Runge-Kutta Integration Algorithm Research Articles

Data analysis of non-linear radiative electro-periodic MHD flow past a stretching sheet with activation energy impact

Electro-periodic Magneto-hydrodynamics (EPMHD), activation energy, and non-linear radiation are pivotal elements in engineering and industry also contribute to advancements in fields like heat transfer, materials science, and chemical engineering. Through similarity transformation, governing partial differential equations simplify into ordinary differential equations. Precise computational solutions are derived using the Nachtsheim-Swigert shooting iteration method (NSSIM) and the reliable 6th-order Runge-Kutta integration algorithm (RKIA). The profiles of velocities, temperatures, concentrations, skin friction, heat and mass transfer rate are presented by graphically. The novel aspect of this research is that thermal performance is enhanced by non-linear radiation by around 2.28%, whereas linear radiation only produces a 0.28% improvement in thermal performance. Also the mass transfer rate experiences a decrease of about 25.76% with increasing chemical reactions in the presence of activation energy and a more substantial 62.22% reduction in its absence, notably showing a comparatively milder decline of 36.46% in the presence of activation energy than in its absence. Furthermore, this study yields three novel linear regression equations, skillfully derived from the numerical and graphical results. This research have practical applications in understanding and optimizing processes involving fluid dynamics, such as the behavior of certain materials or substances subjected to stretching or deformation.

Large-Strain Vibrations of Axially Stretched Hyperelastic Rods

Fast-moving soft robotics usually suffer large-strain deformation with a relatively high velocity. To improve their performance, it is necessary to study the underly dynamical mechanism. However, the inherent strong material nonlinearity brings considerable difficulties in modeling and calculating the soft structure’s nonlinear dynamics. This work focuses on the large-strain vibrations of the hyperelastic rod under an axial tensile force. Several typical hyperelastic models, including the neo-Hookean, Mooney, and Yeoh models, are used to derive the equations governing the rod’s dynamics. The equations based on these models are relatively complex, and their corresponding numerical calculation diverges in some cases. Therefore, a reference stretch ratio(s) polynomial fitting (RSRPF) model is proposed to derive the concise governing equation. Such a governing equation is effectively solved by the combination of the Galerkin discretization and the fourth-order Runge–Kutta integration algorithm. The correctness of the developed model and the employed algorithm is verified by comparing with the previous data of a hyperelastic membrane. In most cases, the rod periodically vibrates under a periodic tensile force. However, when the structural damping is small and the load frequency approaches the resonant frequency, the hyperelastic rod may quasi-periodically vibrate. The revealed large-strain vibration mechanism is supposed to be useful for guiding the dynamical design of soft structures.

On the dynamics of curved magnetoactive soft beams

Recently, hard-magnetic microparticles were embedded into soft matrix to manufacture a new type of magnetoactive soft materials, i.e. hard-magnetic soft (HMS) materials. Due to the promising applications of HMS materials in the areas of soft robotics, flexible electronics, biomedicine, etc., there is a burgeoning trend in investigating the mechanical responses of HMS structures. Special attention has been placed on the static deformations of initially straight HMS beams. However, to fulfill the potential applications of HMS materials, it is also necessary to examine the dynamical behaviors of HMS beams. The present study aims to develop a large-deformation dynamical model of curved HMS beams actuated by a harmonically rotational magnetic field, which is varying periodically with time. This dynamical curved HMS beam model could be easily reduced to the corresponding static one, which is then solved both analytically and numerically. To tackle the dynamic problem, the derived governing equation for nonlinear vibrations of the curved HMS beam is first discretized by using the Galerkin method and then solved by a fourth-order Runge-Kutta integration algorithm. It is found that the curved HMS beam tends to align with the direction of the applied constant magnetic field as the strength of the applied field increases. The curved HMS beam would vibrate periodically if a periodically rotational magnetic field is applied. Furthermore, when the rotational frequency of the magnetic field is close to the natural frequency of the beam, the system resonates and the vibration amplitude of the curved beam becomes extremely large.

