Order Natural Frequency Research Articles

Effect of the transition piece on the natural frequencies of monopile-supported offshore wind turbines

The lateral natural frequency of an offshore wind turbine (OWT) is a crucial factor to consider in the design of OWT foundations. Eighty percent of currently installed OWTs are mounted on monopile foundations, which are connected to the superstructure via a transition piece (TP). To explore the differences in natural frequencies between grouted connection (GC) and TP-less designs, this study incorporates a transition piece into the calculation model of the OWT system. Based on the Timoshenko beam theory and transfer matrix method, the characteristic equations of the lateral natural frequency of the OWT are derived. The study found that the effect of the transition piece on the natural frequency is strongly influenced by the OWT type and the soil stiffness, and the influence of transition piece on different order natural frequencies is different for the same OWT. For the NREL offshore 5-MW baseline wind turbine, the TP-less model reduces the first natural frequency of the OWT by 1.31 % and elevates the second and third natural frequencies by 11.88 % and 2.72 %, respectively, compared to the GC model. The increase in soil shear wave velocity amplifies the influence of the transition piece on the first and third natural frequencies, while weakening its influence on the second natural frequency.

Damage detection of Kevlar woven fabric using optical fiber multimode interferometer

Kevlar woven fabric is crucial for the lightweight design of aerospace structures due to its unique advantages. However, it is susceptible to damage during operation, making effective damage detection essential for ensuring structural reliability. Nevertheless, conventional sensors used for damage detection have limitations such as complex manufacturing processes and weight. To address this issue, this paper proposed a simple singlemode-multimode-singlemode (SMS) optical fiber sensor to collect vibration signals and construct a damage recognition model based on natural frequency in order to identify the size of damages in Kevlar woven fabrics. To achieve a highly linear and accurate sensor, a length determination criterion for multimode optical fiber segments based on quasi-image, notch, and transmission spectrum loss was proposed. An optical sensor was fabricated based on this criterion. The feasibility and sensitivity of the sensor for vibration signal measurement were verified by constructing a vibration signal measurement device. An experimental study was conducted to detect damage in Kevlar tapes, where the natural frequency changes measured by the SMS sensor were integrated with an optimized multi-variable gray model for damage size detection. The experimental results indicate that the estimated error in damage size is 0.71%. The damage identification system proposed in this paper offers a simple and cost-effective solution for measuring damage in flexible structures using fiber optic sensors.

Chirp excitation for natural frequency optical coherence elastography.

Optical coherence elastography (OCE) has recently been used to characterize the natural frequencies of delicate tissues (e.g., the in vivo human cornea) with sub-micron tissue oscillation magnitudes. Here, we investigate broadband spectrum sample stimulation using a contact-based piezoelectric transducer (PZT) chirp excitation and compare its performance with a non-contact, air-pulse excitation for OCE measurements on 1.0-7.5% agar phantoms and an ex vivo porcine cornea under intraocular pressures (IOPs) of 5-40 mmHg. The 3-ms duration air-pulse generated a ∼0-840 Hz excitation spectrum, effectively quantifying the first-order natural frequencies in softer samples (e.g., 1.0%-4.0% agar: 239-782 Hz, 198 Hz/%; porcine cornea: 68-414 Hz, 18 Hz/mmHg, IOP: 5-25 mmHg), but displayed limitations in measuring natural frequencies for stiffer samples (e.g., 4.5%-7.5% agar, porcine cornea: IOP ≥ 30 mmHg) or higher order natural frequency components. In contrast, the chirp excitation produced a much wider spectrum (e.g., 0-5000 Hz), enabling the quantification of both first-order natural frequencies (1.0%-7.5% agar: 253-1429 Hz, 181 Hz/%; porcine cornea: 76-1240 Hz, 32 Hz/mmHg, IOP: 5-40 mmHg) and higher order natural frequencies. A modified Bland-Altman analysis (mean versus relative difference in natural frequency) showed a bias of 20.4%, attributed to the additional mass and frequency introduced by the contact nature of the PZT probe. These findings, especially the advantages and limitations of both excitation methods, can be utilized to validate the potential application of natural frequency OCE, paving the way for the ongoing development of biomechanical characterization methods utilizing sub-micron tissue oscillation features.

