# Journal of Mechanical Design

Print ISSN: 1050-0472
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This paper presents a symbolic formulation for analytical compliance analysis and synthesis of flexure mechanisms with serial, parallel, or hybrid topologies. Our approach is based on the screw theory that characterizes flexure deformations with motion twists and loadings with force wrenches. In this work, we first derive a symbolic formulation of the compliance and stiffness matrices for commonly used flexure elements, flexure joints, and simple chains. Elements of these matrices are all explicit functions of flexure parameters. To analyze a general flexure mechanism, we subdivide it into multiple structural modules, which we identify as serial, parallel, or hybrid chains. We then analyze each module with the known flexure structures in the library. At last, we use a bottom-up approach to obtain the compliance/stiffness matrix for the overall mechanism. This is done by taking appropriate coordinate transformation of twists and wrenches in space. Four practical examples are provided to demonstrate the approach. A numerical example is employed to compare analytical compliance models against a finite element model. The results show that the errors are sufficiently small (2%, compared with finite element (FE) model), if the range of motion is limited to linear deformations. This work provides a systematical approach for compliance analysis and synthesis of general flexure mechanisms. The symbolic formulation enables subsequent design tasks, such as compliance synthesis or sensitivity analysis.

Creating highly articulated miniature structures requires assembling a large number of small parts. This is a very challenging task and increases cost of mechanical assemblies. Insert molding presents the possibility of creating a highly articulated structure in a single molding step. This can be accomplished by placing multiple metallic bearings in the mold and injecting plastic on top of them. In theory, this idea can generate a multi degree of freedom structures in just one processing step without requiring any post molding assembly operations. However, the polymer material has a tendency to shrink on top of the metal bearings and hence jam the joints. Hence, until now insert molding has not been used to create articulated structures. This paper presents a theoretical model for estimating the extent of joint jamming that occurs due to the shrinkage of the polymer on top of the metal bearings. The level of joint jamming is seen as the effective torque needed to overcome the friction in the revolute joints formed by insert molding. We then use this model to select the optimum design parameters which can be used to fabricate functional, highly articulating assemblies while meeting manufacturing constraints. Our analysis shows that the strength of weld-lines formed during the in-mold assembly process play a significant role in determining the minimum joint dimensions necessary for fabricating functional revolute joints. We have used the models and methods described in this paper to successfully fabricate the structure for a minimally invasive medical robot prototype with potential applications in neurosurgery. To the best of our knowledge, this is the first demonstration of building an articulated structure with multiple degrees of freedom using insert molding.

A three degree-of-freedom planar parallel manipulator has been extensively studied as the fundamental example of parallel manipulators. In this work, we explicitly show that this mechanism possesses a completely decoupled compliance characteristic at the object space, which is the important operational requirement for RCC devices. As a first condition to have a RCC point, this mechanism should maintain symmetric configurations. As the second condition, the same magnitude of joint compliance should be symmetrically placed at the same joint location of each chain. An alternative planar configuration with more adjustable feature of compliance is introduced. We also investigate the compliance characteristics of a spherical 3-degree-of-freedom mechanism which has a similar kinematic structure as the planar mechanism. It turns out that the spherical mechanism also has a RCC point at the intersection point of all of nine joint axes in its symmetric configuration. It is expected that these two parallel mechanisms not only can be used as excellent three degree-of-freedom RCC devices, but also can be integrated into the design of a new six degree-of-freedom RCC device

Quasi-static wedging of three point contacts is investigated, wherein the concepts of virtual and redundant wedging due to a resultant force falling into a frictional cone are proposed. We consider the fully started case of a square peg and hole consisting of two point-surface contacts and one line-line contact. An analysis of the wedging diagram for this highly constrained configuration is carried out and compared to the two-dimensional case. An approximate wedging diagram is constructed which shows that wedging of square pegs into square holes is more likely than cylindrical pegs and holes of similar sizes.

