Natural frequencies and stability of an axially-traveling string in contact with a stationary load system

The natural frequencies and stability of a string traveling between two fixed supports and in contact with a stationary load system, which contains such parameters as dry friction, inertia, damping, and stiffness, are investigated both numerically and analytically. After establishing the orthogonality properties between the eigenfunctions of a freely traveling string the eigenvalues of the coupled system are calculated by a numerical procedure based on eigenfunction expansion method. It is found that the stiffness in the load system tends to increase the natural frequencies of the traveling string, while the inertia tends to decrease the natural frequencies. When the load system contains both inertia and stiffness elements with natural frequency omega(z), inertia effect is dominant for the modes with natural frequencies higher than omega(z), and stiffness effect is dominant for the modes with natural frequencies lower than omega(z). Large dry friction causes flutter instability in the high speed range, which cannot be suppressed by the damping element in the load system. The expressions for the derivatives of eigenvalues with respect to various load parameters are derived to verify the numerical results.