THE LIMIT CYCLE BIFURCATIONS OF A WHIRLING PENDULUM WITH PIECEWISE SMOOTH PERTURBATIONS

This paper deals with the problem of limit cycles for the whirling pendulum equation x=y,y=sin x(cosx-r)under piecewise smooth perturbations of polynomials of cos x,sin x and y of degree n with the switching line x=0.The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained using the Picard-Fuchs equations,which the generating functions of the associated first order Melnikov functions satisfy.Furthermore,the exact bound of a special case is given using the Chebyshev system.At the end,some numerical simulations are given to illustrate the existence of limit cycles.

Reduced-Order Observer-Based LQR Controller Design for Rotary Inverted Pendulum

The Rotary Inverted Pendulum(RIP)is a widely used underactuated mechanical system in various applications such as bipedal robots and skyscraper stabilization where attitude control presents a significant challenge.Despite the implementation of various control strategies to maintain equilibrium,optimally tuning control gains to effectively mitigate uncertain nonlinearities in system dynamics remains elusive.Existing methods frequently rely on extensive experimental data or the designer’s expertise,presenting a notable drawback.This paper proposes a novel tracking control approach for RIP,utilizing a Linear Quadratic Regulator(LQR)in combination with a reduced-order observer.Initially,the RIP system is mathematically modeled using the Newton-Euler-Lagrange method.Subsequently,a composite controller is devised that integrates an LQR for generating nominal control signals and a reduced-order observer for reconstructing unmeasured states.This approach enhances the controller’s robustness by eliminating differential terms from the observer,thereby attenuating unknown disturbances.Thorough numerical simulations and experimental evaluations demonstrate the system’s capability to maintain balance below50Hz and achieve precise tracking below1.4 rad,validating the effectiveness of the proposed control scheme.

Astral Actions on Allais’ Pendulum Apparently Inexplicable by Classical Factors: A Point of the Situation

1) The observation by Allais of the precession of pendulums from 1954 to 1960 highlighted regularities of astral origin an in-depth analysis of which showed that, apparently, no classical phenomenon can explain them. These regularities were diurnal waves whose periods are characteristic of astral influence (the main ones being 24 h and 24 h 50 min), annual and semi-annual components, and a multi-annual component of approximately 6 years, an influence of Jupiter being a very good candidate to explain it. 2) Allais had experimentally established that all these astral influences were expressed globally on the pendulum by an action tending to call back its plane of oscillation towards a direction variable in time, and which ovalized its trajectory. In 2019 the observation of 2 pendulums in Horodnic (Romania), thanks to the use of an automatic alidade, made it possible to identify the main mechanism that, very probably, acted on the pendulum to achieve this result. This perturbation model, called “linear anisotropy”, is characterized by its “coefficient of anisotropy” η, and by the azimuth of its “direction of anisotropy”. The composition of 2 linear anisotropies is always a linear anisotropy. 3) In the search for the phenomena which could be at the origin of all what precedes, the fact that they must create an ovalization immediately eliminates some of them. 4) We have calculated the values of η corresponding to the 24 h and 24 h 50 min waves both for the observations in Horodnic and the Allais observations. The order of magnitude (some 10−7) is effectively the same in both cases. 5) Mathematically, the regularities discovered may result of a new force field but also, as Allais proposes, from the creation, under the astral influences, of a local anisotropy of the medium in which the pendulum oscillates. In the first case the length of the pendulum is involved, in the second one not. The data available do not make it possible to decide. 6) The joint exploitation, in mechanics and optics, of Allais observations and of observations by other experimenters provides additional information: a) Allais, and after him several other scientists, discovered also marked anomalies in the precession of pendulums during certain eclipses, and maybe certain other syzygies. For the few eclipses for which both something was observed and sufficient data were available (one of them being a lunar eclipse for which nothing had been published until now), it was always the above perturbation model which acted on the pendulum, but sometimes with quite exceptional magnitude. b) There are quite possible links with optics. During the observation campaign of August 1958, which had implemented both two pendulums and an optical device, all the 24 h 50 min waves were almost in phase. In the precession of the Allais pendulum, in Miller’s interferometric observations in Mont Wilson, and in Esclangon’s observations in Strasbourg, a same peculiarity is found: the extrema of the annual influence are at the equinoxes, not at the solstices.

