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1
Osipov, Vyacheslav S. "Method of calculating the distortion coefficient sinusoidality of the voltage curve created by three-phase straight lines." Vestnik of Samara State Technical University. Technical Sciences Series 29, no. 4 (December 15, 2021): 99–115. http://dx.doi.org/10.14498/tech.2021.4.8.
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The existing calculation methods in various sources represent rectifier installations as sources of higher harmonics. In this case, the current value of the primary winding of the transformer in the form of rectangles is decomposed into a Fourier series and harmonic components of the current are obtained, except for the main harmonic higher 5, 7, 11, 13, 17, 19 The amplitudes of the current of the higher harmonics are multiplied by the inductive resistances of the supply network corresponding to the frequencies, the results are squared and summed up. Obviously, all terms are of equal magnitude, since with an increase in the harmonic number, its amplitude decreases by n times compared to the first harmonic, while the frequency and inductive resistance increase by the same number of times. The non-sinusoidal coefficient Ku is defined as the ratio of the square root of the sum to the magnitude of the phase voltage. The disadvantage of determining Ku is some arbitrariness in the number of harmonics taken into account. If we take, for example, 9 harmonics, we get Ku = 7.22%, if 4 harmonics, then Ku = 4.81%. When the current of the primary winding of the transformer flows, the voltage amplitude on the substation tires decreases by only 1.9 V. At the same time, the coefficient of non-sinusoidality cannot be equal to 7.22%. In all literature sources, when calculating current harmonics, there are no recommendations for which harmonic numbers should be installed at the substation resonant filters. It is no coincidence that in the new GOST-2013, the value of the distortion coefficient of the sinusoidal voltage curve K(U is calculated as a percentage as a result of the i-th observation according to the formula, (that is, no calculation is made by current harmonics). In this paper, another approach is made to determine the distortion coefficient of the sinusoidal voltage curve. Rectifier installations are not sources of harmonics, but are electrical receivers with a nonlinear characteristic of electric current consumption, while the shape of the sinusoid curve of the supply voltage is distorted. This distorted sine wave is the source of the higher harmonics, which in this paper is decomposed into a Fourier series, while the calculation of the integral functions of the coefficients of the Fourier series is performed in the Mathcad program. The solution is made for three rectifier circuits. Harmonics for the installation of resonant filters are determined. As a result, a developed method is proposed for calculating the higher voltage harmonics and determining the distortion coefficient of the sinusoidal voltage curve based on the Fourier series expansion of the distorted voltage sine curve during operation of three-phase uncontrolled rectifiers.
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Yu, Jun-Yao. "Relationship between Mean Value and Fourier Coefficients of a Time-Varying Function with Half-Period Rests: Theory and Application." Journal of Vibration and Acoustics 116, no. 1 (January 1, 1994): 26–30. http://dx.doi.org/10.1115/1.2930392.
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The Fourier coefficients are investigated for such a periodic time function whose magnitude keeps constant during the time of every half-period. In this case the relationship between the mean value and the Fourier coefficients is achieved using appropriate mathematical theory. It was applied successfully to the study of torsional harmonic excitations due to gas pressure in a four-cycle IC engine. A formula relating the truncated Fourier coefficient series to MIP has been established with very little error. The series simulation of Fourier coefficients is utilized in the determination and analysis of excitations.
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BRINGMANN, KATHRIN, and OLAV K. RICHTER. "EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS." International Journal of Number Theory 07, no. 03 (May 2011): 825–33. http://dx.doi.org/10.1142/s1793042111004617.
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In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.
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Luo, Albert C. J., and Bo Yu. "Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator." International Journal of Bifurcation and Chaos 25, no. 13 (December 15, 2015): 1550179. http://dx.doi.org/10.1142/s0218127415501795.
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In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.
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Rodríguez-Maldonado, Johnny, Cornelio Posadas-Castillo, and Ernesto Zambrano-Serrano. "Alternative Method to Estimate the Fourier Expansions and Its Rate of Change." Mathematics 10, no. 20 (October 17, 2022): 3832. http://dx.doi.org/10.3390/math10203832.
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This paper presents a methodology to obtain the Fourier coefficients (FCs) and the derivative Fourier coefficients (DFCs) from an input signal. Based on the Taylor series that approximates the input signal into a trigonometric signal model through the Kalman filter, consequently, the signal’s and successive derivatives’ coefficients are obtained with the state prediction and the state matrix inverse. Compared to discrete Fourier transform (DFT), the new class of filters provides noise reduction and sidelobe suppression advantages. Additionally, the proposed Taylor–Kalman–Fourier algorithm (TKFA) achieves a null-flat frequency response around the frequency operation. Moreover, with the proposed TKFA method, the decrement in the inter-harmonic amplitude is more significant than that obtained with the Kalman–Fourier algorithm (KFA), and the neighborhood of the null-flat frequency is expanded. Finally, the approximation of the input signal and its derivative can be performed with a sum of functions related to the estimated coefficients and their respective harmonics.
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Tupas, M., C. Navacchi, F. Roth, B. Bauer-Marschallinger, F. Reuß, and W. Wagner. "COMPUTING GLOBAL HARMONIC PARAMETERS FOR FLOOD MAPPING USING TU WIEN’S SAR DATACUBE SOFTWARE STACK." International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLVIII-4/W1-2022 (August 6, 2022): 495–502. http://dx.doi.org/10.5194/isprs-archives-xlviii-4-w1-2022-495-2022.
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Abstract. TU Wien’s flood mapping algorithm, used for global operations, utilizes harmonic functions to model the seasonal behavior of backscatter to improve flood classification. In earth observation (EO), temporal harmonic models have been used in various scenarios for vegetation and water mapping in the optical and, recently, synthetic aperture radar (SAR) domains. These models condense EO time series stacks to a few Fourier coefficient images that capture seasonal variability, allowing for variable estimation for each day of the year. TU Wien’s harmonic parameters consist of these coefficients plus the regression standard deviation and number of observations. However, generating harmonic models at large scales and high resolution presents significant logistical and technical challenges. Particularly for SAR, which requires special handling due to acquisition geometry considerations, implementation on a datacube infrastructure is necessary for agile filtering in metadata, temporal and spatial dimensions. In this work, we highlight our harmonic parameter dataset and our software stack of loosely coupled Python packages, which were deployed in a high-performance computing environment to generate these parameters globally.
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Mallat, Stéphane, Sixin Zhang, and Gaspar Rochette. "Phase harmonic correlations and convolutional neural networks." Information and Inference: A Journal of the IMA 9, no. 3 (November 5, 2019): 721–47. http://dx.doi.org/10.1093/imaiai/iaz019.
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Abstract A major issue in harmonic analysis is to capture the phase dependence of frequency representations, which carries important signal properties. It seems that convolutional neural networks have found a way. Over time-series and images, convolutional networks often learn a first layer of filters that are well localized in the frequency domain, with different phases. We show that a rectifier then acts as a filter on the phase of the resulting coefficients. It computes signal descriptors that are local in space, frequency and phase. The nonlinear phase filter becomes a multiplicative operator over phase harmonics computed with a Fourier transform along the phase. We prove that it defines a bi-Lipschitz and invertible representation. The correlations of phase harmonics coefficients characterize coherent structures from their phase dependence across frequencies. For wavelet filters, we show numerically that signals having sparse wavelet coefficients can be recovered from few phase harmonic correlations, which provide a compressive representation.
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Bringmann, Kathrin, Paul Jenkins, and Ben Kane. "Differential operators on polar harmonic Maass forms and elliptic duality." Quarterly Journal of Mathematics 70, no. 4 (July 3, 2019): 1181–207. http://dx.doi.org/10.1093/qmath/haz009.
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Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.
