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In this paper, the periodic solutions of the Burgers equation are expanded in a power series with respect to the viscosity coefficient ε which gives a recurrence relation among the approximate functions. Then, the general form of the Fourier expansions for the approximate functions are given by induction, and it is proved that the Fourier series of each approximate function is uniformly and absolutely convergent. [ABSTRACT FROM AUTHOR]

Based on the Hamiltonian variational principle, the 2D field equations and boundary conditions for flexoelectricity were derived, and the corresponding governing equations were obtained through substitution of the constitutive relation and geometric equations into the field equation. The in-plane tensile deformation, thickness-stretch deformation, symmetric thickness-shear deformation, and their coupled flexoelectric polarization of flexoelectric nanoplates caused by inhomogeneous temperature changes, were studied. The displacement fields and electric potential fields were solved with the double Fourier series method. The results demonstrate that, all fields are sensitive to the temperature load, which raises the prospect of controlling the mechanical and electrical behaviors of flexoelectric nanoplates by means of the temperature field. The effects of the thermal field and mechanical field on the displacement field were compared and examined. The work extends the Mindlin-Medick plate structure analysis theory in view of the flexoelectric and temperature effects, and provides a reference for the structural design of micro- and nano-scale devices. [ABSTRACT FROM AUTHOR]

Objectives: Gaofen-7 (GF-7) is the first 1∶ 10 000 stereo mapping satellite of China that can produce high-resolution fusion products. The fusion products are fused with the 0.65 m-panchromatic images and 2.6 m-multispectral images. However, in some fusion products, halos and shifts are existed. Methods: To deal with this problem, we analyze the cross-sensor error property of push-broom optical satellite and deduce a compensation model that induces a linear term to compensate for the internal and external orientation errors, a quartic piecewise polynomial term to rectify the lens-distortion and the sub-CCD(charge coupled device) placement error, and a trigonometric series term to correct for the errors caused by platform jitter. Results: Several scenes of GF-7 (6 in Guangdong and 3 in Qinghai) were used to demonstrate the compensation model. The results show that the registration RMSE （root mean square error） decreased from 0.6 multispectral-pixels to about 0.07 multispectral-pixels by solving and applying the compensation model, and proportion of areas which have less-than 0.25 multispectral-pixels registration error increased to about 98%, which meets the requirement of producing high-quality fusion products. Conclusions: The quantitative results not only reveal the feasibility to improve the accuracy of GF-7 fusion products used by first-level data and achieve the theoretical limit accuracy, but also demonstrate the stability of the piecewise polynomial term allowing for its repeated application to further enhance compensation efficiency. [ABSTRACT FROM AUTHOR]

The elastic buckling of stepped columns with variable cross sections was studied. Firstly, based on the improved Fourier series method, the displacement function of the column was established in the local coordinate system, then the linear equations for buckling loads were obtained with the constrained variational principle of potential energy. The problem was transformed into a matrix eigenvalue problem and the buckling load was obtained from solution of the matrix eigenvalues. Finally, the parameter values in the method were discussed through numerical examples, and the obtained results were compared with the finite element results and previous literature results so as to verify the accuracy of the method. In the presented model the translational and rotational springs were arranged at the 2 ends and the setback cross sections. The method can determine the buckling loads of stepped columns with various elastic boundary conditions accurately in engineering design. [ABSTRACT FROM AUTHOR]

The dispersive quantization of the 2D linear KdV equation and the 2D linear Schrödinger equation were studied over a bounded rectangle domain in the plane. The research shows that, for the KdV equation, if the period ratio is a rational number, at the rational moments, the solution to the periodic initial boundary value problem will be the linear combination of the initial value conditions; whereas, at the irrational moments, the solution will be continuous and nondifferentiable, and exhibit a fractal-like profile. The same is true for the 2D linear Schrödinger equation. [ABSTRACT FROM AUTHOR]