Nonlinear Flutter Analysis of Porous Functionally Graded Plate in Yawed Hypersonic Flow

An aerothermoelastic analysis of functionally graded plate containing porosities in ‎yawed hypersonic flows is investigated. Due to some incorrect manufacturing processes, two ‎different types of porosity, namely, even and uneven distributions are taken into account. The ‎third-order piston theory is utilized to estimate the unsteady aerodynamic pressure induced by the ‎hypersonic airflow. The material properties of a plate are assumed to vary across the thickness ‎direction according to a simple power law. Based on classical plate theory, the motion equations are ‎developed with geometric nonlinearity taking into consideration of von Karman strains. The ‎formulations are established based on Hamilton’s principle while the generalized differential ‎quadrature method is employed to solve the nonlinear aerothermoelastic equations. Due to lower ‎computational efforts‎‏, ‏the method of generalized differential quadrature (GDQM) is used to obtain ‎accurate results. Moreover, the assumed mode method along with the Runge-Kutta integration ‎algorithm is used as a solution method. The reliability and precision of the obtained results are ‎validated by published literature. Then, the influence of porosity distribution, porosity coefficient, ‎and yawed flow angle are discussed in detail. In general, this paper shows that even porosity ‎distribution would have a more destabilizing effect compared with the ‎uneven porous model.‎ And also, for both porosity distributions, the chaotic behavior appears in higher top surface ‎temperature but even porosity distribution has a profound effect on chaotic motion.‎

Three-dimensional parametric resonance of fluid-conveying pipes in the pre-buckling and post-buckling states

The paper presents a model of describing three-dimension parametric vibrations of a simply supported pipe conveying pulsating fluid. The three-dimensional motion equation of the system is a set of two nonlinear partial differential equations developed on the basis of Euler-Bernoulli beam theory, geometric nonlinearity and Kelvin-Voigt damping model. The three-dimensional motion equation is discretized by the Galerkin method and the nonlinear responses are solved by a fourth order Runge-Kutta integration algorithm. The three-dimensional model is validated for the natural frequencies from the pre-buckling state to the post-buckling state. Results of the nonlinear parametric resonance responses are shown in the form of time histories, spectrograms, phase-plane portraits, Poincaré map sections and motion trajectories. The vibration mode of sub-harmonic resonance is similar to a certain order free vibration mode, and the vibration mode of combination resonance is a superposition of two adjacent modes. The motion trajectories indicate the vibration of sub-harmonic resonance is planar but the vibration of combination resonance is non-planar. The lock-in phenomenon occurs in the pre-buckling and post-buckling states when the natural frequencies are locked to the pulsation frequency. The amplitudes of parametric resonance in the post-buckling state is much larger in comparison to parametric resonance in the pre-buckling state. Jump phenomenon is highlighted for the parametric resonance in the post-buckling state in comparison to the pre-buckling state.

Nonlinear Parametric Vibration of the Geometrically Imperfect Pipe Conveying Pulsating Fluid

In this paper, the novel model of fluid-conveying imperfect pipe supported at both ends is established by considering the geometric imperfection and the geometric nonlinearity induced by mid-plane stretching. The integral-partial differential equation is discretized by the Galerkin method and solved by a fourth-order Runge–Kutta integration algorithm. Compared with the supercritical pitchfork bifurcation of the perfect pipe conveying fluid, the results show that the cusp bifurcation occurs in the imperfect pipe when increasing the flow velocity. Excellent agreement is observed between the numerical results and the analytical results. The two stable asymmetry bifurcation branches bring interesting phenomena in the post-buckling state. The global nonlinear dynamic behaviors of the imperfect pipe are studied by establishing the bifurcation diagrams. The influence of the geometric imperfection amplitude on the nonlinear response is leading to cusp bifurcation comparing with pitchfork bifurcation of the perfect pipe. When pulsation frequency is set as the bifurcation parameter, there are clear nonresonance ranges, low energy resonance ranges and high energy resonance ranges. In the high energy resonance ranges, the first mode vibration coexisting with the sub-harmonic resonance and combination resonance occurs. As the mean flow velocity and pulsation amplitude are set as bifurcation parameters, the vibration of the imperfect pipe becomes more and more complicated. The vibration exhibits far richer dynamic behaviors including periodic, multi-periodic, quasi-periodic, and chaotic motions. The viscoelastic damping can effectively suppress the vibration response and transfer the high energy resonance to the low energy resonance state. The improved model and corresponding results provide useful information for further studying the dynamic behaviors of fluid-conveying pipe with geometric imperfections.

