Bending Natural Frequencies Research Articles

Design of a Two-Degree-of-Freedom Mechanical Oscillator for Multidirectional Vibration Energy Harvesting to Power Wireless Sensor Nodes.

Converting otherwise wasted kinetic energy present in the environment into usable electrical energy to power wireless sensor nodes, is a green strategy to avoid the use of batteries and wires. Most of the energy harvesters presented in the literature are based on the exploitation of a one-degree-of-freedom arrangement, consisting of a tuned spring-mass system oscillating in the main direction of the exciting vibration source. However, if the direction of excitation changes, the efficiency of the harvester decreases. This paper thus proposes the idea of a curved cantilever beam with a two-degree-of-freedom arrangement, where the two bending natural frequencies of the mechanical resonator are designed to be equal. This is thought to lead to a configuration design that can be used in practical circumstances where excitation varies its direction in the plane. This, in turn, may possibly lead to a more effective energy-harvesting solution to power nodes in a wireless sensor network.

Flexoelectric and surface effects on bending deformation and vibration of piezoelectric nanolaminates: Analytical solutions

In this paper, a model of double-layer rectangular piezoelectric nanolaminates with different upper and lower surface properties is established. The bending and vibration behavior of the piezoelectric nanolaminates are evaluated and analyzed, taking into consideration the flexoelectric and surface effects. Utilizing the surface piezoelectric model, flexoelectric theory, and Kirchhoff plate theory, this study provides analytical solutions for the bending deflection and natural frequency of rectangular piezoelectric nanolamination materials under various boundary conditions. The results indicate that the effects of flexoelectric and surface influences on piezoelectric nanolaminates are associated with residual surface stress and flexoelectric coefficients. These effects can enhance the bending stiffness and decrease the natural frequency of the piezoelectric nanolaminates. Furthermore, altering the residual stress on the upper and lower surfaces of the laminates separately will result in different trends in bending deflection, while causing a consistent trend in natural frequency change. When the electric potential direction of piezoelectric nanolaminates is different, the bending deflection of piezoelectric nanolaminates will change accordingly. The natural frequency of piezoelectric nanolaminates is determined by the absolute value of the flexoelectric coefficients, and is independent of their positive or negative values. Both the surface effect and the flexoelectric effect are influenced by size-dependent behaviors. When the thickness of the laminates is sufficiently large, both the surface effect and flexoelectric effect can be disregarded. This paper's research methods and results provide theoretical models and analytical methods for the microstructure design, multi-physical field characterization, and bending deformation behavior of intelligent components containing rectangular piezoelectric nanolaminates.

Analysis of bending and torsional modal of PMSM rotor systems for EV

Abstract The rotor system is crucial in the all-in-one integrated power system of EV, making the study of its dynamic mechanical performance at high speeds imperative. This paper focuses on analyzing the modal characteristics of EV rotor systems and the impact of individual components on bending and torsional vibrations. Using finite element analysis (FEA), a modal analysis of a 100 kW high-speed electric motor rotor is conducted, followed by experimental tests using an exciting method. Comparing simulation with experimental results reveals how end caps and bearings reduce the rotor’s bending natural frequency (BNF), while the core increases it. Additionally, the addition of rotor components leads to varied reductions in the rotor’s torsional natural frequency (TNF). Furthermore, the error between FEA and experimental values for combined rotor modes remains within 3.53%, affirming the accuracy and reliability of the analysis in predicting rotor modal characteristics.

Free Vibration-Based Delamination Detection in Fiber Metal Laminates Composite Beam

Delamination is a typical flaw in fiber-metal laminated composite structures that, when hidden from view, can significantly lower structural stiffness and alter a structure's ability to respond dynamically to forces like natural frequencies. Delamination is undoubtedly an important topic as it causes the performance of the Fiber Metal Laminate structures in the service to worsen. Delamination detection and severity analysis are essential in the aerospace industry for both safety and cost reasons. Although natural frequencies do not directly reveal the location or extent of damage, they are the most dependable metrics for damage detection. Frequency shifts in various modes are used to solve the inverse problem to pinpoint the damage location and extent. The bending natural frequencies shift brought on by delamination is utilized as an input to predict the delamination parameters. This study used an approach based on machine learning and a regression model to find the delamination parameters, i.e., locations and severity, in fiber metal laminated cantilever beams. FEA simulations using ANSYS were used to get the dataset pertaining to the position, severity, and eigen values (or bending natural frequencies) of the delamination. The results of this study indicate that the delamination locations and severity predictions developed using machine learning and regression models are reasonably accurate and demonstrate good agreement with the observed data.

