The input-output displacement relations of two Cardan joints arranged in series on a driveline has been investigated in detail, including the effects of unequal joint angles, the phase angle between the two Cardan joints and also such manufacturing tolerance errors as nonright angle moving link lengths and offset joint axes. A combined Newton-Raphson method and Davidon-Fletcher-Powell optimization algorithm using dual-number coordinate-transformation matrices was employed to perform the analysis. An experiment was conducted to validate the results of the analysis. The apparatus consisted of a double-Cardan-joint driveline whose rotations were measured by optical shaft encoders that were sampled by a computer data-acquisition system. The equipment was arranged so that the phase angle between the joints and the offset angles between the shafts at each of the two joints could be readily varied. The “relative phase angle,” the difference between the phase angle of the two joints and the angle between the planes defined by the input and intermediate and the intermediate and output shafts, was found to be the significant factor. If the offset angles at both Cardan joints are equal, the double-Cardan-joint driveline functions as a constant-velocity coupling when the magnitude of the relative phase angle is zero. If the offset angles at the two Cardan joints are unequal, a condition prevailing in the important front-wheel-drive automobile steering-column application, then fluctuation in output velocity for a constant input velocity is minimized although not eliminated for zero relative phase angle.
Many mechanical and structural design problems encountered in practice require solutions which balance several conflicting objectives. The vector, scalarization, and trade-off-curve methods have been developed to achieve multiobjective solutions. One of the best known methods for generating a compromise solution, based on the concept of Pareto minimum solution, is the cooperative game theory method since it uses a scalarized approach and has a numerical measure of compromise. However, game theory is hard to automate due to a two step optimization process involved. Hence, in this work, a modification to the game theory is introduced in which the two optimization steps are combined and an algorithm for its implementation is developed. The algorithm is tested on two numerical examples, including one dealing with the probabilistic design of an eighteen speed machine tool gear train. The probabilistic theory necessary for the design of the gear train is also introduced. The examples validate the modified game theory.
A new integrated approach to the design of high speed planar mechanisms is presented. The resulting nonlinear programming formulation combines both the kinematic and kinetostatic synthesis aspects of mechanism design. The multiobjective optimization techniques presented in this work facilitate the design of a linkage to meet several kinematic and dynamic design criteria. The method can be used for motion, path, and function generation problems. The nonlinear programming formulation also permits an imposition of constraints to eliminate solutions which possess undesirable kinematic and motion characteristics. To model the vague and imprecise information in the problem formulation, the tools of fuzzy set theory have been used. A novel method of solving the resulting fuzzy multiobjective problem using mathematical programming techniques is presented. The outlined procedure is expected to be useful in situations where doubt arises about the exactness of permissible values, degree of credibility, and correctness of statements and judgements.
The new concept of a mobile screw space is introduced in this paper. A mobile screw space is a set of motors for which the outer product of any two arbitrary elements belongs to the original set. It is shown that the condition for the existence of almost all existing over-constrained mechanisms is related to the mobile screw space, that is, all joint motors of the mechanism either belongs to a mobile screw space or belong to a space which is given as the sum of mobile spaces. Determining an over-constrained mechanism on the basis of this condition makes it possible to generate many mechanisms, including most presently known over-constrained mechanisms.
We have successfully developed a tooth profile which enables a spur gear having zero relative curvature at contact points under the engagement through the concave/convex pattern of contact and further enables a single flank of the tooth profile to have the infinite number of points where the relative curvature and specific sliding are zero values. We have given a name of “LogiX” to this tooth profile. A W-N (Wildhaber-Novikov) tooth profi¨le has been known as that having the concave/convex pattern of contact. The tooth profile, however, is applied only to a helical gear due to its feature of a point contact. While, on the LogiX tooth profile, an improvement has been made so that a pair of spur gears having the tooth profile may contact each other through the concave/convex pattern of a line-contact which cannot be achieved by the W-N tooth profile. Therefore, the LogiX tooth profile has realized a spur gear whose surface durability is as high as that of the W-N tooth profile. Since the results of durability tests show slightly higher than an involute one, it is expected that a new type heavy duty gear of this tooth profile puts into practical use. Furthermore, the tooth profile will have the possibility of creating the new era in the history of tooth profile theory.
