Double Cosine Research Articles

Analytical free vibration solutions of rectangular edge-cracked plates by the finite integral transform method

Analytical free vibration solutions of rectangular edge-cracked thin plates are obtained by the finite integral transform (FIT) method. While the previous studies focus on the mechanical behaviors of intact plates, this study establishes an FIT-based new scheme for analyzing the free vibration characteristics of rectangular thin plates with an edge crack. The plates may have any combined conditions of simply supported, clamped, and free boundaries. In specific solution scheme, an edge-cracked plate is decomposed into four subdomains, and a double cosine FIT is imposed on the fourth-order partial differential governing equations of each subdomain, which yields the relationship between the transformed modal deflections and naturally defined unknowns. By substituting the exterior boundary conditions of the plate, continuity conditions across the subdomains, and free conditions at the crack into the inverse transforms, the natural frequencies and unknowns are successively determined via the established homogeneous linear algebraic equations. The modal deflections are provided by further substituting the obtained unknowns into the inverse transforms. Comprehensive benchmark results for the edge-cracked plates under representative boundary conditions are shown. The results are well verified by other solution methods. A parametric study quantitatively reveals the effects of boundary conditions and crack length on the natural frequencies of edge-cracked plates. Due to the rigorous but easy-to-implement mathematical derivations, this study presents a novel solid way for exploring analytical solutions of vibration problems.

Coupled free vibrations of liquid in a three-dimensional rectangular container with an elastic cover

The coupled free vibration of liquid and its elastic cover, such as a plate or a membrane, in a three-dimensional rectangular tank is investigated through an analytical scheme based on the velocity potential theory for the flow and the linear elastic theory for the cover. For the fluid domain, the velocity potential is expanded into double cosine series along the longitudinal and transverse directions, respectively, with the corresponding eigenvalues determined from the impermeable conditions on the side walls. The vertical modes of the potential are obtained from the Laplace equation. The deflection of the rectangular cover is expanded into the same double cosine series to match the potential, together with additional terms for satisfying the edge conditions. The polynomials are used for these additional terms, which are then expanded into cosine series. For the expansions of the higher-order derivatives of the deflection, the derivatives of these polynomial terms are expanded into cosine series directly, rather than being obtained through differentiating the cosine series of the deflection, to avoid the non-convergent series. Through imposing the boundary conditions on the fluid–plate interface and edge conditions, an infinite matrix equation for the unknown coefficients can be established. The natural frequencies can be obtained when the determinant of the matrix is zero. In practical computation, the infinite matrix equation is truncated into finite size. Results are first provided for natural frequencies. This is followed by the corresponding natural mode shapes and principal strains distribution on the cover. The underlying physics of these results is then provided.

Homotopy Analysis Method and Time-fractional NLSE with Double Cosine, Morse, and New Hyperbolic Potential Traps

A brief outline of the derivation of the time-fractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study time-fractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential $[V(x)=V_{1}\cos x+V_{2}\cos 2x]$, the second one is the Morse potential $[V(x)=V_{1}e^{-2\beta x}+V_{2}e^{-\beta x}]$, and a hyperbolic potential $[V(x)=V_{0}\tanh(x)\sech(x)]$ is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order.

Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method

In this paper, a coupled pure meshless finite pointset method (CFPM) is developed for the first time to numerically predict the inelastic collision process of solitary wave described by the time fractional coupled nonlinear Schrödinger (TF-CNLS) equation. Its construction process is as follows: 1) a high-precision difference scheme is used for the Caputo time fractional derivative; 2) FPM discrete scheme based on Taylor expansion and weighted least square method is adopted for spatial derivatives; 3) The region is locally refined and the double cosine kernel function with good stability is used to improve the numerical accuracy. In the numerical study, the one-dimensional TF-CNLS equations with analytical solutions are solved by CFPM, and the error and convergence rate are analyzed when the nodes are uniformly distributed or locally refined, showing that the proposed method has the approximate second-order accuracy and the flexibility of easy local refinement. Secondly, the inelastic collision process of solitary waves, which is described by the one-dimensional TF-CNLS equation without analytical solution, is numerically predicted by CFPM, and the wave collapse phenomenon depicted above is completely different from the multi-wave phenomenon under the integer order. Meanwhile, by comparing the result with that obtained by finite difference method, it suggests that the CFPM is reliable to predict the complex propagation of the inelastic collision process of the solitary waves under the time fractional order.