An approach consisting of the identification of a minimal mechanical system which is capable to emulate the dynamic effects of the human torso during walking gait is presented. A 3D dynamic simulation of a virtual manikin with 27 degrees of freedom (DOFs) is carried out in order to identify the effort wrench exerted by the torso on the lower limbs. An analysis of the six components of this wrench shows that some components are dependant. A systemic study of two, three and four DOFs mechanisms based on general state equation formalism leads to an interesting result. Indeed, four DOFs are necessary and sufficient to emulate the dynamic effects. We also show the efficiency and generality of the proposed approach

In this investigation a robotic system's dynamic performance is optimized for high reliability under uncertainty. The dynamic capability equations (DCE) allow designers to predict the dynamic performance of a robotic system for a particular configuration and reference point on the end-effector (i.e.,point design). Here the DCE are used in conjunction with a reliability-based design optimization (RBDO) strategy in order to obtain designs with robust dynamic performance with respect to the end-effector reference point. In this work a unilevel performance measure approach (PMA) is used to perform RBDO. This is important for the reliable design of robotic systems in which a solution to the DCE is required for each constraint call. The method is illustrated on a robot design problem

A geometric application of screw theory is used to develop a recursive algorithm for computing all singular configurations of revolute-jointed manipulators with arbitrary geometry and an arbitrary number of joints. The depth of the recursion is linear in the number of joints, n , while the computational burden is proportional to 2 <sup>n-2</sup>. This method does not require explicit construction of the Jacobian matrix elements or a determinant operation. Further, the screw axis of the singular motion is determined at no additional cost. The bifurcations of this algorithm are also explored

Traditional design for reliability is not perfect, because such systems aim at less faults but once a fault happens they hard-fail. To prevent such disasters, one should develop soft-fail machines that can still maintain their functions even if faults happen. The authors propose a framework of a self-maintenance machine (SMM). Two repair strategies are proposed; viz., control type and functional redundancy type that is a strategy to add redundancy to a machine by using potential functions of existing parts. For executing fault diagnosis and repair planning, a model based approach that employs qualitative physics, is proposed. Finally, they illustrate two prototypes of SMM; the control type and the functional redundancy type

Positioning 3R manipulators may have two or four inverse kinematic solutions (IKS). This paper derives a necessary and sufficient condition for 3R positioning manipulators with orthogonal joint axes to have four distinct IKS. We show that the transition between manipulators with 2 and 4 IKS is defined by the set of manipulators with a quadruple root of their inverse kinematics. The resulting condition is explicit and states that the last link length of the manipulator must be greater than a quantity that depends on three of its remaining DH-parameters. This result is of interest for the design of new manipulators.

A spherical wrist of the serial type is said to be isotropic if it can attain a posture whereby the singular values of its Jacobian matrix are all identical and nonzero. What isotropy brings about is robustness to manufacturing, assembly, and measurement errors, thereby guaranteeing a maximum orientation accuracy. In this paper we investigate the existence of redundant isotropic architectures, which should add to the dexterity of the wrist under design by virtue of its extra degree of freedom. The problem formulation leads to a system of eight quadratic equations with eight unknowns. The Bezout number of this system is thus 2^8 = 256, its BKK bound being 192. However, the actual number of solutions is shown to be 32. We list all solutions of the foregoing algebraic problem. All these solutions are real, but distinct solutions do not necessarily lead to distinct manipulators. Upon discarding those algebraic solutions that yield no new wrists, we end up with exactly eight distinct architectures, the eight corresponding manipulators being displayed at their isotropic posture.

Gear endurance tests and rolling-element fatigue tests were conducted to compare the performance of spur gears made from AISI 9310, CBS 600 and modified Vasco X-2 and to compare the pitting fatigue lives of these three materials. Gears manufactured from CBS 600 exhibited lives longer than those manufactured from AISI 9310. However, rolling-element fatigue tests resulted in statistically equivalent lives. Modified Vasco X-2 exhibited statistically equivalent lives to AISI 9310. CBS 600 and modified Vasco X-2 gears exhibited the potential of tooth fracture occurring at a tooth surface fatigue pit. Case carburization of all gear surfaces for the modified Vasco X-2 gears results in fracture at the tips of the gears.

