A beta-type free-piston Stirling cycle engine or cooler is drivingly coupled to a linear alternator or linear motor and has an improved balancing system to minimize vibration without the need for a separate vibration balancing unit. The stator of the linear motor or alternator is mounted to the interior of the casing through an interposed spring to provide an oscillating system permitting the stator to reciprocate and flex the spring during operation of the Stirling machine and coupled transducer. The natural frequency of oscillation, ωs, of the stator is maintained essentially equal to ω p ⁢ 1 - α p k p and the natural frequency of oscillation of the piston, ωp, is maintained essentially equal to the operating frequency, ωo of the coupled Stirling machine and alternator or motor. For applications in which variations of the average temperature and/or the average pressure of the working gas cause more than insubstantial variations of the piston resonant frequency ωp, various alternative means for compensating for those changes in order to maintain vibration balancing are also disclosed.