Structural Equations Of Motion Research Articles

Identification of earthquake ground motion based on limited acceleration measurements of structure using Kalman filtering technique

Identification of earthquake ground motion from structural health monitoring (SHM) data provides a good means to reconstruct seismic loads that are essential for postearthquake safety assessments and disaster simulations of structures. Because the data measured by an SHM system are structural absolute response, they cannot be directly applied to the structural motion equation, which is established in relative coordinate system. As such, this paper originally derives the motion equation in absolute coordinate system and then expands the equation into modal space. In addition, the proposed method allows for identifying earthquake ground motion using incomplete modal information and limited measurements through the standard Kalman filter. Subsequently, a numerical two-dimensional frame is used to validate the feasibility of the proposed method, and the influences of modal parameters and measurement noise on the identification accuracy are also fully investigated. The results show that the proposed method is sensitive to frequency and measurement noise but insensitive to modal shape and damping ratio. It is also found that the identified ground motion subjected to certain measure noise can still be reliably employed for postseismic response calculations of medium- and long-period structures. Finally, a shaking table test performing on a five-floor frame further demonstrates the effectiveness and accuracy of the proposed identification algorithm for practical application.

Fully nonlinear analysis incorporating viscous effects for hydrodynamics of an oscillating wave surge converter with nonlinear power take-off system

WaveRoller can be classified as an effective Oscillating wave surge converter (OWSC) installed in nearshore coastal areas. In this study, the hydrodynamic performance of a two-dimensional (2-D) WaveRoller is investigated based on a fully nonlinear time-domain higher-order boundary element method (HOBEM). The mechanism through which the wave power is extracted, is analyzed in a nonlinear power take-off (PTO) by coupling the time-varying motion of the flap and hydro-electric generator. A simplified Morison model is used to determine the viscosity term in the structural motion equation. The present model is validated against the published experimental and numerical results for an OWSC with its top edge piercing through the water surface. Numerical simulations are undertaken to investigate the mechanism of the nonlinear phenomenon including wave nonlinearity and PTO nonlinearity. The influence of the viscosity is also quantified by comparing the numerical results with and without the quadratic damping term. Numerical results indicate that when the incident wave amplitude is large, the capture efficiency is reduced by the strengthen higher order free wave components, and the PTO nonlinearity becomes more prominent. Furthermore, the maximum discrepancy induced by the viscosity occurs in the resonant zone.

Mode competition in galloping of a square cylinder at low Reynolds number

Galloping is a type of fluid-elastic instability phenomenon characterized by large-amplitude low-frequency oscillations of the structure. The aim of the present study is to reveal the underlying mechanisms of galloping of a square cylinder at low Reynolds numbers ($Re$) via linear stability analysis (LSA) and direct numerical simulations. The LSA model is constructed by coupling a reduced-order fluid model with the structure motion equation. The relevant unstable modes are first yielded by LSA, and then the development and evolution of these modes are investigated using direct numerical simulations. It is found that, for certain combinations of$Re$and mass ratio ($m^{\ast }$), the structure mode (SM) becomes unstable beyond a critical reduced velocity$U_{c}^{\ast }$due to the fluid–structure coupling effect. The galloping oscillation frequency matches exactly the eigenfrequency of the SM, suggesting that the instability of the SM is the primary cause of galloping phenomenon. Nevertheless, the$U_{c}^{\ast }$predicted by LSA is significantly lower than the galloping onset$U_{g}^{\ast }$obtained from numerical simulations. Further analysis indicates that the discrepancy is caused by the nonlinear competition between the leading fluid mode (FM) and the SM. In the pre-galloping region$U_{c}^{\ast }<U^{\ast }<U_{g}^{\ast }$, the FM quickly reaches the nonlinear saturation state and then inhibits the development of the SM, thus postponing the occurrence of galloping. When$U^{\ast }>U_{g}^{\ast }$, mode competition is weakened because of the large difference in mode frequencies, and thereby no mode lock-in can happen. Consequently, galloping occurs, with the responses determined by the joint action of SM and FM. The unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable FM results in the high-frequency vortex shedding in the wake. The dynamic mode decomposition (DMD) technique is successfully applied to extract the coherent flow structures corresponding to SM and FM, which we refer to as the galloping mode and the von Kármán mode, respectively. In addition, we show that, due to the mode competition mechanism, the galloping-type oscillation completely disappears below a critical mass ratio. From these results, we conclude that transverse galloping of a square cylinder at low$Re$is essentially a kind of single-degree-of-freedom (SDOF) flutter, superimposed by a forced vibration induced by the natural vortex shedding. Mode competition between SM and FM in the nonlinear stage can put off the onset of galloping, and can completely suppress the galloping phenomenon at relatively low$Re$and low$m^{\ast }$conditions.

