Vibration and buckling of rotating flexible rods at transitional parameter values

Summary We consider the connection between vibration and buckling problems for a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The rod is assumed off-clamped, i.e. the axis of rotation does not pass through the rod's clamped end. The resulting fourth-order boundary value problem with a turning point for the free vibrations is solved using uniform approximations in a transitional parameter range where high rotation rates balance small off-clampings. Second approximations to the vibration eigenvalues are used to determine critical buckling rotation rates for the slightly off-clamped rods. Buckling is unexpected in this situation as the rod is wholly under tension.     

Transverse buckling of a rotating Timoshenko beam

Summary This work considers a group of problems associated with rotating Timoshenko beams. The beam is not assumed to be hubclamped, i.e. the axis of rotation does not necessarily pass through the beam's clamped end. Cases of physical interest involving off-clamped beams include wobbling rotors, impellor blades, and turbine blades.For clamped-free boundary conditions, we seek solutions of the governing equations which correspond to transverse buckling. For the rotor, it is known that Euler-Bernoulli beams do not have buckled modes. By contrast, the Timoshenko beam will have an infinite number of buckled modes. In the impellor blade case, both Euler-Bernoulli and Timoshenko beams will have an infinite number of buckled modes. However, the Timoshenko beam will buckle at a lower eigenrotation speed. This is also true for the case of a rotating Timoshenko beam with clamped-clamped boundary conditions, e.g. a turbine blade clamped at both the rim and hub of a rotating platform.Analytic results for both the clamped-free and clamped-clamped cases are augmented by results obtained from numerical solution of the corresponding boundary value problems.     

On the slow steady motion of a viscous fluid due to two rotating tori

Summary An approximate solution, correct to order three in a small parameter, is established for the velocity field induced in a viscous fluid by the slow rotation of two tori. The tori possess a common axis and have concentric circular sections. The small parameter is the ratio of the radius of the outer circular section to the distance of the section centre from the axis of rotation. The torque acting on each toroidal surface is also obtained.With 3 Figures     

Asymptotic forms of a simplified version of the non-linear Reissner equations for clamped elastic spherical caps under outward pressure

A new, simplified version of Reissner's equations for the torsionless, axisymmetric deformation of elastically isotropic shells of revolution suffering small strains but large angles of rotation is specialized to clamped spherical caps under uniform outward pressure. The non-dimensional equations contain a thickness parameter, a shallowness parameter, and a load parameter. The latter two are written as powers of the former and the dependent variables scaled so that as the thickness parameter goes to zero, meaningful limit equations emerge. Seventeen distinct sets of simplified equations are found. In thirteen cases these are linear and the solutions are listed. These results should provide a useful set of benchmarks for testing the efficacy of numerical codes which often have difficulties with very thin shells.This research was supported by the National Science Foundation under grant MSM-8412334     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – Part II

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. A new correction procedure for transfer from the embedded crack to surface crack configurations is presented, which is valid for all a/t-values. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions for embedded cracks are describable by one equation only (using Heuman's lambda function). For various a/c-ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.     

Effect of warping-pretwist coupling on the natural vibration of torsionally clamped-pinned thin-walled open-profile bars

Exact natural frequencies are determined for the torsional vibration of pretwisted thin-walled open-profile bars having torsionally clamped and pinned end restraints. The equation of motion and boundary conditions are derived from a multifilament model of a pretwisted thin-walled bar undergoing warp deformation and a moderately large twist about the elastic axis. Closed-form torsional frequencies and mode shapes are derived from the linear terms of the resulting non-linear equations. As the length-to-leg ratio (L/b) and leg-to-thickness ratio (b/t) of typical open-profile bars are varied, the individual and collective effects of warping and pretwist on the non-dimensional torsional frequencies are demarcated. In addition, the closed-form solutions offered here reveal considerable destiffening of the torsional frequencies due to warping-pretwist coupling present in thin-walled bars having pretwisted, unsymmetrical, open-profiles and torsionally clamped and pinned end restraints. This coupling effect, which have been ignored in thin-walled bar vibration theories hitherto, is brought out by the Wagner effect and warping shear stresses acting on the profiles.     

