A free layer damping treatment is formed by applying a coating of a high damping material to one or both sides of a structure. If the coating is perfectly bonded to the structure, the coating may be taken as being subjected to the strain at the interface between structure and coating. Consequently, the dominant component of strain is tensile. Free layer treatments may be symmetric (identical coatings on both sides of the structure), or asymmetric (one side only), and applied to beams, plates or shells in bending or in tension. Two classes of treatments are of interest, those employing thin coatings of a relative stiff damping material, and those employing thick coatings of a relatively soft material. As many materials useful as hard coatings are nominally linear but have inherently nonlinear damping characteristics, provision for this must be made.