Detection, Localisation and Quantification of Structural Damage Using Changes in Modal Characteristics

The article defines and explains methods for detecting and locating damage and quantifying its extent on an example of a finite element model of a simple beam. The defined methods are based on a comparison of the decreases in the bending natural frequencies of the nominally damaged models against the intact model. This comparison assigns natural frequency decreases to curvature of a specific mode shape, which allows creation of an analytical reference model using polynomial regression, which assigns relative natural frequency decreases to the position on the beam. The localisation method assumes that the vector of the relative natural frequency decrease ratios are unique for each position on the beam. The quantification method considers as a relevant comparative quantity, slopes of relative natural frequency decreases, as a function of damage extent. Examples of damage localisation and quantification are defined, and the relative errors of these methods are analysed.

Dynamic Properties of Timber–Concrete Composite Beams with Crossed Inclined Coach Screw Connections: Experimental and Theoretical Investigations

Due to the low density and stiffness of wood, traditional timber floor systems are prone to producing large vibration responses. By combining timber beams with concrete floors, timber–concrete composite (TCC) floor systems show stronger bearing capacity, higher bending stiffness, and better thermal and sound insulation behaviors when compared with traditional timber floor systems. In this study, the vibration performance of TCC beams with crossed inclined coach screw connectors was investigated using dynamic tests. The influence of the screw diameters and slab dimensions on the dynamic performance of the composite beams was evaluated. The test results demonstrated that TCC beams show good dynamic performance when used as a floor component and meet the preliminary requirements of floor vibration comfort for fundamental frequency. The fundamental frequency and damping ratio of TCC beams decreases with the increase in slab dimensions. The bending stiffness and natural frequency of TCC beams decrease smoothly when reducing the screw diameter from 16 to 12 mm. Additionally, two theoretical models were used to predict the natural frequencies of the TCC beams, and the predicted values show good consistency with the measured ones.

High‐speed drive‐by monitoring: field testing with an intercity express train (ICE)

AbstractThe structural condition of bridges in Germany is often affected by aging and increased traffic loads. However, retrofitting or renewing all bridges is not always feasible due to limited resources. Indirect or drive‐by monitoring of bridges with sensors on passing vehicles has been explored as a cost‐effective and scalable alternative to direct monitoring. In this study, we propose a novel High‐Speed Monitoring (HSM) method for determining natural frequencies, based on accelerometers mounted on train axles, developed for speeds under normal operating conditions. In combination with Bayesian optimisation, a resonance curve can be generated using only a few bridge passages, from which the dominant bending natural frequency of a bridge can be determined with an accuracy comparable to direct monitoring. Measurements on an ICE‐TD passaging bridges travelling at high speeds of up to 200 km/h have shown that the natural frequencies can be determined with the same accuracy as direct measurements. Thus, in this study we were able to show that the method is not only accurate, cost‐effective and scalable, but can also be applied at high speeds without affecting operations.

Simulation and Experiment Analysis of Driveshaft

A driveshaft is a small spring coil less than 1mm in diameter, composed of several stainless-steel wire filaments. In intervention, the driveshaft is used to transmit force and motion to the inside body through the existing micro channels (such as arteries, veins, and gastrointestinal tract). The performance of the driveshaft determines the efficiency, stability, and accuracy of force and motion transitions, the ability to pass through tortuous microchannels, and the damage to healthy tissues. To determine the influence of fabrication parameters (filament, wire diameter, and outer diameter) on the mechanical properties (such as bending stiffness and natural frequency) of the driveshaft, a simulation was established in ABAQUS to calculate the deformation displacement under 0.0098N and first-order natural frequency. Then, the bending stiffness is calculated. The results show that the bending stiffness and the first-order natural frequency of the driveshaft increase with the increase of the filament number and wire diameter, and with the outer diameter of the driveshaft increases, the bending stiffness increases, while the first-order natural frequency decreases. Finally, the simulation model is verified by measuring the deformation displacement in the experiment. This study provides a methodology for designing and selecting the driveshaft in Interventional therapy.

