Structural Equations Of Motion Research Articles

Numerical Simulation of Folding Tail Aeroelasticity Based on the CFD/CSD Coupling Method

This paper presents a CFD/CSD coupling method for aeroelastic simulation of folding tail morphing aircraft. The unsteady aerodynamic analysis is based on an in-house computational fluid dynamics (CFD) solver for the Euler equations, and emphasis is made on developing an efficient dynamic mesh method for the tail’s hybrid fold motion/elastic vibration deformation. The structural dynamic analysis is based on the computational structural dynamics (CSD) technique for solving the structural equation of motion in modal space. The aeroelastic coupling was achieved through successive iterations of CFD and CSD computations in the time domain. An adaptive multi-functional morphing aircraft allowing tail fold motion was selected to be studied. By using the developed method, aeroelastic simulation and mechanism analysis for fixed configurations at different folding angles and for variable configurations during the folding process were performed. The influence of folding rate on tail aeroelasticity and its influence mechanism were also analyzed.

Fast Bayesian modal identification with known seismic excitations

AbstractFast and accurate identification of structural modal parameters after an earthquake is crucial for assessing structural conditions and facilitating repair. With the development of modern earthquake observation techniques, the recorded ground motion can be leveraged as extra input information for modal identification, enabling the experimental modal analysis applicable. This study develops a Bayesian modal identification algorithm that aims at estimating the most probable value (MPV) of modal parameters and their identification uncertainty. Incorporating the recorded seismic input, the algorithm utilizes with the structural equation of motion in the frequency domain to formulate the likelihood function and adopts a constrained Laplace method for Bayesian posterior approximation of modal parameters. With the aid of complex matrix calculus, an iterative scheme is developed, allowing a fast search of the MPV of modal parameters and an analytical evaluation of the posterior covariance matrix. The performance of the proposed algorithm is validated by examples with synthetic, laboratory and field data, respectively. In addition, its effectiveness on predicting structural responses under a future earthquake is illustrated, showing its potential for various downstream applications in seismic structural health monitoring.

Displacement‐based partitioned equations of motion for structures: Formulation and proof‐of‐concept applications

AbstractA new formulation for the displacement‐only partitioned equations of motion for linear structures is presented, which employs: the partitioned displacement, acceleration, and applied force (); the partitioned block diagonal mass and stiffness matrices (); and, the coupling projector (), yielding the partitioned coupled equations of motion: ). The key element of the proposed formulation is the coupling projector () which can be constructed with the partitioned mass matrix (), the Boolean matrix that extracts the partition boundary degrees of freedom (), and the assembly matrix () relating the assembled displacements () to the partitioned displacements () via . Potential utility of the proposed formulation is illustrated as applied to six proof‐of‐concept problems in an ideal setting: unconditionally stable explicit‐implicit transient analysis, static parallel analysis in an iterative solution mode; reduced‐order modeling (component mode synthesis); localized damage identification which can pinpoint damage locations; a new procedure for partitioned structural optimization; and, partitioned modeling of multiphysics problems. Realistic applications of the proposed formulation are presently being carried out and will be reported in separate reports.

Design method of frequency similarity relation for shaking table model test

It is very important to determine the dynamic similarity relation in the shaking table model test. The accurate dynamic similarity relation can make the model reflect the dynamic characteristics of the prototype to the greatest extent. This paper describes the principle of similarity design for shaking table model test and reviews the related literature. According to the different research objects and purposes, four common similarity design methods including separation similarity method, artificial mass method, less artificial mass method and ignoring gravity method are summarized. Analysis of the shortcomings of the method of deriving frequency compression ratio from static factors in the design of seismic wave similarity relation. The Duhamel integral analytic expression of the structure motion equation under earthquake is analyzed. The seismic response of the structure is closely related to the natural vibration frequency of the structure. A design method using frequency as input seismic wave similarity design control quantity is proposed. The natural vibration frequencies of the prototype and the model are studied by means of the Gonza landslide field ground pulsation test and the model white noise excitation. A new frequency compression ratio design scheme is given from the perspective of dynamic similarity. It is of great reference significance to optimize the dynamic similarity design of shaking table model test and improve the accuracy of test results.

