A System for Automated Deduction in Engineering Mechanics
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Mathematics in Computer Science
A System for Automated Deduction in Engineering Mechanics Philip Todd
Received: 20 December 2018 / Revised: 15 December 2019 / Accepted: 29 April 2020 © Springer Nature Switzerland AG 2020
Abstract A system is presented for automated formula discovery in engineering mechanics built on a Lagrangian formulation where geometric constraints are treated as physical constraints. While the architecture of the system is described, the main focus of this paper is to highlight the interplay between architecture and user interface in generating formulas for mechanical problems. With this in mind, a number of examples are presented in the kinematics, statics and dynamics of simple mechanisms. Keywords Mechanics · Automated deduction · Symbolic geometry Mathematics Subject Classification Primary 70-04; Secondary 68T15
1 Introduction The techniques of automated deduction in geometry have been applied to mechanics in a number of different settings. In [15] Wu extends his method to differential geomery and to the derivation of Newton’s and Kepler’s Laws of planetary motion. This work is extended by Chou and Gao [2] and Wang [14]. In [3], the theorem prover ISABELLE, along with nonstandard analysis is applied to the automated derivation of the theorems in Newton’s Principia. In addition to celestial mechanics, Chou and Gao apply their methods to more mundane problems in plane kinematics [1]. In [5], Groebner Bases are applied to solve nonlinear constraint problems emerging from the interplay between geometry and mechanics in the statics of trusses. The question of which beams bear zero loads in a specific problem, and always bear zero loads is addressed using the automated deduction system OTTER in [4]. In this paper, we describe the software architecture and user interface for a system which layers Lagrangian mechanics on top of a geometric constraint model to generate formulas in engineering mechanics. The system is designed to be general purpose within its domain of application and to give symbolic solutions for kinematic, static, kinetostatic and dynamic problems involving simple machines of the sort analyzed in engineering texts. An engineering mechanics model has at its root a geometry model. Our mechanics system is built on top of a constraint This material is based upon work supported, in part, by the National Science Foundation under Grant IIP-0750028. P. Todd (B) Saltire Software, 12700 SW Hall Blvd., Tigard, OR 97223, USA e-mail: [email protected]
P. Todd
Fig. 1 A triangular truss with force applied at B. Constraint forces are displayed for the three length constraints
based geometry system Geometry Expressions and intertwines the constraint description with the mechanics model. In a numerical context, this approach is embedded in the product Analytix and described in [10]. The architecture for a symbolic system is discussed in [11]. In this paper, we present a fully realized symbolic mechanics system, discuss user interface and give usage examples. We thus explore its strengths an
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