Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA

DETC2011-4 TRANSIENT FORWARD AND BACKWARD WHIRL OF BEAM AND SOLID ROTORS WITH STIFFENING AND SOFTENING EFFECTS J. S. Rao Fellow ASME Altair Engineering India Bangalore, India

ABSTRACT This paper describe the procedure of achieving unbalance response and stability of general accelerating solid rotor model rotors with variable cross-section on fluid film or rolling element bearings and seals with the casings mounted on foundations including stress stiffening and spin softening effects. The solid rotor analysis results are verified with beam analysis results wherever applicable. . INTRODUCTION Prior to Laval and Parsons steam turbines in early 1880’s Rankine [1] in 1869 made the first attempts to understand Rotor Dynamics. He identified that whirl frequency is same as transverse vibration frequency of a stationary shaft and showed the existence of critical speeds of a rotor. Laval built the first steam turbine in 1883 and identified the equivalence of a single degree of freedom vibrating system for a rotor and therefore some prefer to call one-disk rotor models Laval Rotors. Föppl [2] discussed Laval’s rotor. Jeffcott [3] in 1919 formulated the rotor problem as one of forced vibration. He showed for the first time that the shaft did not primarily rotate about its rest position, but about its own centerline. The whirl of the rotor corresponds to free or forced vibration of a stationary structure. The rotors modeled as beams since then are also referred as Jeffcott Rotors. Though “Dynamics” is same for structures or rotors, the subject matter of “Rotor Dynamics” grew separately by virtue of special characteristics of rotors; they have residual unbalance (imbalance), and have misalignment during assembly. Rotors are supported on rolling or fluid film supports that exhibit cross-coupling and connected to casings and foundations. Rotors are coupled through gear transmissions and may also have looseness between the mounted parts. Rotors are also asymmetric by virtue of their design as in two pole generators giving rise to variable stiffness and instabilities. Because of

these special characteristics the subject matter grew separately from structural dynamics and the significant effects are all considered through the rotors considered as beam models. For example the effect of a disk is considered as a lumped diametral inertia and gyroscopic principles are used to include its effect rather than as a solid. The developments then moved depending on the advances in computational systems, initially by direct application of energy principles then by tabular methods, transfer matrix methods and finally with the availability of recent digital computers the analysis all shifted to finite element methods, [4 and 5]. Most commonly employed finite element is due to Nelson [6]. Beam models cannot include centrifugal loads and their influence on stiffening the rotor raising the frequencies in forward whirl besides a spin softening effect which results in reducing frequencies in backward whirl. The backward whirl has been neglected in traditional beam model approaches but its significance is now being understood with rotors on several supports having simultaneous forward and backward whirls; the first such rotor dynamics study using solid model was provided by Rao [7]. The unbalance response, instabilities, transients during acceleration all of them are affected by solid models with spinning effects. The solid models of today offer further advantages of integrated modeling of rotors with casings and foundations. Further, variable stiffness that may occur in electrical rotors can also be included in stability analysis using solid models. A significant advantage in solid modeling is in avoiding approximate models of rotating parts as inertias to account for gyroscopic effects through which the effect of speed is brought into picture. Solid models remove these approximations, they also offer significant advantage of using the same rotor model as that can be adopted for structural and thermo-mechanical analysis, thus helping the designers in reducing considerable time in preprocessing.