Effect of floater flexibility on global dynamic responses of a 15-MW semi-submersible floating wind turbine

Floater structural flexibility plays a significant role in accurate predictions of global dynamic responses of floating wind turbines (FWTs), especially for ultra-large wind turbines with increasing size and cost-effective floater design. The conventional analysis of FWTs often considers the floater as one rigid body, which cannot capture the correct resonant responses of the floater (especially the column that supports the tower). In this paper, a new approach where the floater is divided into multiple rigid bodies connected by flexible beams is developed for advanced modelling of flexible floaters in combination with coupled time-domain simulations. The hydrostatic and hydrodynmaic loads on each body from a one-body hydrodynamic analysis, together with gravitational loads and inertial loads, are implemented in the beam-based finite element model of the floater to carry out a time-domain analysis. The proposed approach is used to compare the responses of a 15-MW semi-submersible FWT considering a rigid and a flexible floater. The inclusion of floater flexibility reduces the tower natural frequency. It leads to larger bending moment amplitudes at the tower base close to the tower bending natural frequency, particularly in only-wind, irregular wave and wind-wave conditions, but smaller dynamic responses in regular wave conditions.

Equivalent Modeling Method for Screw Joints Based on Stiffness Calculation

Accurate assessment of the mechanical properties of an assembly with screw joints has to rely on the equivalent modeling of the screw joint and accurate kinetic properties analysis. In this paper, an equivalent modeling method for screw joints based on stiffness calculation is proposed. With the solid model of the screw-coupled member, the equivalent virtual material parameters that can characterize the mechanical properties of the joint surface of the coupled member are obtained by calculating the joint surface stiffness, and then the accurate parametric modeling of the screw joints is realized. In addition, the vibration modal test platform for screw joints is built. By comparing the frequency response function obtained from the experiment with that obtained from the equivalent model simulation, it is found that the errors of the 1st-order and 2nd-order bending natural frequencies of the cantilever beam are 3.96% and 5.5%, respectively, and the simulation analysis results are of high accuracy. Furthermore, by comparing the first 6th-order of natural frequencies of the cantilever beam, the maximum frequency deviation between the simulation analysis results and the test results is ≤ 7%, and the average frequency deviation is 4.67%, which can better reflect the mechanical properties of the screw joint.

On 3D exact free torsional-bending vibration and buckling of biaxially loaded isotropic and anisotropic Timoshenko beams with complex cross-section

This research reports on Three Dimensional (3D) exact free torsional-bending vibrations and buckling of Timoshenko beams possessing homogeneous and nonhomogeneous properties and complex cross-section. The proposed exact solution is derived from 3D elasticity equations. The Hamilton’s integral equation is initially employed to establish the 3D exact governing Partial Differential Equations (PDEs) of motion and buckling, taking into account the effect of biaxial load applied to Timoshenko beams possessing complex cross-section. The PDEs are then solved using integral transform and trigonometric series differentiation procedure. The current solution procedure does not require characteristic deformation function to be predetermined and has fast convergence. A root finding algorithm is proposed to screen and filter the discontinuity and breakdown points as well as imaginary roots of the determinant function associated with bending eigenfrequency and eigenbuckling matrices. Two algorithms are further developed to capture the 3D dynamic and buckling mode shapes of biaxially loaded Timoshenko beams without any restriction imposed on the complexity of cross-section and degree of material anisotropy. The accuracy and convergence of the proposed 3D exact solution are first investigated by results comparison with the experimental and theoretical results in the published literature for a particular case in which bending natural frequencies response and modes shapes of a free-free isotropic Timoshenko beam with rectangular cross-section are sought. Finally, the effect of material isotropy/anisotropy, cross-section complexity, and biaxial load on torsional-bending natural frequencies, critical buckling loads, and 3D vibration and buckling mode shapes are thoroughly explored using the proposed exact solution procedure and validated by Abaqus finite element package.

