Spring Rate of Axially Loaded Coil Springs
The mechanics of compression or extension spring can be interpreted from the formulation of forces and stress of just one coil.
For a ring coil as shown in the program, two opposite forces of equal magnitude are applied at the each ends. After the force is applied these two ends move a relative distance e. The load is acting parallel to the coil axial. This load is resisted by the spring through a twisting torsion developed in the coil wire. The coil wire total twisting torque is:
= T * L / (G J) = (F * R) * L / (GJ)
Theta of one loop twisting = Torque * (One loop length)/(G shear modulus * J torsion inertia)
Where T = Torque = F * R = Force * Radius = F * D/2
F = Coil Spring Axial Force
L = Length of One Loop = * D
D = Coil Diameter
J = * d ^ 4 /32
d = wire diameter
= (F * D/2)* ( * D )/ (G * * d ^ 4 /32)
= F * D^2 * / ( G * * d^4 / 16)
= 16 * F * D^2/(G * d^4)
One loops elongation along axial direction = e = R * = D * /2 = 8 * F * D^3/(G * d^4)
Spring Rate for one loop = k = F / e = (G * d^4)/(8 * D^3)
For n loops, total elongation = E = e * n
So the spring rate for the whole spring is F / E = (G * d^4)/(8 * n * D^3)
The spring index is the ratio of (D / d) where D is the mean coil diameter and d is the wire diameter. In general, the spring is designed with the spring index between 5 to 10 for manufacturing feasibility.
Spring End Condition
Compression spring can have many end conditions to achieve stability.
Closed ends: spring wire end contacts the next loop.
Closed and ground ends
Plain ends ground
Number of Coils
Compression spring end coil if grounded is considered ineffective in resist load.
End type No. of inactive coils
Closed and ground 2
Closed and ungrounded 2
Open and ground 1
Taper rolled (forged and hot coiled) 1.5
The load and spring height definition is illustrated in the figure at right.
H0: Theoretical free height
H1: Height at load stage 1
H2: Height at load stage 2
HS: Solid height when coils fully contact each other
Minimum working height: H1 - 0.85(H0 - HS)
Since the spring coils will not be perfectively shaped, the theoretically calculated spring height is less than the actual height.
Stresses and Allowable Stresses
The torsion stress in the compression spring is:
= (8 F D K )/( d ^3)
Where K = ( c + 0.2 )/ ( c -1)
C= D/d = spring index
Compression spring allowable stresses are dependent of the material type and manufacturing process. Ultimately the stress evaluation is part of the checking spring service life cycle using the Modified Goodman Diagram.
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