旋转内接悬臂梁的刚柔耦合动力学特性分析

对固结于旋转刚环上内接柔性梁的刚柔耦合动力学特性进行了研究. 在精确描述柔性梁非线性变形基础上, 利用Hamilton变分原理和假设模态法, 在计入柔性梁由于横向变形而引起的轴向变形二阶耦合量的条件下, 推导出一次近似耦合模型. 忽略柔性梁纵向变形的影响,给出一次近似简化模型,引入无量纲变量, 对简化模型做无量纲化处理. 首先分析在非惯性系下内接悬臂梁的动力学响应, 并与外接悬臂梁进行比较; 其次研究内接悬臂梁的稳定性;最后分析内接悬臂梁失稳临界转速的收敛性. 研究发现, 与外接悬臂梁存在动力刚化效应不同,内接悬臂梁存在着动力柔化效应; 给出了内接悬臂梁无条件稳定的临界径长比以及失稳的临界转速的计算方法; 若第一阶固有频率随转速增大而减小,则该内接悬臂梁处于有条件稳定; 随着模态截断数的增加,内接悬臂梁失稳的临界转速减小且有收敛值. 关键词： 内接悬臂梁 一次近似简化模型 动力柔化 临界转速     

An iterative method for non-stationary response analysis of non-linear random systems

A method for calculating the non-stationary response of non-linear systems subjected to random excitations is formulated. The time-dependent equivalent linear system is considered and an iterative procedure for evaluating the non-stationary mean-square responses is developed. Several examples are presented and applicability of the technique is illustrated.     

Time-dependent variance and covariance of responses of structures to non-stationary random excitations

In this paper techniques for the analysis of non-stationary random responses of linear structures, discretized by the finite element method so that they can be analyzed as multi-degree of freedom systems, subjected to non-stationary random excitation are developed. The non-stationary random excitation is represented as a product of (a) an exponentially decaying function and a white noise process, and (b) a modulating function in the form of an exponential envelope and a white noise process. Closed form expressions for the time-dependent variance and covariance of response of structures are presented. Application of these expressions is made for the analysis of non-stationary random responses of a physical model of a class of mast antenna structures subjected to base excitation. It is concluded that (a) the coupling terms do have a definite influence on the response; the magnitude of the influence is proportional to the amount of damping in the structure and proximity of the modes excited; (b) the non-stationary random excitations considered are general in that the modulating functions are not necessarily identical, and therefore the influence of various modulating functions of the excitations applied to different locations of the structure on responses can be examined quantivatively; and (c) for a given damping parameter the magnitudes of the modulating function parameters cannot be chosen arbitrarily though the shapes of normalized modulating functions can be selected to best fit the excitation realizations.     

Flapwise dynamic response of a wind turbine blade in super-harmonic resonance

The flapwise dynamic response of a rotating wind turbine blade in super-harmonic resonance is studied in this paper, while the blade is subjected to unsteady aerodynamic loads. Coupled extensional–bending vibrations of the blade are considered; the governing equations which are coupled through linear and quadratic terms arising from rotating and geometric effects respectively are obtained by applying the Hamiltonian principle. The lth flapwise linear frequency and the rotational frequency are assumed to be in an almost 3:1 ratio, so super-harmonic resonance occurs when this linear frequency is close to the associated nonlinear frequency. By using the first n, no less than l, linear undamped modal functions as a functional basis and applying the Galerkin procedure, a 2n-degree-of-freedom discrete model with quadratic and cubic terms owing to geometric effect is derived. The generalized displacements corresponding to the discrete system are disintegrated into static and dynamic displacements. Perturbation method is adopted to get analytical solutions of the discrete dynamic system for positive aerodynamic dampings. The coning angle and the inflow ratio are chosen as two control parameters to analyze aeroelastic behaviors of the blade. By assuming that the static and dynamic displacements are of the same order in resonance region, and there is no other resonance except the super-harmonic resonance, the multiple-scales method is employed to obtain a set of amplitude modulation equations whose coefficients depend on two control parameters. The frequency-response equation is derived from the amplitude modulation equations. A method to estimate the functional dependence of the detuning parameter on two control parameters is introduced. The amplitude of the harmonic response is derived from the frequency-response equation after knowing the detuning parameter. Then the stability of the steady motion with respect to control parameters can be determined. The evolution of the dynamic response of the resonance mode with decreasing aerodynamic damping is discussed by means of both perturbation and numerical methods.     

