Gerald V. Brown

Glenn Research Center, Cleveland, Ohio

Albert F. Kascak

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Ralph H. Jansen

University of Toledo, Toledo, Ohio

Timothy P. Dever

QSS Group, Inc., Cleveland, Ohio

Kirsten P. Duffy

University of Toledo, Toledo, Ohio

Stabilizing Gyroscopic Modes in MagneticBearing-Supported Flywheels by Using

Cross-Axis Proportional Gains

NASA/TM—2006-214027

January 2006

ARL–TR–3799

AIAA–2005–5955 U.S. ARMY

RESEARCH LABORATORY
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Gerald V. Brown

Glenn Research Center, Cleveland, Ohio

Albert F. Kascak

U.S. Army Research Laboratory, Glenn Research Center, Cleveland, Ohio

Ralph H. Jansen

University of Toledo, Toledo, Ohio

Timothy P. Dever

QSS Group, Inc., Cleveland, Ohio

Kirsten P. Duffy

University of Toledo, Toledo, Ohio

Stabilizing Gyroscopic Modes in MagneticBearing-Supported Flywheels by Using

Cross-Axis Proportional Gains

NASA/TM—2006-214027

January 2006

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

Guidance, Navigation, and Control Conference and Exhibit

sponsored by the American Institute of Aeronautics and Astronautics

San Francisco, California, August 15–18, 2005

ARL–TR–3799

AIAA–2005–5955 U.S. ARMY

RESEARCH LABORATORY
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Available electronically at http://gltrs.grc.nasa.gov

This work was sponsored by the Fundamental Aeronautics

Progam at the NASA Glenn Research Center.
NASA/TM—2006-214027 1

Stabilizing Gyroscopic Modes in Magnetic-Bearing-Supported 

Flywheels by Using Cross-Axis Proportional Gains 

Gerald V. Brown 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Albert F. Kascak 

U.S. Army Research Laboratory 

Glenn Research Center 

Cleveland, Ohio 44135 

Ralph H. Jansen 

University of Toledo 

Toledo, Ohio 43606 

Timothy P. Dever 

QSS Group, Inc.

Cleveland, Ohio 44135 

Kirsten P. Duffy 

University of Toledo

Toledo, Ohio 43606 

Abstract 

For magnetic-bearing-supported high-speed rotating machines with significant gyroscopic effects, it is necessary 

to stabilize forward and backward tilt whirling modes. Instability or low damping of these modes can prevent the 

attainment of desired shaft speed. We show analytically that both modes can be stabilized by using cross-axis 

proportional gains and high- and low-pass filters in the magnetic bearing controller. Furthermore, at high shaft 

speeds, where system phase lags degrade the stability of the forward-whirl mode, a phasor advance of the control 

signal can partially counteract the phase lag. In some range of high shaft speed, the derivative gain for the tilt modes 

(essential for stability for slowly rotating shafts) can be removed entirely. We show analytically how the tilt 

eigenvalues depend on shaft speed and on various controller feedback parameters. 

Nomenclature 

f effective filter approximately representing all system filters 

gb cross-axis gain for stabilizing backward whirl 

gca general cross-axis gain 

gf cross-axis gain for stabilizing forward whirl 

Ip shaft polar moment of inertia 

It shaft transverse moment of inertia 

r complex tilt displacement 

ω0 tilt frequency for non-spinning shaft 

ωc corner frequency of assumed filters 

λ complex tilt eigenvalue 

ζ damping constant for tilt motion 

Ω adjusted non-dimensionalized shaft speed ≡ Ω’ (Ip/It)/ω0

Ω’ shaft spinning speed 

Ω0 reference shaft adjusted speed for zeta reduction 
NASA/TM—2006-214027 2

I. Introduction and Approach 

High-speed flywheels are being considered for a variety of earth and space applications that require energy 

storage or attitude control. For long life and low energy loss at very high speeds, magnetic bearing support of the 

flywheels is preferred. However, these flywheels exhibit highly gyroscopic behavior which presents magnetic 

bearing control problems that do not arise for shafts with very small polar moments of inertia. We discuss 

approaches to the control of the gyroscopic modes that avoid high control effort and are simple to implement and 

understand. 

We consider only a rigid shaft and treat it as a free body, acted upon by external forces applied by two radial 

magnetic bearings and by the internally-generated gyroscopic torques. The axial degree of freedom is assumed to be 

controlled by an axial magnetic bearing and is ignored. The negative stiffness of linearized magnetic bearings is 

usually treated as part of the plant model. But to attain a “free body” treatment of the shaft, we presume that the 

negative stiffness is first nullified by appropriate proportional gains in the magnetic bearing controller. Then the 

normal modes of the shaft are pure center-of-mass translations and pure tilts about the center of mass. Hence we can 

use a centralized, modal-control approach that treats the rotor’s tilt and center-of-mass-translation modes 

independently. Control of the center-of-mass translation modes does not present any problems, and only the tilt 

modes exhibit speed-dependent gyroscopic effects. Consequently we treat only the tilt modes in this paper. 