Convective Heat Transfer Analysis of Hydromagnetic Micropolar Fluid Flow Past an Inclined Nonlinear Stretching Sheet with Variable Thermo-Physical Properties

The current work examines mixed convection boundary layer flow and heat transfer attributes in hydromagnetic micropolar fluid past a heated inclined sheet which stretches nonlinearly along the direction of flow. The impact of variable thermo-physical characteristics of the fluid together with the influence of magnetic field, thermal radiation and viscous dissipation are also checked on the flow field. The modelled governing equations are translated from partial to ordinary differential equations via relevant similarity transformations and the resulting equations are subsequently solved numerically by means of shooting techniques in company with Runge-Kutta integration algorithms. The reactions of the skin friction coefficient, Nusselt number, dimensionless velocity as well as temperature to variations in the emerging controlling parameters are illustrated through different graphs. In the limiting situations, the results obtained exhibit a strong relationship with the existing related works in literature. The facts emanated from this study also reveal that the thickness of the thermal boundary layer grows widely with a rise in the Eckert number and Biot number parameters whereas increasing the material (micropolar) and thermal conductivity parameters have opposite effects on the rate of heat transfer.

Planar and non-planar vibrations of a fluid-conveying cantilevered pipe subjected to axial base excitation

The main aim of the present study is to explore the nonlinear vibrations of a fluid-conveying cantilevered pipe under an axial base excitation, by considering both two-dimensional (2-D) and three-dimensional (3-D) responses of the pipe. For this purpose, the extended Hamilton’s principle is applied to derive the 3-D nonlinear governing equations of the pipe with an axial base excitation. The resulting partial differential equations are then discretized by employing a Galerkin method. The Floquet theory is adopted to determine the stability regions of the pipe system. A linear analysis shows that the internal flow velocity, excitation amplitude and excitation frequency have great influences on the stability regions of the pipe system, indicating that both subharmonic and combination resonances can occur. The planar and non-planar nonlinear responses of the pipe are calculated by using a fourth-order Runge–Kutta integration algorithm. These nonlinear responses of the pipe are displayed in the form of bifurcation diagrams, phase-plane plots, power spectrum diagrams, Poincare maps, oscillating shapes and oscillation trajectories of the pipe tip. Our numerical results demonstrate that, for a pipe conveying fluid with a subcritical flow velocity, the presence of an axial excitation is able to generate resonant responses, while for a pipe conveying fluid with a supercritical flow velocity, the axial excitation can even make the pipe stable in some specific cases. Furthermore, it is interesting that a self-excited non-planar periodic oscillation can evolve to a planar quasi-periodic or periodic oscillation by adding the axial excitation. In such a fluid–structure interaction system, however, it is shown that a self-excited planar motion cannot be shifted to a non-planar one by adding the axial excitation, at least for the system parameters considered in this work.

