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When the intersection points lie below the 0.5 slopeline， the system is saidto have a ""bending critical speed.'' It is important to identify these points， since they indicate the increasing importance of bending stiffness over sup-port stiffness.
Figures 5-16a and 5-16b show vibration modes of a uniform shaft sup-ported at its ends by flexible supports. Figure 5-16a shows rigid supports and a flexible rotor. Figure 5-16b shows flexible supports and rigid rotors.
To summarize the importance of the critical speedconcept， one should bear in mind that it allows an identification of the operation region of therotor-bearing system， probable modeshapes， and approximate locations of peak amplitudes.
.ritical Speed .alculations for Rotor .earing Systems
Methods for calculating undamped and damped critical speeds thatclosely follow the works of Prohl andLund， respectively， are listed here-in. Computer programs can be developed that use the equations shown in this section to provide estimations of the critical speeds of a given rotor for a range of bearing stiffness and damping parameters.
The method of calculating critical speeds as suggested by Prohl and Lundhas several advantages. By this method， any number of orders of criticalfrequencies may becalculated， and the rotor configuration is not limited in the number of diameter changes or in the number of attached discs. Inaddition， shaft supports may be assumed rigid or may have any values of damping or stiffness. The gyroscopic effect associated with the moment of attached disc inertia may also be taken into account. Perhaps the greatestadvantage of the technique， however， is the relative simplicity with which all the capabilities are performed.
The rotor is first divided into a number of station points， including theends of the shafting， points at which diameter changesoccur， points at whichdiscs are attached， and bearing locations. The shafting connecting the station points are modeled as massless sections which retain the flexural stiffnessassociated with the section'slength， diameter， and modulus of elasticity. The mass of each section is divided in half and lumped at each end of the section where it is added to any mass provided by attached discs or couplings.
The critical-speed calculation of a rotating shaft proceeds with equations to relate loads and deflections from station n 1 to station n. The shaft shear V can be computed using the following relationship:
Vn二Vn 1 + n 1w 2Yn 1 (5-29)
and the bending moment
n二 n 1 +VnZn
The angular displacement can be computed using the following relationship:
 
.n二.n 2 n 1 + 2n +.n 1 (5-30)
where .二flexibility constant. The vertical linear displacement is
 
n 1 n
Yn二.n 3 + 6Zn +.n 1Zn +Yn 1 (5-31)
When crossing a flexible bearing at station n from the left side to the rightside， the following relationships hold:
主xxYn二 [(Vn)Right (Vn)Left](5-32)主...n二[( n)Right ( n)Left](5-33) (.n)Right二(.n)Left (5-34) (Yn)Right二(Yn)Left (5-35)
The initial boundary conditions are V1二 1二0 for a free endand， to assign initial values for Y1 and .1， the calculation proceeds in two parts with the assumptions given as
Pass 1 Y1二1.0 .1二0.0
Pass 2 Y1二0.0 .1二1.0
For each part， the calculation starts at the free end and， using Equations(5-29) through(5-35)， proceeds from station to station until the other end is reached. The values for the shear and moment at the far end are dependent on the initial values by the relationship:
Vn二Vn凡 Pass 1Y1 +Vn凡 Pass 2.1
n二 n凡 Pass 1Y1 + n凡 Pass 2.1 (5-36)
The critical speed is the speed at which both Vn二 n二0， which requires iterating the assumed rotational speed until this condition is observed.
If structural damping is to be considered， then a revised set of relation-ships must be used. For a system allowing vertical and horizontal shaftmotion， the change in shear and moment across a station is given by:
　
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