A novel Udwadia-Kalaba approach for parallel manipulator dynamics analysis is presented. The approach segments a parallel manipulator system into several leg-subsystems and the platform subsystem, which are connected by kinematic constraints. The Udwadia-Kalaba equation is then used to calculate the constraint forces due to the constraints. Based on this, the equation of motion, which is an explicit (i.e., closed) form, can be formulated. The method allows a systematic procedure to generate the dynamic model for both direct dynamics and inverse dynamics without invoking additional variables (such as multipliers or quasi-variables), nor does it require projection. A classical parallel Stewart-Gough platform is chosen to demonstrate the feasibility and advantages of this approach.

Hamel proposed a seemingly intuitive, simple, straightforward, but incorrect, method of formulating the constrained equation of motion. The method has to do with the direct embedding of the constraint into the kinetic energy of the unconstrained motion. His intention was to caution against its possible adoption. Rosenberg echoed Hamel's warning and followed up to explore more insight of this method. He proposed a conjecture that the Hamel's embedding method would work if the constraint was holonomic. It would not work if the constraint was nonholonomic. We investigate the Hamel paradox and Rosenberg conjecture via the use of the Fundamental Equation of Constrained Motion.

An experimental technique for evaluation of the M-integral in an elastic-plastic material containing multiple defects is proposed by using digital image correlation (DIC). This technique makes direct use of the definition of M by experimentally evaluating the integrand of M at various points along a square contour and determining the integral by numerical integration. The nonlinear Ramberg–Osgood model is used to capture the elastic-plastic behavior such as the elastic-plastic stress and the total strain energy density in terms of the measured displacements by DIC used in an ARAMIS 4M instrument. Compared with the previous experimental method proposed by King and Herrmann (King and Herrmann, 1981, “Nondestructive Evaluation of the J and M Integrals,” ASME J. Appl. Mech., 48, pp. 83–87), the present technique could be suitable to measure the M-integral for the various complicated damages, specimen geometries, loading conditions, and material behaviors. The path-independence or path-dependence of the M-integral is investigated under small-scale and large-scale yielding conditions, respectively. It is found that the values of M are path independent when the contours entirely enclose the nonlinear plastic region near the multiple defects. In contrast, the path-dependence is concluded for an elastic-plastic solid under large-scale yielding condition when the contours have to pass through the plastic zone. This interesting path-dependence of the M-integral is consistent with numerical prediction via the finite element method and theoretical analysis developed in this paper.

This paper deals with the surface effect and size dependence on the M -integral representing the energy release due to a nanodefect expansion in plane elasticity. Due to the high surface-to-volume ratio for reinforcing particles in the nanometer scale, the surface effect along the nanosized hole may be induced from the residual surface stress and the surface Lamé constants. The invariant integrals such as the J k -integral vector and the M -integral customarily used in macrofracture mechanics are extended to treat plane elastic materials containing a nanosized hole. It is concluded that both components of the J k -integral vanish when the contour selected to calculate the integral encloses the whole nanosized hole. This leads to the independence of the M -integral from the global coordinate shift. It is concluded that the surface effect and the size dependence on the energy release due to the nanohole expansion are significant especially when the hole size is less than 40 nm. This present study reveals that the discrepancies of the M -integral value with the surface effect from the referenced value M 0 without the surface effect are mainly induced from the residual surface stress τ 0 rather than from the surface Lamé constants μ s and λ s .

