On the optimal shape of compressed rotating rod with shear and extensibility

In this paper, the optimal shape of a compressed rotating rod which maintains stability against buckling is presented. In the rod modeling, extensibility along the rod axis and shear stress is taken into account. Using Pontryagin's maximum principle, the optimization problem is formulated with a fourth order boundary value problem. The optimally shaped compressed rotating (fixed-free) rod has a finite cross-sectional area on the free end. This shape is qualitatively different from that suggested by the Bernoulli-Euler theory with zero cross-sectional area on the free end. In addition, the Bernoulli-Euler theory overestimates the buckling load, and this effect is more significant in the optimally shaped rod than for the corresponding constant cross-sectional rod consisting of the same material volume and length. In order to show this effect, it is necessary to use a generalized constitutive model which takes real material properties, such as axial extensibility and shear stress into account. Particularly, the solution of this generalized problem, obtained for thin rods, approaches the classical solution predicted by the Bernoulli-Euler theory.     

On the Optimal Shape of a Compressed Rotating Rod

By using Pontryagin's maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.     

Optimal shape of a vertical rotating column

We determine the shape of the lightest rotating column that is stable against buckling, positioned in a constant gravity field, oriented along the column axis. In deriving the optimality conditions, the Pontryagin's principle was used. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. For this problem a variational principle and a first integral are formulated. Also a priori estimates of the cross-sectional area at the lower end are presented. The procedure is illustrated by three concrete examples. The problem treated here may be considered as a step in the dynamic optimization procedure of a heavy rotating column.     

Variational principle for a special Cosserat rod

In this paper, we describe the nonlinear models of a rod in three-dimensional space based on the Cosserat theory. Using the pseudo-rigid body method and variational principle, we obtain the motion equations of a Cosserat rod including shear deformations.     

A non-linear rod theory for high-frequency vibrations of thermopiezoelectric materials

In relation to electroelastic media with thermopiezoelectric coupling, the system of one-dimensional equations is consistently derived so as to accommodate the high-frequency vibrations of a rod with temperature-dependent material. In the first part of the paper, a unified variational principle of differential type is presented which describes the fundamental equations of thermopiezoelectricity with second sound, including the physical and geometrical non-linearities. In the second part, the hierarchic system of rod equations is systematically deduced from the three-dimensional fundamental equations by use of Mindlin's method of reduction. The hierarchic system of equations which is derived in both differential and variational forms is capable of predicting the extensional, thickness-shear, flexural and torsional as well as coupled vibrations of the rod of uniform cross-section. All the higher-order effects are taken into account as deemed pertinent in any particular case. In the third part, attention is confined to certain cases involving special motions, materials and geometry. Besides, the uniqueness is investigated in solutions of the linearized system of rod equations and the sufficient conditions are enumerated for the uniqueness of solutions.     

Stability analysis of a substructured model of the rotating beam

One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. Among other approaches, the substructuring technique was proven to be a valid strategy to account for this problem. Later, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated. At the same time, it was found the existence of a critical angular velocity, beyond which the system becomes unstable that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.     

The stability of a compressed rotating rod

The classical stability problem of a compressed hinged elastic rod rotating with constant angular velocity about the axis that passes through the hinges is considered. It is assumed that the compressive force is constant and the line of its action coincides with the axis of rotation of the rod. The stability of a solution of the nonlinear problem that describes deformation of the rod under the action of the compressive force and the distributed centrifugal load is studied within the framework of the stability theory of dynamic systems with distributed parameters. The buckling paramcters of the problem are determined. Calculation results are given. Technology Institute, Altai State Technical University, Biisk 659305. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 190–197, July–August, 2000.     

ANALYTICAL SOLUTIONS FOR RESPONSE OF COLLISION OF PARTICLE WITH CONICAL ROD CAUSED BY LONGITUDINAL VIBRATION

The objective is to present exact analytical solutions of longitudinal impact analysis for slender conical rods struck by a particle and a new method is proposed for conical rod-particle impact analysis, in which the superposition method is used and the response of the rod is presented. These analytical results are exact and can be used to validate the numerical methods or other analytical results. The numerical example shows that one of the advantages of the present method is that the analytical form is very simple. The result is that mass ratio and some variables describing the geometrical shape of rods such as taper, length and radius play an important role in impact dynamic system.     

Fluid flow in a rotating cylindrical container with a rotating disk at the fluid surface

Fluid flow in a rotating cylindrical container of radius R_w and height H with a co-axially rotating disk of radius R_d at the fluid surface is numerically investigated. The container and the disk rotate with angular velocities Ω_w and Ω_d, respectively. We solve the axisymmetric Navier-Stokes equations using a finite-volume method. The effects of the relative directions and magnitudes of the disk and container rotations are studied. The calculations are carried out with various ratios of Ω_w and Ω_d for H/R_w = 2 and R_d/R_w = 0.7. Streamlines and velocity vectors in the meridional plane and azimuthal velocities are obtained. The flow fields in the meridional plane are discussed with relation to azimuthal velocities in the interior of the container. The numerical results are also compared with experimental data.     

The strain solitary waves in a nonlinear elastic rod

Solitary strain waves in a nonlinear elastic rod are analysed in this paper; influence of the physical and geometrical parameters of the rod on the waves are discussed; some main properties of the solitary waves are pointed out.     

