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
The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas...The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement.Four cases of resonance are confirmed by examining the secular terms.The theoretical analysis framework is established for each resonance case based on the method of multiple scales.The original dynamic system is integrated numerically by the Runge–Kutta method.The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results.For the purpose of contrast,modulation equations are also integrated numerically.In all four resonance cases,the theoretical results agree well with the numerical simulation.Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves.Various results are obtained for the rotating blade.A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type,and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter.The quasi-saturation phenomenon of rotating blade only appears with high gas pressure.The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.展开更多
Near-diurnal vertically-standing waves with high vertical wavenumbers k z were observed in the velocity and shear fi elds from a set of 75-d long ADCP moored in the northeastern South China Sea(SCS)away from the“crit...Near-diurnal vertically-standing waves with high vertical wavenumbers k z were observed in the velocity and shear fi elds from a set of 75-d long ADCP moored in the northeastern South China Sea(SCS)away from the“critical”latitude of 28.8°.These enhanced near-diurnal internal waves followed a fortnightly spring-neap cycle.However,they always happened during semidiurnal spring tides rather than diurnal springs although strong diurnal internal tides with the fortnightly spring-neap cycle were prevailing,suggesting that they were generated via subharmonic instability(PSI)of dominant semidiurnal M 2 internal tides.When two semidiurnal internal tidal waves with opposite vertical propagation direction intersected,both semidiurnal subharmonic and super harmonic waves were largely intensifi ed.The observed maximum diurnal velocity amplitudes were up to 0.25 m/s.The kinetic energy and shear spectra further suggested that frequencies of daughter waves were not always perfectly equal to M 2/2.The superposition of two daughter waves with nearly equal frequencies and nearly opposite k z in a PSI-triad leaded to the vertically-standing waves.展开更多
The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose （SV） mode, is studied by a direct numerical simulation （DNS） method in a s...The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose （SV） mode, is studied by a direct numerical simulation （DNS） method in a small spatial-periodic channel. The results show that the SV low-speed streaks are self-sustained at the early stage, and then transform into subharmonic sinuous （SS） low-speed streaks. Initially, the streamwise vortex sheets are formed by shearing, and then evolve into zigzag vortex sheets due to the mutual induction. As the intensification of the SV low-speed streaks becomes prominent, the tilted streamwise vortex tubes and the V-like streamwise vortex tubes can be formed simultaneously by increasing ＋~. When the SV low-speed streaks break down, new zigzag streamwise vortices will be generated, thus giving birth to the next sustaining cycle of the SV low-speed streaks. When the second breakdown happens, new secondary V-like streamwise vortices instead of zigzag streamwise vortices will be generated. Because of the sweep motion of the fluid induced by the secondary V-like streamwise vortices, each decayed low-speed streak can be divided into two parts, and each part combines with the part of another streak, finally leading to the formation of SS low-speed streaks.展开更多
Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions...Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions(that is, all its Floquet multipliers lying on the unit circle on the complex plane C).展开更多
The viscoelasticity and subharmonic generation of a kind of lipid ultrasound contrast agent are investigated. Based on the measurement of the sound attenuation spectrum, the viscoelasticity of the lipid shell is estim...The viscoelasticity and subharmonic generation of a kind of lipid ultrasound contrast agent are investigated. Based on the measurement of the sound attenuation spectrum, the viscoelasticity of the lipid shell is estimated by use of an optimization method. Shear modulus Gs=10MPa and shear viscosity μs=1.49N.S/m^2 are obtained. The nonlinear oscillation of the encapsulated microbubble is studied with Church＇s model theoretically and experimentally. Especially, the dependence of subharmonic on the incident acoustic pressure is studied. The results reveal that the development of the subharmonic undergoes three stages, i.e. occurrence, growth and saturation, and that hysteresis appears in descending ramp insonation.展开更多
The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper.The theoretical results of evolution and...The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper.The theoretical results of evolution and spatial distribution of subharmonic disturbances are compared with experimental data.The re- suits show the important role of the process of route to transition in low-disturbance environments,and indi- cate that spatial mode is more rational than temporal mode.展开更多
Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for th...Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.展开更多
