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A robust approach to computational kinematics intended to cope with algorithmic singularities is introduced in this article. The approach is based on the reduction of the original system of equations to a subsystem of bivariate equations, as opposed to the multivariate polynomial reduction leading to the characteristic univariate polynomial. The effectiveness of the approach is illustrated for the exact function-generation synthesis of planar, spherical, and spatial four-bar linkages. Some numerical examples are provided for the case of the spherical four-bar function generator with six precision points to show the benefits of the proposed method with respect to methods reported in the literature.

The kinetic exchange of SO42−ions with CO32−ions in natural barium sulphate (barite) mineral under alkaline hydrothermal conditions was investigated. The experiments were performed at several temperatures ranging from 423 to 523 K for different reaction times between 1 and 192 h with molar ratios of CO32−/SO42−=1, 5 and 10. Structural characterisation of partially and completely converted witherite (BaCO3) crystals was conducted by XRD and SEM. X-ray diffraction results showed that under hydrothermal conditions the exchange of CO32−ions in barite was complete at 523 K in a Na2CO3solution with a molar ratio of CO32−/SO42−=10 for 192 h. Based on the experimental observations, a mechanism for the conversion of barite under hydrothermal conditions is proposed. As the reaction proceeds, a porous layer of barium carbonate forms on the barite surface. The reaction rate is determined by the diffusion of sulphate ions through the porous layer. The apparent activation energy for the conversion of barite to barium carbonate is 22·5 mol−1.

The formulation of the mathematical model governing the dynamics of parallel Schönflies motion generators (SMGs) is the subject of this paper. These are robotic systems intended to produce motions that entail displacements of the Schönflies subgroup of rigid-body displacements. Such motions are representative of those produced by the serial robots termed SCARA, which involve three independent translations and one rotation about an axis of fixed direction. The main features of SMGs are illustrated with the aid of the McGill Schönflies motion generator. This robot is composed of two limbs, each being a four-joint RΠΠR - R representing a revolute, Π a Π joint, or a parallelogram linkage - kinematic chain, with only two joints actuated. One important feature of the McGill SMG is its mass distribution, as its moving parts account for about 10 per cent of the mass of its drive units, which are (a) fixed to the robot frame and (b) the only steel parts of the whole robot. Moreover, its joints account for roughly 10 per cent of the total mass of its moving parts, fabricated from aluminium, which justifies neglecting the joint inertia in the mathematical model. This feature calls for a formulation of the model in question in terms of the motor displacements, rather than the joint displacements, as the generalized coordinates of the model. Furthermore, in order to derive the model, the robot is decomposed into two subsystems, the driveand the linkage, the drive being decomposed, in turn, into two subsubsystems, the epicyclic gear train and the right-angled gearbox. The robot kinematics is first derived and then the dynamics model is formulated by means of the natural orthogonal complement. In the framework of this methodology, the inertia and what are called the Coriolismatrices of the mathematical model are additive, in the sense that they can be computed as the sum of the contributions of the different subsystems and subsubsystems of a given mechanical system. The contributions of the subsystems and subsubsystems, in turn, can be computed as the sum of the contributions of the individual moving parts of these. Some general results applicable to all SMGs are derived, which leads to the simplification of the mathematical model of such systems.

Cam—roller mechanisms can be used in transmissions and robotic devices as an alternative to their bevel-gear counterparts. As bevel gears are used to couple shafts of intersecting axes, their cam-mechanism replacements are bound to have intersecting axes as well. This means that the contact surface of the cam must be conical, which leads to spherical cam mechanisms. Compared with gear transmissions, cam—roller mechanisms feature low friction, low backlash, and high strength. However, cam mechanisms may end up with a high pressure angle if negative action, a motion in which the cam is driven by mechanism rollers, occurs. This article reports the design of transmissions with spherical multilobe cams (MLCs), as a means to reduce the pressure angle. The kinematics of MLC, including profile generation, undercutting-avoidance, and transmission-quality evaluation, is studied here. A case study has been included, in which the synthesis of a spherical MLC transmission is undertaken as a means to improve the transmission quality of a spherical epicyclic gear train. This train was introduced with the purpose of producing a singularity-free, unlimited-workspace pitch—roll wrist.