Development of Active Wave-induced Overtopping Control System [AWOC] using a Self-adaptive Floating Structure

In this study, an active overtopping control system [AWOC] that could effectively supress the flooding caused by storm surge was presented, and numerical simulations to verify the overtopping control effect of AWOC were also carried out using OlaFlow, an OpenFoam-based toolbox. In the AWOC, the floating structure that works as the overtopping controller is submerged in a mild sea, and when exposed to a storm, it rises when the sea level are getting increased due to a storm surge, and as a result, would effectively supress the wave-induced overtopping. Due to these characteristics, the AWOC has the advantage of controlling overtopping at the minimum opportunity cost to preserve the beautiful beach landscape. In the numerical simulation, the motion of a floating structure that rises or descends coupled with the sea level during a storm surge or harsh waves was described by solving the structure motion equation using the water pressure acting on it as an external force. In doing so, Dynamic Mesh was used for the sake of more accurate numerical simulation, in which the computational mesh is updated whenever floating structure moves. Numerical result shows that even though overtopping flow becomes more energetic compared to that in the rubble-mound breaker due to the emhanced reflection after the deployment of AWOC, AWOC could effectively control the enhanced overtopping. It is also shown that an obliquely descending flow from the first quadrant to the third quadrant is formed in the front of AWOC due to the offshore-directed flow commencing from the floating structure of AWOC.

Computation of gust induced responses of an air taxi by using Navier-Stokes equations

An air taxi's flight in a gust environment is simulated by using the Navier-Stokes equations time-accurately coupled with structural equations of motion. Gusts are numerically simulated using oscillating vanes similar to a wind tunnel experiment. The configuration is modeled using overset grid topology. Computational results of gust modeling are validated with wind tunnel testing. Aerodynamic responses are validated with the linear theory. Gust-induced structural responses are simulated for a Lift+Cruise air taxi with eight lifting propellers and two pushing propellers. The present work extends the capabilities of the current Navier-Stokes solvers to design urban air taxi configurations for gust environments.

Stability analysis for laminar separation flutter of an airfoil in the transitional flow regime

Laminar separation flutter (LSF) is a type of aeroelastic instability phenomenon characterized by small-amplitude low-frequency pitching oscillations of the airfoil. The present study aims to gain insight into the intrinsic dynamics of LSF via data-driven stability analysis. The proposed data-driven approach relies on the autoregressive with exogenous input (ARX) technique to design reduced-order models (ROMs) of unsteady aerodynamics in a state-space format. First, high-fidelity full-order numerical simulations of the LSF phenomenon are performed using the incompressible Unsteady Reynolds-Averaged Navier–Stokes equations and the Shear-Stress Transport k−ω turbulence model with Low-Reynolds-number correction. The calculated LSF responses show good agreement with previous experimental data in the literature. Then, linear stability analysis (LSA) of the aeroelastic system is carried out to reveal the underlying fluid-structure interaction mechanism. The LSA model is developed by coupling the ROM with the structure motion equation. LSA results indicate that the LSF phenomenon is primarily caused by the instability of the structure mode (SM), which is induced by the mutual repulsion effect between one static fluid mode (FM) and the SM. The presence of laminar separation near the trailing-edge of the airfoil can significantly reduce the stability of the static FM, which ultimately strengthens the fluid-structure coupling effect and leads to LSF. We would like to emphasize that LSF is essentially different from other flow-induced vibration phenomena, such as transonic buffeting of an airfoil and vortex-induced vibration of bluff bodies, for which the instabilities are triggered by the coupling between one dynamic FM and the SM. Finally, the effects of the mass ratio, structural damping ratio, and freestream turbulence intensity on the aeroelastic system are also investigated.

Fully-printed metamaterial-type flexible wings with controllable flight characteristics

