Virtual Work Analysis Research Articles

A rotational traveling wave based levitation device – Modelling, design, and control

Described is a device acting on an acoustically levitated object by manipulating the pressure and flow of a thin layer of air such that its rotation can be precisely controlled without mechanical contact. Virtual work analysis assists in simplifying the multi-actuator control problem into one governed by a single controllable parameter. Actuation is accomplished by a vibrating annulus capable of producing ultrasonic standing and traveling waves. Thus creating the acoustic excitation that affects the pressure in a thin, intermediate layer of gas. A distinctive vibration pattern is required to generate the temporal and spatial pressure field of the squeezed air layer that gives rise to both acoustic levitation force and rotational torque. Described are the physical and design development stages leading to an optimized structure, all followed by verifying and dynamics-calibration experiments. Moreover, by precisely controlling the ratio of standing and traveling waves in the annulus, in a closed-loop, one can control the shear forces applied by the squeezed air layer, thus creating a non-contacting levitated object manipulation mechanism. Through an algebraic transformation, the over-actuated setup is converted to a simplified single control-parameter problem. The transformation links the standing waves ratio, and hence the rotational torque, to the amplitudes and phases of the actuators. This arrangement leads to an effective closed loop methodology that was implemented experimentally, showing good performance and exhibiting rapid angular positioning.

Generalized cutting loads decomposition model of five-axis serial machine tools based on the screw theory

There are large volume of existing literatures focused on cutting force modeling, but so far only few research has been reported on cutting loads decomposition model to calculate cutting loads components acted on five axes of the machine, which, however, directly affect five-axis tracking accuracy. To this end, a generalized cutting loads decomposition model of five-axis serial machine tools is proposed, which provides a mathematical basis to evaluate the cutting loads components. A cutting loads decomposition model is first developed based on kinematics and virtual work analysis, then a generalized cutting loads decomposition model is established by differentiating the forward kinematics model based on the screw theory. Kinematic factors that affect the cutting load components on translational and rotary axes are also analyzed, which provides a guidance to optimize the workpiece setup location to reduce the cutting loads components acted on rotary axes. Experiments and simulations on a table-tilting five-axis machine tool verify the effectiveness of the proposed model.

Postbuckling of bilaterally constrained rectangular thin plates

The post-buckling response of bilaterally constrained thin plates submitted to a height reduction is investigated by a joint experimental–theoretical–numerical study. A detailed determination of the shape of plates is compared to predictions based on modelling the plate as an Elastica and a phase portrait of the system is worked out. An integral relationship for unloaded plates is derived and checked experimentally. The variation of the length of flat contacts with compression is first identified from measurements and then introduced in a virtual work analysis so as to determine the plate reaction. The existence of asymmetric solutions is clarified and the robustness of the Elastica to friction is demonstrated. These results improve the predictions of the Euler model in the fully non-linear regime and validate its relevance beyond the ideal limits of frictionless rod or of frictionless infinite plate on which it is established. They should therefore be useful for addressing buckling in different, albeit close, configurations.

Distortional Buckling Solutions for Continuous Composite Beams

The paper presents a theoretical analysis of fixed-ended composite steel-concrete beams, representative of members continuous over an internal support. The analysis is elastic, and allows for the variation of sagging and hogging bending along the member. The stress resultants obtained from the virtual work analysis are used for an extensive finite element study of lateral-distortional buckling of fixed-ended composite beams. The details of the computer program adopted are described elsewhere. Based on the elastic buckling solutions, a design method is proposed to calculate the strength of the composite beam with respect to the buckling limit state. This design method is less conservative than that implied in current design codes, and leads to economies of material. The analysis for continuous composite beams provides a benchmark study for future studies.

Lagrangian Formulation for Penny-Shaped and Perkins-Kern Geometry Models

Summary Basic theories for vertical penny-shaped and Perkins-Kern (PK) geometry models have been developed with a Lagrangian formulation combined with a virtual-work analysis. The Lagrangian formulation yields a pair of nonlinear equations in Rf or Lf and bf, the fracture radius or length and half-width. By introduction of a virtual-work analysis, a simple equation is obtained that can be solved numerically. This equation is written in a form that can be used to determine fracture geometry when the fluid-loss coefficient of the fracturing fluid is known. Also, this equation, coupled with a material-balance equation after shut-in, can be used to analyze pressure-decline data after shut-in to determine the effective fluid-loss coefficient and fracture geometry. Geometry prediction of the new theory agrees well with other methods for the same models. A unique advantage of this approach, however, is that the geometry estimated from prefracturing design is consistent with the geometry determined from the pressure-decline analysis. Thus, it allows the engineers in the field to predict precisely the amount of fracturing fluid required for a designed fracture radius or length in a full-scale treatment from information gained in a minifracture treatment.