A semi-permeable interface crack in infinite elastic dielectric/piezoelectric bimaterials under combined electric and mechanical loading is studied by using the Stroh complex variable theory. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and on the energy release rate (ERR). Thirty five kinds of such bimaterials are considered, which are constructed by five kinds of elastic dielectrics and seven kinds of piezoelectrics, respectively. Numerical results for the interface crack tip singularities are calculated. We demonstrate that, whatever the dielectric phase is much softer or much harder than the piezoelectric phase, the structure of the singular field near the semi-permeable interface crack tip in such bimaterials always consists of the singularity r − 1 ∕ 2 and a pair of oscillatory singularities r − 1 ∕ 2 ± i ε . Calculated values of the oscillatory index ε for the 35 kinds of bimaterials are presented in tables, which are always within the range between 0.046 and 0.088. Energy analyses for five kinds of such bimaterials constructed by PZT-4 and the five kinds of elastic dielectrics are studied in more detail under four different cases: (i) the crack is electrically conducting, (ii) the crack gap is filled with air/vacuum, (iii) the crack gap is filled with silicon oil, and (iv) the crack is electrically impermeable. Detailed comparisons on the variable tendencies of the crack tip ERR against the applied electric field are given under some practical electromechanical loading levels. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR. We also conclude that the previous investigations under the impermeable crack model are incorrect since the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack, whereas the previous investigations under the conducting crack model may be accepted in a tolerant way since the results of the ERR show very small discrepancies from those for the semi-permeable crack, especially when the crack gap is filled with silicon oil. In all cases under consideration the curves of the ERR for silicon oil are more likely tending to those for the conducting crack rather than to those for air or vacuum. Finally, we conclude that the variable tendencies of the ERR against the applied electric field have an interesting load-dependent feature when the applied mechanical loading increases. This feature is due to the nonlinear relation between the normal electric displacement component and the applied electromechanical loadings from a quadratic equation.

A semi-permeable interface crack in dissimilar piezoelectric materials is studied in detail. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the near-tip singularity and the crack tip energy release rate (ERR). The Stroh complex variable theory ( Stroh, A. N., 1958, Philos. Mag. 3, pp. 625–646; Ting, T. C. T., Int. J. Solids Struct., 22, pp. 965–983 ) is used to obtain the solution, from which some useful numerical results for 21 kinds of dissimilar piezoelectric materials are calculated. They are combined from seven kinds of commercial piezoelectric ceramics. The distribution of the normal electric displacement component (NEDC) along the interface crack is assumed to be uniform and the corresponding problem is then deduced to a Hilbert problem with an unknown NEDC. Solving the Hilbert problem and determining the near-tip field for each of the 21 bimaterials, we determine the crack tip singularities and find that the crack-tip singularity for a certain combination of two dissimilar piezoelectric materials can be either oscillatory or nonoscillatory when the poling axes of both piezoelectric materials are perpendicular to the interface crack. Energy analyses for PZT ‐ 4 ∕ Ba Ti O 3 as a typical nonoscillatory class bimaterial and those for PZT - 5 H ∕ Ba Ti O 3 as a typical oscillatory class bimaterial are specially studied in detail under four different conditions: (i) the crack gap is filled with air or vacuum; (ii) the crack gap is filled with silicon oil to avoid discharge; (iii) the crack gap is conducting; and (iv) the electrically impermeable crack. Detailed comparisons are performed among the four cases. We conclude that the different values of the permittivity have no influence on the crack tip singularity but have significant influences on the crack tip ERR under the combined electromechanical loading. We also conclude that the previous investigations under the insulating crack model are incorrect or misleading since the model overestimates the effect of the electric field on the ERR very much and the results of the ERR for the impermeable crack show significant discrepancies from those for the semi-permeable crack. Whereas the previous investigations under the conducting crack model may be accepted in a tolerant, way, the results of the ERR show very small discrepancies from those for the semi-permeable crack model, especially when it filled with silicon oil.

The macrocrack-microcrack interaction problem in transversely isotropic piezoelectric materials is studied. The microcracks near a macrocrack tip in the process zone are assumed to be parallel to the latter, while the poling direction of the piezoelectric materials is assumed to be perpendicular to the cracks. Three kinds of elementary solutions with different crack configurations and under different loading conditions are given, from which the interaction problem is reduced to a system of Fredholm integral equations by using the pseudo-traction electric displacement method (abbreviated PTED). After the equations are solved numerically, the traditional mode I and mode II stress intensity factors and the electric displacement intensity factor are evaluated. In order to confirm the proposed method as well as the numerical results, a consistency check is proposed which is based on the J-integral analysis and provides a powerful tool to examine the numerical results. Thus, any mistakes are avoided since they would certainly lead to unsatisfied numerical results contrary to the check. It is concluded also that the disturbance of the near-tip electric field provides another source of shielding.

