The free vibration of strings with randomly varying mass and stiffness
is considered. The joint probability density functions of the eigenva
lues and eigenfunctions are characterized in terms of the solution of
a pair of stochastic non-linear initial value problems. Analytical sol
utions of these equations based on the method of stochastic averaging
are obtained. The effects of the mean and autocorrelation of the mass
process are included in the analysis. Numerical results for the margin
al probability density functions of eigenvalues and eigenfunctions are
obtained and are found to compare well with Monte Carlo simulation re
sults. The random eigenvalues, when normalized with respect to their c
orresponding deterministic values, are observed to tend to become firs
t order stochastically stationary with respect to the mode count.