角铺设层板弯曲问题的一种半解析法

本文把中的方法用于单连通角铺设层合板的弯曲分析,多组算例表明了此法有较好的收敛性,卽使对板的内力,也往往能以较少的侍定系数及配点获得令人满意的结果。      

Effect of rotation on the stability of the melt during the solidification of a binary alloy

Summary Nonlinear solutal convective flow in the melt under a planar solidifying surface and rotating about the vertical axis is investigated in the limit of small segregation. An evolution equation governing the cellular structure of the flow of a binary alloy is derived. This equation incorporates convective instabilities and rotational effects. The presence of rotation can inhibit the onset of the cellular structure by effectively magnifying the segregation.     

Weight functions and Stress Intensity Magnification factors for elliptical and semi-elliptical cracks under variable normal stress – part I

Surface cracks under peak stresses are investigated. The calculational procedure is based on the general form of the weight function for an elliptical crack embedded in an infinite solid. Two points on the contour of the ellipse are investigated. The superposition method is used for transfer from the embedded crack to surface crack configurations. Weight functions for both points have been found with the crack aspect ratio a/c as parameter. For the point at the end of the minor axis all weight functions are describable by one equation only (Heuman's lambda function). For various a/c ratios of the surface crack under different stress distributions the stress intensity magnification factors are given.     

Flexural vibration behavior of piezoelectric heterogeneous bimorph beams

Bending oscillations of piezoelectric bimorph beams are effective sound sources in gases or fluids, and, therefore, of practical interest. On the basis of the piezoelectric constitutive relations and the elastodynamic equations, the differential equation of flexural vibrations of thin rectangular piezoelectric heterogeneous bimorph beams, consisting of a piezoelectric layer glued onto an elastic substrate, is derived. The piezoelectric layer is polarized in thickness direction and can be excited to thickness vibrations by an electric alternating current voltage applied to electrodes covering the upper and lower surface of the layer. This causes an oscillating transverse contraction in the piezoelectric layer but not in the substrate, and, thus, generates flexural vibrations of the beam. The differential equation is solved analytically for beams of finite length with both ends free, one clamped and one free end, as well as for both ends clamped. Their vibration behavior in viscous fluids is considered. For a piezo-ceramic composite layer joined to a steel plate vibrating in air and in water, the analytical results are evaluated numerically as function of frequency     

Deformation of varying thickness of conical shells subjected to axisymmetric loading with various end conditions

A conical shell subjected to uniform internal pressure is analysed by the finite element displacement method. The end conditions considered are clamped-clamped (CC), small end clamped-big end free (CF), and small end free-and big end clamped (FC). The semi cone angles considered are (=) 30, 45 and 60°. It is shown that by properly varying the thickness the deflections and stresses can be brought down considerably. The results are presented in the form of graphs and tables.     

The unsteady motion of two-dimensional flags with bending stiffness

The motion of a two-dimensional flag at a time-dependent angle of incidence to an irrotational flow of an inviscid, incompressible fluid is examined. The flag is modelled as a thin, flexible, impermeable membrane of finite mass with bending stiffness. The flag is fixed at the leading edge where it is assumed to be either freely hinged or clamped with zero gradient. The angle of incidence to the outer flow is assumed to be small and thin aerofoil theory and simple beam theory are employed to obtain a partial singular integro-differential equation for the flag shape. Steady solutions to the problem are calculated analytically for various limiting cases and numerically for order one values of a non-dimensional parameter that measures the relative importance of outer flow momentum flux and flexural rigidity. For the unsteady problem, the stability of steady solutions depends only upon two non-dimensional parameters. Stability analysis is performed in order to identify the regions of instability. The resulting quadratic eigenvalue problem is solved numerically and the marginal stability curves for both the hinged and the clamped flags are constructed. These curves show that both stable and unstable solutions may exist for various values of the mass and flexural rigidity of the membrane and for both methods of attachment at the leading edge. In order to confirm the results of the linear stability analysis, the full unsteady flag equation is solved numerically using an explicit method. The numerical solutions agree with the predictions of the linear stability analysis and also identify the shapes that the flag adopts according to the magnitude of the flexural rigidity and mass.     