In this paper, a novel bipod parallel grinder with four controlled degrees of freedom is introduced. The moving platform of this 2-leg parallel grinder can always keep moving in horizontal plane by means of four-parallelogram mechanism (Π joints). The closed-form solutions of forward and inverse kinematics are derived.

Summary: A novel technique for the determination of the pose and the twist of rigid bodies using point-acceleration data is proposed. These data are collected from an accelerometer array, which is a kinematically redundant set of triaxial accelerometers. Because orientational error in the installation of the accelerometers can be fatal to the accuracy of the results, a calibration procedure based on the consistency of the point accelerations is outlined. The formulation developed is then utilized in the simulation analysis of two sample motions. The relations required to estimate the pose and the twist are derived in a body-fixed frame. The body angular acceleration and angular velocity, in this order, are determined directly from the acceleration data; the body attitude is then computed through integration.

A laser scanning technique was developed to monitor in situ surface damage on nickel based superalloy turbine components under low cycle fatigue conditions. This technique characterizes a surface state using a discontinuity frequency parameter that was designed to minimize memory requirements and data processing time. As a result, the present technique is capable of scanning speeds that are substantially greater than those achieved with image processing methods. Sections of turbine rotors made of a nickel-based superalloy were tested using a servohydraulic machine at ambient temperatures under load control conditions. The fatigue damage was monitored by scanning a laser beam along the rotor section in situ and during periodic interruptions of the cyclic loading. Acetate replicas of the gage section surface were also made to examine the surface morphology using scanning electron microscopy. Comparisons of the results demonstrate the capabilities of the present light scanning technique for characterizing fatigue damage precursors on the surface of turbine components. In particular, a rapid rise in the mean discontinuity frequency is shown to correspond to surface relief features that correspond to grain boundaries that intersect the surface in the areas of greatest stress. The presence of this surface relief can be attributed to the presence of relatively soft precipitate free zones along the grain boundaries that preferentially deform under fatigue loading conditions, leading to the formation of microcracks. [ABSTRACT FROM AUTHOR]

The dynamics of the swashplate mechanism of the CL-327, anUnmanned Aerial Vehicle (UAV) with a Vertical Take-Off-and-Landing(VTOL) ability is discussed in this paper, the underlying kinematicshaving been discussed in an earlier paper. The CL-327 (Guardian)is currently under production by Bombardier Services-Defence, and isconsidered the world's most versatile and advanced VTOL system inproduction. The steering control mechanism used in all helicopters,including the CL-327, is a complex multiloop kinematic chain oftencalled the swashplate mechanism. In this paper a dynamic analysis of theparticular design used for the swashplate of the CL-327 is discussed.The Newton–Euler formulation is used in conjunction with the naturalorthogonal complement in order to derive the equations of motion of thesystem.

The ion exchange of Cl^- and OH^- with F^- in synthetic calcium chlorapatite and calcium hydroxyapatite single crystals was investigated under hydrothermal conditions. The hydrothermal treatments were conducted using 5 and 10 M KF solutions, at temperatures of 500 and 600 °C under a constant internal pressure of 100 MPa, for intervals between 6 and 336 h. The ion exchange of Cl^- in the chlorapatite crystals proceeded faster than that of OH^- in the hydroxyapatite crystals during hydrothermal treatments at 500 °C in 5 M KF solution. Under these conditions, the chlorapatite crystals were completely converted to fluorapatite crystals. In contrast, the hydroxyapatite crystals were only partially converted to fluorapatite. In both cases, the topotaxial ion-exchange reaction was achieved by a mechanism of dissolution−precipitation of clusters. The partially and completely converted fluorapatite crystals had a characteristic texture, exhibiting tiny tunnels along the c axis of the crystals, and a thin layer of CaF2 was also produced on the surface of the converted crystals. The influence of the reactivity of the individual apatites on the control of the ion exchange rate is discussed.

