Superposition Of Modes Research Articles

Vibration analysis of cylindrical shell discontinuously coupled with annular plate with arbitrary boundary conditions

Discontinuously coupled cylindrical shell-plates or plate-like structures are common components of underwater vehicles, and their vibration characteristics change significantly due to modal coupling. The vibration of discontinuously coupled annular plate-cylindrical shell structures is analyzed based on the wave propagation method(WPM) and virtual spring(VS). Taking uniform annular plates and cylindrical shells as spectral elements, the displacement solutions are described by the trigonometric function and Bessel function respectively, and the vibration responses of the annular plate-cylindrical shell with arbitrary boundaries are obtained. Based on VS and weighted least square norm least square method(WLSNLSM), the discontinuous coupling between the annular plate and cylindrical shell is described as the non-uniform distribution of stiffness of virtual spring by continuity condition, and the dynamic stiffness matrices of continuously and discontinuously coupled structure can be obtained simultaneously. The reason for vibration characteristic change under discontinuous coupling is explained from the perspective of modal superposition. Finally, a finite element model is established to verify the accuracy of the proposed method. The influence of general discontinuous coupling on the dynamic behavior of shell structure is analyzed by numerical examples, which provides useful guidance for structural vibration analysis with discontinuous coupling.

Experimental assessment of square-wave spatial spanwise forcing of a turbulent boundary layer

We present an experimental realisation of spatial spanwise forcing in a turbulent boundary layer flow, aimed at reducing the frictional drag. The forcing is achieved by a series of spanwise running belts, running in alternating spanwise direction, thereby generating a steady spatial square-wave forcing. Stereoscopic particle image velocimetry in the streamwise–wall-normal plane is used to investigate the impact of actuation on the flow in terms of turbulence statistics, drag performance characteristics, and spanwise velocity profiles, for a non-dimensional wavelength of λx+=397\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _x^+ = 397$$\\end{document}. In line with reported numerical studies, we confirm that a significant flow control effect can be realised with this type of forcing. The scalar fields of the higher-order turbulence statistics show a strong attenuation of stresses and production of turbulence kinetic energy over the first belt already, followed by a more gradual decrease to a steady-state energy response over the second belt. The streamwise velocity in the near-wall region is reduced, indicative of a drag-reduced flow state. The profiles of the higher-order turbulence statistics are attenuated up to a wall-normal height of y+≈100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y^+ \\approx 100$$\\end{document}, with a maximum streamwise stress reduction of 45% and a reduction of integral turbulence kinetic energy production of 39%, for a non-dimensional actuation amplitude of A+=12.7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+ = 12.7$$\\end{document}. An extension of the classical laminar Stokes layer theory is introduced, based on the linear superposition of Fourier modes, to describe the non-sinusoidal boundary condition that corresponds to the current case. The experimentally obtained spanwise velocity profiles show good agreement with this extended theoretical model. The drag reduction was estimated from a linear fit in the viscous sublayer in the range 2≤y+≤5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \\le y^+\\le 5$$\\end{document}. The results are found to be in good qualitative agreement with the numerical implementations of Viotti et al. (Phys Fluids 21, 2009), matching the drag reduction trend with A+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^+$$\\end{document}, and reaching a maximum of 20%.Graphical abstract

Semi-analytic modeling and experimental verification of arbitrary aero-engine complex spatial pipeline

In-depth research is imperative for complex spatial pipelines with fluid-structure interaction. A comprehensive and universal three-dimensional semi-analytical modeling method is first established for complex fluid-conveying spatial pipelines under the base excitation with arbitrary connections and configurations under different boundary conditions. The virtual spring technology is used to connect the straight and curved pipeline segments with spatial angles and simulate boundary conditions. For a thorough analysis, the actual stiffness of the bracket-clamp system is experimentally measured for the first time. The multi-level coupling dynamic model of part, component, and system levels under actual boundary conditions is established. An efficient method for solving the fluid-structure coupling pipeline system dynamics considering the base excitation is proposed based on energy method. The mid-plane displacement function of the Fourier series is introduced and solved using the principle of modal superposition. Hammer impact modal experiments, ANSYS software and vibration response experiments are performed on spatial pipelines supported by flexible bracket-clamps; the numerical and experimental results support the semi-analytical results. The degenerate model is verified with the simple plane pipelines in other references, and the errors are less than 5 %. The proposed method exhibits broad applicability and generalizability, making it suitable for various configurations and boundary conditions.