Eigen value analysis of composite hollow shafts using modified EMBT formulation considering the shear deformation along the thickness direction

Composite hollow shafts are used in power transmission applications due to their high specific stiffness and high specific strength. The dynamic characteristics of these shafts are important for transmission applications. Dynamic modelling of these shafts is generally carried out using Equivalent Modulus Beam Theory (EMBT) and Layerwise Beam Theory (LBT) formulations. The EMBT formulation is modified by considering stacking sequence, shear normal coupling, bending twisting coupling and bending stretching coupling. It is observed that modified EMBT formulation is underestimating the shafts stiffness at lower length/mean diameter (l/dm) ratios. In the present work, a new formulation is developed by adding shear deformation along the thickness direction to the existing modified EMBT formulation. The variation of shear deformation along the thickness direction is found using different shear deformation theories, i.e., first-order shear deformation theory (FSDBT), parabolic shear deformation theory (PSDBT), trigonometric shear deformation theory (TSDBT), and hyperbolic shear deformation theory (HSDBT). The analysis is performed at l/dm ratios of 5, 10, 15, 20, 25, 30, 35, and 40 for carbon/epoxy composites, E-glass/epoxy composites, and boron/epoxy composite shafts. The results show that new formulation has improved the bending natural frequency of the composite shafts for l/dm < 15 in comparison with modified EMBT. The effect of new formulation is more significant for the second and third bending modes of natural frequencies.

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.

Analysis and Control of Abnormal Vibration of End Wall on High-Speed Electric Multiple Units

In view of the abnormal vibration of the body end wall during the high-speed operation of electric multiple units (referred hereafter as EMU), the vibration and noise characteristics of the body end wall area are analyzed through the line test. Combined with the end wall modal simulation results, the generation mechanism of the abnormal vibration of the body end wall is analyzed. The results show that when the train is running at a high speed, the aerodynamic excitation of the windshield cavity outside the body end wall acts on the end wall, arousing the first-order bending natural frequency of the body end wall, resulting in resonance of the body end wall, and then causing abnormal vibration and noise in the body end wall area. In order to solve this problem, installing a deflector (guiding plate) above the windshield in the vehicle body end wall area can effectively suppress the aerodynamic excitation acting on the vehicle body end wall. After optimization, the abnormal vibration and noise in the vehicle body end wall area are significantly reduced. The corresponding peak value at 40 Hz of the vehicle body end wall vibration spectrum is reduced by 85%, and the peak noise is reduced by 12%, The correctness of the mechanism analysis of abnormal vibration in the headwall area is verified, which provides a reference basis for guiding the structural optimization and operation and maintenance of rail vehicles.

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.

Modal analysis of spur gears for varied teeth root cracks characteristics: finite element analysis (FEA) simulations

Gear failure is considered the main alarming and undesirable event in gearboxes. Usually, cracks happen by fatigue caused due to cyclic loading. Fatigue stress is focused on the teeth root because of the small tooth fillet radius. This causes progressive damage to gear teeth which causes teeth failure, and hence, a damaged gear. This work shows a numerical framework to identify and quantify cracks' existence at the teeth root of spur gears. The problem is numerically analysed through finite element-based simulation with ANSYS by conducting a modal analysis. There are nine cases of gears with different cracks on their tooth; these cracks are varied by their number (single or multiple), Crack Length Percentage (CLP %), and location. The FEA is used to simulate all nine cases to predict the bending natural frequency of teeth and to investigate the effects of the variation of the crack by looking at the natural frequency of the teeth bending as well as the deformation level. Results revealed that the teeth bending natural frequency decreases as the CLP% increases. In addition, the gear stiffness is calculated based on the natural frequency and gear’s mass and it is indicated that the stiffness decreases as the CLP% increases, while the deformation level increase with CLP%.