Experimental study on sloshing mechanical characteristics in a partially filled storage tank

AbstractDue to excellent performance, hydrogen is treated as the most promising energy carrier. However, during the storage of liquid hydrogen, there are still some thorny issues that need to be addressed urgently, such as fluid thermal stratification and sloshing phenomenon. To efficiently grasp fluid sloshing mechanical characteristics, a simple visual sloshing experiment rig was established by using a rectangle transparent water vessel. The variations of the interface shape and the impact force during sloshing were monitored and analyzed. The effects of sloshing frequency, horizontal acceleration, and initial liquid height on fluid sloshing mechanical mechanism were investigated. The results show that when the external sloshing excitation is close to the first order natural frequency, obvious interface fluctuation and large amplitude sloshing force variation are observed. The present work is of significance to strengthen the understanding of fluid sloshing mechanical performance and may lay a solid foundation for fluid sloshing suppression and long‐term storage of cryogenic fuels.

3D mode shapes characterization under hammer impact using 3D-DIC and phase-based motion magnification

Modal analysis constitutes a fundamental aspect of structural investigation within diverse engineering domains, encompassing sectors such as automotive, wind energy, and aerospace. The prominence of high-frequency excitation loads, exemplified by the combustion phenomena in liquid rocket engines, necessitates an in-depth examination of the high-frequency vibrational response within structural components. However, the complexity of evaluating high-frequency vibrations arises from the negligible displacement associated with these responses. When using an optical full-field measurement system based on a high-speed camera for vibration measurement, it is usually severely affected by noise. Direct analysis of raw data using an optical measurement system (3D-DIC) is not feasible. In this paper, we combine phase-based motion magnification and digital image correlation methods to obtain the high-frequency vibration modes of the structure. 3D-DIC(3D Digital Image Correlation)analysis is performed on the magnified images to quantify the out-of-plane vibration modes of the structure. Using the cantilever beam as an example, the first five out-of-plane vibration mode shapes were separated from the response video under a single hammer excitation. Especially the 5th order natural frequency is as high as 3503 Hz, and the corresponding structural response was below the noise floor of the camera system. The vibration mode results obtained by this method are highly consistent with the vibration modes obtained by the 3D-SLDV(3D Scanning Laser Doppler Vibrometer) method. Finally, this method was applied to identify the out-of-plane vibration modes of real engine pipe. The combination of motion magnification techniques and DIC can enhance the capability of traditional 3D-DIC, which is beneficial for high-frequency structural identification. Future research could concentrate on optimizing motion amplification factors for different structures and loads, and creating automated algorithms for analyzing and visualizing amplified motion data in real time.

Analysis of electromechanical coupling modal and dynamic response characteristics of permanent magnet gear transmission system considering variable load condition

In order to study the influence of electromagnetic (EM) effect and load change on the modal characteristics and dynamic response of gear transmission system driven by permanent magnet synchronous motor (PMSM), a electromechanical coupling dynamic model is established. The composite air gap permeability coefficient is introduced, and the formula of permanent magnet flux linkage under the influence of space harmonic and cogging effect is derived. The variation laws of gear meshing stiffness and bearing support stiffness under the influence of load change are simulated and analyzed. The natural frequency and vibration modal under different load, flux linkage and PI controller gain are revealed. On this basis, the influence of EM effect and load change on the resonance domain is analyzed through frequency sweep analysis and time domain simulation. Results indicate that the 1st order natural frequency and resonance amplitude will increase with the load, and the influence of PMSM EM excitation will gradually dominate. The increase of the flux linkage and speed loop controller gain Kpω will reduce the 1st order natural frequency. When Kpω =123, a new mode appears, and there is a modal transition between the new mode and mode 1 when Kpω =161. The research results can provide theoretical guidance for improving the transmission performance and structure optimization design of the permanent magnet gear transmission system.

