弹簧质量对振子振动周期影响的实验研究

通过对有弹簧质量弹簧振子的振动实验,分析研究弹簧振子振动周期与弹簧质量的关系,并提出"轻质弹簧"实验中的振子质量与弹簧质量关系.     

弹簧振子与摆动

弹簧振子与摆动刘云鹏，郑连琴，陆国庆（东北师范大学物理系长春１３００２４）普通物理实验中的弹簧振子实验，依公式（ｍ为重物质量、ｍ０为弹簧本身的质量、ｋ为劲度系数）改变ｍ测出不同的周期，考查ｍ。对振动周期的影响，即求出系数ｃ之值．实验中当改变ｍ到一定范...     

实数编码自适应遗传算法在弹簧振子实验中的应用

在普通物理实验教学中,提出了用实数编码自适应遗传算法构建弹簧振子振动周期经验公式的新方法。利用遗传算法拟合弹簧振子振动周期的实验数据,不但可以得到高精度的振动周期公式,加深对物理理论的理解,而且有利于培养学生的综合研究能力和科学探索精神。     

弹簧等效质量和劲度系数示波器测量法

介绍了利用圆柱状弹簧作为电路电感元件,把弹簧振动信息转变为电信号,通过数字存储示波器的扫描模式显示,记录弹簧在一段时间内的振动过程,从而可对振动系统的振动周期进行比较直观的测量,进而测出弹簧等效质量、弹簧劲度系数.     

有质量弹簧的振动周期探讨

本文讨论了有质量弹簧的弹簧振子的动力学,运用哈密顿函数解出了振子的运动方程,并推导出振动周期与弹簧质量和阻尼的关系．     

用智能手机传感器测量弹簧的劲度系数

本文提供了一个用智能手机做实验工具的例子, 实验中智能手机既做弹簧的振子, 也可测量弹簧的振 动周期, 用智能手机加速度传感器测量极短暂的振动周期, 从而提高了实验的测量精度, 降低了实验误差     

考虑弹簧质量后弹簧振子的振动周期

考虑弹簧质量后弹簧振子的振动周期曹罗平，邹雪青（贵州省畜牧兽医学校贵阳５５００１８）（贵州省贵阳市第二实验中学）在许多物理场合中，为了使所研究的问题简单、明了，往往忽略一些次要的因素，例如在考虑各大行星绕太阳运动时，就把各行星看成质点，而忽略了它们的...     

弹簧振子实验方法的运用和改进

对弹簧振子振动周期经验公式的实验拟合中所用的实验方法进行了总结,对部分实验方法和实验装置进行了改进。     

弹簧振子系统振动的周期

本文分析了弹簧质量不可忽略时，对振子周期的影响．引用瑞利法讨论系统振动的基频，最后按多自由度理论对精确求解进行了讨论；通过比较，阐明瑞利法的适用范围及精度．文中还提到计算振动周期的简捷方法──能量法．     

垂直悬挂质点弹簧系统的振动

讨论了振动的振幅与周期的变化规律,给出了当质点质量m0与弹簧质量m之比趋于零与趋于无穷大以及系统水平放置时的振动解.     

VIBRATION SUPPRESSION OF A BEAM STRUCTURE BY INTERMEDIATE MASSES AND SPRINGS

A method based on the dynamic Green function has been proposed to determine the optimum values of masses and/or springs and their locations on a beam structure in order to confine the vibration at an arbitrary location. In the analysis, the beam is driven by a harmonic external excitation. The added masses on the beam and the springs attached are modelled as simple reactions that provide transverse forces to the beam. These forces act as secondary forces that reduce the response caused by the external force. Numerical simulation shows that the vibration of the beam can be confined in a certain region by the presence of masses and springs in best arrangement. This method is demonstrated for both a simply supported and a cantilever beam. An experimental set-up was designed in which a simply supported beam is excited by an electrodynamic shaker and the response of the beam is measured using an He-Ne laser system. This assures very accurate measurements and avoids any additional loading effects as in the case of accelerometers. Comparisons of the theoretical and the experimental results show good agreement.     