Measurement of remanent magnetic moment using a torsion pendulum with single frequency modulation method

In Tian Qin spaceborne gravitational-wave detectors, the stringent requirements on the magnetic cleanliness of the test masses demand the high resolution ground-based characterization measurement of their magnetic properties. Here we present a single frequency modulation method based on a torsion pendulum to measure the remanent magnetic moment mr of 1.1 kg dummy copper test mass, and the measurement result is(6.45 ± 0.04(stat) ± 0.07(syst)) × 10^(-8)A · m^(2). The measurement precision of the mr is about 0.9 n A · m^(2), well below the present measurement requirement of Tian Qin. The method is particularly useful for measuring extremely low magnetic properties of the materials for use in the construction of space-borne gravitational wave detection and other precision scientific apparatus.

Inexplicable Multi-Annual Astral Action on the Precession of Allais Pendulum: An Influence of the Solar System (and Especially of Jupiter?)

Between 1954 and 1961, Allais conducted 6 one-month observations of the azimuth of the plane of oscillation of a pendulum installed in his laboratory. That of 1958 also implemented a second pendulum, identical to the first, located 6 km away in an underground quarry. Although, over these 6 years, the average azimuth of each observation, the amplitude of the 24 h 50 min and 24 h waves, as well as certain other quantities, have evolved considerably, in 1958 their values were very close to those of the second pendulum. The analysis shows that these evolutions could only result from an action external to the pendulum, that no classical phenomenon seems to be able to explain, and which appears, at least mainly, to be an astral action. The evolution of the average azimuth of the pendulum and of the amplitudes of the 24 h and 24 h 50 min components can be decomposed into a component associated with the annual revolution of the Earth around the Sun, and a multi-annual component, whose harmonic 1 has a period which was estimated to 5.74 years. An action of Jupiter is an excellent candidate to explain a large part of the multi-annual action: everything happens as if there were an important action of the modulus of its declination on the multi-annual component, and an important daily action of its hour angle on the azimuth of the pendulum. We cannot exclude an action of the solar cycle, whose period was then about 11 years. The main results were obtained by Allais himself, but this was only published in his book “The Anisotropy of Space”, and remained very little known. Starting from the raw data of Allais, the author of this article found them again, and completed them on certain points.

The Extended Non-Elementary Amplitude Functions as Solutions to the Damped Pendulum Equation, the Van der Pol Equation, the Damped Duffing Equation, the Lienard Equation and the Lorenz Equations

In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.

Project-Based Teaching in Control Theory Education Based on V-REP: A Cart Inverted Pendulum Case

This paper provides a teaching concept for control theory education based on Virtual Robot Experimentation Platform(V-REP).A cart inverted pendulum virtual physical model is developed on V-REP.Students must analyze,design,and implement a suitable controller for the cart inverted pendulum system using their knowledge of the control theory.Different from traditional experiment and numerical simulation,virtual experiment is safe and less constrained.Moreover,the experiment results are more intuitive and obvious.This study can improve students’interest in learning the control theory and help students understand the relevant content better.

Pendulum矫治器远中移动磨牙治疗Ⅱ类错的临床及头影测量评价

目的 :探讨Pendulum矫治器治疗Ⅱ类错牙合的临床疗效及对牙颌面的影响。方法 :应用Pendulum矫治器对18例平均年龄为 11.9岁的牙性Ⅱ类错牙合患者进行治疗 ,通过治疗前后的头影测量分析 ,评价其对牙颌面的影响。结果 :Pendulum矫治器能够远中移动上颌磨牙致磨牙关系为Ⅰ类或偏近中 ,未见开牙合及明显前牙支抗丧失。结论

改良Pendulum与改良Jones远移磨牙的对比分析

目的:探讨改良Pendulum矫治器、改良Jones装置推磨牙向远中的疗效。方法:选择22例轻中度前牙拥挤的安氏Ⅱ类错病例,介于拔牙和不拔牙矫治的边缘,所有病例被平均随机分配在2个组,分别采用2种装置推磨牙向远中,在治疗前后进行X线头影测量和模型测量分析,采用SPSS软件对数据进行分析。结果:2组远中移动效果均比较明显,但在移动速度、磨牙中央窝宽度差、前牙覆差、上颌第二前磨牙以及第一磨牙的倾斜度的统计分析上差异有统计学意义。结论:改良Pendulum矫治器J、ones装置均可有效地推上颌磨牙向远中移动,但改良Jones组磨牙向远中更少发生倾斜移动,矫治时间短,移动速度更快,对前牙支抗消耗相对较低。

