Higher Order Vibration Modes Research Articles

A simplified approach for judging and evaluating the contribution of higher-order modes to the wind-induced acceleration of high-rise buildings

Acceleration is the control parameter for the occupant comfort, often solved in the frequency domain with modal decomposition. In general, the first-order mode response is adopted to represent the total acceleration in practice. However, considering only the first-order mode has been demonstrated to yield underestimated results up to 20 %, while no exact criterion exists to determine whether higher-order modes require to be considered or not. Hence, underlying factors, like the structural natural frequencies, the excitation dominant frequencies, and the distributions of building masses/stiffness, which may significantly affect the contribution of higher-order modes to the acceleration, are analyzed first. Notably, the ratio of the natural frequency of the higher-order mode to that of the first-order mode plays a decisive role in the contribution of the higher-order mode's acceleration. Moreover, the third-order and beyond modes can be largely disregarded. Subsequently, an empirical approach is introduced to determine the need for considering high-order modes. Finally, a simplified formula for estimating acceleration response in the frequency domain is derived to consider second-order modal response. This study provides a method for determining the contribution of higher-order vibration modes to acceleration and a simplified formula to consider the higher-order modal response.

Influence of ground motion parameters on seismic response of a large-longitudinal-slope and small-radius curved girder bridge.

The mechanical performance of curved bridges under the action of an earthquake is complex. To obtain the influence of seismic parameters on the seismic response of curved girder bridges, this paper relies on a large slope small-radius curved steel box girder bridge (LSCGB) and selects seismic wave incidence angle, vertical component of ground motion, and site category as seismic parameters to carry out nonlinear time history analysis. Based on the analysis results of the case bridge, it is shown that the torsional vibration of the first 10 modes of LSCGB is significant, the modes are dispersed, and the contribution of high-order modes of vibration cannot be ignored. The most unfavorable seismic wave incidence angle is in the direction of 45°∼60° counterclockwise Angle from the central connection line of Pier No. 1 and Pier No. 4 of the bridge. The seismic response of the curved bridge components increases with the vertical seismic intensity, and the influence on displacement responses is more significant. The basic vibration period of curved girder bridges built on soft soil sites is extended by approximately 18.23%, and the seismic response of key components increases with the softening of the site soil. Therefore, when analyzing the seismic response of LSCGBs, the influence of vertical component of ground motion and site category should not be ignored.

Classification of Time–Frequency Maps of Guided Waves Using Foreground Extraction

Guided waves propagating in mechanical structures have proved to be an essential technique for applications, such as structural health monitoring. However, it is a well-known problem that when using non-stationary guided wave signals, dispersion, and high-order vibrational modes are excited, it becomes cumbersome to detect and identify relevant information. A typical method for the characterization of these non-stationary signals is based on time–frequency (TF) mapping techniques. This method produces 2D images, allowing the study of specific vibration modes and their evolution over time. However, this approach has low resolution, increases the size of the data, and introduces redundant information, making it difficult to extract relevant features for their accurate identification and classification. This paper presents a method for identifying discontinuities by analyzing the data in the TF maps of Lamb wave signals. Singular Value Decomposition (SVD) for low-rank optimization and then perform foreground feature extraction on the maps were proposed. These foreground features are then analyzed using Principal Component Analysis (PCA). Unlike traditional PCA, which operates on vectorized images, our approach focuses on the correlation between coordinates within the maps. This modification enhances feature detection and enables the classification of discontinuities within the maps. To evaluate unsupervised clustering of the dimensionally reduced data obtained from PCA, we experimentally tested our method using broadband Lamb waves with various vibrational modes interacting with different types of discontinuity patterns in a thin aluminum plate. A Support Vector Machine (SVM) classifier was then implemented for classification. The results of the experimental data yielded good classification effectiveness within reasonably low computational time despite the large matrixes of the TF maps used.

