BRYAN’S EFFECT AND NONLINEAR DAMPING · BRYAN’S EFFECT AND NONLINEAR DAMPING Temple H. Fay and...
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BRYAN’S EFFECT AND NONLINEAR DAMPING Temple H. Fay and Stephan V. Joubert: Tshwane University of Technology, South Africa. [email protected] I. I NTRODUCTION If a node of vibration is marked on an annular plate (see Fig. 1) by a spot of paint and then a small inertial rotation rate occurs about the -axis, the node will be seen to move away from the spot. This effect was first observed in 1890 by G.H. Bryan [1] who calculated: Rate of rotation of the vibrating pattern = (1) Consequently, if an ideal vibratory gyroscope is fixed inside a vehicle, the rate of rotation of the vibrating pattern (observed within the vehicle) may be used to determine the inertial rotation rate, once Bryan’s factor is known [2]. Referring to Fig. 1 where we use a Fig. 1. The radial and tangential displacements and (from rest position ) respectively of a point mass in the plate. polar coordinate system, we assume that the displacements satisfy: ( ) = ( )[ () cos + () sin ] (2) ( ) = ( )[ () sin () cos ] (3) where is the circumferential wave number, () and () deter- mine the principal () and quadrature () vibration amplitudes and ( ) and ( ) are eigenfunctions associated with an eigenvalue of the vibration pattern. II.DAMPING Any anisotropic vibration damping causes a departure from the ideal distribution and therefore affects resonator dynamics. It can be shown, as was done for a sphere in [2], that the Lagrangian = ( 2 + 2 ) 0 +2 ( ) 1 2 + 2 3 (4) where 0 1 and 2 are constants determined by inde finite inte- grals of and , density and Young’s modulus We in- troduce anisotropic linear, quadratic, cubic, quartic damping with =1 2 3 4 respectively into the equations of motion using the Lagrange-Euler equations: = F (5) = / (6) where =0 1 if is odd, even respectively and the modi fied Rayleigh dissipation function is / = ( + +1 ) 2 0 ()(( + +1 + + +1 ) (7) where the light damping coef ficient () is the sum of its Fourier series zeroth, 2 and 4 harmonics: ()= 0 + 2 cos 2 + 2 sin 2 + 4 cos 4 + 4 sin 4 (8) As explained in [2], regard () as the “cosine” and () as the “sine” output of a vibratory gyroscope connected to a two- channel oscilloscope. Then the Lissajous figure produced on the oscilloscope screen will resemble the precessing, shrinking, ellipse- like orbit depicted in Fig. 2. Using the methods of [2], we obtain, Fig. 2. () is the amplitude of the principal vibration, () is the amplitude of the quadrature vibration, Θ () is the rotation angle of the mth mode vibration pattern (the precession angle), () is a phase angle and =2 where is an eigenfrequency of the vibration pattern for instance, the following good approximate equations of motion for cubic damping: 3 128 2 (6 30 +4 2 + 4 ) 3 + (2 30 4 ) 2 (9) 3 128 2 (6 30 4 2 + 4 ) 3 + (2 30 4 ) 2 (10) Θ + 3 128 2 2 0 2 2 + 2 2 2 + 0 4 2 + 2 (11) 3 64 2 2 0 2 2 + 2 2 2 + 0 4 (12) where 2 and 4 are (proportional to) the 2 and 4 har- monics respectively, while 0 2 = 1 2 2 and 0 4 = 1 2 4 with 30 30 etc. and 2 = 32 cos 2 + 32 sin 2 (13) 4 = 34 cos 4 + 34 sin 4 (14) Evidently isotropic damping ( 30 ) affects only and whereas anisotropic damping causes changes in all four variables as well as yielding different decay rates for and . Equation (11) confirms Bryan’s Equation (1) for the ideal case and shows that the precession rate Θ deviates from the ideal situation when anisotropic damping is present, as demonstrated by a numerical experiment conducted with M ATHEMATICA 0 NDSolve routine that resulted in Fig. 3 In order to eliminate anisotropic damping (but not necessarily nonlinear damping) materials with a high Q-factor must be used. Fig. 3. The blue and black curves represent the change with time of the precession angle Θ for cubic and quartic anisotropic damping respectively that show significant deviation from the ideal situation, that is, the solid purple line Θ = while the green and red wavy curves (superimposed upon the purple line) represent Θ for linear and quadratic anisotropic damping respectively. REFERENCES [1] G.H. Bryan, "On the beats in the vibrations of a revolving cylinder or bell". Proc. Cambridge Philo. Soc. 7, (1890), 101-111. [2] M.Y. Shatalov, S.V. Joubert, C.E. Coetzee and I. Fedotov, "The influence of mass imperfections on the evolution of standing waves in slowly rotating spherical bodies". JSV, 330 Ω Ω . . . . Ω . . . . . . ∆ . . ∆ ∆ ~ ~ ~ ~ ( C, S ) P A C Θ . . . . ≈ ≈ ≈ ≈ ∆ ∆ ∆ Ω ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∝ ~ ~ ~ ∆ ∆ . 0.2 0.4 0.6 0.8 1.0 t 0.002 0.001 0.001 - Θ Ω ∆ ∆ ∆ ∆ ∆ , (2011), 127-135.