Life tests were conducted at three different loads with three groups of 8.9 cm, pitch diameter spur gears made of vacuum arc remelted AISI 9310 steel. Life was found to vary inversely with load to the 4.3 and 5.1 power at the L-10 and L-50 life levels, respectively. The Weibull slope varied linearly with maximum Hertz contact stress, having an average value of 2.5. The test data when compared to AGMA standards showed a steeper slope for the load-life diagram.

A method of selecting grid size for the finite element analysis of gear tooth deflection is presented. The method is based on a finite element study of two cylinders in line contact, where the criterion for establishing element size was that there be agreement with the classical Hertzian solution for deflection. The results are applied to calculate deflection for the gear specimen used in the NASA spur gear test rig. Comparisons are made between the present results and the results of two other methods of calculation. The results have application in design of gear tooth profile modifications to reduce noise and dynamic loads.

Inherent flaws, as well as the effects of rate and time, are shown by tests on viscoelastic adhesive-bonded single lap joints to be as critical in joint failure as environmental and stress concentration effects, with random inherent flaws and loading rate changes resulting in an up to 40% reduction in joint strength. It is also found that the asymptotic creep stress, below which no delayed failure may occur, may under creep loading be as much as 45% less than maximum adhesive strength. Attention is given to test results for the case of titanium-LARC-3 adhesive single-lap specimens.

An integrated computerized approach for design and stress analysis of low-noise spiral bevel gear drives with adjusted bearing contact has been developed. The computation procedure is an iterative process, requiring four separate steps that provide: (a) a parabolic function of transmission errors that is able to reduce the effect of errors of alignment, and (b) reduction of the shift of bearing contact caused by misalignment. Application of finite element analysis permits the contact and bending stresses to be determined and investigate the formation of the bearing contact. The design of finite element models and boundary conditions is automated and does not require an intermediate CAD computer program. A commercially available finite element analysis computer program with contact capability was used to conduct the stress analysis. The theory developed is illustrated with numerical examples.

A gear tooth temperature analysis was performed using a finite element method combined with a calculated heat input, calculated oil jet impingement depth, and estimated heat transfer coefficients. Experimental measurements of gear tooth average surface temperatures and instantaneous surface temperatures were made with a fast response infrared radiometric microscope. Increased oil jet pressure had a significant effect on both average and peak surface temperatures at both high load and speeds. Increasing the speed at constant load and increasing the load at constant speed causes a significant rise in average and peak surface temperatures of gear teeth. The oil jet pressure required for adequate cooling at high speed and load conditions must be high enough to get full depth penetration of the teeth. Calculated and experimental results were in good agreement with high oil jet penetration but showed poor agreement with low oil jet penetration depth.

The design and visualization of three-dimensional objects with curved surfaces have always been difficult. The paper given below describes a computer system which facilitates both the design and visualization of such surfaces. The system enhances the design of these surfaces by virtue of various interactive techniques coupled with the application of B-Spline theory. Visualization is facilitated by including a specially built model-making machine which produces three-dimensional foam models. Thus, the system permits the designer to produce an inexpensive model of the object which is suitable for evaluation and presentation.

A computer aided design procedure is presented for minimizing dynamic effects on high contact ratio gears by modification of the tooth profile. Both linear and parabolic tooth profile modifications of high contact ratio gears under various loading conditions are examined and compared. The effects of the total amount of modification and the length of the modification zone were systematically studied at various loads and speeds to find the optimum profile design for minimizing the dynamic load and the tooth bending stress. Parabolic profile modification is preferred over linear profile modification for high contact ratio gears because of its lower sensitivity to manufacturing errors. For parabolic modification, a greater amount of modification at the tooth tip and a longer modification zone are required. Design charts are presented for high contact ratio gears with various profile modifications operating under a range of loads. A procedure is illustrated for using the charts to find the optimum profile design.

