Higher-order Natural Frequencies Research Articles

A semi-analytical approach for dynamic responses of monopile-supported offshore wind turbines subjected to accidental loads

Offshore wind energy is leading the way in sustainable energy generation. Offshore wind turbines (OWTs) need to be designed to withstand extreme or accidental loads in Ultimate Limit States (ULS) and Accidental Limit States (ALS), whereby high order eigenmodes may become significant. This paper introduces a semi-analytical approach for efficient and reliable analysis of the dynamic responses of monopile-supported OWTs subjected to accidental loads. A Rayleigh-Ritz solution is initially adopted to determine the high-order natural frequencies and eigenmodes of monopile OWTs, explicitly considering tapered towers and soil-pile interactions. Dynamic responses of OWTs are then calculated depending on the interaction schemes between accidental loads and the structures, e.g. soft contact loads (e.g. slamming, wind) and hard contact loads (e.g. ship collisions, ice impact). For soft contact loading conditions, the loads are considered independent of turbine responses, and the transient dynamic response is computed using the classical modal superposition method. While for hard contact loading conditions, the load actions depend on the contact stiffnesses and responses of the interacting bodies. A numerical contact algorithm is thus developed, and numerical iterations are performed to ensure convergence. The proposed approach is applied to the DTU 10 MW monopile-supported OWT subjected to extreme water slamming and ship collisions, respectively as example applications of soft and hard contact scenarios. The results are verified against nonlinear finite element analysis using USFOS and discussed with respect to the turbine natural frequencies and eigenmodes, contact forces and dynamic responses. A parametric analysis is conducted for ship-OWT collisions, exploring different impact scenarios by varying ship sizes, contact stiffnesses, and initial impact velocities. The proposed approach can serve as a promising tool for accidental load response analysis and the design of monopile OWTs.

Research on Dynamic Characteristics Detection Method of Tall Power Transmission Tower

In addition to the gravity of the structure itself and the supported object, the open-air high-rise supported steel structure tower mainly bears dynamic loads. Take the transmission line tower as an example, such as wind load, conductor galloping, tower vibration and sudden unbalanced tension caused by line breaking. It is necessary to study the dynamic characteristic parameters to know the structural resistance to dynamic load. Therefore, from the perspective of dynamic safety evaluation, this paper designed the acquisition method of dynamic characteristic parameters and established a reliable finite element modal analysis model of transmission tower. Finally, the dynamic characteristic parameters such as natural frequency, vibration mode and amplitude of transmission line steel tower were obtained. It was found that the inclined material of the transmission tower might be unstable when the high-order natural frequency resonance occurred, which had important practical significance for structure design optimization and safety evaluation.

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).

Multiple-Degree-of-Freedom Modeling and Simulation for Seismic-Grade Sigma–Delta MEMS Capacitive Accelerometers

The high-order mechanical resonances of the sensing element in a high-vacuum environment can significantly degrade the noise and distortion performance of seismic-grade sigma–delta MEMS capacitive accelerometers. However, the current modeling approach is unable to evaluate the effects of high-order mechanical resonances. This study proposes a novel multiple-degree-of-freedom (MDOF) model to evaluate the noise and distortion induced by high-order mechanical resonances. Firstly, the MDOF dynamic equations of the sensing element are derived using the principle of modal superposition and Lagrange’s equations. Secondly, a fifth-order electromechanical sigma–delta system of the MEMS accelerometer is established in Simulink based on the dynamic equations of the sensing element. Then, the mechanism through which the high-order mechanical resonances degrade the noise and distortion performances is discovered by analyzing the simulated result. Finally, a noise and distortion suppression method is proposed based on the appropriate improvement in high-order natural frequency. The results show that the low-frequency noise drastically decreases from about −120.5 dB to −175.3 dB after the high-order natural frequency increases from about 130 kHz to 455 kHz. The harmonic distortion also reduces significantly.

