Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton's principle,the nonlinear governing equation of an axially functionally graded(AFG)pipe is established.The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam.Via the direct multi-scale method,the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method(DQEM).The influence of Young's modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions.In general,the pipe in the supercritical region is more sensitive to the pulsating excitation.The nonlinearity changes from hard to soft,and the non-trivial equilibrium configuration introduces more frequency components to the vibration.Besides,the increasing Young's modulus gradient improves the critical pulsating flow velocity of the parametric resonance,and further enhances the stability of the system.In addition,when the temperature increases along the axial direction,reducing the gradient parameter can enhance the response asymmetry.This work further complements the theoretical analysis of pipes conveying pulsating fluid.

Parametric Resonance Analyses for Spar Platform in Irregular Waves

The parametric instability of a spar platform in irregular waves is analyzed. Parametric resonance is a phenomenon that may occur when a mechanical system parameter varies over time. When it occurs, a spar platform will have excessive pitch motion and may capsize. Therefore, avoiding parametric resonance is an important design requirement. The traditional methodology includes only a prediction of the Mathieu stability with harmonic excitation in regular waves. However, real sea conditions are irregular, and it has been observed that parametric resonance also occurs in non-harmonic excitations. Thus, it is imperative to predict the parametric resonance of a spar platform in irregular waves. A Hill equation is derived in this work, which can be used to analyze the parametric resonance under multi-frequency excitations. The derived Hill equation for predicting the instability of a spar can include non-harmonic excitation and random phases. The stability charts for multi-frequency excitation in irregular waves are given and compared with that for single frequency excitation in regular waves. Simulations of the pitch dynamic responses are carried out to check the stability. Three-dimensional stability charts with various damping coefficients for irregular waves are also investigated. The results show that the stability property in irregular waves has notable differences compared with that in case of regular waves. In addition, using the Hill equation to obtain the stability chart is an effective method to predict the parametric instability of spar platforms. Moreover, some suggestions for designing spar platforms to avoid parametric resonance are presented, such as increasing the damping coefficient, using an appropriate RAO and increasing the metacentric height.

Parametric resonances of convection belt system

Based on the Coriolis acceleration and the Lagrangian strain formula, a gen- eralized equation for the transverse vibration system of convection belts is derived using Newton＇s second law. The method of multiple scales is directly applied to the govern- ing equations, and an approximate solution of the primary parameter resonance of the system is obtained. The detuning parameter, cross-section area, elastic and viscoelastic parameters, and axial moving speed have a significant influences on the amplitudes of steady-state response and their existence boundaries. Some new dynamical phenomena are revealed.

Lagrange-Maxwell equation and magnetic saturation parametric resonance of generator set

Lagrange-Maxwell＇s equation is extended firstly. With the theory of electromechanical analytical dynamics, the magnetic complement energy in air gap of generator is acquired. The torsional vibration differential equations with periodic coefficients of rotor shafting of generator which is in the state of.magnetic saturation are established. It is shown that the magnetic saturation may cause double frequency electromagnetic moment. By means of the averaging method, the first approximate solution and corresponding solution of the primary parametric resonance is obtained. The characteristics and laws of the primary parametric resonance excited by the electromagnetism are analyzed and some of new phenomena are revealed.

Active Magnetic Compensation Based on Parametric Resonance Magnetometer

Based on the parametric resonance magnetometer(PRM)theory,this paper establishes an experimen-tal system of PRM.The experimental results are consistent with the theoretical predictions.A PRM has been developed with sensitivity of 0.5 pT/Hz^(1/2),which can detect the magnitude of residual magnetic field;furthermore,a proportion-integration-differentiation(PID)closed-loop magnetic compensation system of the residual magnetic field also has been realized.Compared with open-loop compensation,the PID closed-loop compensation reduces the average value of the residual magnetic field in the z-axis direction from 0.0244nT to-0.0023nT,and the mean-square error from 0.2083 nT to 0.0691 nT.In the same way,the average value of the residual magnetic field in the y-axis direction is reduced from 0.0816nT to-0.0042nT,and the mean-square error from 0.1316nT to 0.0461 nT.The magnitude of residual magnetic fields in both directions is decreased to the order of picotesla(pT).In addition,based on the signal waveforms of the magnetometer,a method of verifying the effect of magnetic compensation is proposed.

Parametric instability in the pure-quartic nonlinear Schrödinger equation

We study the nonlinear stage of modulation instability(MI)in the non-intergrable pure-quartic nonlinear Schrödinger equation where the fourth-order dispersion is modulated periodically.Using the three-mode truncation,we reveal the complex recurrence of parametric resonance(PR)breathers,where each recurrence is associated with two oscillation periods(PR period and internal oscillation period).The nonlinear stage of parametric instability admits the maximum energy exchange between the spectrum sidebands and central mode occurring outside the MI gain band.