Enhanced vibration energy harvesting using dual-mass systems

A type of dual-mass vibration energy harvester, where two masses are connected in series with the energy transducer and spring, is proposed and analyzed in this paper. The dual-mass vibration energy harvester is proved to be able to harvest more energy than the traditional single degree-of-freedom (dof) one when subjected to harmonic force or base displacement excitations. The optimal parameters for maximizing the power output in both the traditional and the new configurations are discussed in analytical form while taking the parasitic mechanical damping of the system into account. Consistent of the previous literature, we find that the optimal condition for maximum power output of the single dof vibration energy harvester is when the excitation frequency equals to the natural frequency of the mechanical system and the electrical damping due to the energy harvesting circuit is the same as the mechanical damping. However, the optimal conditions are quite different for the dual-mass vibration energy harvester. It is found that two local optimums exist, where the optimal excitation frequency and electrical damping are analytically obtained. The local maximum power of the dual-mass vibration energy harvester is larger than the global maximum power of single dof one. Moreover, at certain frequency range between the two natural frequencies of the dual-mass system, the harvesting power always increases with the electrical damping ratio. This suggests that we can obtain higher energy harvesting rate using dual-mass harvester. The sensitivity of the power to parameters, such as mass ratio and tuning ratio, is also investigated.     

谱分析法测量叶尖小翼对风轮旋转时固有频率的影响

风力机叶尖加小翼,在一定范围内可以增加风力发电机的输出功率,提高功率系数.本文通过应用谱分析法测量风力机风轮旋转时机头的振动谱,通过分析得到了风轮动频,应用动频曲线研究了风力机叶尖小翼对风轮固有频率的影响.实验表明,叶尖小翼虽然降低了风轮静频但通过增大离心力和气动力提高了动频.叶尖小翼可以改善旋转风轮的结构动态特性.     

Auto-tuning of a two-degree-of-freedom rotational pendulum absorber

A dynamic vibration absorber is effective in suppressing harmonic excitation by tuning its natural frequency to match the excitation frequency. The rotational pendulum absorber (RPA) has a wide-range of natural frequencies that are continuously tunable by setting a suitable rotational speed. In this paper it is shown how to automatically tune the rotational speed of a two-degree-of-freedom RPA by detecting the phase between the vibration of the primary structure and that of the RPA. For this purpose the speed response of the RPA is introduced in addition to the frequency response. It is seen that if the excitation frequency is above a critical value dependent on the parameters of the RPA, the second vibration mode of the RPA is effective, allowing a relatively low rotational speed for the pendulums. The speed tuning algorithm is tested on a flexible plate that is subject to excitations of around 80 Hz, which do not generate visible oscillations but emit audible noise instead. Experimental results confirm the noise-level reduction effect of the RPA.     

Non-linear structural vibrations under combined parametric and external excitations

A set of second order equations with weak quadratic and cubic non-linearities is considered. Simultaneous parametric and external (forcing) excitations are included. The frequency of the parametric excitation is near a natural frequency of the system, and three cases are analyzed: (i) the external excitation is absent; (ii) the external excitation is present but is not involved in a resonance; and (iii) the external frequency is the same as the parametric frequency. Results are obtained by the method of multiple scales. Frequency-response curves are presented for various combinations of excitation amplitudes, damping coefficients, and phase shift between the excitations. It is found that stable multi-modal responses may exist in the first-order asymptotic solution, even though only one mode is involved in the resonance and no internal resonance condition is present.     

