Order Natural Frequency Research Articles

Analytical Modeling of Density and Young’s Modulus Identification of Adsorbate with Microcantilever Resonator

Density and Young’s modulus are critical parameters in biological research, which can be used to characterize molecules, cells, or tissues in the diagnosis of severe diseases. Microcantilever resonators are ideal tools to measure the physical parameters of small objects at the micro/nanoscale. In this study, a mathematical model was built based on the Rayleigh–Ritz method with the consideration of the first five-order bending natural frequencies. The mathematical model can be used to detect the density and Young’s modulus of an adsorbate on a cantilever resonator with a single measurement. The influence of different order natural frequencies and the adsorbate position on the measurement accuracy and reliability was analyzed. This study revealed that the frequency pairs and the relative position of the adsorbate on the cantilever are two important factors that affect the accuracy and reliability of the measurement. Choosing appropriate frequency pairs can help to improve the accuracy and reliability of measurement. Finally, the results of finite element analysis verified the proposed method.

Diagnostics of a Clamping Joint System

Abstract The article describes a diagnostic method, on the basis of which it is possible to assess whether there is a loose connection in the clamping system (detection) and to determine which of them is loose (localization). The diagnostic method is based on comparing the drops in the natural frequencies of a measured system with respect to a reference system, which simulates the desired state. When analysing the drop in natural frequencies in order to locate the loosening, it is necessary to relate this drop to the position of the clamping joints within the considered mode shape.

Simulation Analysis of Vibration Characteristics of the Submerged-propeller Shaft Coupling System

The vibration characteristics of the submerged-propeller shaft coupling system are studied in this paper. Firstly, Based on the fluid structure interaction theory on acoustics wave equation, the vibration analysis model of the propeller-shaft system under water is established and computed. And the vibration characteristics of coupling system is analysed by comparing the shaft system and the submerged-propeller system. The results show that the natural vibration of the coupling system presents complicated propeller-shaft strong or weak coupling characteristics. Comparing the natural vibration and frequencies of the shaft system and the submerged-propeller system, all of the coupling system modes are corresponding to the shaft system or the submerged-propeller system in low frequency range. And there is strong transverse coupled vibration of coupling system near the propeller’s 1st-order transverse vibration frequency, which appears as propeller’s 1st-order transverse vibration coupling shaft’s 2nd-order transverse vibration. That indicates the strong transverse coupled natural vibration of the coupling system is easy to be excited near the low order natural frequency of propeller. Which may cause propeller’s transverse excitation force to be easily transmitted to the hull through the bearings under non-uniform flow, resulting in the severe vibration of the hull stern.

Dynamic calculation and optimization design of arbitrary planar branch piping system

The absorbing transfer matrix method (ATMM) is applied to dynamics calculation and optimization design of planar branch piping systems combined with the response surface model or method (RSM) and a new proposed hybrid RSM (HRSM). A main path is chosen to absorb the influences of sub-branches, and the branch point transfer matrix in the main transfer path is established and proved by the ANSYS simulation and experimental results. It shows that the errors between the ATMM and the finite element method are less than 0.36% in the calculation of the natural frequencies of the two-branch piping system, and the ATMM also agrees well with the dynamics response experimental results of the cross-branch piping system. Then, based on the initial design model of a double-branch piping system, first order natural frequency analysis samples are selected for the branch point positions optimization using the uniform design method (UD) and calculated by the ATMM; the minimum value of the RSM, obtained by fitting the samples with a second-order polynomial (SOP) function, is solved as the best design scheme using the genetic algorithm (GA). Finally, for the problem of supporting positions optimization of a T-branch piping system, a HRSM using the SOP function to fit the variation law with different supporting positions in the samples of the dynamics response optimization target, which is the total x-directional vibration acceleration level (TAC) of three supporting points in the frequency 1–300 Hz and the Fourier basis function to fit the relationship of SOP function error with independent variables is proposed combining the Minimize Prediction (MP) adding point criterion to gradually update the samples, until the HRSM reaches stability and obtains the wanted design solution.