Comparison of tensioning effect between roll and multi-spot pressure tensioning process of circular sawblade

ABSTRACT Tensioning is an important manufacturing process of circular sawblade, which can effectively improve the critical rotation speed of the sawblade. With the development of tensioning technology, the comprehensive research of each technology is of great significance. In this paper, the elastic-plastic numerical simulation models of roll and multi-spot pressure tensionings were established. The natural frequency and critical rotation speed of the tensioned sawblade were obtained through modal analysis and analytical calculation, using no buckling of the sawblade as the limiting condition. The results show that with the increase of the mold reduction, the anti-plane displacement, critical rotation speed, and high-order natural frequency of the sawblade increased, while the low-order natural frequency decreased. After the multi-spot pressure tensioning process, the critical rotation speed was higher, and the tensile tangential stress at the edge of the sawblade was greater. The post buckling forms of the roll tensioned and multi-spot pressure tensioned sawblades were a dish and saddle, respectively. Through this study, the upper limit critical rotation speed of the sawblade under different tensioning methods could be found, which could be used for the selection of tensioning methods and parametric optimization.

Equivalent orthotropic model for corrugated plates based on simplified constitutive relation

The transverse bending stiffness of a corrugated plate determined using an equivalent orthotropic model based on the Dirichlet boundary conditions and representative volume element method is the upper limit of the real solution, albeit often inaccurate in practice. Based on the Kirchhoff hypothesis, the constitutive relations of corrugated plates were reasonably simplified to propose a new corrugated plate equivalent orthotropic model. The resulting equivalent transverse bending stiffness agreed with the results of the most reasonable simplified expression available. Static response results obtained using the proposed model were more accurate than those obtained using the existing model without a simplified constitutive relation. Furthermore, for dynamic response, the proposed model exhibited improved accuracy when predicting low- and high-order natural frequencies. For the arc-and-tangent corrugated plate, two integral parameter expressions were calculated; the proposed equivalent plate model was verified by comparing the results of a full-scale test of the corrugated metal pipe arch culvert.

Fluid-structure interaction analysis of the rudder vibrations in propeller wake

Severe vibrations on the rudder worsen the vibrations and noise of the vessel. This study examines the rudder vibrations induced by propeller wake considering the fluid-structure interaction. The propeller's turbulent wake is solved with large eddy simulation. The lock-in regime under the different rotational speeds of propeller is discussed. The results show that rudder suffers from span-wise bending under the load induced by the propeller wake. This span-wise bending strengthens when the rotational speed increases. The rudder's vibrations are excited at the excitation frequencies and the natural frequencies. The vibrations of the rudder are locked at the first natural frequency fn1 when the blade passage frequency (BPF) is lower than 1.83 fn1. The vibration mode is similar to the first natural vibration mode in this lock-in regime. Intense vibrations are observed at BPF when the BPF exceeds 1.83 fn1. The vibration at the first natural frequency strengthens again when the shaft frequency exceeds 0.715 fn1. The vibrations at high-order natural frequencies strengthen as the excitation frequencies increase. The vibration at secondary natural frequency fn2 strengthens when the BPF approaches 0.532 fn2.

Shaking Table Test Research on the Influence of Center-Hung Scoreboard on Natural Vibration Characteristics and Seismic Response of Suspen-Dome Structures

The center-hung scoreboard is a large electronic display device, that usually has a significant weight and is flexibly suspended from the roof of the gymnasium. The suspen-dome structure is one of the common roof structures of gymnasiums. In order to investigate the effect of the center-hung scoreboard on the natural vibration characteristics and seismic response, including acceleration, displacement, and strain, of the suspen-dome structure and considering that the dynamic behavior of the structure under seismic action can be realistically reflected, a shaking table test was carried out based on the suspen-dome structure of the Gymnasium of the Lanzhou Olympic Sports Center. Firstly, according to the size of the shaking table, a dynamic scale model with a geometric similarity ratio of 1:20 was established. After that, the white noise excitation test and numerical modal analysis were conducted on the models with and without the center-hung scoreboard to compare the two modes’ natural vibration characteristics. Furthermore, the earthquake simulation test was carried out on the models with and without a center-hung scoreboard, and their various seismic responses, including acceleration, displacement, and strain, were compared and analyzed. Finally, the acceleration response and displacement response of the center-hung scoreboard were investigated. The results show that the higher-order natural frequencies of the suspen-dome structure will increase when the center-hung scoreboard is suspended from the roof, and the swing of the center-hung scoreboard will be excited first in the low-order mode. In addition, the various seismic responses, including acceleration, displacement, and strain, of the model with the center-hung scoreboard are all increased compared to the model without the center-hung scoreboard. Meanwhile, the influence of the center-hung scoreboard on the seismic response of the suspen-dome structure decreases from the inner ring to the outer ring of the structure. Moreover, under the action of an earthquake, the acceleration response and displacement response of the center-hung scoreboard are both extremely high. Considering the center-hung scoreboard in the analysis and design stage of the suspen-dome structure is necessary.