Numerical results are shown in figures and tables. The major features for the traditional stress intensity factors and the electric displacement intensity factor against the microcrack location angle and the distance of the microcrack center from the macrocrack tip are discussed. It is shown that, unlike single-crack problems, the mechanical loading and the electric loading are coupled together since the microcrack not only releases the near-tip stresses, but also disturbs the near-tip electric field. Furthermore, the influence of the electric loading on the mechanical strain energy release rate (MSERR) at the macrocrack tip is discussed in detail. It is found that the variable nature of the MSERR against the normalized electric loading is monotonic and proportional wherever the parallel microcrack is located near the macrocrack tip. However, the slope of the MSERR's curve considering microcracking diverges far from those without considering microcracking. This finding reveals that, besides the two sources of microcrack shielding discussed by Hutchinson (1987) for brittle solids, the disturbance of the near-tip electric field due to microcracking really provides another source of shielding for piezoelectric solids.

We consider a control design problem for a class of uncertain systems. The salient features of the problem are fourfold. First, the uncertainty is time-varying. Second, the system is nonlinear. Third, the state is constrained to be only positive. Fourth, the control input may be constrained to be one-sided (i.e., either positive or negative). We start with a state transformation to release the state constraint. A partial sign-definiteness knowledge of the uncertainty is then shown to be critical to meet the control constraint. Three classes of controls are constructed: one for state constraint and two for the additional control constraint. They render the system globally practically stable.

Machining complex three dimensional surfaces is a challenging task. This paper presents two methods of machining these surfaces on a 4 and 5 axis machine, using a toroidal shaped cutter. The methods propose to align the principal axis of curvature of the machining surface with that of the machined surface in order to increase the volume of material removed. The increase in material removal at a point reduces the scallop height. Thus, fewer passes are required to achieve the same surface finish.

A two-time-scale linear system with uncertain time-varying parameter is to be stabilized. A class of robust composite controls is proposed. The control renders the system practically stable regardless of the true value of the parameter. The control scheme consists of linear and nonlinear parts. The linear part is designed via a two-level optimization setting. No matching condition is needed for the linear control. The nonlinear part is of continuous saturation type. Matching condition is needed for the nonlinear control. However, it can be achieved by choosing the boundary layer model appropriately.

We study the control design problem for uncertain nonlinear systems. A new matching condition is presented. The main idea is to explore the route through which the (worst case) uncertainty may affect the stability. This route is then used to establish the new matching condition. Compared with the previous case, the current matching condition prescribes the route nonlinearly while the early matching condition is a special case of the linear description. A class of robust controls, which guarantees practical stability, can be designed based on this new matching condition. The size of the uniform ultimate boundedness ball can be made arbitrarily small by an appropriate choice of a design parameter.

This paper presents an analytical theory to define the dynamic characteristics of a layered beam which is composed of two parallel beams of uniform properties with a flexible core in-between. This flexible core may be made of a kind of viscoelastic material in order to achieve a high shock-absorbing performance. The dynamic interactions between these two parallel beams are especially studied. The dynamic shape functions and the dynamic stiffness matrix of a layered-beam element are established based on the analytical model of two parallel damped Timoshenko beams, connected to each other by the vertical springs and dashpots uniformly distributed along the beam length. Some simple layered beams are employed as the application examples for demonstrations and discussions.

A class of large-scale linear uncertain systems is under consideration. The uncertainty is time-varying. No a priori deterministic or stochastic information is assumed except its possible bound. Decentralized robust control for each subsystem is proposed. The salient feature of the design is that it decomposes the internal uncertainty and the interconnection. This enables the structural property of the uncertainty to be incorporated in the design. An adaptive version of the decentralized robust control is also proposed as no input matrix uncertainty appears.

We consider a class of nonlinear uncertain interconnected systems with time-varying uncertainty. The uncertainty may arise within each system as well as in the interconnections. The uncertainty is assumed bounded but the bound is unknown. No a priori statistical information is imposed. Decentralized adaptive robust control is proposed for each system. The control has two parts. First, an adaptive scheme for the estimation of the bound is constructed. Second, a robust control, which is based on the adaptive parameter, is adopted for each system.

We consider the tracking control problem of mechanical manipulators in the presence of uncertainty. Two classes of control algorithms are proposed. If the possible bound of the uncertainty is known, a class of nonadaptive robust computed torque control schemes is used. The control guarantees the tracking error to be confined within a specified region after a finite time. If the bound of uncertainty is unknown, a class of adaptive robust computed torque control schemes is used. The control guarantees the tracking error to converge to zero. Both classes of controls are continuous. No statistical information on the uncertainty is ever assumed.