This paper presents a programmable frequency scan algorithm based on harmonic balance.The core idea involves treating systems under perturbation as nonlinear time-periodic(NTP)systems.Steady-state harmonics are first ...This paper presents a programmable frequency scan algorithm based on harmonic balance.The core idea involves treating systems under perturbation as nonlinear time-periodic(NTP)systems.Steady-state harmonics are first solved via Newton-Raphson iteration through a set of nonlinear equations,and then input-output variables are selected to estimate the linear transfer function of the original NTP system without perturbations.The applications and insights of the proposed algorithm are discussed,particularly in guiding existing frequency scan algorithms,which are restricted by time-domain signal generation or measurement.This improvement is achieved through linear stability analysis of NTP systems with perturbations.展开更多
Incorporating asymmetric quadratic and cubic stiffnesses into a time-delayed Duffing oscillator provides a more accurate representation of practical systems,where the resulting nonlinearities critically influence subh...Incorporating asymmetric quadratic and cubic stiffnesses into a time-delayed Duffing oscillator provides a more accurate representation of practical systems,where the resulting nonlinearities critically influence subharmonic resonance phenomena,yet comprehensive investigations remain limited.This study employs the generalized harmonic balance(HB)method to conduct an analytical investigation of the subharmonic resonance behavior in asymmetric stiffness nonlinear systems with time delay.To further examine the switching behavior between primary and subharmonic resonances,a numerical continuation approach combining the shooting method and the parameter continuation algorithm is developed.The analytical and numerical continuation solutions are validated through direct numerical integration.Subsequently,the switching behavior and associated bifurcation points are analyzed by means of the numerical continuation results in conjunction with the Floquet theory.Finally,the effects of delay parameters on the existence range of subharmonic responses are discussed in detail,and the influence of initial conditions on system dynamics is explored with basin of attraction plots.This work establishes a comprehensive framework for the analytical and numerical study on time-delayed nonlinear systems with asymmetric stiffness,providing valuable theoretical insights into the stability management of such dynamic systems.展开更多
A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve...A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve elastic collision.However,since the system is non-smooth,the required Fourier/harmonic truncation order is high in order to achieve the theoretical convergence rate,resulting in expensive computational cost.Furthermore,for rigid body collision,the periodic response of the system cannot be solved with the AFT-HB method due to the discontinuous velocity of the system.In order to accelerate the convergence and solve highly non-smooth systems,an enriched harmonic balance(HB)method is proposed,which is derived from the AFT-HB method in the framework of event-driven Gauss quadrature.The basic idea is to augment the Fourier bases by introducing a non-smooth Bernoulli base such that the non-smooth Bernoulli base compensates for the non-smooth part of the solution and the smooth part of the solution is approximated by the Fourier bases,thus achieving accelerated convergence.Based on the enriched HB method,gear pair systems with gear backlash and oscillator systems with rigid impact are solved,and the dynamic response characteristics are analyzed in this work.Then,based on the Floquet theory,the event-driven monodromy matrix method for non-smooth systems is used to analyze the stability and bifurcation of the periodic solutions.The numerical example shows that the results obtained from the enriched HB method are consistent with those from the Runge-Kutta method,which proves that the presented method is an effective method for analyzing the dynamic response characteristic of the vibro-impact system.展开更多
Quasi-periodic solutions with multiple base frequencies exhibit the feature of 2π-periodicity with respect to each of the hyper-time variables.However,it remains a challenge work,due to the lack of effective solution...Quasi-periodic solutions with multiple base frequencies exhibit the feature of 2π-periodicity with respect to each of the hyper-time variables.However,it remains a challenge work,due to the lack of effective solution methods,to solve and track the quasi-periodic solutions with multiple base frequencies until now.In this work,a multi-steps variable-coefficient formulation is proposed,which provides a unified framework to enable either harmonic balance method or collocation method or finite difference method to solve quasi-periodic solutions with multiple base frequencies.For this purpose,a method of alternating U and S domain is also developed to efficiently evaluate the nonlinear force terms.Furthermore,a new robust phase condition is presented for all of the three methods to make them track the quasi-periodic solutions with prior unknown multiple base frequencies,while the stability of the quasi-periodic solutions is assessed by mean of Lyapunov exponents.The feasibility of the constructed methods under the above framework is verified by application to three nonlinear systems.展开更多
