Need equation for natural frequency of tapered (conical) coil spring

I need the equation for the natural frequency of a coil spring with coil diameters which change along the length of the spring. These types of springs are called tapered or conical. The natural frequency of a coil spring with constant coil diameters is Nat Freq = 1/2 x Square Root ( spring stiffness / spring mass ) This site shows how to calculate the spring rate for a tapered spring: [login to view URL] You will notice that the spring has a larger diameter coil and smaller diameter coil. The spring does not go to a point, with a diameter of zero at 1 end. This is what I need, the natural frequency for a tapered spring with a larger and smaller diameter. I have also attached a technical paper for calculating the natural frequency of a conical spring. There are several cases described. I would need the case where the larger end of the spring fixed "IIB Based Fixed, Vertex Free". IMPORTANT: The solution would likely not be exactly what is described in this paper. I need the case where the "vertex" has a coil diameter which is not zero. I would expect the solution to look very much like: Nat Freq = 1/2 x Square Root ( spring stiffness / spring mass ) but with some modifier based on Diameter 1 and Diameter 2 of the tapered spring. I will need a reference to your solution if it is not based on the attached document. I will also accept a reasonable approximation of the answer if you can provide a source reference.