EQUATION OF MOTION FOR NOVEL FRICTION PENDULUM SYSTEM WITH DUAL ROLLERS

A novel friction pendulum system （FPS） with dual rollers is studied based on the multibody dynamics theory. By analyzing kinematic characteristics of the system, it is reduced to a one degree-of-freedom system. Then the equation of motion for the system is analytically derived by applying the theorem of the relative kinetic energy for a system of particles in differential form in the non-inertial reference system described as a nonlinear differential equation. In the case of the small angular displacement, the natural frequency of the corresponding undamped linear system is obtained, which is consistent with the experimental observation. The derived equation is useful for the study of dynamic characteristics of novel FPS, and its solution directly expedites the simulation of the system in a control loop, and further facilitates the semi-active control process including novel FPS.

Study on a conical bearing for acceleration-sensitive equipment

Seismic isolation effectively reduces seismic demands on building structures by isolating the superstructure from ground vibrations during earthquakes.However,isolation strategies give less attention to acceleration-sensitive systems or equipment.Meanwhile,as the isolation layer’s displacement grows,the stiffness and frequency of traditional rolling and sliding isolation bearings increases,potentially causing self-centering and resonance concerns.As a result,a new conical pendulum bearing has been selected for acceleration-sensitive equipment to increase self-centering capacity,and additional viscous dampers are incorporated to enhance system damping.Moreover,the theoretical formula for conical pendulum bearings is supplied to analyze the device’s dynamic parameters,and shake table experiments are used to determine the proposed device’s isolation efficiency under various conditions.According to the test results,the newly proposed devices have remarkable isolation performance in terms of minimizing both acceleration and displacement responses.Finally,a numerical model of the isolation system is provided for further research,and the accuracy is demonstrated by the aforementioned experiments.

Pendulum矫治器在上颌牙列前段拥挤中的临床应用

目的评价Pendulum矫治器在解除上颌牙列前段拥挤中的临床效果。方法选择安氏Ⅱ类（前牙区拥挤3-5mm）患者11例,其中女7例,男4例,年龄10.6-12.9岁,平均11.5岁,采用Pendulum矫治器推磨牙向远中来解除牙列前段的拥挤,疗程为2-3个月。结果每侧磨牙向后移动2.5-5mm,患者矫治前后侧面型无明显改变,但上颌第二磨牙术前偏颊向位的其间距增加1.5-2.0mm。结论Pendulum矫治器能有效地移动上颌磨牙,快速解除上颌牙前段的轻中度拥挤。但可能导致原来偏颊向位的第二磨牙在矫治后更趋颊向位,临床治疗中须引起注意。

Design of A Hydraulic Power Take-off System for the Wave Energy Device with An Inverse Pendulum

This paper describes a dual-stroke acting hydraulic power take-off （PTO） system employed in the wave energy converter （WEC） with an inverse pendulum. The hydraulic PTO converts slow irregular reciprocating wave motions to relatively smooth, fast rotation of an electrical generator. The design of the hydraulic PTO system and its control are critical to maximize the generated power. A time domain simulation study and the laboratory experiment of the full-scale beach test are presented. The results of the simulation and laboratory experiments including their comparison at full-scale are also presented, which have validated the rationality of the design and the reliability of some key components of the prototype of the WEC with an inverse pendulum with the dual-stroke acting hydraulic PTO system.

Numerical Study of a Novel Procedure for Installing the Tower and Rotor Nacelle Assembly of Offshore Wind Turbines Based on the Inverted Pendulum Principle

