A helical tension spring has a natural frequency f given bywhere k is the spring constant...

A helical tension spring has a natural frequency f given by

where k is the spring constant given by k = d4G/8D3N and the weight W of the spring is W = π2d2DNp/4, N is the number of turns, D is the diameter of the spring, d is the wire diameter, G is the shear modulus, and p is the density. To avoid resonance, the natural frequency of the spring should be 20 times higher than the frequency w of the applied force. If the only random variable is the diameter d of the spring wire, obtain an expression for the tolerance on this diameter so that the probability of resonance occurring is < 0.0001.