torsional analysis

In fact the problem is more complex and the 1/3 is not a rule of thumb.

1- The equivalent mass is in general defined as the mass (or inertia) having same kinetic energy as all the masses it is the replacement for. In the case of the couple piston+connecting rod both participates.

The kinetic energy of piston and connecting rod depend on the relative position which is function of the angle θ and the relative length of the connecting rod L=λ*R. The connecting rod has a plan movement which can be considered as the sum of a translation and a rotation with respect to the COG. As one sees this is variable over the 360° rotation of crankshaft.

2- Since the piston effect is in general small with respect to the connecting rod attention concentrated on this last one. Its translation energy is small s o that the rotation was considered. A further simplification was done the connecting rod was considered as with a constant distribution of mass. In this case the Ec = ∫dm*(x*ω)^2 from 0 to L . With dm=Mcr/L*dx the result is Ec= Mcr/L*ω^2*L^3/3

= Mc*L^2*ω^2/3= Mequivalent*L^2*ω^2 thus Meq= Mcr/3.

3- In reality the mass distribution is NOT uniform so that Meq should be > 1/3Mcr. But the critical frequency computation has not to be very precise and this simplification is welcome.