Non-dimensional Frequency Research Articles

Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory

The main subject of this study is the dynamic and static pull-in analyses of rectangular nanoplates made of functionally graded materials based on the nonlocal strain gradient theory compared with the strain gradient theory, Eringen’s differential method, and classical theory considering disparate boundary conditions (BCs). To this end, the governing equation is derived based on Kirchhoff’s plate theory and then, the Galerkin method is implemented to arrive at an ordinary differential equation of second-order in the time domain. The homotopy analysis method is used as an analytical solution methodology to solve the nonlinear ordinary differential equation which is under different nonlinear forces such as intermolecular, electrostatic and hydrostatic one. Likewise, an efficient technique is exerted for omitting secular terms. Accordingly, the dynamic pull-in voltage is obtained for different non-classical continuum theories, material gradient indices, nonlocal parameters and internal length scale parameters. The dynamic and static pull-in instabilities are also compared in the same situation. Additionally, the effects of the nonlocal parameter and internal length scale parameter on the non-dimensional frequency against the initial amplitude, electrostatic force and hydrostatic actuation are explored in disparate boundary conditions. Communicated by Eleonora Tubaldi.

Exponential and harmonic forced torsional vibration of single-walled carbon nanotube in an elastic medium

In this study, free torsional vibration and forced torsional vibration analysis under the time-dependent exponential and harmonic torsional loadings in single-walled carbon nanotube are investigated. The SWCNT is embedded in an elastic medium. Eringen's theory among the small-scale theories is selected. The nonlocal differential constitutive relation and corresponding boundary condition are derived via Hamilton's principle. Clamped–clamped boundary condition is utilized. The assumed modes method is employed for the dynamic torsional vibration in order to discretize the derived governing equations. The novelty of this work is devoted to the analysis of forced torsional vibration of a carbon nanotube embedded in an elastic medium under the various loadings. The angular displacement for the resonance frequency neglecting the elastic medium is illustrated. For the free analysis, the first three nondimensional natural frequencies with various small-scale parameters and stiffness of the elastic medium are calculated. The results are compared with another study for the first 10 mode numbers. The effects of the nonlocal parameter, length of carbon nanotube, stiffness of the elastic medium, thickness, time constant, and excitation frequency on the nondimensional and dimensional angular displacements are investigated, dynamically. For the greater values of the stiffness of the medium, the nonlocal parameter becomes negligible. When a time-dependent exponential torque is applied to the model, the angular displacement becomes greater and then lower by an increase in the value of the length, but the nondimensional angular displacement decreases continuously by increasing the value of the length under the time-dependent harmonic loading. Moreover, the angular displacement for a determined time becomes lower first and then becomes greater by increasing the time constant.

Locomotion of a self-propulsive pitching plate in a quiescent viscous fluid

The locomotion of a flexible plate pitching in a quiescent viscous fluid is numerically studied by using the lattice Boltzmann method (LBM) for the fluid and a finite element method (FEM) for the plate, with an immersed boundary (IB) method for the fluid–structure interaction (FSI). In the simulation, the leading edge of the plate undergoes a prescribed pitching motion, and the entire plate moves freely due to the fluid–plate interaction. The effects of the pitching amplitude, bending rigidity, plate-to-fluid mass ratio and Reynolds number on the propulsive performance of the flexible plate are examined in a range of parameters. The numerical results show that a certain flexibility can remarkably improve the propulsive speed and efficiency. The optimal parameters for the pitching plate are obtained, i.e. [Formula: see text] ([Formula: see text] is a non-dimensional frequency, with [Formula: see text] means rigid plate and larger [Formula: see text] means more flexible) and 20° ≤ α0 ≤ 25° ( α0 is the pitching amplitude). The comparisons of three plate-to-fluid mass ratios (1.0, 2.5 and 5.0) show that the mass of the plate decreases the propulsive speed, but contrarily increases the efficiency. The results obtained in the present study provide an insight into the understanding of the performance of self-propulsive plate in pitching motion and can further guide the engineering design of micro aerial vehicles.

