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.

     

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.     

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.     

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.     

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.     

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.      

High-velocity operation of piezoelectric inertia motors: experimental validation

Piezoelectric inertia motors use the inertia of a body to drive it by means of a friction contact in a series of small steps. It has been shown previously in theoretical investigations that higher velocities and smoother movements can be obtained if these steps do not contain phases of stiction (“stick-slip” operation), but use sliding friction only (“slip-slip” operation). One very promising driving option for such motors is the superposition of multiple sinusoidal signals or harmonics. In this contribution, the theoretical results are validated experimentally. In this context, a quick and reliable identification process for parameters describing the friction contact is proposed. Additionally, the force generation potential of inertia motors is investigated theoretically and experimentally. The experimental results confirm the theoretical result that for a given maximum frequency, a signal with a high fundamental frequency and consisting of two superposed sine waves leads to the highest velocity and the smoothest motion, while the maximum motor force is obtained with signals containing more harmonics. These results are of fundamental importance for the further development of high-velocity piezoelectric inertia motors.     

On the connection between the principal components of the stress and permeability tensors of porous media

The dependence of the flow properties of rocks on the conditions under which they are loaded is one of the least studied questions in the general problem of the connection between the physical properties of porous systems and the loads acting on them. The experimental [l] and theoretical [2, 3] studies so far made have dealt primarily with the dependence of the permeability of oil and gas collector rocks on the hydrostatic compression to which they are subjected. In the present paper, a connection is established between the parameters of a nonuniform anisotropic load and the permeability tensor of rock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 173–177, January–February, 1984.     

Effects of microscale inertia on dynamic ductile crack growth

The aim of this paper is to investigate the role of microscale inertia in dynamic ductile crack growth. A constitutive model for porous solids that accounts for dynamic effects due to void growth is proposed. The model has been implemented in a finite element code and simulations of crack growth in a notched bar and in an edge cracked specimen have been performed. Results are compared to predictions obtained via the Gurson–Tvergaard–Needleman (GTN) model where micro-inertia effects are not accounted for. It is found that microscale inertia has a significant influence on the crack growth. In particular, it is shown that micro-inertia plays an important role during the strain localisation process by impeding void growth. Therefore, the resulting damage accumulation occurs in a more progressive manner. For this reason, simulations based on the proposed modelling exhibit much less mesh sensitivity than those based on the viscoplastic GTN model. Microscale inertia is also found to lead to lower crack speeds. Effects of micro-inertia on fracture toughness are evaluated.     

Rayleigh's conjecture on the principal frequency of the clamped plate

Communicated by J. Ball     

Combined effects of non-Newtonian couple stresses and fluid inertia on the squeeze film characteristics between a long cylinder and an infinite plate

On the basis of the Stokes micro-continuum theory together with the averaged inertia principle, the combined effects of non-Newtonian couple stresses and convective fluid inertia forces on the squeeze film motion between a long cylinder and an infinite plate are presented. A closed-form solution has been derived for squeeze film characteristics including the film pressure, the load capacity and the response time. Comparing with the Newtonian-lubricant non-inertia case, the combined effects of couple stresses and convective inertia forces provide an increase in the film pressure, the load capacity and the response time. In addition, the quantitative effects of couple stresses and convective inertia forces are more pronounced for cylinder–plate system operating at a larger couple stress parameter and film Reynolds number, as well as a smaller squeeze film height. To guide the use of the present study, a numerical example is also illustrated for engineers when considering both the effects of non-Newtonian couple stresses and fluid convective inertia forces.     

Influence of the surface roughness on the third-order moments of velocity fluctuations

Roughness wall effects in a zero pressure gradient turbulent boundary layers were investigated using hot-wire anemometry. The skewness and diffusion factors of u and v, the longitudinal and normal velocity fluctuations, were measured and represented using wall variables. The results indicate that the wall roughness removes the crossover point between sweep and ejection events to the outer region of the layer for a single Reynolds number Re _θ  > 3,000. This behaviour exhibits that the roughness surface favours the maintaining of sweep events obtained by a quadrant analysis. These results show that communication between the wall region and outer region of a turbulent boundary layer exists and the wall similarity hypothesis for a rough wall is questionable. The effect of the wall roughness on the position of the point crossover from sweep to ejection motions with respect to the wall seems to be the same as that obtained when the Reynolds number is higher. Received: 8 March 2000/Accepted: 15 May 2000