The task of suppressing unwanted vibrations has been one of the most pressing tasks of modern technology for many decades. Increasing demands on the accuracy and efficiency of various mechanisms, instruments, and measuring devices pose new challenges to engineers and researchers. External periodic influence, even of very small amplitude, may cause the disruption of the normal functioning of the system, which leads to unnecessary wear or breakdown of equipment. Among various means of counteracting such vibrations, passive vibration isolation devices occupy an essential place. They have a relatively simple design, are not very expensive, and are somewhat effective in operation. However, a comprehensive study of the dynamics of these systems is not a simple task since the corresponding model is multi-parameter, and the use of numerical methods does not always provide all the answers desired. Therefore, an important part of developing such systems is the parametric analysis of the system's dynamics under study. The present paper considers the basic nonlinear model of a vibration isolator with one degree of freedom. Analytical relationships have been obtained that make it possible to optimize the parameters that determine the operation of the isolator in the vicinity of resonance. The proposed approach is demonstrated using the two examples of mechanical systems.