A flutter prediction method with low cost and low risk from test data

The most common approach to flight flutter testing is to track estimated modal damping ratios of an aircraft over a number of fight conditions. However, the risk is inevitable when this method is utilized since the aircraft must fly near the flutter boundary. In this paper, a method with low cost and low risk for flutter boundary prediction is proposed. Based on structural modal parameters from the ground vibration test (GVT) and the structural response at the sub-critical speed, the generalized aerodynamic force coefficient vector can be solved. The generalized displacement vector is designed as the input and the generalized aerodynamic force coefficient vector as the output. System identification is used to construct the reduced-order aerodynamic model. The aeroelastic model based on experimental results can be constructed by coupling the structural motion equation and the aerodynamic equation in state space. Analyzing the characteristics of the fluid–structure coupling system changing with the dynamic pressure, the flutter onset behaviors can be solved by the eigenvalue method. Wind tunnel tests confirm that this method only needs one wind-on test at the sub-critical speed to predict the flutter onset with a certain precision even the reference dynamic pressure is far from the flutter critical point.

Simulation of Vortex-Induced Vibration of Long-Span Bridges: A Nonlinear Normal Mode Approach

Due to the lack of analytic technique for simulating the vortex-induced vibration (VIV) of long-span bridges, a combination of the VIV semi-empirical model with the structural equation of motion is widely employed to calculate the responses of bridge structures. However, the applicability of this method has seldom been investigated before. In this study, the theoretical defects of the conventional combination strategy (i.e. the finite element procedure or the linear normal mode procedure, LNM) are first discussed, a more theoretically reliable approach (the nonlinear norm mode approach, NNM) is then proposed, and the closed-form expression for the NNM of the VIV system is derived. The accuracy of the proposed method is further illustrated by two case studies. This new approach offers a theoretically reliable tool for analyzing the VIV of long-span bridges. It can also be applied to the process of VIV fatigue analysis or control strategy optimization.

Geometrically nonlinear aeroelastic behavior of pretwisted composite wings modeled as thin walled beams

Geometrically nonlinear aeroelastic behavior of pretwisted composite high aspect ratio wings, structurally modeled as thin walled beams (TWB) is studied. The structural equations of motion are obtained for the circumferentially asymmetric stiffness (CAS) configuration TWB based on the kinematic relations governing the thin walled beams, including the nonlinear strain–displacement relations and utilizing the principles of analytical dynamics. Unsteady aerodynamic loads in the incompressible flow regime are expressed using Wagner’s function in time-domain. The aeroelastic system of equations is augmented by the differential equations governing the aerodynamics lag states to come up with the final coupled fluid–structure equations of motion. The governing equations of the aeroelastic system is solved, for the TWB with CAS composite layup, by means of a Ritz based solution methodology utilizing the mode shapes of the linear structural system to approximate the spatial variation of degrees of freedom in the thin walled beam. Time response of the nonlinear aeroelastic system is obtained via the Runge–Kutta direct integration algorithm. Effects of the fiber angle and pretwist angle of the CAS layup configuration on the nonlinear aeroelastic stability margins and limit cycle oscillation behavior are studied in depth.

