Methodology of modelling the flexural-torsional vibrations in transient states of the rotating power transmission systems

Transcription

1 Methodology of modelling the flexural-torsional vibrations in transient states of the rotating power transmission systems Tomasz Matyja Faculty of Transport Department of Logistics and Aviation Technologies

2 Rotating power transmission systems Turbochargers Ship shaft line Helicopter drive system Much more Dynamic behaviors investigated by methods of Rotordynamics

3 Plan of presentation Motivation for research and work Model of rigid rotor with six degrees of freedom Decoupling equations of the rotor motion Proposed decomposition method of the typical rotating system Simulink library for modeling and simulation of dynamic phenomena in rotating power transmission systems Examples of simulation studies Conclusions 3

4 Motivation for research and work Due: - the use of new materials (eg composites), - lightest and less stiffness shafts, - higher speeds. More important are: - non-linear nature of dynamic phenomena in rotating systems, - couplings between different forms of vibration: flexular, torsional and longitudinal, - transient analysis. 4

5 Methods used in the Rotordynamics Transfer Matrix Method [Prohl] Old method, flexural vibration of linear systems Finite Element Method D Timoshenko beam elements and rigid rotors D axisymmetric elements 3D volume elements Allows to considering couplings Typically flexural vibration only The dynamics of rotating systems is so different from the dynamics of the structure that specialized software is necessary. 5

6 Software used in Rotordynamics There has now been a lot of rotodynamics computer software, both academic and commercial. DYNROT (FEM, MATLAB) [Genta: Dynamics of Rotating Systems. 005] MESWIR [Kiciński: Rotor Dynamics. 005] ANSYS [Rotordynamics with ANSYS Mechanical Solutions.009] MSC NASTRAN [Rotordynamics user s guide,06] DyRoBeS [Chen W.J.: Introduction to Dynamics of Rotor- Bearing Systems] XLRotor [Vance J., Zeidan F., Murphy B.: Machinery Vibration and Rotordynamics] academic commercial 6

7 Motivation for research and work Existing methods of modelling the rotating system and available software not always allow to study coupled vibrations in transient states. Or Simulations of this type are very time-consuming, due to numerical integration of the equations of motion and a large number of degrees of freedom. Hence the idea of building own tools for modeling rotating systems 7

8 Model of rigid rotor with six degrees (ξ, η, ζ) movable coordinate system of freedom C is a geometrical center of a rotor three translational coordinates x c, y c, z c three rotational coordinates {ψ,θ,φ} Angles {ψ,θ} describe inclination of a rotor s plane and must be small. 8 Rotation angle φ connected with the axis ζ may have arbitrary values.

9 Selection of Euler angles (Rot --3) allows to avoid the gimbal lock Transformation matrix R = cosθcosφ sinψsinθcosφ + cosψsinφ cosψsinθcosφ + sinψsinφ = cosθsinφ sinψsinθsinφ + cosψcosφ cosψsinθsinφ + sinψcosφ sinθ sinψcosθ cosψcosθ. lock if θ π R θ= π = 0 sin ψ + φ cos ψ + φ 0 cos ψ + φ sin ψ + φ

10 Static and dynamic unbalance eccentricity e, β δ - inclining angle of principal axes γ- precede angle of principal axes, C - geometrical center P - rotor s mass center 0

11 Inertia matrix of the rotor in the system of principal axes J 3 J = J t J t J p Axisymmetric rotor in local moving frame J ξηζ J t + J p J t cos γ sin δ J p J t sinγcosγ sin δ J p J t cosγsinδcosδ = J p J t sinγcosγ sin δ J t + J p J t sin γ sin δ J p J t sinγsinδcosδ J p J t cosγsinδcosδ J p J t sinγsinδcosδ J p J p J t sin δ More general case J ξηζ = J J J 3 J J J 3 J 3 J 3 J 33