The kinematics of the swashplate mechanism of an Unmanned Aerial Vehicle (UAV) is discussed in this paper. The UAV under study, known as the CL-327 Guardian, is currently under production by Bombardier Services, Defence. It is considered to be the world's most versatile and advanced UAV system capable of Vertical Take Off and Landing (VTOL). The steering control mechanism used in all helicopters, including the CL-327, is a complex closed kinematic chain often called the swashplate mechanism. In this paper a detailed kinematic analysis of the particular design used for the swashplate of the CL-327 is discussed.

Summary: The analysis of arbitrarily-damped linear mechanical systems is the subject of this paper, the main issue being the analysis of nondecouplable systems. It is well known that decouplable systems occur when the damping matrix happens to be a linear combination of the mass and stiffness matrices. Systems with this type of damping are said to have proportional damping, which nevertheless seldom occurs in practice, but many a damped system is analyzed under the assumption that it is proportionally damped. In fact, this property allows the analyst to study these systems using the same approach as that applicable to their undamped counterparts. In this paper, we show that proportional damping need not be assumed in order to analyze the system at hand with the same approach as used to analyze undamped systems. Moreover, we propose an algorithm to determine the natural frequencies, the damped frequencies and the damping ratios of an n-degree-of-freedom damped system, that does not require the casting of the system into first-order form. In this way, the characteristic equation is derived naturally as a 2n-degree polynomial, computing its roots being then straightforward. Furthermore, we propose a semigraphical method to ease this calculation, which should be attractive to practicing engineers.

Isothiocyanatobenzyl imidazoline (IBI) produces characteristic slowly developing contraction of many smooth muscle preparations including the circular smooth muscle of the guinea-pig stomach. Changes in the membrane potential were recorded intracellularly, and the muscle contraction induced by IBI was investigated. IBI at 100 µmol/l slowly produced a sustained depolarization of the membrane with a maximum change of approximately 15 mV. This depolarization could not be blocked by l-hyoscyamine, 100 nmol/l. An imidazoline analogue, oxymetazoline at 1 µmol/l, did not change the resting membrane potential as observed after IBI. Significant membrane depolarization after IBI still occurred in Ca2+-free medium. During IBI-induced depolarization, sudden reduction of Na+to 30 mmol/l in the medium reduced the depolarization slightly. IBI-induced depolarization was additive with that produced by 20 mmol/l K+ in the medium. In the presence of tetraethylammonium chloride or levcromakalim or nifedipine, IBI continued to depolarize the membrane although functional pharmacological experiments showed that the contractile effects of IBI were significantly inhibited by 30 µmol/l levcromakalim and abolished by 100 nmol/l nifedipine. At 100 µmol/l phentolamine (reported by others as an inhibitor of ATP-sensitive potassium channels) completely blocked IBI-induced contraction. Phentolamine (30 µmol/l) blocked the contractile effects of IBI by 50%. On the other hand, S(-)-Bay K 8644, a voltage-dependent calcium channel activator, was additive with the contractile response of IBI. These results indicated that IBI produced membrane depolarization and contraction of the guinea-pig stomach circular muscle, by a mechanism not involving muscarinic receptors or α-adrenoceptors. Even though levcromakalim, an ATP-sensitive potassium channel opener, could not inhibit IBI-induced depolarization, the ATP-sensitive potassium channel and the voltage-dependent calcium channel may be intrinsically linked with the action of IBI.

Summary: The angular-acceleration tensor of rigid-body kinematics is recalled, and its invariant properties are analyzed. An explicit form of its inverse is given, its eigenvalues being calculated in symbolic form. The special case in which the tensor under study becomes singular, is given due attention.