A harmonic balance method for nonlinear fluid structure interaction problems

Over the past decade the Harmonic-Balance technique has been established as a viable alternative to direct time integration methods to predict periodic aeroelastic instabilities. This article reports the progress made in using a frequency updating procedure, based on a coupled fluid-structural solver using the Harmonic-Balance formulation. In particular, this paper presents an efficient implicit time-integrator that accelerates the convergence of the structural equations of motion to the final solution. To demonstrate the proposed approached, the paper includes a detailed investigation of the impact of input parameters and exercises the method for two types of fluid-structural nonlinear instabilities: transonic limit-cycle oscillations and vortex-induced vibrations.

The effects of agglomerated CNTs as reinforcement on the size-dependent vibration of embedded curved microbeams based on modified couple stress theory

ABSTRACTVibration of an embedded nanocomposite curved microbeam is investigated based on the modified couple stress theory and Timoshenko beam model. The microbeam is reinforced by single-walled carbon nanotubes (SWCNTs) considering agglomeration effects. The structure is subjected to axial magnetic field. Applying Hamilton's principle, the motion equations of structure are derived. The generalized differential quadrature method is applied to obtain the frequency of the nanocomposite curved microbeam. The effects of the SWCNTs' volume percent, agglomeration of SWCNTs, elastic medium, magnetic field, boundary conditions, size effects, and central angle of microbeam on the frequency of the structure are shown.

Stable states and mechanism of self-propulsion of two tandem filaments in viscous flow

Fish schooling behavior contains very interesting but complicated hydrodynamics. A simplified and classic fish schooling model was simulated and analyzed in the present study, by using two self-propelled flexible filaments in a tandem arrangment. The two filaments plung harmonically in the vertical direction and are free to move in the horizontal direction in an incompressible viscous fluid flow. The simulations were carried out by using an efficient immersed boundary projection method, where the additional momentum forcing is calculated in an implicit way and an approximate factorization procedure is applied to solve the incompressible Navier-Stokes equations. The coupling between the fluid flow and the flexible structures is realized through the additional momentum forcing. In the immersed boundary projection method, both the pressure and the momentum forcing are considered as the Langrangian multipliers to satisfy the continuity constraint and the no-slip condition at the immersed boundary, respectively. To ensure a long period simulation, a non-inertial frame, which is fixed to the leading edge of the upstream filament, was adopted in the Navier-stokes equations and the structure motion equation, by introducing a translational acceleration in the motion equations. In this way, it significantly reduced the computational cost by avoiding a large compuational domain. Two-dimensional simulations were performed in the present study. The Reynolds number based on the plunging motion was fixed at 200, while the different initial horizontal and vertical distances between the two filaments were considered in simulations. The stable states of the two tandem filaments and the mechanism of the vortex interactions were mainly concerned. The numerical results reveal that three stable states are formed with different horizontal distances and vertical distances between the two filaments, i.e., long-distance tandem state, short-distance tandem state and parallel state. For the long-distance tandem state, the upstream filament performs like the single-filament case and the motion of the downstream filament always tends to be convergent with different horizontal and vertical distances. The downstream filament was found to move through the upstream shedding vortices, and was barely affected by the offset in the vertical direction. For the short-distance tandem state, the shear layer of the upstream filament is merged with the shear layer of the downstream one before it is shed from the filament. This effect generates a strong vortex shedding, and consequently the propulsive force is enhanced with the cruising velocity increased as well. As a result, phase gaps of motions and drag forces between the upstream and downstream filaments arise. For the paralle state, when the initial horizontal distance of the two filamens is small while the vertical offset is large enough, the influence of the upstream vortex on the downstream filament is decreased. So the downstream filament moves forward and the parallel state is formed. In this case, the shear layers of two filaments shed synchronously and strong vortices are formed in the wake. Meanwhile the windward aera are about twice as compared with the tandem states, thus the cruising velocity is reduced. In summary, as a result of the interactions between upstream vortices and downstream vortices/structures, the forward speed of the parellel state is the slowest, and that of the short-distance state is the fastest, while that of the long-distance state is middle among the three stable states.