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This paper outlines the procedure of achieving unbalance response and stability of general accelerating solid rotor model rotors with variable cross-section on fluid film or rolling element bearings and seals with the casings mounted on foundations including stress stiffening and spin softening effects. The solid rotor analysis results are verified with beam analysis results where available. SOLID MODEL ROTOR DYNAMICS ANALYSIS The idea of solid or shell models of stationary structures became evident when the casings and foundations became a part of the rotating machine and test beds became flexible when coupled with rotors that interfered in the validating process. Stephenson and Rouch [8] used axisymmetric solid finite elements with matrix reduction in their analysis; Yu, Craggs and Mioduchowski [9] modeled shafts orbiting with 3-D solid finite elements. Neither of these included the effects of rotation, stiffening and softening effects that go with solid elements. Solid rotor dynamics analysis was first presented by Rao [7]. Rao, Sreenivas and Veeresh [10] provided further details. Beam Models with cross-coupled stiffness Supports We first take a simple Jeffcott type rotor as adopted by Gunter, Choy and Allaire [11]. The rotor of mass 54.432 kg, with its rigid bearing critical speed 4820 rpm, is mounted on two 4 axial groove bearings 2.54 cm dia and 1.27 cm long, with a radial clearance 0.00254 cm and viscosity at operating temperature, 0.0242 N sec/m2. The shaft stiffness K = Mp2 = 1.387 × 107 N/m. The stiffness coefficients for the bearing at 4500 rpm, Sommerfeld Number S = 0.5488 are Kzz = 4.16 × 107 N/m; Kyy = 1.01 × 107 N/m; Kzy = 3.12 × 107 N/m; Kyz = 4.16 × 107 N/m; This rotor has two natural frequencies, 3685 rpm and 4476 rpm. The unbalance u = ma with eccentricity a is expressed as Fx  u 2 cos  cost  iu 2 sin  sin t (1) Fy  u 2 sin  cost  iu 2 cos  sin t The unbalance force is applied at the disk center in real Y and imaginary Z directions. If vc and vs are cosine and sine components of the response vector in Y direction and wc and ws are cosine and sine components of the response vector in Z direction, the major a and minor b axes of the elliptical orbit are, see [4]  w 2  w 2  v 2  v 2 s c s  1  c a, b     2   wc2  ws2  v c2  v s2    





2

  1   2   4wc ws  vc v s 2    

1

2

(2) The phase angle made by the orbit is 2wc ws  v c v s  tan 2t  wc2  ws2  v c2  v s2 The whirl radius r at any instant of time t is

(3)

1 1  r (t )  exp( it )  wc  v s   i vc  ws  2 2  1 1   (4)  exp( it )  wc  v s   i vc  ws  2 2  where the forward r+ and backward r- whirl components are

1 1  r    wc  v s   i v c  ws  2 2 

(5)

1 1  r    wc  v s   i v c  ws  2 2 

(6)

Forward whirl of rotor occurs when r   r  and the rotor will have a backward whirl when r   r  . The rotor can be simulated in any structural code, e.g., with appropriate beam and mass elements and elements for supports. The results are to be post processed and displayed as per one’s choice. If the rotor is not modeled with beam and mass elements, instead a solid model is used, the procedure remains same; and then we have a solid model analysis. Adopting this method we can derive several advantages for rotors on casings and foundations. There are two ways of achieving the above implementation by using existing codes. 1. Ansys [12] allows macros to be written and couple them with the main solver, see [10]. 2. Altair HyperWorks Process Manager [13] is a programmable personal workflow manager that guides users through standard work processes. Programming language C and Tcl/Tk are used for developing the core and Graphical user interface (GUI) respectively. This process is described in [14] for determining life of turbine blades. The advantage in adopting the approaches outlined above allows one to use the rugged solvers and couple them with the missing rotor dynamics technologies. The steady state orbit is a forward whirl for speeds below the first critical speed 3685 rpm and above the second critical speed 4476 rpm. In between the two critical speeds the whirl is backward in nature, see [4]. The transient solutions obtained are given in Figs. 1 and 2 at 3000 and 4000 rpm respectively below the first critical speed and in between the two split critical speeds. These solutions are in agreement with steady state solutions given in [4]. An interesting feature from the transient solution between the two critical speeds is that the whirl initially starts in forward direction as in left side of Fig. 3 but settles down to backward direction on right side. When the rotor is disturbed from the starting position initially it is driven in the forward whirl by virtue of the rotation or spin of the rotor, however it settles to backward whirl once it reaches steady state condition; this is seen whenever backward whirl occurs. The rotor gets back to forward whirl in the post critical condition at 8000 rpm as shown in Fig. 4.

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Figs. 1 to 4 show that any rotor can be modeled as a structure with the unbalance applied as real and imaginary parts along both the real and imaginary axes according to equation (1). The orbital relations given in equations (2) to (6) can be used to post process the results and the steady state response obtained from transient analysis agrees well with closed form solutions. We also understand clearly that the whirl always starts in forward direction before settling to backward whirl at a speed between the criticals. This phenomenon is seen whenever the critical speed splits into two. Figure 1 TRANSIENT SOLUTION OF GUNTER’S ROTOR AT 3000 RPM

Beam and Solid Models - Gyroscopics The gyroscopic effects are introduced into the beam model rotor dynamics so as to account for the kinetic energy of the disk located at a nodal point in a given mode shape. This will not be necessary if we have a solid model with several nodal degrees of freedom. The gyroscopic couple depends on the spin and precessional (slope) velocities, the speed enters into analysis indirectly, otherwise there are no rotational effects in beam models (as in case of turbine blades). These effects are considered by considering a twin spool rotor model of Rajan, Nelson and Chen [15]. Fig. 5 shows this beam model of rotor system with disks. ROTOR 2