Near-inertial waves(NIWs), which can be generated by wind or the parametric subharmonic instability(PSI) of internal tides, are common in the South China Sea(SCS). Moored current observations from the northern SCS hav...Near-inertial waves(NIWs), which can be generated by wind or the parametric subharmonic instability(PSI) of internal tides, are common in the South China Sea(SCS). Moored current observations from the northern SCS have revealed that the PSI of semidiurnal(D_2) internal tides is another source of NIWs. The objective of this study was to examine the energy variance in the PSI of D_2 tides. The PSI of D_2 internal tides generated NIWs and waves with frequencies around the difference frequency of D_2 and f. The observed NIWs induced by PSI could be distinguished clearly from those elicited by typhoon Krosa. Shortly after Krosa entered the SCS, NIWs began to intensify on the surface and they propagated downward over subsequent days. The near-inertial currents were damped quickly and they became relatively weak before the waves were reinforced beneath the mixed layer when wind stress was relatively weak. Rotation spectra indicated an energy peak at exactly the difference frequency D_2–f of the NIWs and D_2, indicating nonlinear wave-wave interaction among D_2, f, and D_2–f. Depth-time maps of band-pass fi ltered velocities of D_2 –f showed the waves amplifi ed when the NIWs were reinforced, and they intensifi ed at depths with strong D_2 tides. The energies of the NIWs and D_2 –f had high correlation with the D_2 tides. The PSI transferred energy of low-mode D_2 internal tides to high-mode NIWs and D_2–f waves. For the entire observational period, PSI reinforcement was observed only when mesoscale eddies emerged and when D_2 was in spring tide, revealing a close connection between mesoscale eddies and NIWs. Mesoscale eddies could increase the energy in the f-band by enhancing the PSI of D_2 internal tides. Thus, this represents another mechanism linking the energy of mesoscale eddies to that of NIWs.展开更多
The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotati...The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.展开更多
Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example,the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated.The approximate anal...Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example,the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated.The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented.By changing the solving process of Krylov-Bogoliubov-Mitropolsky(KBM)asymptotic method for subharmonic resonance of smooth nonlinear system,the classical KBM method is extended to piecewise-smooth nonlinear system.The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved,and the stability of the subharmonic resonance steady-statesolution is also analyzed.It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness.Through a demonstration example,the accuracy of approximate analytical solution is verified by numerical solution,and they have good consistency.Based on the approximate analytical solution,the infuences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail.The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system.And it provides an effective reference for the study of subharmonicr esonance of other piecewise-smooth systems.展开更多
Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic ...Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.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
The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated.The analysis is based on a special Zhuravlev transform...The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated.The analysis is based on a special Zhuravlev transformation,which reduces the system to the one without impacts or velocity jumps,and thereby permits the applications of asymptotic averaging over the period for slowly varying the inphase and quadrature responses.The averaged stochastic equations are exactly solved by the method of moments for the mean square response amplitude for the case of zero offset.A perturbation-based moment closure scheme is proposed for the case of nonzero offset.The effects of damping,detuning,and bandwidth and magnitudes of the random excitations are analyzed.The theoretical analyses are verified by the numerical results.The theoretical analyses and numerical simulations show that the peak amplitudes can be strongly reduced at the large detunings.展开更多
The 1：2 subharmonic resonance of the labyrinth seals-rotor system is inves- tigated, where the low-frequency vibration of steam turbines can be caused by the gas exciting force. The empirical parameters of gas exciti...The 1：2 subharmonic resonance of the labyrinth seals-rotor system is inves- tigated, where the low-frequency vibration of steam turbines can be caused by the gas exciting force. The empirical parameters of gas exciting force of the Muszynska model are obtained by using the results of computational fluid dynamics （CFD）. Based on the multiple scale method, the 1：2 subharmonic resonance response of the dynamic system is gained by truncating the system with three orders. The transition sets and the local bifurcations diagrams of the dynamics system are presented by employing the singular theory analysis. Meanwhile, the existence conditions of subharmonic resonance non-zero solutions of the dynamic system are obtained, which provides a new theoretical basis in recognizing and protecting the rotor from the subharmonic resonant failure in the turbine machinery.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 1/3 subharmonic solution for the Duffing’s equation is investigated by using the methods of harmonic balance and numerical integration. The sensitivity of parameter variation for the transient process and the tra...The 1/3 subharmonic solution for the Duffing’s equation is investigated by using the methods of harmonic balance and numerical integration. The sensitivity of parameter variation for the transient process and the transient process for the perturbance initial conditions are studied. Over and above, the precision of numerical integration method is discussed and the numerical integration method is compared with the harmonic balance method. Finally, asymptotical stability of the pure subharmonic oscillations element is inspected.展开更多
It is shown by this experiment that an appreciable first subharmonic component(one-half of driving frequency)can be observed when a traoelling wave of low-intensity sound goes through the disturbed water in a tank.