SOFIA - A simulation tool for bottom founded and floating offshore structures

This paper presents a recently developed simulation tool, SOFIA (Simulation Of Floaters In Action), suitable for modeling slender bottom founded and moored/freely floating space frame structures exposed to environmental loads. In contrast to traditional rigid body formulations of floating structures, the finite element method is utilized in the implemented numerical approach, which allows for direct output of local section forces and displacements for joint analysis and fatigue calculations. The numerical approach builds upon a partitioned solution procedure, constituted by individual fluid and structure domains, which are coupled through the structural equation of motion. The structural domain is handled by means of the finite element method, while large displacements and stress stiffening effects, exhibited by moored floating structures, are inherently included due to a co-rotational element formulation. The fluid domain is modeled by an appropriate water wave theory, and the hydrodynamic loads are evaluated at the instantaneous fluid-structure interface by means of a relative Morison equation. The equation of motion is solved in time domain, which makes SOFIA capable of handling bottom founded and floating space frame structures that may experience non-linear behavior. To demonstrate the applicability of the simulation tool, numerical examples of a bottom founded and a floating space frame structure are presented.

A method combined immersed boundary with multi-relaxation-time lattice Boltzmann flux solver for fluid-structure interaction

This paper performs a newly developed method, which combines the immersed boundary method (IBM) with multi-relaxation-time lattice Boltzmann flux solver (MRT-LBFS), for solving fluid-structure interaction problems. Finite volume discretization is used to solve the macroscopic governing equations with the flow variables defined at cell centers. Based on the multi-scale Chapman-Enskog expansion analysis, LBFS builds a relationship between the variables and fluxes in incompressible Navier-Stokes equations and density distribution functions in lattice Boltzmann equation. In order to ensure no-slip boundary condition, boundary condition-enforced immersed boundary method is used to treat the fluid-structure interface. The restoring force can be resolved by making a velocity correction in the flow field. The four-stage RungeKutta scheme is used to solve the motion equation of structure. Using the lattice model and immersed boundary method, fluid-structure coupling calculation can be implemented in a Cartesian grid, without generating the body-fitted mesh and using moving mesh technique. Therefore, the computational process is considerably simplified. In order to verify the validity and feasibility of IB-MRT-LBFS to solve fluid-structure interaction problems, both one-and two-degree of freedom vortex-induced vibrations (VIV) of a circular cylinder and two-degree of freedom VIV of two cylinders in a tandem arrangement are simulated by this proposal method. For a VIV cylinder system, the transverse vibration response is much stronger than the axial response. When the vibration occurs in the range of lock-in regime, the shedding vortex frequency of the wake is close to natural frequency of the cylinder so that resonance appears, consequently causing larger amplitude. For two VIV cylinders in a tandem arrangement, the dynamic behavior of each cylinder is significantly different from that of a single cylinder. The gap spacing between the two cylinder centers is a significant parameter which effects vibration characteristics and the spacing is fixed in the simulations of two tandem cylinders. With the effects of upstream cylinder wake, the axial and transverse amplitudes of downstream cylinder obviously increase with adding the reduced velocity. The downstream cylinder is delayed, coming into lock-in regime, and the range of lock-in regime is expanded under the effects of the wake of the upstream cylinder. As the reduced velocity is relatively large, the vibration response of the upstream cylinder is close to a single cylinder and the vibration response of the downstream cylinder is more intense than the upstream cylinder. Compared with the existing literature results, our result illustrates that IB-MRT-LBFS owns the ability to correctly predict the lock-in regime, dynamic response and the forces of vortex-induced vibrations of cylinders. And this method can accurately capture the wake structures.