1.905 cm ri 2.54 cm ro

Figure 2 TRANSIENT SOLUTION OF GUNTER’S ROTOR AT 4000 RPM

26.2795E6N/m

17.519E6 N/m

17.519E6 N/m 10 5 4 6

9 2 3

1 ROTOR 1

7

8

8.7598E6 N/m

1.52 cm r Distances: 1-2=7.62 cm; 2-3=17.78 cm; 3-4=15.24 cm; 4-5=5-6=7-8=9-10=5.08cm;8-9=15.24 cm Masses: 2 = 4.904; 5 =10.02034; 8 = 0.01469; 9 = 2.227 kg IP: 2 = 0.02712; 5 = 0.02304; 8 = 0.01469; 9 =0.0972 kgm2 ID = IP/2 E = 206.9 GPa;  = 8.304 kg/m3 2 = 1.5 1

Figure 3 INITIAL FORWARD WHIRL SETTLES TO BACKWARD WHIRL AT 4000 RPM

Figure 5 TWIN SPOOL ROTOR An equivalent solid rotor model of Fig. 5 is made as shown in Fig. 6. The four rotors with given masses and inertias are replaced by four disks (1) 10.626 cm dia 1.7 cm wide, (2) 9.7344 cm dia 1.38 cm wide, (3) 9.682 cm dia 0.978 cm wide and (4) 9.9954 cm dia 1.66 cm wide consecutively. It may be noted that Fig. 5 can represent in a unique manner an equivalent beam model of the solid model of Fig. 6, even though several other solid models can be derived for Fig. 5 beam model. This in fact is the main limitation of beam model analysis as an equivalent derived beam model may represent the dynamics of different solid models. An actual physical model in solid form eliminates this approximation. Figure 4 TRANSIENT SOLUTION OF GUNTER’S ROTOR AT 8000 RPM

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there are spin softening effects only in the plane of rotation, not normal to the plane of rotation. The eigen-value problem with spin softening effect is

K   

2 s

M    2 M   0

K  2 M   0 Figure 6 EQUIVALENT SOLID TWIN SPOOL ROTORS Any rotating structure is influenced by stress stiffening and spin softening effects. Stress stiffening effect is significant for turbomachine blades with the blade elements far away from shaft center, particularly for last stage steam turbine blades with nearly a meter long on disks of large diameters at 1500 or 3000 rpm (1800 or 3600 rpm for 60 Hz machines) or blades rotating at high speeds but of smaller length. The bending stiffness of these blades increases with speed and the natural frequencies also increase. For rotating shafts in bending, the stress stiffening effect is not that significant as in the case of blades and that explains the reason why beam models without rotational effects do give satisfactory results; however the spin softening effect is significant in backward whirl modes. In traditional beam model rotor dynamics analysis spin softening effect is not accounted in the belief that the backward whirl modes are not important because the unbalance provides only forward whirl. This of course is not entirely true because we do see a whirl starting as forward whirl becoming backward whirl as in Fig. 3 for speeds between the split criticals. Such a backward whirl has been observed in the laboratory by Subbaiah et al [16] and can be excited by an unbalance. Thus the backward whirl critical speed is important in rotor dynamics analysis and is to be determined more accurately by including the spin softening effects.

(9)

where [K] is the stiffness matrix, [M] is the mass matrix,  is the eigen frequency and s is the spin speed and the effective stiffness matrix is denoted by a bar above K. The rotor is still a structure and the eigen values can be determined from (9) to determine the effect of spin softening on the split natural frequencies. Some structural codes provide for accounting the spin softening or we can use any structural solver to determine the split critical speeds according to (9). This spin softening effect is not included in the traditional beam model rotor dynamics and is found to have a significant influence on backward whirl modes. The twin spool rotor of Fig. 6 is modeled as shown in Fig. 7. The Campbell diagram obtained without spin softening and including spin softening effect is given in Fig. 8.