Design, fabrication, and testing of CVI-SiC/SiC turbine blisk under different load spectrums at elevated temperature

Abstract In this article, design, fabrication, and testing of SiC/SiC turbine blisk with the fiber’s preform of Spider Web Structure (SWS) subjected to different load spectrums at elevated temperature are conducted. Micro-CT scans are conducted to show the fibers preform and defects inside the SWS-SiC/SiC turbine blisk. For 2D plain-woven SiC/SiC composite under monotonic tensile loading at an elevated temperature of T = 900°C in air atmosphere, the composite ultimate tensile strength is σ uts = 200 MPa with the fracture strain ε f = 0.36%. For SWS-SiC/SiC turbine blisk under the rotation testing, the first and second order natural frequency of the SWS-SiC/SiC turbine blisk are tested using the laser vibration meter. Relationships between the rotation speed, internal damage, and the natural frequency degradation of the SWS-SiC/SiC turbine blisk are established. Under the maximum rotation speed of n = 17,000 rpm at the exhaust temperature of T = 930°C, no damage occurred in the SWS-SiC/SiC turbine blisk. However, multiple coating spalling occurred due to the thermal-chemical coupling failure of the coating under flame impingement. The first natural frequency of the SWS-SiC/SiC turbine blisk decreases by 5% with the increase in the rotating speed from n max = 85,000 rpm to n max = 105,000 rpm, which indicates that there is an internal damage in the SWS-SiC/SiC turbine blisk, which leads to the decrease in the stiffness of the blisk.

Fluid sloshing hydrodynamics in a cryogenic fuel storage tank under different order natural frequencies

Liquid hydrogen is a promising energy carrier due to its excellent performance. However, liquid hydrogen sloshing usually causes serious issues on its safety production, storage, transportation, application and management. To study fluid sloshing in cryogenic storage tanks, a numerical calculation model is developed with considerations of the environmental heat leakage and phase change occurring at the free interface. The VOF model, coupled with the mesh motion treatment, is adopted to predict fluid sloshing under the first three order natural frequencies. Validated against to the experimental data, the calculation uncertainty is limited within 5.0%. The sloshing force and moment, the phase distribution and the interface shape, the dynamic fluctuation of the liquid-vapor interface and the fluid pressure variations are numerically investigated. The results show that the natural frequency has caused obvious effects on fluid sloshing hydrodynamics. Generally, the sloshing force and moment increase with the natural frequency, and the sloshing force of the first order natural frequency has obvious reductions. Meanwhile, obvious interface fluctuations and large elevation motions take shape in the first order natural frequency case. As the interface variation promotes the heat transfer from the vapor to the liquid, the largest fluid pressure drop also forms and occurs in the first order natural frequency condition. In brief, the first order natural frequency results in serious fluid sloshing and large amplitude interface fluctuation, and should be given enough considerations. The present study is significant to depth understanding on fluid sloshing performance during fuel transportation and may supply some technique supports for the design on cryogenic fuel storage tanks.

Measurement of natural frequencies and mode shapes of transparent insect wings using common-path ESPI.

In this study, a common-path electronic speckle pattern interferometry system which upholds the natural property of transparency of insect's wings has been developed to measure the wings' natural frequencies and mode shapes for the first time. A novel base-exciting method was designed to enable the simultaneous application of sinusoidal and static forces to excite wings and introduce an additional phase. The moiré effect induced by the amplitude modulation was employed to accurately recognize the resonance state. Subsequently, the mode shapes were visualized by phase-shifting and real-time frame subtraction. Eight pairs of forewings from cicadas were investigated. The first three order natural frequencies of the wings are approximately 145 Hz, 272 Hz and 394 Hz, respectively, which are dispersed to prevent modal coupling. The cambered mode shapes exhibit a strongly spanwise-chordwise anisotropy flexural stiffness distribution, generally dominated by bending and twisting deformation. The details of the high-order mode shapes show that the tip exhibits distinct deformation, indicating more flexibility to cope with external impact load, and the nodal lines usually comply with the direction of the wing veins in higher modes, substantiating the fact that the veins play an important role as stiffeners of the membrane. The results are in excellent agreement with the dynamic performance of previous studies, which will potentially affect a broader community of optical measurement specialists and entomologists to enhance our understanding of time-averaged interferograms and insect flights.

Grey relation analysis of natural frequency and semi-active robust optimal control of a multi-dimensional isolator based on parallel mechanism and magneto-rheological damper

For isolating multi-dimensional vibrations experienced by precise facilities carried on a vehicle, a novel isolator is proposed based on 2-RPC/2-SPC parallel mechanism with magneto-rheological dampers. Kinematics and dynamics of the isolator are analyzed by geometrics and the Lagrange method. Grey relation analysis approach is conducted to determine the contributions of geometric parameters on natural frequency conveniently. Through analysis, the first order natural frequency of the isolator is affected by the length of the fixed platform most significantly. Due to manufacturing and assembling errors which could not be avoided in the isolator, robust optimal control algorithm is conducted to ensure control effect and robustness of the isolator at the same time. The gain of robust optimal control algorithm is obtained by deducing and solving linear matrix inequality. Compared to passive control, velocity root mean square values of robust optimal semi-active control decreased obviously in horizontal, longitudinal, vertical, and roll directions.