Analytical solutions for bending, buckling and vibration of nanobeams on Winkler–Pasternak foundation

The nonlocal elasticity theory and the Euler–Bernoulli (EB) beam theory are used to present closed-form analytical expressions for static bending, axial buckling, and free vibration of nanosized beams resting on an elastic foundation. The differential constitutive relations of Eringen are utilized to represent the small-scale effects of the nanobeam’s mechanical response. The governing equation of motion is derived by employing Hamilton’s principle. Utilizing the Laplace transform approach, analytical expressions of the bending displacements, the critical buckling force, and the vibration frequency of nanobeams with simply supported (S-S), clamped, cantilevered, and propped cantilevered boundary conditions are produced. In order to confirm the correctness of the offered closed-form equations, their outputs are compared to those of the available numerical method solutions. The effects of the Winkler parameter, the Pasternak parameter and the nonlocal parameter on bending, buckling, and vibration characteristics of nanobeams have been explained. Presented analytical expressions and graphical representations demonstrate how increasing Winkler and Pasternak parameters reduce bending displacements while raising the critical buckling load and the natural frequency of nonlocal nanobeams. Benchmark numerical results are also presented to investigate and discuss the effects of all parameters on bending deflections, buckling loads, and natural frequencies of nanobeams.

Analytical solutions of bending analysis and vibration of rectangular nano laminates with surface effects

In this paper, a novel continuum mechanics model is developed to investigate the influence of surface effects on the bending and free vibration behaviors of Kirchhoff rectangular nano laminates. The governing equations are established based on the Gurtin–Murdoch theory and the analysis of the equilibrium of forces. Analytical solutions for bending deflections and natural frequencies are presented for selected boundary conditions. Numerical results show that the influences of surface effects on bending deflections and natural frequencies are related to the aspect ratio of the nano laminates and the value of residual surface stress. Moreover, the influence of surface effects can be ignored when the thickness of the nano laminate is large enough.

Gyroscopic effect on a scaled rotor-bearing system

This paper deals with scale-fullscale models of rotor-bearing systems on rotating condition. The investigation is focused on the gyroscopic effect, which causes forward and backward whirl frequencies. When the rotor-bearing system is scaled proportionally in its dimension (height, length, and width), the scaling factor of whirl frequencies can be derived. It depends on the ratio between a transverse and polar moment of inertia on its rotating axis called the gyroscopic factor. The experimental study is conducted to validate it on three scaled rotor-bearing systems, which shifts its disc from the middle of the shaft on the scale of 1:1, 2:1, and 3:1 to clearly show the gyroscopic effect on first bending natural frequency. The scaling factor is then validated using the Campbell diagram by finding its critical speed. From this critical speed, the whirl frequencies along the range of the full-scale model speed can be obtained. The result also shows that the scaling factor remains the same whether it is at rotation or rest condition. Consideration must be made on the effect of the structural design, that is, blade and support, because of its unsymmetric stiffness that can cause backward whirl frequencies. The bearing stiffness must be ensured to be scaled proportionally, especially on journal bearing cases. This finding can be used by engineers to deal with scaling method implementation on rotating machinery design.

Bending Properties of an Extensile Fluidic Artificial Muscle.

Low stiffness, large stroke, and axial force capabilities make Extensile Fluidic Artificial Muscles (EFAMs) a feasible soft actuator for continuum soft robots. EFAMs can be used to construct soft actuated structures that feature large deformation and enable soft robots to access large effective workspaces. Although FAM axial properties have been well studied, their bending behavior is not well characterized in the literature. Static and dynamic bending properties of a cantilevered EFAM specimen were investigated over a pressure range of 5–100 psi. The static properties were then estimated using an Euler-Bernoulli beam model and discrete elastic rod models. The experiments provided data for the determination of bending stiffness, damping ratio, and natural frequency of the tested specimen. The bending stiffness and the damping ratio were found to change fourfold over the pressure range. Experimentally validated bending properties of the EFAM presented insights into structural and control considerations of soft robots. Future work will utilize the data and models obtained in this study to predict the behavior of an EFAM-actuated continuum robot carrying payloads.