Transcript of BRYAN’S EFFECT AND NONLINEAR DAMPING · BRYAN’S EFFECT AND NONLINEAR DAMPING Temple H. Fay and...
BRYAN’S EFFECT AND NONLINEAR DAMPING
Temple H. Fay and Stephan V. Joubert: Tshwane University ofTechnology, South Africa. [email protected]
I. INTRODUCTION
If a node of vibration is marked on an annular plate (see Fig. 1) by aspot of paint and then a small inertial rotation rate occurs aboutthe -axis, the node will be seen to move away from the spot. Thiseffect was first observed in 1890 by G.H. Bryan [1] who calculated:
Rate of rotation of the vibrating pattern = (1)
Consequently, if an ideal vibratory gyroscope is fixed inside a vehicle,the rate of rotation of the vibrating pattern (observed within thevehicle) may be used to determine the inertial rotation rate, onceBryan’s factor is known [2]. Referring to Fig. 1 where we use a
Fig. 1. The radial and tangential displacements and (from rest position ) respectively of a point mass in the plate.
polar coordinate system, we assume that the displacements satisfy:
( ) = () [ () cos+ () sin] (2)( ) = () [ () sin () cos] (3)
where is the circumferential wave number, () and () deter-mine the principal () and quadrature () vibration amplitudesand () and () are eigenfunctions associated with an eigenvalue of the vibration pattern.
II.DAMPING
Any anisotropic vibration damping causes a departure from the idealdistribution and therefore affects resonator dynamics. It can be shown,as was done for a sphere in [2], that the Lagrangian
= (2 + 2)0 + 2 ( )1 2 + 2
3 (4)
where 01 and 2 are constants determined by indefinite inte-grals of and , density and Young’s modulus We in-troduce anisotropic linear, quadratic, cubic, quartic damping with = 1 2 3 4 respectively into the equations of motion using theLagrange-Euler equations:
=
F (5)
=
/ (6)
where = 0 1 if is odd, even respectively and the modifiedRayleigh dissipation function is
/ =
(++1)
20
()((
++1 + ++1) (7)
where the light damping coefficient () is the sum of itsFourier series zeroth, 2 and 4 harmonics:
() = 0 + 2 cos 2+ 2 sin 2
+ 4 cos 4+ 4 sin 4 (8)
As explained in [2], regard () as the “cosine” and () asthe “sine” output of a vibratory gyroscope connected to a two-channel oscilloscope. Then the Lissajous figure produced on theoscilloscope screen will resemble the precessing, shrinking, ellipse-like orbit depicted in Fig. 2. Using the methods of [2], we obtain,
Fig. 2. () is the amplitude of the principal vibration, () is the amplitudeof the quadrature vibration, Θ() is the rotation angle of the mth modevibration pattern (the precession angle), () is a phase angle and = 2where is an eigenfrequency of the vibration pattern
for instance, the following good approximate equations of motion forcubic damping:
3128
2(630 + 4 2 + 4)
3
+(230 4)2 (9)
3128
2(630 4 2 + 4)
3
+(230 4)2
(10)
Θ + 3128
22 0
22+2
2 2 + 04
2 +2 (11)
3642
2 0
22+2
2 2 + 04
(12)
where 2 and 4 are (proportional to) the 2 and 4 har-monics respectively, while 0
2 = 12
2
and 0
4 = 12
4
with
30 30 etc. and
2= 32 cos 2 + 32 sin 2 (13)
4 = 34 cos 4 + 34 sin 4 (14)
Evidently isotropic damping (30) affects only and whereasanisotropic damping causes changes in all four variables as well asyielding different decay rates for and . Equation (11) confirmsBryan’s Equation (1) for the ideal case and shows that the precessionrate Θ deviates from the ideal situation when anisotropic dampingis present, as demonstrated by a numerical experiment conductedwith MATHEMATICA0 NDSolve routine that resulted in Fig. 3 Inorder to eliminate anisotropic damping (but not necessarily nonlineardamping) materials with a high Q-factor must be used.
Fig. 3. The blue and black curves represent the change with time of theprecession angle Θ for cubic and quartic anisotropic damping respectivelythat show significant deviation from the ideal situation, that is, the solid purpleline Θ = while the green and red wavy curves (superimposed uponthe purple line) represent Θ for linear and quadratic anisotropic dampingrespectively.
REFERENCES
[1] G.H. Bryan, "On the beats in the vibrations of a revolving cylinder orbell". Proc. Cambridge Philo. Soc. 7, (1890), 101-111.
[2] M.Y. Shatalov, S.V. Joubert, C.E. Coetzee and I. Fedotov, "The influenceof mass imperfections on the evolution of standing waves in slowlyrotating spherical bodies". JSV, 330
Ω
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.
.
.
.
.
.
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~ ~ ~
( C, S )
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0.2 0.4 0.6 0.8 1.0 t
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, (2011), 127-135.
