Mechanical Engineering Community

Equation for the spring constant of an orthotropic, rectangular wire spring

Recommended Posts

I hope I can get some help with this problem as I've been struggling with it for a long time. I have a helical tension spring with radius R and a square wire cross section and made of orthotropic material. I'm trying to derive an equation for the spring constant k by equating the energy stored by the spring 1/2 k x^2, where x is the distance stretched, with the energy stored U_t by a shaft under torsion of equivalent length so
1/2 k x^2=U_t

As explained in the title, I have a particular situation because the material is orthotropic and the cross-section of the shaft is a rectangle as shown in the attached image. I already have an expression for the torsional rigidity GJ of an orthotropic rectangular shaft with shear moduli G_{zx} and G_{zy}:
GJ=a b^3  G_{zx} [&beta;]
where [&beta;] is a series where you sum from n=1,3,5... to infinity.

When I equate
1/2 k x^2=U_t=(GJ/2L) [&theta;]^2
I can plug in [&theta;]=x/R (the angle the shaft has been twisted by is roughly equal to the angle each coil of the spring gets stretched by) so
k=GJ/(L R^2)
where L=2[&pi;] R N_a and N_a are the number of active coils in the spring. To me the equation looks complete but the problem is that when I enter values like a=0.005m, b=0.0025m, G_{zx}=1159.1MPa, G_{zy}=974.3MPa, N_a=5, and R=0.01m I get k=632.642 instead of k=2780, which is the right answer.

Since I have that the torque on the shaft is T=a b^3 G_{zx} [&theta;] [&beta;] I tried other equations for U_t where U_t=(1/2) T [&theta;] or U_t=(T^2 L)/2GJ but none of the answers come even close, 198.75 and 62.4393. Am I doing something wrong when equating the two energies?

Join the conversation

You can post now and register later. If you have an account, sign in now to post with your account.

×   Pasted as rich text.   Paste as plain text instead

Only 75 emoji are allowed.