Experimental and numerical investigations on the influence of center-hung scoreboard on dynamic characteristics of suspend-dome structure

The LED display, i.e., the center-hung scoreboard, nowadays has become an indispensable display device in large stadiums, and some of which with weight is more than 30t. To investigate the influence of center-hung scoreboard load on the dynamic characteristics of the suspend-dome structure, a 1:20 dynamically scaled model of the Lanzhou Olympic Sports Complex was designed, considering different lifting-lowering heights of the center-hung scoreboard by changing the sling length. Shaking table dynamic characteristics experiment was conducted by using the white noise excitation method for the suspend-dome structure with the sling lengths of 200 mm, 400 mm, and 600 mm individually. The change rule of dynamic characteristic parameters such as natural frequencies and damping ratios was concluded by the Hilbert-Huang Transformation method for modal identification and the transfer function method for checking. With the increase of sling length, the natural frequencies of the center-hung scoreboard become smaller, and the damping ratios become more extensive. Compared with the results of the scaled model without a center-hung scoreboard, the high-order natural frequencies and damping ratios of the overall structure with the center-hung scoreboard are significantly increased, and the coupling effect between the center-hung scoreboard and the structure should not be ignored in the dynamic analysis. The test values were compared with the simulation results of Abaqus as well. It is shown that the test values of the scaled model are essentially consistent with the simulation results, which indicates that the test points are reasonably arranged, the test method is correct, and the finite element modeling method is accurate. The effects of sling stiffness and sling length on the dynamic characteristics of the suspend-dome structure were studied by Abaqus software. The results show that the sling stiffness hardly affect the dynamic characteristics of the overall model. When the sling length is less than 300 mm, the lifting-lowering of the center-hung scoreboard influences the distribution of the natural frequency and the vibration mode of the overall structure. In this range, the lifting-lowering of the center-hung scoreboard should be focused on actual engineering.

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

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.

An improved concentrated mass method for analysis of the mechanics modes of a four-stage pump rotor system

To improve the calculation accuracy for the mechanics mode of a double-support four-stage centrifugal pump rotor system (FCPRS), a simplified mathematical model derived from “four segments-four concentrations” and an ANSYS model were used to solve the model of a D-type FCPRS with balanced mass. Based on ourresults, we present an improved mathematical model combining "five segments-four concentrations" and the modal resistance-bending stiffness (RBS) correction coefficient. Moreover, the calculation results of the impeller mass eccentricity from the optimized model were compared with those from ANSYS simulations. The natural frequencies (NF) from the concentrated mass method (CMM) are slightly lower than those obtained from ANSYS. After optimization, the errors of the first four orders of NFs decreased from 1.13%, 1.24%, 7.62%, and 7.96% to 0.79%, 0.90%, 4.17%, and 4.51%, respectively. The NFs of each order increase slightly in the case of impeller mass eccentricity. The low-order NF increases slightly and the high-order NF decreases when the impeller in the middle is eccentric. The NF error obtained via the optimized CMM for the impeller with eccentric mass is slightly larger than that for the impeller without eccentricity.