Analysis of the Vibrissa Parametric Resonance Causing a Signal Amplification during Whisking Behaviour

The paper deals with the mechanical vibrational motion ofvibrissae during natural exploratory behaviour of mammals. The theoretical analysis is based on a mechanical model of a cylindrical beam with circular natural configuration under an applied periodic force at the tip, which corresponds to the surface roughness of an investigated object. The equation of motion of the beam is studied using the Euler-Bemoulli beam theory and asymptotic methods of mechanics. It is shown that from the me- chanical point of view the phenomenon of parametric resonance of the vibrissa is possible. It means that the amplitude of forced vibrations of a vibrissa increases exponentially with time, if it is stimulated within a specific resonance frequency range, which depends on biomechanical parameters of the vibrissa. The most intense parametric resonance occurs, when the excitation frequency is close to the doubled natural frequency of free vibrations. Thus, it may be used to distinguish and amplify specific periodic components of a complex roughness profile during texture discrimination.

PRINCIPAL RESONANCE IN TRANSVERSE NONLINEAR PARAMETRIC VIBRATION OF AN AXIALLY ACCELERATING VISCOELASTIC STRING

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string,the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string.To derive the governing equation,Newton's second law,Lagrangean strain,and Kelvin's model are respectively used to account the dynamical relation,geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms,closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance.The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance.Some numerical examples are presented to show the effects of the mean transport speed,the amplitude and the frequency of speed variation.

Nonlinear dynamic singularity analysis of two interconnected synchronous generator system with 1:3 internal resonance and parametric principal resonance

The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1：3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus （TMIB） power system.

Suppression strategy for parametrically excited lateral vibration of mass-loaded string

This paper discusses a simple way to suppress the parametrically excited lateral vibration of a mass-loaded string. Supposing that the mass at the lower end of the string is subjected to a vertical harmonic excitation and neglecting the higher order vibration modes, the equation of motion for the mass-loaded string can be represented by a Mathieus equation with cubic nonlinearity. According to the theory of the Mathieus equation, in the mass-loaded string system, when the vertical vibration frequency of the mass approaches twice the natural frequency of the string lateral vibration, once the vertical vibration amplitude of the mass exceeds a critical value, the parametric resonance will occur in the string. To avoid the parametric resonance, a vibration absorber, composed of a thin beam and two mass blocks attached at both sides of the beam symmetrically, is proposed to install with the mass to reduce its vertical vibration, and ultimately suppress the lateral vibration of the string. Such a suppression strategy is finally validated by experiments.

Theoretical Explanation for Lower Order Harmonic Resonance Phenomenon of Three-Ring Gear Transmissions

Dynamic investigations revealed that lower order harmonic resonance phenomenon exists in the three ring gear transmission. That is, when the input speed is close to 1/3, 1/6, 1/9,…, 1/3 n of the primary natural frequency of the transmission, the loads on the bearings and gears are especially high. This paper explained this phenomenon from the viewpoint of parametric resonance in terms of perturbation technique. A conclusion was drawn that the basic reason for this phenomenon is the primary resonance caused by forcing excitation and parametric resonance caused by parametric change.

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node （SN） or Hopf bifurcation （HB） as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

PARAMETRIC SHIMMY ANALYSIS

A more reliable parametric shimmy analysis model taking the elasticity of strut, the elasticity of damper linkage system and the influence of shimmy damping into consideration is set up in this paper. The Multiple Scales Method is utilized to obtain the expressions of stability condition and transition curve under the conditions of non resonant case, principal parametric resonance and combination resonance respectively. The parametric shimmy analysis for a real type of aircraft is carried out based on the perturbation analysis results, and the stability chart and critical taxiing velocities of two types of aircraft in the case of principal parametric resonance are given. It is indicated that there exist two critical taxiing velocities near which parametric shimmy is susceptible to occur, while the landing gear system is stable when taxiing velocity is far from them. This characteristic makes the parametric shimmy distinct from the shimmy in general case. Because few previous studies of nose gear shimmy have dealt with parametric shimmy problem, the results given in this paper provide useful reference for aircraft design and maintenance.