This paper presents a multidisciplinary optimization framework developed by the authors and applied to small-size supersonic aircraft. The multidisciplinary analysis suite is based on the combination of low (empirical) and high-fidelity computational fluid dynamics (CFD) and computational structure mechanics (CSM) tools for predicting the overall aircraft performance and the sonic boom overpressure at supersonic flight, which represents the most challenging environmental constraint for supersonic aircraft. The analysis suite is coupled with a multi-objective optimization strategy for quantifying the trade-off between the maximum take-off weight, mission range, and the sonic boom overpressure. The optimization framework is applied to a small-size supersonic business-jet cruising at Mach number M=1.8 and featuring a double delta wing. The trade-offs between disciplines are well captured and an optimized configuration achieving the target mission range with a lower maximum take-off weight, and a moderate sonic boom signature is obtained through changes in wing dihedral and sweep. A more drastic reduction of the sonic boom signature is also obtained but at the cost of a significant reduction of the aircraft performance.

The fracture strengths of two large batches of A357-T6 cast aluminum coupon specimens were compared by using two-parameter Weibull analysis. The minimum number of these specimens necessary to find the fracture strength of the material was determined. The applicability of three-parameter Weibull analysis was also investigated. A design methodology based on the combination of elementary stress analysis and Weibull statistical analysis is advanced and applied to the design of a spherical pressure vessel shell. The results from this design methodology are compared with results from the applicable ASME pressure vessel code.

This paper describes a prototype software system that implements a form of feature-based design for assembly. It is not an automated design system but instead a decision and design aid for designers interested in Concurrent Design. Feature-based design captures design intent (assembly topology, product function, manufacturing, or field use) while creating part and product geometry. Design for assembly as used here extends existing ideas about critiquing part shapes and part count to include assembly process planning, assembly sequence generation, assembly fixturing assessments, and assembly process costs. This work was primarily interested in identifying the information important to DFA tasks, and how that information could be captured using feature-based design. It was not intended to extend the state of the art in feature-based geometry creation, but rather to explore the uses of the information that can be captured. The prototype system has been programmed in LISP on Sun workstations. Its research contributions comprise integration of feature-based design with several existing and new assembly analysis and synthesis algorithms; construction of feature properties to meet the needs of those algorithms; a carefully chosen division of labor between designer and computer; and illustration of feature-based models of products as the information source for assembly analysis and process design. Some of its functions have been implemented approximately or partially but they give the flavor of the benefits to be expected from a fully functional system.

A suspension system based on a band mechanism is investigated to provide the free-free conditions for ground-based validation testing of flexible space structures. The band mechanism consists of a noncircular disk with a convex profile, preloaded by torsional springs at its center of rotation so that static equilibrium of the test structure is maintained at any vertical location; gravitational force will be directly counteracted during dynamic testing of the space structure. This noncircular disk within the suspension system can be configured to remain unchanged for test articles with different weights as long as the torsional spring is replaced to maintain the originally designed frequency parameter of W/k(sub s). Simulations of test articles which are modelled as lumped-parameter as well as continuous parameter systems are also presented.

Thin rim gears find application in high-power, light-weight aircraft transmissions. Bending stresses in thin rim spur gear tooth fillets and root areas differ from the stresses in solid gears due to rim deformations. Rim thickness is a significant design parameter for these gears. To study this parameter, a finite element analysis was conducted on a segment of a thin rim gear. The rim thickness was varied and the location and magnitude of the maximum bending stresses reported. Design limits are discussed and compared with the results of other researchers.

Face-milled spiral bevel gears with uniform tooth height are considered. An approach is proposed for the design of low noise and localized bearing contact of such gears. The approach is based on the mismatch of contacting surfaces and permits two types of bearing contact either directed longitudinally or across the surface to be obtained. A Tooth Contact Analysis (TCA) computer program was developed. This analysis was used to determine the influence of misalignment on meshing and contact of the spiral bevel gears. A numerical example that illustrates the developed theory is provided.

