the data prediction accuracy. Embedded in a Bayesian framework which exhibits built-in protection against over-fitting problems, the proposed iSBL scheme has high robustness to data noise, especially for data forecasting. The model is verified to be effective on SHM vibration field monitoring data collected from a real-world large-scale cable-stayed bridge. The recorded acceleration data with two different vibration patterns, that is, stationary ambient vibration data and non-stationary decay vibration data, are investigated, returning accurate probabilistic predictions in both the time and frequency domains.

АНАЛИЗ КРУТИЛЬНЫХ КОЛЕБАНИЙ ДЕТАЛЕЙ ТРАНСМИССИИ АВТОМОБИЛЯ С ДВУХМАССОВЫМ МАХОВИКОМ ДВС

The study objective is to analyze the influence of elastic-damping characteristics of the two-mass flywheel of the internal combustion engine on the torsional vibrations of the car transmission components.
 The task is to define the characteristics of forced stationary torsional vibrations of transmission parts under the action of torque fluctuations of the internal combustion engine.
 Methods: mathematical and computer modeling of forced torsional vibrations of transmission parts.
 The novelty of the work is in developing mathematical and computer models of the torsional vibrations of a car transmission with a two-mass flywheel and a double dry clutch, defining the conditions for preventing the resonance of the flywheel torsional vibrations at idling speed. 
 Results: mathematical and computer models are developed for the analysis of torsional vibrations of a car transmission with a two-mass flywheel and a double dry clutch. The analysis of forced stationary torsional vibrations of transmission parts in time is carried out. The conditions for preventing the resonance of the flywheel torsional vibrations at idling speed are defined. It is shown that the use of a two-mass flywheel significantly reduces standard deviation of angular velocities and accelerations of parts (except the crankshaft) in comparison with a single-mass flywheel.

An experimental and numerical comparison of traditional spring DVA in parallel and shape memory alloy actuated DVA in series and parallel for fixed beam vibration control

Shape memory alloy is a desirable smart material that could be used as a stiffness adjustment component in dynamic vibration absorbers (DVA). This research provides an analytical and experimental framework for investigating a stationary beam's vibration powered by a mixture of three SMA springs in series and two in parallel. In order to reduce vibration from a stationary beam, DVA, which is supported by springs composed of shape-memory alloy, is developed. In this study, a fixed beam in the system is held in place by a nut and bolt arrangement. The shaker is used with a function generator to provide vibration in real time. After installing two conventional springs with masses 305.1 mm apart from the fixed beam's ends, the vibration amplitude was measured. Experiments were carried out at various frequencies with three SMA springs with masses in series and two in parallel in place of the conventional springs. The results were recorded using Pulse software, which interfaced with the FFT and accelerometer. According to the experimental arrangement, the system was adequately dampened. Harmonic response analysis results were compared over a range of frequencies using ANSYS Workbench R15.0 to simulate the fixed beam spring-mass mass system.

An investigation on the stochastic thermal vibration behaviors for laminated combined composite plate systems

The present work aims to study the stationary and nonstationary stochastic thermal vibration properties of the laminated combined plate structures using meshless method. The combined system structure consists of several rigidly connected laminated rectangular plates, where the thermal effect of each subplate is considered according to the thermo-elastic theory. The Hamilton’s principle combined with the first-order shear deformation theory (FSDT) is adopted for generating the vibration formulas of the divided plates. The boundary and rigid compatibility conditions of the combined plates are imposed by massless springs with variable stiffness. The displacement components of the subplate system are approximated by 2D Chebyshev meshfree shape functions and the stochastic acceleration excitation is taken into account by introducing the pseudo excitation method (PEM). The modal behaviors and the stochastic dynamic solutions of laminated combined plates subjected to thermal loads are then successfully obtained. A considerable number of computational examples on the free vibration and random responses of the combined laminated plates are carried out to verify the convergence, accuracy and efficiency of the presented meshless model. Finally, systematic parametric studies are conducted to investigate the effects of thermal factor, acceleration excitations and boundary constraints on the free vibration and random dynamic characteristics of the laminated combined plates.