Unified mechanisms for separation control around airfoil using plasma actuator with burst actuation over Reynolds number range of 103–106

We conduct large-eddy simulations of separated airfoil flows with control by a dielectric-barrier-discharge plasma actuator over a wide range of Reynolds numbers. The Reynolds numbers based on the chord length (Re) are set at Re = 5.0 × 103, 1.0 × 104, 6.3 × 104, 2.6 × 105, and 1.6 × 106. These Reynolds numbers cover most of the conditions used in the previous studies on separation control by a plasma actuator. The burst frequency nondimensionalized by the chord length and freestream velocity (F+) is used as the computational parameter, and the effective burst actuation and control mechanisms at each Reynolds number condition are investigated. With regard to cases without the control, the flows separate near the leading edge in the laminar state at the Reynolds number range of 103–105, and a substantial turbulent separation occurs at the Reynolds number of 1.6 × 106. Separation control with a high burst frequency [F+ ≃ O(10)] can cause early flow reattachment through the promotion of turbulent transition of a separation shear-layer for Re = 6.3 × 104 and 2.6 × 105. Flow reattachment is mainly caused by momentum entrainment into the boundary layer by fine-scale turbulent vortices. On the other hand, the large-scale spanwise vortices play an important role at F+ = 1 for Re = 1.0 × 104 and 1.6 × 106. In these cases, the dynamics of the spanwise vortices show similar behavior and the pairing of these vortices significantly contributes to the separation control by increasing the momentum entrainment. The optimum value of F+ changes with a Reynolds number. In contrast, when a nondimensional burst frequency based on the characteristics of the separation shear-layer (Fθs) is considered, a high lift-to-drag ratio is found at Fθs≃O(10−2) for all Reynolds numbers. This demonstrates that one of the effective burst frequencies is closely related to the scale of the separation shear-layer, especially for the spanwise vortex shed from the separation shear-layer.

Free vibration of FGM annular sectorial plates with arbitrary boundary supports and large sector angles

An original work is presented for the free vibration analysis of thin functionally graded material (FGM) annular sectorial plates with arbitrary boundary supports using the weak form quadrature element method (QEM). Novel C 1 quadrature annular sectorial plate element is developed based on the physically neutral surface thin plate theory in cylindrical coordinate system. Three displacement functions are used to avoid the introduction of the mixed second order derivative degree of freedom at corner points. The derived explicit formulations with a varying node number can be applied for any type of node. A way to define the elastic spring constants is proposed to ensure the independent of the non-dimensional frequency parameter on the power-law exponent. The convergence and accuracy of the present model are validated by comparing the obtained results with those available in literatures. It is demonstrated that the proposed model is simple, straightforward and has good performance. A wide range of numerical results are reported to demonstrate the effect of the sector angle, ratio of inner/outer radii, and boundary conditions on the vibration behavior of the thin annular sectorial plate. Some tabulated results should be new which may serve as a reference for verifying numerical solutions obtained by other methods.

Exploring the Dynamics of Base-Excited Structures Impacting a Rigid Stop

This paper explores the nonlinear dynamics of a multidegree of freedom (MDoF) structure impacting a rigid stop. The contact mechanics is simplified by continuous sigmoid function idealisation of a lossless spring. By introducing a smooth nonlinear formulation, we avoid the computational expense of event-driven, piecewise, nonsmooth dynamics. A large parametric study using high-performance computing is undertaken. The nondimensional equations of motion suggest one primary structural parameter, contact-to-storey stiffness ratio, and two excitation parameters, nondimensional ground amplitude and frequency. Bifurcation plots indicate an extremely rich and complex behaviour, particularly in the cases where at least two-floor degrees of freedom (DoFs) impact the stop and when the contact-to-storey stiffness ratio is large. When considering interstorey drift as a performance measure, period-1 impacting solutions are generally favourable when compared to an analogous nonimpacting case. This paper also discusses whether chaotic impacting can be favourable. Finally, we consider the question of whether higher modes are significantly excited, at a linear resonance, for impacting solutions to this system.

On torsional vibrations of triangular nanowire

Abstract The fundamental torsional vibration of the triangular nanowire for three different boundary conditions, including clamped-clamped (C–C), clamped-free (C–F), and clamped-torsional spring (C-T) boundary conditions are analyzed in this study. The equation of motion is derived by Hamilton's principle first, and then the nonlocal elasticity theory based on Eringen model is selected to cover the size-dependent behavior of the nanowire. An analytical method is utilized to solve the governing equation. This work is original and completely novel due to the analysis of vibration of the nanostructures, which possess the noncircular cross-sections. Accordingly, in this study, the effects of the value of the triangle edge and the nonlocal parameter on the variation of the natural frequencies are evaluated. The first three nondimensional natural frequencies are assessed by setting the warping function to zero and comparison with another work. Furthermore, the effect of the stiffness of the boundary torsional spring for C-T boundary condition, and influences of the aspect ratio and nonlocal parameter on the responses are studied.