Sets of constraints on analytic functions of linkage parameters, equivalent to those of Grashof’s mobility criteria, are derived. These inequalities represent necessary and sufficient mobility conditions for the input and the output links of planar 4R linkages, as well as for its coupler link. The application of the foregoing constraint relations to linkage optimization using gradient-dependent methods is shown with an example that is fully solved resorting to Newton-Raphson’s method and an interior penalty function.

Sets of constraints on analytic functions of linkage parameters, equivalent to those of Grashof’s mobility criteria, are derived. These inequalities represent necessary and sufficient mobility conditions for the input and the output links of planar 4R linkages, as well as for its coupler link. The application of the foregoing constraint relations to linkage optimization using gradient-dependent methods is shown with an example that is fully solved resorting to Newton-Raphson’s method and an interior penalty function.

Nonholonomic mechanical systems are revisited. This review article focuses on Lagrangian formulations leading to a system of governing equations free of constraint forces. While eliminating the constraint forces, the number of scalar Lagrange equations is reduced to a number of independent equations lower than the original system with constraint forces. In the process of constraint-force elimination and dimension-reduction, a matrix that appears to play a relevant role in the formulation of the mathematical models of mechanical systems arises naturally. We call this matrix here the holonomy matrix. It is shown that necessary and sufficient conditions for the integrability of the constraints in Pfaffian form are readily derived using the holonomy matrix. In the same vein, a class of nonholonomic systems is identified, of current engineering relevance, that is termed quasiholonomic. Examples are included to illustrate these concepts. This review article contains 40 references.

Discussed in this paper are the issues underlying the mechanical design of a seven-axes isotropic manipulator. The kinematic design of this manipulator was made based on one main criterion, namely, accuracy. Thus, the main issue determining the underlying architecture, defined by its Hartenberg—Denavit (HD) parameters, was the optimization of its kinematic conditioning. This main criterion led not to one set of HD parameters, but rather to a manifold of these sets, which allowed the incorporation of further requirements, such as structural behavior, workspace considerations and functionality properties. These requirements in turn allowed the determination of the link shapes and the selection of actuators. The detailed mechanical design led to heuristic rules that helped in the decision-making process in defining issues such as link sub-assemblies and motor location along the joint axes.

The problem of mobility associated with four-bar linkages is addressed in this paper. The mobility analysis is reduced to finding the global extrema of a quadratic function on a cylinder, which then leads to the geometric problem of finding the intersections of a circle and a hyperbola. The method proposed here produces an efficient mobility analysis that can be readily integrated into any suitable optimization algorithm.

As shown elsewhere, the approximate least-square synthesis of function-generating linkages can be formulated in terms of one single decision variable. Hence, functions representing global properties of linkages such as accuracy and transmission quality, which are terms precisely defined in this paper, as well as mobility type, are all functions of one single variable. The optimization procedure presented here is aimed at the synthesis of function-generating linkages with a minimum least-square design error and a maximum transmission quality, while meeting conditions prescribed on the type of mobility (crank or rocker) of its input and/or output links. Alternative performance indices and/or constraints can be treated likewise.

In this paper, the design of a spherical three-degree-of-freedom parallel manipulator is considered from a kinematic viewpoint. Three different design criteria are established and used to produce designs having optimum characteristics. These criteria are (a) symmetry (b) workspace maximization, and (c) isotropy. The associated problems are formulated and their solutions, one of them requiring to resort to a numerical method, are provided. Optimum designs are thereby obtained. A discussion on singularities is also included.

In this paper, the design of a planar three-degree-of-freedom parallel manipulator is considered from a kinematic viewpoint. Four different design criteria are established and used to produce designs having optimum characteristics. These criteria are (a) symmetry (b) the existence of a nonvanishing workspace for every orientation of the gripper (c) the maximization of the global workspace, and (d) the isotropy of the Jacobian of the manipulator. The four associated problems are formulated and their solutions are derived. Two of these require to resort to numerical methods for nonlinear algebraic systems. Results of optimum designs are also included.

