Optimal estimation of the amplitude of signal with known frequency in the presence of thermal noise

In the torsion pendulum experiments, the thermal noise sets the most fundamental limit to the accurate estimation of the amplitude of the signal with known frequency. The variance of the conventional method can meet the limit only when the measurement time is much longer than the relaxation time of the pendulum. By using the maximum likelihood estimation and the equation-of-motion filter operator, we propose an optimal(minimum variance, unbiased) amplitude estimation method without limitation of the measurement time, where thermal fluctuation is the leading noise. While processing the experimental data tests of the Newtonian gravitational inverse square law, the variance of our method has been improved than before and the measurement time of determining the amplitude with this method has been reduced about half than before for the same uncertainty. These results are significant for the torsion experiment when the measurement time is limited.     

Femto-Newton light force measurement at the thermal noise limit

The measurement of very small light forces has wide applications in many fields of physics. A common measurement method for small force detection is the determination of changes in the dynamic behavior of mechanical oscillators, either in amplitude or in frequency. The detection of slowly varying forces mostly requires long period oscillators, such as a torsion pendulum. We demonstrate the application of a macroscopic, low-noise, torsion balance oscillator for the detection of radiation pressure forces at the femto-Newton level. The system is "precooled" (removing excess seimic noise) to be only thermal noise limited. The demonstrated force sensitivity reaches the thermal limit.     

Correlation method estimation of the modulation signal in the weak equivalence principle test

In a test of the weak equivalence principle(WEP) with a rotating torsion pendulum, it is important to estimate the amplitude of the modulation signal with high precision. We use a torsional filter to remove the free oscillation signal and employ the correlation method to estimate the amplitude of the modulation signal. The data analysis of an experiment shows that the uncertainties of amplitude components of the modulation signal obtained by the correlation method are in agreement with those due to white noise. The power spectral density of the modulation signal obtained by the correlation method is about one order higher than the thermal noise limit. It indicates that the correlation method is an effective way to estimate the amplitude of the modulation signal and it is instructive to conduct a high-accuracy WEP test.     

Quantum limits of cold damping with optomechanical coupling

Thermal noise of a mirror can be reduced by cold damping. The displacement is measured with a high-finesse cavity and controlled with the radiation pressure of a modulated light beam. We establish the general quantum limits of noise in cold damping mechanisms and we show that the optomechanical system allows to reach these limits. Displacement noise can be arbitrarily reduced in a narrow frequency band. In a wide-band analysis we show that thermal fluctuations are reduced as with classical damping whereas quantum zero-point fluctuations are left unchanged. The only limit of cold damping is then due to zero-point energy of the mirror. Received 1st August 2001 and Received in final form 12 October 2001     

Influence of the moment of inertia of the pendulum on the internal friction (loss tangent) measured by free decay

Summary The internal friction (loss tangent) of solids is normally measured, as a function of temperature, with a torsion pendulum operating in free decay. Curves of the internal friction and the oscillation frequency, against temperature, are obtained at various moments of inertia, to extract the parameters characteristic of the relaxation process (relaxation time and strength). In all these experiments only the temperature is considered as an independent variable and the moment of inertia of the pendulum is mainly used to shift the internal friction peaks in the temperature scale. It is pointed out in the paper that the moment of inertia is also an independent variable which can be used to determine, with high accuracy, if the measured peak is of the Debye type or not. A new torsion pendulum, with continuously variable moment of inertia is presented, which allows measurements of the partial derivative of the internal friction and the oscillation frequency with respect to the moment of inertia, at constant temperature. Finally, some measurements of the Snoek relaxation in Nb-O alloys are presented, to show the applicability of the concepts developed in the paper. The authors of this paper have agreed to not receive the proofs for correction.     

Experiments on the bifurcation behaviour of a forced nonlinear pendulum

We investigate a mechanical system (forced nonlinear torsion pendulum). The state diagram is given as function of both the external driving frequency and the damping parameter. A bifurcation diagram is measured showing period doubling, chaos and periodic windows. The results are in qualitative agreement with the recent theory.     

