A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed, which assumes that the nonlinear constraints can be expressed with piecewise-linear force-deflection profiles. Using this assumption, the mode shapes and natural frequencies are found for each state, and a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Several design parameters of the guide are considered: flange clearance, stiffness, symmetry, and the guide’s position. Critical bifurcation thresholds exist, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position.