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Hong, Xiaobin, Yuan Liu, Xiaohui Lin, Zongqiang Luo, and Zhenwei He. "Nonlinear Ultrasonic Detection Method for Delamination Damage of Lined Anti-Corrosion Pipes Using PZT Transducers." Applied Sciences 8, no. 11 (November 14, 2018): 2240. http://dx.doi.org/10.3390/app8112240.
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Lined anti-corrosion pipes are widely used in oil and gas, petrochemical, pharmaceutical industries. However, defects, especially delamination, may occur in the production and service of pipes which result in safety accidents. Based on nonlinear ultrasonic theory, this paper studied the delamination detection method using the nonlinear harmonics for lined anti-corrosion pipes. The response characteristics of the anti-corrosion pipe were obtained through a sweep sine response experiment and the preferred excitation frequency was determined. The Wavelet Packet transform and Hilbert–Huang transform is applied for signal process and feature extraction. Then, a series of experiments were carried out and the results were analyzed and discussed. The results showed that a second-order and third-order nonlinear coefficient increased with the delamination damage. The amplitude of second-harmonic is much stronger than the third-order one. The mean squared error of the nonlinear coefficient, which was processed by Wavelet Packet transform and Hilbert–Huang transform, is smaller than wavelet packet transform and Discrete Fourier transform or processed only Hilbert–Huang transform. The higher harmonics can describe the change of delamination damage, which means that the nonlinear ultrasonic detection method could use for damage detection of anti-corrosion pipe. The nonlinear higher-harmonic is sensitive to delamination damage. The nonlinear ultrasonic method has the potential for damage detection for lined anti-corrosion pipes.
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Hu, Zhengmin, Kai Zhou, and Yong Chen. "Sound Radiation Analysis of Functionally Graded Porous Plates with Arbitrary Boundary Conditions and Resting on Elastic Foundation." International Journal of Structural Stability and Dynamics 20, no. 05 (May 2020): 2050068. http://dx.doi.org/10.1142/s0219455420500686.
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In this paper, the sound radiation behaviors of the functionally graded porous (FGP) plate with arbitrary boundary conditions and resting on elastic foundation are studied by means of the modified Fourier series method. It is assumed that a total of three types of porosity distributions are considered in the present study. The material parameters are determined according to the porosity coefficient used to denote the size of pores in the plate. The governing equations of the FGP plate are derived by utilizing the Hamilton’s principle on the basis of the first-order deformation theory (FSDT). Each displacement component of the FGP plate is expanded as the Fourier cosine series combined with auxiliary polynomial functions introduced to enhance the convergence rate of the series expansions. The acoustic response of the FGP plate due to a concentrated harmonic load is calculated by evaluating the Rayleigh integral. Good agreements are attained by comparing the present results with those in available literatures, which show the accuracy and versatility of the developed method in this paper. Finally, the influences of the porosity distribution type, porosity coefficient, boundary condition and elastic foundation on the sound radiation of the FGP plate are analyzed in detail.
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Abdul-Rahman, Anmar, William Morgan, Ying Jo Khoo, Christopher Lind, Allan Kermode, William Carroll, and Dao-Yi Yu. "Linear interactions between intraocular, intracranial pressure, and retinal vascular pulse amplitude in the fourier domain." PLOS ONE 17, no. 6 (June 28, 2022): e0270557. http://dx.doi.org/10.1371/journal.pone.0270557.
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Purpose To compare the retinal vascular pulsatile characteristics in subjects with normal (ICPn) and high (ICPh) intracranial pressure and quantify the interactions between intraocular pressure, intracranial pressure, and retinal vascular pulse amplitude in the Fourier domain. Materials and methods Twenty-one subjects were examined using modified photoplethysmography with simultaneous ophthalmodynamometry. A harmonic regression model was fitted to each pixel in the time-series, and used to quantify the retinal vascular pulse wave parameters including the harmonic regression wave amplitude (HRWa). The pulse wave attenuation was measured under different ranges of induced intraocular pressure (IOPi), as a function of distance along the vessel (VDist). Intracranial pressure (ICP) was measured using lumbar puncture. A linear mixed-effects model was used to estimate the correlations between the Yeo-Johnson transformed harmonic regression wave amplitude (HRWa-YJt) with the predictors (IOPi, VDist and ICP). A comparison of the model coefficients was done by calculating the weighted Beta (βx) coefficients. Results The median HRWa in the ICPn group was higher in the retinal veins (4.563, interquartile range (IQR) = 3.656) compared to the retinal arteries (3.475, IQR = 2.458), p<0.0001. In contrast, the ICPh group demonstrated a reduction in the median venous HRWa (3.655, IQR = 3.223) and an elevation in the median arterial HRWa (3.616, IQR = 2.715), p<0.0001. Interactions of the pulsation amplitude with ICP showed a significant disordinal interaction and the loss of a main effect of the Fourier sine coefficient (bn1) in the ICPh group, suggesting that this coefficient reflects the retinal vascular response to ICP wave. The linear mixed-effects model (LME) showed the decay in the venous (HRWa-YJt) was almost twice that in the retinal arteries (−0.067±0.002 compared to −0.028±0.0021 respectively, p<0.00001). The overall interaction models had a total explanatory power of (conditional R2) 38.7%, and 42% of which the fixed effects explained 8.8%, and 5.8% of the variance (marginal R2) for the venous and arterial models respectively. A comparison of the damping effect of VDist and ICP showed that ICP had less influence on pulse decay than distance in the retinal arteries (βICP = -0.21, se = ±0.017 compared to βVDist=-0.26, se = ±0.019), whereas the mean value was equal for the retinal veins (venous βVDist=-0.42, se = ±0.015, βICP = -0.42, se = ±0.019). Conclusion The retinal vascular pulsation characteristics in the ICPh group showed high retinal arterial and low venous pulsation amplitudes. Interactions between retinal vascular pulsation amplitude and ICP suggest that the Fourier sine coefficient bn1 reflects the retinal vascular response to the ICP wave. Although a matrix of regression lines showed high linear characteristics, the low model explanatory power precludes its use as a predictor of ICP. These results may guide future predictive modelling in non-invasive estimation of ICP using modified photoplethysmography.
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Owczarek, Mariusz, and Stefan Owczarek. "Heat transport solutions in rectangular shields using harmonic polynomials." MATEC Web of Conferences 282 (2019): 02064. http://dx.doi.org/10.1051/matecconf/201928202064.
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The search for the temperature field in a two-dimensional problem is common in building physics and heat exchange in general. Both numerical and analytical methods can be used to obtain a solution. Here a method of initial functions, the basics of which were given by W.Z. Vlasov i A.Y. Lur’e were adopted. Originally MIF was used for analysis of the loads of a flat elastic medium. Since then it was used for solving concrete beams, plates and composite materials problems. Polynomial half-reverse solutions are used in the theory of a continuous medium. Here solutions were obtained by direct method. As a result, polynomial forms of the considered temperature field were obtained. The Cartesian coordinate system and rectangular shape of the plate were assumed. The governing are the Fourier equation in steady state . Boundary conditions in the form of temperature (τ(x),t(y)) or/and flux (p(x), q(y)) can be provided. The solution T(x, y) were assumed in the form of an infinite power series developed in relation to the variable y with coefficients Cn depending on x. The assumed solution were substituted into Fourier equation and after expanding into Taylor series the boundary condition for y = 0 and y=h were taken into account. Form this condition a coefficients Cn can be calculated and therefore a closed solution for temperature field in plate.
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Huang, Dishan. "Study of the Response Prediction of a Parametric Vibration Based on Modulation Feedback." Advances in Acoustics and Vibration 2010 (March 1, 2010): 1–9. http://dx.doi.org/10.1155/2010/284187.