Elastic connection disk subjected to periodically fluctuating transmitted torque and rotational speed

Involving the intrinsic power transmission torque/speed coupling characteristics of prime-movers, a rotating elastic connection disk subjected to periodically fluctuating transmitted torque and rotational speed generated by the fluctuation of external loads is investigated. Using Galerkin's method, the rotating elastic connection disk is modeled as a parametrically excited gyroscopic system. The effects of the torque/speed coupling, transmitted torque fluctuation amplitude and frequency, and constant parts of the transmitted torque and the rotational speed on the system dynamic stability are explored for the disk modes possessing different nodal diameters. The rotational speed, transmitted torque and their fluctuations can all result in system instability of the elastic connection disk. The instability can be suppressed or avoided by operating at small amplitude and low frequency of the transmitted torque fluctuation, and by operating in the weakly coupled torque/speed regime of the prime-movers. Low rotational speed avoids the instability in the case of a small transmitted torque, but medium rotational speed operation is valuable to suppress the instability induced by a large transmitted torque and its fluctuation. Instability parameter regions for the positive and negative torque/speed coupling coefficient are roughly similar in shape, but there are some differences in the value of the instability coefficient.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Modal parameter identification through Gabor expansion of response signals

A new method of identifying modal parameters by decomposing response signals with Gabor transform is presented in this paper to estimate natural frequencies, damping ratios and mode shapes of linear time invariant systems. According to Gabor expansion theory, responses of a multi-degree-of-freedom system can be decomposed into uncoupled signal components, each vibrating at a single natural frequency. From these uncoupled signals, modal parameters are subsequently extracted with common methods. The proposed method can process stationary and non-stationary responses and requires no input signal except for the response signals generated by unknown excitation acting on a system. In the sense of less restriction on the in-out signals, the approach based on time-frequency decomposition is very general. A simulation study on a simply supported beam under non-stationary excitation has demonstrated that the proposed method is effective in parameter estimation.     

Nonlinear parametric instability of wind turbine wings

Nonlinear rotor dynamic is characterized by parametric excitation of both linear and nonlinear terms caused by centrifugal and Coriolis forces when formulated in a moving frame of reference. Assuming harmonically varying support point motions from the tower, the nonlinear parametric instability of a wind turbine wing has been analysed based on a two-degrees-of-freedom model with one modal coordinate representing the vibrations in the blade direction and the other vibrations in edgewise direction. The functional basis for the eigenmode expansion has been taken as the linear undamped fixed-base eigenmodes. It turns out that the system becomes unstable at certain excitation amplitudes and frequencies. If the ratio between the support point motion and the rotational frequency of the rotor is rational, the response becomes periodic, and Floquet theory may be used to determine instability. In reality the indicated frequency ratio may be irrational in which case the response is shown to be quasi-periodic, rendering the Floquet theory useless. Moreover, as the excitation frequency exceeds the eigenfrequency in the edgewise direction, the response may become chaotic. For this reason stability of the system has in all cases been evaluated based on a Lyapunov exponent approach. Stability boundaries are determined as a function of the amplitude and frequency of the support point motion, the rotational speed, damping ratios and eigenfrequencies in the blade and edgewise directions.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

A study on radial directional natural frequency and damping ratio in a vehicle tire

The measurement of radial directional natural frequency and damping ratio in a vehicle tire has been studied. Natural frequencies and damping ratios in the radial direction of various tires, from passenger car tires to truck bus tires, are reported. The radial direction modal parameters of tires subjected to different levels of inflation pressure, have been determined by using a frequency response function method. To obtain the theoretical natural frequency and mode shape, the plane vibration of a tire has been modeled as though it were that of a circular beam. By using the Tielking method that is based on Hamilton’s principle, theoretical results have been determined by considering the rotational velocity, tangential and radial stiffness, radial directional velocity and tension force which is due to tire inflation pressure. The results show that experimental conditions can be considered as the parameters that shift the natural frequency and damping ratio.     

On the notion of a rotating fluid force induced by swirling flow

The Bently/Muszynska (B/M) model shows that oil whirl and oil whip are both self-sustained vibrations associated with two unstable modes of a rotor–fluid system. The model includes a rotating fluid damping and inertia force. In certain configurations, the rotating damping force overcomes the frictional internal damping of the rotor and pushes the rotor into a stable limit cycle of circular orbiting. Such a notion of a rotating fluid force is based on bulk-flow models of fluid-filled clearances that could be approximated as narrow since the tangential velocity of the fluid then translates to one angular velocity at a certain radial distance defined by an average radius. This paper scrutinizes the assumption of a rotating fluid inertia force and pinpoints the additional inertial effects of the swirling flow as the gap width increases. These effects are clarified by deriving the equation of motion of a body with a mass subjected to motion-induced fluid forces of a confined swirling flow. We show that the inertial effects of the swirling flow counteract the destabilizing effect of the rotating damping force. However, if the body mass is larger than the displaced fluid mass, instability follows. The frequency of the unstable mode is unchanged by the additional inertial effects and is always equal to the frequency of the damping that induces the instability.     

