Static Equilibrium Configuration Research Articles

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Normal contact stress analysis of large-deflection compliant mechanisms using a CPRBM-based method

The contact interaction between compliant mechanisms and external objects is a common occurrence in the field of grasping and manipulation. However, accurately modeling the large deformation and contact force/stress remains a challenging task. This paper presents a contact analysis method to evaluate the deformation and normal contact force/stress of compliant mechanisms with general-shape beams. The general beams are modeled by using the chained pseudo-rigid-body model (CPRBM). On basis of CPRBM framework, linear and distributed contact springs are introduced at the interface to represent the contact force. A numerical method is established to fast calculate the boundary penetrations, and a contact stiffness formulation is proposed based on the theory of elasticity. Then, the effect of contact interaction can be evaluated by the contact potential energy of contact springs, which is part of the system’s total potential energy. By minimizing the potential energy function of the compliant system, the static equilibrium configuration and contact force/stress can be obtained easily. Numerical examples are used to verify the feasibility and accuracy of the proposed method.

Three-dimensional nonlinear vibrations of slightly curved cantilevered pipes conveying fluid

In this paper, nonlinear static and dynamic behaviors of slightly curved cantilevered pipes conveying fluid under transverse base excitations are investigated. Incorporating geometrical and inertial nonlinearities, a new three-dimensional dynamic model is developed for the inextensible cantilevered pipe based on the Euler–Bernoulli beam theory. The resultant partial differential equations are discretized using the Galerkin method and solved utilizing the pseudo-arclength continuation technique and a time-integration method. The static equilibrium configuration, linear stability, and nonlinear response of the pipe are calculated for various cases, which allow us to investigate the effects of flow velocity, gravity coefficient, initial curvature amplitude and shape, and base-excitation amplitude and frequency. An emphasis is put on the importance of modal interactions on frequency-response and force-response relationships, evidencing the possible in-plane and out-of-plane transverse vibrations and identifying the preferred forms of pre- and post-instability behaviors. This study offers an efficient theoretical model for exploring the nonlinear forced response of slightly curved cantilevered pipes conveying fluid and provides a better insight for further investigations on the modal interaction in cantilevered pipe devices.

Stability of hypersurfaces with constant mean curvature trapped between two parallel hyperplanes

Static equilibrium configurations of continua supported by surface tension are given by constant mean curvature (CMC) surfaces which are critical points of a variational problem to extremize the area while keeping the volume fixed. CMC surfaces are used as mathematical models of a variety of continua, such as tiny liquid drops, stars, and nuclei, to play important roles in both mathematics and physics. Therefore, the geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and in a variety of physical sciences. In this paper we examine the stability of CMC hypersurfaces in arbitrary dimensions, possibly having boundaries on two parallel hyperplanes, by investigating the second variation of the area. We determine the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations.

Refined Semi-Analytical Framework to Predict the Natural Vibration Characteristics of Bistable Laminates

Bistable unsymmetrical laminates have received significant attention in morphing applications due to their ability to attain multiple shapes when subjected to thermal loads. Morphing structures in general are subjected to dynamic operating conditions. Also, the highly nonlinear snap-through transition between stable configurations possesses rich dynamic characteristics. Therefore, understanding the dynamic characteristics of bistable laminates is essential for designing morphing structures constituting bistable elements. Thus, the present study aims to explore the dynamics of bistable unsymmetrical laminates by evaluating their natural vibration characteristics associated with small-amplitude dynamic excitation around the static equilibrium configurations. A refined semi-analytical framework is proposed to analyze the natural vibration characteristics of the bistable laminate, where the potential energy is expressed only in terms of the unknown coefficients of the assumed out-of-plane displacement function. The in-plane components are separately evaluated using the in-plane equilibrium equations and compatibility conditions. In the dynamic analysis, perturbations are imposed on the static equilibrium configurations to capture the modal characteristics. A full geometrically nonlinear finite element (FE) model of the bistable laminate has been created in a commercially available FE package to compare semi-analytical solutions. To validate the proposed frameworks, an experimental strategy to capture the natural frequencies of a bistable laminate is presented in this paper. Unsymmetric laminates mounted at its center have been used for the experimental testing, where the vibrations are measured using miniature integrated electronics piezoelectric accelerometer sensors attached at the corners. The semi-analytical and FE results are validated against the experimental observations for the selected unsymmetrical cross-ply laminates. The proposed frameworks are further extended to a family of unsymmetrical variable-stiffness (VS) laminates generated using curvilinear fiber alignments. The selected VS family can generate bistable shapes without any twisting curvature similar to that of an unsymmetrical cross-ply laminate, where the designer can expand the design space with a plethora of multiple configurations. A parametric study is performed by tailoring the VS parameters to investigate the influence of curvilinear fiber alignments on the natural vibration characteristics of bistable VS laminates.