The motion control of mechanical systems with flexible links is investigated. Issues addressed are the modeling and simulations, the design of a feedback control scheme and its implementation on an actual system. The modeling method used is a combination of a spline-based spatial discretization technique and the natural orthogonal complement of the kinematic constraints that eliminates the constraint forces and moments from the mathematical model. The control algorithm consists of two parts, namely, the uncoupling of nonlinear equations of motion and the filtering of consequent nonworking constraint forces. This control scheme is implemented on a prototype four-bar flexible mechanism. Results show that the proposed control scheme provides successful trajectory tracking while suppressing the vibration triggered by a doublet-type of disturbance.

A key issue in the geometric and mechanical analyses of machine elements with complex shapes such as turbine blades and bevel gears is the accurate and economic computation of their volumetric properties, namely, volume, centroid coordinates, and inertia tensor. A method is proposed here for the computation of these properties that is applicable to sweep-generated solids based only on their 2D generating contour and their sweeping parameters. Thus, the direct 3D calculations of the volumetric properties of such solids are reduced to simple 2D calculations. Comparisons of numerical results are presented, which show that the volumetric properties of this class of solids can be effectively performed in a computationally reduced fashion which, additionally, enhances the accuracy of the computed values.

A four-bar linkage having a quadratic input-output equation can admit a constant value of the output variable for any value of the input variable. Such a linkage is termed constant-branch four-bar linkage in this paper. The general conditions governing the existence of this special class of linkages are derived through an in-depth analysis of the input-output equation. Geometric, mobility, and force-transmission properties are discussed with regard to most common types of planar, spherical, and spatial four-bar linkages. As an application example, a clutch mechanism, based on a constant-branch spatial RSSR four-bar linkage, is proposed.

An optimization scheme for four-bar function generators under mobility constraints, which can be applied to both planar and spherical four bar linkages, is presented in this paper. The design error, defined as the residual in the input-output equation, is minimized over the vector of linkage parameters. The mobility constraints, given as a set of inequalities, are converted into equalities by introducing slack variables. The problem is thus formulated as an equality-constrained minimization problem, which is then solved using the orthogonal-decomposition algorithm, an iterative numerical method introduced elsewhere. To reduce the dimensional unbalance, which often occurs in solving a synthesis problem, a penalty function is combined with the original objective function, whose minimization leads to dimensionally balanced linkages. A numerical example is included.

In this paper, the optimization of the spherical RRRR four-bar linkage for the problem of path generation is addressed. The problem is formulated as a two-loop minimization of the error between the path-generating point in the coupler curve and the prescribed position, while decoupling the linkage parameters from the configuration variables, namely, the input angles. The synthesis problem consists of evaluating a set of input angles {ψk}1m defining m linkage configurations and the linkage parameters independently. This leads to a constrained overdetermined system of nonlinear equations. The orthogonal decomposition algorithm, introduced elsewhere, is employed to solve the problem. Continuation and damping techniques are used in the numerical procedure to ensure convergence and speed up its rate. The optimization scheme is developed on a general basis and can handle the problem for any number of given path points. Three numerical examples are included.

In this paper, a novel performance index for the kinematic optimization of robotic manipulators is presented. The index is based on the condition number of the Jacobian matrix of the manipulator, which is known to be a measure of the amplification of the errors due to the kinematic and static transformations between the joint and Cartesian spaces. Moreover, the index proposed here, termed global conditioning index (CGI), is meant to assess the distribution of the aforementioned condition number over the whole workspace. Furthermore, the concept of a global index is applicable to other local kinematic or dynamic indices. The index introduced here is applied to a simple serial two-link manipulator, to a spherical three-degree-of-freedom serial wrist, and to three-degree-of-freedom parallel planar and spherical manipulators. Results of the optimization of these manipulators, based on the GCI, are included.

