Computer aided analysis and optimal design of mechanical systems using vector-network techniques

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Abstract

Stringent tolerances on mechanical components has created increasingly severe demands on the quality of new mechanical designs. The mathematical models used to simulate the various types of mechanical systems these days need to incorporate an optimization algorithm capable of predicting and improving the efficiency of an initial design. Hence, this article presents an extension of the vector-network model for the computer aided simulation and design optimization of dynamic systems. The method is based on a simplistic topological approach which is incorporated into an efficient numerical optimization process used to solve non-linear problems. The procedure casts, simultaneously, the three-dimensional equations of motion and the objective function into a symmetrical format, always yielding an optimum solution. The algorithm serves as a basis for a “self-formulating” computer program which simulates the best system response, given only the system description as input. The effectiveness of this approach is demonstrated in the analysis of a 3-D eight DOF vehicle model. In this example, allowable values for design parameters and suspension types are incorporated in order to minimize the performance variables which are beneficial for comfort without unduly compromising road holding capabilities.

Introduction

Engineers, who are living in a world of rapid change and extensive interaction, must continually improve their own decision-making skills. A good and efficient technique is needed to support the engineer’s experience. The technique must be formal so that it can be learned quickly and applied automatically to new situations. The technique must be efficient so that its cost does not increase in proportion to the complexity of the mechanical system. The central theme behind this work is that computer aided design and simulation is a technique that will fulfill these needs. Hence, a general purpose multibody program has been created which can be adapted to the complexities and change of modern mechanical systems and can, also, be developped and communicated efficiently.

The refinement of formulation techniques for dynamic analysis of mechanisms has led to the development of general purpose computer programs. Several have been described in the literature [1], [2], [3], [4], [5], [6], [7], [8], [9]. All are capable of automatically generating and numerically integrating the differential equations governing the motion of dynamic systems but they do not incorporate automatically an optimization algorithm. In recent years, substantial work has been done in the field of dynamic analysis and optimization of mechanisms and machines. Most mechanical systems have many degrees of freedom, design constraints and a large number of design variables. Several researchers are working on the development of algorithms for three-dimensional systems and several programs were discussed in the 1975 survey paper by Paul [1], and in recent years more researchers have been attracted to this problem and there are probably 10 fairly well-known methods (and computer programs based on the methods), including DRAM and ADAMS [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], IMP [24], [25], [26], [27], DYMAC [1], [28], [29], [30], [31], DADS [32], [33], [34], [35], [36] and of course programs based on the vector-network method [8], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53]. Another survey paper [54] listed 17 programs which are commercially available for kinematics and dynamics of linkages, 10 of which are for kinematic synthesis and analysis; the remaining seven are of the self-formulating dynamic type.

In this work, the methodology is based on the vector-network model which is different from the other methods described in the literature and presents some advantages: (i) the model uses a simple modeling framework to discretize a complex topology of interconnected rigid bodies with holonomic and non-holonomic constraints, (ii) the procedure is very attractive and may prove to be helpful due to its simplicity and fundamental application of Newton’s law, (iii) the vector-network formalism is an ideal algorithm for a CAD system because of the pictorial relationship existing between the physical and mathematical system, (iv) in comparison with work done in the field of optimization of dynamic systems, this approach evaluates the design parameters effect on the system behavior without any kind of derivatives.

One of the most appealing features of vector-network lies in the geometric and pictorial aspect of the method. Given a lumped mechanical system, one can construct the vector diagram by a simple mapping of the system. Each element is represented by a line segment and each joint or connection by an appropriate point such that a user can associate the vector-network diagram to the mechanical system in a direct fashion. The technique was very methodical and well suited for computer implementation. A self-formulating computer program based on this formulation, VECNET (for Vector-Network) [8], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], was implemented and tested. Although restricted to particle masses, VECNET was intensively used as a research tool at the University of Waterloo. It spawned another program called RESTRI [48], [49], [50], [51], [52], [53] for the French version of 3-Dimensional Network (RESeau TRIdimensionnel) which essentially was constructed from VECNET’s fundamental concepts enhanced with three-dimensional rigid body capabilities including kinematic constraints and optimization algorithms. Previous formulation required translational and rotational network in order to ensure consistency in the initial theory. A simplification was perceived in that both diagrams could be assembled by a unique description of the system. Translational and rotational variables were assigned to all system elements allocating zeros to the superfluous variables.

