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//INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 185 ± [178±218/41] 29.10.2001 3:57PM This very special case is known as critical damping. The value of c for this case is given by: c 2 cr 4m 2 k m c 2 cr 4m 2 k m 4mk Thus, c cr 4mk p 2m k m r 2m! n Underdamped system. If c 2 =4m 2 < k=m, then the roots r 1 and r 2 are imaginary, and the solution is an oscillating motion as shown in Figure 5-9. All the previous cases of motion are characteristic of different oscillating systems, although a specific case will depend upon the application. The underdamped system exhibits its own natural frequency of vibration. When c 2 =4m 2 < k=m, the roots r 1 and r 2 are imaginary and are given by r 1;2 Æi k m À c 2 4m 2 r 5-15 Then the response becomes x e Àc=2mt C 1 e i k m À c 2 4m 2 p C 2 e Ài k m À c 2 4m 2 p 45 Figure 5-8. Critical damping decay. Rotor Dynamics 185 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 186 ± [178±218/41] 29.10.2001 3:57PM which can be written as follows: x e Àc=2mt A cos ! d t B sin ! d t5-16 Forced Vibrations So far, the study of vibrating systems has been limited to free vibrations where there is no external input into the system. A free vibration system vibrates at its natural resonant frequency until the vibration dies down due to energy dissipation in the damping. Now the influence of external excitation will be considered. In practice, dynamic systems are excited by external forces, which are themselves periodic in nature. Consider the system shown in Figure 5-10. The externally applied periodic force has a frequency !, which can vary independently of the system parameters. The motion equation for this system may be obtained by any of the previously stated methods. The Newtonian approach will be used here because of its conceptual simplicity. The freebody diagram of the mass m is shown in Figure 5-11. Figure 5-9. Underdamped decay. Figure 5-10. Forced vibration system. 186 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 187 ± [178±218/41] 29.10.2001 3:57PM The motion equation for the mass m is given by: m x F sin !t À kx Àc x 5-17 and can be rewritten as m x c x kx F sin !t Assuming that the steady-state oscillation of this system is represented by the following relationship: x D sin !t À5-18 where: D amplitude of the steady-state oscillation phase angle by which the motion lags the impressed force The velocity and acceleration for the system are given by the following relationships: v x D! cos !t À D! sin !t À 2 5-19 a x D! 2 sin !t À D! 2 sin !t À 2 5-20 Substituting the previous relationships into motion equation (5-17), the following relationship is obtained: mD! 2 sin !t À ÀcD! sin !t À 2 À D sin !t À F sin !t 0 5-21 Inertia force Damping force Spring force Impressed force 0 From the previous equation, the displacement lags the impressed force by the phase angle , and the spring force acts opposite in direction to Figure 5-11. Free body diagram of mass (M). Rotor Dynamics 187 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 188 ± [178±218/41] 29.10.2001 3:57PM displacement. The damping force lags the displacement by 90 and is there- fore in the opposite direction to the velocity. The inertia force is in phase with the displacement and acts in the opposite direction to the acceleration. This information is in agreement with the physical interpretation of harmo- nic motion. The vector diagram as seen in Figure 5-12 shows the various forces acting on the body, which is undergoing a forced vibration with viscous damping. Thus, from the vector diagram, it is possible to obtain the value of the phase angle and the amplitude of steady oscillation D F k À m! 2 2 c! 2 q 5-22 tan c! k À m! 2 5-23 The nondimensional form of D and can be written as D F=k 1 À ! 2 ! 2 n 2 ! ! n 2 s 5-24 Figure 5-12. Vector diagram of forced vibration with viscous damping. 188 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 189 ± [178±218/41] 29.10.2001 3:57PM tan 2 ! ! n 1 À ! ! n 2 5-25 where: ! n k=m p natural frequency c c c damping factor c c 2 m! n critical damping coefficient: From these equations, the effect on the magnification factor (D=F=k)and the phase angle () is mainly a function of the frequency ratio !=! n and the damping factor . Figures 5-13a and 5-13b show these relationships. The damping factor has great influence on the amplitude and phase angle in the region of resonance. For small values of !=! n ( 1:0, the inertia and damping force terms are small and result in a small phase angle. For a value of !