Figure 7 LUMPED MASS AND INERTIA MODEL

3000

Spin Softening The vibration of a spinning body causes relative circumferential motions, which will change the direction of the centrifugal load. To explain this effect, consider a simple spring K mass M system, with the spring oriented radially with respect to the axis of rotation. As a small deflection analysis cannot directly account for changes in geometry, the effect can be accounted for by an adjustment of the stiffness matrix, called spin softening. Ku   s2 M r  u  (7) where r is the spring length and u is the axial extension and s is the spin speed. Rewriting the above (8) Ku  F

WHIRL SPEED (rad/s)

2000

1500

1000

500

0 0

500

1000

1500

2000

2500

3000

ROTOR 1 SPIN SPEED (rad/s)

is K  K   s2 M

the modified stiffness matrix . The stiffness of the system decreases from the original spring stiffness, thus decreasing the natural frequency. We can extend the above equation (8) to three dimensions; it can be shown that

Mode 1B Mode 1F Mode 2B Mode 2F Mode 3B Mode 3F Tmode 1B Tmode 1F Tmode 2B Tmode 2F Tmode 3B Tmode 3F 1*REV(omega1) 1*REV(omega2)

2500

Figure 8 CAMPBELL DIAGRAM Dashed lines – no spin softening; Full lines – spin softening included With the conventional gyroscopic effects included in a beam model without spin softening effect, the backward whirl

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decreases in an almost linear manner. When spin softening effects are included, the backward whirl natural frequency decreases with spin speed and the effective stiffness becomes zero when the spin speed becomes the natural frequency of the stationary shaft as in Fig. 8. Therefore the backward whirl natural frequency drops at a faster rate when spin softening is accounted for and disappears after a spin speed equal to the natural frequency under stationary conditions. This is a new result from the spin softening effects included in the model. As mentioned before, if the stress stiffening effect of rotating shafts can be neglected, equation (9) can be easily applied to include the spin softening effects and determine the backward whirl accurately using appropriate structural solver. It should be noted that a solid model automatically includes the beam model gyroscopic effects and with the spin softening effects included, a solid model analysis can be directly accomplished using a structural solver and adopting equations (1) to (6) and (9) as macros or subroutines that can be called onto a suitable platform and post process the results as desired. Stress stiffening effects can be included if desired from those codes which have this capability besides applying macros for the unbalance force and orbital relations from equations (1) to (6). ROTORS ON CASINGS AND ELASTIC CONNECTIONS In turbomachinery applications an accurate and reliable analysis of the rotor dynamic behavior requires complex and sophisticated modeling of the engine spools rotating at different speeds, static structures like casings and frames, elastic connections simulating bearings and the full drivetrain. Surial and Kaushal [17] developed finite element model of a Rolls Royce industrial gas turbine engine by dividing it into four substructures using super elements. The first super element represents the engine casing, second low pressure rotor (LP rotor), third - intermediate pressure rotor (IP rotor), and the fourth - high pressure rotor (HP rotor). In their model, the engine in Fig. 9 is shown with the LP, IP and HP rotors as in Fig. 10. These models were then integrated into the Whole Engine Model as a separate super element, forming the drivetrain model.

Figure 10 LP, IP AND HP ROTORS Another approach to this complex problem is to adopt directly a solid rotor dynamics model as presented by Rao [18] for a two spool aircraft engine. Fig. 11 shows a typical mode shape obtained for this engine.

Figure 11 TWO SPOOL AIRCRAFT ENGINE This approach avoids substructuring analysis which is time consuming and expensive as hitherto practiced in industry. BEARING NONLINEARITIES Major advantage of solid rotor modeling lies not only in the integration of casings and supports but also on considering nonlinearities of bearings and their influence on unbalance response and shift in frequency where peak whirl amplitudes occur, see [19]. The CAD model of a high speed cryogenic pump employed in a Geo Synchronous Satellite Launch Vehicle is shown in Fig. 12. The rotor and the casing are meshed with eight-noded brick elements, bearings and seals are simulated by 12 by 12 matrix elements accounting for all translational and rotational degrees of freedom between two nodes. The FE mesh model is shown in Fig. 13.