Virtual sensing for real-time strain field estimation and its verification on a laboratory-scale jacket structure under water waves

This study aims to achieve real-time estimation of the full-field strain distribution in a structure by signals measured from several strain gauges attached to the structure. Our virtual sensing procedure is developed based on finite element formulation and employs the mode superposition approach. To verify the feasibility of the proposed procedure, numerical and experimental tests are conducted on a laboratory-scale offshore jacket structure subjected to water waves. Key aspects addressed in this study include the selection of displacement modes and the division of strain signals. The experiments are performed in an ocean basin, and comprehensive explanations are provided for the jacket prototype design, implementations, experimental setup, and wave loading conditions. The performance of the proposed virtual sensing procedure is thoroughly assessed through various evaluation measures, enhancing the understanding of its capabilities and limitations in practical applications.

Multi-sensor data fusion reconstruction method for vibration dynamic responses of aerospace structures

The dynamic responses of key locations are important inputs for the life and reliability assessment of spacecraft structures. Due to the limited sensing resources, most critical responses are difficult to measure directly. A structural dynamic response reconstruction method is necessary. The responses of target locations can be reconstructed based on the empirical mode decomposition (EMD) of measured signals and the modal superposition. However, the structural modal information contained in the measured signal of a single sensor is limited, affecting the reconstruction accuracy. In this paper, a response reconstruction method based on multi-sensor data fusion is proposed. It is applied to a main load-bearing structure of a spacecraft and its typical components to verify its strain response reconstruction effect under random vibration loads. The experimental results show that multi-sensor data fusion improves the strain reconstruction accuracy. The maximum reduction in reconstruction error is from 8.7% to 1.3%. The reconstruction accuracy is further improved with the increase in the number of sensors. The optimal weighted fusion strategy for this problem is the weights defined by the Euclidean distance (EUC) or the dynamic time warping distance (DTW). The fusion results show a better performance with the increase in the power of the defined distance. The proposed multi-sensor fusion method improves the reconstruction accuracy via supplementing structural information to each other and eliminating the instability of single measured signals. More accurate dynamic responses via reconstruction reduce the large input uncertainty in life prediction and lay the foundation for building structural digital twins and managing structural health more effectively.

Heat Capacity and Features of the Phonon Spectrum of Single Crystals of Solid Solutions of Yttrium–Lutetium Aluminum Garnets

The temperature dependences of the heat capacity were measured and the general patterns of the formation of the phonon spectrum of single crystals of solid solutions of yttrium-lutetium aluminum garnets Y3 – xLuxAl5O12 at 0 ≤ x ≤ 3 were investigated in the temperature range from 1.9 to 220 K. According to the data obtained at temperatures below 10 K, the Debye temperatures were calculated. The features of the phonon spectrum in the intermediate temperature range are interpreted as a superposition of optical modes for yttrium and lutetium garnets. It is shown that the low heat capacity values due to the contribution of acoustic phonons for Y2.25Lu0.75Al5O12 correlate with anomalies in the concentration dependences of the phonon transport, the absorption of acoustic waves, and the shape of the aluminum NMR line.

Theoretical modeling and experimental investigation of carbon fiber reinforced plastic-based piezoelectric actuator

Piezoelectric actuators based on non-metallic materials have drawn much attention in recent years. Carbon fiber reinforced plastic (CFRP) is one of the ideal materials for the development of lightweight, high power density piezoelectric actuators because of its low density, high stiffness. However, its anisotropic characteristics pose a challenge in actuator development. In this study, we designed a CFRP-based piezoelectric actuator, which utilizes hybrid modes of first-order longitudinal mode and second-order bending mode. The electromechanical coupling dynamic model for CFRP-based piezoelectric actuator was established a, based on the modal superposition method and energy method, and while taking into account the stress-strain relationships in anisotropic materials. The size of the actuator was calculated through the model and a prototype was processed for experimental research. The experimentally obtained results of frequencies and transient- as well as steady-state vibration characteristics demonstrated excellent agreement with the predictions of our mathematical modeling. Actuator performance evaluation results show that under a single-phase excitation voltage of 200 Vp-p, the CFRP-based actuator can reach maximum speed, thrust force, output power, thrust–weight ratio, power density, and efficiency of 617 mm s−1, 65 N, 1.14 W, 980.3 N kg−1, 223.5 W kg−1, and 10.4%, respectively. These results are satisfactory compared with actuators in other reports, especially the power density, which is nearly tripled. These results demonstrate the superior performance of the CFRP-based actuator and illustrate a new approach for developing lightweight and powerful actuators.