ADDING ONE BIFURCATION IN A ROTOR SYSTEM WITH CLEARANCES

A mathematical model of a rotor system with clearances is analysed by the application of modern nonlinear dynamic theory. From the bifurcation diagrams, it is discovered that the rotor system alternates between periodic and chaotic motions at a supercritical rotational speed, and after undergoing a chaotic region the periodic number of the motion will increase by one. At the same time, the periodic number is equal correspondingly to the integral multiple of the critical rotational speed. At the subcritical rotational speed, it is discovered that the chaotic bands among successive orders of superharmonic responses return to the period one through a reversed period-doubling bifurcation, as a result of a period-doubling bifurcation. It is shown that the increase of damping may reduce the width of the chaotic bands and the amplitude of the periodic response; the increase of nonlinear degree also leads to the reduction of chaotic bandwidth, but makes the amplitude of the subharmonic response increase. So it is suggested that proper damping and correct material selection by considering the dynamic characteristics of the rotor system may reduce the proportion of faults and enhance the dynamic characteristics when designing the rotor system. The working speed should not be selected at N times its natural frequency and should not be set in the chaotic bands among the successive orders of periodic motion for the same purpose.     

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v₁. For v>v₁, there is a value of the internal viscous damping coefficient, C_iv, which depends on the rotational speed and torque, above which destabilization occurs.The value of C_iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C_iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%.For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C_iv critical, parametric instabilities are still found in the practical range of operation.     

Equivalent linear stochastic seismic response of isolated bridges

Stochastic response of bridges seismically isolated by lead-rubber bearings (LRB) is investigated. The earthquake excitation is modeled as a non-stationary random process (i.e. uniformly modulated broad-band excitation). The stochastic response of isolated bridge is obtained using the time-dependent equivalent linearization technique as the force-deformation behavior of the LRB is highly nonlinear. The non-stationary response of isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. For a given isolated bridge system and excitation, it was observed that there exists an optimum value of the yield strength of LRB for which the root mean square (rms) absolute acceleration of bridge deck attains the minimum value. The optimum yield strength of LRB is investigated under important parametric variations such as isolation period and damping ratio of the LRB and the frequency content and intensity of earthquake excitation. It is shown that the above parameters have significant effects on the optimum yield strength of LRB. Finally, closed-form expressions for the optimum yield strength of LRB and corresponding response of the isolated bridge system are proposed. These expressions were derived based on the model of bridge with rigid deck and pier condition subjected to stationary white-noise excitation. It was observed that there is a very good comparison between the proposed closed-form expressions and actual optimum parameters and response of the isolated bridge system. These expressions can be used for initial optimal design of seismic isolation system for the bridges.     

Wave nature of moving striations in a longitudinal magnetic field

Wave nature of stationary moving striations in helium and neon discharges in a uniform longitudinal magnetic field is studied. With the increase of the magnetic field, the frequency of natural striations decreases, while the wave length increases, and they damp out at high field region. Artificial excitations in these gases show that the wave length is proportional to the excitation frequency for given magnetic field and the slope of linear lines increases with the field. These wave nature of striations is explained following the dispersion relation derived from the consideration of waves of ionization and including effects of the magnetic field on the ionization.     

Theoretical and experimental vibration analysis of rotating beams with combined ACLD and Stressed Layer Damping treatment

Active constrained layer damping (ACLD) treatment increases the efficiency of passive constrained layer damping (PCLD) treatment, but in case of circuit failure, only the decreased efficiency of PCLD treatment is available. The efficiency of the ordinary PCLD treatment can be enhanced by adding a stressed poly vinyl chloride (PVC) layers on the base beam instead of using viscoelastic materials. Hamilton Principle in conjunction with finite element method is used to derive the non-linear differential equations of motion for a rotating beam. Using proportional feedback controllers, the complex closed loop eigenvalue problem is developed and solved numerically. The effect of rotational speed of the beam, initial strain and other parameters of the PVC layer is investigated. To prove the effectiveness of the new technique, experimental investigations have also been carried out.