Why should an object even have a natural frequency, why not vibrate at any other frequency? This just sounds nonsensical.
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In everyday terms, commonplace objects are capable of resonance at some natural frequency if they possess 1) mass, 2) elasticity, and 3) little or no internal friction or damping. The mass allows them to temporarily store up energy by being in motion. The elasticity allows them to temporarily store up energy by being deformed. The absence of damping allows these two energy storage mechanisms to trade their energy back and forth repeatedly without losses, thereby establishing oscillation.
The amount of mass present compared to the amount of elasticity present sets the natural frequency that results. The amount of damping determines how long the oscillations persist after being set in motion.