S. N. Kramer

An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and Its Applications to Mechanical Design

An algorithm for solving nonlinear optimization problems involving discrete, integer, zero-one, and continuous variables is presented. The augmented Lagrange multiplier method combined with Powell’s method and Fletcher and Reeves Conjugate Gradient method are used to solve the optimization problem where penalties are imposed on the constraints for integer/discrete violations. The use of zero-one variables as a tool for conceptual design optimization is also described with an example. Several case studies have been presented to illustrate the practical use of this algorithm. The results obtained are compared with those obtained by the Branch and Bound algorithm. Also, a comparison is made between the use of Powell’s method (zeroth order) and the Conjugate Gradient method (first order) in the solution of these mixed variable optimization problems.

A Simple and Accurate Method for Determining Large Deflections in Compliant Mechanisms Subjected to End Forces and Moments

Compliant members in flexible link mechanisms undergo large deflections when subjected to external loads. Because of this fact, traditional methods of deflection analysis do not apply. Since the nonlinearities introduced by these large deflections make the system comprising such members difficult to solve, parametric deflection approximations are deemed helpful in the analysis and synthesis of compliant mechanisms. This is accomplished by representing the compliant mechanism as a pseudo-rigid-body model. A wealth of analysis and synthesis techniques available for rigid-body mechanisms thus become amenable to the design of compliant mechanisms. In this paper, a pseudo-rigid-body model is developed and solved for the tip deflection of flexible beams for combined end loads. A numerical integration technique using quadrature formulae has been employed to solve the large deflection Bernoulli-Euler beam equation for the tip deflection. Implementation of this scheme is simpler than the elliptic integral formulation and provides very accurate results. An example for the synthesis of a compliant mechanism using the proposed model is also presented.

Detection and elimination of mechanism defects in the selective precision synthesis of planar mechanisms

Abstract Possible mechanism defects and errors that may be encountered during the selective precision synthesis of planar mechanisms are discussed in detail and suitable remedial measures recommended. These defects include dyadic assembly error, locking defect, branching defect and the sequential mismatch defect. By solving these problems, the selective precision synthesis method can easily be adapted to solving numerous planar kinematic synthesis problems.

Selective precision synthesis of planar mechanisms satisfying position and velocity constraints

Abstract For many path generating mechanisms used in todays machines, the tracer point velocity may be as critical as its generated position for adequate mechanism performance [1,2]. In this paper the recently developed Selective Precision Synthesis technique has been extended to include the synthesis of mechanisms whose tracer points satisfy velocity as well as position specifications. The mechanism designer now has the capability of determining the planar mechanisms whose tracer points generate the positions and velocities within specified limits of accuracy. The method is applicable to all planar mechanisms constructed from dyads—including 4-bar, slider-crank, multi-loop and multi-link mechanisms. The theory is programmed for digital computation and numerical examples are presented to illustrate the use of the method.

Development of the Tri-Level Variable Rate Trajectory With Discretely Vanishing Shock for Optimum Design

The tri-level variable rate trajectory is a general motion which can be applied to programmable controllers, robotic manipulators, mechanisms, and mechanical devices where the input crank orientation, velocity, and acceleration vary with time. In the work presented here, the tri-level variable rate trajectory is an extension of the variable-rate trans-symmetric motion developed by the first author in 1984. That motion and the one developed here consist of discrete segments of constant and linearly varying accelerations occurring over specified time intervals, thereby providing versatile programmable trajectories with several advantages over the constant acceleration motion, simple harmonic motion, cycloidal motion, and the popular polynomial trajectories used in robotics. The tri-level variable-rate trajectory allows much more control of the acceleration contour of the motion and as a result, there is a decrease in the power required, a decrease in the operating cost, and a decrease in dynamic responses such as shock, vibration, and shaking force and virtual elimination of the overshoot problem that sometimes accompanies the polynomial segment motions. This is a general method which can be applied to many applications. The results of applying this trajectory to a complex machine controller are presented as an example.

Kinematic design and analysis of the rack-and-gear mechanism for function generation

Abstract The complex number method is applied to the kinematic design and analysis of the rack-and-gear function generating mechanism for three finely separated positions. The rack-and-gear mechanism is a useful planar mechanism that can be used to generate many functions that the four-bar mechanism can generate plus a great deal more. Due to the extra complexity of the rack-and-gear both monotonic and non-monotonic functions, as well as non-linear amplified motions, can be generated. A major advantage of the rack-and-gear mechanism is that the transmission angle always remains at its optimal value since the rack is always tangenial to the gear. Both the design and analysis have been programmed for use on the PDP 11/70 and are available to interested readers.

Kinematic Synthesis and Analysis of the Rack and Pinion Multi-Purpose Mechanism

The general procedure for synthesizing the rack and pinion mechanism up to seven precision conditions is developed. To illustrate the method, the mechanism has been synthesized in closed form for three precision conditions of path generation, two positions of function generation, and a velocity condition at one of the precision points. This mechanism has a number of advantages over conventional four bar mechanisms. First, since the rack is always tangent to the pinion, the transmission angle is always 90 deg minus the pressure angle of the rack. Second, with both translation and rotation of the rack occurring, multiple outputs are available. Other advantages include the generation of monotonic functions for a wide variety of motion and nonmonotonic functions for a full range of motion as well as nonlinear amplified motions. In this work the mechanism is made to satisfy a number of practical design requirements such as completely rotatable input crank and others. By including the velocity specification, the designer has considerably more control of the output motion. The method of solution developed in this work uses the complex number method of mechanism synthesis. A numerical example is included.

A Generalized Expert System Shell for Implementing Mechanical Design Applications: Review, Introduction, and Fundamental Concepts

The foundations for an expert system shell for implementing mechanical design applications are presented in this paper. The shell supports facilities for knowledge acquisition, quasireactive planning, design evaluation, and subjective explanation. The underlying philosophy of each of these facilities and some preliminary implementation issues are discussed. A brief summary of a recent research effort and its implications on the development of a generalized expert system shell for implementing mechanical design applications are also presented

Kinematic synthesis and analysis of the rack and pinion mechanism for planar path generation and function generation for six precision conditions

Abstract The rack and pinion mechanism is synthesized for generating both three prescribed path points with input coordination and three positions of function generation. This mechanism has a number of advantages over the four bar linkage. First, since the rack is always tangent to the pinion, the transmission angle is always the same (optimum) value of 90° minus the pressure angle of the pinion. Second, with both translation and rotation of the rack occuring, multiple outputs are available. Other advantages include the generation of monotonic functions for a wide variety of motion and nonmononotomic functions for the full range of motion as well as nonlinear amplified motions. In this work, the mechanism is made to satisfy a number of practical design requirements such as a completely rotatable input crank and others. The method of solution developed in this work utilizes the complex number method of mechanism synthesis and the solution is programmed on a VAX 11 785 computer and is being made available to interested readers.

A Computer-Aided-Design Technique for the Optimum Synthesis of RRSS Path Generating Spatial Mechanisms With Prescribed Input Timing

With the current emphasis on automation, the need for single actuator mechanical devices that can perform simple repetitive tasks much more economically, energy-efficiently and accurately than multiple-degree-of-freedom, multiple-actuator robotic manipulators is greatly felt. This paper presents an optimum synthesis technique for the RRSS path generating spatial mechanism with prescribed input timing. The selective precision synthesis technique is used to formulate the nonlinear constraint equations involving accuracy neighborhoods and corresponding error envelopes and these are then solved using the generalized reduced gradient method of optimization. The mathematical formulation and derivation as well as numerical examples are presented in this paper.