# Automating the Analysis of Uncertainties in Multi-Body Dynamic Systems Using Polynomial Chaos Theory

## Date of Award

Fall 2018

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Mechanical Engineering

## First Advisor

Voglewede, Philip A.

## Second Advisor

Corliss, George

## Third Advisor

Baxter, Sarah C.

## Abstract

Variation occurs in many multi-body dynamic (MBD) systems in the geometry, mass, or forces. This variation creates uncertainty in the responses of an MBD system. Understanding how MBD systems respond to the variation is imperative for the design of a robust system. However, the simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This presentation shows the automation of an uncertainty analysis of an MBD system with variation. The first step to automating the solution is to create a robust algorithm based on the Constrained Lagrangian formulation for deriving the equations of motion. Using the Constrained Lagrangian algorithm as a starting point, the new process presented uses polynomial chaos theory (PCT) to embed the stochastic parameters into the equations of motion. To accomplish this, the concept of Variational Work is derived and implemented in the solution. Variational Work applies PCT to the energy terms and Principle of Virtual Work of the Constrained Lagrangian rather than applying PCT on the equations of motion. Using an automated process for applying PCT to an MBD system, some example problems are solved. Each of these problems is compared to a Monte Carlo analysis using the deterministic automation process. Some of the examples are non-textbook based problems, which show limitations in the application of PCT to an MBD system. The limitations and the possible solutions to overcoming them are discussed.