U-transformation-finite element method in 3-dimensional analysis of orthotropic laminates

For an orthotropic laminate, an equivalent system with doubly cyclic periodicity is introduced. Then a 3-dimensional finite element model for the equivalent system is transformed into the unitary space, where the large finite element matrix equation is decoupled into some small matrix equations. Such a decoupling very efficiently reduces the computational effort. For an orthotropic laminate with four clamped edges, no exact elasticity solution is available, and the deflection values predicted by different methods have a considerable difference each other for a small length-to-thickness ratio. The present predictions are the largest because the present method is a full 3-dimensional finite element analysis without superfluous constraints. Illustrative numerical examples are presented to observe the distributions of stresses through the thickness of the laminates.     

Free in-plane vibrations of highly buckled beams carrying a lumped mass

Summary Free undamped in-plane vibrations of shear undeformable beams around their highly buckled configurations are investigated neglecting rotary inertia effects. The beams are inertially nonuniform since a lumped mass is rigidly clamped along the span. Two mechanical models are considered depending on the boundary conditions in the post-buckling phases. First, the beam is considered inextensible because it is hinged at one end and is acted upon by an axial compressive force on the other end, a roller support, both in the buckling and post-buckling phases. In the second model, the beam is extensible in the post-buckling phase because the roller support boundary is changed into a fixed hinged end. Free undamped vibrations are governed, in the first case, by a homogeneous integral-partial-differential equation and, in the second case, by two coupled partial-differential equations with variable coefficients. The solutions of the associated eigenvalue problems are found employing two approaches: a semi-analytical method based on Galerkin discretization and a finite element method. A close agreement in the outcomes is found. The leading differences relating to the natural frequencies and linear normal modes of the two pre-stressed curved beam models are discussed; in particular, the occurrence of the veering phenomenon and the crossovers are outlined.     

Buckling analysis of bimaterial beams with single asymmetric delamination

Delamination is one of the most common failure modes of composite laminates. A one-dimensional mathematical model is developed to analyze the buckling behavior of a two-layered beam with single asymmetric delamination for simple supported and clamped boundary conditions. The characteristic equation is developed by using Bernoulli–Euler beam theory. Nondimensionalized parameters, axial and bending stiffnesses are introduced and used as input parameters to the derived characteristic equation, rather than the young's moduli and thickness ratios of the individual layers. A new parameter effective-slenderness ratio is introduced which gives a measure of local, mixed and global buckling behavior. The analysis is validated with previously published data. A parametric study is carried out.     

Elliptic integral solution of the extensible elastica with a variable length under a concentrated force

In problems of beams that may move relative to one of their spatially fixed supports, the length of the reference configuration of the beam that is currently located in between the supports depends on the deformation and therefore on the intensity of the external loads. In the present paper, the large deformation of a slender beam under a concentrated force is considered, for which one end is clamped while the other may slide through another clamping device in horizontal direction. The geometric symmetry properties allow the complete structure to be partitioned into four similar parts, each of which represents a cantilever beam under a tip force. In the analysis, a beam theory based on Reissner??s geometrically exact relations for the plane deformation of beams is utilized, from which the equation of a cantilever beam with an extensible axis is obtained by adopting a linear dependence of the stress resultants on the generalized strain measures. For this non-linear equation, a closed-form solution in terms of elliptic integrals is presented, from which the equilibrium shape of the entire structure is constructed afterward. The mid-span deflection and the additional length, by which the reference configuration exceeds the distance between the supports, are discussed and compared with the relations of the inextensible elastica, which exhibits bending deformation only. Moreover, a critical load is found for which no equilibrium configuration exists.     