The motivation to use cross-axis proportional gains stems from a desire to avoid the increase in noise and control 

effort associated with same-axis derivative gains or with cross-axis derivative gains, the latter of which arise in 

gyroscopic-cancellation techniques (ref. 1). In a conventional PID magnetic bearing controller, the shaft 

displacement is differentiated to get the shaft velocity so that the actuator can oppose that velocity and hence damp 

vibration. (Obtaining the velocity by differentiation or from an observer controller is required because velocity is 

rarely actually measured.) Extra derivative gain is required to counteract control system phase lags at high 

frequencies. But because of the special whirling nature of gyroscopic modes, it is possible to obtain the tilt velocities 

without differentiation. In a pure whirl mode the shaft tilt velocity in one plane is equal (to within a sign) to the tilt 

displacement in the perpendicular plane times the mode frequency. Since both displacements are always being 

measured, it would appear that differentiation can be avoided entirely. This would be straightforward to implement 

for a single whirl direction, but the sign of the required gain is opposite for the two whirling directions. The crossaxis proportional gain that stabilizes forward-whirl destabilizes backward whirl and vice versa. 

The key to controlling both tilt modes at high shaft speeds is to utilize the considerable difference in their 

frequencies that develops at higher shaft speed. The forward-whirl mode frequency is asymptotic to (Ip/It)* Ω’, 

where Ω’ is the shaft angular speed and Ip and It are the shaft polar and transverse moments of inertia. The frequency 

of the backward-whirl mode is approximately inversely proportional to that of the forward-whirl. The mode 

frequencies can differ by more than two orders of magnitude in high-speed flywheels. We can therefore utilize two 

parallel paths through the controller which are separated by low- and high-pass filters. One path, containing a lowpass filter, stabilizes the backward-whirl tilt mode with an appropriately-signed cross-axis proportional gain. The 

other path, containing a high-pass filter and an oppositely-signed cross-axis gain, stabilizes the forward-whirl mode. 

We therefore achieve very independent action on the two whirl modes at high shaft speed. Because the physical 

gyroscopic torques are proportional to the spinning speed of the shaft, it is convenient to gain-schedule the crossaxis control terms by making them proportional to shaft speed. 

Phase lags in the closed control loop place a limit on the shaft speed at which the above simple method is 

sufficient. The forward-whirl mode frequency can become high enough that the system phase lags are appreciable. 

Hence that mode’s stability is degraded and eventually lost as shaft speed increases. (The lags come from the finite 

bandwidths of power amplifiers, position sensors, anti-alias and smoothing filters, as well as from controller 

sampling rate and latency and magnetic bearing core eddy currents.) In order to maintain stability to higher speed 

the frequency of the forward-whirl mode and the phase lags at that frequency can be measured or estimated as a 

function of shaft speed. With this information a “phasor” lead can be introduced to extend the benefits of cross-axis 

proportional gains to much higher shaft speeds. At a given shaft speed the tilt displacements in two planes can be 

expressed as a single complex “phasor” displacement. The control output can also be expressed as a complex 

phasor. The effect of the total system phase lag θ (at the forward-whirl frequency) on the complex control force is 

simply to multiply the intended force by exp (–i θ). The effect of the lag would be nullified if the control output 

command is multiplied by exp (i θ). Any smaller phasor advance that might be applied is better than none. In some 

range of high shaft speed, the derivative gain for the tilt modes (essential for stability for slowly rotating shafts) can 

be removed entirely. 
NASA/TM—2006-214027 3

II. Analysis and Results 

A. Infinite Bandwidth without Cross-Axis Gains 

Let the tilt angular displacements in the two perpendicular planes through the non-displaced shaft axis be 

designated by the real and imaginary parts of a complex variable r. Then the governing differential equation of shaft 

tilt motion, with only proportional and derivative feedback in the magnetic bearing controller and for infinite closedloop bandwidth, is (in agreement with the axes and sign conventions of refs. 2 and 3) 

'( ) 2 0 2

0 0

2 2 d r dt + iΩ I p It dr dt + ζω dr dt + ω r = (1) 

where Ω’ is the shaft angular rotation speed, r is the complex tilt displacement, ω0 is the tilt frequency for the nonrotating shaft and ζ is the tilt damping factor for the non-rotating shaft. (Note that the terms after the first two arise 

from magnetic bearing action; ω0 and ζ are determined by the proportional and derivative gains in the magnetic 

bearing control law, as well as by Ip/It.) The characteristic equation for the complex tilt eigenvalue λ’, obtained by 

assuming solutions of the form r = A exp (λ’ t), is 

( ) 2 0 2

0 0

2 λ′ + iΩ′ I p It λ′ + ζω λ′ + ω = (2) 

The imaginary part of λ’ is the modal angular frequency and the real part is the absolute growth rate (negative of 

damping) for the mode. It is convenient to non-dimensionalize by dividing each term of equation (2) by 2 ω0 , 

yielding 

2 1 0 2λ + iΩλ + ζλ + = (3) 

where λ = λ’/ω0 and Ω ≡ Ω’ (Ip/It)/ω0. Thus all frequencies are measured in units of the non-spinning tilt frequency 

ω0, which is set by the proportional gain in the magnetic bearing controller. The negative of the real part of the 

eigenvalue, divided by ω0, is analogous to the damping factor ζ. The frequency ω0 may typically be on the order of 