Aeroelastic response and limit cycle oscillations for wing-flap-tab section with freeplay in tab

A linear and nonlinear theoretical and experimental aeroelastic investigation of a wing-flap-tab typical section model undergoing two-dimensional incompressible airflow is described. The linear flutter velocity (LFV) and frequency are predicted using linear analysis. Then a freeplay structural nonlinearity is considered in the tab. The structural equations of motion have been coupled with Theodorsen aerodynamic theory to produce the theoretical aeroelastic model which is analyzed by a state space method to predict the LFV and flutter frequency. Linear piecewise function has been used to introduce the tab spring stiffness in the freeplay state. The ground vibration test is used to measure the model structural dynamic characteristics. Then the experimental aeroelastic model is placed in a low speed wind tunnel to measure the LFV and the limit cycle oscillation (LCO) of the physical model induced by freeplay. The root main square amplitude value of the pitch, flap pitch, tab pitch and plunge degrees of freedom of the tab nonlinearities are normalized with freeplay gap size to produce a bifurcation diagram with normalized airflow velocity as the bifurcation parameter. The results show that the LCO frequency jumps from low to high frequency at a yet higher flow velocity. At the same flow velocity, the pitch and plunge motion response amplitudes drop while the flap pitch and tab pitch degrees of freedom response amplitude increase. In general the experimental measured LCO is more complicated than the theoretically calculated LCO in terms of the harmonic content of the response. On the other hand there is good agreement between the theoretical and experimental result of the linear system as well the LCO for the tab freeplay nonlinearities.

Layered reduced-order models for nonlinear aerodynamics and aeroelasticity

A layered reduced-order modeling approach for nonlinear unsteady aerodynamics comprising both linear and nonlinear characteristics is developed. The constructed reduced-order model (ROM) with a parallel connection structure is composed of a linear surrogate model and a nonlinear one. The linear autoregressive with exogenous input (ARX) model is constructed by a training case at small amplitude, while the nonlinear radial basis function neural network (RBFNN) model is identified to model the residual nonlinear responses from the second training case under large amplitude motions. Outputs from both models are superimposed to obtain the total aerodynamic coefficients. Either quasi-steady or unsteady layered model can be constructed by introducing only input delay or considering both input and output delayed effects. The present approach is tested by predicting the unsteady aerodynamic forces, limit cycle oscillations and flutter behaviors of a transonic NACA 64A010 airfoil. Results show that both the quasi-steady and the unsteady models agree well with those from high fidelity simulations within a wide range of amplitude and frequency, and the LCO trends are reasonable after the models are coupled with the structural equations of motion. A better agreement is achieved by the unsteady layered model, even though this model may sometimes become unstable.

High-Fidelity Flexibility-Based Component Mode Synthesis Method with Interface Degrees of Freedom Reduction

The partitioned structural equations of motion via the method of localized Lagrange multipliers (PALM) consist of internal and interface variables. The model reduction achieved based on PALM is called flexibility-based component mode synthesis (F-CMS), and it thus far has concentrated on reducing the internal degrees of freedom (DOFs) or reducing the internal and interface DOFs at once. In this work, well-defined three-level reduction formulations of the F-CMS method are proposed for better precision: internal DOFs, the localized Lagrange multipliers, and the interface boundary displacement. The proposed formulations include three different reduction techniques for the interface boundary displacement for better computational efficiency. Numerical experiments indicate that the proposed formulations can substantially reduce the overall reduced-order models further without compromising the accuracy, thus achieving a substantial additional reduction of computational cost compared with the previous works. In addition, the performance of a novel error estimator in the overall reduction levels is tested.

Aerothermoelastic-Acoustics Simulation of Flight Vehicles

This paper describes a novel computational-fluid-dynamics-based numerical solution procedure for effective simulation of aerothermoacoustics problems with application to aerospace vehicles. A finite element idealization is employed for both fluid and structure domains, which fully accounts for thermal effects. The accuracies of both the fluid and structure capabilities are verified with flight- and ground-test data. A time integration of the structural equations of motion , with the governing flow equations, is conducted for the computation of the unsteady aerodynamic forces, which uses a transpiration boundary condition at the surface nodal points in lieu of the updating of the fluid mesh. Two example problems are presented herein to that effect. The first one relates to a cantilever wing with a NACA 0012 airfoil. The solution results demonstrate the effect of temperature loading that causes a significant increase in acoustic response. A second example, the hypersonic X-43 vehicle, is also analyzed; and relevant results are presented. The common finite element-based aerothermoelastic-acoustics simulation process, its applicability to the efficient and routine solution of complex practical problems, the employment of the effective transpiration boundary condition in the computational fluid dynamics solution, and the development and public domain distribution of an associated code are unique features of this paper.

