Physical and mathematical foundations of probabilistic engineering design with application to rotors

Conventional statistical models for engineering systems rely on data intensive methods. In the design phase, little data may be available. This thesis derives probability models for engineering systems using laws of physics and engineering knowledge. In doing so, design insight is gained prior to observing data.; Chapter 2 gives several illustrative examples involving rotors. The models show how engineering parameters in the manufacturing process affect the probability distribution of rotor imbalance. From this, design decisions can be made to effectively improve the manufacturing process.; In rotor design, the mathematical objects involved are more complicated than scalar quantities. This is true in other engineering systems as well. For instance, the position of a drill hole is a vector. The inertial properties of a rotor are given by a 3 x 3 tensor. Existing techniques in probabilistic design are not equipped to handle these mathematical objects.; Chapter 3 gives a generalized mathematical framework for deriving engineering probability models involving these objects. A rich physical structure is shown to underlie the relation between physical tensors and small manufacturing errors. This structure gives derived probability models a substantial amount of predictive insight in the absence of data. Conventional probability models, on the other hand, such as the multinormal, are shown to be physically inconsistent. Chapter 3 shows how the models can be used to effectively predict and improve rotor imbalance.; Physically based probability models are most naturally represented using differential geometry. Chapter 4 shows how the geometry of differential forms can be used to draw probability densities. The new method of drawing probability densities has numerous advantages over existing methods. In particular, the pictures can be used to eyeball design decisions in engineering problems.; Chapter 5 applies the theory and methods in Chapters 2-4 to the design of a microgyroscope. The chapter shows quantitatively and pictorially how effective improvements can be made to the fabrication process of the gyroscope.