International Conference
20th EURO Mini Conference
“Continuous Optimization and Knowledge-Based Technologies” (EurOPT-2008) May 20–23, 2008, Neringa, LITHUANIA
ISBN 978-9955-28-283-9 L. Sakalauskas, G.W. Weber and E. K. Zavadskas (Eds.): EUROPT-2008 Selected papers. Vilnius, 2008, pp. 93–98
© Institute of Mathematics and Informatics, 2008 © Vilnius Gediminas Technical University, 2008
MATHEMATICAL MODELING OF DYNAMICS OF AIR BLOWER ROTOR Marius Vasylius1, Jolanta Janutėnienė2, Jurgita Grigonienė3
Klaipeda University, Department of Mechanical Engineering, Bijunų g. 17, LT-91225 Klaipeda, Lithuania E-mail: 1
[email protected], 2
[email protected],
[email protected] Abstract: In this paper vibration of a high power air blower machine which rotating on journal bearings was estimated, modelled and simulated and experimentally tested in situ. The influence of dynamic stiffness of bearing support and gyroscopic effect of air blower rotor vibrations are analyzed. Theoretical modelling of rotor dynamics and experimental investigations in this article are presented. Theoretical and experimental research’s results are compared. The authors designed the air blower rotor‘s dynamical model with four degrees of freedom. The equations of rotor motion by using the second type of Lagrange equations are constructed. These equations are investigated by numerical methods with software Matlab Simulink. Keywords: differential equation, gyroscopic effect, rotor dynamics, stability, vibration.
1.
Introduction
2.
Air blower dynamic model
One of most occurring rotoring system exploitations problems is damaging vibrations. The general causes of these vibrations are unbalance, insufficient dynamic stiffness, rotating elements wear and etc. Gyroscopic effect has influence on high vibration. Vibratory wheel gyroscopes have been studied extensively in the recent years. The design considerations, operation principle, mathematical model governing dynamic characteristics and simulation for a novel dual-axis sensing decoupled vibratory wheel gyroscope were presented in the work [5]. Modeling framework is presented for the development of the frequency equation of a microgyroscope, which is modeled as a suspended cantilever beam with a tip mass under general base excitation in the work [6]. Technological critical machinery is expensive. They can not be reserved. The stoppage or breakdown of these machines are destructive to all technological process and associated with big economical losses. That is the main reason to estimate the technical condition of the machinery by using technical condition monitoring, protection and failure diagnostics and particular unexpected breakdown preventions systems [2–4]. The purpose of this paper is to design the mathematical model of air blower rotor’s system, which describes the vibrations of the system and to identify the reasons of the vibrations: dynamic stiffness of bearing support, the rotor’s systems resonance frequency and identify the influence of gyroscopic effect of rotor vibration. The air blower machine SF01-18 comprises electric motor EM, gear box GB and blower rotor BR, as shown in Fig. 1. All rotors are rotating in the hydrodynamics tilting-pad journal bearings.
Fig. 1. The air blower and proximity probes location scheme: 1, 2 – journal bearings of EM; 3-6 – journal bearings of gearbox (GB) 7, 8 – tilting-pad journal bearings of blower rotor (BR); C1, C2 – flexible couplings; 1X, 1Y – proximity probes fixed at the 7th bearings.
93
M. Vasylius, J. Janutėnienė, J. Grigonienė
3.
The mathematical model
The purpose of the mathematical modeling is to confirm a gyroscopic effect hypothesis and to estimate the influence of dynamics forces to bearings durability. The second type of Lagrange equations to construct the equations of system vibration for complicated vibration system is recommended to use. These equations are applied for air blower vibration system:
d ∂T dt ∂qi
∂T ∂R ∂V + + = Fi ( t ) (i = 1,2,..., n ) ; (1) − ∂qi ∂qi ∂qi Where T, V are kinetic and potential energies respectively, R is Rayleigh dissipative function, qi are system generalized coordinates, Fi(t) are external forces. The dynamic model of all system in quiet state is showed in Fig. 2.
Fig. 2. Dynamic model of the air blower rotor: l = 0.844 m, a = 0.5 m
Generalized coordinates are taken: ξ, η - coordinates, defining position of point O2 in the plane O1X1Z1 ϕy, ϕz – coordinates, defining small angles about axes Y2, Z2.
