Peter, the steel rod broke near the end of the dowel rod. The dowel did not break nor did the steel rod pull out of the dowel. Your method sounds good but I don't think it will solve the problem other than maybe change the resonance frequency.

A friend suggested a possible fix. Use several layers of heat shrink tubing on the steel rod to build it up near the diameter of the dowel and then put a length of heat shrink tubing over the dowel and the rod. This combo should give a shock absorbing effect to the assembly to keep down the resonance vibration.

Ken