Free vibration analysis of Timoshenko pipes with fixed boundary conditions conveying high velocity fluid

Sever vibration will be induced due to the high flow velocity in the pipe. When the flow velocity exceeds the critical value, the static equilibrium configuration of the pipe loses its stability, and the vibration properties change accordingly. In this paper, the free vibration characteristics of the pipe with fixed-fixed ends are revealed in the supercritical regime. Based on the Timoshenko beam theory, the governing equations of the nonlinear vibration near the non-trivial static equilibrium configuration are established. The influences of system parameters on equilibrium configuration, critical velocity, and free vibration frequency is analyzed. The effects of supercritical velocity in different ranges on the natural frequencies are revealed. In addition, the comparison with the Euler-Bernoulli pipe model shows that the differences in critical velocity, equilibrium configuration, and frequency are still significant even the length-diameter ratio is large. The increase of the flow velocity reduces the difference of non-trivial static equilibrium configurations, but eventually aggravates the difference of natural frequencies. Within a certain supercritical velocity range, the vibration difference between the two pipe models is small, beyond this range, the vibration difference increases significantly.

Phase transition and stiffer core fluid in neutron stars: effects on stellar configurations, dynamical stability, and tidal deformability

In this work, we investigate the influence of the phase transition and a stiffer fluid in neutron stars’ cores on the static equilibrium configuration, dynamical stability, and tidal deformability. For this aim, it is taken into account that the fluid in the core and the envelope follow the relativistic polytropic equation of state. We find that the phase transition and a stiffer fluid in the core will reflect in the total mass, radius, speed of sound, core radius, radial stability with a slow and rapid conversion at the interface, and tidal deformability. We also investigate the dimensionless tidal deformability varLambda _1 and varLambda _2 for a binary neutron stars system with chirp mass equal to GW170817. Finally, we contrast our results with observational data to show the role that phase transition and a stiffer core fluid could play in the study of neutron stars.

Rapidly rotating neutron stars with realistic nuclear matter equation of state

AbstractWe performed the comparison of three different numerical codes for constructing equilibrium models; (I) a code for static equilibrium configurations, (II) an implementation of the Hartle–Thorne slow‐rotation approximation, (III) a numerical solution of the full Einstein equations by LORENE. We aimed to construct sequences of uniformly rotating configurations at various rotation frequencies up to the Keplerian frequency for a hybrid hadronic–quark matter EOS where a smooth transition is provided between two separate phases. We investigated the difference between the results computed by the implementation of Hartle–Thorne slow‐rotation approximation and by LORENE/nrotstar, respectively. We have concluded that the codes can increase the difference between the slow‐rotating and the fast‐rotating approach exponentially, reaching 6.67% for the maximal mass configuration rotating at the Keplerian frequency.

Weakly nonlinear two dimensional wave scattering in incompressible pre-stressed materials: Far field approximation

We develop a series expansion approach to the calculation of the scattering coefficients of in-plane transverse elastic waves. The waves are scattered by a cylindrical obstacle as they propagate through an incompressible nonlinear material which is inhomogeneously pre-stressed. We deal with the calculation using a distorted wave Born approximation in the far field. A neo-Hookean strain energy function is considered for the incompressible elastic material, and a rigid cylinder is considered as an obstacle. From the static equilibrium configuration, a small-on-large approach is constructed via a series expansion of the wave equation which allows us to compute both the scattered wave and the induced pressure field, and their dependence on the applied pre-stressed. In the far field limit, the pre-stress effect on the scattering coefficients is calculated via partial waves. The formalism can be expanded directly to three-dimensional media with different obstacle symmetries.

