An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams

An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions

Influence of third-degree geometric nonlinearities on the vibration and stability of pretwisted, preconed, rotating blades

The governing coupled flapwise bending, edgewise bending, and torsional equations are derived including third-degree geometric nonlinear elastic terms by making use of the geometric nonlinear theory of elasticity in which the elongations and shears are negligible compared to unity. These equations are specialized for blades of doubly symmetric cross section with linear variation of pretwist over the blade length. The nonlinear steady state equations and the linearized perturbation equations are solved by using the Galerkin method, and by utilizing the nonrotating normal modes for the shape functions. Parametric results obtained for various cases of rotating blades from the present theoretical formulation are compared to those produced from the finite element code MSC/NASTRAN, and also to those produced from an in-house experimental test rig. It is shown that the spurious instabilities, observed for thin, rotating blades when second degree geometric nonlinearities are used, can be eliminated by including the third-degree elastic nonlinear terms. Furthermore, inclusion of third degree terms improves the correlation between the theory and experiment

Vibration and buckling of rotating, pretwisted, preconed beams including Cooriolis effects

The effects of pretwist, precone, setting angle and Coriolis forces on the vibration and buckling behavior of rotating, torsionally rigid, cantilevered beams were studied. The beam is considered to be clamped on the axis of rotation in one case, and off the axis of rotation in the other. Two methods are employed for the solution of the vibration problem: (1) one based upon a finite-difference approach using second order central differences for solution of the equations of motion, and (2) based upon the minimum of the total potential energy functional with a Ritz type of solution procedure making use of complex forms of shape functions for the dependent variables. The individual and collective effects of pretwist, precone, setting angle, thickness ratio and Coriolis forces on the natural frequencies and the buckling boundaries are presented. It is shown that the inclusion of Coriolis effects is necessary for blades of moderate to large thickness ratios while these effects are not so important for small thickness ratio blades. The possibility of buckling due to centrifugal softening terms for large values of precone and rotation is shown

Improved methods of vibration analysis of pretwisted, airfoil blades

Vibration analysis of pretwisted blades of asymmetric airfoil cross section is performed by using two mixed variational approaches. Numerical results obtained from these two methods are compared to those obtained from an improved finite difference method and also to those given by the ordinary finite difference method. The relative merits, convergence properties and accuracies of all four methods are studied and discussed. The effects of asymmetry and pretwist on natural frequencies and mode shapes are investigated. The improved finite difference method is shown to be far superior to the conventional finite difference method in several respects. Close lower bound solutions are provided by the improved finite difference method for untwisted blades with a relatively coarse mesh while the mixed methods have not indicated any specific bound

Improved finite-difference vibration analysis of pretwisted, tapered beams

An improved finite difference procedure based upon second order central differences is developed. Several difficulties encountered in earlier works with fictitious stations that arise in using second order central differences, are eliminated by developing certain recursive relations. The need for forward or backward differences at the beam boundaries or other similar procedures is eliminated in the present theory. By using this improved theory, the vibration characteristics of pretwisted and tapered blades are calculated. Results of the second order theory are compared with published theoretical and experimental results and are found to be in good agreement. The present method generally produces close lower bound solutions and shows fast convergence. Thus, extrapolation procedures that are customary with first order finite-difference methods are unnecessary. Furthermore, the computational time and effort needed for this improved method are almost the same as required for the conventional first order finite-difference approach

Ultrastructure of the epithelial cells and oleo-gumresin secretion in Boswellia serrata (Burseraceae)

The ultrastructure of epithelial cells of oleo-gumresin ducts in Boswellia serrata, the source of Indian olibanum, is described. Oleo-gumresin ducts are present in primary and secondary phloem. The duct lumen forms an enlarged apoplastic space surrounded by epithelial cells. The epithelial cells are rich in dictyosomes, lipid bodies, mitochondria with dilated cristae, multivesicular bodies, osmiophilic materials, plastids and vesicIes. Plastids have poorly developed internal membranes. Dictyosomes and plastids are possible sites of resin synthesis. The gum component of the exudate is formed in dictyosomes and from the outer layers of the inner tangential wall (wall facing the duct lumen). This wall is replenished from inside by the activity of dictyosomes. The secretory materials are transported to the apoplast by granulocrine and eccrine secretion. They migrate through the loose microfibrils of the inner tangential wall into the duct lumen. Rarely, epithelial cells of young ducts have rudimentary plasmodesmata on the inner tangential wall which may be channels for passage of secretory materials into the duct lumen

Effect of tool pin diameters on friction stir welding of Aluminium alloy 5083

Friction stir welding is a solid-state welding process invented by the welding institute (TWI) of UK in 1991. The main objective is to study the effect of rotational speed and tool pin diameters of straight cylindrical tool on mechanical properties of Aluminium Alloy5083. Three different tool pin diameters (2.5mm,4mm,6mm) have been used to fabricate the joints at three different tool rotational speeds (710,1120 and 1400rpm) by keeping the constant welding speed and axial force. Tensile properties and micro hardness of the joints have been evaluated