=! n 1:0, the phase angle is 90 . The amplitude at resonance approaches infinity as the damping factor approaches zero. The phase angle undergoes nearly a 180 shift for light damping as it passes through the critical frequency ratio. For large values of !=! n ) 1:0, the phase angle approaches 180 ,and the impressed force is expended mostly in overcoming the large inertia force. Design Considerations Design of rotating equipment for high-speed operation requires careful analysis. The discussion in the preceding section presents elementary analy- sis of such problems. Once a design is identified as having a problem, it is an altogether different matter to change this design to cure the problem. The following paragraphs discuss some observations and guidelines based on the analysis presented in the previous sections. Natural frequency. This parameter for a single degree of freedom is given by ! n k=m p . Increasing the mass reduces ! n , and increasing the spring constant k increases it. From a study of the damped system, the damped natural frequency ! d ! n 1 À 2 p is lower than ! n . Unbalances. All rotating machinery is assumed to have an unbalance. Unbalance produces excitation at the rotational speed. The natural fre- quency of the system ! n is also known as the critical shaft speed. From the study of the forced-damped system, the following conclusions can be drawn: Rotor Dynamics 189 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 190 ± [178±218/41] 29.10.2001 3:57PM (1) the amplitude ratio reaches its maximum values at ! m ! n 1 À 2 2 p , and (2) the damped natural frequency ! d does not enter the analysis of the forced-damped system. The more important parameter is ! n , the natural frequency of the undamped system. In the absence of damping the amplitude ratio becomes infinite at ! ! n . For this reason, the critical speed of a rotating machine should be kept away from its operating speed. Small machinery involves small values of mass m and has large values of the spring constant k (bearing stiffness). This design permits a class of machines, which are small in size and of low speed in operation, to operate in a range below their critical speeds. This range is known as subcritical operation, and it is highly desirable if it can be attained economically. Figure 5-13a. Amplitude factor as a function of the frequency ratio r for various amounts of viscous damping. 190 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 191 ± [178±218/41] 29.10.2001 3:57PM The design of large rotating machineryÐcentrifugal compressors, gas and steam turbines, and large electrical generatorsÐposes a different problem. The mass of the rotor is usually large, and there is a practical upper limit to the shaft size that can be used. Also, these machines operate at high speeds. This situation is resolved by designing a system with a very low critical speed in which the machine is operated above the critical speed. This is known as supercritical operation. The main problem is that during start-up and shut-down, the machine must pass through its critical speed. To avoid dangerously large amplitudes during these passes, adequate damping must be located in the bearings and foundations. The natural structural frequencies of most large systems are also in the low- frequency range, and care must be exercised to avoid resonant couplings between the structure and the foundation. The excitation in rotating machinery comes from rotating unbalanced masses. These unbalances result from four factors: 1. An uneven distribution of mass about the geometric axis of the system. This distribution causes the center of mass to be different from the center of rotation. 2. A deflection of the shaft due to the weight of the rotor, causing further distance between the center of mass and the center of rotation. Add- itional discrepancies can occur if the shaft has a bend or a bow in it. Figure 5-13b. Phase angle as a function of the frequency ratio for various amounts of viscous damping. Rotor Dynamics 191 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 192 ± [178±218/41] 29.10.2001 3:57PM 3. Static eccentricities are amplified due to rotation of the shaft about its geometric center. 4. If supported by journal bearings, the shaft may describe an orbit so that the axis of rotation itself rotates about the geometric center of the bearings. These unbalance forces increase as a function of ! 