Numerical simulation in time domain to study cross-flow VIV of catenary riser subject to vessel motion-induced oscillatory current

The present study proposes a time domain model for the Vortex-induced Vibration (VIV) simulation of a catenary riser under the combination of the current and oscillatory flow induced by vessel motion. In this model, the hydrodynamic force of VIV comprises excitation force, hydrodynamic damping and added mass, which are taken as functions of the non-dimensional frequency and amplitude ratio. The non-dimensional frequency is related with the response frequency, natural frequency, lock-in range and the fluid velocity. The relatively oscillatory flow induced by vessel motion is taken into account in the fluid velocity. Considering that the added mass coefficient and the non-dimensional frequency can affect each other, an iterative analysis is conducted at each time step to update the added mass coefficient and the natural frequency. This model is in detail validated against the published test models. The results show that the model can reasonably reflect the effect of the added mass coefficient on the VIV, and can well predict the riser’s VIV under stationary and oscillatory flow induced by vessel motion. Based on the model, this study carries out the VIV simulation of a catenary riser with harmonic vessel motion. By analyzing the bending moment near the touchdown point, it is found that under the combination of the ocean current and oscillatory flow the vessel motion may decrease the VIV response, while increase the excited frequencies. In addition, the decreasing rate of the VIV under vessel surge is larger than that under vessel heave at small vessel motion velocity, while the situation becomes opposite at large vessel motion velocity.

Analysis of porous micro sandwich plate: Free and forced vibration under magneto-electro-elastic loadings

In this study, the free and forced vibration analysis of micro sandwich plate with porous core layer and magnetoelectric face sheets based on modified couple stress theory and first order shear deformation theory under simply supported boundary conditions is illustrated. It is noted that the core layer is composed from balsa wood and also piezo magneto-electric facesheets are made of BiTiO3-CoFe2O4. Using Hamilton's principle, the equations of motion for micro sandwich plate are obtained. Also, the Navier's method for simply support boundary condition is used to solve these equations. The effects of applied voltage, magnetic field, length to width ratio, thickness of porous to micro plate thickness ratio, type of porous, coefficient of porous on the frequency ratio are investigated. The numerical results indicate that with increasing of the porous coefficient, the non-dimensional frequency increases. Also, with an increase in the electric potential, the non–dimensional frequency decreases, while and with increasing of the magnetic potential is vice versa.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

Observations of Typhoon Waves in a Reef Lagoon of the South China Sea

AbstractBased on wave measurements lasting three and a half years, typhoon wave characteristics in a lagoon of the South China Sea are discussed in this paper. According to the upper boundaries of wind speed and significant wave height Hs, the maximum wave height has a similar rate of change as the maximum wind speed. The peak frequencies decrease in the approaching stage and are concentrated with an average of 0.1415 Hz in the leaving stage. The distributions of the wind and wave directions indicate that the local wind waves and the waves from the open sea propagated to the site in the approaching and leaving stages, respectively. Wind waves generated inside the lagoon are fetch-trapped, and the spectra can be described by αf(−4) or βk(−2.5) when f > fp. In addition, the measurements show that the nondimensional maximum spectrum can be estimated by the nondimensional fetch. The spectra of the waves from the open sea can be described by . Parameters a and c are determined by the nondimensional peak frequency. In addition, it is found that the peak energy density S(fp) is a linear function of for all waves.

Numerical Prediction of Wall Pressure Fluctuations in a Turbulent Boundary Layer on a Cylinder in Axial Flow

Flow noise originating in the turbulent boundary layer (TBL) often severely limits the performance of towed sonar cylinder and therefore it is necessary to predict this noise for the design of efficient towed cylinder. This paper presents large eddy simulation methodology to establish the TBL properties and wall pressure fluctuations on a 3 m long cylinder with length to diameter ratio of 315 in the operating speed of 11.4 m/s in air. The computed flow induced sound is compared with experimental measurement available in the literature successfully. The effectiveness of scaling the flow noise spectra with diameter and tow speed is discussed and non-dimensional wall pressure spectra presented with respect to non-dimensional frequency. The overall sound pressure levels are also compared with experimental data that show good accuracy achieved by the proposed numerical methodology.