To accurately simulate the temperature field distribution of rolling solid tires, a simulation method of viscoelastic force field of rolling tire was proposed to accurately calculate the heat generation rate of rubber compound. To study the tire of target car, the influence of Mullins effect on the viscoelastic behavior of rubber was considered, and the Mooney-Rivlin model was used to simulate the viscoelastic field of rolling tire. The heat generation rate was accurately calculated by the Fourier coefficient to simulate the temperature field of the rolling tire, and the temperature of the tire was experimentally tested to verify the simulation method .The test results show that the calculation of the heat generation rate by the viscoelastic field of rolling tire can simulate the steady temperature field of tire more reasonably and accurately. Based on the proposed simulation method, the relationship between the highest temperature and the structure and material of tire can be investigated, and the temperature resistance limit of the tire can be improved by changing the process, the rubber material and the tire structure. [ABSTRACT FROM AUTHOR]

Ｔｈｅ ｖｉｂｒａｔｉｏｎ ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ ｏｆ ｃｏｎｔｉｎｕｏｕｓ ｍｕｌｔｉ⁃ｓｅｇｍｅｎｔ ｂｅａｍｓ ｗｅｒｅ ｓｔｕｄｉｅｄ． Ｏｔｈｅｒ ｔｈａｎ ｃｌａｓｓｉｃａｌ ｂｏｕｎｄａｒｙ ｃｏｎｄｉｔｉｏｎｓ （ｓｕｃｈ ａｓ ｓｉｍｐｌｅ ｓｕｐｐｏｒｔｓ）， ｔｈｅ ｅｌａｓｔｉｃ ｃｏｎｓｔｒａｉｎｔｓ ｗｅｒｅ ｃｏｎ⁃ ｓｉｄｅｒｅｄ ｔｏ ａｎａｌｙｚｅ ｔｈｅ ｆｒｅｅ ｖｉｂｒａｔｉｏｎ ｃｈａｒａｃｔｅｒｉｓｔｉｃｓ ｏｆ ｍｕｌｔｉ⁃ｓｅｇｍｅｎｔ ｂｅａｍｓ． Ｆｉｒｓｔｌｙ， ａｃｃｏｒｄｉｎｇ ｔｏ ｔｈｅ ｓｐｅｃｔｒｏ⁃ｇｅｏｍｅｔｒｉｃ ｍｅｔｈｏｄ， ４ ａｕｘｉｌｉａｒｙ ｆｕｎｃｔｉｏｎｓ ｗｅｒｅ ａｄｄｅｄ ｏｎ ｔｈｅ ｂａｓｉｓ ｏｆ ｔｈｅ ｔｒａｄｉｔｉｏｎ⁃ ａｌ Ｆｏｕｒｉｅｒ ｓｅｒｉｅｓ ｔｏ ｃｏｎｓｔｒｕｃｔ ｔｈｅ ｌａｔｅｒａｌ ｄｉｓｐｌａｃｅｍｅｎｔ ｆｕｎｃｔｉｏｎ ｆｏｒ ｅａｃｈ ｓｅｇｍｅｎｔ ｏｆ ｔｈｅ ｂｅａｍ． Ｓｅｃｏｎｄｌｙ， ｎｅｗ ｅｘｐｒｅｓｓｉｏｎｓ ｏｆ ｔｈｅ Ｌａｇｒａｎｇｉａｎ ｆｕｎｃｔｉｏｎｓ ｆｏｒ ｔｈｅ ｂｅａｍ ｓｔｒｕｃｔｕｒｅ ｗｅｒｅ ｏｂｔａｉｎｅｄ ｂｙ ｓｕｂｓｔｉｔｕｔｉｏｎ ｏｆ ｔｈｅ ｓｕｐｐｏｓｅｄ ｓｐｅｃｔｒｏ⁃ｇｅｏｍｅｔｒｉｃ ｆｏｒｍ ｉｎｔｏ ｔｈｅ Ｌａｇｒａｎｇｉａｎ ｆｕｎｃｔｉｏｎｓ． Ｔｈｅ ｆｒｅｅ ｖｉ⁃ ｂｒａｔｉｏｎ ｐｒｏｂｌｅｍ ｗａｓ ｔｒａｎｓｆｏｒｍｅｄ ｉｎｔｏ ｔｈｅ ｓｔａｎｄａｒｄ ｍａｔｒｉｘ ｅｉｇｅｎｖａｌｕｅ ｆｏｒｍ ｆｒｏｍ ｔｈｅ Ｈａｍｉｌｔｏｎｉ⁃ ａｎ ｐｒｉｎｃｉｐｌｅ， ａｎｄ ｔｈｅ ｎａｔｕｒａｌ ｆｒｅｑｕｅｎｃｉｅｓ ａｎｄ ｍｏｄｅｓ ｏｆ ｔｈｅ ｂｅａｍ ｕｎｄｅｒ ａｒｂｉｔｒａｒｙ ｂｏｕｎｄａｒｙ ｃｏｎ⁃ ｄｉｔｉｏｎｓ ｗｅｒｅ ｏｂｔａｉｎｅｄ． Ｆｏｒ ４ ｎｕｍｅｒｉｃａｌ ｅｘａｍｐｌｅｓ， ｔｈｅ ｎａｔｕｒａｌ ｆｒｅｑｕｅｎｃｉｅｓ ａｎｄ ｍｏｄｅｓ ｏｆ ｔｈｅ ｃｏｎ⁃ ｔｉｎｕｏｕｓ ｍｕｌｔｉ⁃ｓｅｇｍｅｎｔ ｂｅａｍｓ ｗｅｒｅ ｃａｌｃｕｌａｔｅｄ ｕｎｄｅｒ ｄｉｆｆｅｒｅｎｔ ｂｏｕｎｄａｒｙ ｃｏｎｄｉｔｉｏｎｓ ｏｆ ｖａｒｉａｎｔ ｓｐｒｉｎｇ ｓｔｉｆｆｎｅｓｓ ｖａｌｕｅｓ． Ｃｏｍｐａｒｉｓｏｎ ｏｆ ｔｈｅ ｒｅｓｕｌｔｓ ｂｅｔｗｅｅｎ ｅｘｉｓｔｉｎｇ ｌｉｔｅｒａｔｕｒｅｓ ａｎｄ ｔｈｉｓ ｗｏｒｋ ｓｈｏｗｓ ｃｏｒｒｅｃｔｎｅｓｓ， ｓｔａｎｄａｒｄｉｚａｔｉｏｎ ａｎｄ ｅｆｆｉｃｉｅｎｃｙ ｏｆ ｔｈｅ ｐｒｏｐｏｓｅｄ ｍｅｔｈｏｄ． [ABSTRACT FROM AUTHOR]