Mechanical hands have become of greater interest in robotics due to the advantages they offer over conventional grippers in tasks requiring dextrous manipulation. However, mechanical hands also tend to be more complex in construction and require more sophisticated analysis to determine the forces in the system. A mechanical hand can be described as a kinematic chain with time-varying topology which becomes redundantly actuated when an object is grasped. When this occurs, care must be exercised to avoid crushing the object or generating excessive forces within the mechanism. In the present work, this problem is formulated as a constrained quadratic optimization problem. The forces to be minimized form the objective, the dynamic equations of motion form the equality constraints, and the finger-object contacts yield the inequality constraints. The quadratic-programming approach is shown to be advantageous due to its ability to minimize “internal forces.” A technique is proposed for smoothing the discontinuities in the force solution which occur when the topology changes.

The concept of generalized coupler curves of linkages is introduced in this paper. Using this concept, the study of path generation and rigid-body guidance can be done within the same framework for analysis, synthesis and optimization of linkages. Moreover, a unified method of performance evaluation for rigid-body guidance is presented, that is applicable to four-bar linkages of any type, as long as the linkages have quadratic input-output or input-coupler functions. The method is an extension of the performance-evaluation method for path generation introduced in a previous paper. That method is based on the usual coupler curves, and hence, is only applicable to path-generation problems. The present method is based on generalized coupler curves, and is, thus, applicable to the problems of both path-generation and rigid-body guidance. It is oriented towards the iterative optimization of rigid-body-guidance tasks with four-bar linkages, and hence, considers numerical complexity as a determining issue. Numerical examples in which the method is applied to a planar four-bar linkage and a spherical four-bar linkage are included.

In this paper, the kinematic inversion of redundant parallel manipulators in the presence of incompletely specified tasks is formulated as an optimization problem. The performance index used is the condition number of the Jacobian matrix of the manipulator which is a measure of Jacobian invertibility. In order to optimize this index along a partially prescribed Cartesian trajectory, the concept of trajectory map is introduced. It is also shown that the optimum value of the free parameter that minimizes the condition number is not a continuous function of the prescribed Cartesian coordinates. An on-line algorithm producing continuous joint histories is then discussed. This method has been implemented and tested, as illustrated with the results presented here.

A series of trimetoquinol (1, TMQ) analogs were designed and synthesized based on the lead compound 2, a diiodinated analog of trimetoquinol which exhibits improved selectivity for β2-versus β1-adrenoceptors (AR). To determine the influence of 1-benzyl substituents of trimetoquinol on β2-AR binding affinity and selectivity, we replaced and/or removed the 3‘-, 4‘-, and 5‘-methoxy substituents of trimetoquinol. Replacement of the 4‘-methoxy group of 2 with an amino (21c) or acetamido (15) moiety did not significantly alter β2-AR and thromboxane A2/prostaglandin H2 (TP) receptor affinity. Substitution with a 4‘-hydroxy (18) or -iodo (21b) group did not significantly alter β2-AR affinity, but greatly reduced TP receptor affinity (380- and 1200-fold, respectively). Further, the β2-AR can accommodate larger substituents such as a benzamide at the 4‘-position (26b). Other monoiodo derivatives (24, 26a) have similar or slightly lower affinity to both β2-AR and TP receptor compared to their diiodo analogs. Interestingly, removal of the 4‘-substituent of 3‘,5‘-diiodo analogs increased β2-AR affinity with little or no effect on β1-AR and TP binding. Thus, analog 21a displayed highly potent (pKi 9.52) and selective (β2/β1 = 600) binding affinity for β2-AR. On the other hand, trifluoromethyl substituents at the 3‘- and 5‘-positions (27) essentially abolished binding affinity at β2-AR and TP receptors. The differential binding effects of the aforementioned trimetoquinol modifications on the receptor systems may reflect differences in the binding pocket that interacts with the benzyl portion of trimetoquinol analogs. Thus, manipulation of the 1-benzyl moiety of trimetoquinol (1) has resulted in analogs that exhibit potent β2-AR binding affinity and significantly lower β1-AR and TP receptor affinities.