Analysis of the effect of surface topography on lubrication using heterogeneous multiscale method

In this paper, flow factors considering the roughness of the different orientation angles and different wavelengths are obtained from the derived form of the homogenization coefficient, which is calculated by a heterogeneous multiscale inverse approach. The influence of anisotropic roughness on the flow factors is investigated. When the angle is equal to 45°, the span flow is maximum. The flow factor curves happen to overlap when the wavelength ratio is less than 4 with double cosine roughness. To verify the flow factors, film pressures are calculated by a finite element heterogeneous multiscale method (FE-HMM) without considering the flow factors. Then, by comparing the results with solutions of a finite difference method, the FE-HMM method gets the results that are needed, and with a shorter time of calculation. The pressure curve is greatest when the orientation angle is 45° with a double cosine roughness, which is caused by the asperity number and the span flow. The pressure distributions of the cosine roughness and double cosine roughness are different as the orientation angle changes.

Semi-resolved CFD-DEM modeling of gas-particle two-phase flow in the micro-abrasive air jet machining

Abrasive air jet (AAJ) machining is attractive for micromachining hard and brittle materials, while it is usually a big challenge to numerically investigate the particle velocity and concentration distribution in the particle erosion process in the AAJ. In this work, a recently developed semi-resolved CFD-DEM approach which bridges the simulation gap between the resolved and unresolved CFD-DEM, is further improved to reconstruct the background information with a double cosine kernel function. The semi-resolved CFD-DEM is then employed to numerically investigate the gas-particle two-phase flow in the AAJ and numerical results show that this semi-resolved CFD-DEM is more accurate in modeling particulate flow in the fine AAJ nozzle than the conventional unresolved CFD-DEM. We further conduct mechanism investigations on the AAJ micromachining process including particle flow characteristics inside the cylindrical nozzle, velocity, and concentration distribution over the nozzle exit, which are essential jet characteristic features. We identified the particle flow patterns, analyzed the particle distribution, and its correlation with air pressure, abrasive mass flow rate, and turbulence effects. The present simulation results and analyses can be great helpful in understanding the erosion mechanism and optimizing the setting parameters to improve the cutting performance of AAJ.

Properties of Sums of Double Trigonometric Series with Multiply Monotone Coefficients

Sums of double sine and cosine series with multiply monotone coefficients are studied. Sufficient conditions for such sums to belong to Orlich’s weighted classes are obtained.

Convergence of Double Cosine Series

Convergence of Double Cosine Series

Deriving priorities from pairwise comparison matrices with a novel consistency index

It is important to measure the inconsistency level of a pairwise comparison matrix (PCM) and derive the priority vector in the analytic hierarchy process (AHP). In the present study, a new consistency index is proposed by using the cosine similarity measures of two row/column vectors in a PCM. It is called as the double cosine similarity consistency index (DCSCI) since the row and column vectors are all considered. Some interesting properties of DCSCI are investigated and the thresholds for inconsistency tolerance level are discussed in detail. Then following the idea of DCSCI, we provide a new method for obtaining the priority vector from a PCM. Through maximizing the sum of the cosine similarity measures between the priority vector and the row/column vectors, the priority vector is derived by solving the constructed optimization problem. By analyzing the proposed double cosine similarity maximization (DCSM) method, it is found that the existing cosine maximization (CM) method can be retrieved. By carrying out numerical examples, some comparisons with the existing methods show that the proposed index and method are effective and flexible.

Bubble Size Distribution Characteristics of a Jet-Stirring Coupling Flotation Device

In this study, a new jet-stirring coupling flotation device that incorporates the advantages of three conventional flotation machines (specifically, Jameson cell, mechanical flotation cell, flotation column) was designed based on jet suction. The suction capacity of a double cosine self-aspirated nozzle utilized by the device was analyzed under different feeding pressures, and the effects of frother concentration, feeding pressure, suction capacity, and height of sampling location on the bubble size distribution (BSD) were investigated using a high-speed video system. It was found that a large amount of air was sucked into the flotation cell by the self-aspirated nozzle arranged in a non-submerged manner, which met the requirements of flotation in terms of the suction amount of air. The suction capacity showed a positive linear correlation with negative pressure inside the nozzle. When the Methyl isobutyl carbinol (MIBC) concentration reached the critical coalescence concentration (CCC), the bubble size stabilized at approximately 0.31 mm, which was smaller than the bubble size produced by the conventional flotation machine. This indicated that bubbles suitable for flotation were generated. D32 linearly decreased with increasing of feeding pressures and conversely increased with increasing suction capacities and sampling location heights, independent of the frother concentration.

