Dimensional Tolerance Allocation of Stochastic Dynamic Mechanical Systems Through Performance and Sensitivity Analysis

Uncertainties due to random dimensional tolerances within stochastic dynamic mechanical systems lead to mechanical errors and thus, performance degradation. Since design standards do not exist for these systems, analysis and design tools are needed to properly allocate tolerances. This paper presents probabilistic models and methods to allocate tolerances on the link lengths and radial clearances such that the system meets a probabilistic and time dependent performance criterion. The method includes a general procedure for sensitivity analysis, using the effective link length model and nominal equations of motion. Since the sensitivity analysis requires only the nominal equations of motion and statistical information as input, it is straight forward to implement. An optimal design problem is formulated to allocate random tolerances. Examples are presented to illustrate the approach and its generality. This paper provides a solution to the tolerance allocation problem for stochastic dynamically driven mechanical systems.

A Multi-Objective Optimization Strategy With Priority Ranking of the Design Objectives

Generally, two types of priorities are considered among multiple objectives in the design of machine structures. One of these objectives is named the “hard objective”, which is the absolutely indispensable design requirement. The other is called the “soft objective”, which has lower priority order. This paper proposes a multi-objective structural optimization strategy with priority ranking of those design objectives. Further, this strategy is demonstrated on the actual example of a motorcycle frame structural design which has three design objectives, (1) an increase in static torsional rigidity, (2) a reduction of dynamic response level, and (3) a decrease in the weight of the motorcycle frame.

Dynamic Analysis of Mechanical Systems With Time-Varying Topologies: Part 1 — Dynamic Model and Stability Analysis

A model is developed for analyzing mechanical systems with a pair of bodies with topological changes in their kinematic constraints. It is built upon the concept of Poincaré map rather than following the traditional methods of differential equations. The model provides a set of well-defined and naturally-discrete equations of motion and is capable of giving physical insights of dynamic characteristics of deadbeat convergence of multiple collisions and periodic or chaotic responses. The development of dynamic model and a local stability analysis are presented in Part 1, and the global analysis and numerical simulation are discussed in Part 2.

Design Optimization of Machine Structures Based on Multiphase Structural Modeling of Ideal, Intermediate, and Detailed Models

This paper proposes a design optimization method consisting of the multiphase structural modeling of ideal, intermediate, and detailed models for machine structures. In this method, the ideal characteristics are first obtained for a specific ideal model. Then, the detailed designs are determined so that the characteristics in the detailed model are as close to the ideal characteristics as possible. For easily and surely obtaining the final detailed designs, an intermediate model is introduced between the ideal model and the detailed model. This method not only effectively generates optimum detailed designs of machine structures but also brings about an easy realization of the optimum characteristics in practical manufactured machine products. The proposed method is applied to a machine-tool structural model for demonstrating the effectiveness of the method.

Optimal Packaging of Complex Parametric Solids According to Mass Property Criteria

A solution to the 3-D optimal packaging problem is presented. The positions and orientations of arbitrarily shaped objects in arbitrarily shaped 3-D containers are found. Mass property and spatial criteria are addressed. The robustness and speed of the optimal packaging solution is increased by supplying explicit gradients for the optimization search routines.

Shape Synthesis and Optimization Using Intrinsic Geometry

The present paper outlines a method of shape synthesis using intrinsic geometry to be used for two-dimensional shape optimization problems. It is observed that the shape of a curve can be defined in terms of intrinsic parameters such as the curvature as a function of the arc length. The method of shape synthesis, proposed here, consists of selecting a shape model, defining a set of shape design variables and then evaluating Cartesian coordinates of a curve. A shape model is conceived as a set of continuous piecewise linear segments of the curvature; each segment defined as a function of the arc length. The shape design variables are the values of curvature and/or arc lengths at some of the end-points of the linear segments. The proposed method of shape synthesis and optimization is general in nature. It has been shown how the proposed method can be used to find the optimal shape of a planar Variable Geometry Truss (VGT) manipulator for a pre-specified position and orientation of the end-effector. In conclusion, it can be said that the proposed approach requires fewer design variables as compared to the methods where shape is represented using spline-like functions.

Experimental Validation of a Dynamic Simulation

Using forward and inverse dynamic analysis, the dynamic simulation of a backhoe has been compared with experiments. In the experiment, recorded were the configuration and force histories; that is, velocity and position, and force output from the hydraulic cylinder-all were measured in the time domain. When the experimental force history is used as driving force in the simulation, forward dynamic analysis produces a corresponding motion history. And when the experimental motion history is used as if a prescribed trajectory, inverse dynamic analysis generates a corresponding force history. Therefore, these two sets of motion and force histories — one set from experiment, and the other from the simulation that is driven forward and backward with the experimental data — are compared in the time domain. The comparisons are discussed in regard to the effects of variations in initial conditions, friction, and viscous damping.

Efficient Dynamic Computer Simulation of Simple-Closed Spatial Linkages

A new method of formulating the generalized equations of motion for simple-closed (single loop) spatial linkages is presented in this paper. This method is based on the generalized principle of D’Alembert and the use of the transformation Jacobian matrices. The number of the differential equations of motion is minimized by performing the method of generalized coordinate partitioning in the joint space. Based on this formulation, a computational algorithm for computer simulation the dynamic motions of the linkage is developed, this algorithm is not only numerically stable but also fully exploits the efficient recursive computational schemes developed earlier for open kinematic chains. Two numerical examples are presented to demonstrate the stability and efficiency of the algorithm.

Remarks on Conditions for the Validity of Parametric Decomposition in Optimal Design

Decomposition strategies are used in a variety of practical design optimization applications. Such decompositions are valid, if the solution of the decomposed problem is in fact also the solution to the original one. Conditions for such validity are not always obvious. In the present article, we develop conditions for two-level parametric decomposition under which: (1) isolated minima at the two levels imply an isolated minimum for the original problem; (2) necessary conditions at the two-levels are equivalent to the necessary conditions for the original problem; and, (3) a descent algorithm for computing Karush-Kuhn-Tucker points in decomposition formulations is globally convergent. Since no special problem structure is assumed, the results are general and could be used to evaluate the suitability of a variety of approaches and algorithms for decomposition strategies.

Analysis and Synthesis of Mechanical Error in Universal Joints

The modeling of manufacturing errors in mechanisms is a significant task to validate practical designs. The use of probability distributions for errors can simulate manufacturing variations and real world operations. This paper presents the mechanical error analysis of universal joint drivelines. Each error is simulated using a probability distribution, i.e., a design of the mechanism is created by assigning random values to the errors. Each design is then evaluated by comparing the output error with a limiting value and the reliability of the universal joint is estimated. For this, the design is considered a failure whenever the output error exceeds the specified limit. In addition, the problem of synthesis, which involves the allocation of tolerances (errors) for minimum manufacturing cost without violating a specified accuracy requirement of the output, is also considered. Three probability distributions — normal, Weibull and beta distributions — were used to simulate the random values of the errors. The similarity of the results given by the three distributions suggests that the use of normal distribution would be acceptable for modeling the tolerances in most cases.