2π times 50 Hz, as was roughly the case for the flywheel in reference 4. Equation (1) and subsequent characteristic 

equations were solved numerically for the eigenvalues by using commercial software. The two nondimensional 

eigenvalues, λ+ with positive frequency (forward whirl) and λ– with negative frequency (backward whirl) are 

plotted versus Ω in the appendix, along with other plots that show some interesting symmetries in the whirl 

frequencies and damping. It is shown there that a plot of log (Im (λ+)) and log (|Im (λ–)|) has a symmetry not 

revealed by a linear plot. Furthermore the loci of λ+ and λ– in the complex plane with Im (λ+) and |Im (λ–)| plotted 

on a log scale show more clearly how the eigenvalues are affected by the control law modifications discussed below. 

Such a plot of the solutions of equation (3) is shown in figure 1 for ζ = 0.05. In all eigenvalue plots we are plotting 

the real and imaginary parts of λ, a non-dimensional quantity. The axes of the plots should strictly be labeled “NonDimensional Growth Rate” and “Non-Dimensional Frequency,” but we use “Growth Rate” and “Frequency” for 

simplicity. Note that the values of Ω run up to 100, which, for ω0 = 2π*50 Hz and Ip/It = 0.8, corresponds to a speed 

of 375,000 rpm, or for Ip/It = 1.2 to a speed of 250,000 rpm. Few flywheels run to such speeds; the large range is 

chosen to clearly show the asymptotic behavior. (Points are plotted for Ω = (0, 0.333, and 0.666) and for 

Ω = 100(n/10), n = 0, 1, ..., 20, in order to give roughly even point spacing on the log scale.) Note that the eigenvalue 

for Ω = 0 is ~ (–ζ, 1), the point plotted in green. As shaft speed increases from zero, the non-dimensional forwardwhirl frequency increases from 1 and the backward decreases from the same value. (We henceforth speak of the 

backward-whirl frequency as a positive number.) As noted in the appendix, the forward-whirl eigenvalues appear to 

have higher damping than the backward. However, this is true only in the sense that their growth rate has a more 

negative value, reflecting their higher frequencies. When the real part of the eigenvalues are divided by the 

frequency (absolute value of the imaginary part), the two branches of the eigenvalue plot reveal the same damping 

ratio (or damping factor, loss factor or percent of critical damping). 
NASA/TM—2006-214027 4

0.01

0.1

1

10

100

-0.15 -0.10 -0.05 0.00 0.05

Growth Rate

Frequency

Forward Whirl

Non- Spinning

Backward Whirl

Figure 1.—Non-dimensional eigenvalue loci for ζ = 0.05 and Ω values from 

0 to 100. The non-spinning eigenvalue (for Ω = 0) is shown in green. 

Forward-whirl eigenvalues are shown in red and backward-whirl eigenvalues 

in blue. Ω = 0, 0.333, and 0.666 and 100(n/10), for n = 0, 1, ..., 20.

B. Infinite Bandwidth with Cross-Axis Gains 

Now we introduce a cross-axis gain gca into the controller so that the differential equation for shaft tilt motion 

becomes 

' 2 '( ) 0 2

0 0

2 2 d r dt + iΩ I p It dr dt + ζω dr dt + ω r + iΩ I p It gca r = (4) 

where we have inserted the cross-axis term with the same sign as the gyroscopic term. We have also chosen to 

explicitly insert the factor Ω’ (Ip/It) in the cross-axis term, since the gyroscopic term contains that same factor. The 

characteristic equation becomes (after non-dimensionalizing, as before) 

2 1 0 2 λ + iΩλ + ζλ + + iΩgca = (5) 

The resulting eigenvalues for ζ = 0.05, Ω = 100 (n/10), n = 0, 1, ..., 10 and for gca = –0.025, 0.0, and 0.025 are 

shown in figure 2. One can see that a positive value of gca improves backward-whirl stability but degrades forwardwhirl stability. Conversely a negative value of gca improves forward-whirl stability but degrades backward-whirl 

stability. Note that gca = ζ gives uniform values of growth rate (= –ζ) for all Ω. 

To improve the stability of both modes, we can provide two cross-axis terms, corresponding to two separate 

parallel paths through the controller. In one path through the controller we place a forward-whirl cross-axis gain, gf, 

an explicit negative sign, and a first-order high-pass filter. In a second path we place a backward-whirl cross-axis 

gain, gb, an explicit positive sign, and a first-order low-pass filter. We choose the non-spinning tilt frequency as the 

corner frequency of both filters. The characteristic equation in this case becomes 

2 1 (1 ) (1 ) 0 2 λ + iΩλ + ζλ + + iΩgb + λ − iΩg f λ + λ = (6) 

where gf, a positive number, is the gain that improves forward-whirl stability and gb, also a positive number, is the 

gain that improves backward-whirl stability. (We have changed the sign of the forward-whirl stabilizing term to 

allow both gains to have positive values.) The resulting eigenvalues for gf = gb = 0.025, ζ = 0.05, Ω = 100(n/10), 

n = 0, 1, ..., 10, and Ω = 0, 0.333, and 0.666 are shown in figure 3 along with the eigenvalues for gf = gb = 0. Note 

that in the high-speed regions the damping of each mode has been improved by the value of the cross-axis gain. 