This paper discusses the fundamental geometrical characteristics of spiral bevel gear tooth surfaces. The parametric representation of an ideal spiral bevel tooth is developed. The development is based on the elements of involute geometry, differential geometry, and fundamental gearing kinematics. A foundation is provided for the study of nonideal gears and the effects of deviations from ideal geometry on the contact stresses, lubrication, wear, fatigue life, and gearing kinematics.

A modeling method for analyzing the three-dimensional thermal behavior of spiral bevel gears has been developed. The model surfaces are generated through application of differential geometry to the manufacturing process for face-milled spiral bevel gears. Contact on the gear surface is found by combining tooth contact analysis with three-dimensional Hertzian theory. The tooth contact analysis provides the principle curvatures and orientations of the two surfaces. This information is then used directly in the Hertzian analysis to find the contact size and maximum pressure. Heat generation during meshing is determined as a function of the applied load, sliding velocity, and coefficient of friction. Each of these factors change as the point of contact changes during meshing. A nonlinear finite element program was used to conduct the heat transfer analysis. This program permitted the time- and position-varying boundary conditions, found in operation, to be applied to a one-tooth model. An example model and analytical results are presented.

An analysis of the surface geometry of spiral bevel gears formed by a circular cutter is presented. The emphasis is upon determining the tooth surface principal radii of curvature of crown (flat) gears. Specific results are presented for involute, straight, and hyperbolic cutter profiles. It is shown that the geometry of circular cut spiral bevel gears is somewhat simpler than a theoretical logarithmic spiral bevel gear.

A procedure is presented for performing three-dimensional stress analysis of spiral bevel gears in mesh using the finite element method. The procedure involves generating a finite element model by solving equations that identify tooth surface coordinates. Coordinate transformations are used to orientate the gear and pinion for gear meshing. Contact boundary conditions are simulated with gap elements. A solution technique for correct orientation of the gap elements is given. Example models and results are presented.

The design of spiral bevel and hypoid gears that have a shaft extended from both sides of the cone apex (straddle design) is considered. A main difficulty of such a design is determining the length and diameter of the shaft that might be undercut by the head cutter during gear tooth generation. A method that determines the free space available for the gear shaft is proposed. The approach avoids collision between the shaft being designed and the head cutter during tooth generation. The approach is illustrated with a numerical example.

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods: (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations, (2) projection operator methods in a function space which includes the frequencies of vibration and the control function, (3) epsilon-technique penalty function transformation resulting in a highly nonlinear programming problem, (4) finite difference discretization of the state equations again resulting in a nonlinear program, (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff, (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

Fracture data from five series of four point bend tests of beam and spin tests of flat annular disks were reanalyzed. Silicon nitride and graphite were the test materials. The experimental fracture strengths of the disks were compared with the predicted strengths based on both volume flaw and surface flaw analyses of four point bend data. Volume flaw analysis resulted in a better correlation between disks and beams in three of the five test series than did surface flaw analysis. The Weibull (moduli) and characteristic gage strengths for the disks and beams were also compared. Differences in the experimental Weibull slopes were not statistically significant. It was shown that results from the beam tests can predict the fracture strength of rotating disks.

The summary of a complete analytical design evaluation of an existing parallel flow compressor is presented and a field vibration problem that manifested itself as a subsynchronous vibration that tracked at approximately 2/3 of compressor speed is reviewed. The comparison of predicted and observed peak response speeds, frequency spectrum content, and the performance of the bearing-seal systems are presented as the events of the field problem are reviewed. Conclusions and recommendations are made as to the degree of accuracy of the analytical techniques used to evaluate the compressor design.

A uniform clearance pumping ring, as opposed to the conventional taper clearance one, is described. The uniform clearance concept eliminates complex elastohydrodynamic problems and enables a simple analytical treatment to be made. An analytical expression is derived for the pumping rate showing the effect of various design parameters on the pumping ring's performance. An optimum clearance is found by which the pumping rate is maximized and a numerical example is presented to demonstrate the potential of the uniform clearance design.