A new parameter transformation alpha = alpha (epsilon, n omega (0)/m, omega (l)) was defir2ed for extending the applicable range of the modified Lindstedt-Poincare method. It is suitable for determining subharmonic an...A new parameter transformation alpha = alpha (epsilon, n omega (0)/m, omega (l)) was defir2ed for extending the applicable range of the modified Lindstedt-Poincare method. It is suitable for determining subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems. The 1/3 subharmonic and 3 ultraharmonic resonance solutions of the Duffing equation and the 1/2 subharmonic resonance solution of the Van der Pol-Mathieu equation were studied. These examples show approximate solutions are in good agreement with numerical solutions.展开更多
The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend ...The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0〈ρ〈1, then the existence of the limit limr→∞logu（r）/N（r）, where N（r） is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.展开更多
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This project is supported by the National Natural Science Foundation of China(Grant Nos.11702033 and 11872159)the Fundamental Research Funds for the Central Universities,CHD(Grant Nos.300102120106,300102128107)the Innovation Program of Shanghai Municipal Education Commission(No.2017-01-07-00-09-E00019).
文摘The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance,including two subharmonic resonances and two combination resonances.The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement.Four cases of resonance are confirmed by examining the secular terms.The theoretical analysis framework is established for each resonance case based on the method of multiple scales.The original dynamic system is integrated numerically by the Runge–Kutta method.The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results.For the purpose of contrast,modulation equations are also integrated numerically.In all four resonance cases,the theoretical results agree well with the numerical simulation.Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves.Various results are obtained for the rotating blade.A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type,and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter.The quasi-saturation phenomenon of rotating blade only appears with high gas pressure.The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.
基金Supported by the Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory(Guangzhou)(No.GML2019ZD0304)the National Natural Science Foundation of China(Nos.41630970,41876016,41676022,41521005)+2 种基金the Natural Science Foundation of Zhejiang(No.LR20D060001)the Instrument Developing Project of the CAS(No.YZ201432)the State Key Laboratory of Tropical Oceanography,South China Sea Institute of Oceanology,Chinese Academy of Sciences(No.LTO1915)。
文摘Near-diurnal vertically-standing waves with high vertical wavenumbers k z were observed in the velocity and shear fi elds from a set of 75-d long ADCP moored in the northeastern South China Sea(SCS)away from the“critical”latitude of 28.8°.These enhanced near-diurnal internal waves followed a fortnightly spring-neap cycle.However,they always happened during semidiurnal spring tides rather than diurnal springs although strong diurnal internal tides with the fortnightly spring-neap cycle were prevailing,suggesting that they were generated via subharmonic instability(PSI)of dominant semidiurnal M 2 internal tides.When two semidiurnal internal tidal waves with opposite vertical propagation direction intersected,both semidiurnal subharmonic and super harmonic waves were largely intensifi ed.The observed maximum diurnal velocity amplitudes were up to 0.25 m/s.The kinetic energy and shear spectra further suggested that frequencies of daughter waves were not always perfectly equal to M 2/2.The superposition of two daughter waves with nearly equal frequencies and nearly opposite k z in a PSI-triad leaded to the vertically-standing waves.
基金supported by the National Natural Science Foundation of China(Nos.11372140 and11202102)the Innovation Project for College Graduates of Jiangsu Province(No.CXZZ13-0189)
文摘The evolution of two spanwise-aligned low-speed streaks in a wall turbulent flow, triggered by the instability of the subharmonic varicose （SV） mode, is studied by a direct numerical simulation （DNS） method in a small spatial-periodic channel. The results show that the SV low-speed streaks are self-sustained at the early stage, and then transform into subharmonic sinuous （SS） low-speed streaks. Initially, the streamwise vortex sheets are formed by shearing, and then evolve into zigzag vortex sheets due to the mutual induction. As the intensification of the SV low-speed streaks becomes prominent, the tilted streamwise vortex tubes and the V-like streamwise vortex tubes can be formed simultaneously by increasing ＋~. When the SV low-speed streaks break down, new zigzag streamwise vortices will be generated, thus giving birth to the next sustaining cycle of the SV low-speed streaks. When the second breakdown happens, new secondary V-like streamwise vortices instead of zigzag streamwise vortices will be generated. Because of the sweep motion of the fluid induced by the secondary V-like streamwise vortices, each decayed low-speed streak can be divided into two parts, and each part combines with the part of another streak, finally leading to the formation of SS low-speed streaks.