Insect wings are an outstanding example of how a proper interplay of rigid and flexible materials enables an intricate flapping flight accompanied by sound. The understanding of the aerodynamics and acoustics of insect wings has enabled the development of man-made flying robotic vehicles and explained basic mechanisms of sound generation by natural flyers. This work proposes the concept of artificial wings with a periodic pattern, inspired by metamaterials, and explores how the pattern geometry can be used to control the aerodynamic and acoustic characteristics of a wing. For this, we analyzed bio-inspired wings with anisotropic honeycomb patterns flapping at a low frequency and developed a multi-parameter optimization procedure to tune the pattern design in order to increase lift and simultaneously to manipulate the produced sound. Our analysis is based on the finite-element solution to a transient three-dimensional fluid–structure interactions problem. The two-way coupling is described by incompressible Navier–Stokes equations for viscous air and structural equations of motion for a wing undergoing large deformations. We 3D-printed three wing samples and validated their robustness and dynamics experimentally. Importantly, we showed that the proposed wings can sustain long-term resonance excitation that opens a possibility to implement resonance-type flights inherent to certain natural flyers. Our results confirm the feasibility of metamaterial patterns to control the flapping flight dynamics and can open new perspectives for applications of 3D-printed patterned wings, e.g. in the design of drones with target sound.

A strongly coupled immersed boundary method for fluid-structure interaction that mimics the efficiency of stationary body methods

Strongly coupled immersed boundary (IB) methods solve the nonlinear fluid and structural equations of motion simultaneously for strongly enforcing the no-slip constraint on the body. Handling this constraint requires solving several large dimensional systems that scale by the number of grid points in the flow domain even though the nonlinear constraints scale only by the small number of points used to represent the fluid-structure interface. These costly large scale operations for determining only a small number of unknowns at the interface creates a bottleneck to efficiently time-advancing strongly coupled IB methods. In this manuscript, we present a remedy for this bottleneck that is motivated by the efficient strategy employed in stationary-body IB methods while preserving the favorable stability properties of strongly coupled algorithms—we precompute a matrix that encapsulates the large dimensional system so that the prohibitive large scale operations need not be performed at every time step. This precomputation process yields a modified system of small-dimensional constraint equations that is solved at minimal computational cost while time advancing the equations. We also present a parallel implementation that scales favorably across multiple processors. The accuracy, computational efficiency and scalability of our approach are demonstrated on several two dimensional flow problems. Although the demonstration problems consist of a combination of rigid and torsionally mounted bodies, the formulation is derived in a more general setting involving an arbitrary number of rigid, torsionally mounted, and continuously deformable bodies.

Nonlinear vibrations and stability of rotating cylindrical shells conveying annular fluid medium

This study aims to investigate the nonlinear dynamic characteristics of annular cylinders coupled with a fluid medium to identify the effective parameters on stability margins at different rotation speeds. To achieve this, stability and nonlinear vibrations of rotating cylindrical shells containing incompressible annular fluid are considered. The fluid medium is bounded by a rigid external cylinder. Sanders–Koiter kinematic assumptions are utilized to determine the geometrically nonlinear structural equations of motion. These equations are then used to investigate finite amplitude vibrations of various fluid-loaded shells at different states. A penalty approach is introduced by using boundary springs with arbitrary stiffness to simulate any desired set of end conditions. Both driven and companion modes of vibration are included in the solution algorithm to capture the traveling wave response. Fluid equations are determined by using the linearized Navier–Stokes equations. The two sets of governing equations are coupled through the flow impermeability condition on the internal shell’s interface and the zero fluid particle velocity condition on the external boundary. Nonlinear numerical solutions to rotating shells with extremal fluid loading are conducted by the Runge–Kutta direct integration technique. The minimum number of driven/companion modes is determined and the results are verified with the available literature. Results show a significant exchange of energy between the driven and companion modes which consequently initiates a traveling wave response in rotating shell configurations. Moreover, the centrifugal loading and external excitation are shown to be effective on the oscillation amplitude as rotation speed changes leading to alteration of the response quality from periodic to chaotic.

Aeroelastic Simulation of Stall Flutter Undergoing High ‎and Low Amplitude Limit Cycle Oscillations