Hard coatings have potential for increasing gear surface fatigue lives. Experiments were conducted using gears both with and without a metal-containing, carbonbased coating. The gears were case-carburized AISI 9310 steel spur gears. Some gears were provided with the coating by magnetron sputtering. Lives were evaluated by accelerated life tests. For uncoated gears, all of fifteen tests resulted in fatigue failure before completing 275 million revolutions. For coated gears, eleven of the fourteen tests were suspended with no fatigue failure after 275 million revolutions. The improved life owing to the coating, approximately a six-fold increase, was a statistically significant result.

The design of a standard gear mesh is treated with the objective of minimizing the gear size for a given ratio, pinion torque, and allowable tooth strength. Scoring, pitting fatigue, bending fatigue, and the kinematic limits of contact ratio and interference are considered. A design space is defined in terms of the number of teeth on the pinion and the diametral pitch. This space is then combined with the objective function of minimum center distance to obtain an optimal design region. This region defines the number of pinion teeth for the most compact design. The number is a function of the gear ratio only. A design example illustrating this procedure is also given.

The optimal design of compact spur gear reductions includes the selection of bearing and shaft proportions in addition to gear mesh parameters. Designs for single mesh spur gear reductions are based on optimization of system life, system volume, and system weight including gears, support shafts, and the four bearings. The overall optimization allows component properties to interact, yielding the best composite design. A modified feasible directions search algorithm directs the optimization through a continuous design space. Interpolated polynomials expand the discrete bearing properties and proportions into continuous variables for optimization. After finding the continuous optimum, the designer can analyze near optimal designs for comparison and selection. Design examples show the influence of the bearings on the optimal configurations.

This paper presents procedures for designing compact spur gear sets with the objective of minimizing the gear size. The allowable tooth stress and dynamic response are incorporated in the process to obtain a feasible design region. Various dynamic rating factors were investigated and evaluated. The constraints of contact stress limits and involute interference combined with the tooth bending strength provide the main criteria for this investigation. A three-dimensional design space involving the gear size, diametral pitch, and operating speed was developed to illustrate the optimal design of spur gear pairs. The study performed here indicates that as gears operate over a range of speeds, variations in the dynamic response change the required gear size in a trend that parallels the dynamic factor. The dynamic factors are strongly affected by the system natural frequencies. The peak values of the dynamic factor within the operating speed range significantly influence the optimal gear designs. The refined dynamic factor introduced in this study yields more compact designs than AGMA dynamic factors.

The magnitude and variation of tooth pair compliance with load position affects the dynamics and loading significantly, and the tooth root stressing per load varies significantly with load position. Therefore, the recently developed time history, interactive, closed form solution for the dynamic tooth loads for both low and high contact ratio spur gears was expanded to include improved and simplified methods for calculating the compliance and stress sensitivity for three involute tooth forms as a function of load position. The compliance analysis has an improved fillet/foundation. The stress sensitivity analysis is a modified version of the Heywood method but with an improvement in the magnitude and location of the peak stress in the fillet. These improved compliance and stress sensitivity analyses are presented along with their evaluation using test, finite element, and analytic transformation results, which showed good agreement.

A method for calculating a component's design survivability by incorporating finite element analysis and probabilistic material properties was developed. The method evaluates design parameters through direct comparisons of component survivability expressed in terms of Weibull parameters. The analysis was applied to a rotating disk with mounting bolt holes. The highest probability of failure occurred at, or near, the maximum shear stress region of the bolt holes. Distribution of failure as a function of Weibull slope affects the probability of survival. Where Weibull parameters are unknown for a rotating disk, it may be permissible to assume Weibull parameters, as well as the stress-life exponent, in order to determine the disk speed where the probability of survival is highest.