Semi-analytical dynamic modeling and fluid-structure interaction analysis of L-shaped pipeline

The traditional semi-analytic modeling is mainly aimed at the straight pipeline with simple boundary conditions. In contrast, the actual aero-engine pipeline needs to consider the effects of complex configuration, boundary and multiple clamps, so proposing an improved semi-analytic method to model an arbitrary fluid-conveying L-shaped pipeline is challenging. Based on the Hamiltonian principle, the differential equations of the L-shaped pipeline are derived. Then, the modal shape functions are determined under the conditions that the elastic support points and the joints of the straight pipeline and the curved pipeline meet the deformation coordination requirements. The proposed semi-analytical model is verified by comparing the natural frequencies and mode shapes obtained from the finite element method using ANSYS and experiment; the error of each order natural frequency is less than 5 %. In addition, the model is verified under various working conditions: different center angle and boundary conditions (clamp position and pipeline end constraints). The effects of the fluid velocity and pressure on the natural characteristics of the L-shaped pipeline are also analyzed. The results show that under the critical pressure of 60.51 MPa and the critical velocity of 247 m/s, the natural frequency decreases with the increase of fluid velocity and pressure. The proposed method can provide theoretical support for efficient and accurate modeling of complex spatial pipeline systems.

Multiscale topology optimization framework for natural frequency maximization of multi-morphology lattice structures

This paper proposes a multiscale topology optimization framework for maximizing the natural frequency of multi-morphology lattice structures (MMLSs). The proposed framework addresses the challenges of computational efficiency, design space, numerical convergence, and compatibility between adjacent microstructures in multiscale topology optimization for natural frequency problems. The macrostructural topology, the morphologies categories, distribution regions, volume fractions of different lattice materials (LMs), and the relative densities of lattice unit cells (LUCs) are simultaneously optimized to enhance the structural natural frequency. Specifically, level set functions are utilized to generate prototype LUCs, enabling obtaining graded LUCs by configuration interpolation. Multi-morphology LUCs with smooth characteristics are achieved using the Kriging-assisted morphological post-process and sigmoid function (SF) based hybrid transition strategy. Kriging metamodel-assisted Uniform Multiphase Materials Interpolation (KUMMI) schemes are constructed for the mechanical properties estimation of macro elements to promote the multiscale topology optimization procedure. Distinct processing methods for LUCs' elasticity tensor and density are incorporated within the KUMMI schemes, by which the local mode problem is avoided. The objective order natural frequency is accurately optimized with the modal assurance criterion (MAC) based mode-tracking technique. Numerical examples demonstrate that the developed design framework can efficiently maximize the natural frequency of MMLSs, while also ensuring microstructural connectivity.

Static bending and forced vibration analyses of a piezoelectric semiconductor cylindrical shell within first-order shear deformation theory

This paper examines the mechanically induced electric potential and charge redistribution in a piezoelectric semiconductor cylindrical shell employing first-order shear deformation theory. The two-dimensional governing equations and corresponding boundary conditions are simultaneously derived through a principle of virtual work method and the fundamental lemma of the calculus of variation. For the application of electronic devices, static bending and forced vibration analyses of an open cylindrical shell under locally distributed normal forces are analytically solved with the derived theoretical model. The effects of doping level on electric potentials and mechanical displacements are presented. An interesting result shows that doping levels can alter the peak position of the zeroth-order electric potential. In addition, the first three order natural frequencies of the shell under time-harmonic load are identified, and the doping level is found to have an inhibiting effect on the first natural frequency. All the numerical results are beneficial for optimizing the design and performance of cylindrical shell-shaped sensors and energy harvesters.

Design of three-roll bending machine tool and research on compensation algorithm

With the rapid development of profile processing industry, the processing shape, efficiency, and consistency of profiles are more and more demanding. In this case, the progress of bending machine tools was very important. In this study, the disadvantages and shortcomings of traditional three-roll bending machine are investigated, and a new design of three-roll bending machine is proposed, including a new mechanical structure, hydraulic pressure, and control system. ABAQUS was used to analyze the constrained mode of the table, and the first six order natural frequencies and natural vibration modes were obtained. A mathematical model was established to analyze the relationship between the pressure displacement and the forming radius under ideal working conditions, and the limit position of the downward movement of the lower press roll and the speed relationship between the lower press roll and the support roll are discussed. Based on the experimental data, the nonlinear curve fitting was carried out on the rebound compensation algorithm of asymmetric rolling bending forming. The fitting rebound compensation algorithm was carried out on the symmetric rolling bending forming experiment. The experimental results show that this algorithm can be applied to both symmetric rolling bending and asymmetric rolling bending.