The aeroelastic behaviour of an airfoil oscillating in large and small pitch amplitudes due to nonlinearity ‎in aerodynamics is examined. The phenomenon of stall flutter resulted in the limit cycle oscillations of ‎NACA 0012 at low to intermediate Reynolds number is investigated numerically through the unsteady ‎two-dimensional aeroelastic simulation. The simulations employed unsteady Reynolds Average Navier ‎Stokes shear stress transport k-ω turbulent model with the low Reynolds number correction. The ‎simulations of the fluid-structure interaction were performed by coupling the structural equation of ‎motion with a fluid solver through the user-defined function utility. Numerical simulations were executed ‎at three different elastic axis positions; the leading-edge, 18% and 36% of the airfoil chord length. The ‎airfoil chord measures 0.156 m. The simulations were executed at the free stream velocity ranging from ‎‎5.0 m/s to 13 m/s corresponding to the Reynolds number between 51618 and 134207. Two types of ‎oscillation amplitudes were observed at each elastic axis position. At the leading-edge and 18% case, ‎small amplitude oscillations were observed while at 36%, the system underwent high amplitude ‎oscillations. The analysis revealed the cause for small oscillation amplitude is due to the separation of the ‎laminar boundary layer on the suction side of the airfoil starting at the trailing edge. High amplitude ‎oscillations occurred due to the existence of the dynamic stall phenomenon beginning at the leading-edge. ‎Small amplitude LCOs only occurred within a limited range of airspeed before it disappeared due to ‎increasing airspeed‎‎.

State-Input System Identification of Tall Buildings under Unknown Seismic Excitations Based on Modal Kalman Filter with Unknown Input

The seismic safety evaluation of tall building structures is necessary, but the multidegree-of-freedom feature of tall buildings increases the difficulty of joint identification of structural states, structural parameters, and unknown seismic excitations. Therefore, the state-input-system identification of tall buildings under unknown seismic excitations is the focus of this paper. To avoid the complex work of establishing a structural motion equation in absolute coordinate system, the simple structural motion equation in relative coordinates is directly adopted, but the measured absolute accelerations are used to establish the observation equation. In this way, the establishment of approximate assumptions is avoided, and direct feedback is not reflected in the observation equation. Also, the modal expansion technique is adopted to reduce the dimension of the motion equations and the size of the structural state to be identified. Different from the previous methods based on modal Kalman filtering, the unknown seismic excitation is treated as an unknown input instead of unknown modal forces in this paper, so the dimension of unknown forces in the identification process is not increased. When structural parameters of tall buildings are known, the generalized modal Kalman filtering with unknown input (GMKF-UI) proposed by the authors can simultaneously identify structural states and unknown seismic excitations by observing partial absolute acceleration responses. When extended to the case of unknown structural parameters, a generalized modal extended Kalman filtering with unknown input (GMEKF-UI) is proposed in this paper to simultaneously identify structural states, the unknown seismic inputs, and tall building systems using only partial absolute acceleration responses. Two numerical examples are used to verify the effectiveness of the proposed method.

On the stability of rotating pipes conveying fluid in annular liquid medium

This study provides a stability analysis of flexible rotating pipes taking into account the simultaneous effects of internal and external fluid loading. Using the Euler-Bernoulli beam assumptions, governing equations for flexural vibrations of rotating pipes are obtained. The internal flow characteristics and the double gyroscopic effect are taken into account when deriving the structural equations coupled with the internal flow. External fluid loading is determined by a special linearization of the Navier-Stokes equations. Considering the circular wall of the pipe as an impermeable boundary to the flow, fluid-induced forcing functions are obtained and then applied to the structural equations of motion to get a full description of the coupled field problem. Both analytical and semi-analytical solutions are utilized to get the stability results for the coupled equations of motion. Interesting findings are reported by providing stability results for two separate categories of pipes: with purely external fluid, and simultaneous internal-flow and external-fluid loading, and some conclusions are drawn.

Three dimensional aeroelastic analyses considering free-play nonlinearity using computational fluid dynamics/computational structural dynamics coupling

Most of the previous studies on free-play aeroelasticity were based on the linear aerodynamic theory, which cannot consider the aerodynamic nonlinear effects. This paper proposes a framework of computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling approach that can deal with the three dimensional aeroelastic problems with both the free-play nonlinearity and the aerodynamic nonlinearity. The fictitious mass method is used to construct the reduced structural equations of motion and the switching point is detected using the bisection method. The adaptive time step obtained by the bisection method is returned to the CFD solver so that both the structural and the fluid equations are integrated using the same time step. An all-movable wing with free-play at the root is considered for numerical studies. Results demonstrate the CFD/CSD coupling method can predict the stable limit cycle oscillation (LCO) effectively. The initial condition study shows that the LCO behavior is subcritical and the hysteresis response can be predicted in time domain effectively by the presented method. The viscous effect is shown to increase the LCO boundary and shift the LCO amplitude to a larger velocity in transonic regime. The LCO boundary is determined from subsonic to transonic Mach numbers. The transonic dip in LCO boundary is found by the CFD/CSD coupling method, but the equivalent linearization with doublet lattice method fails to predict this phenomenon. From the results of this study, the LCO boundary is shown to be 43.5% below the flutter boundary at most.