Figure 9 FINITE ELEMENT MODEL OF THE ENGINE Figure 12 CAD MODEL OF A HIGH SPEED PUMP

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Turbine_Bending Mode_backward

1200

Turbine_Bending Mode_forward

1000

Mid_span_ Bending_backward

800

Mid_span_Bending _forward

600

Inducer_Bending_ Mode_backwad

400

Inducer_Bending Mode_backward

Hz

1400

200

1xx

0 0

10000

20000

Figure 13 FE MODEL OF HIGH SPEED PUMP

30000 40000

50000

60000

rpm

Figure 15 CAMPBELL DIAGRAM OF ROTOR + CASING The rotor alone with bearings at locations 1 to 6 and the rest eight locations seals is shown in Fig. 14. The bearing (deep groove ball) stiffnesses are estimated at 6, 5.6, 5.6, 5.04, 5.04 and 3.27×104 N/mm for bearing numbers 1 to 6 respectively based on static bearing reactions. There are eight seals and their properties are determined using bulk flow models. The stiffness and damping values are curve fitted; typically the properties of first seal are given by

Following the unbalance response calculations in equation (1) the unbalance distribution in gm-mm is taken as given in Fig. 16. Bearing reaction forces obtained for the rotor alone case in the Y direction are given in Fig. 17. 8

5

5

4

K xxlb1  4.738452  10 -12 rpm 3  2.261960  10 -6 rpm 2 

1

2.659990  10 - 3 rpm  2.958842  10 2 N/mm

6

5

4

2 3

Figure 16 UNBALANCE VALUES AT FOUR LOCATIONS Bearing Reaction FY

2500

Bearing 1 Bearing 2 Bearing 3 Bearing 4 Bearing 5 Bearing 6

Figure 14 ROTOR WITH BEARINGS AND SEALS

Reaction Force FY (N)

2000

1500

1000

500

1400 0 15000

1200

25000

30000

35000

800

T urbine_Bending Mode_forward

600

Mid_span_ Bending_backward Mid_span_Bending _forward

400

Inducer_Bending_ Mode_backward

200

Inducer_Bending Mode_forward

0 0

10000 20000

30000 40000 50000 60000 rpm

Figure 14 CAMPBELL DIAGRAM OF ROTOR ALONE Following the process outlined earlier, the Campbell diagram for the rotor alone case is obtained and given in Fig. 14. It is observed that the first forward critical speed of the rotor is at 28000 rpm and the second forward critical speed is at 55000 rpm. The first and second backward critical speeds are observed at 20000 and 44000 rpm respectively. The Campbell diagram with the rotor + casing case is given in Fig. 15. The critical speeds dropped significantly to 23500 rpm and 39000 rpm when the casing effect is included.

40000

45000

50000

55000

60000

rpm

T urbine_Bending Mode_backward Hz

20000

1xx

1000

Figure 17 BEARING REACTIONS FOR ROTOR ALONE It is observed that peak responses occur at critical speeds as predicted in the rotor alone case Campbell diagram Fig. 14. Bearing 1 suffers maximum reaction force 2150 N at the first forward critical speed. The response in this region is not correct because the bearing stiffness is significantly affected by the response. In reality the bearing stiffness is non linear and given by k  1.289n z 0.666d 0.333F 0.333N/m where nz = number of balls in the present ball bearing = 9, d = diameter of the ball = 11.12 mm and F = Reaction force at the desired frequency = 2150 N. Substituting these values, we get k  16.363  107 N/m. This is 2.727 times more than the stiffness used before. This calls for consideration of bearing non-linearity in the analysis. Conventionally the nonlinearity of the spring is expressed as a function of displacement; here it is based on the magnitude of

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force experienced by the bearing. An iterative process can be adopted based on the present approach of working on a suitable platform to determine the response in the critical speed region to match the starting value of stiffness and the bearing reaction force with the results in the final step. First, it is assumed that the average value of starting stiffness is 6×107 N/m and the evaluated stiffness is 16.36×107 N/m; and repeat the process to arrive at applicable value of 12×107 N/m. With this stiffness value, the Campbell diagram obtained is shown in Fig. 18. It should be noted that the response in the neighborhood of the peak amplitude is governed by the stiffness estimated at the speed corresponding to peak amplitude of the force through iteration, unlike the case of a Duffing equation where the stiffness is dependent on the frequency. The first forward critical speed is now at 34,000 rpm and the second forward critical speed is above 55,000 rpm. Thus, the first forward critical speed increased from 28,000 rpm by 21.4%. It is reported that the test bed results have shown that peak response was obtained in this speed region. 1400