马氏体式相变过程中所引起的内耗变化

用扭摆测量两种跌锰合金(Mn17.5％和12.8％)和一种铜铝合金(Al13％)的内耗,在发生正和反的马氏体相变的温度范围内各出现一个内耗峰。这种内耗峰出现的条件是必须伴随着马氏体式相变过程的进行。用含Mn17.5％的铁锰合金作了系统实验,观察到内耗峰的高度随升温(或降温)速度和应力的增加而增高,随振动频率和含碳量的增加而减低。可以用振动一周内试样中转变量愈多内耗也愈大的关系得到统一的解释。讨论了关于产生内耗峰的机构。认为主要由于马氏体相变是突然间完成,此时扭转的外力可视为常数,从应力和不均匀物质交互作用能的计算,可以证明外力所做的功必须损失一半,因而引起内耗。此外应力感生相变也可以引起很小一部分内耗。     

精密扭摆研究

本研究旨在提高扭摆精度。扭摆采用大理石——这取自地球的天然材料作摆盘;通过试验,挑选了Q值高、内耗小、剪切模量温度系数接近于零的恒弹性合金丝作为吊丝。特别是,我们把Saxl类型的每周期吸合一次的间断振荡摆回复到小阻尼情况下的自由振荡摆,减少对摆振动的人为干预;增加真空条件,以减少周围环境干扰,延长了阻尼时间;同时在摆盘中央安放多面反射镜环,增加测量点,变一周期仅测一次时间为同一周期测四次以上。这就提供了用计算机求阻尼振荡方程θi=Ae^-t_i/τsin(ωt_i+φ)数值解的条件,不但可直接测得周期T的变化,而且可望计算得振幅A、阻尼时间τ、初相角φ的变化。另外,在光电接收方面也作了一些改进。通过这些措施,我室的精密扭摆对周期的探测精度已比美国Saxl 1964年发表第一篇文章时的扭摆精度提高100倍;比其运转17年,经一再改进的扭摆精度提高10倍。测量有效数字6位至7位,周期相对误差一般达2×10^-6,最佳状况可达8×10^-7,基本达到预期设计要求,为将来探测引力异常,“区分长期间与短期间内频率起伏”提供了可能性。 关键词：     

Precise Determination of Period of a Torsion Pendulum in Measurement of Gravitational Constant

The period of a torsion pendulum would vary under the disturbances of environmental noise factors. In order to subtract the period of the pendulum from external influence, we employ the correlation method to determine the period with a high precision. Theoretical analysis shows that the relative precision is improved to be proportional to 1/m^3/2 with the number of the period m, compared with the conventional statistical mean that is proportional to 1/m^1/2, which is significant for the determination of gravitational constant with the swing time method.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

Abatement of thermal noise due to internal damping in 2D oscillators with rapidly rotating test masses

Mechanical oscillators can be sensitive to very small forces. Low frequency effects are up-converted to higher frequency by rotating the oscillator. We show that for 2-dimensional oscillators rotating at frequency much higher than the signal the thermal noise force due to internal losses and competing with it is abated as the square root of the rotation frequency. We also show that rotation at frequency much higher than the natural one is possible if the oscillator has 2 degrees of freedom, and describe how this property applies also to torsion balances. In addition, in the 2D oscillator the signal is up-converted above resonance without being attenuated as in the 1D case, thus relaxing requirements on the read out. This work indicates that proof masses weakly coupled in 2D and rapidly rotating can play a major role in very small force physics experiments.     

Dynamic torsion of a two-layer viscoelastic beam

The complex dynamic shear modulus of soft polymeric materials may be determined in principle at low and audio frequencies from torsion pendulum and torsional resonance experiments on metal strips coated with the polymer. In order to determine the polymer shear properties from such experiments, it is necessary to know the torsional rigidity of the two-layer compound beam. This is calculated in the paper by using classical elasticity theory, for the particular case when the metal shear modulus is much greater than that of the soft coating. The theory of dynamic torsion tests is then briefly reviewed and experiments are suggested for determining the polymer dynamic shear properties. A discussion is also given of the effectiveness of polymer coatings as a damping treatment for torsional vibrations.     

Occurrence of stable periodic modes in a pendulum with cubic damping

Dynamical systems with nonlinear damping show interesting behavior in the periodic and chaotic phases. The Froude pendulum with cubical and linear damping is a paradigm for such a system. In this work the driven Froude pendulum is studied by the harmonic balancing method; the resulting nonlinear response curves are studied further for resonance and stability of symmetric oscillations with relatively low damping. The stability analysis is carried out by transforming the system of equations to the linear Mathieu equation.     

