The grad school does not impose coursework requirements for a PhD (see this page).
The ME field imposes one coursework requirement (pass MAE 7999 twice) for a PhD. (see ME field rules here).
Your Special Committee will presumably impose a number of course requirements chosen to equip you to effectively complete your research. Your committee members each represent Major and minor Subjects (to a first approximation, this means Fields) and concentrations within those fields (these are your major and minor areas of study); they may also, if they choose, specify course requirements to satisfy their expectations for study in those minor areas of study.