A novel multi-scale modeling strategy based on variational asymptotic method for predicting the static and dynamic performance of composite sandwich structures

To establish a universal and convenient mathematical model for predicting static and dynamic performance of composite sandwich structures, a novel three-dimensional equivalent homogenized model (3D-EHM) is proposed based on variational asymptotic method. The multiscale mechanical analysis of composite hexagonal and re-entrant honeycomb sandwich structures is conducted by 3D-EHM, enabling reasonable transmission of mechanical information at different scales and facilitating accurate predictions of mechanical responses in sandwich structures. The comparison analysis with 3D printing experimental results and 3D finite element model results demonstrates that the 3D-EHM exhibits remarkable precision and computational efficiency in forecasting static displacement distributions and higher-order vibration modes, eliminating the need for time-consuming and expensive experiments. Moreover, the effects of the ply angle of the face sheet and core geometric parameters on the effective properties of composite hexagonal honeycomb sandwich structures are systematically investigated. In summary, 3D-EHM is an effective applied mathematical model for designing and analyzing composite sandwich structures, fully combining the systematicity of variational methods and the asymptotic convergence of asymptotic methods.

FRF-based modal testing of the highest timber building in the world in operation using wireless accelerometers

Abstract This paper describes practical application of a novel and oven-controlled crystal oscillator (OCXO) high-precision synchronised wireless data acquisition system for a modal testing of the tallest all-timber building in the world – the 18-storey, 88.8m tall Mjøstårnet (the Mjøsa Tower) in Brumunddal, Norway. The modal testing was challenging as it was conducted in an occupied building in normal operation and was based on measuring both dynamic excitation and response yielding a set of frequency response functions (FRFs) across four different floors of the building. To the best knowledge of the authors, this was the first time that an FRF-based modal test was attempted on such a large building structure in operation. Moreover, although the structure was in operation, the testing was completed in only 48h, including commissioning and de-commissioning of all of the instrumentation, yielding a full set of point- and transfer accelerance FRFs. The paper presents the equipment used, key logistical challenges and quality assurance steps made to assure measurement of good quality FRF data in such short period of time demonstrating feasibility of the whole exercise. A total of seven modes of vibration, including higher order modes of vibration were estimated by curve-fitting the measured FRF data. This includes the 3rd bending and 2nd torsion modes difficult to measure via the standard ambient vibration testing of tall buildings. These modes could be used to investigate a number of uncertain modelling features in tall timber buildings, such as the stiffness of timber joints.

Simplification of Electrode Profiles for Piezoelectric Modal Sensors by Controlling Gap-Phase Length

This paper presents a method to optimize and simplify the electrode profile of a piezoelectric modal sensor. At the same time that the electrode profile is optimized to maximize the charge, a null-polarity phase is introduced. This gap-phase is modeled using the normalized norm of the spatial gradient of the polarity profile along with a two-step filtering and projection technique. High-order vibration modes generate a complex polarization profile that makes the manufacturing process difficult. The novelty of the proposed method is the addition of a constraint on the length of the interface in the topology optimization problem. This constraint simplifies the optimal designs and facilitates the fabrication process. Several examples show the simplified electrode profile that maximizes the electrical charge produced by a vibration mode, while reducing the number of different polarization regions by means of the gap-length constraint.

Symmetric/Asymmetric Vibrational Characteristics of Bi-Directional Functionally Graded Annular Microplates