An approach for design and generation of low-noise helical gears with localized bearing contact is proposed. The approach is applied to double circular arc helical gears and modified involute helical gears. The reduction of noise and vibration is achieved by application of a predesigned parabolic function of transmission errors that is able to absorb a discontinuous linear function of transmission errors caused by misalignment. The localization of the bearing contact is achieved by the mismatch of pinion-gear tooth surfaces. Computerized simulation of meshing and contact of the designed gears demonstrated that the proposed approach will produce a pair of gears that has a parabolic transmission error function even when misalignment is present. Numerical examples for illustration of the developed approach are given.

An integrated single-step design approach to the synthesis of a mechanical device and an active controller is presented for slew maneuvers of flexible space structures. A mechanical device consists of a noncircular gear pair that acts as a varying gear ratio transmission between an electro-mechanical actuator and a flexible structure. Such a system has been introduced as an analytical application in minimizing flexural vibrations in the slewing maneuver of flexible space structures such as large solar panels and satellites in space. Only a simple regulator-type feedback controller is called for to control this complex electro-mechanical and -structural system and the purpose is to design this complex system in an integrated procedure to achieve low flexural vibrations of the structure. Numerical simulations of the slewing control tasks for a planar articulated double-beam structure are presented.

The penetration depth onto the tooth flank of a jet of oil at different velocities pointed at the pitch line on the outgoing side of mesh was determined. The analysis determines the impingement depth for both the gear and the pinion. It includes the cases for speed increasers and decreasers as well as for one to one gear ratio. In some cases the jet will strike the loaded side of the teeth, and in others it will strike the unloaded side of the teeth. In nearly all cases the top land will be cooled regardless of the penetration depth, and postimpingement oil spray will usually provide adequate amounts of oil for lubrication but is marginal or inadequate for cooling.

This paper presents the experimental evaluation of the Structure-Based Connectionist Network (SBCN) fault diagnostic system introduced in the preceding article. For this vibration data from two different helicopter gearboxes: OH-58A and S-61, are used. A salient feature of SBCN is its reliance on the knowledge of the gearbox structure and the type of features obtained from processed vibration signals as a substitute to training. To formulate this knowledge, approximate vibration transfer models are developed for the two gearboxes and utilized to derive the connection weights representing the influence of component faults on vibration features. The validity of the structural influences is evaluated by comparing them with those obtained from experimental RMS values. These influences are also evaluated ba comparing them with the weights of a connectionist network trained though supervised learning. The results indicate general agreement between the modeled and experimentally obtained influences. The vibration data from the two gearboxes are also used to evaluate the performance of SBCN in fault diagnosis. The diagnostic results indicate that the SBCN is effective in directing the presence of faults and isolating them within gearbox subsystems based on structural influences, but its performance is not as good in isolating faulty components, mainly due to lack of appropriate vibration features.

Robust gear designs consider not only crack initiation, but crack propagation trajectories for a fail-safe design. In actual gear operation, the magnitude as well as the position of the force changes as the gear rotates through the mesh. A study to determine the effect of moving gear tooth load on crack propagation predictions was performed. Two-dimensional analysis of an involute spur gear and three-dimensional analysis of a spiral-bevel pinion gear using the finite element method and boundary element method were studied and compared to experiments. A modified theory for predicting gear crack propagation paths based on the criteria of Erdogan and Sih was investigated. Crack simulation based on calculated stress intensity factors and mixed mode crack angle prediction techniques using a simple static analysis in which the tooth load was located at the highest point of single tooth contact was validated. For three-dimensional analysis, however, the analysis was valid only as long as the crack did not approach the contact region on the tooth.