It has become obvious that interrelated advancements in the field of dynamics and computer science have seized the interest of many researchers. This work naturally does not comprise an exhaustive list of all the general multibody programs. Surely, many more computer programs exist and the number is growing, for instance, NEWEUL [55], [56], [57], [58], [59], [60], [61], [62] which combines the principles of d’Alembert and Jourdain into a symbolic approach, COMPAMM [63], [64], [65] which uses natural coordinates and programs based on Kane’s algorithm like UCIN [66], [67], [68], SD-EXACT [69] and AUTOLEV [70]. Also, we have MEDUSA [71] assembled from a chemical background perspective, MEDYNA [72], [73] used for vehicles and NBOD 2 [74] conceived at the NASA Space Flight Center, just to name a few. More multibody programs are discussed and analyzed in the survey papers [4], [5], [6].

In the design analysis area and optimization of mechanical systems, Haug and Arora [75] and Barman [76] exploited the adjoint variables technique to perform a design sensitivity analysis. The implementation of this approach is simple but it involves a large number of operations and contains interpolation errors. Krishnaswami and Bhatti [77] proposed the direct differentiation technique which can be integrated simultaneously with the equations of motions by ensuring a positive error control. However this method can create some numerical instability in the optimization process. Ashrafiuon and Mani [78] used this method combined with a symbolic manipulation program MACSYMA [79] to generate the necessary equations for the dynamic and design sensitivity analysis.

There are two approaches used to find the derivatives of the functions with respect to the design variables. The first one is the finite difference method which is fairly simple to implement but there is a risk of numerical instability in the optimization process. The other method for computing the derivatives is the analytical method or the exact method which takes less time in computing than the first method but the analytical one is not helpful for a complex design problem where several derivatives must be computed analytically. In this algorithm, the optimum design problem starts with an initial estimate of the design parameters. Furthermore the achieved solution has to satisfy certain constraints to ensure physical and constructional feasibility. The vector-network approach is extended to the simulation and design optimization of spatial multibody systems subjected to a wide variety of kinematic joints. The constrained optimization problem is converted in an unconstrained form using a penalty method and the resulting problem is solved by the Hooke and Jeeves [80] method.

Section snippets

Vector-network model

Many researchers have studied the theory of graphs [81], [82], [83], [84], [85], [86], [87] and bond-graphs [88], [89], [90] mainly due to the fact that among all the fields of human interest, there are few where graph theory cannot be applied to the process of analyzing or synthesizing problems. In order to extract the kinetic properties resting within lumped-parameter mechanical systems, it is convenient to discretize the system into a schematic diagram composed of nodes or vertices

Dynamical formalism

Consider a mechanical system with n rigid bodies interconnected with springs, dampers and mechanical joints. In this model, we consider ideal joints with no friction. The reference frames linked to different rigid bodies are Cartesian with their origin located at the center of mass, and their axes preferably oriented in the principle axes of inertia of the rigid bodies. The inertial reference frame will be written OXYZ and the reference frame linked to the rigid body i will be written Gixiyizi.

The design optimization problem

The equations of motion for mechanical systems and the constraint equations are both dependent on design variables where each variation in design changes the system behavior. The goal of this procedure consists in optimizing specified objective functions subjected to performance constraints and physical restrictions in design. These criteria are dependent on positions, velocities, accelerations, constraints, design variables and time. The general mathematical model for the vector optimization

Numerical example

A 3-D vehicle model with eight degrees of freedom is analyzed. It is illustrated in Fig. 1 and the vehicle behavior is expressed by vertical, pitching and rolling motions in terms of acceleration, velocity and movement for the various vehicle solid components. In this analysis, we do not consider yaw motion because it has been shown that its effect is negligible on vehicle comfort and stability.

The vector-network algorithm is exploited to obtain the differential equations of motion of the

Conclusion

In this paper, the vector-network algorithm has been extended to perform the design optimization of multibody systems. From this study, the optimization procedure was found to be easy to implement, effective and efficient for many different classes of mechanical design problems. With this method, the attainability of a feasible solution does not depend on the continuity and differentiability of the objective functions and the performance constraints as does many other methods.

The most exciting

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