Some considerations on the experimental determination of moments of inertia

The experimental determination of the moments of inertia of rigid bodies, which is a common practice in many fields of technology, is usually performed using torsional or multifilar pendula. Different experimental techniques, based on the study of the periodic or non-periodic motion of the test object on which known forces act, are described and critically analysed. In particular, the effect of damping and, in the case of devices which can be assimilated to a pendulum, that of the non-linearity due to the finite amplitude of the motion, are studied in detail.The equations of motion of the multifilar pendulum are studied in detail to assess how the assumptions usually considered affect the accuracy of the results. An example of experimental determination of the moment of inertia of a flywheel, employing the trifilar pendulum currently used at the Laboratory of the Mechanics Department of Politecnico di Torino, shows how the theoretical considerations apply to a practical case.

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Sommario La determinazione sperimentale dei momenti d'inerzia di corpi rigidi, prassi consueta in molti settori della tecnologia, si avvale normalmente dell'utilizzo di pendoli di torsione o di pendoli multifilari. Nel presente lavoro vengono descritti ed esaminati criticamente vari metodi, basati sullo studio di moti periodici o non periodici dell'oggetto in prova sottoposto a forze note. Viene studiato in particolare l'effetto dello smorzamento e, nel caso di metodi basati su dispositivi assimilabili ad un pendolo, quello delle nonlinearità dovute all'ampiezza delle oscillazioni sui risultati della misura.Vengono studiate in dettaglio le equazioni del moto del pendolo multifilare allo scopo di valutare in quale misura le ipotesi semplificative normalmente accettate influiscano sulla precisione dei risultati. Un esempio di misura sperimentale effettuata mediante il pendolo trifilare correntemente utilizzato presso il laboratorio del Dipartimento di Meccanica del Politecnico di Torino conclude il lavoro e permette di verificare le conclusioni teoriche in un caso concreto.

     

求解截面主惯性矩的矩阵特征值法

提出求解截面形心主惯性矩的一种新算法,通过构造惯性矩阵,将主惯性矩和主惯性轴的计算归结为矩阵的特征值问题,算法简单,便于应用.     

Determining the moments of inertia of large bodies from vibrations in elastic suspension

To ensure the maneuvering capabilities of aircraft and high-speed sea vessels, designers should know the moments of inertia of their massive parts. But since the structure of some elements such as power units is very complicated, it is impossible to determine their moments of inertia analytically. Thus the problem of measuring the moments of inertia of massive large bodies arises. To this end, a measuring bench was designed in N. E. Zhukovskii Central Institute for Aerohydrodynamics (TsAGI) on the basis of a new method for determining the body moments of inertia from vibrations in the elastic suspension [1]. In this connection, it is necessary to develop the corresponding mathematical algorithms for determining the moments of inertia.In this paper, we develop mathematical algorithms for determining the body moments of inertia by using methods for identification of linear systems in the state space [2–5]. We present three versions of solving the problem of determining the body moments of inertia depending on the information about the method for exciting the vibrations or about the body parameters and the rigidity of the bench springs. We study the influence of damping on the accuracy of determining the moments of inertia. Numerical results are given for a specific system.     

On the dynamical symmetry points and the orientations of the principal axes of inertia of a rigid body

For an arbitrary rigid body, all dynamical symmetry points are found, and the directions of the axes of dynamical symmetry are determined for these points. We obtain conditions on the principal central moments of inertia under which the Lagrange and Kovalevskaya cases can be realized for the rigid body. We also analyze the set of orientations of the bases formed by the principal axes of inertia for various points of the rigid body.     

由三平行轴惯量求重心的一种方法

为了求得复杂刚体重心的位置, 根据转动惯量平行轴定理, 由已知轴转动惯量和一定的几何 关系推导出用于求重心的关系式, 从理论上给出两个可能重心坐标值, 再实际判断取舍.只要 用实验仪器先测量刚体对3个平行转轴的转动惯量, 就可由该关系式计算得到刚体重心的位 置. 这一方法有时要比直接测量重心更为简单.     

G. I. Taylor revisited: The cone of unextended directions in double slip

Recently developed general equations for the cone of unextended directions in arbitrary d double slip in f.c.c. crystals, in each of uniaxial tension and compression, are applied to the interpretation of finite deformation measurements from the classic series of papers (1923–27) by G. I. Taylor on cubic metal crystals. Measurements of comparable completeness are not found readilu elsewhere in the literature. From the equations and the experimental data (including lattice positions from X-ray diffraction analysis), the respective amounts of finite double slip are calculated for several cases from comparisons of theoretical and experimentally determined cones in stereographic projection. Apparent misconceptions in the literature about the subsequent deformations of Taylor's crystals after the loading axis reached a crystallographic symmetry line are addressed.     

CENTRIFUGAL INERTIA EFFECTS OF THE FLUID FLOW BETWEEN STEPPED SPHERICAL SURFACES

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认识惯性的过程与惯性定律

对惯性被认识的过程, 以及惯性定律的形成历史进行了探讨.     