There are other branches of eigenvalues (introduced by the filters) with considerably higher damping. We do not 

plot those. For more extreme selections of the various gains, these branches can become less stable and must be 

reckoned with. It is seen that with low- and high-pass filters, the stability of both forward-whirl and backward-whirl 

modes can be improved, as indicated by the filled symbols in figure 3. 
NASA/TM—2006-214027 5

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05

Growth Rate

Frequency

gca = - 0.025

gca = 0.0

gca = 0.05

gca = 0.025

Non- spinning

Figure 2.—Effect of Cross-Axis Gain. Positive gain improves stability of 

backward whirl but reduces stability of forward-whirl. ζ = 0.05. Ω = 0, 0.333, 

0.666 and 100(n/10), for n = 0, 1, ..., 10. 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05

Growth Rate

Frequency

N. S. (no c.a.)

F. W. (no c.a.)

B. W. (no c.a.)

F. W. (c.a. & h.p. filter)

B. W. (c.a. & l.p. filter)

Figure 3.—Effect of High- and Low-Passed Cross-Axis Proportional Gains. 

Open points are for cross axis (c.a.) gains equal to 0. Filled points are for 

gf = gb = 0.025 and ζ = 0.05. 

C. Effect of Finite Bandwidth with Filtered Cross-Axis Gains 

The appendix lists a number of sources of delays and phase lags that are typically found in a digitally controlled 

magnetic bearing system. We now consider the effect of these destabilizing influences. To qualitatively explore 

these influences, we arbitrarily pick 5 first-order filters, in series in the closed control loop, and all with the same 

corner angular frequency, ωc, measured in units of ω0. Thus every term in the controller is considered to be 

multiplied by the following factor, f: 

( )

5 1 1 c f ≡ + λ ω (7) 

The characteristic equation becomes 
NASA/TM—2006-214027 6

[ ] 1 2 (1 ) (1 ) 0 2 λ + iΩλ + f + ζλ + iΩgb + λ − iΩg f λ + λ = (8) 

Some representative solutions for various values of ωc are shown in figure 4. We now plot values of Ω ranging only 

up to 40, which, if the non-spinning tilt mode is set at 50 Hz, would represent a shaft speed of 150,000 rpm for 

Ip/It = 0.8 or 100,000 rpm for Ip/It = 1.2. The points for ωc = 800,000 (effectively an infinite bandwidth, since ωc is 

measured in units of ω0) are for the same gains as the filtered cases plotted in figure 3. The value of ωc = 80 yields a 

total phase lag around the closed loop of about 70° at 1 kHz (for our usual example with ω0 = 314 rad/sec (50 Hz)). 

This is close to the phase lag at 1 kHz in the actual system described in reference 4. Note that the forward-whirl 

mode becomes unstable at about Ω = 20, or 75,000 rpm for Ip/It = 0.8 or 50,000 rpm for Ip/It = 1.2. Note that even 

the damping of the non-spinning tilt mode is reduced by more than half by the filters. 

Examine now to what extent increasing the tilt damping constant ζ can improve the stability of the forward-whirl 

mode. In figure 5, we show the eigenvalues for 3 values of ζ, namely ζ = 0.025, 0.05, and 0.075. We see that higher 

tilt damping ζ improves the forward-whirl stability until the frequency of the mode is somewhat over 20. Then more 

tilt damping is actually destabilizing. 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

ωc = 800000

ωc = 200

ωc = 80

Figure 4.—Effect of Finite Bandwidth on Eigenvalues. Ω = 0.333, 0.666 

and 40(n/10), n = 0, 1, ..., 10. gf = gb = 0.025, ζ = 0.05. 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

zeta = 0.1

zeta = 0.075

zeta = 0.05

Figure 5.—Effect of zeta on forward-whirl eigenvalues at high speed. Higher zeta 

improves forward stability up to about Ω = 20. gf = gb = 0.025. ωc = 80. 
NASA/TM—2006-214027 7

Consider now whether higher forward-whirl cross-axis proportional gain can help. Eigenvalues are shown in 

figure 6 for ζ = 0.05 and gb = 0.025 for three values of gf, namely (0.025, 0.05, and 0.075). Higher values of gf

improve the stability of the forward whirl mode for lower speeds, but the same “barrier” appears at about the same 

frequency. The reason for this becomes clear from a look at the phase lag of the filter function f as a function of 

frequency, plotted in figure 7. We see that at a non-dimensional frequency of about 26, the phase lag is 90°. 

Damping can advance phase by at most 90°, so when the non-dimensional forward-whirl frequency reaches 26, the 

beneficial phase advance desired from damping is entirely obliterated by the system phase lag. But we can 

counteract the phase lags beyond this point with a phasor advance, as we now show. 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

gfw = 0.075

gfw = 0.05

gfw = 0.025

Figure 6.—Effect of forward-whirl cross-axis gain on eigenvalues with 

significant phase lags present. gb = 0.025, ζ = 0.025, ωc = 80. 