Free Vibration of Functionally Graded Sandwich Shallow Shells on Winkler and Pasternak Foundations with General Boundary Restraints

In this investigation, an exact method based on the first-order shear deformation shallow shell theory (FSDSST) is performed for the free vibration of functionally graded sandwich shallow shells (FGSSS) on Winkler and Pasternak foundations with general boundary restraints. Vibration characteristics of the FGSSS have been obtained by the energy function represented in the orthogonal coordinates, in which the displacement and rotation components consisted of standard double Fourier cosine series and several closed-form supplementary functions are introduced to eliminate the potential jumps and boundary discontinuities. Then, the expansion coefficients are determined by using Rayleigh-Ritz method. The proposed method shows good accuracy and reliability by comprehensive investigation concerning free vibration of the FGSSS. Numerous new vibration results for FGSSS on Winkler and Pasternak foundations with various curvature types, geometrical parameters, and boundary restraints are provided, which may serve as benchmark solutions for future research. In addition, the effects of the inertia, shear deformation, and foundation coefficients on free vibration characteristic of FGSSS are illustrated.

L1-convergence of double trigonometric series

In this paper we study the pointwise convergence and convergence in L1-norm of double trigonometric series whose coefficients form a null sequence of bounded variation of order (p,0),(0,p) and (p,p) with the weight (jk)p-1 for some integer p > 1. The double trigonometric series in this paper represents double cosine series, double sine series and double cosine sine series. Our results extend the results of Young [9], Kolmogorov [4] in the sense of single trigonometric series to double trigonometric series and of M?ricz [6,7] in the sense of higher values of p.

An analytical study of the free and forced vibration response of a ribbed plate with free boundary conditions

An analytical study to predict the vibration response of a ribbed plate with free boundary conditions is presented. The analytical solution was derived using a double cosine integral transform technique and then utilized to study the free and forced vibration of the ribbed plate, as well as the effect of the rib on the modal response of the uniform plate. It is shown that in addition to the three zero-frequency rigid body modes of the plate, the vibration modes of the uniform plate can be classified into four mode groups according to the symmetric properties of the plate with respect to the two orthogonal middle lines parallel to the plate edges. The four mode groups correspond to a double symmetric group, a double anti-symmetric group and two symmetric/anti-symmetric groups. Whilst the inclusion of the rib to the plate is shown to cause distortion to the distribution of vibration modes, most modes can still be traced back to the original modes of the uniform plate. Both the mass and stiffness of the rib are shown to affect the modal vibration of the uniform plate, whereby a dominant effect from the rib mass leads to a decrease in the modal frequency of the plate, whereas a dominant effect from the rib stiffness leads to an increase in plate modal frequency. When the stiffened rib behaves as an effective boundary to the plate vibration, an original plate mode becomes a pair of degenerate modes, whereby one mode has a higher frequency and the other mode has a lower frequency than that of the original mode.

Hourly Air Temperature Modeling Based on Atmospheric Pressure, Global Solar Radiation and Relative Humidity Data

This paper is focusing on hourly air temperature estimation model (MAT) using available meteorological measured data located in Laghouat (Algeria). The hourly air temperature defined by the present model can be calculated at any time of the night or the day period based on atmospheric pressure, global solar radiation and relative humidity data. This work was compared with three published models from the literature as: Wave, Idliman and Double cosine. Fifteen months of hourly atmospheric pressure, global solar radiation, relative humidity and air temperature data collected during the period (January 2015 to March 2016) were used to test the accuracy of the various models studied. The analysis of the days selected randomly showed that the MAT model gave substantially good fit to the observed data. The RMSE of the MAT model is less than 0.5 oC during all the period of study than the other models studied ranged in the interval (2 oC, 4 oC). The estimated results are compared to the measured ones by using statistical parameters tests such as the mean bias error (MBE), the mean percentage error (MPE), the mean absolute error (MAE), the root mean square error (RMSE) and the coefficient of determination (R2).

Uniform convergence of double trigonometric integrals

We study the uniform convergence of double cosine integrals, sine-cosine integrals and double sine integrals of double general monotone functions. We extend the results of F. Móricz and of A. Debernardi concerning the uniform convergence of double sine in

Vibration analysis of the functionally graded carbon nanotube reinforced composite shallow shells with arbitrary boundary conditions

The aim of this paper is to firstly present the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow shells with arbitrary boundary conditions. The first-order shear deformation theory and the artificial spring boundary technique are introduced to achieve the general theoretical modeling and arbitrary boundary conditions, respectively. Then according to the energy variational principle, the energy expression of the FG-CNTRC shallow shells is obtained. Furthermore, the admissible displacement functions of the FG-CNTRC shallow shells are chosen as an improved Fourier series which combines the standard double cosine Fourier series and several auxiliary functions which are introduced to remove any potential discontinuity of the displacement function and its derivatives at the edges. In the current framework, the vibration analysis of the FG-CNTRC shallow shells with arbitrary boundary conditions can be conducted by a unified model, and the calculation core code derived from the MATLAB software does not need changing while the boundary conditions or geometric parameters change. The good accuracy, reliability and efficiency of the current approach are validated by several numerical examples. Simultaneously, a comprehensive parametric investigation concerning the effects of elastic restraint parameters, shear deformation and rotary inertia, shallowness and material properties is performed.

