Balancing a Rotor with Two Coupled Perfectly Rigid Bodies

The steady motions of a vertical rotor balanced by two coupled perfectly rigid bodies (PRBs) are analyzed for stability. The rotor executes plane-parallel motions. The PRBs can rotate together about the rotor axis and about its transverse axes through equal angles but in opposite directions. It is established that when the mass of the PRBs is much less than that of the rotor and the rotor spins with a superresonant speed, only the basic steady motions, which are those of the best-balanced rotor, are asymptotically stable.     

Comment on: I-Shih Liu: Constitutive theory of anisotropic rigid heat conductors

In I-Shih Liu’s paper (J Math Phys 50:083506, 2009), the compatibility of anisotropy and material frame indifference of a rigid heat conductor is investigated. For this purpose, the deformation gradient is introduced into the domain of the constitutive mapping. Because of the presupposed rigidity, the deformation gradient is here represented by an orthogonal tensor. The statement, that the usual procedure—not to introduce the deformation gradient into the state space of rigid heat conductors—causes isotropy because of material frame indifference, is misleading.     

Dynamics of a cylindrical body in a liquid-filled sector of a cylindrical layer under rotational vibration

The dynamics of a rigid cylindrical body subjected to high-frequency rotational vibration about its own axis in a liquid-filled sector of a horizontal cylindrical layer are investigated experimentally and theoretically. It is found that the vibrations generate an average force on the body which in the case of a body denser than the fluid is directed toward the axis of rotation. Under certain conditions this force exceeds the gravity force, causing the body to float. This effect is analyzed theoretically in the high-frequency low-amplitude vibration approximation. It is shown that the force detected acts on the body over the entire fluid volume. Perm’. Paris. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 29–39, July–August, 1998. The work partially supported by the Russian Foundation for Basic Research (project No. 96-01-00932) and Grant No. 920-5208/95 of the US National Aviation and Space Administration (US NASA).     

Pore space network characterization with sub-voxel definition

Morphological measurements in 3D for pore space characterization (connectivity pore-body/throat classification, shape factors, virtual fluid intrusion) are based on computed intensive digital-thinning operations for skeletonization and medial axis extraction from 3D digital images. We present an alternative method that is measurably faster and allows sub-voxel definition of the pore space network. The method allows extracting—based on morphological considerations only—the centered and shortest stream-lines—i.e., the paths—to follow in order to go through the pore space from one given point to another and to exit. In addition the method penalizes long and narrow pore-throats in favor of short stubby/ones—i.e., it has a built-in exemplification capacity. It exploits well-established mathematical methods successfully applied in medical endoscopy.     

On the validity of planar, thick curved beam models derived with respect to centroidal and neutral axes

Most dynamic analyses of planar curved beams found in the literature are carried out based on a curved beam model which assumes that the neutral axis coincides with the centroidal axis of the curved beam. This assumption leads to governing equations of motion which are relatively simple with analysis results that have acceptable accuracy for shallow curved beams. However, when a curved beam is not shallow and/or its cross section is not doubly symmetric, the offset distance between the neutral and centroidal axes may be large enough to influence the in-plane dynamics of the curved beam even for small motion. In this paper, the validity of this underlying assumption for modeling a linear curved beam is examined. To this end, two sets of equations of motion governing the in-plane dynamics of a planar curved beam are derived, in a consistent manner for comparison, based on the linear strain-displacement relations and Hamilton’s principle. The first set of equations is derived from the displacement components measured with reference to the neutral axis of the curved beam while the second set is derived with respect to the centroidal axis of the cross section. The curved beam is considered extensional and the effects of rotary inertia and radial shear deformation are included. In addition to the curvature parameter that characterizes the wave motion for both curved beam models, an eccentricity parameter is introduced in the first model to account for the offset between the neutral and centroidal axes. The dynamic behavior predicted by each curved beam model is compared in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and modeshapes, and frequency responses. In order to ensure that the comparison is accurate, the wave propagation technique is applied to obtain exact wave solutions. It is shown that, when the curvature parameter is not small, the underlying assumption has a substantial impact on the accuracy of the linear dynamic analysis of a curved beam.     

Thermovibrational convective instability of mechanical equilibrium of an inclined fluid layer

The convective instability of mechanical equilibrium of an inclined plane layer of fluid developing under the action of a static gravity field and high-frequency vibration is studied. Configurations corresponding to four directions of the equilibrium temperature gradient — vertical, longitudinal, horizontal, and transverse — are considered for an arbitrary orientation of the vibration axis. The stability limits and the characteristics of the critical perturbations are determined. Perm’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 8–15, January–February, 1998. This investigation was carried out with partial support form RSA-NASA (contract No. 920/18 — 5208/96).     

Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body

We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations.     

The twisting tennis racket

This paper describes, analyzes, and explains a novel twisting phenomenon which occurs in a triaxial rigid body (such as a tennis racket) when it is rotating about an axis initially near its unstable intermediate principal axis.     