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The problem of response prediction is investigated for parametric vibration in terms of a new concept. The response solution is presented in the special form of Fourier series for signal degree freedom of parametric vibration based on modulation feedback. By applying harmonic balance and limitation operation, all coefficients of a harmonic component are fully determined with a set of series. Meanwhile, some important dynamic behaviors are exposed through mathematical deduction, and an instability phenomenon can be discussed through given frequency factors. The investigation result shows that the new approach has an advantage in the complete response expression, and it is very significant for the theoretical research and engineering application concerning parametric vibration.
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Valdez, Marc Andrew, Alex J. Yuffa, and Michael B. Wakin. "On-grid compressive sampling for spherical field measurements in acoustics." Journal of the Acoustical Society of America 152, no. 4 (October 2022): 2240–56. http://dx.doi.org/10.1121/10.0014628.
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We derive a compressive sampling method for acoustic field reconstruction using field measurements on a pre-defined spherical grid that has theoretically guaranteed relations between signal sparsity, measurement number, and reconstruction accuracy. This method can be used to reconstruct band limited spherical harmonic or Wigner D-function series (spherical harmonic series are a special case) with sparse coefficients. Contrasting typical compressive sampling methods for Wigner D-function series that use arbitrary random measurements, the new method samples randomly on an equiangular grid, a practical and commonly used sampling pattern. Using its periodic extension, we transform the reconstruction of a Wigner D-function series into a multi-dimensional Fourier domain reconstruction problem. We establish that this transformation has a bounded effect on sparsity level and provide numerical studies of this effect. We also compare the reconstruction performance of the new approach to classical Nyquist sampling and existing compressive sampling methods. In our tests, the new compressive sampling approach performs comparably to other guaranteed compressive sampling approaches and needs a fraction of the measurements dictated by the Nyquist sampling theorem. Moreover, using one-third of the measurements or less, the new compressive sampling method can provide over 20 dB better de-noising capability than oversampling with classical Fourier theory.
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Schreyer, Frederic, and Remco I. Leine. "A Mixed Shooting – Harmonic Balance Method for Unilaterally Constrained Mechanical Systems." Archive of Mechanical Engineering 63, no. 2 (June 1, 2016): 297–314. http://dx.doi.org/10.1515/meceng-2016-0017.
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Abstract In this paper we present a mixed shooting – harmonic balance method for large linear mechanical systems on which local nonlinearities are imposed. The standard harmonic balance method (HBM), which approximates the periodic solution in frequency domain, is very popular as it is well suited for large systems with many degrees of freedom. However, it suffers from the fact that local nonlinearities cannot be evaluated directly in the frequency domain. The standard HBM performs an inverse Fourier transform, then calculates the nonlinear force in time domain and subsequently the Fourier coefficients of the nonlinear force. The disadvantage of the HBM is that strong nonlinearities are poorly represented by a truncated Fourier series. In contrast, the shooting method operates in time-domain and relies on numerical time-simulation. Set-valued force laws such as dry friction or other strong nonlinearities can be dealt with if an appropriate numerical integrator is available. The shooting method, however, becomes infeasible if the system has many states. The proposed mixed shooting-HBM approach combines the best of both worlds.
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Venkat, N. K., and M. Spaulding. "Influence of Vibration Amplitude on Laminar Flow Over a Plate Vibrating at Low Strouhal Number." Journal of Fluids Engineering 115, no. 3 (September 1, 1993): 411–19. http://dx.doi.org/10.1115/1.2910154.
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The spectral and hydrodynamic response of laminar flow over a flat plate with a vibrating section forced in sinusoidal motion with a dimensionless amplitude ratio, H0 (vibration amplitude divided by plate length) varying in the range 0.0 < H0 < 0.1 is analyzed using numerical simulations. The Reynolds number, Re, based on the length of the vibrating plate, is fixed at 1000. The flow is simulated for Strouhal number, St, = 0.25 (low frequency). The spectral characteristics are obtained by performing Fast Fourier Transform (FFT) on the pressure coefficient time series data. The hydrodynamic analysis is performed by plotting stream function contour plot in the vicinity of the vibrating section for one vibration cycle. The model predicted results show that the friction and pressure coefficients over the vibrating body vary with vibration amplitude. For low amplitude ratios, the interaction of the external flow with the vibrating section is linear and there is little up or downstream influence. For high H0, there is considerable downstream influence of the disturbance. Nonlinear energy transfer, as evidenced by the existence of a significant first harmonic in the pressure wave, takes place between the vibrating plate and the flow field. Energy transfer to the higher harmonics is less significant.
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Failla, G., P. D. Spanos, and M. Di Paola. "Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique." Journal of Applied Mechanics 70, no. 5 (September 1, 2003): 708–14. http://dx.doi.org/10.1115/1.1599916.
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This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation of Newton’s method. The proposed technique is also applicable for arbitrary nonlinearities via a cubicization procedure over the interval T. Upon determining the Fourier coefficients, estimates of the response power spectral density matrix are constructed by averaging their squared moduli over the samples ensemble. Examples of application prove the reliability of the technique by comparison with digital simulation data.
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Gimeno, Encarnación, and Augusto Beléndez. "Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations." Zeitschrift für Naturforschung A 64, no. 12 (December 1, 2009): 819–26. http://dx.doi.org/10.1515/zna-2009-1207.
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This paper presents a new approach for solving accurate approximate analytical solutions for nonlinear phenomena governed by pendulum-like differential equations. The new approach couples Taylor series expansion with rational harmonic balancing. An approximate rational solution depending on a small parameter is considered. After substituting the approximate solution into the governing differential equation, this equation is expanded in Taylor series of the parameter prior to harmonic balancing. The approach gives a cubic equation, which must be solved in order to obtain the value of the small parameter. A method for transforming this cubic equation into a linear equation is presented and discussed. Using this approach, accurate approximate analytical expressions for period and periodic solutions are obtained. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This allowed us to compare the coefficients for the different harmonic terms in these solutions. These analytical approximations may be of interest for those researchers working in nonlinear physical phenomena governed by pendulum-like differential equations in fields such as classical mechanics, vibrations, acoustics, electromagnetism, electronics, superconductivity, optics, gravitation, and others
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Dzyuba, Anatolii, Inha Safronova, and Larysa Levitina. "Algorithm for reducing computational costs in problems of calculation of asymmetrically loaded shells of rotation." Strength of Materials and Theory of Structures, no. 105 (November 30, 2020): 99–113. http://dx.doi.org/10.32347/2410-2547.2020.105.99-113.
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The problem of calculating the shells of rotation of a variable along the meridian of rigidity under asymmetric loading is reduced o a set of systems of one-dimensional boundary value problems with respect to the amplitudes of decomposition of the required functions into trigonometric Fourier series.
A method for reducing the number of one-dimensional boundary value problems required to achieve a given accuracy in determining the stress-strain state of the shells of rotation with a variable along the meridian wall thickness under asymmetric load. The idea of the proposed approach is to apply periodic extrapolation (prediction) of the values of the decomposition coefficients of the required functions using the results of calculations of previous coefficients of the corresponding trigonometric series, thus replacing them with some prediction values calculated by simple formulas.
To solve this problem, we propose the joint use of Aitken-Steffens extrapolation dependences and Adams method in the form of incremental component, which is quite effective in solving the Cauchy problem for systems of ordinary differential equations and is based on Lagrange and Newton extrapolation dependences.
The validity of the proposed approach was verified b the results of a systematic numerical experiment by predicting the values of the expansion coefficients in the Fourier series of known functions of one variable.
The approach is quite effective in the calculation of asymmetrically loaded shells of rotation with variable along the meridian thickness, when the coefficients of decomposition of the required functions into Fourier series are functions of the longitudina lcoordinate and are calculated by solving the corresponding boundary value problem. In this case, the approach allows solving solutions of differential equations for the amplitudes of decompositionin to trigonometric series only for individual "reference" harmonics, and the amplitudes for every third harmonic can be calculated by interpolating their values for all node integration points of the corresponding boundary value problem. This significantly reduces the computational cost of obtaining the solution as a whole.