超细长弹性杆的分析力学问题

超细长弹性杆作为DNA等生物大分子链的力学模型，其平衡和稳定性问题已成为力学与分子生物学交叉的研究热点．虽然在Kirchhoff动力学比拟的基础上，用分析力学方法讨论弹性杆的文章已见诸文献，但尚未形成弹性杆分析力学的严格理论．本文研究了超细长弹性杆分析力学的若干基础性问题．对杆截面的自由度、虚位移、约束方程及约束力等基本概念给出严格的定义和表达式．建立弹性杆平衡的D’Alembert-Lagrange原理、Jourdain原理和Gauss原理；从D’Alembert-Lagrange原理导出Hamilton原理．从变分原理出发导出Lagrange方程、Nielsen方程、Appell方程和Hamilton正则方程；对于受约束的弹性杆，导出了带乘子的Lagrange方程．讨论了Lagrange方程的首次积分．对于杆中心线存在尖点的情形，导出了微段杆平衡的近似方程。     

Film flow around a fast rotating roller

In this study, the film thickness around the roller is numerically estimated using the volume of fluid (VOF) method to clarify the film-formation process around the rotating roller. Parametric studies were performed to compare the effects of ink properties (viscosity, surface tension) and operational conditions (roller rotation speed, initial immersed angle) on film thickness. The viscosity of the ink and the speed of rotation of the roller were found to be the dominant factors that determine the ink film thickness. In addition, a correlation equation is proposed to predict the thickness of the ink film around a printing roller rotating at a speed of 20–30 rad/s, as a function of angular position, angular velocity, and viscosity.     

Viscoelastic flow between eccentric rotating cylinders: unstructured control volume method

This paper reports a convergent numerical algorithm for the Upper-Convected Maxwell (UCM) fluid between two eccentric cylinders at various eccentricity ratios (?); the outer cylinder is stationary, and the inner one rotating. The problem is solved by an unstructured control volume method (UCV), which is designed for a general viscoelastic flow problem with an arbitrary computational domain. A self-consistent false diffusion technique and an iteration scheme are used in combination to solve the problem. The computations of the UCM fluid using the numerical algorithm are carried out to a higher value of the Deborah number (De) at each eccentricity tested than hitherto possible with previous numerical simulations. The solutions are compared with previous numerical results, confirming the effectiveness of the UCV method as a general technique for solving viscoelastic flow problems.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Fluid flow in a rotating curved rectangular duct

The fluid flowing in a rotating curved duct is subjected to both the Coriolis force due to a rotation and the centrifugal force due to a curvature. In this paper, the combined effects of the two forces on the flows in rotating curved rectangular ducts are examined numerically. According to the aspect ratio of the cross-section, the rectangular ducts are divided into three types: η>1, η=1, η<1, where η is the aspect ratio. The variations of the flow structures with the force ratio F (the ratio of the Corislis force to the centrifugal force) are studied in detail and many hitherto unknown flow patterns are found. The effects of the force ratio and the aspect ratio of the cross-section on the friction factor are also examined. Present results show both the characteristics of the secondary flow, axial flow and the natures of the friction factor.     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Body wave propagation in rotating elastic media

We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation Ω. Wave propagation is shown to be directly related to the Kibel number Ki=ω/Ω, where ω is the wave frequency. Two dispersive waves W₁ and W₂ are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W₁ shows a cutoff frequency ω_c=Ω below which it does not propagate. The case of small angular rotation Ω is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order O(Ki⁻¹).     

Nonparallel stability of the flow in an axially rotating pipe

The linear stability of the developing flow in an axially rotating pipe is analyzed using parabolized stability equations (PSE). The results are compared with those obtained from a near-parallel stability approximation that only takes into account the axial variation of the basic flow. Though the PSE results obviously coincide with the near-parallel ones far downstream, when the flow has reached a Hagen-Poiseuille axial velocity profile with superimposed solid-body rotation, they differ significantly in the developing region. Therefore, the onset of instability strongly depends on the axial evolution of the perturbations. The PSE results are also compared with experimental data from Imao et al. [Exp. Fluids 12 (1992) 277], showing a good agreement in the frequencies and wavelengths of the unstable disturbances, that take the form of spiral waves. Finally, a simple method for detecting one of the conditions to characterize the onset of absolute instability using PSE is given.     

The effect of side walls on particles mixing in rotating drums

In this paper, we study the effects of the presence and shape of side walls and of the overall length of rotating cylindrical drums on the mixing of particles with differing sizes by application of the discrete element method (DEM). By varying the semi-axis of the spheroidally shaped side walls and the length of the overall drum, we observe the formation of circulation patterns near the side walls. Although there is a vast amount of literature studying mixing regimes in rotating drums, little is known about the effect of the side walls of the drum on particle mixing. The results of our study demonstrate that introducing curved side walls induces a strong circulation pattern near these side walls, but has, paradoxically, a negative impact on mixing and actually promotes segregation. The cause for this segregation is the difference in velocity of differently sized particles near the curved side walls. Large particles accumulate at the curved side walls, whereas small particles move away from the curved side walls. When the length of the drum is increased, the overall effect of the side walls is decreased, although it does remain observable, even in very large drums.     

Flow produced in a conical container by a rotating endwall

Numerical calculations have been carried out for flow in a truncated cone generated by rotation of one endwall. For both convergent (radius increasing with approach to the rotating endwall) and divergent geometries, vortex breakdown is suppressed beyond a certain angle of inclination of the sidewall. At the same time Moffat eddies of increasing strength and extent appear in the corner between the sidewall and the non-rotating endwall. For the divergent geometry, a zone of recirculation appears on the sidewall and eventually merges with the Moffat eddies. The flow phenomena identified from streamline patterns are consistent with the calculated variation of pressure around the periphery of the computational domain.