We consider a linear undamped vibrating system with one degree of freedom, excited by a sinusoidal force of constant amplitude. To this is attached a secondary system by means of a spring whose load-deflection characteristic is the sum of a linear and cubic term. It is desired to find optimum coefficients k and v for this coupling spring, such that one obtains as large a band of exciting frequencies as possible within which the vibration amplitude of the primary system is kept below unity. This choice is subject to the condition that the fundamental of the primary system response have zero amplitude at a preassigned expected excitation frequency f~0. The first approximation by the Duffing Iteration Method is used to obtain the response in terms of the system parameters. Optimum values of k * and v are expressed in terms of ft0 2. Using the optimum values corresponding to a particular value of fl0, the results of the synthesis are compared with those obtained by a more exact analysis and by an electronic differential analyzer. It is found that by the synthesis criterion used, the nonlinear absorber offers a significant advantage over the corresponding linear absorber. NOMENCLATURE response amplitudes for the linear case approximate response amplitudes by Duffing's method viscous damping coefficient for damping between main mass and inertial frame of reference damping ratio exciting force on main mass maximum value of f(t) spring constant for the linear spring between main mass and inertial frame of reference masses of main mass and absorber suppression band width displacements from equilibrium of main mass and absorber dimensionless main mass displacement dimensionless relative displacement of absorber coefficient of the linear part of ~/~ mass ratio coefficient of nonlinear part of sv edge of forbidden region in a ~* phase plane * Portions of a dissertation submitted to the Department of Applied Mechanics, Washington University, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.