Low noise,continuous-wave single-frequency 1.5-μm laser generated by a singly resonant optical parametric oscillator

We report a low noise continuous-wave （CW） single-frequency 1.5-μm laser source obtained by a singly resonant optical parametric oscillator （SRO） based on periodically poled lithium niobate （PPLN）. The SRO was pumped by a CW single-frequency Nd：YVO4 laser at 1.06μm. The 1.02 W of CW single-frequency signal laser at 1.5 μm was obtained at pump power of 6 W. At the output power of around 0.75 W, the power stability was better than ±l.5% and no mode-hopping was observed in 30 min and frequency stability was better than 8.5 MHz in 1 min. The signal wavelength could be tuned from 1.57 to 1.59 μm by varying the PPLN temperature. The 1.5-μm laser exhibits low noise characteristics, the intensity noise of the laser reaches the shot noise limit （SNL） at an analysis frequency of 4 MHz and the phase noise is less than 1 dB above the SNL at analysis frequencies above 10 MHz.

High-Efficiency Mid-Infrared Picosecond MgO:PPLN Single Resonant Optical Parametric Oscillator

We demonstrate a high-emciency mid-infrared picosecond optical parametric oscillator （OPO） based on MgO doped periodically poled lithium niobate （MgO：PPLN） with a laser diode array （LDA） pumped Innoslab amplifier as the pumping source. Under a 16 W synchronously pumping power, 4.5 W of idler light at 2896nm is obtained. A tuning range of idler light from 2688nm to 3016nm is achieved, within which the highest optical-optical conversion ettlciency from pump power to OPO output is 35.1%. Moreover, a signal light of -500mW from 1644 to 1700nm with a repetition rate of 233.8 MHz is generated.

DYNAMIC STABILITY OF A BEAM-MODEL VISCOELASTIC PIPE FOR CONVEYING PULSATIVE FLUID

The dynamic stability in transverse vibration of a viscoelastic pipe for conveying puisative fluid is investigated for the simply-supported case. The material property of the beammodel pipe is described by the Kelvin-type viscoelastic constitutive relation. The axial fluid speed is characterized as simple harmonic variation about a constant mean speed. The method of multiple scales is applied directly to the governing partial differential equation without discretization when the viscoelastic damping and the periodical excitation are considered small. The stability conditions are presented in the case of subharmonic and combination resonance. Numerical results show the effect of viscosity and mass ratio on instability regions.

Chaotic Roll Motions of Ships in Regular Longitudinal Waves

Parametric resonance can lead to dangerously large rolling motions, endangering the ship, cargo and crew. The QR-faetorization method for calculating （LCEs） Lyapunov Characteristic Exponents was introduced; parametric resonance stability of ships in longitudinal waves was then analyzed using LCEs. Then the safe and unsafe regions of target ships were then identified. The results showed that this method can be used to analyze ship stability and to accurately identify safe and unsafe operating conditions for a ship in longitudinal waves.

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is pre- sented. The governing equation and simple support boundary condition are derived with the extended Hamilton＇s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscos- ity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method （DQM） is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.

Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions

The dynamic stability of axially accelerating plates is investigated. Longitudi- nally varying tensions due to the acceleration and nonhomogeneous boundary conditions are highlighted. A model of the plate combined with viscoelasticity is applied. In the viscoelastic constitutive relationship, the material derivative is used to take the place of the partial time derivative. Analytical and numerical methods are used to investigate summation and principal parametric resonances, respectively. By use of linear models for the transverse behavior in the small displacement regime, the plate is confined by a viscous damping force. The generalized Hamilton principle is used to derive the govern- ing equations, the initial conditions, and the boundary conditions of the coupled planar vibration. The solvability conditions are established by directly using the method of mul- tiple scales. The Routh-Hurwitz criterion is used to obtain the necessary and sufficient condition of the stability. Numerical examples are given to show the effects of related parameters on the stability boundaries. The validity of longitudinally varying tensions and nonhomogeneous boundary conditions is highlighted by comparing the results of the method of multiple scales with those of a differential quadrature scheme.

Stability of columns with original defects under periodic transient loadings

To study the influence of original defects on the dynamic stability of the columns under periodic transient loadings,the approximate solution method and the Fourier method of the stable periodic solution are adopted while considering the influence of original defects on columns.The dynamic stability of the columns under periodic transient loadings is analyzed theoretically.Through the study of different deflections,the dynamic instability of the columns is obtained by Maple software.The results of theoretical analysis show that the larger the original defects,the greater the unstable area,the stable solution amplitude of columns and the risk of instability caused by parametric resonance will be.The damping of columns is a vital factor in reducing dynamic instability at the same original defects.On the basis of the Mathieu-Hill equation,the relationship between the original defects and deflection is deduced,and the dynamic instability region of the columns under different original defects is obtained.Therefore,reducing the original defects of columns can further enhance the dynamic stability of the compressed columns in practical engineering.