Optimization of spatial pipeline with multi-hoop supports for avoiding resonance problem based on genetic algorithm.

The reasonable layout of hoops can effectively avoid the excitation frequency of engine rotors and greatly reduce the vibration level of pipeline systems. In this study, a spatial pipeline supported by multi-hoops was taken as the object, the method of using genetic algorithm to efficiently obtain the optimal layout of hoops to avoid resonance was investigated. The finite element model of the pipeline system was created as the basic model of optimization, spring elements were applied to simulate the mechanical characteristics of hoop and the influence of spring element direction on the vibration characteristics of pipeline system were mainly described. In the optimization of avoiding resonance for spatial pipelines, the optimization goal was to maximize the first-order natural frequency, the positions of the hoops were converted into the node number as design variables, and the final optimization model of pipeline to avoid resonance was determined on the premise of reasonably setting of constraint conditions for design variables. The genetic algorithm was utilized to solve the optimization model, and two optimization methods were proposed, which were named as "genetic algorithm calling finite element model" and "genetic algorithm updating stiffness matrix" respectively. Finally, a case study was carried out to display the proposed methods. The maximum deviation between the calculation and the test results is less than 1.5% for the first three order natural frequencies, which proves the rationality of the created finite element model of spatial pipeline. Furthermore, the optimization practices show that the reasonable hoop layout of the pipeline system can be obtained by the two optimization methods, but the efficiency of the optimization performed by "genetic algorithm updating stiffness matrix" is much higher than that of "genetic algorithm calling finite element model".

Measurement method of natural frequencies and tension forces for cables based on elasto-magnetic sensors calibrated by frequencies

Vibration-based (VB) method and elasto-magnetic (EM) method are usually used to measure cable forces of cable-supported bridges. For the VB method, it is difficult to accurately identify each order natural frequency of the cable disturbed by random excitations, and there are also no precise selection criteria between the taut string model and the hinged beam model. For the EM method, it is not convenient to calibrate EM sensors on bridges in service due to unknown cable forces. To address these issues, a vibration-based elasto-magnetic (VBEM) method is proposed. In this method, a numerical model describing the tie between each order natural frequency and induced voltage was constructed first, and then, a new cable force formula with nominal flexural stiffness was derived. To verify the VBEM method, a steel strand experimental platform was built and the load applied to the steel strand was achieved by a jack. At 18 °C, the first three order natural frequencies of the steel strand and corresponding induced voltage were recorded for each load. According to the obtained experimental data, the VBEM method is tested and analyzed. The results show that the VBEM method exhibits the ability to identify each order natural frequency of the steel strand with high precision; the introduction of nominal flexural stiffness makes the hinged beam model cover the taut string model, resulting in tension force measurement with satisfactory accuracy; the constructed models do not contain tension forces, and this will make it very beneficial to calibration of EM sensors on bridges in service.

A Rayleigh-Ritz Solution for High Order Natural Frequencies and Eigenmodes of Monopile Supported Offshore Wind Turbines Considering Tapered Towers and Soil Pile Interactions

A Rayleigh-Ritz Solution for High Order Natural Frequencies and Eigenmodes of Monopile Supported Offshore Wind Turbines Considering Tapered Towers and Soil Pile Interactions

Numerical analysis of combined optimization of turbine runner flow pattern and shafting System

This paper attempts to solve the turbine failures reported by a hydropower station, namely, the violent vibration in the runner region under a special working condition and the blade cracking on the outlet edge near the lower ring. For this purpose, the entire flow channel of the turbine was simulated by computational fluid dynamics (CFD) on ANSYS, and the runner strength and mode of shaft assembly system (SAS) were computed by the liquid-solid coupling algorithm. The calculation results show that severe low (negative) pressure appeared on the outlet edge near the lower ring, excess stress was observed in that area, and the resonance occurred as the fifth and sixth order natural frequencies of the SAS were the same with the rotation frequencies of the blade. On this basis, the original blade was modified repeatedly. Through the modification, the flow field distribution in the runner region and the blade strength were both greatly improved, and the SAS natural frequencies were kept away from the various external excitation frequencies, laying a solid basis for the safe and stable operation of the turbine.