Dynamic Performance Analysis of Ancient Buildings Based on Soil-Structure Interaction

This paper takes Guangyue Tower as the research object and uses ANSYS software to establish three finite element models of its upper timber structure, timber structure and high platform foundation and wooden structure, high platform foundation and soil mass. Through modal analysis, it is concluded that when the soil-structure interaction is considered, the natural frequency of the structure is reduced, and its influence on the high-order natural frequency of the structure is greater than the influence on the low-order natural frequency. Select El-Centro wave, Taft wave and artificial wave to analyze the dynamic performance. The results show that only considering the wood structure or wood structure and the high platform will make the calculated displacement smaller, so the influence of the existence of the foundation soil on the superstructure should be considered.

Multi-Objective Optimization Design of Natural Frequency of Two-Degree-of-Freedom Fast Steering Mirror System

In the design of two-degree-of-freedom (2-DOF) fast steering mirror (FSM) system, in order to improve the control bandwidth of the system, the low-order natural frequency in the working direction should be reduced as much as possible, and the high-order natural frequency in the non-working direction should be increased. In this paper, a deep-cut flexure hinge mirror system was studied. Firstly, the motion direction of the first to third order natural vibration mode of the system was analyzed. Next, the working stiffness in the third vibration mode direction was deduced to solve the problem that the traditional stiffness calculation method is not suitable for the third vibration mode direction. Then, the energy method and the Castigliano’s second theorem were used to analyze the working stiffness of the deep-cut flexure hinge. Next, the formula for calculating the relationship between the thickness of the mirror and the moment of inertia in the direction of the vibration mode was derived. Combined with the calculation formula of working stiffness in the vibration mode direction, the formula of the first to third order natural frequencies was derived, and the finite element verification and sensitivity analysis of structural parameters were carried out. Finally, by using NSGA-II, a multi-objective optimization design was carried out on the first to third natural frequencies of the system with hinge structure parameters and reflector thickness as independent variables, and the optimization results were analyzed by theoretical calculation and finite element simulation verification. The results show that the first and second-order natural frequencies of fast steering mirror system are reduced by 7.8% and 7.11%, and the third-order natural frequencies are increased by 139.8%. It proves that the optimized structure is much better than the original structure, and the optimal calculation can effectively increase the control bandwidth of the system

Characteristics of wave propagation through rock mass slopes with weak structural planes and their impacts on the seismic response characteristics of slopes: a case study in the middle reaches of Jinsha River

Time-frequency joint analysis was used to investigate the seismic response of rock slopes containing weak structural planes. Four three-dimensional finite element models containing infinite element boundaries were modelled: homogeneous slope, anti-dip slope, bedding slope, and block slope models. The material of the models was elastic material. The influence of the structural planes on the seismic responses of the slopes was systematically analyzed. The results of time-frequency joint analysis show that the structural planes impact the propagation characteristics of the waves and the dynamic amplification effect of the slopes. The wave propagation characteristics and peak ground acceleration distribution of the block slope model are more complex than those of the other models. The influence of structural planes on the amplification effect of the slopes is discussed. Additionally, according to frequency-domain analysis, structural planes have a small impact on the natural frequencies and the peak Fourier spectrum amplitude distribution of the slope but have a significant influence on its dynamic deformation characteristics. The natural frequencies of slopes can be obtained by using modal analysis and Fourier spectrum analysis. The relationships between the natural frequencies of the slopes and their dynamic deformation characteristics were analyzed; in particular, the impacts of low-order and high-order natural frequency on the deformation of the surficial slope were investigated. The applicability of the energy spectrum in the evaluation of the dynamic deformation characteristics of slopes was discussed. The effects of structural planes on the seismic failure mode of the slopes are identified according to the analyses of the dynamic responses of the slopes.