Natural Frequency and Modes of the Longitudinal and Torsional Vibrations in the Bars with Variable Cross Section

The paper is focused on the problem of natural frequencies and modes determination based on perturbation theory for longitudinal and torsional vibrations in bars with variable cross section. The mechanical properties and cross section geometry of the bar are changing small from the average value with regard to longitudinal coordinate. Based on the theory of small perturbations the analytical solution is obtained for natural frequencies and modes of the stationary harmonic vibrations in bars. The efficiency of the proposed method is supported by comparison and good agreement of the obtain results with a sharp solution for a given cross section profiles. It was established that the approximate solution is working good up to the ratio 2.5–3 between maximum and minimum diameter of cross section. The results of numerical simulations are aimed to estimate the geometry and elastic behavior of the metallic specimens for very high cycle fatigue experimental investigation under axial tension-compression and torsion loadings. The piezoelectric fatigue testing system and procedure is based on stationary vibration excitation in the metallic specimen at the first mode natural frequency.

A Volterra-PEM approach for random vibration spectrum analysis of nonlinear systems

In this paper, the Volterra series and the pseudo-excitation method (PEM) are combined to establish a frequency domain method for the power spectral density (PSD) analysis of random vibration of nonlinear systems. The explicit expression of the multi-dimensional power spectral density (MPSD) of the random vibration response is derived analytically. Furthermore, a fast calculation strategy from MPSD to physical PSD is given. The PSD characteristics analysis of the random vibration response of nonlinear systems is effectively achieved. First, within the framework of Volterra series theory, an improved PEM is established for MPSD analysis of nonlinear systems. As a generalized PEM for nonlinear random vibration analysis, the Volterra-PEM is used to analyse the response MPSD, which also has a very concise expression. Second, in the case of computation difficulties with multi-dimensional integration from MPSD to PSD, the computational efficiency is improved by converting the multi-dimensional integral into a matrix operation. Finally, as numerical examples, the Volterra-PEM is used to estimate the response PSD for stationary random vibration of a nonlinear spring-damped oscillator and a non-ideal boundary beam with geometrical nonlinearity. Compared with Monte Carlo simulation, the results show that by constructing generalized pseudo-excitation and matrix operation methods, Volterra-PEM can be used for input PSD with arbitrary energy distribution, not only restricted to broadband white noise excitation, and accurately predict the secondary resonance phenomenon of the random vibration response of nonlinear systems in the frequency domain.

Evaluating bolt connecting faults in ring-type structures with nonlinear fault-induced loads and transmissibility functions

Existing vibration-based methods often encounter some obvious limitations when they are applied to evaluate bolt connecting faults in ring-type structures, e.g., steel truss bridges in civil engineering and copper interlaced pipes in aerospace engineering. To overcome these limitations like ignorance of structural nonlinear boundaries, consideration of entire structural dynamics, and requirement of benchmark vibration data etc., a novel fault evaluation method is thus proposed. To this aim, the dynamic behaviour of ring-type structures is described by a discrete ring-type multiple degree-of-freedom (MDOF) model with nonlinear connecting components simulating bolt connecting faults and structural nonlinear boundaries. By exciting the ring-type structure three times with appropriate amplitudes of harmonic excitations, stationary vibration data are collected and evaluation features based on nonlinear fault-induced loads and transmissibility functions are defined. Finally, an effective evaluation method with sensitive and local indexes is established for complex ring-type structures. The method works with only three-time excitations for measured output data, and it is easy to implement to many other similar structures with complex, coupling and inter-connected nonlinearities. Experimental results with comparison to other two general nonlinear transmissibility function-based methods vindicate that the proposed method is more reliable and sensitive, demonstrating a totally new approach to such a challenging issue in engineering practices.