Two-dimensional problem of a uniform flow of a two-layer fluid of finite depth past a circular cylinder

The linear steady problem of an irrotational uniform flow past a horizontal circular cylinder located in the upper or in the lower layer of a two-layer fluid is solved by the multipole-expansion method. The flow is perpendicular to the axis of the cylinder. The fluid is assumed to be inviscid and incompressible, and the flow in each layer is assumed to be potential. The upper layer can be bounded by a free surface or a solid lid, and the lower layer by a rigid horizontal bottom. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 91–101, November–December, 1998.     

Effect of Rotation on Rayleigh—Lamb Waves in an Isotropic Generalized Thermoelastic Diffusive Plate

The present investigation is aimed at studying the effect of rotation on propagation of Rayleigh—Lamb waves in a homogeneous isotropic thermoelastic diffusive plate of finite width in the framework of different theories of thermoelasticity, including the Coriolis and centrifugal forces. The medium is subjected to stress-free, thermally insulated, isothermal, and chemical potential boundary conditions and is rotating about an axis perpendicular to its plane. Secular equations corresponding to the symmetric and skew-symmetric modes of the plate are derived. Phase velocities and attenuation coefficients of various possible modes of wave propagation are computed from the secular equations. Amplitudes of displacements, temperature, and concentration for symmetric and skew-symmetric modes of plate vibrations are computed numerically. The computed results are presented graphically.     

Measurement on particle rotation speed in gas–solid flow based on identification of particle rotation axis

When a two-dimensional (2D) imaging system is used to visualize particle motion in a 3D gas–solid flow, the particle rotation speed was found extremely difficult to be accurately measured due to the fact that the direction of rotation axis was usually random and hard to be distinguished. The paper presents a method to calculate the particle rotation speed from particle images based on the identification of its rotation axis using two or more characteristic points on its surface. The idea was analyzed and realized in a mathematical way and based on which a calculation program was given. The measurement method was verified with an experiment using a small sphere with known rotation axis and rotation speed. The effects of several factors, including the direction of the particle rotation axis, the particle image resolution, the types and positions of characteristic points, etc., on the measurement error are discussed. The error is found to be acceptable for most cases. The measurement method was finally applied to those small glass beads in a real 3D gas–solid flow inside a cold circulating fluidized bed (CFB) riser, which indicates that the problems of 2D imaging system applying to 3D particulate system could be solved by using this mathematical method.     

Elastic bending analysis of bilayered beams by an alternative two-variable method

An alternative two-variable method is used to reanalyze thermoelastic bending problems of bilayered beams subjected to external moments and internal stresses. The differences among zero-stress axis, zero-strain axis (i.e. neutral axis), bending axis, centroidal axis, and the parameter conditions for null/single/dual zero-stress axes are investigated analytically and numerically. Comparisons of thermoelastic stress predictions by the present model with Stoney's model and Hsueh's model are discussed in a representative case of GaAs top coat/Si substrate wafers. Results showed that the neutral axis does not coincide with the zero-stress axis in the general case, and the numbers and the locations of zero-strain or zero-stress axes depend on not only elastic modulus, thickness and/or thermal expansion coefficient ratios between the film and the substrate but also mechanical/thermal loading ratio.     

Viscosity of Cryoprotective Agents Near Glass Transition: A New Device,Technique, and Data on DMSO,DP6, and VS55

The low strain-rate viscosity of glass-forming cryoprotective agents (CPAs) in the vicinity of the glass transition is studied experimentally. Data on the mechanical behavior in this regime is necessary to the long-term goal of developing planning tools for cryopreservation via vitrification (vitreous means glassy in Latin); such tools will provide guidelines for reducing thermal stress with its devastating effects. While the flow behavior of some glass-forming CPAs is well documented in the literature for the upper part of the cryogenic temperature range (where the CPA has a comparatively low viscosity), it is the flow behavior near the glass transition temperature (where the CPA behaves as nearly a solid with an extremely high viscosity) which is critical to the analysis of stress that develops in the cryopreserved material. If the elevated viscosity limits the material’s ability to flow—in order to accommodate the thermal strain resulting from large temperature gradients, especially at the high cooling rates necessary to form glass—structural damage may follow. Information on the behavior of the CPA in the lower part of the cryogenic temperature range is largely unavailable. A new measurement device is presented in this study, in which a solid rod is pulled from a long narrow cup containing a CPA, producing an essentially one-dimensional and isothermal field of flow. The viscosity and relaxation time of the CPA is inferred from measurements of the resulting load on the rod when extracted at a constant velocity. The current study reports on experimental data near glass transition of 7.05 M DMSO, a reference CPA solution, and the CPA cocktails VS55 and DP6.     

The contact problem for a class of orthotropic elastic solids

The contact problem between two orthotropic solids is examined. The problem is solved by using Lodge's method, which permits the transformation of the boundary-value problem of an anisotropic solid to a form identical with the corresponding problem of an isotropic medium. The proposed solution is then compared with known results of certain cases and it is observed that it producesHertz's solution when used for an isotropic case,Lodge's solution when applied to contact between an orthotropic solid and a rigid plane and, finally,Love's solution if the solid is transversely isotropic with the axis of material symmetry perpendicular to the rigid plane of contact.     

