Vortex-induced Force Research Articles

Mapping vortex-induced forces of oscillating bluff bodies from subcritical to critical Reynolds numbers

Mapping vortex-induced forces of oscillating bluff bodies from subcritical to critical Reynolds numbers

Aerodynamic interference between road vehicles and bridge deck subjected to vortex-induced vibration

An accurate assessment of the driving comfort and safety of road vehicles moving on a long-span bridge subjected to vortex-induced vibration (VIV) is essential for bridge administrators to decide whether the bridge should be closed to traffic. However, previous assessments often ignore the aerodynamic interference between the road vehicles and the bridge deck subjected to VIV. In this study, a specific wind tunnel model is developed to explore the aerodynamic interference between road vehicles and twin-box bridge deck during VIV. The vortex-induced force (VIF) and vortex-induced response (VIR) of the twin-box bridge deck and the aerodynamic forces on the vehicle were simultaneously measured. The influence of the vehicles on the VIV of the deck was investigated, and the influence of the deck vibration on the aerodynamic forces of the vehicle was also explored. The results show that the VIR and VIF of the bridge deck were generally reduced, depending on the type, position, and number of vehicles. The aerodynamic forces of vehicles could be amplified due to the deck vibration. These findings supplement the database of vehicle aerodynamic coefficients for assessing the driving comfort and safety of road vehicles moving on a long-span bridge subjected to VIV.

Aerodynamic force correlation of a foundational streamlined box girder under vortex-induced vibration in turbulent flow fields

Study on the vortex-induced vibration (VIV) performance of structures under vibration states in turbulent flow fields is closer to the essence of bridge VIV in real service environments, which has been lacking in existing research. The key parameter used in the VIV performance evaluation, vortex-induced force (VIF) correlation, was focused on in this paper. A foundational streamlined box girder was selected, and free vibration sectional model tests were conducted in turbulent flow fields using synchronous pressure-vibration measurement. The results indicate that turbulence have an impact on the VIV performance of streamlined box girder. The presence of turbulence is likely to decrease the VIV amplitude, but sometimes it may also increase it. The phase difference between the leading and trailing edges of the structural upper surface is approximately 100°, and the vortices on the upper surface do not drift continuously. The streamwise correlation of VIFs decreases to varying degrees with the increase in turbulence intensity. The aerodynamic force spanwise correlation when the structure is stationary is significantly weaker than when it is in a vibrational state. When the structure experiences sustained VIV, the spanwise correlation of VIFs decreases with increasing turbulence intensity, increases with increasing amplitude, and decreases with increasing spanwise distance and tends to be constant at a certain distance. Finally, an extended model based on Ricciardelli's correlation function was proposed, which can better describe the VIF spanwise correlation of streamlined box girders under different amplitudes.

An amplitude-dependent nonlinear approach for vortex-induced vibration evaluation of long-span bridges with inhomogeneous cross-sections

In the design of modern long-span bridges, inhomogeneous cross sections are increasingly used to meet various requirements, posing challenges in the investigation of vortex-induced vibration (VIV). This paper introduces a nonlinear approach with a focus on amplitude dependency in VIV for inhomogeneous bridges. It presents an inhomogeneous vortex-induced force model with amplitude-dependent parameters to capture spanwise variations in aerodynamics. These parameters are identified through multiple sectional wind tunnel tests. VIV amplitudes are determined by searching for solutions under the zero-damping condition for VIV. An illustrative bridge with an inhomogeneous fairing design is employed as an engineering application. The approach is rigorously validated with experimental data and applied to assess the prototype VIV behavior of the bridge with different fairing lengths. An optimal fairing length, constituting 40% of the main girder length in the mid-span, is identified to strike a balance between VIV performance and material usage. Discussion for the prediction results provides insights into the effects of fairing length inhomogeneity on VIV. In practice, deploying fairings in the large mode shape sections of the main girder achieves VIV mitigation comparable to a full-span installation.