In this study, a free vibration model of bi-directional functionally graded (BD-FG) annular microplates is established based on the modified couple stress theory (MCST) and the first-order shear deformation theory (FSDT), and the corresponding governing equations are derived from Hamilton’s principle. The annular microplates are assumed to be composed of metal and ceramic materials, and the equivalent material properties are a combination of power-law function and exponential ones. The generalized differential quadrature method (GDQM) is used to obtain the vibration frequencies and mode displacements for the C–C, S–C, C–S, and S–S (S: simply supported; C: clamped) annular microplates. The convergence and validity of the proposed model are verified through selective numerical examples. Further, the effects of gradient index, material length scale parameter, boundary conditions, and geometrical dimensions on the symmetric/asymmetric vibration frequencies of the annular microplate are explored. The modal assurance criterion is applied to estimate the influence of the radial gradient index and material length scale parameter on different-order mode displacements of the annular microplate. The results show that: (1) the vibration frequencies of the annular microplate are sensitive to changes in the geometry and boundary conditions; (2) compared with the axial gradient index, the radial gradient index not only affects the vibration frequencies of the annular microplate, but also impacts the distribution of the highest and lowest points of the vibration mode displacements as well as the vibration mode nodal lines; (3) as the material length scale parameter increases, the vibration frequencies are significantly increased, and the higher-order vibration modes of the annular microplate also change.

Experimental study of vibration modes switching based triple frequency-up converting energy harvesting with pre-biased displacement

In this study, a novel piezoelectric energy harvesting (PEH) system featuring triple frequency-up converting is proposed. The system comprises a piezoelectric cantilever and two stoppers. When in contact with the stoppers, the cantilever transitions into a high-order vibration mode, leading to the reversal of voltage outputs. The frequency-up converting effect is achieved by multiple mode switches occurring during a single cycle of motion. Experimental results demonstrated the significant ability of the proposed system to achieve triple frequency-up converting across a broad frequency range. Furthermore, three pre-bias displacements were introduced to assess the frequency-up converting characteristics of the PEH. The system exhibited a maximum ratio of triple and essential frequency components reaching up to 8.86. A maximum root-mean-square power output of 3.03 mW was achieved.

Numerical analysis on two-phase flow-induced vibrations at different flow regimes in a spiral tube

Spiral tubes are used in a wide range of applications and it is significant to understand the vibration introduced by two-phase flow in spiral tubes. In this paper, the numerical method is used to study the vibration induced by the gas-liquid two-phase flow in a spiral tube with different flow regimes. The pressure fluctuation characteristics at the pipe wall and the solid vibration response characteristics are obtained. The results show that the motion of small bubbles in bubbly flow leads to small pressure fluctuations with low-frequency broadband (0–50 Hz). The motion of the gas plug in the plug flow causes small amplitude periodic pressure fluctuation with a shortened low-frequency broadband (0–15 Hz) compared to the bubbly flow. The motion of the gas slug in the slug flow causes large periodic fluctuations in pressure with a significant dominant frequency (6–7 Hz). The wavy flow is very stable and has a distinct main frequency (1–2 Hz). The vibration regime in the bubbly flow and wave flow are close to the first-order mode, and the vertical vibrating component is dominant. The plug flow and slug flow excite higher-order vibration modes, and the lateral vibration component plays more important part in the vibration response.

Analysis of Dynamic Characteristics of Attached High Rise Risers

The exhaust chimney of third-generation nuclear power units is a typical attached high-rise riser structure. In this paper, the simplified mechanical model and dynamic model of China’s third-generation nuclear power Hualong-1 VNA system, including multiple nonlinear factors, are established for the first time. The DTM (differential transformation method) was first applied to solve the natural vibration characteristics of a multi-point constrained variable cross-section riser structure, and the effects of variable cross-section, variable mass, variable axial force, and different elastic constraint parameters on the natural vibration characteristics of the system were studied. The dynamic behavior of the VNA system under the combined action of internal flow velocity, vortex excitation, and foundation excitation was studied. The results show that the outer diameter function of the VNA system pipeline should be designed as a quadratic function or a near quadratic multi-segment constant value function. The “limiting” effect of constraining large stiffness can force low-order vibration modes with high constraint stiffness to jump to high-order vibration modes with low constraint stiffness. The elastic constraint arrangement scheme with near center symmetry can make the system vibration mode present a half stable and half-curved form. A new optimization design scheme has been proposed regarding the layout and stiffness parameters of the VNA system guide bracket. This can enable the VNA system pipeline to avoid severe oscillations near the response extreme values caused by multiple frequency excitations of seismic loads under design and accident conditions and ensure the service life of the equipment.