Nonlinear forced vibrations of supported pipe conveying fluid subjected to an axial base excitation

The present work focuses on the nonlinear forced vibrations of a supported pipe conveying fluid under an axial base excitation, devoting to exploring the effects of internal flow velocity and axial base excitation on the nonlinear dynamical behaviors of the pipe. Based on Hamilton's principle, the nonlinear coupled governing equations of the pipe system are introduced and discretized into a set of coupled ordinary differential equations by the aid of a Galerkin method. The resulting equations are then solved by a fourth-order Runge-Kutta integration algorithm. Numerical results indicate that the excitation amplitude and internal flow velocity have a combined effect on the dynamic responses of the pipe. When the internal flow velocity is below the critical value for buckling instability, an axial base excitation with small, moderate or large amplitude can only induce oscillations of the pipe in the axial direction, while an axial excitation with very large amplitude, interestingly, could induce the oscillations of the pipe in both axial and transverse directions. When the internal flow velocity is beyond the critical value for buckling instability, however, an axial base excitation with small amplitude can make the pipe oscillate around a positive or a negative static/dynamic equilibrium position in a wide range of excitation frequencies. In addition, a strong axial base excitation with large amplitude can make the pipe conveying a supercritical fluid oscillate around both positive and negative equilibrium positions for relatively large excitation frequencies. Indeed, various dynamical behaviors including periodic, multi-periodic, quasi-periodic and chaotic oscillations can be observed in many cases.

Nonlinear dynamics of viscoelastic pipes conveying fluid placed within a uniform external cross flow

This paper investigates the nonlinear dynamic response of a viscoelastic pipe conveying fluid subjected to a uniform external cross flow based on the Euler-Bernoulli theory. The main objective of this work is to find the proper viscoelastic coefficients to mitigate the dynamic response of a marine riser. A nonlinear oscillator is utilized to simulate the mean drag force and the vortex-induced lift force. Also, the pipe material is assumed to be viscoelastic and consisted of the Kelvin-Voigt type. The extended Hamilton's principle along with the Galerkin discretization are employed to construct the nonlinear model of the coupled fluid-structure system. Moreover, the assumed mode method along with the Runge-Kutta integration algorithm is used as a solution method. The results from the developed numerical model are verified with those available in the literature. Then, the effects of internal fluid velocity, external fluid reduced velocity and viscoelastic damping on the vibration amplitude of the pipe are examined. It is noted that, when lock-in phenomena occur, the internal fluid velocity and viscoelastic damping have vital effects on the vibration amplitude and the lock-in region range of the pipe. It is however demonstrated that employing a proper viscoelastic material may lead to eliminating jump phenomenon.

Nonplanar post-buckling analysis of simply supported pipes conveying fluid with an axially sliding downstream end

In this study, the nonplanar post-buckling behavior of a simply supported fluid-conveying pipe with an axially sliding downstream end is investigated within the framework of a three-dimensional (3D) theoretical model. The complete nonlinear governing equations are discretized via Galerkin’s method and then numerically solved by the use of a fourth-order Runge-Kutta integration algorithm. Different initial conditions are chosen for calculations to show the nonplanar buckling characteristics of the pipe in two perpendicular lateral directions. A detailed parametric analysis is performed in order to study the influence of several key system parameters such as the mass ratio, the flow velocity, and the gravity parameter on the post-buckling behavior of the pipe. Typical results are presented in the form of bifurcation diagrams when the flow velocity is selected as the variable parameter. It is found that the pipe will stay at its original straight equilibrium position until the critical flow velocity is reached. Just beyond the critical flow velocity, the pipe would lose stability by static divergence via a pitchfork bifurcation, and two possible nonzero equilibrium positions are generated. It is shown that the buckling and post-buckling behaviors of the pipe cannot be influenced by the mass ratio parameter. Unlike a pipe with two immovable ends, however, the pinned-pinned pipe with an axially sliding downstream end shows some different features regarding post-buckling behaviors. The most important feature is that the buckling amplitude of the pipe with an axially sliding downstream end would increase first and then decrease with the increase in the flow velocity. In addition, the buckled shapes of the pipe varying with the flow velocity are displayed in order to further show the new post-buckling features of the pipe with an axially sliding downstream end.