In-plane free vibrations of small-sag inclined cables considering bending stiffness with applications to cable tension identification

This study proposed an analytical model to characterize in-plane free vibration of a cable considering sags, inclined angles, and non-negligible bending stiffness to enhance the accuracy and efficiency of vibration analysis and health monitoring of cable structures under complex engineering environments. The static equilibrium configuration of the cable is derived, and the singularity at boundary regions is elucidated in detail. The analytical solution of the dynamical control equation is obtained using the successive approximation method. The influences of the sags, inclined angles, and bending stiffness on eigenfrequencies and mode shapes are individualy studied via introducing three non-dimensional parameters. The analytical model is further examined through parametric anaysis. As a typical extension, the model is used to establish a cable tension identification method, in which an inverse problem is solved by the trust-region-dogleg technique. The identification method is verified through comparisons with existing methods established based on classical string vibration theory and various numerical methods, e.g., finite element method and finite difference method. Validations are made by using experimental outcomes, on which the data collected from both an experiment in laboratory and a field experiment in the Poyanghu bridge are used for comparison. It is found that the proposed model along with the method is able to capture precise vibration features and identify cable tensions accurately in favor of the improvement of cable condition monitoring and health management in fields such as bridge engineering.

Bifurcation and stability analysis of static equilibrium configuration of curved pipe conveying fluid

We present bifurcation and stability analysis of static equilibrium configurations of a clamped-clamped curved pipe conveying fluid in this paper. To capture large deformations of the pipe, this pipe is modeled with an absolute nodal coordinate formulation (ANCF). A technique of parameter continuation is used to solve the equilibria of the governing equation of this pipe system. Effects of external force, flow velocity and arc angle on the nonlinear responses of the curved pipe conveying fluid are discussed in detail. In the case of the curved pipe without any external loadings, the system admits two stable equilibria and an unstable equilibrium when the flow velocity exceeds a critical value on which a saddle-node bifurcation occurs. This critical flow velocity that characterizes the onset of the multistability is increased as the arc angle of the pipe increases. In addition, a pitchfork bifurcation occurs along the branch of the unstable equilibria. When the curved pipe is subjected to a concentrated force at its midpoint, the pipe deforms only in symmetrical mode shapes and undergoes a snap-through buckling under variations in the concentrated force. In contrast, when the curved pipe is subject to a gravity and the arc angle of the pipe exceeds a critical value, a pitchfork bifurcation buckling is observed under variations in the gravity. Along the secondary branch of this pitchfork bifurcation, equilibria in asymmetric mode shapes are found. Finally, effects of system parameters on these critical buckling loads are carefully explored.

Critical velocity and supercritical natural frequencies of fluid-conveying pipes with retaining clips

Although retaining clips are widely used to constrain high-velocity fluid-conveying pipes, there are few studies on the influence of retaining clips on the supercritical vibration characteristics of the pipes. In this study, the vibration characteristics of the fluid-conveying pipe with retaining clip are firstly presented in the supercritical regime. According to the generalized Hamiltonian principle, the governing equation of the fluid-conveying pipe with the retaining clip is derived. When the flow velocity exceeds the critical velocity, the equilibrium configuration of the pipe becomes the non-trivial static equilibrium configuration. By coordinate transformation, the governing equation of the supercritical pipe around the non-trivial static equilibrium configuration is obtained. Emphasis is put on the effects of the clip stiffness and position on the critical flow velocity, equilibrium configuration and natural frequencies. It is found that variations in the retaining clip stiffness can cause different types of non-trivial equilibrium configurations. The natural frequencies will not be monotonic with the increasing clip stiffness in the supercritical flow. Furthermore, the two-span model of the pipe with the clip is obtained for verification. In conclusion, this work lays a foundation for the study of the dynamic response of the supercritical fluid-conveying pipe with retaining clip and can provide a reference for the analysis of the vibration characteristics of the clips and pipes system in engineering.