12 Lagrange equations of rotor motion Mass matrix, generalized coordinates, generalized forces, velocities vector M = m m m J J J 3 J J J 3 J 3 J 3 J 33, q = x C y C z C ψ θ φ, Q = Q xc Q yc Q zc Q ψ Q θ Q φ, V = V P Ω ξηζ. total kinetic energy of the rotor E k = VT MV. V q T M V + d dt V q T V q T MV = Q q. q {x C, y C, z C, ψ, θ, φ

13 Inertia coupling equations For example equation of rotational motion due to the first rotational coordinate. Linearization: sin α α cos α α ψ, θ - small angles. ψ[j cos φ J sinφ + J sin φ + θ(j 3 cosφ J 3 sinφ) + θ J 33 ] + θ (J J )sinφcosφ + J (cos φ sin φ) + θ(j 3 sinφ + J 3 cosφ) + φ [J 3 cosφ J 3 sinφ + θj 33 ] + φ θ (J J )cosφ J sinφ + θ(j 3 cosφ J 3 sinφ) + J 33 + φ ψ (J J )sinφ J cosφ θ(j 3 sinφ + J 3 cosφ) + ψ θ [J 3 cosφ J 3 sinφ + θj 33 ] + (θ φ )(J 3 sinφ + J 3 cosφ) = Q ψ Q zc e sin(β + φ ). 3

14 Decoupling the equations of the rotor motion Implicit form: Φ q, q, q = 0 a very difficult task Explicit form: q = Φ ( q, q) The decoupling simplifies numerical integration. 4 Standard form: q = q, q q = Φ 3 (q )

15 Four models of the rotor Model Form of the equations Inertia matrix Level of the linearization α {ψ, θ Numeric efficiency Explicit form Simplifying assumptions J = J 3 = 0 sin α α cos α Faster, the smallest number of arithmetic operations Implicit matrix form Full without simplifying assumptions sin α α cos α Matrix multiplication required 3 Implicit matrix form Full without simplifying assumptions sin α α cos α α Additional cost of calculating the correction matrix 4 Implicit matrix form Full without simplifying assumptions Without linearization Additional cost of calculating sines and cosines of angles ψ, θ 5

16 First method of decupling Model n0. of the rotor J cos φ + J sin φ J J sinφcosφ J 3 cosφ + θj 33 J J sinφcosφ J sin φ + J cos φ J 3 sinφ J 3 cosφ + θj 33 J 3 sinφ J 33 F ψ (ψ, θ, φ, ψ, θ, φ, Q xc, Q yc, Q zc, Q ψ, Q θ, Q φ ) = F θ (ψ, θ, φ, ψ, θ, φ, Q xc, Q yc, Q zc, Q ψ, Q θ, Q φ, F φ (ψ, θ, φ, ψ, θ, φ, Q xc, Q yc, Q zc, Q ψ, Q θ, Q φ ψ θ φ H ψ θ φ = F ψ F θ F φ. ψ θ φ = H φ F ψ F θ F φ. x C yc z C = m F xc F yc F zc, 6

17 Other method of decupling Model n0. of the rotor V = A(q) q A T MA q + (A T M A + BMA) q = Q. q = C Q (A T M A + BMA) q. A = e sin β + φ e cos β + φ 0 0 e sin β + φ e cos(β + φ) e ψ cos β + φ + θ sin β + φ cosφ sinφ sinφ cosφ θ 0 B = φ sinφ φ cosφ θ φ cosφ θcosφ + ψsinφ 7 φ sinφ θsinφ + ψ ψcosφ 0.

18 Three methods used to verify the model mathematical verification - comparison of equations verification by simulation methods - comparison of vibration signals from simulations experimental verification- comparison of measured vibration signals with simulation results 8

19 Mathematical verification - comparison of equations Proposed 6 DOF model with static and dynamic imbalance 6 DOF ideal rotor turning off imbalance linearization Simplified 6 DOF model v. From Euler-Newton equations switching off torsional and longitudinal vibrations 4 DOF model from literature 9

20 Verification by simulation methods - comparison of vibration signals from simulations Jetcrofft s rotor 6 DOF Developed by the author Jetcrofft s rotor 4 DOF 0

21 The distance between the center of mass of the rotor and the axis of rotation Verification by simulation methods - comparison of vibration signals from simulations Slower startup - less torsional vibrations - greater compatibility

22 Experimental verificationcomparison of measured vibration signals with simulation results Frequency o the flexural vibrations measured on the bearings were compared. More comprehensive comparison is planned.