0

30

60

90

120

0 10 20 30 40

Frequency

Phase Lag (deg)

ωc = 80, five 1st order

Figure 7.—Phase lag in closed-loop as a function of frequency. Product of five 

first order filters with corner non-dimensional frequencies of 80. 
NASA/TM—2006-214027 8

D. Effect of Phasor Advance with Finite Bandwidth and Filtered Cross-Axis Gains 

Now consider trying to counteract the phase lags discussed in the last section. If we multiplied the cross-axis term 

in the controller that contains the high-pass filter by a phasor, p, of unit magnitude, defined by 

{ [ ( ) ]} npoles c p ≡ exp − iArg 1 1− iΩ ω (9) 

where npoles is the order of the filter, then we would expect to approximately counteract the system phase lags. An 

adequate approximation of equation (9) for high shaft speed is 

p [ i( )( )npoles] ≈ exp − π 4 Ω ωc (10) 

For Ω >> 1 this gives an approximation of the phase lags caused by the function f. That is, at high shaft speed, for 

which the forward-whirl frequency is asymptotic to Ω, this approximately nullifies the phase lags introduced by the 

function f, at least as far as the forward-whirl cross-axis feedback term is concerned. The characteristic equation then 

becomes 

[ ] 1 2 (1 ) (1 ) 0 2 λ + iΩλ + f + ζλ + iΩgb + λ − ipΩg f λ + λ = (11) 

An eigenvalue plot for this case is shown in figure 8, along with the eigenvalues for no phasor correction. A plot of 

just the damping of the forward-whirl mode as a function of Ω is shown in figure 9. Inspection of the figures 8 and 9 

reveals that applying the phase correction gives little change in stability for Ω less than 10, but an increasing 

improvement at higher speed, raising the instability speed from 22 to 35 (which, for our example with the nonspinning tilt frequency set at 50 Hz, would be from 82,500 rpm to 131,250 rpm for Ip/It = 0.8 or from 55,000 rpm to 

87,500 rpm for Ip/It = 1.2). However, as shown in figure 10, the magnitude of the filter f falls substantially below 1 

as Ω approaches 40. Therefore, a magnitude correction for the effect of the filter function f is warranted. We 

therefore replace equation (10) by 

p M ( ) i( )( )npoles ≡ exp − π 4 Ω ωc (12) 

where 

( ( ) ( ) ) npoles c M ≡ abs 1 1+ i Ω ω (13) 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

no phasor

unit magnitude phasor

full phasor

N. S.

full phasor on both

Figure 8.—Effect of phasor corrections for system filters. gf = 0.075, gb = 0.5, 

ζ = 0.05, ωc = 80. Phasor applied to forward-whirl term only—red circles. Full 

phasor applied to both forward-whirl and damping term—blue triangles. 
NASA/TM—2006-214027 9

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 10 20 30 40

Shaft Non-Dimensional Speed, Ω

Damping

no phasor

unit-magnitude phasor

full phasor

N. S.

full phasor on both terms

Figure 9.—Damping as a function of shaft non-dimensional speed for various 

phasor corrections. gf = 0.075, gb = 0.5, ζ = 0.05, ωc = 80. Phasor applied to 

damping term only—red symbols. Squares—no phasor. Open circles—unitmagnitude phasor. Closed circles—full phasor. Blue triangles—full phasor 

applied to both forward-whirl and damping terms. 

0.1

1

1 10 100

Shaft Non-Dimensional Speed Ω

Phasor Magnitude

Figure 10.—Magnitude of assumed lumped filters in magnetic bearing 

closed loop. The magnitude M of the phasor is plotted. ωc = 80. 

In figures 8 and 9, it can be seen that the magnitude correction provides an increasing benefit with speed above 

about Ω = 6. Next consider the effect of applying the phasor correction to the term in equation (11) that contains zeta 

(which came from the first derivative term in the differential equation), rather than to the forward-whirl term. The 

characteristic equation then becomes 

(1 2 (1 ) (1 )) 0 2 λ + iΩλ + f + pζλ + iΩgb + λ − iΩg f λ + λ = (14) 
NASA/TM—2006-214027 10

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

no phasor

unit- magnitude phasor

full phasor

N. S.

full phasor on both terms

Figure 11.—Eigenvalues with phasor correction applied to derivative term and to 

both derivative and forward-whirl terms. gf = 0.075, gb = 0.5, ζ = 0.05, ωc = 80. 

The eigenvalues for this case are shown in figure 11 for the case where M is given by equation (13) (full phasor) and 

the case where M is set equal to 1 (phase correction only). Comparison of figures 10 and 11 indicates that a greater 

stability benefit is obtained by correcting the phase lag of the derivative term than by correcting the phase lag of the 

forward-whirl term, at least for the values of the other parameters used here. Correcting both has a still larger 

benefit, as is shown in figures 10 and 11 by the points plotted in blue. However, it should be remembered that the 

use of the derivative term entails noise amplification. 