As an example, the results of the calculation of the stress-strain state of a steel annular plate under asymmetric transverse loading are given.
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Zeinali, Reza, Dave Krop, and Elena Lomonova. "Modeling the dynamic behavior of laminated steels using a Fourier-based approach." European Physical Journal Applied Physics 92, no. 1 (October 2020): 10905. http://dx.doi.org/10.1051/epjap/2020200020.
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A new magneto-dynamic model is proposed to approximate the dynamic hysteresis effect in laminated steels considering the static hysteresis, eddy-current field, and excess field. An accurate congruency-based hysteresis model is used to predict the static hysteresis field. The eddy-current is determined from the 1D diffusion equation and the well-known Bertotti empirical equation is utilized to model the excess-field effect. The dynamic lamination model obtained from coupling three field components is solved using a Fourier-based approach. In this approach, the flux density across the lamination thickness is approximated by a cosine-based Fourier series. The coefficients of the Fourier series are determined by solving a system of nonlinear equations through an iterative procedure. Owing to the employed congruency-based static hysteresis model, the proposed magneto-dynamic model offers high accuracy for arbitrary magnetization regimes. To validate the model accuracy, the model results are compared with sinusoidal and multi-harmonic measurements. The comparison shows that the proposed model predicts the dynamic hysteresis phenomenon in laminated steels with a relative energy error of less than 7%.
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Zhou, Feng Xi, and Qiang Ma. "Dynamic Response of Two-Dimensional Fluid-Saturated Porous Beam." Applied Mechanics and Materials 580-583 (July 2014): 169–74. http://dx.doi.org/10.4028/www.scientific.net/amm.580-583.169.
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The dynamic response of the two dimensional fluid-saturated porous beam has been studied based on the linear elastic theory and the Biot’s model for saturated porous media, in which the compressibility of solid particles and fluid and the viscosity of pore fluid are taken into account. By adopting Fourier series expansion and solving ordinary differential equations, the calculation formula for dynamic response of a two dimensional simply-supported fluid-saturated porous beam is established. The dynamic response characteristics of the forced vibration under uniformly distributed harmonic load are analyzed as a numerical example. Moreover, the solid displacement and pore fluid pressure of the two-dimensional fluid-saturated porous beam would be affected by the surface in filtration conditions, pore fluid permeability coefficient and loading frequency parameters etc.
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Laneev, Evgeniy B., and Obaida Baaj. "On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images." Discrete and Continuous Models and Applied Computational Science 30, no. 4 (December 26, 2022): 342–56. http://dx.doi.org/10.22363/2658-4670-2022-30-4-342-356.
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The paper considers mathematical methods of correction of thermographic images (thermograms) in the form of temperature distribution on the surface of the object under study, obtained using a thermal imager. The thermogram reproduces the image of the heat-generating structures located inside the object under study. This image is transmitted with distortions, since the sources are usually removed from its surface and the temperature distribution on the surface of the object transmits the image as blurred due to the processes of thermal conductivity and heat exchange. In this paper, the continuation of the temperature function as a harmonic function from the surface deep into the object under study in order to obtain a temperature distribution function near sources is considered as a correction principle. This distribution is considered as an adjusted thermogram. The continuation is carried out on the basis of solving the Cauchy problem for the Laplace equation - an ill-posed problem. The solution is constructed using the Tikhonov regularization method. The main part of the constructed approximate solution is presented as a Fourier series by the eigenfunctions of the Laplace operator. Discretization of the problem leads to discrete Fourier series. A modification of the Hamming method for summing Fourier series and calculating their coefficients is proposed.
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Sun, Huiyuan, Thushara D. Abhayapala, and Prasanga N. Samarasinghe. "Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording." Applied Sciences 11, no. 3 (January 25, 2021): 1074. http://dx.doi.org/10.3390/app11031074.
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Spherical harmonic analysis has been a widely used approach for spatial audio processing in recent years. Among all applications that benefit from spatial processing, spatial Active Noise Control (ANC) remains unique with its requirement for open spherical microphone arrays to record the residual sound field throughout the continuous region. Ideally, a low delay spherical harmonic recording algorithm for open spherical microphone arrays is desired for real-time spatial ANC systems. Currently, frequency domain algorithms for spherical harmonic decomposition of microphone array recordings are applied in a spatial ANC system. However, a Short Time Fourier Transform is required, which introduces undesirable system delay for ANC systems. In this paper, we develop a time domain spherical harmonic decomposition algorithm for the application of spatial audio recording mainly with benefit to ANC with an open spherical microphone array. Microphone signals are processed by a series of pre-designed finite impulse response (FIR) filters to obtain a set of time domain spherical harmonic coefficients. The time domain coefficients contain the continuous spatial information of the residual sound field. We corroborate the time domain algorithm with a numerical simulation of a fourth order system, and show the proposed method to have lower delay than existing approaches.
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Starikov, Alexsandr V., Sergey L. Lisin, Olga S. Belyaeva, and Viktor A. Kirdyashev. "Method for reducing the amplitudes of higher harmonics in the output voltage of the frequency converter." Vestnik of Samara State Technical University. Technical Sciences Series 29, no. 1 (April 23, 2021): 120–32. http://dx.doi.org/10.14498/tech.2021.1.9.
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Modern low-voltage frequency converters, built on the principle of stand-alone inverters with pulse-width modulation, have large amplitudes of higher harmonics in the output voltage. This adversely affects the operation of an asynchronous or synchronous motor connected to such a converter, since it causes a decrease in the efficiency of the AC machine. The analysis of the reasons for the poor harmonic composition of the output voltage of traditional inverters with sinusoidal pulse-width modulation is carried out and it is noted that the main one is the introduction of "dead" time when switching half-bridges every modulation period. The proposed method of sinusoidal modulation does not require the introduction of "dead" time. The modes of operation of power transistors with this switching method and control signal diagrams are considered. Analytical expressions are found that make it possible to determine the effective value of the output voltage of the frequency converter and the coefficients of higher harmonics for the proposed method of switching power transistors. The main reason for the low effective value of the phase voltage at the output of the inverter at sinusoidal pulse-width modulation is determined - the ineffective use of the switching period. The proposed method of sinusoidal modulation does not require the introduction of "dead" time, increasing the effective value of the output voltage and reducing the total harmonic component by 75 times. The diagrams of signals that control power transistors and provide a decrease in the amplitudes of higher harmonics at the output of the frequency converter are given. An approximation by the harmonic Fourier series of the output voltage of the inverter, obtained in the case of using the proposed method of switching power transistors, is made. It is noted that for the technical implementation of sinusoidal modulation, which provides small amplitudes of higher harmonics, only two pulse-width modulators are needed.
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Li, Yu, and Jun-Sheng Duan. "The periodic response of a fractional oscillator with a spring-pot and an inerter-pot." Journal of Mechanics 37 (December 18, 2020): 108–17. http://dx.doi.org/10.1093/jom/ufaa009.
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Abstract The fractional oscillation system with two Weyl-type fractional derivative terms $_{ - \infty }D_t^\beta x$ (0 < β < 1) and $_{ - \infty }D_t^\alpha x$ (1 < α < 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 < β < 1) and α (1 < α < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude–frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.
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Cai, Li Ming, Hui Qi, and Xiang Nan Pan. "The Scattering of Circular Cylindrical Cavity with Time-Harmonic SH Waves in Infinite Strip Region." Applied Mechanics and Materials 580-583 (July 2014): 3083–88. http://dx.doi.org/10.4028/www.scientific.net/amm.580-583.3083.