This paper presents a new method for describing the complexity of geographic line elements. Firstly, the Fourier series is used to transform the geographic line elements from the spatial domain to the frequency domain for analysis. Fourier expansions are performed on different geographic line elements to obtain different Fourier descriptors, then we use the method of setting the area threshold to control the deviation of the geographic elements before and after the Fourier expansion to a certain extent, so that the curve reduced by the Fourier descriptor can replace the original geographic line elements. Secondly, based on Shannon’ s information entropy theory, the amount of information of the fitted curve after Fourier expansion is calculated. Then, by combining the frequency domain with the fractal theory, the data of the amount of information is further processed based on the Head‐tail data break and Koch’ s Two‐Eighth Law, and the concept of the distribution index p is proposed. Finally, the distribution index and the square root model are combined, and the approximate expression of the distribution index of the map at different scales is obtained by the least squares method. We select the contour data of a region to conduct experiments, and experiment results show that the distribution index has a good expression effect on the complexity of geographic line elements at different scales, and can be applied to multi‐scale expression of line elements. [ABSTRACT FROM AUTHOR]

The natural frequencies and modes of liquid sloshing in cylindrical container with multiple rigid baffles were analyzed by the fluid sub-domain method. Introducing the generalized time-dependent coordinates, the velocity potential and the surface wave height were expanded into the form of generalized Fourier series. Using the multi modal method, the finite dimensional modal system was deduced. Based on the modal system, the nonlinear sloshing responses were investigated, and the effects of baffle parameters on the nonlinear sloshing of liquid were discussed. The results show that the inhibitory effect of baffles on the liquid sloshing is more significant when the inner radius of the baffles is small with baffles near to the free surface. The annular baffle is considered to be a common technique to reduce the liquid sloshing. The nonlinearity of liquid sloshing is very strong, and the design of anti-sloshing baffle should not ignore the effect of non-linearity. [ABSTRACT FROM AUTHOR]