基金supported by NSFC(11471170,11790271)innovation and development project of Guangzhou University
文摘Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions(that is, all its Floquet multipliers lying on the unit circle on the complex plane C).
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10274032 and 320200265）, National Natural Science Foundation of Jiangsu Province, China （Grant No BK2004081）.
文摘The viscoelasticity and subharmonic generation of a kind of lipid ultrasound contrast agent are investigated. Based on the measurement of the sound attenuation spectrum, the viscoelasticity of the lipid shell is estimated by use of an optimization method. Shear modulus Gs=10MPa and shear viscosity μs=1.49N.S/m^2 are obtained. The nonlinear oscillation of the encapsulated microbubble is studied with Church＇s model theoretically and experimentally. Especially, the dependence of subharmonic on the incident acoustic pressure is studied. The results reveal that the development of the subharmonic undergoes three stages, i.e. occurrence, growth and saturation, and that hysteresis appears in descending ramp insonation.
基金Project supported by the National Natural Science Foundation of China
文摘The general equations of secondary instability with respect to three-dimensional subharmonic disturbances are derived and applied to Blasius boundary layer in the present paper.The theoretical results of evolution and spatial distribution of subharmonic disturbances are compared with experimental data.The re- suits show the important role of the process of route to transition in low-disturbance environments,and indi- cate that spatial mode is more rational than temporal mode.
文摘Bifurcations of subharmonic solutions of order m of a planar periodic perturbed system near a hyperbolic limit cycle are discussed. By using a Poincare map and the method of rescaling a discriminating condition for the existence of subharmonic solutions of order m is obtained. An example is given in the end of the paper.
基金Supported by the Natural Science Foundation of Shandong Province of China(No.ZR2014DM017)the Natural Science Foundation of Zhejiang Province(No.LY15D060001)+4 种基金the National High Technology Research and Development Program of China(863 Program)(No.2013AA09A502)the National Natural Science Foundation of China(Nos.41606006,41371496)the National Key Technology Research and Development Program(No.2013BAK05B04)the 111 Project of Ministry of Education of China(No.B07036)the China Postdoctoral Science Foundation(No.2017M611979)
文摘Near-inertial waves(NIWs), which can be generated by wind or the parametric subharmonic instability(PSI) of internal tides, are common in the South China Sea(SCS). Moored current observations from the northern SCS have revealed that the PSI of semidiurnal(D_2) internal tides is another source of NIWs. The objective of this study was to examine the energy variance in the PSI of D_2 tides. The PSI of D_2 internal tides generated NIWs and waves with frequencies around the difference frequency of D_2 and f. The observed NIWs induced by PSI could be distinguished clearly from those elicited by typhoon Krosa. Shortly after Krosa entered the SCS, NIWs began to intensify on the surface and they propagated downward over subsequent days. The near-inertial currents were damped quickly and they became relatively weak before the waves were reinforced beneath the mixed layer when wind stress was relatively weak. Rotation spectra indicated an energy peak at exactly the difference frequency D_2–f of the NIWs and D_2, indicating nonlinear wave-wave interaction among D_2, f, and D_2–f. Depth-time maps of band-pass fi ltered velocities of D_2 –f showed the waves amplifi ed when the NIWs were reinforced, and they intensifi ed at depths with strong D_2 tides. The energies of the NIWs and D_2 –f had high correlation with the D_2 tides. The PSI transferred energy of low-mode D_2 internal tides to high-mode NIWs and D_2–f waves. For the entire observational period, PSI reinforcement was observed only when mesoscale eddies emerged and when D_2 was in spring tide, revealing a close connection between mesoscale eddies and NIWs. Mesoscale eddies could increase the energy in the f-band by enhancing the PSI of D_2 internal tides. Thus, this represents another mechanism linking the energy of mesoscale eddies to that of NIWs.
基金This project is supported by National Key Project of China(No.PD9521901).