Design and experiment of a compact quasi-zero-stiffness isolator capable of a wide range of loads

This paper proposes the design and experiment of a vibration isolator capable of isolating a wide range of loads. The isolator consists of two oblique springs and one vertical spring to achieve quasi-zero stiffness at the equilibrium position. The quasi-zero-stiffness characteristic makes the isolator attenuate external disturbance more at low frequencies, when compared with linear isolators. Unlike previous studies, this paper focuses on the analysis of the effect of different loads and the implementation of an adjustment mechanism to handle a wide range of loads. To ensure zero stiffness under imperfect stiffness matching, a lateral adjustment mechanism is also proposed. Instead of using coil springs, special planar springs are designed to realize the isolator in a compact space. Static and dynamic models are developed to evaluate the effect of key design parameters so that the isolator can have a wide isolation range without sacrificing its size. A prototype and its associated experiments are presented to validate the transmissibility curves under three different loads. The results clearly show the advantage of quasi-zero-stiffness isolators against linear isolators.     

Dynamic stiffness vibration analysis of an elastically connected three-beam system

An exact dynamic stiffness method is developed for predicting the free vibration characteristics of a three-beam system, which is composed of three non-identical uniform beams of equal length connected by innumerable coupling springs and dashpots. The Bernoulli-Euler beam theory is used to define the beams’ dynamic behaviors. The dynamic stiffness matrix is formulated from the general solutions of the basic governing differential equations of a three-beam element in damped free vibration. The derived dynamic stiffness matrix is then used in conjunction with the automated Muller root search algorithm to calculate the free vibration characteristics of the three-beam systems. The numerical results are obtained for two sets of the stiffnesses of springs and a large variety of interesting boundary conditions.     

EXPRESSIONS FOR PREDICTING FUNDAMENTAL NATURAL FREQUENCIES OF NON-CYLINDRICAL HELICAL SPRINGS

Numerical and analytical studies are performed for the free vibration analysis of non-cylindrical (conical, barrel and hyperboloidal types) helical springs. The stiffness matrix method is used in the numerical analysis. A total of 12 degrees of freedom (six displacements and six rotations) is described for an element. The exact element stiffness matrix and the exact concentrated element inertia matrix are used in the formulation. The rotary inertia, the shear and extensional deformation effects are considered in the analysis. Comparison of the numerical results with the reported results obtained numerically and experimentally gives satisfactory values. After verification of the numerical frequencies, the non-dimensional fundamental frequencies of fixed-fixed non-cylindrical helical springs with circular section are expressed in a simple formula with a maximum absolute relative error of 5% using those numerical values for the constant helix pitch angles (5°, 10°, and 15°). These expressions restricted to the fundamental frequencies are also verified with ANSYS results.     

单摆振动实验数字化演示的定量分析

单摆振动实验的数字化传输和演示是实验教学中的一个值得讨论的基本问题．当前单摆振动实验的研究以定性建立理论模型为主，只能得到单摆振动的示意图像，且实验效果不好．本文针对当前的单摆振动实验进行改进，并通过定量计算，给出较精确的实验设计要求，得到了很好的实验效果，为该实验提供完善的理论依据和技术指导．除此之外，本文提出的实验方案不仅适用于单摆的简谐振动还适用于各种阻尼振动。     

THE EFFECT OF ROTATION UPON THE NATURAL FREQUENCIES OF A MASS-SPRING SYSTEM

When a mass-spring system vibrates it does so with frequencies characteristic of the system. If the system as a whole now undergoes a rotational motion then these characteristic frequencies will change from their non-rotational values. It is the purpose of this paper to show how these changes may be calculated for a specified system and, in particular, to investigate the role in these changes of both the system and the rotational parameters. A system of N masses linked sequentially by springs in tension is allowed to vibrate about an equilibrium configuration both radially and transversely upon a smooth turntable. If the turntable is stationary then the radial and transverse vibrations are independent of each other, provided the amplitudes of vibration are sufficiently small. There are then N natural frequencies of vibration for each mode. However, when the turntable rotates then the Coriolis effects give rise to an interaction between the two modes of vibration, and there are now 2 N natural frequencies for the combined vibrations. If the rate of rotation is “small” then the two modes are almost separated and it is possible to discuss the “essentially radial” or “essentially transverse” mode of vibration each of which has N natural frequencies. It is these natural frequencies which are considered in this work, in particular their dependence upon the rotation rate and upon the tension in the springs (when in the static configuration). In a previous paper, it was shown that if only radial vibrations are allowed (by admitting say a guide rail) then all the natural frequencies decrease, with increasing rotation rate, from their static values. It is shown that the opposite is the case here in that the “essentially radial” natural frequencies increase with increasing rotation rate. This is due to the Coriolis interaction with the transverse vibrations. The “essentially transverse” frequencies are also found and the nature of their dependence discussed. Also included in the analysis is the effect on the frequencies of the (weak) coupling between the motion of the masses and the rotation of the turntable as a consequence of the conservation of angular momentum. In addition to treating N being finite the limiting case of an infinite number of masses is considered to determine the natural frequencies of vibration of a continuous stretched string undergoing rotation.     

