The influence of adding a geometrically nonlinear viscous damper to a system of coupled oscillators with essential nonlinear stiffness will be discussed. All nonlinear terms are restricted to the coupling terms between a linear oscillator and light attachment. We show that the addition of the nonlinear damper introduces dynamics not observed with linear damping. In fact, we find the surprising result that the nonlinear damper introduces new dynamics into the problem, and its effect on the dynamics is far from being purely parasitic - as one would expect in the case of weak linear viscous dissipation. Similar to essential nonlinear stiffness, geometrically nonlinear damping of the type considered in our work is physically realizable by means of linear viscous damping elements. Numerical work examining this problem will be discussed.