Analysis of the Drag-Reduction Ability of the Layout and Cross-Sectional Shapes of Subsea Structures in the Critical Flow Mode

Introduction. Slender structures of subsea energy production systems are under constant influence of currents and waves. Hydrodynamic loads result from the interaction of subsea pipelines, umbilicals, equipment supports with fluid flows, and lead to the vortex formation in the area behind the structures. Vortex-induced forces are the sources of the cyclic loading. They accelerate gradually the fatigue damage, which may result in a failure. One of the ways to reduce the loads on subsea structures is to alter the shape of a cross-section, taking into account the flow regime. Dependence of the resulting hydrodynamic loads on the cross-sectional shape and relative position of structures has not been studied in details for the uniform flow in the critical mode. The current work is aimed at filling this gap. The research objective is to consider the impact of the distance between the structures, and also, the presence of a D-shaped structure, placed upstream relative to the group of three cylinders of different cross-sectional shapes.Materials and Methods. The computational fluid dynamics approach was used in this work for numerical simulations of vortex-induced forces in the ANSYS Fluent software for cylinder with D = 0.3 m. Modelling was conducted with the Detached Eddy Simulation (DES) method, which combined advantages of the Reynolds-averaged Navier-Stokes equation (RANS) method and the Large Eddy Simulation (LES) method. The object of the research was the system of four structures in the 2D computational domain, which included the upstream D-shaped cylinder and the main group of three cylinders with the circular, squared and diamond shapes of the cross-section. The transient process was considered, where structures were under the influence of the uniform flow in the critical regime at Re = 2.5×10⁵.Results. Five sets of data were obtained in simulations for the time-dependent coefficients of the lift and drag forces: for the main system — of the D-shaped, circular, square and diamond structures, and also for the four systems — of only D-shaped, only circular, only square and only diamond shaped structures. Additional analysis was conducted for the effect of the distance between the structures on the amplitude of fluctuating hydrodynamic force coefficients. The obtained results are presented as time histories of coefficients of the lift and drag forces, frequency analysis and contours of velocity, pressure and vorticity fields. The results indicate a positive effect of the upstream D-shaped structure on reducing the drag force, acting on the central structure in the group of three cylinders located downstream.Discussion and Conclusion. The results of the performed studies facilitate the informed decisions regarding the arrangement of subsea structures in a group of four objects, depending on the cross-sectional shape and the distance between the structures. The upstream D-shaped structure provides reducing the hydrodynamic drag force acting on the central structure in the downstream group of three structures, thereby slowing the fatigue accumulation and increasing the time of safe operation.

Numerical investigation of vortex-induced forces on a wind turbine tower segment at very high Reynolds numbers

The vortex-induced forces on an extruded cylinder with a span of two diameters representative of a finite segment of a non-tapered wind turbine tower at a very high Reynolds number (Re = 8.0×106) are numerically investigated using an incompressible Navier-Stokes flow solver with an Improved Delayed Detached Eddy Simulation (IDDES) turbulence model and correlation-based boundary layer transition modelling. The solution shows spanwise correlated structured vortex shedding with the Strouhal number St = 0.48. The boundary layer transition is found to occur at θ transition = 70 ◦ , and boundary layer separation is found to occur at θ separation = 120 ◦ . Results from the grid dependency study strongly imply that when using IDDES, the Strouhal number converges to higher values than previously reported by the literature as the grid is refined, with results ranging from St ∼ 0.44 using a grid with 4.2 × 106 cells, to St = 0.48 for the finest considered grid with 33 × 106 cells. This behaviour is not seen for URANS, where St = 0.33 for the finest grid.

Vortex-shedding and bistability in cylinder-flexible plate assembly in a channel

This study numerically investigates the fluid–structure interaction dynamics of a stationary circular cylinder with an attached flexible splitter plate in a confined fluid domain. We explore how variations in Reynolds number (Re), plate length, offset from the channel centerline, and blockage ratio influence the vortex-shedding and oscillatory behavior of the plates in laminar flow. Results indicate that at low Reynolds numbers (Re = 100), there are no vortex shedding or oscillations, while at higher Reynolds (Re = 200 to 300), the shorter plate exhibits first-mode oscillations and symmetry breaking, whereas the longer plate shows only transient second-mode oscillations at Re = 300 that diminish in amplitude, suggesting that increased plate length reduces vortex-induced forces. Dynamic Mode Decomposition (DMD) analysis confirms consistent primary flow structures across different Reynolds numbers, with initial modes encapsulating the majority of the system’s dynamics. In addition to determining the critical Re for vortex-shedding, vortex-induced oscillations and bistability, the study also examines the critical blockage ratios on the symmetry breaking phenomenon, and also shows that plate stability is influenced by domain asymmetry. These insights enhance understanding of the biomechanics of fish-like robots and inform the design and control of bioinspired flexible structures in fluid flows.