Current installation costs of offshore wind turbines(OWTs) are high and profit margins in the offshore wind energy sector are low, it is thus necessary to develop installation methods that are more efficient and practical. This paper presents a numerical study(based on a global response analysis of marine operations) of a novel procedure for installing the tower and Rotor Nacelle Assemblies(RNAs) on bottom-fixed foundations of OWTs. The installation procedure is based on the inverted pendulum principle. A cargo barge is used to transport the OWT assembly in a horizontal position to the site, and a medium-size Heavy Lift Vessel(HLV) is then employed to lift and up-end the OWT assembly using a special upending frame. The main advantage of this novel procedure is that the need for a huge HLV(in terms of lifting height and capacity) is eliminated. This novel method requires that the cargo barge is in the leeward side of the HLV(which can be positioned with the best heading) during the entire installation. This is to benefit from shielding effects of the HLV on the motions of the cargo barge, so the foundations need to be installed with a specific heading based on wave direction statistics of the site and a typical installation season. Following a systematic approach based on numerical simulations of actual operations, potential critical installation activities, corresponding critical events, and limiting(response) parameters are identified. In addition, operational limits for some of the limiting parameters are established in terms of allowable limits of sea states. Following a preliminary assessment of these operational limits, the duration of the entire operation, the equipment used, and weather-and water depth-sensitivity, this novel procedure is demonstrated to be viable.

Equivalent series system to model a multiple friction pendulum system with numerous sliding interfaces for seismic analyses

Current structural analysis software programs offer few if any applicable device-specifi c hysteresis rules or nonlinear elements to simulate the precise mechanical behavior of a multiple friction pendulum system(MFPS) with numerous sliding interfaces.Based on the concept of subsystems,an equivalent series system that adopts existing nonlinear elements with parameters systematically calculated and mathematically proven through rigorous derivations is proposed.The aim is to simulate the characteristics of sliding motions for an MFPS isolation system with numerous concave sliding interfaces without prior knowledge of detailed information on the mobilized forces at various sliding stages.An MFPS with numerous concave sliding interfaces and one articulated or rigid slider located between these interfaces is divided into two subsystems: the fi rst represents the concave sliding interfaces above the slider,and the second represents those below the slider.The equivalent series system for the entire system is then obtained by connecting those for each subsystem in series.The equivalent series system is validated by comparing numerical results for an MFPS with four sliding interfaces obtained from the proposed method with those from a previous study by Fenz and Constantinou.Furthermore,these numerical results demonstrate that an MFPS isolator with numerous concave sliding interfaces,which may have any number of sliding interfaces,is a good isolation device to protect structures from earthquake damage through appropriate designs with controllable mechanisms.

Diterpenoid Alkaloids and One Lignan from the Roots of Aconitum pendulum Busch

Diterpenoid alkaloids have neroprotective activity.Herein,three napelline-type diterpenoid alkaloids 1-3,two aconitine-type diterpenoid alkaloids 4-5,and one isoquinline-type alkaloid 6,as well as one lignan glycoside 7,have been isolated from the roots of Aconitum pendulum Busch.Compounds 1 and 7 were new compounds,and their chemical structures were determined on the basis of nuclear magnetic resonance(NMR)spectra and mass spectrometry analysis.A ThT assay revealed that compound 2 showed significant disaggregation potency on the Aβ_(1−42)aggregates.

Optimal feedback gains of a delayed proportional-derivative(PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a controlled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a systematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost characteristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial values around the origin are fixed. The main results are checked by numerical simulation.

前方牵引和Pendulum在临床上的联合应用

[目的 ]对适合做前方牵引矫正的患者在牵引后存在牙列不齐的问题进行研究。 [方法 ]对替牙期Ⅲ类错牙合的患者共计 2 8例 (男性 1 6例 ,女性 1 2例 ) ,用前方牵引的方法进行矫正。在解除反牙合后 ,6 6 ┼6 6 为远中关系的患者 8例 ,使用Pendulum调整6 6 ┼6 6 的咬牙合关系。余下的患者6 6 ┼6 6 为远中关系的患者 1 0例对照观察。 [结果 ]用前方牵引矫正反牙合 ,X光头影测量分析可见A点明显前移SNA明显增大 ,ANB改善明显。不做Pendulum的 1 0例患者6 6 ┼6 6 的位置关系和拥挤度无明显变化。做Pendulum的患者可维持6 6 ┼6 6 中性关系 ,拥挤度明显减少。 [结论 ]前方牵引后或多或少出现6 ┼6 的前移 ,造成牙性Ⅱ类错牙合。Pendulum是以上颌前部齿槽骨和前牙为支抗 ,后移磨牙来矫正牙源性Ⅱ类错牙合。这样牙齿的拥挤量减少或消失 ,在使用固定矫正器进行治疗时减少了减数拔牙的可能性 ,在排齐牙列时降低前牙的唇倾度。