Dynamic response of a thin-walled curved beam with a mono-symmetric cross-section under a moving mass

This paper presents analytical solutions for the dynamic response of thin-walled curved beams induced by moving vehicles. The solutions encompass vertical, torsional, radial, and axial motions. A comprehensive set of governing equations for the dynamic response of thin-walled curved beams is established by considering the mass inertia of moving vehicles, rotary inertia, and warping resistance. The solutions are obtained for the four-directional response of curved beams, involving vertical, torsional, radial, and axial motions, based on the Fourier finite integral transformation, the Laplace–Carson transformation, their inverse transformations, and the Galerkin approach. A comparison of the results calculated by the analytical solutions in this study are compared with those calculated by the moving-force solutions using the Galerkin approach reported in a related literature, demonstrates the reliability and superiority of the analytical solutions. Hence, an extensive parametric study is carried out to investigate the effect of the mass inertia of moving vehicles, velocities, higher-order modes of vibrations, and radii of curvature on the dynamic responses of beams. As per the results, the mass and velocity of the moving vehicles play important roles in the dynamic response of curved beams. Furthermore, the dynamic response for thin-walled curved beams increases significantly with heavier vehicles and higher velocities, confirming that it is necessary to consider the mass inertia effect to study the dynamic response of curved beams.

Maximisation of Bending and Membrane Frequencies of Vibration of Variable Stiffness Composite Laminated Plates by a Genetic Algorithm

PurposeIn comparison to traditional, constant stiffness laminates, variable stiffness composite laminates (VSCL) with curvilinear fibres represent an extra analysis effort. It is the purpose of this work to present and test a relatively simple optimisation procedure, in order to find the maximum fundamental frequency of vibration in bending and in in-plane vibrations. It is also intended to explain why certain fibre paths lead to higher frequencies.MethodsThe optimisation is performed using a genetic algorithm (GA), which is described in detail. The bending and the in-plane plate models are based on the p-version Finite Element Method. Each model requires a small number of degrees of freedom, an important feature because applying the GA involves the solution of a large number of eigenvalue problems. In order to support the physical interpretation of the optimal designs, mode shapes and stress fields corresponding to some optimal solutions are illustrated.ResultsSingle- and multi-layer plates with different boundary conditions and fibre path types are studied. Fibre paths that lead to maximum fundamental frequencies are found and justified. The consequences that maximising the first frequency has on the higher-order modes of vibration are studied.ConclusionThe proposed optimisation and modelling methods are effective. Curvilinear fibres with the characteristics considered led to the maximum first natural frequency of vibration in a few cases, but not all. Particularly in in-plane vibrations, curvilinear fibres can provide major gains in comparison to straight fibres. The increase in the vibration frequency is accompanied by, overall, larger stresses.

Shaking table test of a new special-shaped arch bridge

In this work, a shaking table test was conducted on a new special-shaped arch bridge consisting of an inclined arch and a curved beam called BIACB. In order to meet a landscape requirement, the design of BIACB is unique and complex. Therefore, different from conventional bridges, it exhibits a prominent space effect and unpredictable complex seismic responses. To check the seismic performance of the BIACB, a 1:40 scaled test model was designed and fabricated to achieve a shaking table test by using simulated earthquake. The seismic responses of the tested BIACB model under three typical seismic excitations (El-Centro wave, Taft wave and Artificial wave) were explored. It is found that: (1) The scaled test bridge model can well reflect the seismic response characteristics of the prototype bridge verified by the corresponding finite element analysis. (2) Obvious dynamic amplification phenomena in the transverse and vertical directions of the bridge were observed, showing that the vertical response of the curved beam is greater than the transverse one, while the transverse response of the leaning arch is much more apparent than the vertical one. Generally, the seismic responses of BIACB under transverse seismic excitation are more obvious than those under longitudinal seismic excitation. (3) Higher-order vibration mode of the BIACB has a substantial participation under the longitudinal seismic excitation. (4) The inclined arch significantly amplifies the impact of vertical seismic excitation on the BIACB. (5) The relative variation amplitudes of suspender force of short suspenders on the BIACB are greater than that of long suspenders under longitudinal seismic excitation. The research findings are of great potential to critically guide the seismic design of relevant bridges.

