Abstract: Nearly all of examples in vibration textbooks are characterized by the intrinsic frequency not implicated by the spring's static deformation. However，there exist indeed counter examples. We consider the case of the single-degree-of-freedom system and investigate its if-and-only-if conditions for the spring's static deformation vanishing from the intrinsic frequency expression. The generalized coordinates rendering a constant generalized mass are used，consequently，only the potential energy is analyzed. After systematically theoretical analysis，we find that the vanishing conditions pertain to parameters of the static balance position ( SBP) . The if-and-only-if conditions are that，firstly，the spring axis must be tangential to its moving end trajectory at the SBP，and secondly，if the displacement arc of the spring moving end is chosen as the generalized coordinate，the derivative of the generalized mass with respect to the arc coordinate must be zero at the SBP.

Key words: vibration, spring, static deflection, potential energy, intrinsic frequency, generalized coordinates