Study on dynamic response characteristics of high and steep layered rock mass slopes using a modal analysis method

The common geological bodies found in southwest China are layered mass slopes. The modal analysis method is performed on four finite element numerical models of rock slopes, including the homogeneous slope, horizontally layered slope, bedding slope, and toppling slope to investigate the dynamic deformation characteristics of the layered slopes. Their dynamic deformation characteristics are systematically studied according to the modal characteristics of slopes. By using modal analysis, the numerical results show that a series of natural frequencies and vibration modes of the slopes can be obtained. The type of structural planes affects the natural frequency of the slopes. The values of the natural frequency of the slopes are as follows: homogeneous slope > toppling slope > horizontally layered slope > bedding slope. Weak structural planes also have an amplification effect on the dynamic deformation of the slopes according to the analyses of the relative displacement (U) of the first vibration modes of the slopes. The natural frequency has a significant impact on the dynamic deformation characteristics of the slopes. Low-order natural frequency mainly induces the overall deformation of the top slope and slope surface area, whereas the high-order natural frequency mainly induces the local deformation of the slope. Modal analysis can provide a new idea for further study of dynamic characteristics of rock slopes.

A simplified methodology for finding the natural frequencies and mode shapes of the machine tool structures

Finding the natural frequencies of the machine tool structures is essential so that these can be kept away from the operational frequencies. Modelling and analysis of the actual structures is expensive and time-consuming. This paper presents a simplified, FEM based methodology for approximating the natural frequencies and mode shapes of the machine tools structures. This methodology was used to find the vibration characteristics of a vertical milling machine under different loading conditions. The results showed that an increase in the mass of the workpiece, causes a decrease in its lower-order natural frequencies, but the higher-order natural frequencies remain largely unaffected.

Seismic response analysis of a bedding rock slope based on the time-frequency joint analysis method: a case study from the middle reach of the Jinsha River, China

To investigate the seismic response of a bedding rock slope, a time-frequency joint analysis method was proposed using the finite element method (FEM) based on the time, frequency, and time-frequency domains. According to the time-frequency joint analysis, the results show that the topographic and geological conditions have great impacts on the dynamic response characteristics of the slope. An obvious slope surface and elevation amplification effect can be observed during earthquake excitation. The bedding structural plane has a great influence on the characteristics of wave propagation through the slope. The impacts of the structural plane and wave propagation directions on the seismic response of the slope are discussed. In addition, dynamic deformation characteristics and the natural frequency of the slope are clarified according to the Fourier spectrum and modal analysis. Low-order and high-order natural frequencies mainly cause the overall and local dynamic deformation of the surface slope, respectively. Moreover, the seismic energy of the Hilbert energy spectrum and marginal spectrum is mainly concentrated in the low-order and high-order natural frequency components, respectively. The difference in the seismic response between the slip mass and slip bed is the main triggering factor of a landslide based on the time-frequency joint analysis. The dynamic failure mechanism of the slope is identified: cracks first occur in the structural planes; then, with the earthquake excitation continuing, cracks further expand, deepen and penetrate, thus gradually forming the slip plane; finally, the overall shear failure of the surface slope appears.

Enhancing multiple harmonics in tapping mode atomic force microscopy by added mass with finite size

A method to enhance the harmonics of tapping mode atomic force microscopy is proposed in this study through an attached mass with finite size. It is demonstrated that ratios between higher-order natural frequencies and the fundamental one can be tuned to be specified integers with this method. The first three eigenmodes could be excited simultaneously, defining a highly-sensitive multi-harmonic cantilever, which can be utilized to resolve the sample properties effectively. The established theoretical model is validated by finite element analysis. This method can also be developed for the determination of mass, position and geometry of micro-particle in mass sensing application.

Natural frequencies of full-height panel reinforced soil walls of variable cross-section

Natural frequency has a significant influence on the dynamic response of structures. In this study, a new transfer matrix method is proposed to calculate the natural frequencies of reinforced soil retaining walls by modeling the facing and reinforcement layers as a beam-and-spring assembly using beam-on-elastic foundation theory. Among the advantages of the proposed method are that the method can be applied to both the uniform and tapering facing cross-sections, and it can be used to calculate higher-order natural frequencies of reinforced soil retaining walls. The accuracy of the proposed method is demonstrated by comparing its predicted frequency results with those using analytical and numerical methods reported in related literature. A parametric analysis is carried out, which indicates that the size and geometry of the facing panel have more significant influences on the natural frequencies of the full-height panel reinforced soil walls than their reinforcement properties.