Nonplanar flow-induced vibrations of a cantilevered PIP structure system concurrently subjected to internal and cross flows

Pipe-in-pipe (PIP) structures are widely used in offshore oil and gas pipelines to settle thermal insulation issues. A PIP structure system usually consists of two concentric pipes and one softer layer for thermal insulation consideration. The total response of the system is related to the dynamics of both pipes and the interactions between these two concentric pipes. In the current work, a theoretical model for flow-induced vibrations of a PIP structure system is proposed and analyzed in the presence of an internal axial flow and an external cross flow. The interactions between the two pipes are modeled by a linear distributed damper, a linear distributed spring and a nonlinear distributed spring along the pipe length. The unsteady hydrodynamic forces due to cross flow are modeled by two distributed van der Pol wake oscillators. The nonlinear partial differential equations for the two pipes and the wake are further discretized by the aid of Galerkin’s technique, resulting in a set of ordinary differential equations. These ordinary differential equations are further numerically solved by using a fourth-order Runge–Kutta integration algorithm. Phase portraits, bifurcation diagrams, an Argand diagram and oscillation shape diagrams are plotted, showing the existence of a lock-in phenomenon and figure-of-eight trajectory. The PIP system subjected to cross flow displays some interesting dynamical behaviors different from that of a single-pipe structure.

Nonplanar multi-modal vibrations of fluid-conveying risers under shear cross flows

Risers/pipes conveying fluid are a typical kind of slender structures commonly used in marine engineering. It is of great academic significance and application value for us to evaluate and understand the vibration characteristics and nonlinear responses of these risers under the combined action of internal and external fluid flows. In this paper, the nonplanar vibrations and multi-modal responses of pinned-pinned risers in shear cross flow are numerically studied. With this objective in mind, the van der Pol wake oscillators are used to simulate the dynamical behavior of the vortex shedding in the wake. Two nonlinear equations of motion of the riser are proposed to govern the lateral responses of the riser structure. The nonplanar nonlinear equations for the riser and wake are then discretized by employing Galerkin's method and solved by using a fourth-order Runge–Kutta integration algorithm. Theoretical results show that the coupled frequencies for cross-flow (CF) and in-line (IL) motions and the corresponding coupled damping ratio could be influenced by the external and/or internal fluid velocities. Based on extensive calculations, the dynamical behavior of the riser with various internal and external flow velocities are presented in the form of bifurcation diagrams, time traces, phase portraits, oscillation trajectories and response spectrum curves. It is shown that some interesting dynamical phenomena, such as ‘lock-in’ state, ‘figure-of-eight’ trajectory and quasi-periodic oscillation, could occur in such a fluid-structure interaction system. Our results also demonstrate that the shear parameter can significantly affect the dynamic responses of the riser. When the shear parameter of the cross flow is large, multi-modal quasi-periodic responses of the riser can be excited, showing some new features undetected in the system of fluid-conveying risers in uniform cross flow.

Nonplanar vortex-induced vibrations of cantilevered pipes conveying fluid subjected to loose constraints

In this paper, a theoretical model for vortex-induced vibrations (VIVs) of a cantilevered pipe conveying fluid subjected to loose constraints is proposed and analyzed. The unsteady hydrodynamic forces associated with the wake dynamics are modeled by two distributed van der Pol wake oscillators. Taking into account the coupling between the structure and fluid, the full nonlinear partial differential equations of the flexible pipe are derived by using Hamilton's principle. The nonlinear partial differential equations for the pipe and wake are further discretized by the aid of the Galerkin's technique, resulting in a set of ordinary differential equations. These ordinary differential equations are further numerically solved by using a fourth-order Runge-Kutta integration algorithm. Via varying the internal and cross-flow velocities, the derived high-dimensional reduced-order model displays some interesting dynamical behaviors of the coupled system. Phase portraits, bifurcation diagrams, Argand diagram and oscillation shape diagrams are plotted, showing the existence of lock-in phenomenon and figure-of-eight trajectory. When the internal fluid velocity is small and below the critical value for flutter instability, the oscillation of the pipe is mainly induced by the cross flow. For internal flow velocity beyond the critical value for flutter instability, interestingly, the effect of low cross-flow velocity is not pronounced. For relatively high cross-flow velocity, both internal and external fluid flows can significantly contribute to the responses of the pipe. This work is expected to be useful for predicting the responses of marine oil-conveying pipes under the combined effects of internal and external fluid flows.

