Critical rotational speed of the two-support ponderable shaft without disc
A two-support rotor with continuously and uniformly distributed weight has infinite number of critical rotational speeds, whose values are related as quadrates of numbers of the natural series:
The critical rotational speed of the weighty non-fastened shaft without disc with continuously and uniformly distributed weight is determined by the formula:
(8.3)
where J = (D4 - d4) /64 is an equatorial inertia moment of the shaft section, m4; L is a shaft length without a console part, m; msh = (D2 - d2) /4 is linear shaft weight, kg/m; is a shaft material density, kg/m3.
8.6. Critical rotational speeds of the ponderable shaft with several discs
Academician А. N. Krylov offered solution to a problem of determination of critical rotational speed of a multy-disc rotor with depending on shaft weight.
In practical calculations of a rigid rotor it is usually necessary to know only the critical rotational speed of the first order. It’s advisable to determine the critical speed of the second order when calculating flexible shaft in order to fined out how close the operational speed to the critical speed of the second order is.
To determine critical rotational speed of the first order the approximated methods are successfully used.
8.6.1. Method of decomposition into elementary systems
According to this method critical rotational speed of a composite system (shaft with several discs) is expressed through critical rotational speeds of elementary systems:
(8.4)
(8.5)
where ncr is a required critical rotational speed of a composite system; ncr sh is a critical rotational speed of the powerful shaft without discs, determined by the formula (8.3); ncr dm is a critical rotational speed of the weightless shaft with one i‑th disc, determined by the formula (8.1).
From the formula (8.5) it follows, that the critical rotational speed of a system with the powerful shaft is always less than a critical rotational speed of the same system with the weightless shaft, and that the increase in disc number results in critical rotational speed reduction.
For a critical rotational speed of the ponderable shaft with one disc we get the following formula:
or
Given, that for the weightless shaft with one disc
the formula (8.5) can be written as follows:
Now the problem of critical rotational speed determination is reduced to determination of shaft static sags under actions of separate discs weights affixed to the shaft sequentially. To simplify calculation the evenly distributed load of shaft weight can with adequate accuracy be substituted by several concentrated forces.
In this case the critical rotational speed of a system with the ponderable shaft can be determined by the formula
(8.5a)
As the research showed, formulas (8.4) and (8.5) are approximated and yield deviation from real values of the critical rotational speed is up to 4 %, and at multistage shafts and composite configuration shafts – up to 10 %.