1xx

1200

Turbine_Bending Mode_backward

Hz

1000

Turbine_Bending Mode_forward

800

Mid_span_ Bending_backwad

600

Mid_span_Bending _forward

400

However if the frequency of excitation is increased rapidly with high acceleration or speed is increased with high angular acceleration, the system is not allowed to reach a steady value while accelerating. Before a peak value is reached the frequency gets changed and this becomes important while reaching a resonance. As the rotor accelerates and reaches resonance frequency (or speed) it senses magnification and before peak value at resonance is reached it goes past the dangerous speed. Therefore the peak value occurs at a frequency (speed) to the right of resonant condition. In a similar way the bearing nonlinearity (hard spring) shifts the peak condition to the right. The rotor acceleration has also an influence on the peak amplitude of whirl. While resonant response under quasi-steady conditions is determined by damping alone, during acceleration it cannot reach the peak value. As it moves across past the resonance there is no time for the rotor to build full response value and once it is past resonance it begins to rapidly fall. Therefore peak value of whirl reached is less than the resonant value; if we go past resonance with high acceleration then the peak value will also be lowered further with increased acceleration. In general it is reported that the amplitudes of whirl observed are much lower than values predicted by quasisteady methods. This is advantageous in preventing rubbing between blades and casing. We therefore need accurate methods of determining the response under accelerating rotor conditions. Rao [20] used Gunter’s rotor considered before with damped steady state unbalance response as shown in Fig. 19.

Inducer_Bending_ Mode_backward 0.35

200

Inducer_Bending Mode_forward

0

10000

20000

30000

40000

50000

60000

rpm

Figure 18 CAMPBELL DIAGRAM WITH NONLINEAR BEARING 1 PROPERTY ROTOR ACCELERATION Steam turbines are started gradually with stops at several speeds to allow the rotors reach thermal stability; may take hours or even days. Land based gas turbines are also started with slow acceleration to allow for rotor safety. These engines are heavy (a steam turbine rotor may weigh as much as 80 tons) and therefore their starting cycle is determined by thermomechanical considerations. Aircraft engines are started quickly, particularly defense applications and take less time to reach full speed, few seconds to a few minutes. In aerospace application turbomachinery starting time may be crucial, e.g., the total coasting time may be just 2-3 seconds to reach 50000 rpm or more. These high accelerations play significant role in rotor dynamics behavior. Classical forced vibration or rotor whirl analysis is always conducted for a steady frequency or speed excitation to study frequency response assuming quasi-steady conditions are valid. That is we allow the rotor to reach steady conditions for each excitation and then plot the results.

Amplitude - m

0.28

0

0.21

0.14

0.07

0 0

20

40

60

80

100

120

140

160

Frequency - Hz

Figure 19 QUASI-STEADY UNBALANCE RESPONSE Next, a beam based finite element model of the rotor system has been adopted. The unbalance force in equation (1) is modified to take into account angular acceleration as

1   Fx  u 0  t 2 cos  cos 0t  t 2   2   1    iu0  t 2 sin  sin  0t  t 2  2   1   F y  u 0  t 2 sin  cos 0t  t 2   2   1    iu0  t 2 cos  sin  0t  t 2  2  

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(10)

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An integration time step of 0.0005 seconds is used with acceleration rates of 60 rad/s2 as well as 600 rad/s2. A plot of the excitation expression for 60 rad/s2 is shown in Fig. 20. Amplitude - m

0.20

Excitation Force - N

400000

0.00

-0.20

0

0.0

0.5

1.0

1.5

2.0

2.5

Time - sec

Figure 22 COAST UP RESPONSE (600 rad/s2) -400000 0

2

4 Time - sec

6

8

10

Figure 20 EXCITATION FORCE The transient response obtained at 60 rad/s2 acceleration is shown in Fig. 21.

Amplitude - m

0.20

With the help of macros and the template process we can use any solver to obtain the transient response. This transient response also helps in finding whether there is instability in the system. We can study these for the space pump considered in Figs. 12, 13 and 16 with an acceleration of 1800 rad/s2. The residual unbalance during coasting up is taken 0.0032 Kg.mm applied to the turbine rotor of the pump. The pump is allowed to coast up from about 10,000 to about 30,000 rpm. Fig. 23 shows the time domain response of the rotor, while coasting up through its first critical speed of 228 Hz which occurs at around 0.795 seconds.