2 , making the design and operation of high-speed machinery a complex and exacting task. Balan- cing is the only method available to tame these excitation forces. Application to Rotating Machines Rigid Supports The simplest model of a rotating machine consists of a large disc mounted on a flexible shaft with the ends mounted in rigid supports. The rigid supports constrain a rotating machine from any lateral movement, but allow free angular movement. A flexible shaft operates above its first critical. Figures 5-14a and 5-14b show such a shaft. The mass center of the disc ``e'' is displaced from the shaft centerline or geometric center of the disc due to manufacturing and material imperfections. When this disc is rotated at a rotational velocity !, the mass causes it to be displaced so that the center of Figure 5-14a. Rigid supports. Figure 5-14b. Flexible supports. 192 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 193 ± [178±218/41] 29.10.2001 3:57PM the disc describes an orbit of radius r , from the center of the bearing centerline. If the shaft flexibility is represented by the radial stiffness (K r ), it will create a restoring force on the disc of K r r that will balance the centrifugal force equal to m! 2 ( r e). Equating the two forces obtains K r m! 2 r e Therefore, r m! 2 e K r À m! 2 !=! n 2 e 1 À!=! n 2 5-26 where ! n K r /m, the natural frequency of the lateral vibration of the shaft and disc at zero speed. The previous equation shows that when !<! n , r is positive. Thus, when operating below the critical speed, the system rotates with the center of mass on the outside of the geometric center. Operating above the critical speed (!>! n ), the shaft deflection r tends to infinity. Actually, this vibration is damped by outside forces. For very high speeds (!>>! n ), the amplitude r equals Àe, meaning that the disc rotates about its center of gravity. Flexible Supports The previous section discussed the flexible shaft with rigid bearings. In the real world, the bearings are not rigid but possess some flexibility. If the flexibility of the system is given by K b , then each support has a stiffness of K b =2. In such a system, the flexibility of the entire lateral system can be calculated by the following relationship: 1 K t 1 K r 1 K b K b K r K r K b K t K r K b K b K r 5-27 Therefore, the natural frequency ! nt K t m r K r K b K b K r 0 m s K r m r  K b K b K r r ! n K b K b K r r 5-28 Rotor Dynamics 193 //INTEGRA/B&H/GTE/FINAL (26-10-01)/CHAPTER 5.3D ± 194 ± [178±218/41] 29.10.2001 3:57PM It can be observed from the previous expression that when K b ( K r (very rigid support), then ! nt ! n or the natural frequency of the rigid system. For a system with a finite stiffness at the supports, or K b > < K r , then ! n is less than ! nt . Hence, flexibility causes the natural frequency of the system to be lowered. Plotting the natural frequency as a function of bearing stiffness on a log scale provides a graph as shown in Figure 5-15. When K b ( K r , then ! nt ! n K b =K r . Therefore, ! nt is proportional to the square root of K b , or log ! nt is proportional to one-half log K b . Thus, this relationship is shown by a straight line with a slope of 0.5 in Figure 5-15. When K b ) K r , the total effective natural frequency is equal to the natural rigid-body frequency. The actual curve lies below these two straight lines as shown in Figure 5-15. The critical speed map shown in Figure 5-15 can be extended to include the second, third, and higher critical speeds. Such an extended critical speed map can be very useful in determining the dynamic region in which a given system is operating. One can obtain the locations of a system's critical speeds by superimposing the actual support versus the speed curve on the critical speed map. The intersection points of the two sets of curves define the locations of the system's critical speeds. Figure 5-15. Critical speed map. 194 Gas Turbine Engineering Handbook [...]... capacitance C, and resistance R //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 199 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM Rotor Dynamics Figure 5- 17 Forced vibration with viscous damping Figure 5- 18 A force-voltage system 199 //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 0 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 20 0 Gas Turbine Engineering Handbook Figure 5- 19 Force-current analogy represent the mass, spring... Group Engineering Division, May 11, 19 72. ) //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 8 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 20 8 Gas Turbine Engineering Handbook Figure 5 -22 b Dry friction whirl (Ehrich, F.F., ``Identification and Avoidance of Instabilities and Self-Excited Vibrations in Rotating Machinery,'' Adopted from ASME Paper 72- DE -21 , General Electric Co., Aircraft Engine Group, Group Engineering. ..