Semi-analytical solution for static and quasi-static analysis of an inextensible cable

The static and quasi-static analysis of cable structure and cable-driven systems is of significant interest to the researchers. In this paper, a semi-analytical solution for static equilibrium analysis of an inextensible cable is developed to know its initial configuration and tension. The analytical expression for the static equilibrium configuration of a cable, fixed at the two ends is generally available in the literature in terms of unknown reactions at the boundary nodes. Conventionally, researchers use numerical methods to solve for those unknown reactions at the boundary nodes, where a solution is sensitive to the initial guess. The proposed semi-analytical solution avoids such difficulty faced by an analyst or a designer of cable systems. The results obtained here were validated with a numerical solution for the static profile and cable tension. For the quasi-static analysis, these numerical methods are not suitable to compute the cable tension and its equilibrium profile due to the dependency of the solution on the initial guess and computation cost. To illustrate the need for a semi-analytical solution, the design problem for a cable support system and quasi-static analysis of an underwater tether connected between parent ship and a remotely operated vehicle (ROV) are presented. For the quasi-static analysis of the underwater tether, the proposed results are compared with those using the Newton-Raphson method in terms of computational cost and accuracy. In addition, guidelines are provided in this paper to choose an initial guess for the numerical approach so that a solution would be converged much faster.

Bubble dynamics in Liquid Hole Multipliers

In bubble-assisted Liquid Hole Multipliers (LHM), developed for noble-liquid radiation detectors, the stability of the bubble and the electro-mechanical properties of the liquid-to-gas interface play a dominant role in the detector performance. A model is proposed to evaluate the static equilibrium configurations of a bubble sustained underneath a perforated electrode immersed in a liquid.For the first time bubbles were optically observed in LAr; their properties were studied in contact with different material surfaces. This permitted investigating the bubble-electrodynamics via numerical simulations; it was shown that the electric field acts as an additional pressure term on the bubble meniscus.The predictions for the liquid-to-gas interface were successfully validated using X-ray micro-CT in water and in silicone oil at STP. The proposed model and the results of this study are an important milestone towards understanding and optimizing the parameters of LHM-based noble-liquid detectors.

Rock and roll of an ellipse

An ellipse on a non-slip inclined plane can rock, roll, or jump. Below a threshold energy, it rocks about its static equilibrium configuration. Above this energy, it rolls, and, if it rolls, it will eventually jump off the incline after some number of rolls. It is shown that jumping can only occur in certain configurations of the ellipse.

New insight into the stability and dynamics of fluid-conveying supported pipes with small geometric imperfections

In several previous studies, it was reported that a supported pipe with small geometric imperfections would lose stability when the internal flow velocity became sufficiently high. Recently, however, it has become clear that this conclusion may be at best incomplete. A reevaluation of the problem is undertaken here by essentially considering the flow-induced static deformation of a pipe. With the aid of the absolute nodal coordinate formulation (ANCF) and the extended Lagrange equations for dynamical systems containing non-material volumes, the nonlinear governing equations of a pipe with three different geometric imperfections are introduced and formulated. Based on extensive numerical calculations, the static equilibrium configuration, the stability, and the nonlinear dynamics of the considered pipe system are determined and analyzed. The results show that for a supported pipe with the geometric imperfection of a half sinusoidal wave, the dynamical system could not lose stability even if the flow velocity reaches an extremely high value of 40. However, for a supported pipe with the geometric imperfection of one or one and a half sinusoidal waves, the first-mode buckling instability would take place at high flow velocity. Moreover, based on a further parametric analysis, the effects of the amplitude of the geometric imperfection and the aspect ratio of the pipe on the static deformation, the critical flow velocity for buckling instability, and the nonlinear responses of the supported pipes with geometric imperfections are analyzed.

Numerical analysis of the Cargo Transfer Vessel offloading operation in a Brazilian oil field

In a new concept of offloading operation, the oil from a spread moored FPSO is transferred to a conventional Tanker with an intermediate Cargo Transfer Vessel (CTV) equipped with a DP system. A comprehensive numerical analysis of the CTV offloading operation is here developed considering the sea states in the Libra Field, Brazil. First, the environmental loads on the vessels were estimated. Current and wind forces coefficients were obtained using the CFD software StarCCM+, while wave coefficients were obtained using WAMIT. For the 10-year metocean data, the multi-body static equilibrium configurations of the system – composed of the CTV, Conventional Tanker and Tugboat - were calculated. With those results, it was possible to estimate the downtime considering different offloading sectors and to compare with the DP Shuttle Tanker operation. Finally, a 5-day campaign of real-time simulations was carried out with the presence of DP Operators and captains, to verify critical conditions, considering the human factor. Each vessel was simulated in an individual cabin. For the simulations, the real DP System was integrated with the numerical simulator in a Hardware-in-the-loop configuration, resulting in a more realistic simulation.