Dispersion equation of Rayleigh waves in transversely isotropic nonlocal piezoelastic solids half-space

This study is devoted to investigate the propagation of Rayleigh-type waves in transversely isotropic nonlocal piezoelastic half-space. When the stress-free boundary is maintained at charge-free condition, the dispersion equation for the propagation of Rayleigh waves at the free surface of transversely isotropic piezoelastic solids has been obtained. Based on the obtained dispersion equation, the effect of the nonlocality on the speed of Rayleigh wave is numerically considered. The dependence of velocities of plane waves in transversely isotropic nonlocal piezoelastic medium on the direction of propagation as well as non-dimensional frequency parameter has been also illustrated.

Large-Scale Forcing of a Turbulent Channel Flow Through Spanwise Synthetic Jets

The investigation focuses on the forcing of a fully developed turbulent channel flow through a linear array of synthetic jets injected tangentially to the wall and orthogonal to the mean flow direction. Forcing configurations are varied by differently combining the number of actuated jets working in an opposing blowing–suction configuration. Instantaneous wall shear stress and streamwise velocity fluctuations evidence drag reductions as well as turbulence attenuation up to 20%. The forcing effects are persistent up to at least a 150 half-channel height downstream of the injection section. Particle image velocimetry investigations in planes perpendicular to the channel axis highlight the presence of a large-scale streamwise vortical structure covering the whole height of the channel. This structure is thought to be responsible for the significant drag reduction, which is similar to the typical behavior evidenced in the case of colliding jets. The nondimensional forcing frequency of the synthetic jets producing the maximum drag reduction and turbulence attenuation is 0.0074 for the investigated Reynolds number (). A statistical analysis of the near-wall structures demonstrates that the control mechanism acts in a way to reduce them in the forced configuration. It is conclude that the effect of the forcing is such that the near-wall structures merge and become less prone to inducing new structures, thus effectively reducing their number, and consequently the near-wall turbulence activity.

Moving axial load on dynamic response of single-walled carbon nanotubes using classical, Rayleigh and Bishop rod models based on Eringen’s theory

The main objective of the present work is devoted to the study of both free and time-dependent forced axial vibration simultaneously in single-walled carbon nanotubes subjected to a moving load. The governing equation is derived via Hamilton’s principle. Classical theory, along with the Rayleigh and Bishop theories, is used to analyze the nonlocal vibrational behaviors of single-walled carbon nanotubes. A Galerkin method is established to solve the derived equations. The boundary conditions are assumed to be clamped-clamped and clamped-free. Firstly, the variation of nondimensional natural frequencies is calculated based on the classical theory, and the effect of the nonlocal parameter, the mode number and the length is illustrated and schematically compared for clamped-clamped and clamped-free boundary conditions. Besides, the obtained nondimensional responses are compared with the results of another study to validate the accuracy of the used method. Ultimately, the dynamic axial displacement due to the moving load in the time domain has been studied for the first time. Furthermore, the effects of the thickness, length, velocity of the moving load, excitation frequency, and the nonlocal parameter based on the classical, Rayleigh, and Bishop theories are investigated in this paper. Also, the influence of the nonlocal parameter on the variations of maximum axial displacement with respect to the velocity parameter for the aforementioned boundary conditions and theories is evaluated relative to each other.

New combinational active control strategy for improving aerodynamic characteristics of airfoil and rotor

In order to improve the aerodynamic characteristics of rotor, a new active flow control strategy by combining a synthetic jet actuator and a variable droop leading-edge or a trailing-edge flap has been proposed. Their control effects are numerically investigated by computational fluid dynamics (CFD) method. The validated results indicate that variable droop leading-edge and synthetic jet can suppress the formation of dynamic stall vortex and delay flow separation over rotor airfoil. Compared with the baseline state, Cdmax and Cmmax are significantly reduced. Furthermore, parametric analyses on dynamic stall control of airfoil by the combinational method are conducted, and it indicates that the aerodynamic characteristics of the oscillating rotor airfoil can be significantly improved when the non-dimensional frequency ( k*) of variable droop leading-edge is about 1.0. At last, simulations are conducted for the flow control of rotor by the combinational method. The numerical results indicate that large droop angle of variable droop leading-edge can better reduce the torque coefficient of rotor and the trailing-edge flap has the capability of increasing the thrust of rotor. Also, the synthetic jet could further improve the aerodynamic characteristics of rotor.