E. Removal of Derivative Feedback at High Shaft Speed 

In figure 12 we plot the forward whirl eigenvalues (in open blue triangles, as in figs. 8 and 11, and using the same 

data as in those figures) that result from applying the full phasor correction, equations (12) and (13), to both the 

forward-whirl feedback term and the damping term, and the eigenvalues (in open orange symbols) with the full 

phasor correction applied to the forward-whirl term but with the damping constant ζ set equal to zero. It can be seen 

that above a non-dimensional speed of 3, the system is stable with ζ = 0. Thus we can expect that a transition from 

the open orange eigenvalues at low speed to the open blue eigenvalues at high speed would yield the benefit of 

stability at all speeds, but low noise generation at high speed where the system may be more vulnerable. As one 

illustration of this approach we choose to gain schedule the value of ζ in the following manner: 

( ( ) ) n ζ = ζ 0 1+ Ω Ω0 (15) 

where ζ0 is the initial value of ζ, Ω0 sets the shaft speed around which the transition to ζ = 0 takes place, and the 

exponent n governs the rapidity of the transition. As a specific example, plotted with filled red symbols in figure 12, 

we choose ζ0 = 0.05, Ω0 = 8, and n = 4. The transition between the limiting loci in blue and orange is clearly seen. 
NASA/TM—2006-214027 11

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05 0.1

Growth Rate

Frequency

Phasor on f.w. & zeta

Zeta = 0

Zeta zeroed with speed

N. S. zeta = 0

N. S. zeta ≠ 0

Figure 12.—Removal of damping feedback at high speed. Blue triangles—full 

phasor advance applied to both forward-whirl and damping terms with gf = 0.075, 

gb = 0.5, ωc = 8, ζ = 0.05; orange circles—gf = 0.075, gb = 0.5, ωc = 80, ζ = 0; Red 

circles—transition from ζ = 0.05 to ζ = 0.0 according to equation (15).

III. Experimental Use of Methods 

The effectiveness of cross-axis proportional gains was experimentally demonstrated (ref. 4) on the “D1” energy 

storage flywheel unit (with Ip/It = 0.8) at Glenn at speeds up to 60,000 rpm and on predecessor flywheels. The 

qualitative effect of a single unfiltered cross-axis gain on forward-and backward whirl stability was easily seen by 

measuring the spectral density versus frequency of the position sensor signals. When the controller was of the type 

described by equations (4) and (5), one sign of gca raised the amplitude of the forward whirl peak in the spectral 

density and reduced the backward whirl peak. The opposite sign of gca had the opposite effect. The sharpening of 

each peak in turn was useful for accurate frequency measurements of the whirl modes. 

Filtered cross-axis gains, described by equation (6), had the predicted qualitative effect of reducing the spectral 

density peaks of both the forward and backward whirl modes. 

The phase advance method described by equations (10) to (14) was applied in a limited sense. Rather than 

attempting to use the full angular advance of equation (10), smaller angular advances were applied to the forward 

whirl controller term and/or to the controller’s damping term while the forward-whirl peak in the sensor spectral 

density was monitored. The advance was beneficial, but only enough advance was applied to render the peak size 

non-threatening. The magnitude correction of equations (12) and (13) was not attempted. The elimination of the 

damping term described at high shaft speed described by equation (15) has not been implemented. The reason for 

caution in applying these latter methods stems from the omission of flexible modes from the present analysis. The 

inclusion of these modes is beyond the intended scope of our analysis. However the flywheel of reference 4 had a 

flexible mode which was actually crossed by the synchronous shaft speed near the maximum design speed. Thus we 

could not expect the present analysis to adequately describe the system stability in that speed range. 

IV. Concluding Remarks 

We have approached the control of gyroscopic modes of a high-speed flywheel from the standpoint of classical 

control theory. We have assumed the use of a centralized modal controller that allows us to work with the tilt modes 

only, excluding motions not affected by gyroscopic influences. We have excluded flexible modes from the 

discussion. We have desired to get as simple and physical an understanding of the modal behavior and the effects of 

the various control approaches as possible. We have been motivated to produce a control strategy that generates low 

noise in the closed loop. A modern control approach could achieve this as well, but with less transparency. 
NASA/TM—2006-214027 12

Stability of both forward- and backward-whirl modes was shown to be improved by the use of cross-axis 

proportional gains, aided by high- and low-pass filters that permit the controller to affect the modes independently 

when the rotor speed is high. The forward-whirl mode stability was seen to be degraded at high rotor speed by 

system phase lags, because the forward-whirl frequency increases indefinitely with rotor speed. It was shown that 

the severity of the stability reduction can be reduced by applying a phasor advance in the controller to counteract the 

system phase lags. Lastly it was shown that the tilt damping feedback in the controller could be entirely removed at 

a sufficiently high shaft speed. 
NASA/TM—2006-214027 13

-5

0

5

10

15

20

25

30

35

40

-0.15 -0.10 -0.05 0.00 0.05

Growth Rate

Frequency

Forward Whirl

Non- Spinning

Backward Whirl

Figure A1.—Eigenvalues of equation (3), plotted 

without any alteration. Ω = 0, 0.333, 0.666 

and 100(n/20), n = 0, 1, ..., 16. 