Longitudinal encoding of focused vector vortex beams empowered by silicon metasurfaces

Vector vortex beams (VVBs) have attracted considerable applications due to their unique properties, such as vortex filtering, particle trapping, and quantum information. Recently, information coding methods based on metasurface strategies show excellent performance in remote sensing and communication, but the implementation of the longitudinal dimension of the optical field in information coding remains to be developed. Here, we provide a strategy for constructing all-dielectric terahertz (THz) metasurface designs for longitudinal codecs, with the individual focal planes carrying specific encrypted information. Benefiting from the joint modulation of the geometrical and propagation phases, the physical mechanism of spin-polarization multiplexing realizes the coherent superposition of multidimensional focused orbital angular momentum (OAM) modes. As a bridge between Poincaré sphere (PS) and the higher-order Poincaré sphere (HOPS), the proposed design produces a longitudinally controllable variation of OAM mode superposition states. A series of metasurface examples for generating specific coding sequences are characterized by manipulating topological charges and phase factors embedded within orthogonal circularly polarized channels. The longitudinal dimension, as a new coding degree of freedom, can be further expanded for meta-optical applications with the assistance of the VVB pattern.

A reconstruction method for structural stress distribution of wind turbine tower using optimised mathematical model

Wind turbine towers experience complex dynamic loads during actual operation, and these loads are difficult to accurately predict in advance, which may lead to inaccurate structural strength assessment during the structural design phase, thereby posing safety risks to the wind turbine tower. Therefore, a reconstruction method for structural stress distribution of wind turbine tower was developed based on modal expansion and linear superposition theory to estimate the full-field stress distribution of a tower, which can be used for online monitoring of structural strength. The dynamic stress distribution of the tower is assumed to be the superposition of several structural modal stress distributions, in the reconstruction method for structural stress distribution developed in this study. The modes with the greatest contribution to the wind turbine structural response were selected based on the modal effective mass, and used for the base functions of an initial mathematical model for the reconstruction of structural stress distribution of the tower, in which many candidate strain gauge locations and directions were considered. Then, the initial mathematical model was expressed as a linear system of equations. Two numerical examples were used to verify the accuracy and stability of the initial mathematical model for the reconstruction of structural stress distribution of the tower, and the reconstruction results indicate that the initial mathematical model combined with the Moore–Penrose inverse algorithm can provide stable and accurate reconstruction results. However, the initial mathematical model uses too many sensors, which is not conducive to engineering applications. Therefore, D-optimal and C-optimal design methods were used to reduce the dimensions of the initial mathematical model, optimise the system of equations, and determine the location, direction and number of strain gauges. Finally, the numerical examples of stress distribution reconstruction show that dimensionality reduction of the mathematical model leads to higher accuracy, in which the C-optimal design algorithm providing a more robust reconstruction outcome.

Low-frequency vibration reduction of an underwater metamaterial plate excited by a turbulent boundary layer

Flow-induced structural noise is an important component of hydrodynamic noise of underwater structures. Local resonance metamaterials are considered to have excellent performance and enormous potential in the field of low-frequency vibration and noise control. To verify its potential, the paper derived the underwater band gap of a lateral local resonance (LLR) plate through the plane wave expansion (PWE). Then, utilizing the modal superposition approach and Rayleigh integral technique, the vibro-acoustic response of a LLR plate under a turbulent boundary layer (TBL) excitation is obtained. Finite element certification is also conducted through an uncorrelated wall plane wave technique. Parametric study is conducted to analyse the factors which influence the control effects. The result shows that the plate exhibits excellent suppression performance for flow-induced vibration at band gap frequencies. The band gaps and suppression ranges generated by the underwater metamaterial plate, are dramatically narrowed due to the thick fluid load. The paper provides theoretical guidance for the control of flow-induced structural vibration and the application of acoustic metamaterials.