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The scattering of time harmonic SH waves by arbitrary positions of circular cylindrical cavity is studied in continuous, homogeneous, isotropic, elastic strip region. In this paper, the completely analytical expression of total wave field is explicitly presented and the dynamic stress distribution is symbolically visualized in the strip region. The total wave field is divided into four sub wave fields, incident wave field and scattering wave field by the upper bound, the lower bound and the cylindrical bound, on big arc supposition. Specific wave functions are employed for each wave field expansion in series, such as circular cylindrical functions, respectively. Corresponding infinite linear algebraic equations are constructed by means of solving coefficients of Fourier series expansion on each sub wave field. Coefficients of cylindrical function expansion of each sub wave field are determined by truncated equations, which are reduced number of coefficients on pre-given computational accuracy. Numerical results graphically describe the dynamic stress concentration factor around the circumference of the cavity and the normalized dynamic stress along the cross section directly above the cavity.
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Qi, Hui, Gen Chang Zhang, and Jing Fu Nan. "Ground Motion of Non-Circular Alluvial Valley for Incident Plane SH-Wave." Applied Mechanics and Materials 105-107 (September 2011): 2092–97. http://dx.doi.org/10.4028/www.scientific.net/amm.105-107.2092.
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The seismic ground motions are studied in an infinite half-space with a non-circular alluvial valley under time harmonic incident anti-plane shear waves. Based on the conformal mapping method and Fourier series expansions, the conditions of displacement continuity and stress equilibrium at the interface of alluvial valley are set as semi-circular alluvial valley in conformal plane, then the result is obtained by constructing a set of infinite linear algebraic equations with boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a semi-analytical solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples for semi-elliptic alluvial valley are given to test our program. The research indicates that great interaction exists between the alluvial valley and the horizontal surface, which will bring on great influence on ground motion. Therefore enough importance must be attached to the existing of subsurface cavity while finishing seismic design.
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Segundo-Ramírez, Juan, Rafael Peña-Gallardo, Aurelio Medina, Ciro Núñez-Gutiérrez, and Nancy Visairo-Cruz. "A Comprehensive Modeling of a Three-Phase Voltage Source PWM Converter." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/426245.
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This contribution reports the development of a time domain model of a three-phase voltage source converter (VSC) that can be used in the transient and steady state analysis of nonlinear power systems including their associated closed-loop control schemes. With this proposed model, the original discontinuous nonlinear power system can be transformed into a continuous system, while keeping the underlying harmonic nature of the VSC and avoiding typical and undesirable numerical problems associated with the large derivatives during the switching transitions. The development of this model was based on the dynamic Fourier series of the switching functions under a sinusoidal PWM modulation scheme, which require the calculation of the switching instants at each integration step; the switching instants and the dynamic Fourier series coefficients are calculated by explicit mathematical formulas. The proposed model of the VSC is suitable for the fast computation of the periodic steady state solution through the application of Newton method. Simulations were carried out in order to illustrate the benefits of the proposed VSC model.
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Li, Hong Liang, and Cun Zhang. "Green's Function Solution of the Semi-Space with Double Shallow-Buried Cavities." Key Engineering Materials 417-418 (October 2009): 145–48. http://dx.doi.org/10.4028/www.scientific.net/kem.417-418.145.
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In mechanical engineering and modern municipal construction, shallow-buried cavity structure is used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with double shallow-buried cavities while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by the circle cavities comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress-free condition of the circle cavities in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the circle cavities and the location of the line source force. Based on this solution, the problem of interaction of double circular cavities and a linear crack in semi-space can be investigated further.
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Wu, Guo Hui, and Hong Liang Li. "Green's Function Solution of the Semi-Space with Double Shallow-Buried Inclusions." Key Engineering Materials 462-463 (January 2011): 518–23. http://dx.doi.org/10.4028/www.scientific.net/kem.462-463.518.
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In mechanical engineering and modern municipal construction, shallow-buried inclusion structure is used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with double shallow-buried inclusions while bearing anti-plane harmonic line source force at any point. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by the circle inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition of the circle inclusions in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the circle inclusions and the location of the line source force. Based on this solution, the problem of interaction of double circular inclusions and a linear crack in semi-space can be investigated further.
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Gulaj, A. V., and V. M. Zajcev. "Digital technology for spectral analysis of oscillation parameters." «System analysis and applied information science», no. 1 (June 8, 2022): 4–8. http://dx.doi.org/10.21122/2309-4923-2022-1-4-8.
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Performing the Fourier analysis of the harmonic composition of oscillations is an effective approach to the study of vibration processes in machines and mechanisms. This method of express control remains relevant and highly demanded in the acoustic diagnostics of machines due to its relatively high informativeness. The expansion of its analytical capabilities is implemented by introducing additional means for digital processing of vibration signals, including smart systems. The objective of this work consists in creating a digital method for spectral analysis of vibration signals, the implementation of which would be possible with the use of smart devices. This work presents a method of polyharmonic approximation of the parameters of the signals reflecting oscillatory processes in machines based on the additive combinations of sinusoidal functions. The technology of truncation of the time discretised decomposition of the above signals into a Fourier series is implemented using the Rayleigh theorem and the Lyapunov – Parseval equality. The distortion of the frequency spectrum of oscillations is eliminated by suppressing its side lobes using window filters. To compare the efficiency of different windows, the following special parameters are used: width of the main lobe of the frequency spectrum; coefficient of relative expansion of the main lobes; maximum relative level of side lobes; coefficient of relative attenuation of signal energy. In so doing, the windows of Hanning, Nuttall, Hamming, Barlett were considered, and the parameters of a rectangular window were taken as basic values. The limitations of the digit capacity of analogto-digital signal converters depending on the required accuracy of maintaining the energy balance when truncating the discrete Fourier expansions have been determined. An example of implementation of the digital technology of spectral analysis of the parameters of oscillatory processes is given.
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Xiong, Yangshou, Kang Huang, Fengwei Xu, Yong Yi, Meng Sang, and Hua Zhai. "Research on the Influence of Backlash on Mesh Stiffness and the Nonlinear Dynamics of Spur Gears." Applied Sciences 9, no. 5 (March 12, 2019): 1029. http://dx.doi.org/10.3390/app9051029.
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In light of ignoring the effect of backlash on mesh stiffness in existing gear dynamic theory, a precise profile equation was established based on the generating processing principle. An improved potential energy method was proposed to calculate the mesh stiffness. The calculation result showed that when compared with the case of ignoring backlash, the mesh stiffness with backlash had an obvious decrease in a mesh cycle and the rate of decline had a trend of decreasing first and then increasing, so a stiffness coefficient was introduced to observe the effect of backlash. The Fourier series expansion was employed to fit the mesh stiffness rather than time-varying mesh stiffness, and the stiffness coefficient was fitted with the same method. The time-varying mesh stiffness was presented in terms of the piecewise function. The single degree of freedom model was employed, and the fourth order Runge–Kutta method was utilized to investigate the effect of backlash on the nonlinear dynamic characteristics with reference to the time history chart, phase diagram, Poincare map, and Fast Fourier Transformation (FFT) spectrogram. The numerical results revealed that the gear system primarily performs a non-harmonic-single-periodic motion. The partially enlarged views indicate that the system also exhibits small-amplitude and low-frequency motion. For different cases of backlash, the low-frequency motion sometimes shows excellent periodicity and stability and sometimes shows chaos. It is of practical guiding significance to know the mechanisms of some unusual noises as well as the design and manufacture of gear backlash.
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Ying, Hongwei, Lisha Zhang, Kanghe Xie, and Dazhong Huang. "Pore Pressure Response to Groundwater Fluctuations in Saturated Double-Layered Soil." Mathematical Problems in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/389089.