0

20

40

60

80

100

-0.15 -0.10 -0.05 0.00 0.05

Growth Rate

Frequency

Forward Whirl

Non- Spinning

Backward Whirl

Figure A2.—Eigenvalues of equation (3), plotted using 

absolute values of imaginary parts. Ω = (0, 0.333, 

0.666) and 100(n/20), n = 0, 1, ..., 20. 

0.01

0.1

1

10

100

-0.15 -0.1 -0.05 0 0.05

Growth Rate

Frequency

Forward Whirl

Non-Spinning

Backward Whirl

Figure A3.—Eigenvalues of equation (3), with absolute 

values of frequencies plotted on log scale. Ω = (0, 

0.333, 0.666) and 100(n/20), n = 0, 1, ..., 20.

Appendix 

A. Relation between Forward- and 

Backward-Whirl Eigenvalues 

Consider a magnetically supported flywheel with 

infinite bandwidth in its closed-loop control system. 

The eigenvalues of the system are the solutions of 

equation (3). The imaginary parts of the solutions to 

equation (3) for Ω ≠ 0 have two signs. A plot of these 

eigenvalues on linear scales is shown in figure A1. The 

limits of the two solutions for Ω → 0 are (–ζ±i), plotted 

in figure A1 as the two green points. (Note that ζ = 0.05 

for this case.) As Ω increases, the forward-whirl 

frequency increases and the backward-whirl frequency, 

which has a negative value, decreases in magnitude. A 

plot of the eigenvalues with the frequency replaced by 

its absolute value is shown in figure A2, which gives a 

slightly better picture of modes developing from a 

common single non-spinning frequency. But the most 

illuminating plot of the eigenvalues is one in which the 

absolute value of the frequency is plotted on a 

logarithmic scale, as in figure A3. This reveals a 

symmetry that might have been unsuspected. The 

product of the absolute values of the frequencies is 

approximately constant and the sum of the absolute 

damping values (negative of the growth rates) is fixed. 

The absolute values of the frequencies of the two 

modes are plotted as a function of the spinning speed in 

figure A4. Now consider figure A3 again. It appears 

from the figure that the forward-whirl modes are better 

damped than the backward. This is true in the sense that 

they damp out faster in time; the absolute damping is 

higher. However, if the real part of each eigenvalue in 

figure 3 is divided by the absolute value of the 

imaginary part to obtain the fraction of critical 

damping, then the curve of figure A5 is obtained. See 

that the two modes are equally damped as a fraction of 

critical damping, that is, the amplitudes of forward and 

backward-whirl oscillations, initiated by some 

disturbance, will decay to a given fraction of their 

initial amplitudes in the same number of cycles. 

However, the elapsed time for decay to a given fraction 

of initial amplitude is shorter for forward whirl than for 

backward whirl. 

B. Sources of Phase Lags and Delays in 

Magnetic-Bearing-Supported Flywheel Systems 

Every component in a closed loop magnetic bearing 

system has a bandwidth which limits the highfrequency response of the system and introduces phase 

lags. These phase lags may not severely affect rigidrotor backward whirl because of the low frequency of 

those modes. However, the frequency of the forwardwhirl mode is asymptotic to (Ip/It) * Ω’, and therefore 

may be on the order of a kilohertz for high speed 
NASA/TM—2006-214027 14

0.001

0.01

0.1

1

10

100

1000

0 20 40 60 80 100

Non-Dimensional Shaft Speed

Frequency

Forward Whirl

Non-spinning

Backward Whirl

Figure A4.—Frequencies of forward- and backward-whirl as 

functions of the non-dimensional spinning speed. 

0.01

0.1

1

10

100

-0.1 -0.05 0 0.05

Growth Rate

Frequency

Forward Whirl

Non-Spinning

Backward Whirl

Figure A5.—Eigenvalues with real parts plotted in terms of 

fraction of critical damping. Ω = (0, 0.333, 0.666) 

and 100 (n/20), n = 0, 1, ..., 100. 

energy storage flywheels. Phase lags of several 

components can be appreciable at such a frequency 

and lead to forward-whirl instability. The sources of 

phase lag or delay include the following: 

(1) The lowest bandwidth in a magnetic bearing 

system is often that of the power amplifiers, which are 

commonly configured as transconductance amplifiers 

(voltage command input, current output), in 

combination with their inductive load. This bandwidth 

is usually only one or two kilohertz or less. We pick 

2 kHz for definiteness here, with a phase lag of the 

order of 22° at 1 kHz. 

(2) A digital controller implemented on a digital 

signal processor (DSP) can introduce serious phase 

lags. One of these (sometimes worse than the power 

amplifiers) comes from the anti-aliasing input filter in 

front of the processor that runs the control code. If the 

sample rate is 25,000/sec, the filter is usually set to 

cut off frequencies above the Nyquist frequency of 

12.5 kHz. The filter may be 6th order and produce a 

phase lag of the order of 30° at 1 kHz. 

(3) A digital signal processor board can also have a 

smoothing filter on the output that produces as much 

phase lag as the input filter. We suppose for this paper 

that there is only a first order filter at the Nyquist 

frequency of 12.5 kHz, giving a phase lag at 1 kHz of 

about 4°. 

(4) The sampling itself introduces a low-pass 

filtering effect with a corner at the Nyquist frequency. 