A distance coefficient-multi objective information fusion algorithm for optimal sensor placement in structural health monitoring

Optimal sensor placement (OSP) plays a key role in the construction and implementation of an effective structural health monitoring system (SHM). In this study, a novel and effective method named the distance coefficient-multi objective information fusion algorithm (D-MOIF), which is different from the conventional method and easier to be implemented, is developed to select the best sensor location for large-scale structures. An integrated information matrix including mode independence, damage sensitivity and modal strain energy is deduced from the structural motion equation to meet multiple needs of SHM. A European distance derived from the analytic geometry is proposed to overcome the information redundancy between sensors. Based on the principle of information entropy, an optimized objective function is constructed, which could balance the sensitivity and robustness of the algorithm. A computational case of a high arch dam is implemented to demonstrate the effectiveness of the modified algorithm, and three classical evaluation criteria are used to estimate the comparison between the D-MOIF algorithm and four traditional OSP methods. Finally, the optimization of the number of sensors based on different algorithms is discussed in detail. Results indicate that the proposed D-MOIF algorithm could generate more applicable sensor configurations for large-scale structures.

A unified algorithm for fully-coupled aeroelastic stability analysis of conical shells in yawed supersonic flow to identify the effect of boundary conditions

Supersonic flutter analysis of truncated conical shells with arbitrary classical boundary conditions is investigated in the present study. Structural equations of motion are obtained in their integral representation using the Flügge's first-order shear deformable assumptions along with the first-order piston theory for the aerodynamic model. Effects of curvature correction and airflow's yaw angle are also included in the aerodynamic model. Using a Rayleigh-Ritz based solution algorithm, the aeroelastic system of equations is solved by a standard eigenvalue solver with quadruple precision. After verification of the presented algorithm and solution procedure, a comprehensive parametric study is performed and effects of various parameters, namely the boundary conditions, semi-vertex angle, length and thickness ratios and yaw angle of the airflow on flutter boundaries of truncated conical shells are examined.

Nonlinear dynamics and flutter of plate and cavity in response to supersonic wind tunnel start

The transient response of a plate and a cavity is investigated in a supersonic wind tunnel start experiment where the freestream flow inside the test section reaches turbulent flow at Mach 2. Experimentally measured plate displacement time history shows flutter onset, transition to limit cycle oscillation, and stabilization at a static deformed state during the 30 s run. To analyze and interpret the measured plate response, a fully coupled aero-thermal-acousto-elastic analysis is carried out. A theoretical–computational model is formulated with a nonlinear structural plate model, acoustic pressure equation for the stationary fluid in a cavity, and the first-order Piston Theory aerodynamics. A linear stability analysis is performed that includes the nonlinear added stiffness due to an initial deformation to investigate the combined effects of freestream coupling and temperature differential on system stability. Also, direct time integration of the nonlinear fluid structural equations of motion is performed using experimentally measured flow parameters as inputs. All stability transitions are captured using the theoretical model with good agreement with experiment for transitions from no flutter to flutter/limit cycle oscillations (LCO) although the theoretical LCO amplitude is approximately $$50\%$$ larger than measured. The system’s sensitivity to cavity coupling, temperature differential, thickness calibration, static pressure differential, and turbulent pressure fluctuations are investigated. Lastly, snap-through buckling analyses in response to periodic and quasi-static excitations are conducted.