Investigation of a cutting state-independent method for identifying in-process frequency response functions of the micro milling system

Conveniently obtaining the in-process frequency response functions (FRFs) of micro milling tool is a very difficult task due to the small diameter. This article presents a method to identify the in-process FRFs of micro milling tool based on actual cutting tests, which can be at either stable or chatter cutting state. According to mode superposition principle, the in-process FRFs are predicted by integrating the mode shape with the pole, which includes damping ratio and natural frequency. The in-process mode shape of tool is modeled as being consistent with that corresponding to static state, and thus, it is calculated from the tool’s static state by combining receptance coupling substructure analysis (RCSA) with Timoshenko beam theory. The poles of in-process FRFs are acquired by modal analysis of the response signals from actual micro milling tests. To do this, cutting forces are used as excitation signals, and the velocities of tool are selected as response signals and they are measured from actual micro milling tests. With the aid of editing cepstrum and hampel outlier processing methods, a filtering method is established to remove the harmonic and useless chatter components involved in the response signals. As a result, the finally obtained FRFs are not influenced by the noises in the measured response signals, and the effectiveness of this method is verified by a series of modal tests and micro milling tests.

Multi‐mode inversion of dispersive surface‐wave free fields in layered media

AbstractInversion of surface‐wave free fields is of great importance in providing seismic excitations for the finite element (FE) analysis of soil–structure interaction (SSI) systems. In the case of layered media, the propagating surface waves are characterized by their unique dispersion property, resulting in dispersive multi‐mode surface wavefields. Available studies primarily focus on only the fundamental mode and neglect the influence of higher modes, which fail to identify the real structural response characteristics. To enhance the precision of seismic excitations for SSI systems situated in layered media, a multi‐mode inversion method for dispersive surface‐wave free fields is proposed in this study. The proposed method employs the energy flow density to characterize the participation volume of dispersive modes, and further calculates corresponding modal participation factors by combining with the dispersion curve and dynamic‐stiffness matrix (DSM) method. The modal participation factors are then utilized to assign the ground surface components of each mode and invert respective single‐mode surface‐wave free fields. Based on the mode superposition theory, all the single‐mode surface‐wave free fields are finally superimposed to construct the multi‐mode wavefields. The accuracy of the proposed method is thoroughly verified and validated across various types of layered media. Furthermore, the engineering application of the proposed method is illustrated by performing the FE analysis on an SSI example. The analysis results highlight the significant participation of surface waves' higher modes in structural seismic responses, particularly in the weak‐interbed type media.

Seismic analyses of single-layer dome structures with random geometrical imperfections under stochastic ground motions

In this paper, seismic analyses of single-layer dome structures are carried out, with a particular emphasis on the incorporation of randomness in both geometrical imperfections and ground motions. First, the paper introduces models for random geometrical imperfections and stochastic ground motions, employing the Stochastic Imperfection Mode Superposition Method (SIMSM) for imperfections and utilizing the Spectral Representation Method (SRM) for ground motions. These models are designed to comprehensively account for the inherent uncertainties in real-world structural behavior. Subsequently, the Probability Density Evolution Method (PDEM) is applied to investigate the probabilistic response and seismic reliability of dome structures, which allows for a detailed examination of key performance indicators, such as displacement, plastic ratio, and strength damage, under the influence of random factors. Both deterministic and stochastic analyses are conducted to gain insights into the impact of randomness in geometrical imperfections and ground motions. The computational results unveil a crucial finding: an unfavorable distribution of imperfections can significantly compromise the dynamic performance and safety of single-layer dome structures. Furthermore, the paper delves into parametric analyses, with a specific focus on imperfection amplitude, the rise-span ratio and roof load as pivotal parameters. These analyses provide valuable insights into the sensitivity of the structural response to variations in these critical design factors.

