The frequency response of a cylinder in a viscously damped fluid is a problem of fundamental engineering interest, with applications ranging from microsystems to offshore structures. The analytic solution for the drag in a vibrating cylinder in the laminar flow regime is combined with the equations of motion for forced vibration of a cylinder attached to a spring. The resulting model gives an analytic solution for the dynamic response of the system, including the gain, frequency lag, resonant frequency, quality factor, and stability of the system. The results show that the response of the system is nonlinear, with the phase of the system differing from the phase predicted by linear models. The gain, quality factor, resonant frequency, and crossover frequency all increase with the nondimensional natural frequency and decrease with the ratio of the fluid density to the resonator density.