Effect of inertia on the hydrodynamic interaction between two liquid capsules in simple shear flow

Three-dimensional numerical simulations using front-tracking method are performed to study the hydrodynamic interaction between two liquid capsules suspended in simple shear flow in presence of inertia. Capsules are modeled as liquid drops surrounded by neo-Hookean elastic membranes. In the limit of zero inertia, it has been known from past research that the hydrodynamic interaction between two deformable particles (drops/capsules) suspended in shear flow results in an irreversible shift in the trajectories of the particles as one particle rolls over the other. In this article, we show that the presence of inertia can significantly alter the capsule trajectories. When inertia is small but finite, the capsules do undergo an irreversible displacement, but the lateral separation between them first decreases before they roll over each other, unlike in Re ? 1. For moderate to high inertia, the capsules reverse their directions of motion before coming close to each other. The reversal of motion occurs progressively earlier in time (that is, the capsules come less closer to each other) with increasing inertia. The long-time behavior of the capsule–capsule interaction at finite inertia showed that the capsules engage in spiraling motions. Based on our simulations, four different regimes of capsule–capsule interaction at finite inertia are identified: (i) a self-diffusive type interaction, (ii) an outwardly spiraling motion, (iii) a fixed-orbit spiraling motion, and (iv) an inwardly spiraling motion in which the capsules settle with zero relative velocity. The reversal of motion, and the spiraling trajectories at finite inertia have no analogy in the limit of zero inertia. Such motions are explained by analyzing the flow field around a deformed capsule which shows reverse flow regions and off-surface stagnation points, similar to those previously reported in case of rigid spheres and cylinders under torque-free condition.     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

Issues of stability and accuracy in identifying the principal axis of inertia of an inhomogeneous rigid body with a rotating Hooke’s joint

The paper addresses issues of real-world operation of a device designed to experimentally identify the principal central axes of inertia of an arbitrary inhomogeneous rigid body. The effect of external and internal dissipation on the stability and accuracy of the device is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 131–143, November 2006.     

Effect of inertia on thermoelastic flow instability

Thermal effects induced by viscous heating cause thermoelastic flow instabilities in curvilinear shear flows of viscoelastic polymer solutions. These instabilities could be tracked experimentally by changing the fluid temperature T₀ to span the parameter space. In this work, the influence of T₀ on the stability boundary of the Taylor–Couette flow of an Oldroyd-B fluid is studied. The upper bound of the stability boundary in the Weissenberg number (We)–Nahme number (Na) space is given by the critical conditions corresponding to the extension of the time-dependent isothermal eigensolution. Initially, as T₀ is increased, the critical Weissenberg number, We_c, associated with this upper branch increases. Increasing T₀ beyond a certain value T^* causes the thermoelastic mode of instability to manifest. This occurs in the limit as We/Pe → 0, where Pe denotes the Péclet number. In this limit, the fluid relaxation time is much smaller than the time scale of thermal diffusion. T₀ = T^* represents a turning point in the We_c–Na_c curve. Consequently, the stability boundary is multi-valued for a wide range of Na values. Since the relaxation time and viscosity of the fluid decrease with increasing T₀, the elasticity number, defined as the ratio of the fluid relaxation time to the time scale of viscous diffusion, also decreases. Hence, O(10) values of the Reynolds number could be realized at the onset of instability if T₀ is sufficiently large. This sets limits for the temperature range that can be used in experiments if inertial effects are to be minimized.     

Dynamic pre-strain and inertia effects on the fracture of metals

A combined experimental and theoretical study is described which examines the influence of strain-rate and dynamic pre-strain on the ductile fracture of thin cylinders. The thin-cylinder configuration is particularly important in this case because it allows inertia terms to be directly incorporated into the theory of plastic instability. A series of quasi-static and dynamic tests is conducted on three materials with differing degrees of strain-rate sensitivity and strain-hardening. The experimental observation that fracture is inhibited at high strain-rates is in accord with the theory when inertia can no longer be considered insignificant. It is also shown that dynamic pre-strain has little or no effect on the flow stress or the strain at fracture in materials which-are essentially strain-rate insensitive, but does reduce the fracture strain in the strain-rate sensitive materials.     

单向板与双向板的划分依据与设计中若干问题的讨论

仍采用简单条带法, 通过分析载荷及弯矩在长、短跨方向的分配及随跨长比的变化情况, 明确提出当按弹性理论分析时, 宜采用长短跨长比为3作为划分单、双向板的界限. 然后从塑性铰线法设计双向板的基本方程入手, 经分析讨论后得出当按塑性理论设计时, 单双向板的分界实质上是相对的, 矩形板甚至正方形板究竟是单向板还是双向板, 完全取决于配筋方式这一结论. 最后给出了双向板按单向板设计的建议.     

Rossby inertia gravity solitary wave and the remote correlation between the East- and the West-Pacific subtropical high

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On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis.We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable.     

梁的载荷、内力微分关系推导方法的改进

在推导梁截面上的剪力、弯矩与分布载荷集度之间的关系时, 流行的《材料力学》教材 采用的推导过程在数学上失于严密. 根据导数定义和洛必达(L'Hospital)法则(或积分中 值定理)给出了逻辑严密的推导.     

The effects of confinement and inertia on the production of droplets

Recent experiments of Sibillo et al., Phys. Rev. Lett. 97:054502, (2006) investigate the effect of walls on flow-induced drop deformation for Stokes flow. The drop and the fluid in which it is suspended have the same viscosities. The capillary numbers vary from 0.4 to 0.46. They find that complex start-up transients are observed with overshoots and undershoots in drop deformation. Drop breakup is inhibited by lowering the gap. The ratio of initial drop radius to wall separation is fixed at 0.34. We show that inertia can enhance elongation to break the drop by examining Reynolds numbers in the range of 1 to 10. The volumes of the daughter drops can be larger than in the unbounded case, and even result in the production of monodisperse droplets.