A comprehensive study on seismic dynamic responses of stochastic structures using sparse grid-based polynomial chaos expansion

The seismic response is a critical aspect of structural engineering, as it directly influences the design and safety evaluation of such systems. Structural seismic response becomes more complicated when unavoidable uncertainties in the structure are considered. Therefore, solving and analyzing the stochastic dynamic response of structures efficiently and accurately is of great significance. Various uncertainty quantification methods such as the stochastic finite element method, Monte Carlo method, and polynomial chaos expansion method have made significant progress in this field. However, these methods do not provide a comprehensive study of the seismic response of stochastic structures and efficiency problems rapidly emerge as the number of random variables increases. This paper delivers a comprehensive study of the seismic dynamic response of space truss and reticulated shell structure considering uncertainties in material properties, geometry, and loading. To this end, the finite element model of the structure was first modeled and the deterministic solutions were obtained. Then, polynomial surrogates are modeled for the natural frequency, mode participation factor, mode superposition response, and transient response of the structure under stochastic uncertainties. The sparse grid technique is used to select sparse quadrature points, which overcomes the curse of dimensionality caused by full grid integration. The results demonstrate that the sparse grid-based polynomial chaos expansion is more efficient. Uncertainty quantification reveals the uncertainty of modal analysis, response spectrum analysis, and transient response analysis of space truss and reticulated shell under seismic excitation. In summary, this study provides valuable insights and an efficient and accurate uncertainty quantification tool for the seismic dynamic analysis of stochastic structures, ultimately contributing to safer and more robust structural designs.

Modal reduction for exterior acoustic and vibroacoustic problems

Modal analysis, mode superposition and modal reduction are considered standard approaches for interior acoustic problems as well as for pure vibration problems. Things are different for exterior problems. Only a few methods are known to formulate a linearized eigenvalue problem for unbounded acoustic problems. Even fewer techniques on modal superposition and reduction are found in the literature. This talk will review such techniques which are either based on conjugate Astley-Leis infinite elements or on boundary elements for the unbounded region. While the results appear to be promising, a number of open questions need to be solved in this context. Among others, it is yet unclear which modes are actually required for a modal reduction and how to find a method to just evaluate these modes and the related eigenvalues. The situation is different for vibroacoustic radiation problems with clear resonances. In such cases, a modal reduction is able to substantially accelerate the solution process. Future work might lead to modal reduction using modes from non-linear eigenvalue problems which are known from the numerous papers on the boundary element methods published over the last decade.

Investigating the entry of symmetric/asymmetric elastic wedge into water

Understanding the impact of fluid–structure interaction during the entry of a marine vessel into water is crucial for its design. While numerous models have been proposed to investigate the penetration of symmetric elastic wedges into water, in this study, we propose a numerical model that combines the fully nonlinear boundary element method with a mode superposition method to analyze the penetration of symmetric and asymmetric elastic wedges into water. We derive the boundary conditions of the acceleration potential within the flow field for the problem of fluid–structure interaction and use auxiliary functions to decompose the acceleration potential and extract the instantaneous added mass. Following this, the differential equations of the modes of rigid motion and elastic deformation of the body are established and solved. This approach enables the simultaneous solution of the motion and vibrations of the wedge as well as the hydrodynamic pressure. The proposed model also accounts for flow separation while the wedge is submerged, thereby prolonging the simulation. We confirmed the validity of the model through comprehensive examinations based on semi-analytical, computational, and experimental data on the problem of the entry of a symmetric body into water. We then extended our study to encompass the free-falling entry of an asymmetric elastic wedge into water. The results revealed notable discrepancies in the evolution of deformation and jet flow between the right and left boundaries during the entry of the wedge into water.

Study on the Generation of 1.9 μm Mode Superposition Conversion Laser by Double-End Off-Axis Pumping

In this paper, the Laguerre–Gaussian (LG) mode superposition is obtained by using the technology of double-end off-axis pumping Tm:YLF crystal, and the LG mode superposition is achieved by combining the extra-cavity conversion method. The impact of changing the off-axis distance on the order of Hermite–Gaussian (HG) mode and the topological charge of LG mode is studied. The results show that when the off-axis distance of the pump source at both ends is tuned, when the off-axis distance is in the range of 260 μm~845 μm, the single-ended 0~10 order HG mode can be obtained. Subsequently, the mode converter is placed to obtain the LG mode beam, and the double-end simultaneously pumps the crystal to obtain the superimposed LG mode. The tuning off-axis quantity changes the topological charge number. When P = 0, l1=l2, the superimposed LG mode is a single-ring spot, and the vortex beam center’s dark hollow area increases with the topological charge number. When P = 0, l1=−l2, the superimposed LG mode is a petal-like spot. The number of petals differs from the topological charges of two opposite numbers. Finally, in the case of changing the topological charge number of the double-ended LG mode, the output of the vortex array structured beams of the tuning mode order 1.9 μm Tm:YLF is completed in the case of conversion and superposition.