0.00

-0.20 0.00

2.00

4.00

6.00

8.00

10.00

Time - sec

Figure 21 COAST UP RESPONSE (60 rad/s2) Figure 23 COAST UP RESPONSE OF PUMP IN FIG. 12 WITH 1800 rad/s2 ACCELERATION

The critical speeds were found to increase for increasing acceleration rates and the amplitude of vibration was found to reduce with increase in acceleration rates. The peak value of transient response occurs at a time of 8.513 seconds and at an excitation frequency of 510.78 rad/sec (81.29 Hz). It is clearly seen that the frequency at which the peak response occurs has marginally shifted to the right with respect to the peak at the steady state response of this rotor system, which was occurring at 80.54 Hz. It may also be noted that the peak amplitude in the transient response reduced to more than half of its value from the steady state response analysis. The transient response of the rotor, with acceleration rate 600 rad/s2 is shown in Fig. 22. The peak amplitude occurs at 0.826 seconds. Here again, there is a considerable shift of the frequency at which peak amplitude occurs (87.58 Hz), with respect to the resonant frequency obtained during steady state response of the rotor system. The peak amplitude is also seen to decrease with increased acceleration.

Figure 24 MODE SHAPE SPACE PUMP

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The mode shape at peak response speed obtained is shown in Fig. 24 Land based power plants are mounted on foundations and play a significant role on rotor dynamics. A solid rotor model procedure described here can be adopted as presented by Rao [21]. The rotor system of turbines and generator are modeled as beams and the foundation as solid model connected by support stiffnesses from oil film bearings. The mode shape of such a rotor foundation system is given in Fig. 25 with the foundation participating in the vibration. Rotor-alone analysis cannot predict such frequencies.

Figure 25 TURBOGENERATOR ON SOLID FOUNDATION – COMBINED MODE SHAPE BEAM AND SOLID MODEL COMPARISON The advantages and disadvantages of the beam and solid rotor modeling techniques can be summarized as follows. Beam Model Advantages  Is simple containing discrete masses and massless beam elements  Takes less computational time  Support conditions can be included  Disk effects are included by discrete rotary inertias and implementing gyroscopics  The rotor is a stationary structure to which the speed dependent unbalance forces are applied and the whirl response quantities determined  Spin softening effect can be predominant and beam models can include this (not included in traditional modeling) and predict the backward whirl directly rather than through gyroscopics modeling alone. Backward whirl disappears at speeds beyond the stationary shaft critical speed Beam Model Disadvantages  Derivation of a beam model from actual structure is approximate and time consuming  Coupling of rotors, casings, foundations is difficult



Speed effects that come in through gyroscopics do not address stress stiffening and spin softening effects; these effects can alter the rotor dynamics characteristics significantly

Solid Model Advantages  Making the model is easier and is in tune with the modern procedure of model, mesh and analyze  No approximation in making the model and also saves considerable skilled engineer time  Support conditions are easily included  Stationary parts of the system are easily modeled as structures  There is no need to model separately gyroscopics as all the nodes have three degrees of freedom (or more) that brings in the rotational kinetic energy of disks and other attached parts to the shaft  The spin of the rotor can be included and stress stiffening effect accounted; this effect is responsible in increasing the frequency particularly when the disk and shaft modes are close and get coupled.  If the shaft and disk modes are well separated the centrifugal load or stress stiffening effect can be excluded making the model simpler  Spin softening effects are easily included as in beam models  Transient response at a steady speed or accelerating rotor conditions can be accounted  Bearing and other nonlinearities can be included by iteration methods Solid Model Disadvantages  Takes more computational time

SUMMARY In summary, we have a procedure for rotor dynamic analysis of complex rotor-casing-foundation systems with support stiffnesses and dampings connecting the rotors with stationary structure. The traditional Jeffcott beam model with gyroscopic moments of disks is getting modified to account for the needs of new engines. It is shown that any structural solver that allows to be called on a platform along with suitable preprocessors and postprocessors and macros that are developed for unbalance forces, spin softening modifications, iterative procedures for bearing nonlinearities, acceleration effects on instantaneous rotor position, velocity and unbalance forces can be suitably integrated to form a template process, e.g. the way in which TurboManager works [14] for lifing. The stationary parts of the structure can be retained as they are instead of adopting substructuring methods. Stress stiffening effects are minimal and can be included by using such a solver that allows the pre-stressing stiffness matrix to be included.

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ACKNOWLEDGMENTS The author wishes to acknowledge support from Altair Engineering India Ltd. Several rotor dynamics engineers have churned out a lot of results in solid rotor model work and earlier beam model rotor dynamics. Their dedication to the subject is appreciated and the author thanks all of them from academia and industry for their contributions.