Honeycomb-cored plates are widely used in aircraft, high-speed trains and other fields, with clear scientific value and engineering significance worthy of vibration analysis. Other than classical boundary condition assumptions (such as simple supports), the elastic constraints were considered to analyze the free vibration characteristics of honeycomb-cored plates. Specifically, the sandwich plate problem was transformed into a 3-layer structure with the honeycomb core layer simplified as an equivalent anisotropic layer. Furthermore, the displacement field function for the structure was expressed in the form of the improved 2D Fourier series, and the natural frequencies and modal shapes of the structure were derived with the Rayleigh-Ritz method based on the energy principle. The theoretical results are in good agreement with the numerical ones. The proposed theoretical model can be used to systematically discuss the effects of boundary constraints on the free vibration characteristics of honeycomb-cored plates, and provides a theoretical basis for the design of constraint schemes for this kind of structures. [ABSTRACT FROM AUTHOR]

In aero-engines, downstream blades are forced to vibrate by the upstream wake, which often severely affects the flutter and fatigue performance of blades. Hence, an efficient method is needed for the analysis of this complex fluid-structure interaction phenomenon. A reduced-order model (ROM) method for blade vibration was proposed based on the harmonic balance method. The upstream wake was firstly decomposed into a number of harmonic waves by FFT ( fast Fourier transform) , and the aerodynamic forces on the blade were obtained through calculation of the amplitudes of aerodynamic forces on the blade due to each harmonic wave； then the blade vibration was fast analyzed by means of the structure dynamic equations for the blade coupled with the aerodynamic ROM. The results show that this method can analyze the flutter characteristics of the blade under wake excitation quickly and accurately. [ABSTRACT FROM AUTHOR]

For the current RLC series resonant circuit experiment, its content and requirements are relatively simple and limited to the reproduction of the basic principles of resonance positioning； therefore，there is a lack of a “starting point” to guide students to think deeply about the circuit. A new cycle of research and development has resulted in the building of a non-sinusoidal signal decomposition synthesis and harmonic generation experiment box. This box is designed to have a series resonant circuit for comprehensive experiment ( Part I) which is the periodic non-sinusoidal signals decomposition and synthesis, including experimental circuit structures, experimental operation of signal decomposition and synthesis, and model validation. This guides students to explore their own tasks and enhance their own supplementary experiments. Teaching practice shows that the integrated circuit can help students improve their experiments from a different perspective. Understanding the basic principles and the value of the series resonant circuit can stimulate students to hand on experience and describe the differences between the ideal circuit model and the actual circuit. This will lead students to developing an interest in learning more. [ABSTRACT FROM AUTHOR]

Typical three-wheel disk structure was chosen in order to improve the accuracy of the stress dis-tribution of 2D finite element model of the engine fan disk.Three-wheel fan disk structure of 3D finite ele-ment modeling was completed by using ANSYS 12.0.And the three-wheel fan disk of 2D axisymmetric fi-nite element model was established.The non axisymmetric load was decomposed into Fourier series.Solu-tion of radial stress,circumferential stress and equivalent stress was done subsequently.Compared with typ-ical 2D analysis load,2D non axisymmetric load and 3D analysis load show that the stress distribution of 2D model applying the uniform load is far away from the stress distribution of 3D model,and the stress dis-tribution of 2D model applying the non axisymmetric load was close to the stress distribution of 3D model. Compared with applying the uniform load,this method can improve the accuracy of the stress distribution of 2D finite element model of the engine fan disk. [ABSTRACT FROM AUTHOR]

In order to achieve the ambition of GOCE, a Fortran program is compiled for recovering Earth's gravitational field model. A filter approach is designed for gravity gradients based on Fourier series. Several gravity field models are recovered by using GOCE PKI orbits and gravity gradients. The results show that the recovering ability of GOCE orbits is up to 120 degree and order. The gravity gradients are influenced by polar gaps more serious than orbits. An Earth's gravitational field model up to degree and order 200 named SWJTU2013GO has been resolved by combining 70 days GOCE orbits and gravity gradients between 2009-11-02 ~ 2010-01-10. The precision of the model is well, but slightly less than the precision of GO_CONS_GCF_2_TIM_R3 as shown by internal and external validation. [ABSTRACT FROM AUTHOR]

