In this paper, the author considers an approximation to the Rayleigh equation,y¨+&egr;(13y˙3−y˙)+y=A cos&ohgr;t, and its equivalent form under the transformationy˙=v,y=&egr;&xgr;. A cubic function13v3−v+&xgr;that appears in the altered equation is replaced by a piecewise linear function. For large &egr;, the existence of periodic solutions of the approximating equation is demonstrated. Strong orbital stability properties of neighboring solutions is also shown.