{
qi = η, ζ, ϕ y , ϕ z
}
T
Fig. 3. The scheme of coordinates and angular velocity vectors deployment which describe the rotor position
94
(2)
MATHEMATICAL MODELING OF DYNAMICS OF AIR BLOWER ROTOR
There are used three types of coordinate systems of axes to localize rotor position: 1) Fixed coordinate system O1 X1 Y1 Z1. Point O1 in steady state coincident with point where shaft is attached to rotor. Weight of rotor is not evaluated. 2) In the coordinate system O2 X2 Y2 Z2 point O2 is tightly connected with the rotor. When rotor rotates, center O2 moves in plane O1, Y1, Z1. Then displacement of O2 in system of axes O1X1Y1Z1 is described by generalized coordinates ζ and η. 3) System O2XYZ tightly connected to rotor, moving with it about point O2. Rotor in movement can incline by small angles ϕy, ϕz about axes Y2, Z2 and large angle ϕx about axis X2. Coordinates of center of rotor mass is x0, y0, z0. The kinetic energy when rotors shaft rotating in constant angular velocity is described by equation (3) [1]. The influence of gyroscopic effect isn’t estimated here. 1 T = m η 2 + ζ 2 + I x ω2 + I y ϕ y 2 + I z ϕ z 2 − 2 (3) − I xy cos ωt + I xz sin ωt ωϕ y + I xz cos ωt − I xy sin ωt ωϕ z + I yz ϕ z ϕ y + + mη [ z0 ω cos ωt − x0ϕ z − y0ωt sin ωt ] + mζ x0ϕ y − y0ω cos ωt − z0 ω sin ωt Where: ω – shaft (not rotor) angular velocity; ω = constant; Ix, Iy, Iz, Ixy, Ixz, Iyz – moments of inertia of rotor; m – weight of rotor. To estimate the influence of gyroscopic effect some new values must be included to equation (3). They were described in equation (4): ω = ω x = ϕ x ≅ ω + ϕ z ϕ y ; (4) ϕ y = ϕ y ; ϕ z ≅ ϕ z ;
(
{(
)
)
}
(
)
Kinetics energy equation is differentiate by generalized coordinates velocities η , ζ , ϕ y , ϕ& z and by displacements η , ζ , ϕ y , ϕ z .
Received eight differential equations of motion of the rotating system are described in compact matrix form: [ A]{q&&} + [ B ]{q&} + [C ]{q} = {F (t )} . (5)
Where {q} = {ξ, η, ϕy, ϕz} – vector of generalized coordinates, [A] – matrix of inertia, [C] – matrix of stiffness, [B] – matrix of damping and gyroscopic forces; {F(t)} – generalized vector of external excitation, components of which are functions of time t, angular speed ω, rotor moment of inertia. Using eight differential equations, which are received after differentiation matrix of inertia [A] is designed: 0 mx0 0 ζ m 0 m 0 −mx0 η ; (6) [ A]{qi } = mx 0 Iy − I yz ϕ y 0 − mx0 − I yz I z z 0 ϕ Matrix of damping [B] is designed by Rayleigh dissipative function R, which consists of systems friction elements. [B] matrix consists of coefficients of rotor friction in to flowing air: b11, b22, b33, b44. Matrix [B] is diagonal. The elements of gyroscopic forces I x ω + ϕ& z ϕ y are included in this matrix. In
(
)
order to get linear differential equation with constant coefficients, Ixω must be replaced. Then I x ω = I x ω x ≅ I x ω + ϕ& z ϕ y . Therefore:
(
)
b11 0 0 b [ B ]{qi } = 0 22 0 0 0
95
0 0 ζ 0 0 η ; b33 − I x ω ϕ y I x ω b44 ϕ z
(7)
M. Vasylius, J. Janutėnienė, J. Grigonienė
It is very difficult to receive stiffness coefficients incoming to potential energy, so influence coefficients are found and therefore influence coefficient matrix [D] is designed, which is received by following: [ D ]
−1
= [C ] . Therefore matrix [C] is in this form:
0 ζ c24 η ; 0 ϕ y c44 ϕz External excitation forces are evaluated by following equations: F = − My ω2 sin ωt + Mz ω2 cos ωt − Mg 0 0 ζ F = − I ω2 sin ωt + I ω2 cos ωt − Mgx xy xz 0 ϕy ; Fη = My0ω2 cos ωt + Mz0 ω2 sin ωt Fϕ = − I xz ω2 sin ωt + I xy ω2 cos ωt z c11 0 0 c 22 [C ]{qi } = c 0 31 0 c42
c13 0 c33 0
(8)
(9)
Here is added gravitation force G = mg, which has influence to generalized coordinate ζ and gravitation force moment G . x0 = mgx0, which has influence to generalized coordinate ϕy. Equations of rotating systems vibration were performed in to equations system, which will be used in mathematical modeling with software Matlab Simulink: + mx ϕ mζ 0 y + b55ζ + c11ζ + c12ϕ y = Fζ y − I yz ϕ z + b77 ϕ y − I x ωϕ z + c12ζ + c22ϕ y = Fϕ mx0 ζ + I y ϕ y ; (10) − mx0 ϕ z + b66η + c33η + c34ϕ z = Fη mη − I yz ϕ y − mx0η + I z ϕ z + I x ωϕ y + b88ϕ z + c34η + c44 ϕ z = Fϕ z 4.