Experimental investigation on hydrodynamic characteristics of water intake riser undergoing vortex-induced vibration in uniform flow

With the development of ocean engineering, freely hanging water intake riser (WIR) has aroused widespread interests. The WIR with one free end would be easier to deflect remarkably, along with more complex vortex-induced vibration (VIV) responses and hydrodynamic forces. To elucidate the VIV characteristics of WIR under uniform flow, a series of VIV experiments for a scaled WIR model are carried out. Then a hydrodynamic identification method for VIV along with large offset is developed. The response and hydrodynamic characteristics of the WIR in cross-flow (CF) and in-line (IL) directions are further analyzed. Results indicate that VIV displacements with higher exciting modes are unstable when uniform flow velocity gets large. The IL-to-CF dominant frequency ratio of VIV is found to be 1.0, not 2.0 in conventional VIV knowledge and the IL harmonic at nearly twice the dominant CF frequency is observed more complex for flow velocity greater than 0.3 m/s. The vortex-induced force coefficients are found to be time-varying. Moreover, it is found that within the spectrum of Reynolds number from 2 × 103 to 1.1 × 104, the drag coefficient distribution changes along the WIR model, and the average drag coefficient is in range of 1.0–1.5, which is slightly smaller than the results predicted by the other published empirical formulae.

Simulation and prediction of vortex-induced vibration of a long suspension bridge using SHM-based digital twin technology

Although wind tunnel tests have already become a design basis for improving the stability of long-span bridges under the action of wind loads, vortex-induced vibration (VIV) events of in-service long-span bridges have still been frequently reported in recent years. The reasons behind this could be attributed to the limitations and uncertainties involved in wind tunnel tests as well as the changes and degradations of in-service bridges. In consideration that most long-span bridges have now been equipped with long-term structural health monitoring (SHM) systems and that digital twins could provide a new solution for simulating and predicting the reality, this study aims at developing a digital twin for VIV of a long suspension bridge to simulate and predict its VIV. The design document-based finite element (FE) model of the bridge is first updated using the measured dynamic characteristics. The numerical model for VIV of the bridge is then established using the vortex-induced force (VIF) model established based on wind tunnel test results. The numerical model together with the VIF model are finally updated against the measured vortex-induced responses (VIR) to produce a digital twin for VIV of the bridge. The relationships between the parameters in the VIF model and wind characteristics are further investigated to enable the digital twin to predict future VIV events. The accuracy of the digital twin for predicting the future VIV and the influence of the VIF parameters obtained from wind tunnel tests on the prediction accuracy both are confirmed through comparisons between the predicted and measured results.

Study on vortex-induced vibration mitigation of parallel bridges by multi-tuned mass damper inerter

This study proposes a novel Multi-Tuned Mass Damper Inerter (MTMDI) system for mitigating the vortex-induced vibration (VIV) responses of parallel bridges. In the proposed system, two individual bridges are linked using two Tuned Mass Damper Inerters (TMDIs). A model of parallel bridges with MTMDI under vortex-induced force is constructed, and two additional schemes wherein each bridge is equipped with a TMDI and a Tuned Mass Damper (TMD) separately are considered. The VIV responses of the parallel bridges are estimated through full bridge aeroelastic model wind tunnel tests. The performance of the MTMDI system with the optimal parameters obtained by the multi-objective optimization technique is compared with the performance of the other aforementioned two schemes. The results indicate that the optimal MTMDI system with smaller attached mass and inertance effectively mitigates the vertical VIV responses when applied to the parallel bridges. The frequency response functions and displacement time histories further demonstrate the effectiveness of the MTMDI system as an attractive alternative to the other schemes. Finally, the sensitivity of performance against variations in each parameter in the MTMDI system is discussed.

Vortex-Induced Force Identification of a Long-Span Bridge Based on Field Measurement Data

Vortex-induced force (VIF) identification and modelling of a long-span bridge are often conducted in terms of aeroelastic sectional model tests in wind tunnels. However, there are uncertainties inherent in wind tunnel model tests so that vortex-induced vibration (VIV) still occurs in real long-span bridges designed according to wind tunnel test results. This paper presents a framework for VIF identification of a long-span bridge based on field-measured wind and acceleration data. The framework is composed of the four steps: (1) decompose field-measured acceleration response time histories using variational mode decomposition (VMD) method; (2) obtain velocity and displacement response time histories using frequency domain integration (FDI) method; (3) establish and update the finite element model and identify the generalized VIF time histories of the bridge; and (4) identify the parameters in the polynomial VIF models and decide the most suitable VIF model. The proposed framework is finally applied to a real suspension bridge with a recent VIV event. The results show that the proposed framework can accurately identify the generalized VIF acting on the bridge from the field-measured acceleration and wind data, and the derived most suitable VIF model can produce almost the same vortex-induced response (VIR) as the measured ones.