Investigation of planar translational and rotational stationary non-Gaussian random vibration test

Multi-shaker random vibration control with coordinate transformation technique is an advanced mechanical environment test method that translational and rotational vibrations of structure can be controlled simultaneously. This work presents theoretical and experimental investigations of multi-shaker planar translational and rotational non-Gaussian random vibration tests. First, the coordinate transformation technique is used to obtain the translational and rotational vibration responses of the test structure. Then the control procedure is formulated theoretically with matrix presentation in which a new nonlinear transformation method for generating non-Gaussian random signals is proposed. The correlation between translational and rotational vibrations is explored briefly from the perspective of common square control and transformation control. The influence of elastic characteristics of the test structure on the response spectral densities is investigated and a level definition method for translational and rotational vibrations is proposed. Finally, a planar translational and rotational non-Gaussian random vibration test is carried out to verify the validity of the proposed methods.

Fatigue analysis of coil springs in the primary suspension of a railway vehicle based on synthetic spectrum for time-varying vibration load

The fatigue life of coil springs is usually predicted with a stationary Gaussian vibration load and deterministic structural parameters. However, the obtained fatigue life is inconsistent with the observed fatigue life of fractured springs which varies within a wide range. The work aims to propose a method to predict the fatigue life of the coil spring by considering the time-varying vibration load, i.e., root mean square (rms) varies with time and the uncertainties of geometric parameters. First, a synthetic method for time-varying vibration loads is proposed. The time-varying load is decomposed into multiple stationary Gaussian short samples represented by their power spectral density (PSD). These PSDs are synthesized according to the distribution characteristics of spectral values, in which data that are non-Gaussian are processed with the Johnson system. Second, the influence of parameter uncertainties in the coil spring is studied by a Monte Carlo analysis of the stress frequency response function. Finally, the fatigue life is calculated and compared with the results predicted by using the measured stress. The results show that the synthetic spectrum has almost the same damage potential as the measured time-varying load. In comparison with results predicted from the measured stress, the synthetic spectrum gives much better estimates of the fatigue life of the coil spring than the average spectrum. Parameter uncertainties of coil springs significantly affect fatigue life and should be taken into account.

Generation of vibration load spectrum for fatigue analysis of equipment mounted on bogie frame of railway vehicle based on fatigue damage spectrum

The equipment mounted on the bogie frame of railway vehicles is subjected to non-stationary vibrations from the bogie frame. Currently, the vibration load spectrum for fatigue analysis of equipment is expressed with the power spectral density, which is generally obtained with the power values of measured vibrations or stationary Gaussian vibrations collected at the specific operating conditions. However, this does not have the equivalent damage potential as the non-stationary vibrations. Thus, a generation method of the vibration load spectrum is proposed based on the fatigue damage spectrum (FDS) to obtain the equivalent damage potential as the one from the non-stationary vibrations in this paper. That is, the FDS of non-stationary vibration is calculated with the time domain method, and the vibration load spectrum is generated in the frequency domain with this FDS. Then the method is verified by the measured vibration and dynamic strain of equipment. Finally, the vibration load spectrum of equipment mounted on the bogie frame is generated with the measured vibrations in a maintenance cycle and the fatigue analysis is performed. The results show that the generated vibration load spectrum can validly contain the high power in non-stationary vibrations and express the distribution of power over frequency, which gives much better estimates of the fatigue damage of equipment than other load spectra. The generated vibration load spectrum is suitable for quantifying the actual non-stationary vibrations and can be used for the anti-fatigue design of equipment.

浮体式洋上風車の風と波による応答および構造損傷発生確率評価に関する研究

An evaluation method of structural damage probability of floating offshore wind turbines under wind and wave actions is presented. The structural damage probability of the floating wind turbines is analyzed by using the first order reliability method in the structural reliability evaluation. In the evaluation of the probability, the response analysis results of the floating wind turbine for the wind and wave actions are used. The response for the action by the mean wind speed is analyzed by using the statically equation of equilibrium. On the other hand, the response for the dynamic actions by the variable wind speed and the wave is analyzed by using a mode superposition method and a stationary random vibration theory. Numerical model of the floating wind turbine is discretized by finite elements, whereas interface of the floating wind turbine and the sea water is discretized by boundary elements. Moreover, variation of the damage probability of the floating wind turbine for different wind speed is discussed by using the proposed method.