NOMENCLATURE a a, b d F F x, F y K, k Kzz, Kyy Kyz, Kzy K M nz p r r+ rS t u v c, v s wc, ws

   s

Eccentricity Major and minor axes of elliptical orbit Ball diameter Bearing reaction Dynamic forces Stiffness Direct stiffnesses Cross-coupled stiffnesses Modified stiffness Mass Number of balls in bearing Frequency Whirl radius, Length of Spring Forward whirl radius Backward whirl radius Sommerfeld number Time Unbalance, Axial extension Cosine and Sine components of response in Y direction Cosine and Sine components of response in Z direction Angular acceleration Phase angle Eigen frequency, Whirl frequency Spin frequency

REFERENCES 1. Rankine WJM, (1869) On the Centrifugal Force of Rotating Shafts, Engineer, v. 27, p. 249 2. Föppl O, (1895) Das Problem der Lavalschen Turbinenwelle, Der Civilingenieur, vol. 4, p. 335 3. Jeffcott HH, (1919) The Lateral Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed – The Effect of Want of Balance, Phil. Mag., Series 6, vol. 37, p. 304 4. Rao JS, (1983) Rotor Dynamics, John Wiley & Sons, New Age International 1996 5. Krämer E, (1993) Dynamics of Rotors and Foundations, Springer-Verlag, Berlin

6. Nelson HD, (1980) Finite Rotating Shaft Element using Timoshenko Beam Theory, J of Mechanical Design, ASME, v. 102, p. 793 7. Rao JS, (2002) Rotor Dynamics Comes of Age, Keynote address, Sixth IFToMM International Conference Rotor Dynamics, Sydney, September 30-October 3, vol I, p. 15 8. Stephenson RW and Rouch KE, (1993) Modeling rotating shafts using axisymmetric solid finite element with matrix reduction, ASME J. of Vib. & Acoustics, vol. 115, p. 484 9. Yu J, Craggs A and Mioduchowski A (1999) Modeling of shaft orbiting with 3-D solid finite elements, Int. J. of Rotating Machinery, vol.5, p.53 10. Rao JS, Sreenivas R and Veeresh CV, (2002, 2003) Solid Rotor Dynamics, Fourteenth U.S. National Congress of Theoretical and Applied Mechanics, Blacksburg, VA, 23-28 June 2002, Advances in Vibration Engineering, Journal of Vibration Institute of India, vol. 2, No. 4, 2003, p. 305 11. Gunter, E. J., Choy, K. C. and Allaire, P. E., (1978) Modal Analysis of Turborotors using Planar Modes Theory, Journal of Franklin Institute, vol. 305, p. 221 12. Ansys Inc., (2011) Canonsburg, Pennsylvania, USA 13. Altair Inc., (2011) Troy, Michigan, USA 14. Rao JS, Narayan R and Ranjith MC, (2008) Lifing of Turbomachinery Blades – A Process Driven Approach, GT2008-50231 15. Rajan M, Nelson HD and Chen WJ, (1986) Parameters Sensitivity in the Dynamics of Rotor-Bearing Systems, J Vib. Acoustic. Stress Rel. Des., ASME, v. 108, p. 197 16. Subbaiah R, Bhat RB, Sankar TS and Rao JS, (1985) Backward Whirl in a Simple Rotor Supported on Hydrodynamic Bearings, NASA Conf. Publication 2409, p. 145 17. Surial A and Kaushal A, (2005) Dynamic Analysis of a Variable Speed Industrial Gas Turbine Engine and Drivetrain Analysis and Testing, Advances in Vibration Engineering, Journal of Vibration Institute of India, vol. 4, no. 3, p. 279 18. Rao JS, (2003) Recent Developments in Structural Design Aspects of Aircraft Engines, National Conference on Association of Machines and Mechanisms, IIT, New Delhi, 1819 December 2003 19. Rao JS, Sreenivas R and George PP, (2004) Dynamics of High Speed Cryo Pump Rotors, 8th International I Mech E Conf Vibrations in Rotating Machinery, C623/103/2004, p.467 20. Rao JS, (2006) Transient Dynamics of Solid Rotors under high angular accelerations, Advances in Vibration Engineering, Journal of Vibration Institute of India, vol. 5, No. 1, 2006, p. 25 21. Rao JS (2005) Comprehensive Approach in Analytical Design of Advanced Modern Day Machinery, Proceedings of the International Conference Advances In Structural Dynamics and its Applications, ICASDA-2005, Gandhi Institute of Technology and Management, Visakhapatnam

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