Sound Propagation in Rigid Porous Media: Non-local Macroscopic Effects Versus Pores Scale Regime

This paper investigates how the regime (quasi-static, transient, out of equilibrium) of the phenomena occurring at pores scale determine the nature of the (non)-local effect—in time and/or space—involved in the macroscopic behavior of a porous medium. The study focuses on sound propagation examining—through the homogenization method of periodic media—situations of single porosity, Rayleigh scattering and double porosity. Non-locality effects reveals the loss of a perfect quasi-static equilibrium free of volume loading at the local scale. The non-locality in time is due to phenomena in transient regime at the ERV scale, while non-locality in space is due to the non-homogeneity in space of the macrofields. The generality of the arguments lead to infer that the conclusions about non-locality versus pores scale regime, could be extended to other physical phenomena in heterogeneous media.     

Study of the tensile fracture of plates with a rigid linear inclusion

The limit equilibrium of elastoplastic body is studied under the conditions of a plane problem. The body contains a linear inclusion, which is rigid but of finite rupture strength. The plastic or prefracture zones develop near the ends of the inclusion and are modeled by slip cracks along the matrix—inclusion interface. A new interpretation of the boundary conditions is proposed to solve a model problem for such a composition, and its analytical solution is derived. Two possible mechanisms of local fracture are considered: (a) fracture of the inclusion and (b) separation of the inclusion. The critical length of the inclusion is determined. This length together with the elastic and strength parameters of the composition determines the mechanism of local fracture. The limit loads are found for each mechanism of fracture. State Academy of Water Industry, Rovno, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 7, pp. 123–129, July, 2000.     

Direct determination of curvatures of bent plates using a double-glass-plate shearing interferometer

A single-exposure reflective moiré technique for the direct recording of curvature contours of laterally loaded plates is proposed. The specular reflection of a coherent, collimated beam of light by the loaded specimen is collected by a field lens and focussed onto a shearing interferometer. The shearing interferometer consists of two thin, identical plane-parallel glass plates, placed successively in the optical path near the focal plane and aligned at 45 deg to the optical axis. The two plates shear the narrow convergent beam from the lens twice optically, producing three sets of images—superposing one over the other—at the image plane. Curvature fringes result in the region overlapped by all three images.     

On the Arnold Stability of a Solid in a Plane Steady Flow of an Ideal Incompressible Fluid

We study the stability of a rigid body in a steady rotational flow of an inviscid incompressible fluid. We consider the two-dimensional problem: a body is an infinite cylinder with arbitrary cross section moving perpendicularly to its axis, a flow is two-dimensional, i.e., it does not depend on the coordinate along the axis of a cylinder; both body and fluid are in a two-dimensional bounded domain with an arbitrary smooth boundary. Arnold's method is exploited to obtain sufficient conditions for linear stability of an equilibrium of a body in a steady rotational flow. We first establish a new energy-type variational principle which is a natural generalization of the well-known Arnold's result (1965a, 1966) to the system “body + fluid.” Then, by Arnold's technique, a general sufficient condition for linear stability is obtained. Received 21 February 1997 and accepted 23 June 1997     

A Rigid Elliptical Disc-Inclusion, in an Elastic Solid, Subjected to a Polynomial Normal Shift

The article investigates the elastostatic problem of axial translation of a rigid elliptical disc-inclusion embedded into an elastic solid. The case of constant translation was previously studied by Kassir and Sih. The present study focuses on the more general case in which the translation is characterized by an arbitrary-order polynomial in the Cartesian coordinates of the points of the disc. By means of Fourier transforms, the problem is reduced to a two-dimensional integral equation. Ferrers–Dyson's and Galin's theorems and some of their refinements by the present author are then employed to derive a closed-form solution of the integral equation, permitting the displacements and the stresses in the entire solid to be expressed in explicit analytical form. In particular, the behavior of the stress field in the immediate vicinity of the edge of the disc is analyzed in great detail and closed-form expressions for the stress intensity factors are deduced for an arbitrary-order polynomial normal shift. Furthermore, these results are generalized, based on an idea by Tada and Paris, to flat rigid inclusions of an arbitrary shape having an axis of symmetry. This revised version was published online in June 2006 with corrections to the Cover Date.     

Derivation of Plate and Rod Equations for a Piezoelectric Body from a Mixed Three-Dimensional Variational Principle

We use a mixed 3-dimensional variational principle to derive 2-dimensional equations for an anisotropic plate-like piezoelectric body and one-dimensional equations for an anisotropic beam-like piezoelectric body. The formulation accounts for double forces without moments which may change the thickness of the plate and deform the cross-section of the rod. The dependence of the bending rigidities of a transversely isotropic plate upon the angle between the normal to the midsurface and the direction of transverse isotropy is exhibited. The plate equations are used to study the cylindrical deformations of a transversely isotropic plate due to equal and opposite charges applied to its top and bottom surfaces. It is also found that a piezoelectric circular rod with axis of transverse isotropy not coincident with its centroidal axis and subjected to electric charges at the end faces is deformed into a non-prismatic body.