Composite Shafts - Series Connection:

If two or more shafts of different types of materials, lengths & diameters are connected in such a way that each carries the same torque, after that a composite shaft linked in series is produced. Figure shows such a shaft in which shafts AB, BC and CD are connected in series and their characteristics such as diameter, length and modulus of rigidity are explained in figure. The composite may also be regarded as stepped shaft.

Figure

As each of AB, BC and CD is subjected to same torque T, then from Eq.

T = J₁  G₁ θ₁/ l₁ = J ₂ G₂ θ₂/      l₂ = J₃ G₃ θ₃/ l₃

where, θ₁ = θ_BA , θ₂  = θ_CB , θ₃ = θ_DC

where θ_BA means angle of twist of section A with respect to B.

Therefore,

 θ_DA  = θ₁ + θ₂  + θ₃

=  T l₁ / J₁ G₁ +   T l₂ / J ₂ G₂  +   T l₃ / J₃ G₃

Or θ_DA /T = l₁ /J₁G₁ + l₂ /J₂G₂+ l₃/J₃G₃

We have described the ratio T as torsional stiffness which is understood as torque per unit twist. Also, Torsional stiffness is referred to as spring constant of the shaft and denoted by k. It is obvious that shaft and for others T / θ_DA is the spring constant of the composite shaft and for others

K₁  = T/ θ₁= J₁ G₁/ l₁, k₁  =  T/ θ₂ = J ₂ G₂/ l₂ , k =  T/ θ₃  = J₃ G₃/ l₃

Therefore, Eq. (i) can be written as

1 / k  =  1 / k₁ +  1/ k₂  +  1/k₃

Here, k is referring to spring constant or torsional stiffness of the composite shaft.