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Revisiting the space-time gradient method:A time-clocking perspective, high order difference time discretization and comparison with the harmonic balance method

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Chinese Journal of Aeronautics 2022年第11期35卷 45-58页

作者： Boqian WANG Dingxi WANG Mohammad RAHMATI Xiuquan HUANG School of Power and Energy Northwestern Polytechnical UniversityXi'an 710072China Department of Mechanical and Construction Engineering Northumbria UniversityNewcastle upon Tyne NEI 8STUnited Kingdom

This paper revisits the Space-Time Gradient(STG) method which was developed for efficient analysis of unsteady flows due to rotor–stator interaction and presents the method from an alternative time-clocking perspective. The STG method requires reordering of blade passages according to their relative clocking positions with respect to blades of an adjacent blade row. As the space-clocking is linked to an equivalent time-clocking, the passage reordering can be performed according to the alternative time-clocking. With the time-clocking perspective, unsteady flow solutions from different passages of the same blade row are mapped to flow solutions of the same passage at different time instants or phase angles. Accordingly, the time derivative of the unsteady flow equation is discretized in time directly, which is more natural than transforming the time derivative to a spatial one as with the original STG method. To improve the solution accuracy, a ninth order difference scheme has been investigated for discretizing the time derivative. To achieve a stable solution for the high order scheme, the implicit solution method of Lower-Upper Symmetric GaussSeidel/Gauss-Seidel(LU-SGS/GS) has been employed. The NASA Stage 35 and its blade-countreduced variant are used to demonstrate the validity of the time-clocking based passage reordering and the advantages of the high order difference scheme for the STG method. Results from an existing harmonic balance flow solver are also provided to contrast the two methods in terms of solution stability and computational cost.

关键词： harmonic balance method High order difference scheme Passage reordering Space-time gradient method Unsteady flows

Almost Periodic Solutions by the harmonic balance method

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Acta Mathematicae Applicatae Sinica 2022年第4期38卷 943-954页

作者： Ji-feng CHU Zuo-huan ZHENG Zhe ZHOU Department of Mathematics Shanghai Normal UniversityShanghai 200234China Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China College of Mathematics and Statistics Hainan Normal UniversityHaikou 571158China

We consider non-autonomous ordinary differential equations in two *** is the one-dimensional case that admits a condition of hyperbolicity,and the other one is the higher-dimensional case that admits an exponential *** differential equations of this kind,we give a rigorous treatment of the harmonic balance method to look for almost periodic solutions.

关键词： almost periodic solution harmonic balance method exponential dichotomy

Bifurcation in a 3-DOF Airfoil with Cubic Structural Nonlinearity

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Chinese Journal of Aeronautics 2011年第3期24卷 265-278页

作者： Saied IRANI Hamid SARRAFZADEH Mohammad Reza AMOOZGAR Department of Aerospace Engineering K.N.Toosi University of Technology

Limit cycle oscillations （LCOs） as well as nonlinear aeroelastic analysis of a 3-DOF aeroelastic airfoil motion with cubic restoring moments in the pitch degree of freedom are investigated. Aeroelastic equations of an airfoil with control surface in an incompressible potential flow are presented in the time domain. The harmonic balance （HB） method is utilized to calculate the LCO frequency and amplitude for the airfoil. Also the semi-analytical method has revealed the presence of stable and unstable limit cycles, along with stability reversal in the neighborhood of a Hopf bifurcation. The system response is determined by nu- merically integrating the governing equations using a standard Runge-Kutta algorithm and the obtained results are compared with the HB method. Also the results by the third order HB （HB3） method for control surface are consistent with the other numerical solution. Finally, by combining the numerical and the HB methods, types of bifiarcation, be it supercritical, subcritical, or diver- gent flutter area are identified.