A study of the vibration and lay-up optimization of rotating cross-ply laminated nanocomposite blades

A comprehensive understanding of the vibration characteristics of rotating blades is critical to avoid resonance fracture. This paper presents the vibration analysis and lay-up optimization of rotating pre-twisted laminated functionally graded CNTs reinforced composite (FG-CNTRC) shallow conical shells. A numerical shallow conical shell model is proposed, and the displacement fields of the shell model are described via the first-order shear deformation theory. Considering the rotating speed, the governing equation of the rotating pre-twisted laminated conical shell is derived, and then solved using the meshless kp-Ritz method. We carry out the comparison studies to demonstrate the accuracy and efficiency of the present model in dealing with the vibrations of this kind of structures. Detailed parametric studies are then performed and demonstrate that the CNT configuration, pre-twisted angle, and stacking sequence play essential roles in vibration behaviors of rotating FG-CNTRC blade. With the nondimensional frequency parameter being the optimization objective function, a genetic algorithm is employed to search for the optimal layering sequence of the rotating shallow conical shells. The optimized results demonstrate that the GA serves as a useful tool for sequence optimization of rotating blade in terms of improving the vibration characteristics.

An investigation on external airflow around low-rise building with various roof types: PIV measurements and LES simulations

In this study, the influence of different roof shapes on the airflow around isolated low-rise buildings is investigated with particle imaging velocity (PIV) wind tunnel experiments and large eddy simulations (LESs). PIV measurements are conducted in an atmospheric boundary layer wind tunnel for obtaining the mean velocity, fluctuating velocity component and time-averaged vortical structure around three low-rise buildings. The results provide a useful database for validation of an LES numerical model. Next, LES simulations are employed for solving the flow fields. Then, the performance of LES prediction is quantitatively evaluated based on the PIV database. Finally, the influences of the various roofs on the flow characteristics around buildings are analysed. The results illustrate that the LES numerical predictions show good agreement with the experimental results for all models. However, the predicting performance is largely related to the roof shape: A flat roof with a simpler geometry results in better prediction performance than more complex geometries do. The roof shapes remarkably affect the surrounding wind conditions at the pedestrian level. The area of low wind velocity (Ri < 0.2) in the wake region of the stepped model is by far the smallest at a wind direction of 180°. The size of a horseshoe vortex formed at the front of the model is positively correlated with the upwind height. The turbulence energy downstream is approximately concentrated around the non-dimensional frequency of 0.10, and the probability density functions of the velocity component at the monitor points approximately obeyed a normal distribution, with a weak skewness and kurtosis fatness.

On the modeling of dynamic behavior of composite plates using a simple nth-HSDT

In the present paper, a simple refined nth-higher-order shear deformation theory is applied for the free vibration analysis of laminated composite plates. The proposed displacement field is based on a novel kinematic in which include the undetermined integral terms and contains only four unknowns, as against five or more in case of other higher-order theories. The present theory accounts for adequate distribution of the transverse shear strains through the plate thickness and satisfies the shear stress-free boundary conditions on the top and bottom surfaces of the plate, therefore, it does not require problem dependent shear correction factor. The governing equations of motion are derived from Hamilton\'s s principle and solved via Navier-type to obtain closed form solutions. The numerical results of non-dimensional natural frequencies obtained by using the present theory are presented and compared with those of other theories available in the literature to verify the validity of present solutions. It can be concluded that the present refined theory is accurate and efficient in predicting the natural frequencies of isotropic, orthotropic and laminated composite plates.

A lumped parameter model to explain the cause of the hysteresis in OWC-Wells turbine systems for wave energy conversion

A Wells turbine is an axial-flow turbine consisting of a rotor usually with symmetric (uncambered) blades staggered at a 90 degree angle relative to the machine’s axis. This turbine is used within oscillating water column systems: during its normal operation, the blades experience a continuous change in incidence angle, that according to many authors is at the origin of a hysteresis in its force coefficients. Aerodynamic hysteresis in rapidly moving airfoils is a well-known phenomenon, but happens only at non-dimensional frequencies significantly larger than the ones encountered in Wells turbines. This work presents a re-examination of the two phenomena, that shows the unlikeliness of the presence of any aerodynamic hysteresis in Wells turbines. A simple yet effective lumped parameter analysis is used to prove how the real cause of the hysteresis is to be found in a different phenomenon. Results are compared to experiments and CFD analyses for the same problem, with an excellent agreement.