Dynamic response amplification of resonant microelectromechanical structures utilizing multi-mode excitation

This work presents an analytical and experimental study to enhance the dynamic response of the higher-order vibration modes of resonant microstructures. The detection of higher order vibration modes is usually very difficult due to their low amplitude response, which gets buried in noise. Higher input voltages can be used to enhance the response; however, it is highly energy demanding, can result in electronics problems, and may lead to pull-in of the lower participating modes. In this paper, we present a multi-mode excitation (MME) technique to enhance the vibration response of the higher order modes of an electrostatically actuated micro cantilever beam. First, we derive the dynamic equations of motion of the micro resonator accounting for two electrostatic excitation sources. Then, we apply the Galerkin method to analyze the dynamics of the system analytically. We investigate the effect of using various dynamic AC combinations on the involved modes to yield optimal conditions of the amplitude magnification. Then, an experimental investigation is conducted based on a micro resonator made of polyimide. The results showed that, compared to the single mode excitation (SME), the MME can significantly raise the level of the response above noise and amplify the displacement amplitude to yield a higher signal-to-noise ratio. The maximum displacement amplitude can be amplified multiple times as determined by the AC loads. Hence, the improved displacement amplitude with the proposed technique makes it promising for future signal processing technologies and for realizing high-sensitivity sensors.

Electrode pattern definition in ultrasound power transfer systems

We use a high pattern-fidelity technique on piezoelectric electrodes to selectively excite high-order vibration modes, while isolating other modes, in multi-layered through-wall ultrasound power transfer (TWUPT) systems. Physical mechanisms, such as direct and inverse piezoelectric effects at transmitting and receiving piezoelectric elements, as well as wave propagation across an elastic barrier and coupling layers, all contribute to TWUPT. High-order radial modes in a TWUPT system feature strain nodes, where the dynamic strain distribution changes sign in the direction of disks' radii. This study explains theoretically and empirically how covering the strain nodes of vibration modes with continuous electrodes results in substantial cancelations of the electrical outputs. A detailed analysis is given for predicting the locations of the strain nodes. The electrode patterning for creating the transmitter and receiver shapes is determined by the regions where local force and charge cancelation do not occur, i.e., the two modal principal stress components have the same sign. Patterning for creating the electrode shapes is performed by high-fidelity numerical modeling supported by experiments. Using differential excitation on the transmitter side while monitoring transmitted power and efficiency on the reception side at various vibration modes is made possible by the unique nature of TWUPT systems. Due to an improvement in system quality and power factors, it is determined that employing the proposed electrode pattern designs enhances overall device efficiency and active power. The suppression of other modes makes up a filter feature that is paired with the enhancement at the mode under consideration.

Novel FRF-based fast modal testing of multi-storey CLT building in operation using wirelessly synchronised data loggers