Discrete sizing, cross-sectional shape, topology optimization, and material selection of a framed automotive body

This study proposed a discrete structural optimization method for a framed automotive body. Up to four types of discrete design variables are considered simultaneously, that is, the sizing, cross-sectional shape, topology, and material variables. Firstly, to solve the nonconvex and nonlinear optimization problem, the original non-dominated sorting genetic algorithm, the third version (NSGA-III), is adapted. An improved extreme points identification scheme and a new mutation operator are proposed to stabilize the normalization of the population and accommodate the manufacturing constraints, respectively. Two test problems demonstrate that the modified NSGA-III can handle continuous and discontinuous multiple objective optimization. Subsequently, the classical 10-bar truss is used to illustrate the proposed method. A weight reduction of 4.5 kg is achieved as compared to previous optimal designs in the literature. Finally, a framed automotive body is optimized for maximizing the first order natural frequency and minimizing the total mass, the maximum stresses and the maximum displacements in different load cases and the manufacturing cost. The results obtained by different optimization procedures are presented and discussed. The results demonstrate the feasibility and effectiveness of the proposed method. A weight reduction of 17.59% is achieved while other structural performances satisfy the design requirements.

Phononic Band Gap and Free Vibration Analysis of Fluid-Conveying Pipes with Periodically Varying Cross-Section

Phononic crystals (PCs) are a novel class of artificial periodic structure, and their band gap (BG) attributes provide a new technical approach for vibration reduction in piping systems. In this paper, the vibration suppression performance and natural properties of fluid-conveying pipes with periodically varying cross-section are investigated. The flexural wave equation of substructure pipes is established based on the classical beam model and traveling wave property. The spectral element method (SEM) is developed for semi-analytical solutions, the accuracy of which is confirmed by comparison with the available literature and the widely used transfer matrix method (TMM). The BG distribution and frequency response of the periodic pipe are attained, and the natural frequencies and mode shapes are also obtained. The effects of some critical parameters are discussed. It is revealed that the BG of the present pipe system is fundamentally induced by the geometrical difference of the substructure cross-section, and it is also related to the substructure length and fluid–structure interaction (FSI). The number of cells does not contribute to the BG region, while it has significant effects on the amplitude attenuation, higher order natural frequencies and mode shapes. The impact of FSI is more evident for the pipes with smaller numbers of cells. Moreover, compared with the conventional TMM, the present SEM is demonstrated more effective for comprehensive analysis of BG characteristics and free vibration of PC dynamical structures.

The effect of curing deformation on the vibration behavior of laminated composite beams

In this paper, the effect of curing deformation on the vibration behavior of a laminated composite beam is studied. The analysis model of the vibration of the laminated composite beam with arbitrary boundary conditions considering the curing deformation is established based on the first-order shear deformation theory. Curing deformation is considered as one kind of geometric imperfections and introduced into the analysis model. The geometric imperfection model in the form of parabolic functions is developed to describe the curing deformation in which curing temperature, ply stacking sequence, and tool-part interaction are taken into account. The Rayleigh-Ritz method is used to obtain the natural frequencies of the laminated composite beam considering curing deformation. The results reveal that the initial curing deformation has a great influence on the vibration behavior of laminated composite beam. Due to the existence of curing deformation, the first order natural frequencies of laminated composite beam with clamped–clamped (C–C) boundary condition have increased. The influences of the ply stacking sequence, the curing temperature, boundary conditions and the tool-part interaction on the free vibration behavior of the laminated beam are discussed.

Parametric Vibration Analysis of a Six-Degree-of-Freedom Electro-Hydraulic Stewart Platform

Electro-hydraulic Stewart 6-DOF platform is a 6-DOF parallel mechanism combined with the electro-hydraulic servo control system, which is widely used in the field of construction machinery. In actual working conditions, the flow and pressure pulsation of the hydraulic oil output from the hydraulic leg of the electro-hydraulic Stewart platform are inevitable, so the equivalent stiffness of the platform leg will change, and the stiffness parameters of the transmission system will change, resulting in vibration, which will affect the accuracy of the platform. This paper considering the fluid unit equivalent stiffness cyclical fluctuations and leg, on the basis of the relationship between hydraulic stiffness, constructs the electric hydraulic Stewart platform machine vibration dynamics equation, fluid coupling parameters of vibration parameters using the method of the multiscale approximate analytic formula of the main resonance and combination resonance are derived, and the system parameters vibration time-domain response and frequency response under two different poses are discussed. Results show that the system first to six order natural frequency and the first to the sixth order natural frequency and frequency of hydraulic oil equivalent stiffness of the combination of frequency will have an effect on the parameters of the system vibration. In the main resonance, the dominant frequency is mainly the first to sixth order natural frequency of the system; in the combined resonance, the dominant frequency is the combined frequency. Through the parameter vibration analysis of two different positions of the platform, it is concluded that when the platform is in an asymmetric position, each leg of the system is more involved in vibration. This study can provide the reference for the subsequent dynamic optimization and reliability analysis of the electro-hydraulic Stewart platform.