低分析频率压缩光的实验制备

基于PPKTP晶体的阈值以下光学参量振荡(OPO)过程,制备了共振于铷原子D1线795 nm的压缩真空态光场,研究了分析频率处于千赫兹范围的主要噪声来源,特别是795 nm激光及其二次谐波397.5 nm激光在晶体内吸收引起的非线性损耗增加和系统热不稳定的问题(397.5 nm激光处于PPKTP晶体透光范围边缘,具有高于其他波长数倍的吸收系数).以795 nm和1064 nm为例,分析了非线性损耗及晶体内热效应对压缩度的影响.受限于以上因素,795 nm压缩光很难得到1064 nm波段同样的压缩度.探测系统中的噪声耦合则限制了压缩频带.实验上对分析频率为千赫兹的经典噪声进行了有效控制,通过使用真空注入的OPO、垂直偏振及反向传输的腔长锁定光、低噪声的平衡零拍探测器、高稳定度的实验系统及量子噪声锁定等方法,最终在2.6—100 kHz的分析频段得到了约2.8 dB的795 nm压缩真空.该压缩光可用作磁场测量系统的探测光以提高测量灵敏度.     

Damping in metals under combined stress loading

Studies have been made of damping behaviour under conditions of combined stress (i.e., biaxial loading) for six high damping metals—two alloys of manganese-copper and four grades of cast iron. Measured values of damping are presented from tests covering a range of combined stress states and the damping behaviour is interpreted in terms of dependence on stress state. The measured values were obtained from a new experiment in which various combined stress conditions are generated by coupled torsion/bending vibration of a cantilever. The range covered is from all torsion to all bending, i.e., principal stress ratios from ?1 to 0. Damping is determined from energy input during steady state resonant vibration, but frequency response and free decay methods can also be used. Details of the apparatus are given.     

The decay of low-frequency elastic oscillations in a Ba2NaNb5O15 crystal

The internal friction and shear modulus of a barium sodium niobate crystal were studied using the torsion pendulum technique at a frequency of ～25 Hz in the temperature range from 80 to 800 K. An internal friction relaxation peak of the domain nature was observed at 228 K. A relaxation process responsible for this internal friction peak is explained through compensation of the electric charge (induced by a piezoelectric effect at the 180° domain walls) by charged point defects diffusing toward the domain walls.     

The internal friction of iron whiskers

In this paper of the measurements internal friction of iron whiskers are described; a torsion pendulum was especially adapted for this purpose. It was found that the internal friction of whiskers having larger diameter than 10 m was usually amplitude dependent. This amplitude dependence approximatively fits the Granato-Lücke theory of dislocation damping.The effects of plastic deformation and its recovery in iron whiskers were also studied. The results and their discussion lead to the assumption that during the recovery an important role is played either by the interaction of dislocations with vacancies, or by the interaction of dislocation dipoles with the impurity atoms.     

Experimental control for initiating and maintaining rotation of parametric pendulum

In this paper, the authors have studied experimentally the control methods of a parametric pendulum excited harmonically to initiate and maintain a period one rotation – the most superior response for energy harvesting. For initiating the period one rotation inherent in the system, first the bang-bang method is applied. Then a new method where velocity is monitored is proposed and applied and finally the time-delayed feedback method with multi-switching is considered. Ultimately the problem of maintaining the rotation of the pendulum is addressed. For first time, robustness and sensitivity of the latter method to change of frequency and amplitude of excitation and added noise are studied. Finally, it has been demonstrated how the delayed feedback method can be applied in a system of two pendula to ensure synchronized rotation.     

周期场中非各态历经布朗运动

将自由状态下呈弹道扩散的非各态历经系统置于周期场中,进而将非各态历经布朗运动分为两类.第一类是阻尼核的Laplace变换的低频为零的系统,当温度远大于势垒高度时,系统平均能量的动能部分依赖粒子的初始速度分布;随温度降低,系统的各态历经性得到恢复.然后将第一类系统的稳定速度变量作为一个内部噪声,再去驱动一个自由布朗粒子,则阻尼核的Laplace变换在零频时为无穷大.结果发现,粒子扩散系数随温度的增加而趋于零,显示一种经典局域化特征,系统的渐进分布依赖于初始坐标分布.这是第二类非各态历经性运动,不能通过外加势而恢复. 关键词： 非各态历经 非Markov布朗运动 扩散系数 噪声谱     

Work fluctuation theorem for coloured noise driven open systems

We have studied work fluctuation behaviour in the presence of internal thermal noise as well as external coloured noise. The external coloured noise may have both Gaussian or non-Gaussian characteristics. We have investigated the dependence of position and work fluctuations on the properties of both the environments. For thermal noise driven systems, there is a maximum in the variation of mean square fluctuation of work (MSFW) as a function of damping strength at intermediate times, while at asymptotic long times MSFW monotonically increases in the same damping regime. But for external noise, MSFW monotonically decreases as a function of damping strength at intermediate times, whereas at long times it becomes almost independent of damping strength.Another interesting observation is that for the external noise driven systems, noise correlation time and damping strength have similar roles in the dynamics.