This gives about 5° at 1 kHz for a 25 kHz sampling 

rate. 

(5) Another lag, distinct from the sampling lag, is 

the latency delay in the controller. A signal introduced 

at the input (after the anti-aliasing filter) and sent directly to the output (before the smoothing filter) arrives there 

after some elapsed time, which is often on the order of the time between samples, but can be worse by the amounts 

of input and output settling times. This might add another 5° of lag at 1 kHz. 

(6) Another lag is due to eddy currents in the magnetic bearings. The magnetic field in the working gap lags 

behind the current in the windings by an amount that depends on the size of the bearing poles and the thickness of 

the laminations (mainly the latter). It can contribute 10 to 20° phase lag at 1 kHz.

(7) The last obvious component that can introduce phase lag is the position sensor. In most cases the bandwidth of 

a sensor is 20 kHz or better, yielding no more than a degree or two phase lag at 1 kHz. 

To bring out general principles in this non-dimensional study without having to be specific on these phase lags, 

we lump all the filtering effects into a simple product of first order filters. We use a product of five first order filters, 

all with the same corner frequency, and measure that frequency in units of ω0, as is done with all frequencies. 
NASA/TM—2006-214027 15

TABLE I.—EXAMPLE PHASE LAGS IN A 

MAGNETIC-BEARING-SUPPORTED FLYWHEEL SYSTEM. 

Delay Source Phase Lag @ 1 kHz 

Power amplifiers (1st order @ 2 kHz) 22 

Anti-alias filter (6th order @ 12.5 kHz) 24 

Smoothing filter (1st order @ 12.5 kHz) 4 

Sampling @ 25 kHz 5 

Latency @ 25 kHz 5 

Magnetic core eddy currents 10 

Position sensors 1 

Total phase lag @ 1 kHz 71 

For a 25 kHz sampling rate, the above listed phase lags add up to 71° at 1 kHz. A product of five first-order filters 

will produce approximately that phase lag if the corner frequency ωc is set at about 4 kHz. For a 60,000 rpm 

(1000 Hz) flywheel this is 4 times the spinning frequency. In terms of an assumed non-spinning tilt frequency of 

50 Hz, the value of ωc would be 80. In the main text a number of results are presented for ωc = 80. 

References 

1. Ahrens, M., Kucera, L. and Larsonneur, R.: “Performance of a Magnetically Suspended Flywheel Energy 

Storage Device,” IEEE Transactions on Control Systems Technology, vol. 4, no. 5, September 1996. 

2. Vance, J.M., Rotordynamics of Turbomachinery, John Wiley & Sons, New York, 1988, pg. 125. 

3. Kascak, A.F., “Stability Limits of a PD Controller for a Flywheel Supported on Rigid Rotor and 

Magnetic Bearings,” AIAA Guidance, Navigation and Controls Conference, San Francisco, CA, 2005, 

AIAA–2005–5956. 

4. Dever, T.P., Brown, G.V., Duffy, K.P., and Jansen, R.H.: “Modeling and Development of a Magnetic Bearing 

Controller for a High Speed Flywheel System,” Second International Energy Conversion Engineering 

Conference, Providence, RI, Aug. 16–19, 2004, paper no. AIAA–2004–5626. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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ARL–TR–3799

AIAA–2005–5955

E–15382

21

Stabilizing Gyroscopic Modes in Magnetic-Bearing-Supported Flywheels

by Using Cross-Axis Proportional Gains

Gerald V. Brown, Albert F. Kascak, Ralph H. Jansen, Timothy P. Dever,

and Kirsten P. Duffy

Flywheels; Magnetic bearings; Gyroscopic; Energy storage

Unclassified -Unlimited

Subject Category: 37

WBS 561581.02.08.03

1L161102AF20

Prepared for the Guidance, Navigation, and Control Conference and Exhibit sponsored by the American Institute of

Aeronautics and Astronautics, San Francisco, California, August 15–18, 2005. Gerald V. Brown, NASA Glenn Research

Center; Albert F. Kascak, U.S. Army Research Laboratory, NASA Glenn Research Center; Ralph H. Jansen and Kirsten P.

Duffy, University of Toledo, 2801 W. Bancroft Street, Toledo, Ohio 43606; and Timothy P. Dever, QSS Group, Inc., 21000

Brookpark Road, Cleveland, Ohio 44135. Responsible person, Gerald V. Brown, organization code RSS, 216–433–6047.

For magnetic-bearing-supported high-speed rotating machines with significant gyroscopic effects, it is necessary to

stabilize forward and backward tilt whirling modes. Instability or low damping of these modes can prevent the attainment

of desired shaft speed. We show analytically that both modes can be stabilized by using cross-axis proportional gains and

high- and low- pass filters in the magnetic bearing controller. Furthermore, at high shaft speeds, where system phase lags

degrade the stability of the forward-whirl mode, a phasor advance of the control signal can partially counteract the phase

lag. In some range of high shaft speed, the derivative gain for the tilt modes (essential for stability for slowly rotating

shafts) can be removed entirely. We show analytically how the tilt eigenvalues depend on shaft speed and on various

controller feedback parameters.