A novel vibration isolation system designed for superior performance in low-frequency environments is proposed in this work.The isolator is based on a unique hexagonal arrangement of linear springs,allowing for an adj...A novel vibration isolation system designed for superior performance in low-frequency environments is proposed in this work.The isolator is based on a unique hexagonal arrangement of linear springs,allowing for an adjustable geometric configuration via the initial inclination angle.Based on the principle of Lagrangian mechanics,the equation of motion governing the structural dynamics is rigorously derived.The system is modeled as a strongly nonlinear single-degree-of-freedom dynamical system,loaded with a normalized payload and subject to harmonic base excitation.To analyze the steady-state response,the harmonic balance method is employed,providing accurate predictions of the payload's vibration amplitude and displacement transmissibility as functions of both the base excitation amplitude and frequency.The analysis reveals a direct relationship between the isolator's geometric and stiffness parameters and its load-bearing capacity,leading to the identification of three distinct operational regimes.Depending on the unloaded initial inclination angle,the equivalent stiffness ratio,and the payload design configuration,the system can exhibit one of three vibration isolation modes:(i)the quasizero stiffness(QZS)isolation mode,(ii)the zero linear stiffness with controllable nonlinear stiffness,and(iii)the full-band perfect zero stiffness.The vibration isolation performance of the proposed structure is thoroughly discussed for all three oscillation modes in terms of frequency response curves,displacement transmissibility,and time-domain responses.The key novel finding is that this structure can operate as a full-band,high-performance vibration isolator when the initial inclination angle is designed to be a right angle,enabling full isolation of the maximum possible payload.Moreover,the analytical results and numerical simulations demonstrate that the isolator's displacement transmissibility T with the unit dB tends to-∞as the air-damping coefficient approaches zero,enabling ideal vibration isolation across the entire excitation frequency range.These analytical insights are validated through comprehensive numerical simulations,which show excellent agreement with the theoretical predictions.展开更多
Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear ...Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.展开更多
In this paper, period-doubling bifurcation in a two-stage power factor correction converter is analyzed by using the method of incremental harmonic balance （IHB） and Floquet theory. A two-stage power factor correcti...In this paper, period-doubling bifurcation in a two-stage power factor correction converter is analyzed by using the method of incremental harmonic balance （IHB） and Floquet theory. A two-stage power factor correction converter typically employs a cascade configuration of a pre-regulator boost power factor correction converter with average current mode control to achieve a near unity power factor and a tightly regulated post-regulator DC-DC Buck converter with voltage feedback control to regulate the output voltage. Based on the assumption that the tightly regulated postregulator DC-DC Buck converter is represented as a constant power sink and some other assumptions, the simplified model of the two-stage power factor correction converter is derived and its approximate periodic solution is calculated by the method of IHB. And then, the stability of the system is investigated by using Floquet theory and the stable boundaries are presented on the selected parameter spaces. Finally, some experimental results are given to confirm the effectiveness of the theoretical analysis.展开更多
As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for ach...As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for achieving high controlling precision is to search for an appropriate time delay.This paper proposes a simple yet effective approach,based on incremental harmonic balance method,to determine the optimal time delay in the delayed feedback controller.The time delay is adjusted within the iterative scheme provided by the proposed method,and finally converges to the period of the target UPO.As long as the optimal time delay is fixed,moreover,the attained solution makes it quite convenient to analyze its stability according to the Floquet theory,which further provides the effective interval of the feedback gain.展开更多
We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial va...We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.展开更多
The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to ...The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.展开更多
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 perspecti...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.展开更多
The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered...The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.展开更多
The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability c...The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.展开更多