Eigenvalues and eigenfunctions of nonuniform beams with a new method based on perturbation

By using a new method based on perturbation method and Fredholm theory, the eigenvalues and eigenfunctions of nonuniform Euler-Bernoulli beam vibration are solved analytically. The eigenvalues and eigenfunctions under boundary conditions are given for the tension beam problem which considers the variation of tension in vertical marine riser from top to bottom caused by gravity effect. The asymptotic analytical results are compared with classical results to verify the effectiveness of the new method based on perturbation method. The influence of the change of the bottom tension on the perturbation method is analyzed. The results show that: for each order of natural frequency of riser, the natural frequency will gradually increase with the increase of bottom tension, and the obtained natural frequency will become more and more accurate; when the bottom tension is constant, the new method based on perturbation method in this paper can give good accuracy for higher order natural frequency rather than lower order.

Analysis for a novel folding frame tensegrity tent

This study presents a novel tensegrity tent based on the parameters of a certain military tent. Firstly, we derive the framework of the tent with the optimal complexity and the geometric and physical parameters of the elements are determined based on the certain military tent. Secondly, the dynamics, statics and modal analysis equation of the tensegrity tent are introduced. The prestresses of the elements to meet the first order natural frequency for the requirement of the International Building Code are given. The simulation method of membrane covering which applying strings instead of membranes is proposed. Then, the wind and snow loads analysis of the tensegrity tent are completed. Finally, we propose a folding scheme for the tent, and the joints required are also presented. The results show that the total mass and effective volume of the tensegrity tent is 45.4 kg and 37.1 m3 which are 12.1 kg less and 0.3 m3 more than the military tent, while the tensegrity tent can take the dynamic wind load with velocity of 22.1 m/s and the static snow load with thickness of 0.14 m, which are 1.4 m/s and 0.08 m more than the military tent. Compared with a certain military tent, the tensegrity tent has a more reasonable layout of the framework to take external dynamic and static loads while using less structural mass and maintaining the slightly larger effective volume.

Free vibration properties of beetle elytron plate: Composite material, stacked structure and boundary conditions

Beetle elytron plate (BEP), which is derived from the bionics of Trypoxylus dichotomus's elytra, is a typical lightweight and high-strength structure with superior and comprehensive mechanical properties. To promote the engineering application of BEPs, this paper explores the vibration characteristics of foam-filled fiber composite BEPs. Through the finite element method (FEM), structural parameters such as the length-to-width ratio, core height-to-thickness ratio, skin thickness and foam density influence the natural frequency and mode. Furthermore, the influence pattern of parameters (such as boundary conditions and the number of stacked layers) toward the first two orders of natural frequencies and modes of the BEP were investigated under controlled total height and controlled single layer height. The results indicate that 1) the ranking of structural parameters on the 1st order frequency from a strong to weak sequence is as follows: length-to-width ratio, core height-to-thickness ratio, foam density and skin thickness; for the 2nd order natural frequency: length-to-width ratio, skin thickness, foam density and core height-to-thickness ratio. 2) Generally, an increase in the length-to-width ratio will reduce the frequency of the BEP, and increases in the core height-to-thickness ratio and skin thickness will raise its frequency. Additionally, foam filling plays a decisive role in local deformation. 3) When the total height of the sandwich plate is controlled, the improvement of the boundary conditions can effectively boost its 2nd-order frequency (approximately 200%), but the 1st-order frequency can be significantly increased (approximately 140%) only when all four sides are improved. 4) In the case of controlling the height of the single layer of the sandwich plate, its frequency gradually increases with the increase in the number of stacked layers. However, when the constraints are weak or there are few stacked layers, increasing the number of stacked layers can effectively boost the vibration frequency. This paper provides references and suggestions for the design and application of sandwich structures, enriching the research on the mechanical properties of BEPs and establishing groundwork for promoting BEPs in engineering practices.