Does elasticity stabilize a magnetic neutron star?

ABSTRACT The configuration of the magnetic field in the interior of a neutron star is mostly unknown from observations. Theoretical models of the interior magnetic field geometry tend to be oversimplified to avoid mathematical complexity and tend to be based on axisymmetric barotropic fluid systems. These static magnetic equilibrium configurations have been shown to be unstable on a short time-scale against an infinitesimal perturbation. Given this instability, it is relevant to consider how more realistic neutron star physics affects the outcome. In particular, it makes sense to ask if elasticity, which provides an additional restoring force on the perturbations, may stabilize the system. It is well known that the matter in the neutron star crust forms an ionic crystal. The interactions between the crystallized nuclei can generate shear stress against any applied strain. To incorporate the effect of the crust on the dynamical evolution of the perturbed equilibrium structure, we study the effect of elasticity on the instability of an axisymmetric magnetic star. In particular, we determine the critical shear modulus required to prevent magnetic instability and consider the corresponding astrophysical consequences.

Static equilibrium configuration and nonlinear dynamics of slightly curved cantilevered pipe conveying fluid

In engineering applications, fluid-conveying pipes usually have geometric imperfections or initially curved configurations. Unlike the initially curved pipe supported at both ends, a slightly curved cantilevered pipe is capable of displaying some interesting behavior because it is a nonconservative system of fluid-structure interactions. In the present study, nonlinear static and dynamic behaviors of cantilevered pipes conveying fluid are explored, with four different initial shapes being considered. To this end, the strongly nonlinear governing equation is derived by employing the extended Lagrange equations written for dynamical systems containing non-material volumes. The static (steady) equilibrium configurations, stability, and nonlinear dynamics of the slightly curved cantilevered pipes are obtained with the aid of absolute nodal coordinate formulation (ANCF). Based on extensive numerical calculations, some interesting and sometimes unexpected results are displayed. The first unexpected feature in this dynamical system is that the flow-induced static deformation of the pipe can be extremely large even if the initial geometric imperfection of the pipe is quite small. The second unexpected result is that the critical flow velocity for flutter instability of the slightly curved pipe conveying fluid may be either lower or higher than that of a straight pipe, mainly depending on the static equilibrium configuration when the critical flow velocity is just reached. Moreover, it is demonstrated that the slightly curved pipe oscillates about the static equilibrium position instead of the initially curved centerline, and the preferred form of post-instability behavior is periodic motion within a wide range of system parameters considered.

A bending-active twisted-arch plywood structure: computational design and fabrication of the FlexMaps Pavilion

Bending-active structures are able to efficiently produce complex curved shapes from flat panels. The desired deformation of the panels derives from the proper selection of their elastic properties. Optimized panels, called FlexMaps, are designed such that, once they are bent and assembled, the resulting static equilibrium configuration matches a desired input 3D shape. The FlexMaps elastic properties are controlled by locally varying spiraling geometric mesostructures, which are optimized in size and shape to match specific bending requests, namely the global curvature of the target shape. The design pipeline starts from a quad mesh representing the input 3D shape, which defines the edge size and the total amount of spirals: every quad will embed one spiral. Then, an optimization algorithm tunes the geometry of the spirals by using a simplified pre-computed rod model. This rod model is derived from a non-linear regression algorithm which approximates the non-linear behavior of solid FEM spiral models subject to hundreds of load combinations. This innovative pipeline has been applied to the project of a lightweight plywood pavilion named FlexMaps Pavilion, which is a single-layer piecewise twisted arch that fits a bounding box of 3.90x3.96x3.25 meters. This case study serves to test the applicability of this methodology at the architectural scale. The structure is validated via FE analyses and the fabrication of the full scale prototype.