In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, whi...In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.展开更多
The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The a...The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.展开更多
The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with b...The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle.A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method.A high-dimensional model of the buckled beam is derived,concerning nonlinear coupling.The incremental harmonic balance(IHB)method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve,and the Floquet theory is used to analyze the stability of the periodic solutions.Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited.Bifurcations including the saddle-node,Hopf,perioddoubling,and symmetry-breaking bifurcations are observed.Furthermore,quasi-periodic motion is observed by using the fourth-order Runge-Kutta method,which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.展开更多
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 ...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 matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In 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 JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.展开更多
The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components, the nonlinear dynamic equation of an elli...The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components, the nonlinear dynamic equation of an elliptical sandwich plate under a harmonic force is derived. A superpositive-iterative harmonic balance (SIHB) method is presented for the steady-state analysis of strongly nonlinear oscillators. In a periodic oscillation, the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as fundamental differential equation with fundamental harmonics and incremental differential equation with derived harmonics. The 1/3 subharmonic solution of an elliptical sandwich plate is investigated by using the methods of SIHB. The SIHB method is compared with the numerical integration method. Finally, asymptotical stability of the 1/3 subharmonic oscillations is inspected.展开更多
The response of visco_elastic system to combined deterministic harmonic and random excitation was investigated. The method of harmonic balance and the method of stochastic averaging were used to determine the response...The response of visco_elastic system to combined deterministic harmonic and random excitation was investigated. The method of harmonic balance and the method of stochastic averaging were used to determine the response of the system. The theoretical analysis was verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increase, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions and jumps may exist.展开更多
This paper analyzes the nonlinear dynamic characteristics and stability of Aero-Engine Dual-Rotor(AEDR)systems under high-frequency excitation,based on the Adaptive Harmonic Balance with the Asymptotic Harmonic Select...This paper analyzes the nonlinear dynamic characteristics and stability of Aero-Engine Dual-Rotor(AEDR)systems under high-frequency excitation,based on the Adaptive Harmonic Balance with the Asymptotic Harmonic Selection(AHB-AHS)method.A finite element dynamic equation for the AEDR system is introduced,considering complex nonlinearities of the intershaft bearing,unbalanced excitations,and high-frequency excitation.A solving strategy combining the AHB-AHS method and improved arclength continuation method is proposed to solve highdimensional dynamic equations containing complex nonlinearities and to track periodic solutions with parameter variations.The Floquet theory is used to analyze the types of bifurcation points in the system and the stability of periodic motions.The results indicate that high-frequency excitation can couple high-order and low-order modes,especially when the system undergoes superharmonic resonance.High-frequency excitation leads to more combination frequency harmonics,among which N_(f)ω_(1)-2ω_(2)dominates.Furthermore,changing the parameters(amplitude and frequency)of high-frequency excitation widens or shifts the unstable regions of the system.These findings contribute to understanding the mechanism of high-frequency excitation on aero-engines and demonstrate that the proposed AHB-AHS method is a powerful tool for analyzing highdimensional complex nonlinear dynamic systems under multi-frequency excitation.展开更多
基金supported by China Southern Power Grid Corporation(036000KC23090005(GDKJXM20231027)).
文摘This paper presents a programmable frequency scan algorithm based on harmonic balance.The core idea involves treating systems under perturbation as nonlinear time-periodic(NTP)systems.Steady-state harmonics are first solved via Newton-Raphson iteration through a set of nonlinear equations,and then input-output variables are selected to estimate the linear transfer function of the original NTP system without perturbations.The applications and insights of the proposed algorithm are discussed,particularly in guiding existing frequency scan algorithms,which are restricted by time-domain signal generation or measurement.This improvement is achieved through linear stability analysis of NTP systems with perturbations.
