The single-degree-of-freedom (SDOF) system is the most widely used model for vibration isolation systems. The SDOF system is a simple but worthy model because it quantifies many results of an isolation system. For instance, a SDOF model predicts that the high frequency transmissibility increases when the isolator has passive damping although this does not occur for an isolator implementing active damping. A severe limitation of this system is that it cannot be used when the base and/or equipment are flexible. System flexibility has been considered in previous literature but the flexibility has always been approximated which leads to truncation errors. The analysis used in this work is more sophisticated in that it can model the system flexibility without the use of any approximations. Therefore, the true effects of system flexibility can be analyzed analytically. Current literature has not fully explored the choice of mount frequency or actuator placement for flexible systems either. It is commonly suggested that isolators should be designed with a low-frequency mount. That is, the isolator frequency should be much lower than any of the system frequencies. It is shown that these isolators tend to perform best in an overall sense; however, mount frequencies designed between system modes tend to have a coupling effect. That is, the lower frequencies have such a strong interaction between each other that when isolator damping is present, multiple system modes are attenuated. Also, when the base and equipment are flexible, isolator placement becomes a critical issue. For low-frequency mount designs, the first natural frequency can shift as much as 15.6% for various isolator placements. For a mid-frequency mount design, the shift of the first three modes can be as high as 34.9%, 26.6% and 11.3%, respectively, for varying isolator placements. NOTE: (03/2011) An updated copy of this ETD was added after there were patron reports of problems with the file. / Ph. D.