Numerical study of natural convection heat transfer of Al2 O3/Water nanofluid in a Γ-shaped microchannel

Finite-volume procedure is presented for solving the natural convection of the laminar nanofluid flow in a Γ shaped microchannel in this article. Modified Navier-Stokes equations for nanofluids are the basic equations for this problem. Slip flow region, including the effects of velocity slip and temperature jump at the wall, are the main characteristics of flow in the slip flow region. Steady state equations were solved by using time marching method. In provided FORTRAN code, the finite volume method and an explicit fourth-order Runge–Kutta integration algorithm were applied to find the steady state solutions. Also an artificial compressibility technique was used to couple the continuity to the momentum equations as it is simpler and converges faster. The Grashof numbers from to were considered. The results showed that Nusselt number increases with the Grashof number and the parameter R (the ratio of minimum diameter of nanoparticles and maximum one).. As the parameter R increases, the distortion of the isotherm lines increases to some extent.

Non-planar responses of cantilevered pipes conveying fluid with intermediate motion constraints

In this paper, the nonlinear responses of a loosely constrained cantilevered pipe conveying fluid in the context of three-dimensional (3-D) dynamics are investigated. The pipe is allowed to oscillate in two perpendicular principal planes, and hence its 3-D motions are possible. Two types of motion constraints are considered. One type of constraints is the tube support plate (TSP) which comprises a plate with drilled holes for the pipe to pass through. A second type of constraints consists of two parallel bars (TPBs). The restraining force between the pipe and motion constraints is modeled by a smoothened-trilinear spring. In the theoretical analysis, the 3-D version of nonlinear equations is discretized via Galerkin’s method, and the resulting set of equations is solved using a fourth-order Runge–Kutta integration algorithm. The dynamical behaviors of the pipe system for varying flow velocities are presented in the form of bifurcation diagrams, time traces, power spectra diagrams and phase plots. Results show that both types of motion constraints have a significant influence on the dynamic responses of the cantilevered pipe. Compared to previous work dealing with the loosely constrained pipe with motions restricted to a plane, both planar and non-planar oscillations are explored in this 3-D version of pipe system. With increasing flow velocity, it is shown that both periodic and quasi-periodic motions can occur in the system of a cantilever with TPBs constraints. For a cantilevered pipe with TSP constraints, periodic, chaotic, quasi-periodic and sticking behaviors are detected. Of particular interest of this work is that quasi-periodic motions may be induced in the pipe system with either TPBs or TSP constraints, which have not been observed in the 2-D version of the same system. The results obtained in this work highlight the importance of consideration of the non-planar oscillations in cantilevered pipes subjected to loose constraints.

Nonlinear Forced Vibration of Cantilevered Pipes Conveying Fluid

The nonlinear forced vibrations of a cantilevered pipe conveying fluid under base excitations are explored by means of the full nonlinear equation of motion, and the fourth-order Runge–Kutta integration algorithm is used as a numerical tool to solve the discretized equations. The self-excited vibration is briefly discussed first, focusing on the effect of flow velocity on the stability and post-flutter dynamical behavior of the pipe system with parameters close to those in previous experiments. Then, the nonlinear forced vibrations are examined using several concrete examples by means of frequency response diagrams and phase-plane plots. It shows that, at low flow velocity, the resonant amplitude near the first-mode natural frequency is larger than its counterpart near the second-mode natural frequency. The second-mode frequency response curve clearly displays a softening-type behavior with hysteresis phenomenon, while the first-mode frequency response curve almost maintains its neutrality. At moderate flow velocity, interestingly, the first-mode resonance response diminishes and the hysteresis phenomenon of the second-mode response disappears. At high flow velocity beyond the flutter threshold, the frequency response curve would exhibit a quenching-like behavior. When the excitation frequency is increased through the quenching point, the response of the pipe may shift from quasiperiodic to periodic. The results obtained in the present work highlight the dramatic influence of internal fluid flow on the nonlinear forced vibrations of slender pipes.

