The random vibrations of a truss were minimized analytically by placing viscous dashpots where they would be most effective. It was assumed that there was a limited amount of damping available and the question was, where can the damping be used most effectively? Dashpots were placed parallel to the truss members, which deform only axially. The problem was developed as a nonlinear optimization problem. The response of the structure is represented in terms of its complex normal modes. The sensitivities of the individual modal natural frequencies, loss factors, and complex eigenvectors were obtained for nonproportional damping. The excitation was white noise applied as concentrated forces at nodal points. The mean‐square displacement was minimized using a modification of the method of steepest descent. An example is given for a 10‐bar truss. The problem was also formulated for the optimization programCONMINand the results are summarized for several constraint situations. The work is related to a previous paper dealing with the minimization of settling time of free vibrations by maximizing the energy dissipation rate [V. H. Neubert, Proc. DAMPING89, Int. Symp., W. Palm Beach. FL, February 1989].