Comparison of Dynamic Amplification Factor of Deflection and Bending Moment of Highway Continuous Box-Girder Bridges by Mode Superposition.

There are differences between the dynamic deflection and bending moment (strain) in the same section of continuous girder bridges. However, the selection of the response for calculating dynamic amplification factors (DAFs), which are essential for bridge health monitoring and safety assessment, remains controversial. Modes may play a role in the relationship between the deflection DAF and the bending moment DAF in both numerical analysis and field tests. To investigate the distinctions between the DAFs of the deflection and bending moment in a continuous girder bridge, functional expressions of the DAFs were derived, taking into account multi-factor coupling under concentrated forces. The interaction effects of the mode and road surface condition (RSC), vehicle speed, bridge span length, and span number on the deflection DAF, the bending moment DAF, and the ratio of the deflection DAF to the bending moment DAF (RDM) of precast continuous box-girder bridges were analyzed using vehicle-bridge interaction. To ensure the accuracy of the DAF in numerical computations and experimental tests, two types of accuracy indexes and the corresponding cut-off modes were provided. Validation was conducted by performing dynamic load tests on two field bridges. The results indicate that different modes have a significant effect on the RDM of the mid-span section of a bridge. When considering multiple factors, the deflection DAF and bending moment DAF of the mid-span section increased rapidly with the considered modes and then stabilized. Statistically, the RDM of all nine bridges ranged from 1.00 to 1.12, indicating that the deflection DAF was greater than the bending moment DAF. The suggested cut-off modes can be utilized for efficient and accurate calculation of the DAF and response signal fidelity.

A refined mode superposition method for dynamical responses of an underwater cylindrical shell with substructures

To overcome the difficulties of time-consuming and inefficient in the response calculations of the submerged cylindrical shell with internal substructures, this paper firstly presents a refined mode superposition method. In view of the necessity and difficulties of directly and quickly obtaining the frequencies and modes of structures submerged in fluid (named the wet modes), the wet modes are expanded by the modes in vacuum (dry modes) and solved by the energy method. The added kinetic energy of the fluid is calculated via boundary integration, and the Lagrange equations of the second kind is applied to the fluid-structure coupling equations. Then the wet modes are solved by eigenvalue calculation and the modal mass and stiffness of each order wet mode are obtained. Finally, they are used for establishing a mode superposition approach for response calculations. The accuracy of the present method is verified by ANSYS software. In this method, all the required data are obtained from the structural analysis and the traditional complicated fluid force modeling is no longer required. Thus it has the advantages of high computational efficiency and applicability. Compared to the traditional semi-analytic model, this modeling methodology has broad application potential for vibration problems of complex underwater structures since the structural dry modes can be solved efficiently by commercial software. It also has practical value as a theoretical reference for developing mode-superposition-based calculations for fluid-structure problems using commercial software.

Fluid Structure Interaction Using Modal Superposition and Lagrangian CFD

This study investigates the impact of fluid loads on the elastic deformation and dynamic response of linear structures. A weakly coupled modal solver is presented, which involves the solution of a dynamic equation of motion with external loads. The mode superposition method is used to find the dynamic response, utilizing predetermined mode shapes and natural frequencies associated with the structure. These essential parameters are pre-calculated and provided as input for the simulation. Integration of the weakly coupled modal solver is accomplished with the Lagrangian Differencing Dynamics (LDD) method. This method can directly use surface mesh as boundary conditions, so it is much more convenient than other meshless CFD methods. It employs Lagrangian finite differences, utilizing a strong formulation of the Navier–Stokes equations to model an incompressible free-surface flow. The elastic deformation of the structure, induced by fluid forces obtained from the flow solver, is computed within the modal coupling algorithm through direct numerical integration. Subsequently, this deformation is introduced into the flow solver to account for changes in geometry, resulting in updated flow pressure and velocity fields. The flow particles and vertices of the structure are advected in Lagrangian coordinates, resulting in Lagrangian–Lagrangian coupling in spaces with weak or explicit coupling in time. The two-way coupling between fluid and structure is successfully validated through various FSI benchmark cases. The efficiency of the LDD method is highlighted as it operates directly on surface meshes, streamlining the simulation setup. Direct coupling of structural deformation eliminates the conventional step of mapping fluid results onto the structural mesh and vice versa.