Optimization of frequency separation of laminated shells carrying transversely distributed mass using genetic algorithm

Shell structures are primarily used in the aerospace and maritime sectors. Such structures often oscillate, and undesirable oscillations can cause structural damage. So, to eliminate resonance and prevent damage to vibrating structures, it is necessary to build optimal shells. Maximizing the frequency gap between the first two eigenfrequencies is essential so that they do not coincide with the resonance frequency. Depending on the requirements of the design, the external frequencies might be placed between zero and the first natural frequency or between the first and second natural frequencies in order to prevent failure. This study proposes an efficient finite element model based on first-order shear deformation theory in combination with a genetic algorithm to maximize the frequency separation of laminated cylindrical shell panels. The maximum first natural frequency determined by the current FE-GA formulation is verified by previously published data, and it is found that the present results are better than the literature for 56% of the test cases. Moreover, for real-world situations like a clamped shell or a simply supported shell, the present solutions are approximately 0.6 to 2% better than the literature. Effect on the optimal design due to changes in mass ratio, curvature ratios of the shell panels, carrying dispersed mass over the entire shell panel, and dispersed patch mass at the middle is analyzed. The ideal stacking sequence for greatest frequency separation values is presented for a variety of curvature ratios and mass ratios.

A Rayleigh-Ritz solution for high order natural frequencies and eigenmodes of monopile supported offshore wind turbines considering tapered towers and soil pile interactions

Offshore wind turbines may be exposed to extreme or accidental loads during the service life, such as extreme water slamming, ship collisions, earthquakes, extreme winds, ice loads, etc. Under the action of such loads, they will respond not only in the primary eigenmode, but also high order modes. This paper presents a Rayleigh-Ritz solution for high order natural frequencies and eigenmodes of monopile supported offshore turbines. The model considers explicitly tapered wind turbine towers and soil-pile interactions. Different soil conditions are considered using equivalent cross coupled springs at the mudline. The proposed Rayleigh-Ritz solution is applied to the DTU 10 MW wind turbine mounted on a monopile foundation. The predicted natural frequencies and eigenmodes are compared with nonlinear finite element simulations using USFOS. The results show excellent agreement for the first 3 eigenmodes. Even higher eigenmodes than the 3rd order are also calculated but are not presented. They can be readily obtained if needed.High order natural frequencies and eigenmodes using the proposed Rayleigh-Ritz solution can, together with the Duhamel's integral, be well utilized in the buckling and collapse analysis and design of monopile supported offshore wind turbines in Ultimate Limit States (ULS) and Accidental Limit States (ALS). In addition, the high accuracy of the predicted primary natural frequency makes it a useful design tool to help avoid undesired resonance from external excitation loads and to reduce resonance related fatigue effects in Fatigue Limit States (FLS).

Influence of pier-water interaction on natural vibration characteristics of bridge with complex piers in water

In order to clarify the influence of water on the natural vibration of bridge with complex piers, based on a continuous beam with 4-column pier, the numerical analysis model is established. Single column circular pier is taken to discuss the range of waters. Then the influences of water on the natural vibration are analyzed. The research shows that waters reduce the natural frequency. When waters area width is less than 10 m, the natural frequency of the pier decreases. And the first-order longitudinal bending frequency is reduced by 3.36%. When waters area width is more than 10 m, the vibration frequencies tend to be stable gradually. Therefore, the waters 10 m can be regarded as an infinite boundary. The natural frequencies of single column pier and 4-column pier decrease with the increase of water depth. When the water depth is less than 10 m, the changes of natural frequency of the first four orders of single column pier are relatively small, and the changes of 5–10 order natural frequency are large. The maximum effect of the first ten orders is 14.84%. The natural vibration frequency of the bridge decreases gradually with the increase of water depth. The maximum effect of the first five orders is 3.33%.

Dynamic performance of a membrane-based variable focus lens with a large aperture.