基金Project supported by the National Natural Science Foundation of China(Nos.U24B2062,520754285247051087)+1 种基金the Two-chain Fusion High-end Machine Tool Project of Shaanxi Province of China(No.2021LLRh-01-02)the Youth Fund of the National Natural Science Foundation of China(No.52205281)。
文摘Incorporating asymmetric quadratic and cubic stiffnesses into a time-delayed Duffing oscillator provides a more accurate representation of practical systems,where the resulting nonlinearities critically influence subharmonic resonance phenomena,yet comprehensive investigations remain limited.This study employs the generalized harmonic balance(HB)method to conduct an analytical investigation of the subharmonic resonance behavior in asymmetric stiffness nonlinear systems with time delay.To further examine the switching behavior between primary and subharmonic resonances,a numerical continuation approach combining the shooting method and the parameter continuation algorithm is developed.The analytical and numerical continuation solutions are validated through direct numerical integration.Subsequently,the switching behavior and associated bifurcation points are analyzed by means of the numerical continuation results in conjunction with the Floquet theory.Finally,the effects of delay parameters on the existence range of subharmonic responses are discussed in detail,and the influence of initial conditions on system dynamics is explored with basin of attraction plots.This work establishes a comprehensive framework for the analytical and numerical study on time-delayed nonlinear systems with asymmetric stiffness,providing valuable theoretical insights into the stability management of such dynamic systems.
基金Project supported by the National Natural Science Foundation of China(No.12372028)the Guangdong Basic and Applied Basic Research Foundation(No.2022A1515011809)。
文摘A vibro-impact system is a hot topic in the study on nonlinear dynamics due to its generality and importance in engineering.In general,the alternating frequency-time harmonic balance(AFT-HB)method can be used to solve elastic collision.However,since the system is non-smooth,the required Fourier/harmonic truncation order is high in order to achieve the theoretical convergence rate,resulting in expensive computational cost.Furthermore,for rigid body collision,the periodic response of the system cannot be solved with the AFT-HB method due to the discontinuous velocity of the system.In order to accelerate the convergence and solve highly non-smooth systems,an enriched harmonic balance(HB)method is proposed,which is derived from the AFT-HB method in the framework of event-driven Gauss quadrature.The basic idea is to augment the Fourier bases by introducing a non-smooth Bernoulli base such that the non-smooth Bernoulli base compensates for the non-smooth part of the solution and the smooth part of the solution is approximated by the Fourier bases,thus achieving accelerated convergence.Based on the enriched HB method,gear pair systems with gear backlash and oscillator systems with rigid impact are solved,and the dynamic response characteristics are analyzed in this work.Then,based on the Floquet theory,the event-driven monodromy matrix method for non-smooth systems is used to analyze the stability and bifurcation of the periodic solutions.The numerical example shows that the results obtained from the enriched HB method are consistent with those from the Runge-Kutta method,which proves that the presented method is an effective method for analyzing the dynamic response characteristic of the vibro-impact system.
基金supported by the National Natural Science Foundation of China(Grant Nos.12172267 and 12302014).
文摘Quasi-periodic solutions with multiple base frequencies exhibit the feature of 2π-periodicity with respect to each of the hyper-time variables.However,it remains a challenge work,due to the lack of effective solution methods,to solve and track the quasi-periodic solutions with multiple base frequencies until now.In this work,a multi-steps variable-coefficient formulation is proposed,which provides a unified framework to enable either harmonic balance method or collocation method or finite difference method to solve quasi-periodic solutions with multiple base frequencies.For this purpose,a method of alternating U and S domain is also developed to efficiently evaluate the nonlinear force terms.Furthermore,a new robust phase condition is presented for all of the three methods to make them track the quasi-periodic solutions with prior unknown multiple base frequencies,while the stability of the quasi-periodic solutions is assessed by mean of Lyapunov exponents.The feasibility of the constructed methods under the above framework is verified by application to three nonlinear systems.