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Analytical solutions are developed for one-dimensional consolidation of double-layered saturated soil subjected to groundwater fluctuations. The solutions are derived by an explicit mathematical procedure using Duhamel’s theorem in conjunction with a Fourier series, when groundwater fluctuation is described by a general time-dependent function and assumed to be the pore water pressure variations at the upper boundary. Taking as an example the harmonic groundwater fluctuation, the relevant response of the excess pore water pressure is discussed in detail, and the main influencing factors of the excess pore pressure distribution are analyzed. A dimensionless parameterθhas been introduced because it significantly affects the phase and the amplitude of excess pore pressures. The influences of the coefficients of permeability and compressibility of soil on the excess pore pressure distribution are different and cannot be incorporated into the coefficient of consolidation in double-layered soil. The relative permeability ratio of two clayey soils also plays an important role on the curves of the distributions of the excess pore pressures. The effects of the thickness of the soil layer on the excess pore pressure distribution should be considered together with the dimensionless parameterθand the permeability and compressibility of the double-layered soil system.
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Stern, F., H. T. Kim, V. C. Patel, and H. C. Chen. "A Viscous-Flow Approach to the Computation of Propeller-Hull Interaction." Journal of Ship Research 32, no. 04 (December 1, 1988): 246–62. http://dx.doi.org/10.5957/jsr.1988.32.4.246.
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A model is developed to simulate two-dimensional laminar flow over an arbitrarily shaped body, a part of which is subjected to simple harmonic motion. The vibration amplitude ratio, Ho, and the Reynolds number, Re, are maintained at 0.1 and 1000, respectively. The Strouhal number, St, is varied in the range 0.0 ≤ St ≤ 1.0. The computer code is tested for the flow in a square cavity and also over a flat plate. The friction and pressure coefficients over the vibrating portion of the body are determined. Fast Fourier Transforms are performed on the time series data of these coefficients. For low-frequency vibrations (low Strouhal number) the pressure and friction coefficients match the steady-state results for flow over a sinusoidal bump. A small-amplitude pressure wave generated by the oscillating plate propagates downstream with the flow. For high-frequency vibrations (high Strouhal number) the pressure and friction coefficients over the vibrating portion of the body deviate from the steady-state results and a high-amplitude pressure wave propagates downstream. The pressure at one chord length upstream is also affected. As St increases, the flow becomes highly nonlinear and higher harmonics appear in the downstream flow. Subsequent analysis indicates that the nonlinearity is controlled by the term v(Əu/Əy).
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Venkat, N. Kolluru, and Malcolm Spaulding. "Numerical Simulation of Unsteady Laminar Flow Over Vibrating Bodies (Planar Flow)." Journal of Ship Research 35, no. 03 (September 1, 1991): 230–49. http://dx.doi.org/10.5957/jsr.1991.35.3.230.
Full textAbstract:
A model is developed to simulate two-dimensional laminar flow over an arbitrarily shaped body, a part of which is subjected to simple harmonic motion. The vibration amplitude ratio, Ho, and the Reynolds number, Re, are maintained at 0.1 and 1000, respectively. The Strouhal number, St, is varied in the range 0.0 ≤ St ≤ 1.0. The computer code is tested for the flow in a square cavity and also over a flat plate. The friction and pressure coefficients over the vibrating portion of the body are determined. Fast Fourier Transforms are performed on the time series data of these coefficients. For low-frequency vibrations (low Strouhal number) the pressure and friction coefficients match the steady-state results for flow over a sinusoidal bump. A small-amplitude pressure wave generated by the oscillating plate propagates downstream with the flow. For high-frequency vibrations (high Strouhal number) the pressure and friction coefficients over the vibrating portion of the body deviate from the steady-state results and a high-amplitude pressure wave propagates downstream. The pressure at one chord length upstream is also affected. As St increases, the flow becomes highly nonlinear and higher harmonics appear in the downstream flow. Subsequent analysis indicates that the nonlinearity is controlled by the term v(Əu/Əy).
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Li, Hong Liang. "Green's Function Solution of the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions." Advanced Materials Research 261-263 (May 2011): 863–67. http://dx.doi.org/10.4028/www.scientific.net/amr.261-263.863.
Full textAbstract:
In mechanical engineering, earthquake engineering and modern municipal construction, semi-cylindrical gap and shallow-buried inclusion structure are used widely. In this paper, Green's Function is studied, which is the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple shallow-buried inclusions while bearing anti-plane harmonic line source force at any point. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by semi-cylindrical gap and multiple cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition of the cylindrical inclusion in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. Green's function, that is, the total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gap , the cylindrical inclusions and the location of the line source force. Based on this solution, the problem of interaction of semi-cylindrical gap , multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.
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Zhang, Rui, and Hong Liang Li. "Interaction of Multiple Semi-Cylindrical Gaps and a Shallow-Buried Cavity." Advanced Materials Research 163-167 (December 2010): 3910–13. http://dx.doi.org/10.4028/www.scientific.net/amr.163-167.3910.
Full textAbstract:
In modern municipal construction and earthquake engineering, semi-cylindrical gap and shallow-buried cavity structure are used widely. In this paper, the solution of displacement field for elastic semi-space with multiple semi-cylindrical gaps and a shallow-buried cavity while bearing anti-plane harmonic line source force at any point is studied. In the complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field impacted by multiple semi-cylindrical gaps and a cylindrical cavity comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the stress-free condition of the gaps and the cylindrical cavity in the radial direction. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane harmonic line source force and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of semi-cylindrical gaps , the cylindrical cavity and the location of the line source force. Based on this solution, the problem of interaction of multiple semi-cylindrical gaps , a cylindrical cavity and a linear crack in semi-space can be investigated further.
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Harlander, U. "Towards an analytical understanding of internal wave attractors." Advances in Geosciences 15 (March 12, 2008): 3–9. http://dx.doi.org/10.5194/adgeo-15-3-2008.
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Abstract. Time harmonic inviscid internal wave motions constrained to fully closed domains generically lead to singular velocity fields. In spite of this difficulty, several techniques exist to solve such internal wave boundary value problems. Recently it has been shown that for a domain with the shape of a trapezium, solutions can be written in terms of a double sine Fourier series. However, the solutions were found by a numerical technique and thus not all coefficients of the series are available. Unfortunately, for questions related e.g. to regularization of the inviscid singular solutions, the knowledge of the asymptotic behavior of the spectrum for large wave numbers is essential. Here we discuss solutions of internal wave boundary value problems for which the spectra are known, at least asymptotically. We further describe shortcomings of the found solutions that need to be overcome in the future. Finally, we sketch applications of the solutions in the context of viscous energy dissipation.
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Mondal, M. Shahjahan, and Jahir Uddin Chowdhury. "Generation of 10-day flow of the Brahmaputra River using a time series model." Hydrology Research 44, no. 6 (December 5, 2012): 1071–83. http://dx.doi.org/10.2166/nh.2012.242.
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Time series models are used in hydrology for the generation of river flow data. The development of such a model, namely deseasonalized Autoregressive Moving Average (ARMA), for the generation of 10-day flows of the Brahmaputra River in Bangladesh is described. The model was fitted following systematic stages of identification, estimation and diagnostic checking of model building. A negative power transformation for the Brahmaputra flow was found to be necessary for model construction. The seasonality of the flow was removed by Fourier analysis using five harmonics for 10-day means and 13 harmonics for standard deviations. The fitted model was ARMA (1, 3) having one autoregressive parameter and three moving average parameters. The validation forecasts made with the model indicated that the deseasonalized ARMA model could capture the 10-day variability of the Brahmaputra flow reasonably well. To further validate and verify the model 200 synthetic flow sequences, each with a length of 50 years, were generated. The fitted ARMA model was found to be capable of preserving both short-term statistics (variance and autocorrelation) and long-term statistics (Hurst coefficient and rescaled adjusted range) of the historic Brahmaputra flow.
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Menshykov, Oleksandr, Vasyl Menshykov, and Olga Kladova. "NONLINEAR PROBLEM OF FRACTURE MECHANICS OF AN INTERFACE CRACK SUBJECTED TO SHEAR WAVE." Bulletin of the National Technical University "KhPI". Series: Mathematical modeling in engineering and technologies, no. 2 (November 30, 2021): 55–64. http://dx.doi.org/10.20998/2222-0631.2021.02.07.