关键词： flutter bifurcation cubic nonlinearity limit cycle oscillations harmonic balance method

Forced response of quadratic nonlinear oscillator: comparison of various approaches

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Applied Mathematics and Mechanics(English Edition) 2015年第11期36卷 1403-1416页

作者： Wenan JIANG Guoce ZHANG Liqun CHEN Shanghai Institute of Applied Mathematics and Mechanics Shanghai University College of Sciences Shanghai University Shanghai Key Laboratory of Mechanics in Energy Engineering Shanghai University

The primary resonances of a quadratic nonlinear system under weak and strong external excitations are investigated with the emphasis on the comparison of dif- ferent analytical approximate approaches. The forced vibration of snap-through mecha- nism is treated as a quadratic nonlinear oscillator. The Lindstedt-Poincar method, the multiple-scale method, the averaging method, and the harmonic balance method are used to determine the amplitude-frequency response relationships of the steady-state responses. It is demonstrated that the zeroth-order harmonic components should be accounted in the application of the harmonic balance method. The analytical approximations are com- pared with the numerical integrations in terms of the frequency response curves and the phase portraits. Supported by the numerical results, the harmonic balance method pre- dicts that the quadratic nonlinearity bends the frequency response curves to the left. If the excitation amplitude is a second-order small quantity of the bookkeeping parameter, the steady-state responses predicted by the second-order approximation of the Lindstedt- Poincar method and the multiple-scale method agree qualitatively with the numerical results. It is demonstrated that the quadratic nonlinear system implies softening type nonlinearity for any quadratic nonlinear coefficients.

关键词： forced vibration quadratic nonlinearity primary resonance multiple-scalemethod Lindstedt-Poincar~ method averaging method harmonic balance method

Nonlinear vibration of functionally graded graphene platelet-reinforced composite truncated conical shell using first-order shear deformation theory

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Applied Mathematics and Mechanics(English Edition) 2021年第7期42卷 981-998页

作者： Shaowu YANG Yuxin HAO Wei ZHANG Li YANG Lingtao LIU College of Mechanical Engineering Beijing Information Science and Technology UniversityBeijing 100192China Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures College of Mechanical EngineeringBeijing University of TechnologyBeijing 100124China

In this study,the first-order shear deformation theory(FSDT)is used to establish a nonlinear dynamic model for a conical shell truncated by a functionally graded graphene platelet-reinforced composite(FG-GPLRC).The vibration analyses of the FG-GPLRC truncated conical shell are *** the graphene platelets(GPLs)of the FG-GPLRC truncated conical shell with three different distribution patterns,the modified Halpin-Tsai model is used to calculate the effective Young’s ***’s principle,the FSDT,and the von-Karman type nonlinear geometric relationships are used to derive a system of partial differential governing equations of the FG-GPLRC truncated conical *** Galerkin method is used to obtain the ordinary differential equations of the truncated conical ***,the analytical nonlinear frequencies of the FG-GPLRC truncated conical shell are solved by the harmonic balance *** effects of the weight fraction and distribution pattern of the GPLs,the ratio of the length to the radius as well as the ratio of the radius to the thickness of the FG-GPLRC truncated conical shell on the nonlinear natural frequency characteristics are *** study culminates in the discovery of the periodic motion and chaotic motion of the FG-GPLRC truncated conical shell.

关键词： nonlinear free vibration harmonic balance method functionally graded graphene platelet-reinforced composite(FG-GPLRC) truncated conical shell chaos

Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations

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Acta Mechanica Sinica 2018年第3期34卷 561-577页

作者： Fei Fang Guanghui Xia Jianguo Wang School of Civil and Hydraulic Engineering Hefei University of Technology

The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler-Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton＇s principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency-response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency-response curves. We also study the difference between the nonlinear lumped-parameter and distributed- parameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested. We also illustrate that the damping and load resistance affect the initiation excitation threshold.