A continuous contact force model for the impact analysis of a two-particle collision is presented. The model uses the general trend of the Hertz contact law. A hysteresis damping function is incorporated in the model which represents the dissipated energy in impact. The parameters in the model are determined, and the validity of the model is established. The model is then generalized to the impact analysis between two bodies of a multibody system. A continuous analysis is performed using the equations of motion of either the multibody system or an equivalent two-particle model of the colliding bodies. For the latter, the concept of effective mass is presented in order to compensate for the effects of joint forces in the system. For illustration, the impact situation between a slider-crank mechanism and another sliding block is considered.
The present paper deals with an analysis of the kinematics and geometry of the tooth engagement of the harmonic drive, and presents a new method for determining tooth profiles of the harmonic drive. First, two assumptions are set up: (1) Though there is relative motion among the respective teeth of the flexspline during flexing, the individual teeth themselves can be considered to be rigid. (2) The pitch between the respective teeth of the flexspline is constant on the neutral line. Experiments confirm the existence of a neutral line along the flexspline rim which maintain the same length as that prior to bending. Then, tooth profiles of the circular spline and the flexspline are determined according to the theory of gear mechanism that the common tooth normal at the meshing point passes through the pitch point. The analysis is carried out in a plane mechanism.
A general purpose algorithm for the solution of nonlinear mathematical programming problems containing integer, discrete, zero-one, and continuous design variables is described. The algorithm implements a branch and bound procedure in conjunction with either an exterior penalty function or a quadratic programming method. Variable bounds are handled independently from the design constraints which removes the necessity to reformulate the problem at each branching node. Examples are presented to demonstrate the utility of the algorithm for solving design problems.
A new equilibrator design approach based on system potential energy functions is presented. This approach was used to discover a group of spring equilibrators which perfectly balance a rotatable rigid link at every orientation angle through 360 deg of link rotation. Springs are connected between a rotatable link and ground, where one end of each spring is connected to the rigid link and the other end of each spring is connected to ground. The rigid link is connected to ground by a pin joint and is free to rotate about that joint. The conditions for existence and the design equations for all equilibrators which fall into this category are developed and presented. Three designs appear to offer unique advantages over the infinite number of design options available.
Four-bar linkages which generate approximate straight-line motion are investigated for use in vehicle suspension. The particular suspension geometry studied is the beam-axle type, leading to systems with two degrees of freedom of body roll motion relative to the axle. Kinetostatic analyses of these spring-restrained systems are carried out for a number of cases involving different combinations of loads and linkage geometry. From the predicted orientation of the body relative to the axle various vehicle dynamics parameters can be calculated, including roll center location and roll stiffness. It is found that these parameters, which are normally considered constant, are functions of the suspension linkage, jounce/rebound, and roll angle. Conditions under which the linkage types produce comparable roll performance are established.
The traditional one-component transmission error of parallel-axis helical gears is generalized to a three-component transmission error which characterizes the composite displacement in the plane-of-contact resulting from arbitrary small deviations in the positions of both gears of a meshing pair from the positions of their rigid perfect involute counterparts. A set of linear algebraic equations is derived for the contribution to the three generalized transmission error components arising from elastic deformations of the teeth and gear bodies and deviations of the tooth running surfaces from equispaced perfect involute surfaces. It is shown how to combine this set of equations with the generalized transmission error definition and the equations of motion of a gear system to predict the dynamic response of gear elements in the system. For the case of negligible gearbody and bearing/bearing support inertial forces, an additional set of algebraic equations that includes the effects of bearing flexibility and misalignment is derived. Combining the solution of this set of equations with the above-mentioned generalized transmission error equations yields the three-component generalized static transmission error.
The use of independent actuation of suspension and locomotion degrees of freedom offers significant potential for improved performance vehicles. In this paper, the mathematics of active coordination of such systems is explored. Two different, actively coordinated mobility system configurations are used as examples to elucidate the principles involved.
A method for the simultaneous calculation of optimal tooth tip relief and tooth crowning for spur and helical gears is presented in this paper. The tooth profile modification is described by a linear function. Two types of crowning are introduced: linear and parabolic. The optimization of the tooth modifications is based on the following conditions: (1) The teeth are entering in mesh smoothly, without interference. (2) The load distribution factor is minimized. A computer program is developed for the calculation of the optimal tooth tip relief and crowning for spur and helical gears. By using this program the influence of type and length of optimal crowning and length of tooth tip relief on load distribution factor is investigated. Also, the influence of gear parameters on optimal tooth profile modification is discussed. On the basis of the obtained results, by regression analysis an equation is derived for the calculation of the optimal tooth tip relief.