When the high-order discretization scheme is applied for solving the hyperbolic conservation laws, the Gibbs oscillations emerging around the discontinuities may cause the fields to take unphysical values, which could result in the undesired interruption of calculation process. This problem is especially severe for the discontinuous Galerkin scheme. Therefore, suitable limiting calculators are supposed to be designed in order to damp the appearance of unphysical oscillations. However, the existed limiters for the discontinuous Galerkin scheme originate mostly from the finite volume methods, thus for the unstructured grids they can only limit the spatial solutions of the solutions, namely the lower-order derivatives, rather than offer an general criterion for the higher order derivatives. Due to this background, the discontinuous finite element solutions are firstly processed into general Fourier expansion series for the realization of spectral decomposition in different frequency domains. Afterwards, a modal coordinate system is originally proposed, describing the changing patterns of the directional derivatives in different orders in a new form. Finally, after the combination of the information of the current mode and its adjacent mode, and the limitation of the directional derivatives in different orders for different levels, the unphysical solutions are successfully damped. Based on this method, the Euler equations are solved for the numerical simulations of the 2-D Riemann problems, the flow field of the wind-tunnel with a front step, the reflection of shockwaves in the inner field of scramjet engines as well as the subsonic and transonic flow surrounding airfoils. The obtained results demonstrate that the new limiter is relatively reliable and it can be generalized to high-order schemes. [ABSTRACT FROM AUTHOR]

A harmonic balance method for the analysis of nonlinear unsteady flow around oscillating cascades is developed to improve computational efficiency for the simulation of periodic flow in turbomachinery. The periodic unsteady flow parameters can be represented by a Fourier series. In this way, the unsteady problem can be solved using conventional computational fluid dynamic method for steady problem. The analysis exploits the fact that, calculated results by harmonic balance are in good agreement with those from dual-time stepping method and linear method, and present method is highly efficient; it is about one order of magnitude faster than conventional dual-time stepping method in the present computation. The effects of the number of harmonics are also studied. It shows that lower order harmonic balance methods are sufficient to simulate periodic unsteady flow accurately except the flow fields including serious separated domain. So, lower order harmonic balance methods could be adopted to analysis unsteady flow in turbomachinery to reduce the computational cost. In addition, the characteristic of unsteady flow around a compressor cascade is investigated, and the mechanism of generation of separated vortex and the vortex shedding frequency are also analyzed. [ABSTRACT FROM AUTHOR]

A method of rotor wake stability analysis in hovering state by Fourier method is established. By the present method, the interval of harmonic wavelength is given when any disturbance on the rotor wake is separated into Fourier series, by which the theoretical foundation of rotor wake stability analysis is improved. Furthermore, a methodology, which can extend the stability analysis for single collocation point to the whole rotor wake by extracting and analyzing the upper and lower bound of the divergence rate relative to the vortex age, has been presented. Taking the rotor of H-34 helicopter in hovering state as a numerical example, the influence of the number of wakes (blades), distribution of vortex core and distortion of wake on rotor wake stability is investigated by this method. The results show that, as for a single collocation point on the rotor wake, it is intrinsically unstable, while as for the whole rotor wake, its stability relative to the vortex age is affected by many factors and exhibits some variation characteristics. Moreover, by comparing with the experimental data of a periodicity of the hovering rotor wake, it is proved that this method can effectively predict the growth characteristics of any disturbance on the rotor wake. [ABSTRACT FROM AUTHOR]

The unsteady flow around helicopter rotor in forward flight is simulated by using harmonic balance method based on Euler equations. For the periodical characteristics in forward flight, the flow variables can be represented by Fourier series in time. Thus, the unsteady governing equations in time domain can be converted to steady equations in frequency domain, so the well-developed steady computational fluid dynamics(CFD) techniques can be used to solve the steady equations in frequency domain, and the unsteady solution is obtained. The main advantage of the present method is that only one blade needs to be simulated by implementation of the periodical condition. Compared with the traditional dual-time stepping method, harmonic balance method improves unsteady calculation efficiency greatly. The calculated pressure distributions and thrust coefficients agree well with the results of experimental data or dual-time stepping method. Message passing interface(MPI) parallel computing technique is utilized to improve the computational efficiency further. [ABSTRACT FROM AUTHOR]