This paper presents a novel input–output frequency-response function (FRF) based field modal testing (MT) of an operational and fully occupied tallest cross-laminated timber (CLT) building in the U.K. A custom-built MT system and testing protocol were developed to facilitate exceptionally fast field testing work lasting only 10 h, including all instrumentation and field testing work.This yielded eight fundamental and higher-order modes of vibration with natural frequencies up to 12 Hz. The higher order modes are normally not possible to measure well enough using the standard output-only operational modal analysis (OMA). An FE model was developed prior to the testing to assure quality and facilitate the fast testing process. The FE model, based on the best engineering judgement, proved to be able to predict very well the key features of the test building. This includes close matching and correct clustering of the FE-calculated and MT-estimated natural frequencies, as well as a very reasonable prediction of the static stiffness at the top of the building. The in-situ measured horizontal static stiffness at the top of the CLT building is a considerable benefit of the field FRF measurements and is not possible in the standard OMA. It was shown that the preliminary best practice FE model was over-predicting the static stiffness in the two orthogonal directions by only up to 22% of the measured values.Curve-fitting of the good quality FRF data yielded damping ratio values for the higher order modes of vibration, typically above 3%. This is quite high for a full-scale multi-storey residential building.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

Direct Observation of Thermal Vibration Modes Using Frequency-Selective Electron Microscopy

Thermal vibration properties of nanometer-scale objects are critical for their application in devices such as nanomechanical resonators. An imaging method has been developed which allows the direct visualization of higher-order thermal vibration modes at room temperature, which have so far been inaccessible to observation due to their subangstrom amplitudes and the much stronger overlapped first mode. This technique, combining aberration-corrected scanning transmission electron microscopy with broad-band signal acquisition in the time domain, can display the amplitude distribution of several thermal vibration modes simultaneously by selecting specific frequency windows. This is showcased by mapping the first six thermal vibration modes of a singly clamped nanowire and comparing them to natural vibration mode profiles obtained by finite element calculations. This implementation furthers our understanding of the collective Brownian motion of nanostructures and extends the analysis capabilities of electron microscopy.

Higher-order vibration of thick composite and sandwich plates based on an alternative higher-order model

The transverse stretching vibration of thick sandwich plates, which is attributed to largely different stiffness at the adjacent layers, is a challenging issue, and efficient approach for such issue is less reported in the published literature. Thus, natural frequencies corresponding to stretching vibration modes are generally neglected in engineering design, which might impact structural safety as frequencies of the exciting force are close to transverse stretching vibration frequencies. This paper proposes an alternative higher-order model for dynamic analysis corresponding to the higher-order vibration modes. The proposed model is classified in the displacement-based equivalent single-layer theory, as the number of displacement parameters in the proposed model is independent of the layer number. The continuity of displacements and transverse shear stresses can be fulfilled at the interfaces between the adjacent layers of structures. To demonstrate the capability of the proposed model, typical examples are analyzed by utilizing the proposed model, the three-dimensional finite element method and the chosen higher-order models. By comparing with the exact three-dimensional elasticity solutions, it is found that the proposed model can yield more accurate natural frequencies corresponding to the higher-order displacement modes than the selected models. Moreover, the factors influencing reasonable prediction of the higher-order frequencies are investigated in detail, which can provide a reference for the accurate prediction of the higher-order frequencies.

Vortex-induced vibration of flexible riser transporting high-speed spiral flow in deep-sea mining

According to Hamilton's principle and a mechanical decomposition model proposed for high-speed spiral flow, vibration governing equations of flexible riser transporting high-speed spiral flow are developed for vortex-induced vibration (VIV) analysis for lifting pipe system in deep-sea mining. The classic van der Pol oscillator is used to simulate the dynamic characteristics of VIV. The governing equations are then discretized and solved by the finite element method and the Runge-Kuta method. Furthermore, dynamic characteristics of VIV of the flexible riser transporting straight flow and spiral flow are investigated and the effects of the spiral flow incidence velocity and angle controlled by the jet device on VIV responses are evaluated. Also, the VIV dynamics considering different damping ratios are examined. The results show that the rotational angular velocity of the spiral flow would amplify the mean deflection of the in-line (IL) direction, and the incidence velocity has a significant nonlinear effect on the dominant frequency of the riser, which would induce the high-order dominant vibration modes when the incidence velocity is large sufficiently. However, the incidence angle has relatively less effects on the VIV responses of flexible riser. Moreover, the locked interval of VIV is less influenced by damping ratio below 0.1.