Dynamic performance is one of the most important characteristics of a variable focus lens. However, there are few studies investigating the dynamic response of a membrane-based variable focus lens. In this paper, we present a mathematical model to describe spring-damping phenomena in theory. The first order natural frequencies with different scales were confirmed via finite element analysis. We also built a dynamic response experiment platform with changeable optical apertures, which was driven by a high-speed piezo stack actuator. A photodiode module was placed behind the lens to measure the variation of light luminance as the lens changed, and a laser displacement sensor was used to measure the deformation of the membrane. A series of data was collected with different optical apertures (20mm, 30mm, 50mm) and different pre-stretching ratios (200%, 300%) under different driving frequencies (from 5Hz to 25Hz in every 5Hz step). The experimental results were consistent with the mathematical model, which showed that the first order natural frequency increased as the aperture decreased or the membrane stiffness increased. This frequency-dependent characteristic of the variable focus lens provides a basis for further research on its dynamic performance.

Dynamic performance prediction of an annular tensegrity structure based on similitude method

Recently, the deployable antenna structure is designed and built extra larger to address the lack of observation accuracy and high resolution. However, this will bring new challenges for the performance testing of the prototype in a terrestrial environment. Similitude theory is a branch of methodology method that can reflect the basic configuration and characteristic of a reduced scale model. Therefore, this paper briefly introduces a conventional annular tensegrity that can be used by space deployable antennas. The Discrete Similitude Method is proposed to decompose a complex structure into equivalent beam or cable members. The similitude compatibility criteria of members is derived using the Discrete Similitude Method, along with the criteria of consistency of similitude scale factor and conservation of dimension. The criteria of consistency of similitude scale factor and conservation of dimension of the whole structure are also deduced to obtain the scale relationship of dynamic characteristics when the geometrical nonlinearity of beam and cable members are considered. Then, a similitude correction method to address the problem of the distortion in the size, load, and material properties of beam and cable member is proposed to study the dynamic similitude of annular tensegrity structures. The criteria of consistency of distorted coefficients of different beam and cable members are derived. Finally, the comparison of numerical and FEM results is presented to show that the dynamic performance prediction method has a high precision and can guide the design of reduced scale prototype. A 4 m prototype is fabricated and a modal experiment is tested. It is demonstrated that the 1st order natural frequency of the prototype can accurately predict for that of a 100 m oversized prototype.

Effect of the Stiffness of the Turntable Bearing Joint on the Dynamic Characteristics of the Five-Axis Machine Tool Rotary System

The positioning accuracy and machining performance of the five-axis machine tool are significantly impacted by the dynamic characteristics of the rotary system of the two-axis rotary table machine. In this study, first, the stiffness of the joint bearing of the rotary table system is calculated, and the effect of bearing clearance and external load on the stiffness is analyzed. Second, considering the stiffness characteristics of the joint, the natural frequency and mode shapes of the turntable system are calculated. Finally, the influence of turntable angle and bearing stiffness on the dynamic characteristics of the turntable system is analyzed. The results show that the natural frequency of the rotary table system does not change obviously with the axial stiffness of the turntable bearing joint, but increases significantly with the increase in the radial stiffness. The first order natural frequency of the turntable decreases with the increase in the swing angle, and the change in the first order natural frequency is 77.43 Hz. The research results provide theoretical basis and guidance for machine tool design and use.

An effective procedure for extracting mode shapes of simply-supported bridges using virtual contact-point responses of two-axle vehicles

This paper proposed an effective procedure using virtual contact-point responses to extract mode shapes of a simply-supported bridge. The theoretical closed-form solution of virtual contact-point response using two connected two-axle vehicles is derived based on the vehicle-bridge interaction theory. Variational mode decomposition and band-pass filter (VMD-BPF) method is adopted to decompose the virtual contact-point responses and extract the bridge mode shapes. And a mode shape splicing method is proposed to reduce the influence of vehicle length on the bridge mode shape identification. The accuracy of the identified mode shapes is verified by numerical simulations, indicating that the virtual contact-point responses of the front axle or rear axle in the two-axle vehicle can be used for mode shape extraction. Furthermore, parametric studies such as track irregularity, vehicle speed, and noise are conducted to analyze the parameters that affect the identification accuracy of the mode shapes. The results show that the first four order natural frequencies and corresponding mode shapes of bridges can be well identified by the virtual contact-point responses even when the track irregularity of grade 4 is used. It is also demonstrated that the proposed procedure has a good performance and robustness in extracting the mode frequencies and mode shapes of simply-supported bridges.