value of the critical speed ratio Ω increases, and the mid span shaft deflection r increases. If the eccentricity e between the mass centerline G, and the shaft cen-terline S is assumed to be some realistic value such as +2, Fig. 3-32 may be plot-ted. As expected, the midspan deflection becomes quite large as the critical speed (Ω=1) is approached.
Stated in another way, as the ratio of ω/ωc approaches unity, the denomina-tor of equation (3-79) becomes smaller, and the value of r becomes increasingly large. When the shaft speed ω is equal to the critical speed ωc, the Ω = ωc/ωc=1, and the denominator of equation (3-79) becomes zero. Obviously, division by zero will result in infinity. This is consistent with the previous logic, and the defini-tion of an undamped system.
The response plot in Fig. 3-32 was constructed between Ω values of 0 and 0.94 to describe the amplitude characteristics below the resonance. If the plot
Fig. 3–32 Plot Of Jeffcott
Frequency Ratio(Ω=ω/ωc) r e
range is extended from 0 to 3.0, the diagram shown in Fig. 3-33 evolves. Again, a numeric value of +2 was selected for the eccentricity e, and it is noted that the midspan shaft deflection indeed moves off towards infinity. In a real machine, infinitely large shaft deflections are not possible. Only two possibilities for the Jeffcott (or any other) rotor are feasible. The first option is for the midspan amplitudes to increase to the point where the machine destroys itself (only choice for an undamped system). The other alternative for a real machine is to have the displacement amplitudes at the critical speed restrained by damping.
In accordance with the earlier discussion within this chapter, positive damping is an energy dissipater. It will limit the vibration amplitude through the resonance. For a system with low damping, such as a structural resonance, the amplitude at the resonance will be high, and the resonance bandwidth will be small (high Q). Conversely, for a system with large damping, such as a fluid film bearing with viscous damping, the resonance peak will be lower, and the bandwidth will be wider (small Q). Thus, without damping in the system, the machine could not survive a resonance. This discussion also identifies the logic associated with the amplitude increase at the critical speed. It is hopefully clear from the preceding explanation, and the general equations, that the displace-ment at the resonance must increase. Furthermore, the amount of the vibration increase at the critical is dependent upon the available system damping.
Unfortunately, the characteristics of the amplitude response through the resonance are not completely defined, because one other peculiarity must be rec-onciled. Specifically, Fig. 3-33 reveals a midspan amplitude r that migrates off towards positive infinity (+∞) as the resonance is approached. In true mathemat-ical fashion, the amplitude above the critmathemat-ical returns from negative infinity (-∞).
That is particularly disturbing when it is also recognized that the amplitudes above the resonance are all negative. For example, if e=2, and Ω=2, then by equa-tion (3-79), r is computed to be minus 2.67. Hence, the plotted curve is mathe-matically correct, but it does not describe a true physical situation. That is, a negative vibration amplitude is incomprehensible.
Fig. 3–33 Plot Of Jeffcott
Critical Speed Transition 127
In retrospect, equation (3-79) used for computation of this deflection r only contains two terms. The eccentricity e, plus the frequency ratio between the run-ning speed and the natural frequency Ω =ω/ωc. Above the resonant frequency ωc
it is obvious that this speed ratio must be a positive number that is greater than one. Hence, there is no variation of the frequency ratio Ω that would reverse the sign of the midspan deflection r.
The only other alternative resides with the eccentricity e. As previously stipulated, the magnitude of e is fixed by the geometry of the disk, and the loca-tion of the mass on the disk. However, the original definiloca-tion of the Jeffcott rotor did not restrict the direction (i.e., the ± sign) of this eccentricity e. Hence, if the direction of e was reversed, the plotted response curve should flip over into the positive domain.
In fact, that is exactly what happens with a real machine, as well as the Jef-fcott model. To prove this point, the response data above the resonance will be replotted in Fig. 3-34, and e=+2 will be replaced by e=-2. In order to maintain reasonable amplitudes, a frequency ratio range extending from 1.08 to 3.0 will be used. It is noted that the resultant plot illustrates the proper positive deflection r, and it also is indicative of normal behavior on the back slope of a resonance.