Consideration set formation using non-compensatory screening rules is a vital component of real purchasing decisions with decades of experimental validation. Marketers have recently developed statistical methods that can estimate quantitative choice models that include consideration set formation via non-compensatory screening rules. But is capturing consideration within models of choice important for design? This paper reports on a simulation study of a vehicle portfolio design when households screen over vehicle body style built to explore the importance of capturing consideration rules for optimal designers. We generate synthetic market share data, fit a variety of discrete choice models to the data, and then optimize design decisions using the estimated models. Model predictive power, design "error", and profitability relative to ideal profits are compared as the amount of market data available increases. We find that even when estimated compensatory models provide relatively good predictive accuracy, they can lead to sub-optimal design decisions when the population uses consideration behavior; convergence of compensatory models to non-compensatory behavior is likely to require unrealistic amounts of data; and modeling heterogeneity in non-compensatory screening is more valuable than heterogeneity in compensatory trade-offs. This supports the claim that designers should carefully identify consideration behaviors before optimizing product portfolios. We also find that higher model predictive power does not necessarily imply better design decisions; that is, different model forms can provide "descriptive" rather than "predictive" information that is useful for design.

A contact fatigue life analysis method for multiroller traction drives is presented. The method is based on the Lundberg-Palmgren analysis method for rolling element bearing life prediction, and also uses life adjustment factors for materials, processing, lubrication, and effect of traction. The analysis method is applied in an optimization study to the multiroller traction drive, consisting of a single-stage planetary configuration with two rows of stepped planet rollers of five rollers per row. The drive was approximately 25 centimeters in diameter by 11 centimeters long, having a nominal ratio of 15:1. The theoretically predicted drive life was 2510 hours at a nominal continuous power and speed of 74.6 kW (100 hp) and 75,000 rpm.

Helical gears with localized bearing contact of tooth surfaces achieved by profile crowning of tooth surfaces are considered. Profile crowning is provided by application of two imaginary rack-cutters with mismatched surfaces. The goal is to determine the dimensions and orientation of the instantaneous contact ellipse that requires the determination of principle curvatures of pinion-gear tooth surfaces. A simplified solution to this problem is proposed based on the approach development for correlation of principal curvatures and directions of generating and generated tooth surfaces. The obtained equations are applied for profile crowning where the normal profiles of the rack-cutters are either a circular arc or a straight line.

A loaded gear drive with point contact between tooth surfaces is considered. The principal curvatures and directions at a current point of tangency, the contact paths on tooth surfaces, and the transmission errors caused by misalignment we consider as known. In this paper the following topics are covered: (1) determination of the contact force and its distribution over the contact ellipse; (2) determination of the tooth deflection, the load share, and the real contact ratio; and (3) stress analysis by application of the finite element method. The discussed approach is illustrated with a numerical example.

A substructure synthesis method is proposed to account for contact-impact effects in flexible components of mechanical systems. Components that may come into contact is divided into substructures, on each of which local deformation modes are defined to described deformation fields of components. Constraint modes and fixed interface normal modes are used to account for elastic deformation within each substructure. A constraint addition-deletion technique is used to account for contact between impacting bodies. Lagrange multipliers associated with the constraints, which represent constraint forces, are used to determine separation of contacting nodes. Use of the method is illustrated on problems of longitudinal and transverse impact of bodies.

An analysis and computer program were developed for calculating the dynamic gear tooth loading and root stressing for high contact ratio gearing (HCRG) as well as LCRG. The analysis includes the effects of the variable tooth stiffness during the mesh, tooth profile modification, and gear errors. The calculation of the tooth root stressing caused by the dynamic gear tooth loads is based on a modified Heywood gear tooth stress analysis, which appears more universally applicable to both LCRG and HCRG. The computer program is presently being expanded to calculate the tooth contact stressing and PV values. Sample application of the gear program to equivalent LCRG (1.566 contact ratio) and HCRG (2.40 contact ratio) revealed the following: (1) the operating conditions and dynamic characteristics of the gear system an affect the gear tooth loading and root stressing, and therefore, life significantly; (2) the length of the profile modification affect the tooth loading and root stressing significantly, the amount depending on the applied load, speed, and contact ratio; and (3) the effect of variable tooth stiffness is small, shifting and increasing the response peaks slightly from those for constant tooth stiffness.

Top-cited authors
• Northwestern University
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