Analysis of Nanosatellite Formation Establishment and Maintenance Using Electric Propulsion

Nanosatellite clusters are one of the current and future trends in space technology. To maintain a satellite cluster over a long period of time, the nanosatellites need to mitigate the alongtrack drift created by the initial orbit injection. In the mass range of 1–10 kg, CubeSats have strict constraints on allowed mass, volume, and electrical power; and they are equipped with only limited sensor and actuator capabilities. A state-of-the-art miniaturized electric propulsion system is one option to realize the required orbit control capability. Due to the low thrust provided by an electric propulsion system, long orbit control maneuvers are required. Therefore, mission design is highly affected by long-term attitude and power constraints. Assuming a cyclic thrust controller, four methods are developed to allocate the satellite power and attitude resources for orbit control. Two time division methods are presented that allocate dedicated time slots for each of the satellite’s main tasks. Alternatively, two cosine methods are presented that use an error cone, around the required attitude, as a criterion for selecting when to operate the thruster. It is demonstrated that the proposed methods are capable of maintaining a realistic CubeSat formation-flying mission using low-power electric propulsion. For a CubeSat with deployable solar panels, the double cosine method shows the best performance.

A unified formulation for free vibration of functionally graded carbon nanotube reinforced composite spherical panels and shells of revolution with general elastic restraints by means of the Rayleigh–Ritz method

In this article, the free vibration of the functionally graded carbon nanotube reinforced composite spherical panels and shells of revolution with general elastic restraints is analyzed on the basis of the first‐order shear deformation theory. First, the general boundary condition is simulated by using a set of artificial springs along all the edges and the “general” is obtained by changing the stiffness of the corresponding kinds of springs. Then, the admissible function is expressed by the improved Fourier series which is the combination of a standard double cosine Fourier series and several auxiliary functions introduced to remove all potential discontinuities of the original displacement and its derivatives at the edges. Finally, the equations of motion of the FG‐CNT reinforced composite spherical panels and shells of revolution with general boundary conditions are derived by extending the Rayleigh–Ritz method. For the validation of the present method, several comparison studies are conducted and its convergence is also checked meanwhile. In addition, the effect of the volume fraction of CNTs, distribution type of CNTs, boundary restraint parameters on the vibration characteristics are also presented. POLYM. COMPOS., 39:E924–E944, 2018. © 2017 Society of Plastics Engineers

Explicit integration scheme of a subloading surface plasticity model for unsaturated soils

An adaptive substepping explicit integration scheme is proposed for a subloading surface plasticity (SSP) model for unsaturated soils. The SSP model is referred to as the modified σ Θ model, the constitutive laws of which are established in the space of Bishop's effective stress ( σ ) and effective degree of saturation (Θ ). The modified σ Θ model includes a convex subloading surface in the p-q plane and a nonconvex subloading surface in the p- Θ plane. The convex/nonconvex subloading surface in the model may cause problems of incorrect loading-unloading decisions during the stress update. A Double Cosine loading-unloading decision method (DC method) is revised and embedded into the explicit integration scheme for the modified σ Θ model. The local error of the effective degree of saturation ( Θ ) is introduced in the error control for each substep. The performance of the DC method for the modified σ Θ model is discussed. The importance of involving the effective saturation ( Θ ) in the error control is also demonstrated via numerical examples.

Explicit integration scheme for a non-isothermal elastoplastic model with convex and nonconvex subloading surfaces

An adaptive substepping explicit integration scheme with a novel loading-unloading decision method is developed here for the non-isothermal unified hardening (UH) model. The non-isothermal UH model includes a convex subloading surface in the $$p$$p---$$q$$q plane and a nonconvex subloading surface in the $$p$$p---$$T$$T plane. Because of the convex/nonconvex subloading surfaces, the conventional loading-unloading decision method used in stress integration schemes may lead to incorrect elasticity/elastoplasticity judgements. In addition, the conventional loading-unloading decision method is unable to determine the division point that separates the elastic segment from the elastoplastic segment. A simple but robust method, the double cosine (DC) method, is proposed in this paper to solve loading-unloading decision problems. The proposed DC method is then embedded into an adaptive substepping explicit integration scheme to implement the non-isothermal UH model. The accuracy and efficiency of the DC method are discussed by comparing the method with the conventional loading-unloading decision method (the CV method) and the root-finding loading-unloading decision method (the RF method). The performance of the proposed scheme with the DC method is also discussed.