BEM–FEM coupling for the analysis of flexible propellers in non-uniform flows and validation with full-scale measurements

The first part of the paper presents a partitioned fluid–structure interaction (FSI) coupling for the non-uniform flow hydro-elastic analysis of highly flexible propellers in cavitating and non-cavitating conditions. The chosen fluid model is a potential flow solved with a boundary element method (BEM). The structural sub-problem has been modelled with a finite element method (FEM). In the present method, the fully partitioned framework allows one to use another flow or structural solver. An important feature of the present method is the time periodic way of solving the FSI problem. In a time periodic coupling, the coupling iterations are not performed per time step but on a periodic level, which is necessary for the present BEM–FEM coupling, but can also offer an improved convergence rate compared to a time step coupled method. Thus, it allows to solve the structural problem in the frequency domain, meaning that any transients, which slow down the convergence process, are not computed. As proposed in the method, the structural equations of motion can be solved in modal space, which allows for a model reduction by involving only a limited number of mode shapes.The second part of the paper includes a validation study on full-scale. For the full-scale validation study a purposely designed composite propeller with a diameter of 1 m has been manufactured. Also an underwater measurement set-up including a stereo camera system, remote control of the optics and illumination system has been developed. The propeller design and the underwater measurement set-up are described in the paper. During sea trials blade deflections have been measured in three different positions. A comparison between measured and calculated torque shows that the measured torque is much larger than computed. This is attributed to the differences between effective and nominal wakefields, where the latter one has been used for the calculations. To correct for the differences between measured and computed torque the calculated pressures have been amplified accordingly. In that way the deformations which have been computed with the BEM–FEM coupling for non-uniform flows became very similar to the measured results.

Generalized algorithms for the identification of seismic ground excitations to building structures based on generalized Kalman filtering under unknown input

The exact information of seismic excitation and structural state is a prerequisite for structural seismic safety assessment and vibration control. When the seismic excitation to a structure is not measured, the seismic excitation can be identified as an inversed problem from measured structural responses. Although some relevant approaches have been developed, there are certain limitations or drawbacks in the existing approaches. To circumvent these problems, two generalized algorithms are proposed for the identification of seismic ground excitation to multi-story and tall buildings, respectively. When the seismic ground excitation to a structure is not measured, the data measured by a structural health monitoring system are structural absolute responses. So the structural motion equation in the absolute coordinate system is derived, in which the unknown seismic ground excitation is treated as unknown external force acting on the structure. First, the identification of unknown seismic excitations to multi-story building structures is studied. A generalized Kalman filtering under unknown input is proposed for the identification of structural state and unknown seismic excitation without the observation of structural absolute acceleration responses at the location of unknown external force. The derivation of the proposed generalized Kalman filtering under unknown input is based on the classical Kalman filter, but is more general than the existing identification approaches based on Kalman filter with unknown input in the deployments of accelerometers in the building structure. Then, it is extended to explore the identification of unknown seismic excitations to tall building structures. To avoid substructural identification from the top to bottom in a sequential manner, the motion equation in absolute coordinate system is reduced by modal expansion. Moreover, instead of the identification of unknown modal forces in previous approaches, the seismic excitation is directly identified without increasing the number of unknown forces. To demonstrate the proposed algorithms, numerical examples of identifying seismic excitations to a 6-story shear building and an 18-story tall building are investigated.

Reduced order nonlinear aeroelasticity of swept composite wings using compressible indicial unsteady aerodynamics

Nonlinear dynamic aeroelasticity of composite wings in compressible flows is investigated. To provide a reasonable model for the problem, the composite wing is modeled as a thin walled beam (TWB) with circumferentially asymmetric stiffness layup configuration. The structural model considers nonlinear strain displacement relations and a number of non-classical effects, such as transverse shear and warping inhibition. Geometrically nonlinear terms of up to third order are retained in the formulation. Unsteady aerodynamic loads are calculated according to a compressible model, described by indicial function approximations in the time domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to derive the final explicit form of the coupled fluid-structure equations of motion. The final nonlinear governing aeroelastic system of equations is solved using the eigenvectors of the linear structural equations of motion to approximate the spatial variation of the corresponding degrees of freedom in the Ritz solution method. Direct time integrations of the nonlinear equations of motion representing the full aeroelastic system are conducted using the well-known Runge–Kutta method. A comprehensive insight is provided over the effect of parameters such as the lamination fiber angle and the sweep angle on the stability margins and the limit cycle oscillation behavior of the system. Integration of the interpolation method employed for the evaluation of compressible indicial functions at any Mach number in the subsonic compressible range to the derivation process of the third order nonlinear aeroelastic system of equations based on TWB theory is done for the first time. Results show that flutter speeds obtained by the incompressible unsteady aerodynamics are not conservative and as the backward sweep angle of the wing is increased, post-flutter aeroelastic response of the wing becomes more well-behaved.