In this paper it is shown that if the hysteresis loop for a material has a particular shape the damping can be considered adequately by multiplying the modulus of elasticity of the material by the complex number e2bi where 2b is called the complex damping factor. For small values of b it is shown that both for free and forced vibrations of a simple spring-mass system the motion in the case of complex damping is the same as in the case of viscous damping, with b = c/ccr, except that in the steady-state case the phase angles are slightly different. Also, it is shown how complex damping may be applied to cases of forced vibrations of uniform rods and beams. The greatest advantage of using complex damping, however, is in numerical calculations of forced vibrations of engine crankshafts, airplane wings, and other types of structures; and for such calculations it already has been extensively used.