基金Project supported by the National Key R&D Program of China(No.2023YFE0125900)。
文摘A novel vibration isolation system designed for superior performance in low-frequency environments is proposed in this work.The isolator is based on a unique hexagonal arrangement of linear springs,allowing for an adjustable geometric configuration via the initial inclination angle.Based on the principle of Lagrangian mechanics,the equation of motion governing the structural dynamics is rigorously derived.The system is modeled as a strongly nonlinear single-degree-of-freedom dynamical system,loaded with a normalized payload and subject to harmonic base excitation.To analyze the steady-state response,the harmonic balance method is employed,providing accurate predictions of the payload's vibration amplitude and displacement transmissibility as functions of both the base excitation amplitude and frequency.The analysis reveals a direct relationship between the isolator's geometric and stiffness parameters and its load-bearing capacity,leading to the identification of three distinct operational regimes.Depending on the unloaded initial inclination angle,the equivalent stiffness ratio,and the payload design configuration,the system can exhibit one of three vibration isolation modes:(i)the quasizero stiffness(QZS)isolation mode,(ii)the zero linear stiffness with controllable nonlinear stiffness,and(iii)the full-band perfect zero stiffness.The vibration isolation performance of the proposed structure is thoroughly discussed for all three oscillation modes in terms of frequency response curves,displacement transmissibility,and time-domain responses.The key novel finding is that this structure can operate as a full-band,high-performance vibration isolator when the initial inclination angle is designed to be a right angle,enabling full isolation of the maximum possible payload.Moreover,the analytical results and numerical simulations demonstrate that the isolator's displacement transmissibility T with the unit dB tends to-∞as the air-damping coefficient approaches zero,enabling ideal vibration isolation across the entire excitation frequency range.These analytical insights are validated through comprehensive numerical simulations,which show excellent agreement with the theoretical predictions.
基金supported by the National Natural Science Foundation of China(No.10632040)
文摘Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.
基金supported by the National Natural Science Foundation of China (Grant No.51007068)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20100201120028)+1 种基金the Fundamental Research Funds for the Central Universities of Chinathe State Key Laboratory of Electrical Insulation and Power Equipment of China (Grant No.EIPE10303)
文摘In this paper, period-doubling bifurcation in a two-stage power factor correction converter is analyzed by using the method of incremental harmonic balance （IHB） and Floquet theory. A two-stage power factor correction converter typically employs a cascade configuration of a pre-regulator boost power factor correction converter with average current mode control to achieve a near unity power factor and a tightly regulated post-regulator DC-DC Buck converter with voltage feedback control to regulate the output voltage. Based on the assumption that the tightly regulated postregulator DC-DC Buck converter is represented as a constant power sink and some other assumptions, the simplified model of the two-stage power factor correction converter is derived and its approximate periodic solution is calculated by the method of IHB. And then, the stability of the system is investigated by using Floquet theory and the stable boundaries are presented on the selected parameter spaces. Finally, some experimental results are given to confirm the effectiveness of the theoretical analysis.
基金supported by the National Natural Science Foundation of China(Grants 11702333 and 11672337)Natural Science Foundation of Guangdong Province(Grant 2018B030311001).
文摘As a classical technique for chaos suppression,the time-delayed feedback controlling strategy has been widely developed by stabilizing unstable periodic orbits(UPOs)embedded in chaotic systems.A critical issue for achieving high controlling precision is to search for an appropriate time delay.This paper proposes a simple yet effective approach,based on incremental harmonic balance method,to determine the optimal time delay in the delayed feedback controller.The time delay is adjusted within the iterative scheme provided by the proposed method,and finally converges to the period of the target UPO.As long as the optimal time delay is fixed,moreover,the attained solution makes it quite convenient to analyze its stability according to the Floquet theory,which further provides the effective interval of the feedback gain.