Relationship between wake and cylinder dynamics for a cylinder undergoing modulated vortex-induced vibrations

Vortex-induced vibrations (VIV) in the initial branch are investigated for a 2-degree-of-freedom circular cylinder placed near a plane boundary (Re=UDν=200, where U is the inflow average streamwise velocity, D is the cylinder diameter, and ν is the kinematic viscosity) with imposed sinusoidal perturbations of the free stream at resonant, 2fo, and near-resonant conditions, 2.2fo (fo is the natural shedding frequency). The cylinder exhibits a quasi-periodic response, which challenges the comprehension of its relationship with the wake dynamics obtained through conventional VIV models. The total force acting on the cylinder is decomposed into a vortex-induced force, FV, linearly coupled to VIV, and a force induced by the effective mass of the cylinder, FS, which is non-linearly coupled to VIV. The proposed semi-empirical model reveals that the time-varying nature of the effective mass in FS drives the non-linear response. The model's physical consistency is verified against simulation results. While focusing on VIV in the initial branch, the validity of the proposed model is expected to extend to other branches of response, offering a promising avenue for developing a robust predictive model for VIV under various flow conditions.

Aerodynamic interference mechanism of vertical vortex-induced vibration of a ∏-shaped girder paralleled with a truss girder

In this study, the aerodynamic interference characteristics and mechanism of vertical vortex-induced vibration (VIV) of a ∏-shaped steel–concrete composite girder of a highway cable-stayed bridge and a steel truss girder of a railway cable-stayed bridge are investigated via experimental and numerical simulation methods, respectively. To this end, firstly, wind tunnel tests are conducted on section models with a geometry scale of 1:60, consisting of the ∏-shaped girder paralleled with truss girders to investigate their aerodynamic interference behaviors. Moreover, the fluid–structure interaction (FSI) approach based on dynamic mesh and user defined functions (UDF) is applied to study the wind-induced vibrations (WIV) of the single ∏-shaped girder and two parallel main decks for wind attack angle of 0°, respectively. Furthermore, the characteristics of vorticity, surface pressure, and vortex-induced force (VIF) distributions of the ∏-shaped girder and the truss girder are analyzed to investigate the aerodynamic interference mechanism. The results reveal that, the significant vertical vortex-induced vibrations (VIV) of the single ∏-shaped girder are observed, however, no obvious vertical VIV occurs for the single truss girder. Moreover, when the truss girder locates in the windward of the ∏-shaped girder, no obvious vertical VIV responses are observed. However, the ∏-shaped girder and the truss girder both exhibit significant vertical VIV responses for wind from the ∏-shaped girder side, and the maximum vertical VIV amplitude of the ∏-shaped girder is amplified by 32% compared with the single ∏-shaped girder. The vortices shed from the truss girder section disrupt the regular vortex shedding of the ∏-shaped girder section, thus suppressing the VIV of the ∏-shaped girder section. The WIV of the truss girder section is due to the forced vibration caused by the vortex impact of the dominant vortices shed from the ∏-shaped girder section. As the downstream flow field is blocked by the truss girder section, the scale and intensity of the dominant vortices is increased, leading to an amplification of the VIF and the VIV amplitude of the ∏-shaped girder section.