Effects of a Free-to-Roll Fuselage on Wing Flutter: Theory and Experiment

A theoretical study of a wing–fuselage combination with a rigid-body roll degree of freedom for the fuselage and elastic modes for the wing is presented along with a companion wind-tunnel test. The full-span wing dynamics are modeled using a linear-plate wing structure theory. A component modal analysis is used to derive the full structural equations of motion for the wing–fuselage combination system. A three-dimensional time-domain vortex-lattice aerodynamic model is also used to investigate flutter of the linear aeroelastic system. The experimentally observed flutter mode is antisymmetric, although theory suggests that the symmetric-mode flutter velocity is only modestly higher than that for the antisymmetric modes. Correlation between theory and experiment for flutter velocity and frequency is good. The experimentally observed postflutter response is a limit-cycle oscillation, and this is worthy of further study using an appropriate nonlinear theory.

Limit-Cycle Oscillation of High-Aspect-Ratio Wings

In this paper structural equations of motion based on nonlinear beam theory and the unsteady aerodynamic forces are gained to study the effects of geometric nonlinearity on the aerodynamic response of high-aspect-ratio wings. Then the Galerkin’s method is used to discretize the equations of motion. The results of HALE wing show good agreement with references. And other results investigate the effects of geometric structural nonlinearity on the response of a wing. Also the complex changes of the limit-cycle oscillation with speed increasing is carefully studied.

Discussion of paper: “Estimating optimum parameters of tuned mass dampers using harmony search” [Eng. Struct. 33 (9) (2011) 2716–2723

In the aforementioned paper, optimum design of tuned mass dampers (TMD) installed on multiple degree of freedom (MDOF) structures under seismic excitations is proposed. Equations of motion for structures with and without TMD were solved by the modal analysis procedure. Results of numerical studies in this discussion show that using modal method for solving the differential equations governing the motion of controlled structures by TMD dampers usually is not corrected because such systems essentially have not classical damping.

Numerical Prediction of Fluid-Elastic Instability in Normal Triangular Tube Bundles With Multiple Flexible Circular Cylinders

Prediction of fluid-elastic instability onset is a great matter of importance in designing cross-flow heat exchangers from the perspective of vibration. In the present paper, the threshold of fluid-elastic instability has been numerically predicted by the simulation of incompressible, unsteady, and turbulent cross flow through a tube bundle in a normal triangular arrangement. In the tube bundle under study, there were single or multiple flexible cylinders surrounded by rigid tubes. A finite volume solver based on a Cartesian-staggered grid was implemented. In addition, the ghost-cell method in conjunction with the great-source-term technique was employed in order to directly enforce the no-slip condition on the cylinders' boundaries. Interactions between the fluid and the structures were considered in a fully coupled manner by means of intermittence solution of the flow field and structural equations of motion in each time step of the numerical modeling algorithm. The accuracy of the solver was validated by simulation of the flow over both a rigid and a flexible circular cylinder. The results were in good agreement with the experiments reported in the literatures. Eventually, the flow through seven different flexible tube bundles was simulated. The fluid-elastic instability was predicted and analyzed by presenting the structural responses, trajectory of flexible cylinders, and critical reduced velocities.

Flutter analyses of complete aircraft based on hybrid grids and parallel computing

A tightly coupled method was developed to analyse aeroelasticity by constructing subiterative schemes for fluid and structural equations of motion, respectively. With MPI partition parallel computing, the fluid was solved by Navier-Stokes equations based on hybrid grids. A new unstructured background grid deformtion method was used for the CFD grid deformation. The transonic flutter wind tunnel model of a complete aircraft was simulated to validate the developed method. The flutter characteristics of the aircraft was analysed and compared with the test results. It indicates that the devoloped method has a relatively higher precision and can be used for aeronautical engineering application.