This is a lighter literature review than some applications call for. I am excusing myself from anything more deptailed until I understand the math behind most Mechanical Engineering. Currently I lack a background in Physics, Statics, Dynamics, Calculus (differential and partial differential equations), Linear Algebra, and applications there-of.

I am taking steps to rectify these gaps in my knowledge. I have enrolled in a Calculus course at my local community college. I am enrolled in an online course for Linear Algebra and its applications (specifically in programming). I am completing a course in programming (CS50 through edX.org) after which I will be approaching Physics.

And there is probably more that I haven't mentioned. Keeping all this straight is a task unto itself and it can be rather time-consuming.

Complaining aside; here is what you've skipped everything above for:

Hung, Jui Pin, Yuan Lung Lai, Tzuo Liang Luo, and Hsin Chuan Su. 2013. “Analysis of the Machining Stability of a Milling Machine Considering the Effect of Machine Frame Structure and Spindle Bearings: Experimental and Finite Element Approaches.” International Journal of Advanced Manufacturing Technology 68 (9-12): 2393–2405. doi:10.1007/s00170-013-4848-6.

Overview

Analysis of the Machining Stability of a Milling Machine Considering the Effect of Machine Frame Structure and Spindle Bearings: Experimental and Finite Element Approaches (hereafter "the study" or "the paper") specifically considers the combined dynamics of a vertical milling machine frame and spindle. An application of finite element analysis was used to predict the dynamics of the milling machine and spindle and an experiment was used to confirm the analysis.

How is the study useful to this project?

In the introduction the author's clearly state how this application of experimental and mathematical analysis could be directly useful to an end-user, saying that, "The prediction [...] of the abnormal vibration is of great importance for the selection of the free chatter machining conditions." This study aims to specifically address the dynamic conditions that result from using a high-speed spindle in combination with the mill's frame. A complete solid model is used for the analysis, rather than a simplified model (as other studies have done).

Items that are specifically useful are the mathematics and derived equations, the mathematic modeling methodology used for the spindle unit and mill unit.

The math presented will allow me to not have to experiment to find the modal frequency by experimentation and will thus allow a program to handle the heavy lifting (iterative design via algorithm) once the design phase has begun.

1. Huo, Dehong, Kai Cheng, and Frank Wardle. 2010a. “Design of a Five-Axis Ultra-Precision Micro-Milling Machine-UltraMill. Part 1: Holistic Design Approach, Design Considerations and Specifications.” International Journal of Advanced Manufacturing Technology 47 (9-12): 867–77. doi:10.1007/s00170-009-2128-2.
2. ———. 2010b. “Design of a Five-Axis Ultra-Precision Micro-Milling Machine-UltraMill. Part 2: Integrated Dynamic Modelling, Design Optimisation and Analysis.” International Journal of Advanced Manufacturing Technology 47 (9-12): 879–90. doi:10.1007/s00170-009-2129-1.

Overview

These studies walk through the design and implementation process of an ultra-precise ("nano-meter finish") 5-axis CNC milling machine. The first article focuses on the existing ultra-precision machines, component options, and the specifications for the proposed "UltraMill".

The second article focuses on the modeling and simulation of the "UltraMill" with an emphasis on machine dynamics. The simulation allows the authors to predict the performance of the machine over its entire performance envelope.

How is the study useful to this project?

The second portion of the paper is far more useful than the first. While the first sheds some light on the goals of the entire project, the second shows the process by which specific design requirements were met analytically.

I will be posting a more in depth review of these last two papers (as well as an outline with my notes) for the purpose of adapting the design process to this project's needs.