Vibration teaching practice employs a plenty of such kind of examples with the gravity and spring’s static deformation not contributing to intrinsic frequencies. However，there exist indeed counter examples with the gravity and spring’s static deformation playing a role. We investigate the if-and-only-if conditions for the spring’s static deformation and gravity acceleration vanishing from the intrinsic frequency expression. Theoretical analysis shows that the spring’s static deformation vanishes from the intrinsic frequency expression if the spring axis is tan gential to the particle trajectory at the static balance position. With this condition satisfied，the gravity acceleration vanishes from the intrinsic frequency expression if the particle trajectory transgressing the static balance position is 　　either vertical or zero curvature. The conclusion is not only benefit to comprehend the significance of the gravity to the intrinsic frequency，but also useful in compiling and creating new examples for vibration teaching practice.