Effect of large blowing rates on the compressible boundary layer flow at the stagnation point of a rotating sphere

Summary The effect of a large surface blowing (injection) rate on the steady laminar compressible boundary-layer flow at the forward stagnation point of a rotating sphere has been studied. The resulting coupled nonlinear ordinary differential equations have been solved using two methods, namely, the method of matched asymptotic expansion and the implicit finite difference scheme in combination with the quasilinearization technique. It is found that the boundary layer thickness increases considerably with the blowing rate. The location of the dividing streamline moves away from the surface with increasing blowing rates, but moves towards the surface when the total enthalpy at the wall or the rotation parameter increases. For large blowing rates and small rotation parameter the surface heat transfer and the surface shear stress in the tangential direction tend to zero, but the longitudinal shear stress remains finite but small. Also, for this case, the longitudinal shear stress at the wall is approximately found to be directly proportional to the sum of the total enthaply at the wall and to the square of the rotation parameter and inversely proportional to the blowing rates. The rotation parameter induces overshoot in the longitudinal velocity, and the magnitude of the velocity overshoot increases significantly with rotation and blowing parameters. However, there is no overshoot in the longitudinal velocity in the absence of rotation whatever may be the values of the blowing parameter.     

Stability of a thin viscous fluid film flowing down a rotating non-uniformly heated inclined plane

A thin viscous liquid film flowing down a rotating non-uniformly heated inclined plane is investigated. It is assumed that the rotation is small and the region of investigation is very far from the axis of rotation, so that the centrifugal force has a dominant role in the instability of the investigated region. Therefore, we have derived a Benney-like free surface evolution equation on the basis of a long-wave (small wave number) approximation and not included the Coriolis effects in the expansion of the dependent variables. Further, a linear and a weakly nonlinear study have been carried out. The linear study reveals that the growth of linear perturbation is independent of the Rossby number Ro, that is invariant with the Coriolis effect, while the linear phase speed c _r depends on Ro as well as on the Taylor number Ta. It also reveals that as Ta increases the stable zone decreases, and the influence of Ta is stronger for greater inclination of the rotating inclined plane. Again, it is found that the Marangoni number Mn has similar qualitative (destabilizing) behavior as Ta, but the destabilizing behavior of Ta is more at high Mn than at low Mn. The relation between the parameters Ta and Ro gives a unified parameter Ta_rot, which reflects the effect of rotation, and we found that the linear phase speed c _r first decreases with Ta_rot up to a critical value and then increases with Ta_rot. This is due to the fact that the Coriolis force is dominant at very small rotation, while for relatively large rotation the centrifugal force dominates the flow field. The weakly nonlinear study reveals that the effect of rotation appears in the form of both Coriolis and centrifugal force into the growth of finite amplitude perturbations in contrast to the growth of the infinitesimal perturbations where the rotation arrives in the form of centrifugal force only. Also, it plays significant role in the different stability zones, amplitudes of sub/super critical disturbances and nonlinear phase speed.     

Approximate predictions of ice accretion along conductors of finite torsional stiffness and control of rotation using counterweights

A mathematical/computational model for the approximate prediction of ice accretion on a conductor of finite torsional stiffness is constructed. The model, which is adapted to allow for the attachment of counterweights along the span, is three-dimensional and time-dependent. In the absence of counterweights, axial growth is predicted near the anchors, and cylindrical-sleeve growth at the centre of the conductor span. The latter is caused by eccentric ice loading on the windward side of the conductor which then induces the conductor to rotate. The ‘icing-clock’ model, based on the use of the large relaxation time approximation for particle impaction, yields an upper bound to the mass of ice accreted. At the end of each deposit time the torque density along the conductor span due to ice loading is calculated. On solving the non-linear torsion equation for the conductor the rotation of the conductor and location of the evolving ice surface as a function of the distance from the tower and the accretion time are then deduced. Numerical results are given for conductor with varying mechanical properties, which are represented in terms of a single dimensionless torque parameter, with and without the attachment of counterweights. For a specific case illustrative results are given for the mass of ice accreted and conductor rotation when one, two, three and four counterweights are attached per span.