A Two-Dimensional Theory of Fracture Propagation

Summary A basic theory of two-dimensional (2D) fracture propagation has been developed with a Lagrangian formulation combined with a virtual work analysis. Fluid leakoff is included by the assumption that an incompressible filtrate produces a piston-like displacement of a compressible reservoir fluid with a moving boundary between the two. Poiseuille flow is assumed in the fracture. We consider both Newtonian and non-Newtonian fluids with and without wall building. For non-Newtonian fluids, we assume the usual power-law relation between shear stress and shear rate. The Lagrangian formulation yields a pair of nonlinear equations in Lf and bj, the fracture length and half-width. By introducing a virtual work analysis, we obtain a single equation that can be solved numerically. For non-wall-building fluids, it predicts much higher leakoff rates than existing methods. The Lagrangian method also allows nonelastic phenomena, such as plasticity, to be included. A practical computer program developed from this theory has been used for more than 10 years to design fracturing treatments in oil and gas reservoirs in Canada, California, the midcontinent and Rocky Mountain areas, the U.S. gulf coast, the North Sea, and in northern Germany. In most of these applications, it has predicted fracture dimensions that have been in line with production experience. Optimization methods based on this program led to very large fracturing treatments in low-permeability gas sands that were forerunners of massive fracturing treatments in tight gas sands. Specific examples in which this method was used to design fracturing programs in large gas fields in Kansas and Texas are discussed.

Fracture Penetration Through an Interface

Abstract This paper presents a combined theoretical and experimental investigation of fracture penetration through an interface. The results have application to hydraulic-fracture containment in sandstone or limestone reservoir strata bounded by shale layers. Several simplifying assumptions and approximations are made. We assume that the interface separates dissimilar but adhering materials that are elastic to first order. We consider only differences in material properties and take stresses to be locally uniform. Approximations are made that reduce the fourth-order equation of plane strain elasticity to the second-order Laplace equation. Crack shape is taken to be sinusoidal and fluid leakoff is ignored. Additional simplifying assumptions are made about the shape of the crack tip as it passes through the interface. Using a virtual work analysis, we derive a relation for internal fluid pressure required to extend the crack through the interface. This pressure is related to the equilibrium pressure needed to hold the crack barely open. A simple relation is obtained involving only the shear modulus and surface energy of the materials on either side of the interface. The theory was tested in laboratory experiments with Plexiglas™/resin bonds. Test blocks of several configurations were used with various kinds of external loading. Cracks were initiated by applying grease pressure in thin notches. Pressures required to hold cracks in equilibrium were compared with those required to penetrate interfaces. Results were consistent with the theory within limits of precision in measuring surface energy. We conclude that the theory explains field observations of containment in a number of reservoirs that have been fractured hydrualically. However, practical value is limited by inability to estimate surface energies from logging or other wellbore data.

Upper-bound anisotropic yield locus calculations assuming 〈111〉-pencil glide

Deformation by 〈111〉-pencil glide has been analyzed by an upper-bound model which combines a least-shear analysis and Piehler's maximum virtual work analysis. The least-shear analysis gives exact solutions if three 〈111〉 slip systems are active, while the maximum work analysis provides solutions for the case of four active slip systems. Independent checks are used to determine which solution method is appropriate. Computer calculations using this model have been made to determine; (1) the orientation dependence of the Taylor factor for axisymmetric deformation; (2) the yield loci for textured materials having [100], [110] and [111] sheet metals and rotational symmetry; (3) the isotropic yield locus for randomly oriented materials; and (4) flow stresses along critical loading paths for various assumed textures with rotational symmetry. The latter calculations indicate that anisotropic yield loci of textured bcc metals with rotational symmetry are much better approximated by σ x a + σ y a + R¦σ x − σ y¦ a = (R + 1)Y a where R is the strain ratio and Y is the tensile yield strength with an exponent a = 6 rather than with a = 2 as postulated by Hill. It is not known how well upper-bound calculations like these represent actual yielding behavior.