基金supported by the National Natural Science Foundation of China (10772202)Doctoral Program Foundation of Ministry of Education of China (20050558032)Guangdong Province Natural Science Foundation (07003680, 05003295)
文摘We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.
基金supported by National Key Research and Development Program of China(No.2021YFA0717100)NationalNatural Science Foundation of China(Nos.12072270,U2013206).
文摘The harmonic balance(HB)method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems.However,it is confined to low-order approximations due to complex symbolic operations.Many variants have been developed to improve the HB method,among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation.So far,there are two problems remaining to be addressed.i)A dozen of different versions of HB-like methods,in frequency domain or time domain or in hybrid,have been developed;unfortunately,misclassification pervades among them due to the unclear borderlines of different methods.ii)The time domain HB-like methods suffer from non-physical solutions,which have been shown to be caused by aliasing(mixture of the high-order into the low-order harmonics).Although a series of dealiasing techniques have been developed over the past two decades,the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community.This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences.In particular,the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed.
基金co-supported by the National Natural Science Foundation of China(No.51976172)the National Science and Technology Major Project of China(No.2017-Ⅱ-0009-0023)。
文摘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.
基金Project supported by the Ph. D. Programs Foundation of Ministry of Education of China (No.20050558032) the Natural Science Foundation of Guangdong Province of China (No.05003295) the Foundation of Sun Yat-sen University Advanced Research Center (No.06M8) the Young Teacher Scientific Research Foundation of Sun Sat-sen University (No.1131011)
文摘The incremental harmonic balance method was extended to analyze the flutter of systems with multiple structural strong nonlinearities. The strongly nonlinear cubic plunging and pitching stiffness terms were considered in the flutter equations of two-dimensional airfoil. First, the equations were transferred into matrix form, then the vibration process was divided into the persistent incremental processes of vibration moments. And the expression of their solutions could be obtained by using a certain amplitude as control parameter in the harmonic balance process, and then the bifurcation, limit cycle flutter phenomena and the number of harmonic terms were analyzed. Finally, numerical results calculated by the Runge-Kutta method were given to verify the results obtained by the proposed procedure. It has been shown that the incremental harmonic method is effective and precise in the analysis of strongly nonlinear flutter with multiple structural nonlinearities.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11872254 and 11672191)
文摘The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.
文摘In this paper, the fractional-order Genesio-Tesi system showing chaotic behaviours is introduced, and the corresponding one in an integer-order form is studied intensively. Based on the harmonic balance principle, which is widely used in the frequency analysis of nonlinear control systems, a theoretical approach is used to investigate the conditions of system parameters under which this fractional-order system can give rise to a chaotic attractor. Finally, the numerical simulation is used to verify the validity of the theoretical results.
基金Project supported by the National Natural Science Foundation of China(Nos.11972129 and12372008)the National Major Science and Technology Projects of China(No.2017-IV-0008-0045)+3 种基金the Natural Science Foundation of Heilongjiang Province of China(No.YQ2022A008)the Fundamental Research Funds for the Central Universities of China(No.HIT.OCEF.2023006)the Polish National Science Centre of Poland under the OPUS 18 grant(No.2019/35/B/ST8/00980)the Tianjin University Independent Innovation Foundation of China(No.2023XJS-0038)。
文摘The harmonic balance method(HBM)is one of the most widely used methods in solving nonlinear vibration problems,and its accuracy and computational efficiency largely depend on the number of the harmonics selected.The adaptive harmonic balance(AHB)method is an improved HBM method.This paper presents a modified AHB method with the asymptotic harmonic selection(AHS)procedure.This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response,by which the additional calculation is avoided.A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters,and then all solution branches of the amplitude-frequency response are obtained.Numerical experiments are carried out to verify the performance of the AHB-AHS method.Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples.Compared with the classical HBM and Runge-Kutta methods,the proposed AHB-AHS method is of higher accuracy and better convergence.The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.