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Solution for the problem for an interface crack under the action of a harmonic shear wave in normal direction is presented. The contact of the crack faces is put into consideration. The problem is solved by the boundary integral equations method, the vector components in the boundary integral equations are presented by Fourier series. The unilateral Signorini boundary conditions are involved to prevent the interpenetration of the crack faces and the emergence of tensile forces in the contact zone. Amonton-Coulomb Friction Law included allows to put into consideration relative resting of the crack opposite faces or their relative motion when one crack face is slipping or sliding across another face. The contact boundary restrictions are implemented using the iterative correction algorithm. The mathematical model adequacy is checked by comparing with classical model solution for statics problems that takes into account the crack faces contact. Numerical researches of friction influence on the displacement and contact forces distribution, size of contact zone are carried out. Influence of the faces contact and value of the friction coefficient on the distribution of stress intensity coefficients of normal rupture and transverse shear, which are the parameters of the biomaterial fracture, are presented and analyzed. It is shown that the nature of change in the distribution of the stress intensity coefficients for the conditions of tensile and shear waves is fundamentally different. It is concluded that it is possible to extend the approach proposed to the problems of three-dimensional fracture mechanics for composites with interfacial cracks at arbitrary dynamic loading.
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Lee, Cheng-Hsien, Jia-You Chen, Fang-Shou Lee, and Li-Chiu Chang. "A Combined O/U-Tube Oscillatory Water Tunnel for Fluid Flow and Sediment Transport Studies: The Hydrodynamics and Genetic Algorithm." Water 14, no. 11 (May 31, 2022): 1767. http://dx.doi.org/10.3390/w14111767.
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This study proposes an oscillatory water tunnel (O/U-tube) equipped with two impellers to drive flows. The O/U-tube consists of two modes: U-tube mode and O-tube mode. The former can generate oscillatory flow, while the latter can produce both oscillatory and unidirectional flows with velocities up to 1.6 m/s. The hydrodynamics of the U-tube and O-tube modes were examined analytically. For the U-tube mode, a sine-varying rotational speed for the impeller caused higher harmonic components for the velocity, owing to the local inertia and gravity forces. In the steady state of the O-tube mode, owing to the resistance force, the flow velocity was proportional to the rotational speed. To generate the target flow conditions, two open-loop control schemes were proposed according to the analytic approach for the U-tube mode and a genetic algorithm for the O-tube mode. The analytic approach was based on a hydrodynamic model with a given flow condition. In the genetic algorithm, the rotational speed was represented by a square root of the Fourier series. The optimal coefficients of the Fourier series to generate the target flow were determined by using the genetic algorithm and the hydrodynamic model. Both approaches were experimentally validated. Consequently, the O/U-tube with the open-loop schemes can be used to generate the desired oscillatory and unidirectional flows.
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Sutcliffe, P. R. "The development of a regional geomagnetic daily variation model using neural networks." Annales Geophysicae 18, no. 1 (January 31, 2000): 120–28. http://dx.doi.org/10.1007/s00585-000-0120-0.
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Abstract. Global and regional geomagnetic field models give the components of the geomagnetic field as functions of position and epoch; most utilise a polynomial or Fourier series to map the input variables to the geomagnetic field values. The only temporal variation generally catered for in these models is the long term secular variation. However, there is an increasing need amongst certain users for models able to provide shorter term temporal variations, such as the geomagnetic daily variation. In this study, for the first time, artificial neural networks (ANNs) are utilised to develop a geomagnetic daily variation model. The model developed is for the southern African region; however, the method used could be applied to any other region or even globally. Besides local time and latitude, input variables considered in the daily variation model are season, sunspot number, and degree of geomagnetic activity. The ANN modelling of the geomagnetic daily variation is found to give results very similar to those obtained by the synthesis of harmonic coefficients which have been computed by the more traditional harmonic analysis of the daily variation.Key words. Geomagnetism and paleomagnetism (time variations; diurnal to secular) · Ionosphere (modelling and forecasting)
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Knaack, R., and J. O. Stenflo. "Large Scale Solar Magnetic Fields: Temporal Variations." Symposium - International Astronomical Union 219 (2004): 552–56. http://dx.doi.org/10.1017/s0074180900182580.
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We have investigated the temporal evolution of the solar magnetic field during solar cycles 20, 21 and 22 by means of spherical harmonic decomposition and subsequent time series analysis. A 33 yr and a 25 yr time series of daily magnetic maps of the solar photosphere, recorded at the Mt. Wilson and NSO/Kitt Peak observatories respectively, were used to calculate the spherical coefficients of the radial magnetic field. Fourier and wavelet analysis were then applied to deduce the temporal variations. We compare the results of the two datasets and present examples of zonal modes which show significant variations, e. g. with a period of approx. 2.0—2.5 years. We provide evidence that this quasi-biennial oscillation originates mainly from the southern hemisphere. Furthermore, we show that low degree modes with odd l — m exhibit periods of 29.2 and 28.1 days while modes with even l — m show a dominant period of 26.9 days. A resonant modal structure of the solar magnetic field (apart from the 22 yr cycle) has not been found.
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Tyshko, S., O. Lavrut, V. Smolar, O. Zabula, and Yu Chernichenko. "Application of spectral analysis for determine the phase shift of signals with equal amplitudes using full-wave transformation when measuring the characteristics of weapons." Military Technical Collection, no. 25 (December 8, 2021): 12–23. http://dx.doi.org/10.33577/2312-4458.25.2021.12-23.
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The article defines the list of technical characteristics of armaments and military equipment (ARM), the value of which is measured using phase methods. An analysis of known methods that have found wide application in measuring technology, which is designed to determine the technical characteristics associated with the measurement of phase shift during the development, manufacture and operation of weapons. Based on this analysis, it was determined that the measuring systems are designed to determine the phase shift of two harmonic signals in their composition have two channels of information transmission. This architecture of the implementation of measuring systems leads to the fact that a significant impact on the accuracy of the proposed measurement problem, makes a component of the error due to the phase symmetry of the signal transmission channels, as well as internal and external noise. As an alternative approach to solving the measurement problem of determining the phase shift of two harmonic signals, which will significantly reduce the error component due to phase asymmetry of information transmission channels, it is proposed to use the signal obtained by summing harmonic signals after full-wave transformation followed by spectral analysis. In order to implement the above approach, a measurement problem was set to determine the phase shift of two harmonic signals, using spectral analysis of the signal obtained by summing the harmonic signals after their full-wave transformation. A list of assumptions required for the synthesis of analytical relations that establish the relationship between the spectra of phases and amplitudes (power) of the signal obtained by summing harmonic signals after their full-wave transformation and phase shift of two harmonic signals. Analytical relationships are proposed that establish the relationship between the above characteristics. It is shown that the values of the spectrum of phases and amplitudes, which are calculated using the proposed expressions, differ from the values obtained in the calculations using the Fourier series coefficients, not more than 0.1%.
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Yudin, Anatoly S. "Effective Method for Constructing Amplitude-Frequency Shell Characteristics." UNIVERSITY NEWS. NORTH-CAUCASIAN REGION. NATURAL SCIENCES SERIES, no. 4-1 (216-1) (December 28, 2022): 65–74. http://dx.doi.org/10.18522/1026-2237-2022-4-1-65-74.
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Timoshenko type equations are used, taking into account the transverse shift. The shell has the properties of constructive anisotropy. Forced oscillations in the frequency range, including the first significant low-frequency resonances, are considered. The internal scattering of vibrational energy is taken into account by the method of complex amplitudes. The influence of a number of parameters on the amplitude-frequency characteristics (frequency response) of input compliance is investigated. A semi-analytical method of constructing a solution is used. In the case of a cylindrical shell freely supported at the ends, Fourier series are used in two coordinates. The problem of forced oscillations is solved by decomposing the amplitudes of displacements according to their own forms of oscillations. At the same time, the separation of equations for determining coefficients by harmonic numbers is provided. The algorithm on this basis allows you to quickly build the amplitudefrequency characteristics necessary for analysis.