Numerical investigation of natural convection heat transfer of nano-fluids in a micro-channel

An enhanced cell-centered finite-volume procedure has been presented for solving the natural convection of the laminar Al O /Water nanofluid flow in a Γ shaped micro-channel in the slip flow region, including the effects of velocity slip and temperature jump at the wall, which are the main characteristics of flow in the slip flow region. In provided FORTRAN code, an explicit fourth-order Runge–Kutta integration algorithm has been applied to find the steady state solutions. Also an artificial compressibility technique has been to couple the continuity to the momentum equations. The Grashof numbers from 10 to 10 have been considered. Results show that decreasing the mean diameters of nanoparticles, increases the Nusselt number. Increasing the mean nanoparticle volume fraction increases the Nusselt number. Also by increasing the Knudsen number, the Nusselt number decreases. Keywords : velocity slip; temperature jump; Knudsen number; nanoparticle.

Modified Runge-Kutta Integration Algorithm for Improved Stability and Accuracy in Real Time Hybrid Simulation

Stability in Real Time Hybrid Simulation (RTHS) has been shown to be largely affected by system dynamics and associated phase lags. This lag typically originates in the physical components and considerable research has been conducted to compensate for it. Within the computational component of RTHS, different time integration algorithms are employed to achieve a more stable and accurate solution, mostly focusing on dissipating the high frequency content in the model. However, in RTHS, an inherent computational delay exists in the force measurement due to the sequential nature of communication between the numerical and experimental sub- structures. In this article, it is demonstrated that this computational delay affects performance and stability of closed loop RTHS even when no other delays or phase lags are present. This finding is validated through theoretical derivation and simulation results. A modified Runge-Kutta (MRK) integration algorithm is proposed to reduce the effect of computational delay.The MRK integration involves a three-stage computation: (1) the pseudo response is calculated using the delayed force measurement; (2) feedback force from the physical component for the next step is predicted using the pseudo response; and (3) the corrected structural response is then computed using the predicted feedback force. Both analytical and simulation results confirm that the MRK integration scheme is stable and accurate for a wide range time steps and is robust with respect to modeling error and nonlinearity in the experimental substructure. A moment-resisting frame is used as the experimental substructure in different cases of RTHS to validate the MRK integration method. This approach can also be adapted to other existing numerical integration schemes by applying the proposed three-stage computation process approach.

Active Flutter Control of a Supersonic Honeycomb Sandwich Beam Resting on Elastic Foundation with Piezoelectric Sensor/Actuator Pair

In this paper, by using the piezoelectric material, the active aeroelastic flutter characteristics and vibration control of supersonic sandwich panels with different honeycomb interlayers, resting on an elastic foundation, are studied. Classical beam theory along with the Winkler–Pasternak foundation model, and the quasi-steady first order supersonic piston theory are employed in the formulation of the structural theory and aerodynamic loading, respectively. Hamilton's principle in conjunction with the generalized Fourier expansions and Rayleigh–Ritz (RR) method are used to develop the dynamical model of the structural systems in the state–space domain. The aeroelastic flutter bounds are obtained via the p-method. The classical Runge–Kutta integration algorithm is then used to calculate the open-loop aeroelastic response of the system under different loading excitations. Finally, two classical control strategies, including direct proportional feedback and linear-quadratic regulator (LQR) optimal control scheme, are used to actively suppress the closed loop system response, while increasing the flutter aerodynamic pressure.