基金Project supported by the National Natural Science Foundation of China(Nos.11972381 and 11572354)the Fundamental Research Funds for the Central Universities(No.18lgzd08)。
文摘The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated.The nonlinear partial integrodifferential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle.A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with the Galerkin method.A high-dimensional model of the buckled beam is derived,concerning nonlinear coupling.The incremental harmonic balance(IHB)method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and the nonlinear amplitude response curve,and the Floquet theory is used to analyze the stability of the periodic solutions.Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the antisymmetric modes being excited.Bifurcations including the saddle-node,Hopf,perioddoubling,and symmetry-breaking bifurcations are observed.Furthermore,quasi-periodic motion is observed by using the fourth-order Runge-Kutta method,which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.
基金support from the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No.2019319)support from the Start-up Foundation of Suzhou Institute of Nano-Tech and Nano-Bionics,CAS,Suzhou (Grant No.Y9AAD110)。
文摘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 matching.Because the variation of these capacitors and inductors can be detrimental to their high-frequency characteristics,the fabrication of a JTWPA typically necessitates precise processing equipment.To guide the fabrication process and further improve the design for manufacturability,it is necessary to understand how each electronic component affects the amplifier.In 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 JTWPA.The results provide insightful and scientific guidance for device design and fabrication processes.
文摘The problem of nonlinear forced oscillations for elliptical sandwich plates is dealt with. Based on the governing equations expressed in terms of five displacement components, the nonlinear dynamic equation of an elliptical sandwich plate under a harmonic force is derived. A superpositive-iterative harmonic balance (SIHB) method is presented for the steady-state analysis of strongly nonlinear oscillators. In a periodic oscillation, the periodic solutions can be expressed in the form of basic harmonics and bifurcate harmonics. Thus, an oscillation system which is described as a second order ordinary differential equation, can be expressed as fundamental differential equation with fundamental harmonics and incremental differential equation with derived harmonics. The 1/3 subharmonic solution of an elliptical sandwich plate is investigated by using the methods of SIHB. The SIHB method is compared with the numerical integration method. Finally, asymptotical stability of the 1/3 subharmonic oscillations is inspected.
文摘The response of visco_elastic system to combined deterministic harmonic and random excitation was investigated. The method of harmonic balance and the method of stochastic averaging were used to determine the response of the system. The theoretical analysis was verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increase, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions and jumps may exist.
基金the financial support from the National Key R&D Program of China(No.2023YFE0125900)National Natural Science Foundation of China(Nos.12372008 and 12102234)+1 种基金Natural Science Foundation of Heilongjiang Province,China(No.YQ2022A008)Taif University,Saudi Arabia,for supporting this work through Project number(TU-DSPP-2024-73).
文摘This paper analyzes the nonlinear dynamic characteristics and stability of Aero-Engine Dual-Rotor(AEDR)systems under high-frequency excitation,based on the Adaptive Harmonic Balance with the Asymptotic Harmonic Selection(AHB-AHS)method.A finite element dynamic equation for the AEDR system is introduced,considering complex nonlinearities of the intershaft bearing,unbalanced excitations,and high-frequency excitation.A solving strategy combining the AHB-AHS method and improved arclength continuation method is proposed to solve highdimensional dynamic equations containing complex nonlinearities and to track periodic solutions with parameter variations.The Floquet theory is used to analyze the types of bifurcation points in the system and the stability of periodic motions.The results indicate that high-frequency excitation can couple high-order and low-order modes,especially when the system undergoes superharmonic resonance.High-frequency excitation leads to more combination frequency harmonics,among which N_(f)ω_(1)-2ω_(2)dominates.Furthermore,changing the parameters(amplitude and frequency)of high-frequency excitation widens or shifts the unstable regions of the system.These findings contribute to understanding the mechanism of high-frequency excitation on aero-engines and demonstrate that the proposed AHB-AHS method is a powerful tool for analyzing highdimensional complex nonlinear dynamic systems under multi-frequency excitation.