Mxn À Myn Z 2 n
Vxn À Vyn =2 H H H H n1 n C1 Zn
Myn Mxn Z 2 n
Vyn Vxn =2 H H Mx;n1 Mxn Zn Vxn H H My;n1 Myn Zn Vyn H Vx;n1 Vxn H Vy;n1 Vyn //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 198 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 198 Gas Turbine Engineering Handbook where: C1 1=
EIn C2 p 1 2 2 Zn
Z EIn À 6
GAn 5- 38 where: E Young's... //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 6 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 20 6 Gas Turbine Engineering Handbook Figure 5 -21 Characteristics of instabilities or self-excited vibration in rotating machinery (Ehrich, F.F., ``Identification and Avoidance of Instabilities and Self-Excited Vibrations in Rotating Machinery,'' Adopted from ASME Paper 72- DE -21 , General Electric Co., Aircraft Engine Group, Group Engineering. .. forced or resonant instability dependent on outside mechanisms in frequency of oscillations; //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 2 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 20 2 Gas Turbine Engineering Handbook Table 5 -2 Forces Acting on Rotor Bearing Systems Source of Force 1 2 Description Forces transmitted Constant, to foundations, casing, unidirectional force or bearing pedestals Constant... is calculated and found to be 20 0 Hz From the Campbell diagram figure, it //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 21 2 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 21 2 Gas Turbine Engineering Handbook Table 5- 4 Characteristics of Rotor Instabilities Type of Instability Onset Frequency Response Forced Vibration Unbalance Any speed Nf N Shaft misalignment Any speed Nf 2N Self-Excited Vibration Hysteretic... 5 -23 b The resultant force from the cross-coupling of angular motion and radial forces may destabilize the rotor and cause a whirl motion The aerodynamic cross-coupling effect has been quantified into equivalent stiffness For instance, in axial-flow machines Kxy ÀKyx T DP H 5- 42 //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 21 0 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 21 0 Gas Turbine Engineering Handbook. .. //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 5 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM Rotor Dynamics 20 5 Figure 5 -20 Characteristic of forced vibration or resonance in rotating machinery (Ehrich, F.F., ``Identification and Avoidance of Instabilities and Self-Excited Vibrations in Rotating Machinery,'' Adopted from ASME Paper 72- DE -21 , General Electric Co., Aircraft Engine Group, Group Engineering Division, May 11, 19 72. )... critical is exactly 20 N On a Campbell diagram the previous Figure 5 -26 Accelerometer locations on impeller tested //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 21 5 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM Rotor Dynamics 21 5 FPO Figure 5 -27 Impeller showing nodal points example will correspond to an exact intersect of the running speed line, 1000 Hz frequency line, and the line of slope 20 N A shrouded impeller... output  (0±10,000 Hz) Accelerometers can be mounted at various positions on the //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 21 4 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM 21 4 Gas Turbine Engineering Handbook impeller to obtain the frequency responses in conjunction with a spectrum analyzer (Figure 5 -26 ) Initially, tests are run to identify the major critical frequencies of the impeller Mode shapes are then . drive. table continued on next page 20 2 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 20 3 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM and (2) the self-excited instabilities. m! 2 2 c! 2 q 5 -22 tan c! k À m! 2 5 -23 The nondimensional form of D and can be written as D F=k 1 À ! 2 ! 2 n 2 ! ! n 2 s 5 -24 Figure. of Figure 5- 14a. Rigid supports. Figure 5- 14b. Flexible supports. 1 92 Gas Turbine Engineering Handbook //INTEGRA/B&H/GTE/FINAL (26 -10-01)/CHAPTER 5. 3D ± 193 ± [178 21 8/41] 29 .10 .20 01 3 :57 PM the
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