NATURAL FREQUENCIES AND MODE SHAPES OF A FREE-FREE BEAM WITH LARGE END MASSES

An analytical solution is presented for the natural frequencies, mode shapes and orthogonality condition, of a free-free beam with large off-set masses connected to the beam by torsion springs. Results are given for a range of masses with various fixed orientations and the validity of the method is confirmed against established results for natural frequencies of beams with five different boundary conditions. The study lays the foundation for investigations into the dynamics and vibration control of multi-link articulated systems such as the Space Shuttle Remote Manipulator.     

FORCED VIBRATIONS OF TRIPLY COUPLED, PERIODICALLY AND ELASTICALLY SUPPORTED, FINITE, OPEN-SECTION CHANNELS

An exact analytical method is presented for the analysis of forced vibrations of uniform, open-section, single- and multi-bay periodic channels. The centre of gravity and the shear centre of the channel cross-sections do not coincide, and hence the flexural vibrations in two mutually perpendicular directions and the torsional vibrations are all coupled. The ends of the channels and the periodic intermediate supports are modelled with springs having finite flexural and torsional stiffnesses. Single-point force excitation has been considered throughout the study, although the developed method is also capable of dealing with multi-point excitation. The channels are assumed to be of Euler-Bernoulli type beams. The study also takes the effects of cross-sectional warping into consideration. The developed method is suitable for structurally damped analysis and in addition to yielding forced vibration characteristics; it also straightforwardly reveals the free vibration properties like the mode shapes.     

Dynamic stability of weakly damped oscillators with elastic impacts and wear

The dynamics of non-linear oscillators comprising of a single-degree-of-freedom system and beams with elastic two-sided amplitude constraints subject to harmonic loads is analyzed. The beams are clamped at one end, and constrained against unilateral contact sites near the other end. The structures are modelled by a Bernoulli-type beam supported by springs using the finite element method. Rayleigh damping is assumed. Symmetric and elastic double-impact motions, both harmonic and sub-harmonic, are studied by way of a Poincaré mapping that relates the states at subsequent impacts. Stability and bifurcation analyses are performed for these motions, and domains of instability are delineated. Impact work rate, which is the rate of energy dissipation to the impacting surfaces, is evaluated and discussed. In addition, an experiment conducted by Moon and Shaw on the vibration of a cantilevered beam with one-sided amplitude constraining stop is modelled. Bifurcation observed in the experiment could be captured.     

A simplified nonlinear dynamic model for the analysis of pipe structures with bolted flange joints

Bolted flange joints are widely used in engineering structures; however, the dynamic behavior of this connection is complex in nature. In this paper, a simplified nonlinear dynamic model with bi-linear springs is proposed and validated for pipe structures with bolted flange joints. First, static mechanical properties of the bolted flange joint are investigated. The analytical solution reveals that the axial stiffness of the bolted flange joint is different in tension and compression. Then, nonlinear springs with different stiffness in tension and compression are employed to represent the bolted flange joint. A special type of dynamic behavior, coupling vibration in the transverse and longitudinal directions, is observed in analytical derivation. Finally, relevant physical experiments and numerical simulations are performed. The physical experiments confirm the existence of the coupling vibration behavior. The relationship of longitudinal and transverse vibration frequencies is discussed. The numerical solutions reveal that the simplified nonlinear dynamic model better fits the physical response than conventional reduced linear beam model.