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Zhao, Qin, Xiaona Fan, Qing Wang, Guochen Sang, and Yiyun Zhu. "Research on Energy-Saving Design of Rural Building Wall in Qinba Mountains Based on Uniform Radiation Field." Mathematical Problems in Engineering 2020 (September 2, 2020): 1–16. http://dx.doi.org/10.1155/2020/9786895.
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How to create a healthy and comfortable indoor environment without causing a substantial increase in energy consumption has become a strategic problem that the development of all countries must face and solve. According to the climatic conditions of Qinba Mountains in China, combined with the characteristics of local rural residential buildings and residents’ living habits, the field survey and theoretical analysis were used to study the thermal environment status and the heating energy consumption condition of local rural residential buildings. The thermal design method of walls for the local rural energy-saving buildings based on the indoor uniform radiation field was explored by using the outdoor comprehensive temperature function expressed by the fourth-order harmonic Fourier series as the boundary condition of the wall thermal analysis. ANSYS CFX was adopted to study the suitability of the energy-saving wall structure designed by the above method. The results show that the indoor thermal environment of local rural residential buildings in winter is not ideal and the heating energy consumption is high, but this area has the geographical advantage to develop solar energy buildings. It is proposed that the indoor thermal comfort temperature of local rural residential buildings in winter should not be lower than 14°C. When the internal surface temperature of the external walls in different orientations are equally based on the design principle of uniform radiation field, the heat transfer coefficient of the east wall, the west wall, and the north wall of the local rural residential buildings is 1.13 times, 1.06 times, and 1.14 times of the south wall heat transfer coefficient, respectively. The energy-saving structural wall with KPI porous brick as the main material and the south wall heat transfer coefficient of 0.9 W/(m2·K) is the most suitable energy-saving wall for local rural residential buildings.
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Maury, C., G. Delhommeau, M. Ba, J. P. Boin, and M. Guilbaud. "Comparison Between Numerical Computations and Experiments for Seakeeping on Ship Models With Forward Speed." Journal of Ship Research 47, no. 04 (December 1, 2003): 347–64. http://dx.doi.org/10.5957/jsr.2003.47.4.347.
Full textAbstract:
The recent progress of computers and numerical algorithms enables us today to use the translating and pulsating Green function in panel methods for seakeeping calculations. Two different methods of calculations of this function, its derivatives and their integrations over panels or segments, are briefly presented and have been introduced into the seakeeping codes Aquaplus developed at Ecole Centrale de Nantes and Poseidon at Laboratoire d'Etudes Aerodynamiques (LEA), Centre d'Etudes Aerodynamiques et Thermiques (CEAT). Both codes interchange the Fourier and boundary integrals on panels or waterline segments, the last part being performed analytically. These methods have been used to compute flows around Wigley or Series 60 model ships. To check the numerical results, an experimental setup has been developed at the CEAT that measures forces and moments on a model in forced harmonic oscillations of pitch or heave. Tests have been performed in the recirculating water channel of Ecole Centrale de Nantes on two L = 1.2 m Series 60 models of CB = 0.6 and 0.8 block coefficients. Unsteady wave patterns have been recorded using a resistive wave probe. The experimental results are compared with the numerical ones.
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Lalanne, Bernard, and Maurice Touratier. "Aeroelastic Vibrations and Stability in Cyclic Symmetric Domains." International Journal of Rotating Machinery 6, no. 6 (2000): 445–52. http://dx.doi.org/10.1155/s1023621x00000403.
Full textAbstract:
This paper deals with rotating cyclic symmetric structures, immersed in light fluid flow. Firstly, general and usual cyclic symmetric properties are recovered from the Floquet theorem for differential equations, having space periodic coefficients in conjunction with a discrete space Fourier series development. The approach for aeroelastic problem having cyclic symmetry is then formulated based on the twin mode approach. In addition to modal one, rotating and stationary wave bases are introduced to derive the equilibrium equations for a non-dissipative system subjected to aerodynamic loading. Rotating wave basis is the natural one, and it also permits consistently to prescribe the aerodynamic pressure on the boundary between the fluid and the structure. The aerodynamic load is then derived from a harmonic analysis of the fluid flow extending to turbomachinery as in the case of aeroplane wing. In this way, aerodynamic forces may be obtained as general as required, depending on successive time derivatives of degrees of freedom in addition to themselves. Finally, some special cases are given and stability is studied for a cyclic periodic blade assembly, even when mistuning between sectors can occur.
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Gandilyan, D. V. "The influence of surface effects in the problem of theory of elasticity for a circular hole in a half-plane." PNRPU Mechanics Bulletin, no. 4 (December 15, 2020): 250–59. http://dx.doi.org/10.15593/perm.mech/2020.4.21.
Full textAbstract:
Surface effects are important for modeling structures, such as nanofilms, nanoporous materials, and other nanoscale constructions. In the current study, we consider the problem of the theory of elasticity - the problem of a half-plane containing a circular hole, stretched by constant stresses applied at infinity, and take into account surface effects such as surface elasticity and surface stresses. The problem solution has been obtained by expanding the Fourier series with the variables written in the bipolar coordinate system (which simplifies the problem solution because one of the coordinates becomes a constant on the hole contour), where the stress components are expressed through a bi-harmonic stress function. The parametric coefficients involved in the solution, namely in the Fourier series, are determined in order to satisfy the boundary conditions on the hole contour. To solve the problem, in addition to the equations of the theory of elasticity, the equations of surface elasticity were used, in particular, by applying the generalized Young-Laplace’s law and the Shuttleworth’s law; the surface stress on the hole contour has been calculated directly. Using recurrence relations for the stress components at the boundary, stress concentration values have been obtained. The resulting expressions can be considered as a generalized solution of the problem in case of the classical elasticity. The stress concentrations are compared for the cases with and without taking into account surface effects at various points on the hole contour. The contribution caused by the surface effects depending on the relative distance between the hole and the half-plane boundary is studied. It is shown that despite a quite simple geometry, owing to the fairly small distance between the hole and the half-plane boundary, the stress concentration with and without taking into account the surface stress are significantly different from each other, due to the significant contribution of surface effects.
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Mondal, M. Shahjahan, and Jahir Uddin Chowdhury. "Synthetic Stream-Flow Generation with Deseasonalized ARMA Model." Journal of Hydrology and Meteorology 8, no. 1 (August 30, 2016): 32–45. http://dx.doi.org/10.3126/jhm.v8i1.15570.
Full textAbstract:
Time series models are used in river hydrology for synthetic generation. The development of such a model, namely deseasonalized Autoregressive Moving Average (ARMA), for generation of decadal (10-day) flows of the Brahmaputra River in Bangladesh is described. The model was fitted following systematic stages of identification, estimation and diagnostic checking of model building. A negative power transformation for the Brahmaputra flow was found to be necessary for model construction. The seasonality of the flow was removed by Fourier analysis using 5 harmonics for decadal means and 13 harmonics for standard deviations. The fitted model was ARMA (1, 3) having one autoregressive parameter and three moving average parameters. The validation forecasts made with the model indicated that the deseasonalized ARMA model could capture the decadal variability of the Brahmaputra flow reasonably well. Two hundred synthetic flow sequences, each with a length of 50 years, were generated using this model to further validate and verify the model. The fitted ARMA model was found to be capable of preserving both short-term statistics (variance and autocorrelation) and long-term statistics (Hurst coefficient and rescaled adjusted range) of the historic Brahmaputra flow.Journal of Hydrology and Meteorology, Vol. 8(1) p.32-46