The sign reversal of the eccentricity e is physically equivalent to a reversal in the positions of the shaft S, and mass G centerlines. For instance, compare the Jeffcott rotor running below the critical in Figs. 3-30 and 3-31, with the rotor operating above the critical as shown in Figs. 3-35 and 3-36. Below the critical, the mass M was on the outside of the rotor. Whereas, above the critical speed, the mass M is tucked away underneath the curvature of the shaft — and the light side of the rotor is now on the outside. This reversal of heavy side out to light side out is the mechanism behind the nominal 180° phase shift across a resonance.
Another interesting point from the response plot in Fig. 3-34 is that the value of r steadily diminishes with an increasing frequency ratio Ω. At high shaft speeds, the magnitude of r approaches the magnitude of e. Physically, this char-acteristic is described in Fig. 3-36 of the Jeffcott rotor at a high speed condition.
Fig. 3–34 Plot Of Jeffcott
At High Speeds "r" approaches "e"
(Ω=ω/ωc)
and for the condition of :
-2 2
In this state (i.e., well above the critical speed), the values of e and r are essen-tially equal. Since e carries a negative sign (above the critical), the centerline for the mass G is now coincident with the bearing centerline B. This direction rever-sal eliminates the centrifugal force, since the radius to the mass is now equal to, or very close to zero, and the rotor has self balanced itself.
This self-balancing characteristic is the mechanism that allows so many machines to operate successfully at speeds in excess of the rotor balance reso-nance, or critical speed. In this high speed condition, the system is in equilib-rium, and the shaft actually rotates around the mass center M, which is equivalent to the centerline of gravity G for the rotor. For further discussions within this chapter, and especially the balancing chapter 11, reference will be made to rotation about the geometric axis below the critical speed, and rotation about the mass or inertial centerline above the critical. These continuing refer-ences are based upon the fundamental behavior and understanding of the tradi-tional Jeffcott rotor.
At the beginning of this discussion, it was stated that the damping was intentionally set at zero. That was a convenient and necessary presumption to maintain simplicity of the analytical model. The inclusion of damping into the system does substantially complicate the response of the Jeffcott rotor. For instance, consider Fig. 3-37 of this rotor with damping. Note that the centerlines B, S, and G are no longer collinear. An angular phase change φ has been invoked upon the mass centerline G. This implies that the centrifugal force, and the shaft
Fig. 3–35 Jeffcott Rotor Operating Slightly Above The Critical Balance Resonance
Speed
Fig. 3–36 Jeffcott Rotor Operating In A Self Balanced Condition Well Above The
Critical Balance Resonance Speed
Critical Speed Transition 129
spring restoring force are no longer equal and opposite. Actually, a new force must now be included to achieve a force balance. This new force will be a damp-ing term that may be tangential to the disk, and may vary in magnitude with the shaft surface velocity.
The addition of this damping does not invalidate the previous discussion, but it does complicate the scenario. For simplicity, the damping force might be assumed to be viscous, and the resultant force equal to the tangential velocity times the damping coefficient. This type of force could be included, and the previ-ous analysis repeated. However, this inclusion would not necessarily improve the overall understanding of the critical speed phenomena. Furthermore, damping characteristics of real machines are not a simple linear function of rotor velocity, and a proper analysis would be substantially more intricate.
In essence, it must be recognized that all rotors have some amount of damp-ing. The oil film in the bearings, oil in the shaft seals, or the process fluid itself may provide the damping. In all cases, the presence of damping will influence the shaft behavior. There will be positive contributions to the addition of damp-ing, such as lower vibration amplitudes through the critical speed region. How-ever, damping will also open the door to a variety of mechanisms such as non-synchronous whirl. In all cases, the real world is always more complicated than the models that we can build to explain physical events. Fortunately, the undamped Jeffcott model may be used to explain the fundamental characteris-tics associated with the balance resonance or critical speed phenomena.
Fig. 3–37 Jeffcott Rotor With Damping
B S G
Shaft & Disk Center
Mass Center
Bearing
Center ω
φ