Study on vortex-induced multimodal coupled vibration of arch bridge suspenders

The objective of this paper was to investigate the 2-DOF suspender response under the condition of vortex-induced force and reveal the multimodal coupling mechanism. Firstly, an elastically mounted segmental model of suspender was established. Wind tunnel experiments were conducted to obtain the time history displacement signals in cross-flow and in-line directions, the empirical parameters were obtained by identify those signals. The nonlinear partial differential motion equation of suspender with two degrees of freedom was established based on Hamilton's principle, which considered the dynamic strain, bending stiffness and vortex-induced force of suspender. The Galerkin method was used to discretize the nonlinear partial differential equations into nonlinear motion governing equations, which considered the first three order modes. A numerical method was used to solve the motion governing equation, the time history displacement curve, phase diagram, amplitude spectrum and Poincare map of the multi-modal responses in the cross-flow and in-line directions were obtained. With the weak coupling effect of nonlinear stiffness and vortex-induced force, the cross-flow and in-line responses demonstrate period doubling bifurcation, quasi-periodic vibration, period doubling vibration, crises and transient chaotic phenomena. However, with the strong coupling effect, the crises and transient chaos in response amplitude disappear. More importantly, planar/non-planar multimodal internal resonance were found in this work. With the weak multimodal coupling effect in suspender response, the low-order modal response is dominant; as the multimodal coupling effect increases, the high-order modal response becomes predominant. In addition, the system energy not only transfers from high-order modal response to low-order modal response, but also the system energy transfer from low-order modal response to high-order modal response. These conclusions are useful for the design and practical engineering of suspenders, which have high theoretical research value in understanding the mechanism of multimodal coupling effects on suspender response.

Experimental study on the influence of maintenance track position on vortex-induced vibration performance of a box girder suspension bridge

To study the influence of maintenance track on the vortex-induced vibration (VIV) performance of main girder, the VIV response and time history of surface pressure data of a section model were obtained by wind tunnel vibration and pressure measurements for a large-span steel box girder suspension bridge. The VIV performance of the main girder was tested at ±5° attack angles of various maintenance track positions, including 1, 2.5, and 5 m away from the outer edge of the girder bottom plate. The mean values, root variances and amplitude spectra of vortex-induced force and the correlation and contribution coefficients of local aerodynamic force to overall aerodynamic force were analyzed. The results show that when the maintenance track is 1 m away from the outer edge of the bottom plate, the main girder exhibits the worst VIV performance with a maximum amplitude of 0.457 m, far beyond the allowable value of the specification. The VIV performance of the main girder was greatly improved by moving the maintenance track inward. The pressure analysis indicates that the large pressure fluctuation at the front and rear parts of the upper surface is attributed to the strong VIV of the main girder. In this sense, the improved VIV performance is mainly contributed by the weakening of pressure fluctuation in these two areas and the reduced local aerodynamic force. When the distance between the maintenance track and bottom plate is adjusted to 2.5 m, a 3.5 m wind barrier with a ventilation rate of 30% effectively inhibits the VIV of the box girder. The main reason for the suppression is that the elimination of pressure fluctuation on the upper surface of the girder disturbs the correlation between local aerodynamic force and overall aerodynamic force, consequently diminishing the contribution of local aerodynamic force to the vortex-induced force.

Modelling of vortex-induced force and prediction of vortex-induced vibration of a bridge deck using method of multiple scales

With the increase of span length, long-span bridges become more flexible and susceptible to vortex-induced vibrations (VIV). Several models for vortex-induced force (VIF) and various methods for predicting VIV have been developed accordingly but without consensus. This paper uses the method of multiple scales, from a different angle of view, to analyze the general polynomial model for VIF and present a concise polynomial model for VIF and a simple method for predicting VIV of a bridge deck. The aeroelastic vibration of a deck section is simulated by the computational fluid dynamics (CFD) and the results are used to verify the concise polynomial model of VIF. The forced vibration of a deck section is also simulated by CFD to yield the simple method for predicting VIV together with the frequency response equation and the critical equation of VIV derived by the method of multiple scales. The results show that the even and cross terms of aerodynamic damping and stiffness in the general polynomial VIF model are small compared with the odd terms and can be thus omitted. The lock-in region and the amplitude of VIV of the deck section can be accurately predicted by the proposed simple method.

Turbulence Effects on Vortex-Induced Dynamic Response of a Twin-Box Bridge and Ride Comfort of the Vehicle

The rapid growth of suspension bridges’ span makes vortex-induced vibration (VIV) appears more and more frequently, and once it occurs the closure of the bridge results in considerable economic losses. Investigating the dynamic behavior of the bridge experiencing VIV and the vehicles running on it is thus imperative for providing a reliable guidance for the managers to make operation decisions. Nevertheless, most of the existing studies focus on VIV of bridges subjected to smooth winds, but a certain level of turbulence always exists in reality. The effects of turbulence on vortex-induced dynamic response of the bridge and ride comfort of the vehicles are not clear. This study thus develops a coupled vortex-vehicle-bridge system applicable to the multi-mode lock-in regions of a twin-box deck subjected to both vortex-induced forces and buffeting forces in a turbulent flow. The system is then applied to a real long suspension bridge with three types of vehicles subjected to either smooth or turbulent winds. The results from the case study show that the increasing turbulence mitigates vortex-induced responses of both the bridge and the vehicles and reasonably improves the vehicles’ ride comfort. However, the buffeting forces induced by turbulent wind component should not be ignored when turbulence intensity becomes high.