Results of experimental investigation
The experimental testing of the blower rotor vibration made at different running modes: at nominal loading 100 % and 50 % and at free run. The blower rotor resonance speeds measured at run up and coast down regimes. The 7th bearing shaft vibration displacements and kinetic orbits at full loading and coast down running mode are presented in Fig. 4a and Fig. 4b. The peak-to-peak vibration displacement values at resonance speed 1436 rpm reached inaccessible values in both X and Y orthogonal directions as shown in vibration displacements plots and kinetic orbit of the shaft, Fig. 4b. These vibration displacements sp-p values became 6–8 times higher at the resonance in comparison with values at the maximum load at 3119–3132 rpm (Fig. 4a). The blower rotor high vibration displacements at resonance mode damage the journal bearings. The valuable damages are checked at the upper segment of the 7th bearing and the lower segment of the 8th bearing. The dynamic forces that damages bearings depend not only by resonance phenomenon but by rotors gyroscopic effect too. The rotors shaft position in the 7th bearings is described by the gap value. When machine runs in coast down mode, the gap between the vertical sensor 7Y tip and shaft surface decreases about ~200 µm value reference to nominal position at maximum rotation speed.
96
MATHEMATICAL MODELING OF DYNAMICS OF AIR BLOWER ROTOR
Fig. 4. The vibration displacements plots and kinetic orbits of blower rotor measured at bearing 7th at full loading (a): smax = 22,5 µm at 3132 rpm and at coast down (b): µm and smax = 123 µm at 1436 rpm
5.
Numerical analysis
Equations system (10) with software Matlab Simulink was calculated. These equations system solutions were found: – ξ, η – coordinates, defining position of point O2 in the plane O1X1Z1 – ϕy, ϕz – coordinates, defining small angles about axes Y2, Z2. Some calculation results are presented in Fig. 5. Calculations approved that high vibration displacement of the rotor are caused not only by unbalance, but by gyroscopic effect too. Resonance frequency 23,4 Hz was found by calculation (experimental 23,9 Hz).
a)
b)
c)
Fig. 5. The simulation results of rotor with elastic supports: a,b–the shaft vibration displacement sp-p(t) plots; b–kinetic orbit of shaft center O2 orbit at nominal rotation speed 3132 rpm
97
M. Vasylius, J. Janutėnienė, J. Grigonienė
The simulation results acquired when: coefficients of stiffness of support ks = 2·108 N/m, kv = 1.5·108 N/m; coefficients of damping hs=40 000 Ns/m, hv=20000 Ns/m; shaft diameter d=0.25 m; rotor weight m=1 660 kg. In Fig. 6 vibration amplitudes of 7th bearing support in vertical direction are shown by experimental investigation and theoretical calculation. Scientific calculations confirm experimental measurement results. Difference of curves cause that is in mathematical modeling wasn’t evaluated shaft bearing supports mathematical model.
Fig. 6. Vibration amplitudes of 7th bearing supports in vertical direction
Conclusions
1. Air blower rotor’s systems dynamic and mathematical models are composed to analyze rotors dynamic forces influence bearings. The second type of Lagrange equations are selected to compose vibration equations of rotating system. Mathematical model have four first order differential equations: four degrees of freedom of blower rotor. 2. Elevated hypothesis that air blower rotors affect gyroscopic moment, which decrease rotor unbalance and gravitation forces influence at nominal rotating speed. But when rotating speed is degreasing its negative influence for bearing tilting pad wear intensity increase. Therefore rotor is kept in stable position. 3. Air blower rotating system shaft absolute vibrations experimental investigations are performed, then machine working at stationary and transitional regimes. 4. Air blower dynamic equations are solved using scientific methods and software Matlab Simulink. Results of scientific investigation confirm results of experimental vibration measuring. References
Bat, M. I.; Dzanelidze, G. J. 1973. Theoretical mechanics in examples and problems. III. Moscow: Nauka. 528 p. (in Russian). Bently, D. E. 2002. Fundamentals of Rotating Machinery Diagnostics. Library of Congress Control Number 2002094136, ISBN 0-9714081-0-6. Bently Pressurized Bearing Company, printed in Canada, first printing. 726 p. Muszynska, A. 2005. Rotordynamics. CRC. 1116 p. ISBN 0824723996. Barzdaitis, V.; Bogdevičius, M.; Gečys, S. 2003. Vibration Problems of High Power Air Blower Machine, in 2nd International Symposium on Stability Control of Rotating Machinery, ISCORMA-2. Proceedings. Gdansk, Poland; Minden, Nevada, USA, 2003, 606–616. ISBN 83-913028-4-9. Deng-Horng, Tsai; Weileun, Fang. 2006. Design and simulation of a dual-axis sensing decoupled vibratory wheel gyroscope. Sensors and Actuators A 126: 33–40. Mehdi, Esmaeilia;, Nader, Jalilib; Mohammad, Dualia. 2007. Dynamic modeling and performance evaluation of a vibrating beam microgyroscope under general support motion, Journal of Sound and Vibration 301: 146–164.
98