文摘The 1/2 subharmonic resonance of a shaft with unsymmetrical stiffness is studied. By means of the Hamilton's principle the nonlinear differential equations of motion of the rotating shaft are derived in the rotating rectangular coordinate system. Transforming the equations of motion from rotating coordinate system into stationary coordinate system and introducing a complex variable, the equation of motion in complex variable form is obtained, in which the stiffness coefficient varies periodically with time. It presents a nonlinear oscillation system under parametric excitation. By applying the method of multiple scales (MMS) the averaged equation, the bifurcating response equations and local bifurcating set are obtained. Via the theory of singularity, the stability of constant solutions is analyzed and bifurcating response curves are obtained. This study shows that the rotating shaft has rich bifurcation phenomena.
基金the National Natural Science Foundation of China(Grants 11872254,U1934201 and 11790282).
文摘Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example,the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated.The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented.By changing the solving process of Krylov-Bogoliubov-Mitropolsky(KBM)asymptotic method for subharmonic resonance of smooth nonlinear system,the classical KBM method is extended to piecewise-smooth nonlinear system.The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved,and the stability of the subharmonic resonance steady-statesolution is also analyzed.It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness.Through a demonstration example,the accuracy of approximate analytical solution is verified by numerical solution,and they have good consistency.Based on the approximate analytical solution,the infuences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail.The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system.And it provides an effective reference for the study of subharmonicr esonance of other piecewise-smooth systems.
文摘Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.
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supported by the National Natural Science Foundation of China (Nos. 10772046 and 50978058)the Natural Science Foundation of Guangdong Province of China (Nos. 7010407 and 05300566)
文摘The subharmonic response of a single-degree-of-freedom linear vibroimpact oscillator with a one-sided barrier to the narrow-band random excitation is investigated.The analysis is based on a special Zhuravlev transformation,which reduces the system to the one without impacts or velocity jumps,and thereby permits the applications of asymptotic averaging over the period for slowly varying the inphase and quadrature responses.The averaged stochastic equations are exactly solved by the method of moments for the mean square response amplitude for the case of zero offset.A perturbation-based moment closure scheme is proposed for the case of nonzero offset.The effects of damping,detuning,and bandwidth and magnitudes of the random excitations are analyzed.The theoretical analyses are verified by the numerical results.The theoretical analyses and numerical simulations show that the peak amplitudes can be strongly reduced at the large detunings.
基金supported by the National Natural Science Foundation of China (No. 10632040)
文摘The 1：2 subharmonic resonance of the labyrinth seals-rotor system is inves- tigated, where the low-frequency vibration of steam turbines can be caused by the gas exciting force. The empirical parameters of gas exciting force of the Muszynska model are obtained by using the results of computational fluid dynamics （CFD）. Based on the multiple scale method, the 1：2 subharmonic resonance response of the dynamic system is gained by truncating the system with three orders. The transition sets and the local bifurcations diagrams of the dynamics system are presented by employing the singular theory analysis. Meanwhile, the existence conditions of subharmonic resonance non-zero solutions of the dynamic system are obtained, which provides a new theoretical basis in recognizing and protecting the rotor from the subharmonic resonant failure in the turbine machinery.
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 1/3 subharmonic solution for the Duffing’s equation is investigated by using the methods of harmonic balance and numerical integration. The sensitivity of parameter variation for the transient process and the transient process for the perturbance initial conditions are studied. Over and above, the precision of numerical integration method is discussed and the numerical integration method is compared with the harmonic balance method. Finally, asymptotical stability of the pure subharmonic oscillations element is inspected.
文摘It is shown by this experiment that an appreciable first subharmonic component(one-half of driving frequency)can be observed when a traoelling wave of low-intensity sound goes through the disturbed water in a tank.
文摘A new parameter transformation alpha = alpha (epsilon, n omega (0)/m, omega (l)) was defir2ed for extending the applicable range of the modified Lindstedt-Poincare method. It is suitable for determining subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems. The 1/3 subharmonic and 3 ultraharmonic resonance solutions of the Duffing equation and the 1/2 subharmonic resonance solution of the Van der Pol-Mathieu equation were studied. These examples show approximate solutions are in good agreement with numerical solutions.
文摘The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n≥3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0〈ρ〈1, then the existence of the limit limr→∞logu（r）/N（r）, where N（r） is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.