Vibration characteristics of functionally graded corrugated plates by using differential quadrature finite element method

This article proposes a general method to investigate the vibration characteristics of functionally graded corrugated plate structure (FGCPS) according to first-order shear deformation theory (FSDT) by adopting differential quadrature finite element method (DQFEM). The FGCPS can be regard as the coupling of multiple functionally graded circular arc shell elements in the form of share node by coordinate transformation. The boundary conditions of FGCPS including free, simply supported and clamped boundary conditions are considered in this paper and the boundary conditions of FGCPS are applied by using the penalty factor of penalty function. The vibration behaviors consisting of free and forced vibration behaviors of FGCPS are studied systematically from the perspective of model parameters after the convergence, stability, accuracy and generality of the presented approach are verified by finite element software ABAQUS. Meantime, it is worth mentioning that the forced vibration behaviors are investigated the steady-state response of FGCPS. The above model parameters of FGCPS are mainly composed of power law index, thickness, length, radius, radian and boundary condition. At the same time, it is necessary to point out that the investigations of the vibration characteristics of FGCPS are realized by studying the influences the above model parameters of FGCPS on the first order natural frequency and the displacement response of FGCPS respectively.

Size Effect of a Piezoelectric Patch on a Rectangular Plate with the Neural Network Model.

Artificial neural networks have been widely used in many studies, such as the prediction of the piezoelectric effect of the plate of engineering structures in vibration and noise reduction. In this paper, an artificial neural network (ANN) model was employed to explore the piezoelectric patch size and thickness’s effect on the first order natural frequency and displacement amplitude of a plate. With the finite element method (FEM), a rectangular plate actuated by a piezoelectric patch was analyzed with various patch sizes. The FEM data was later used to build an ANN model. The dynamic response of the plate was predicted by the ANN model and validated with FEM in terms of 1st order natural frequency and displacement amplitude. Results from case studies showed that with the input of patch length, width and thickness, ANN model can accurately predict both natural frequency and displacement amplitude. When the input of ANN model was simplified to patch size and thickness or the volume of the patch, the accuracy became worse and worse. The influence of the patch size and thickness on the first order natural frequency was coupled and the maximal and minimal values were predicted based on the ANN model.

The influence of time-varying mesh stiffness on natural characteristics of planetary gear system

Modal property is the crucial factor that can influence the vibration characteristics of the planetary gear system. Previous studies used time invariant mesh stiffness (TIMS) to study the natural characteristics. However, with the rotation of the gear, the meshing stiffness will change periodically which will lead to changes of modal properties. Based on these properties which are ignored in previous studies, a translational-torsional free vibration model base on time-varying mesh stiffness (TVMS) rather than time invariant mesh stiffness (TIMS) is established in this paper to study the modal properties of the planetary gear system, the effects of TVMS on the natural characteristics are investigated. The results show that the meshing phase difference and the change regulation of the time varying natural frequencies are correlated, and there is a dramatic variation of higher order natural frequencies at the contact teeth change points. Furthermore, a useful conclusion can be drawn that takes TVMS to calculate higher order frequencies has certain advantages over traditional methods.

Discrete structural optimization of a passenger car rear seat frame using aluminium alloy

To reduce the weight and improve the structural performances of a passenger car rear seat frame, this study uses a discrete structural optimization method to design an aluminium alloy seat frame to replace the original steel one. The optimization problem aims to minimize the total mass, manufacturing cost, the maximum displacements and tensile stresses under certain load conditions and maximize the first order natural frequency. The cross-sectional dimensions and material types of members in the seat frame are considered as the design variables. A modified non-dominated sorting genetic algorithm, the third version (mNSGA-III) which is adapted at handling problems with four or more objective functions, is used to solve the optimization problem. To benchmark the performance of the proposed method, a multidisciplinary design optimization (MDO) method is also utilized to solve the optimization problem. The comparison between the solutions obtained by the proposed method and the MDO method shows that the accuracy and the effectiveness of the proposed method are better. A weight saving of 35.1% is achieved by the aluminium alloy seat frame as compared to the original steel seat frame.