Tuned Mass Damper Design for Vortex-Induced Vibration Control of a Bridge: Influence of Vortex-Induced Force Model

Existing studies optimized the tuned mass damper (TMD) for vortex-induced vibration (VIV) control of bridges based on an empirical linear model or Scanlan’s nonlinear model, which cannot accurately simulate the effect of the vortex-induced force and hence results in a TMD design that lacks effectiveness or robustness. Four vortex-induced force models are considered to systematically analyze the influence of force models on TMD design for VIV control. The design results based on various vortex-induced force models, as well as three design formulas, are compared in terms of effectiveness and robustness. Numerical results suggest that the empirical linear model and Scanlan’s nonlinear model significantly underestimate the TMD mass ratio required to suppress the VIV amplitude to be lower than a threshold. The empirical linear model may overestimate the robustness of the TMD to changing the frequency ratio, while Scanlan’s nonlinear model may overestimate or underestimate the robustness depending on the threshold VIV amplitude. The TMD design based on the polynomial model is considered more accurate since it can accurately predict VIV responses at various effective damping levels. The aerodynamic envelope model can be used as a convenient model for TMD optimization. The results generated using the free vibration formula are the closest to that of the polynomial model. An appropriate safety factor should be considered to convert the TMD design based on sectional model experiments to real bridges.

Study of Vortex-Induced Vibration of a Pipe-in-Pipe System by Using a Wake Oscillator Model

A pipe-in-pipe (PIP) system is an important engineering structure in the petroleum industry. Under the excitation of ocean fluids, the PIP system produces vortex-induced vibration (VIV), which can affect the oil recovery efficiency of large offshore drilling platforms and cause various engineering accidents. In this work, a fluid–structure coupled vibration model of the PIP system excited by a vortex-induced force is established based on an Euler–Bernoulli double-beam model and a wake oscillator model. The harmonic balance method is used to solve the coupled vibration model. Influences of the connection layer stiffness, tension, mass ratio, phase angle, and frequency ratio on displacements of the PIP system are studied. It is shown from numerical results that displacements of inner and outer pipes increase with the slenderness ratio. Displacements of inner and outer pipes decrease with the connection layer stiffness, but the frequency ratio and traveling wave velocity increase with it. In addition, the frequency ratio of the PIP system exhibits a multivalued characteristic, which is the characteristic of a nonlinear system. A small stiffness of the connection layer can induce unstable vibrations. Displacements of inner and outer pipes decrease with the tension, but the frequency ratio increases with it. The collision time of the inner and outer pipes gradually increases with the tension. The pipe displacements decrease with the mass ratio.

Nonlinear dynamics of viscoelastic fluid-conveying pipe installed within uniform external cross flow by pipe clamps

Nonlinear dynamic response of a viscoelastic fluid-conveying pipe excited by uniform external cross flow is investigated, which is installed by pipe clamps. A pipe clamp theoretical model is proposed for the actual pipe clamp instead of the commonly used simply supported or clamped boundary conditions. Based on the model, equivalent support stiffness and torsional stiffness are obtained from structure and material parameters of the pipe clamp. The vortex-induced lift force and the mean drag force are described by van der Pol oscillator model. The nonlinear equations of motion with the boundary conditions of different constraint stiffness are discretized by Differential Quadrature Method (DQM) and then solved by Runge-Kutta algorithm. The results are verified by Galerkin method for the pinned-pinned pipe, and by ANSYS for the pipe with two arbitrary constraint stiffnesses. The dependence of critical external fluid velocity upon constraint stiffness and pipe length is investigated. Moreover, the effects of support stiffness, torsional stiffness, internal fluid velocity, and viscoelastic coefficient on the amplitude response of the pipe are studied in detail. The influence of constraint stiffness on characteristics of lock-in region is elucidated. Consequently, pipe clamp spacing, material and structure parameters of the rubber mat can be designed to ensure the requirement of economy and safety, which are helpful to guide the layout and design of pipe clamps.