关键词： Parametric and external excitations Nonlinear distributed parameter model Nonlinear dynamic response Energy harvesting harmonic balance method

Nonlinear vibration analysis of pipeline considering the effects of soft nonlinear clamp

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Applied Mathematics and Mechanics(English Edition) 2022年第10期43卷 1555-1568页

作者： Weijiao CHEN Yiming CAO Xumin GUO Hui MA Bangchun WEN Bo WANG School of Mechanical Engineering and Automation Northeastern UniversityShenyang110819China Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China Northeastern UniversityShenyang110819China

Soft nonlinear support is a major engineering project,but there are few relevant *** this paper,a dynamic pipeline model with soft nonlinear supports at both ends is *** considering the influence of the Coriolis force and centrifugal force,the dynamical coupling equation of fluid-structure interaction is derived with extended Hamilton’s ***,the approximate analytical solutions are sought via the harmonic balance *** amplitude-frequency response curves show that different effects can be determined by approximate *** is demonstrated that the increase in the fluid velocity can increase the amplitude of the pipeline *** frequency range of unstable response increases when the fluid pressure *** combination of the soft nonlinear clamp and the large geometrical deformation of the pipeline affects the nonlinear vibration characteristic of the system,and the external excitation force and damping have significant effects on the stability.

关键词： clamp soft nonlinear support harmonic balance method fluid-structure interaction pipeline vibration analysis

harmonic balance simulation of the influence of component uniformity and reliability on the performance of a Josephson traveling wave parametric amplifier

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Chinese Physics B 2024年第4期33卷 339-343页

作者：郑煜臻熊康林冯加贵杨辉 School of Nano-Tech and Nano-Bionics University of Science and Technology of ChinaHefei 230026China Suzhou Institute of Nano-Tech and Nano-Bionics Chinese Academy of SciencesSuzhou 215123China

A Josephson traveling wave parametric amplifier(JTWPA),which is a quantum-limited amplifier with high gain and large bandwidth,is the core device of large-scale measurement and control systems for quantum computing.A typical JTWPA consists of thousands of Josephson junctions connected in series to form a transmission line and hundreds of shunt LC resonators periodically loaded along the line for phase *** the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing *** guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the *** this paper,we use the harmonic balance method to conduct a comprehensive study on the impact of nonuniformity and fabrication yield of the electronic components on the performance of a *** results provide insightful and scientific guidance for device design and fabrication processes.

关键词： Josephson traveling wave parametric amplifier(JTWPA) harmonic balance method yields uniformity

A methodology for Assessing Axial Compressor Stability with Inlet Temperature Ramp Distortion

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Journal of Thermal Science 2024年第3期33卷 856-871页

作者： SUN Dakun GU Benhao NING Fangfei FANG Yibo DONG Xu XU Dengke SUN Xiaofeng School of Energy and Power Engineering Beihang UniversityBeijing102206China Research Institute of Aero-engine Beihang UniversityBeijing102206China

Based on a small perturbation stability model for periodic flow,the effects of inlet total temperature ramp distortion on the axial compressor are investigated and the compressor stability is quantitatively *** the beginning,a small perturbation stability model for the periodic flow in compressors is proposed,referring to the governing equations of the harmonic balance *** stability model is validated on a single-stage low-speed compressor TA36 with uniform inlet ***,the unsteady flow of TA36 with different inlet total temperature ramps and constant back pressure is simulated based on the harmonic balance *** on these simulations,the compressor stability is analyzed using the proposed small perturbation ***,the Dynamic Mode Decomposition method is employed to accurately extract pressure *** two parameters of the temperature ramp,ramp rate and Strouhal number,are discussed in this *** results indicate the occurrence and extension of hysteresis loops in the rows,and a decrease in compressor stability with increasing ramp *** performance is divided into two phases,stable and limit,based on the ramp ***,the model predictions suggest that a decrease in period length and an increase in Strouhal number lead to improved compressor *** DMD results imply that for compressors with inlet temperature ramp distortion,the increase of high-order modes and oscillations at the rotor tip is always the signal of decreasing stability.

关键词： axial compressor stability small perturbation model inlet temperature ramp distortion harmonic balance method dynamic mode decomposition method