23 Interactions between the rigid rotor and the shaft 3

24 Proposed decomposition method of typical rotating system 4

25 Interactions between the inertial and compliance elements 5

26 Simulink library for modeling and simulation of dynamic phenomena in rotating power transmission systems From: Model of thin rigid rotor To: Forces kinematic output 6

27 Thin vs. Long Rotor Presented model can be easily generalized to the case of a long rotor. thin rotor long rotor Two additional outputs are necessary 7

28 Block modelling the shaft force acting to left inertia element kinematic signal from left inertia element kinematic signal internal DOF force acting to right inertia element kinematic signal from right inertia element Block modeling shaft must have external DOF (6 on each end). 8 8

29 FEM model of the shaft FEM - Timoshenko beam elements better! Rigid Finite Element Method (beam elements) faster! 9

30 Rigid Finite Element Method 30

31 A special beam element Coordinate system axes do not have to be the principal axes of inertia N M x M y T y T x M z = EA ES x ES y ES x EI x ED xy ES y ED xy EI y χ y GA 0 χ y GS y χ x GA χ x GS x χ y GS y χ x GS x G(χ x I x + χ y I y ) ε κ x κ y β x β y κ z. N = N uu N uv 0 N uψ N uθ N uφ N uu N uv 0 N uψ N uθ N uφ N vu N vv 0 N vψ N vθ N vφ N vu N vv 0 N vψ N vθ N vφ N wu N wv N ww N wψ N wθ N wφ N wu N wv N ww N wψ N wθ N wφ N ψu N ψv 0 N ψψ N ψθ N ψφ N ψu N ψv 0 N ψψ N ψθ N ψφ N θu N θv 0 N θψ N θθ N θφ N θu N θv 0 N θψ N θθ N θφ N φφ N φφ. Mq + η V K + ΩG q + K + η V ΩK C + Ω G q = Q + Q u Ω f φ, 3

32 Universal dynamic model of the bearing F B = K(q b q h ) + K n (q b q h ) n + D(q b q h), 3

33 Universal dynamic model of the clutch F W = F W (q L, q L, q R, q R), 33

34 Broadband torsional vibration damper Rubber damper Viscous damper 34

35 Influence of dynamic imbalance on the vibrations Machine startup simulation ω n = 000 rad/s t n =.5 s Four cases of the main rotor dynamic imbalance: 0 0 ; 5 0 ; 0 0 ;

36 Simulink model 36

37 Trajectory of the mass center main rotor 37

38 Trajectory of the mass center external rotor 38

39 X component of transverse vibration 39

40 Gyroscopic effect - inclination angle of the rotor 40

41 Torsional vibration velocity difference between the main rotor speed and drive speed d /dt [rad/s] t [s]

42 Torsional vibration acceleration 4

43 Gyroscopic effect dynamic imbalance

44 Distance from the geometric axis of rotation 44

45 RMS of accelerations of torsional vibrations 45

46 Scheme of the rotating system with transverse crack RFEM elements K = K f ψ ΔK f ψ = + cos ψ - crak open 0- crak closed 46

47 Selected results of simulations 47

48 Spectrogram of the gyroscopic acceleration of the central rotor without crak 48

49 Spectrogram of the gyroscopic acceleration of the central rotor with crak 49

50 Conclusions Presented method of modeling the rotating systems is an alternative to existing systems based on FEM. Its advantage is the relatively small number of degrees of freedom which reduces simulation time and allows to study non-steady states with low cost of computer hardware. The authors library of Simulink blocks can be freely expanded with new elements. Another advantage is ability to easily integrate with tools and models available in Simulink and Matlab. 50

51 Thank you for your attention 5