In this paper, the design of a spherical three-degree-of-freedom parallel manipulator is considered from a kinematic viewpoint. Three different design criteria are established and used to produce designs having optimum characteristics. These criteria are (a) symmetry (b) workspace maximization, and (c) isotropy. The associated problems are formulated and their solutions, one of them requiring to resort to a numerical method, are provided. Optimum designs are thereby obtained. A discussion on singularities is also included.

In this paper, the design of a planar three-degree-of-freedom parallel manipulator is considered from a kinematic viewpoint. Four different design criteria are established and used to produce designs having optimum characteristics. These criteria are (a) symmetry (b) the existence of a nonvanishing workspace for every orientation of the gripper (c) the maximization of the global workspace, and (d) the isotropy of the Jacobian of the manipulator. The four associated problems are formulated and their solutions are derived. Two of these require to resort to numerical methods for nonlinear algebraic systems. Results of optimum designs are also included.

As shown elsewhere, the approximate least-square synthesis of function-generating linkages can be formulated in terms of one single decision variable. Hence, functions representing global properties of linkages such as accuracy and transmission quality, which are terms precisely defined in this paper, as well as mobility type, are all functions of one single variable. The optimization procedure presented here is aimed at the synthesis of function-generating linkages with a minimum least-square design error and a maximum transmission quality, while meeting conditions prescribed on the type of mobility (crank or rocker) of its input and/or output links. Alternative performance indices and/or constraints can be treated likewise.

The problem of mobility associated with four-bar linkages is addressed in this paper. The mobility analysis is reduced to finding the global extrema of a quadratic function on a cylinder, which then leads to the geometric problem of finding the intersections of a circle and a hyperbola. The method proposed here produces an efficient mobility analysis that can be readily integrated into any suitable optimization algorithm.

A general scheme for the optimization of planar, spherical and spatial bimodal linkages for function generation is proposed. The problem is solved here following two basic steps: (i) planning input-output ((I/O) curves based on design requirements and selecting data from the planned curve; and (ii) setting up an optimization procedure to minimize a performance index.

In this paper the ruled surfaces of two rigid bodies that are in contact while moving with a prescribed time-varying angular-velocity ratio are generated. These are then used as the contact surfaces of indexing cam mechanisms. In this way, planar, spherical, and spatial mechanisms are synthesized in a unified framework, the pressure angle, for all these cases, being analyzed. The approach is illustrated with various examples.

The solid modeling of bevel gears has been recognized as a major design challenge. Various approaches have been proposed to approximate the theoretical involute-generated contact surface of bevel gears. As a means to accurately represent contact surfaces of these gears, the exact spherical involute is introduced here. The solid models of straight and spiral bevel gears are obtained by applying simple sweeping techniques to their tooth profiles, which are described by the exact spherical involute. Futhermore, the volumetric properties of the modeled bevel gears are readily computed from their piecewise-linear approximation.

A novel method of synthesis of the surface geometry of spherical cam-oscillating roll-follower mechanisms is introduced. The approach aims at zeroing the sliding velocity between the cam and the roller, their respective geometries being thus obtained as conical surfaces. The pressure angle is analyzed in detail. Specific results are presented for the application of this synthesis technique to indexing cam mechanisms.

This paper follows an accompanying one on the constrained minimization of cam profiles with translating flat-face followers. Here, the corresponding problem for oscillating flat-face followers is discussed. The unconstrained optimization of this type of cam mechanisms, aimed at the minimization of the area of the cam disk, has been reported elsewhere. In this paper we recognize that unconstrained cam-size minimization can lead to profiles with curvature discontinuities, i.e., cusps, which should be eliminated, for they give rise to inadmissibly large contact stresses. The conditions under which curvature discontinuities arise are derived. These conditions lead to a simple constrained minimization procedure, whose application is illustrated with a fully solved numerical example.