The well-known pictorial drafting technique of isometric drawing is here generalized in two ways, called “isoclinal” and “axial,” or collectively, “symmetric.” Although the isoclinal projection preserves the useful property of foreshortening adjacent edges equally, and the axial projection foreshortens adjacent normals equally, the directions for these projections usually differ from that for isometric projection. Formulas are derived for the isoclinal and axial directions and their foreshortening factors and rotation matrices. Although intended for three-dimensional computer-aided design, the symmetric projections also can be performed on a drawing board with a protractor. Graphic examples involving design of nonrectangular connectors for three skew structural members and adjacent plates in space are presented.
The design of a turbocharged, gasoline fueled, four-stroke engine is considered with the goal of selecting design and operating variables to minimize fuel consumption. The development of the engine simulation code and the effect of model assumptions on the results are presented. The optimization includes constraints on detonation, exhaust emissions, and torque. Variables are bounded to assure the validity of the simulation. A number of observations about the interaction between the thermo-fluid model and the nonlinear programming algorithm are made and general strategies to enhance the optimization under such circumstances are discussed. The method is illustrated by exploring the design of a turbocharged Buick V-6 engine on an IBM PC/AT personal computer. Stock design variables, and operating variables that provided a design away from the constraints imposed by torque, emission, and detonation were chosen as the starting point for the optimization. Application of the optimization strategy resulted in an 18 percent reduction in predicted fuel consumption at 50 miles per hour. Significant specific recommendations included a reduction in combustion chamber volume, an increase in intake manifold pressure, an increase in intake duration, a decrease in exhaust duration, and relatively small changes in valve geometry. The paper clearly demonstrates that it is feasible to do relatively sophisticated engineering design and optimization on personal computers, and it sets the stage for further work in this area.
The positional errors due to clearances in journal bearings intended to approximate revolute joints are analyzed. Equations are presented which relate joint geometry, external loads, and errors. The paper treats the case common in serial chain manipulators where the revolute joint’s nominal rotational angle is controlled, and it is desired to determine the errors that result from the clearances.
A basic hypothesis of this paper is that the multiplier methods can be effective and efficient for dynamic response optimization of large scale systems. The methods have been previously shown to be inefficient compared to the primal methods for static response applications. However, they can be more efficient for dynamic response applications because they collapse all time-dependent constraints and the cost function to one functional. This can result in substantial savings in the computational effort during design sensitivity analysis. To investigate this hypothesis, an augmented functional for the dynamic response optimization problem is defined. Design sensitivity analysis for the functional is developed and three example problems are solved to investigate computational aspects of the multiplier methods. It is concluded that multiplier methods can be effective for dynamic response problems but need numerical refinements to avoid convergence difficulties in unconstrained minimization.
Single-closed-loop N-bar mobility criteria are presented. These criteria provide a simple and effective method (1) to determine the full rotatability of any single closed-loop linkages, (2) to predict the revolvability between two adjacent links, (3) to explain and identify the existence of singular positions, (4) to classify linkages, and (5) to identify the difference of singular positions between linkages of different types. These criteria are fundamental tools toward a better understanding on the mobility of more complicated linkages.
A method is developed to resolve the redundancy of serial robotic manipulators. The main characteristics of this method are as follows: (a) The solution is conservative (unique); (b) It globally (over the task period) optimizes the objective function; (c) It is computationally very fast. In fact, it is about three orders of magnitude faster than solving for the exact boundary value problem.
The use of a new material is a hopeful strategy for breaking through the barrier of improving the product performance and/or the lowering of the product manufacturing cost. However, if the definite decision making method for selection of materials has not been established, use of a new material may not result in enhancement of the product performance and / or reduction of the product manufacturing cost. This paper proposes a methodology for decision making of a material used in machine structures from the standpoint of structural dynamics. First, purposes for use of new materials and evaluative parameters for decision making of material choice are described, and fundamental knowledge and theorems for decision making are established. Then, general decision making procedures for material choice are constructed. Finally, for model cases of purposes for use of new materials, detailed decision procedures are explained using also some numerical examples.
