1. Flexible gear dynamics modeling in multi-body analysis Alberto Cardona Cimec-Intec (UNL/Conicet) and UTN-FRSF, Santa Fe, Argentina and Didier Granville Samtech SA, Liège, Belgium

2. V USNCCM, Boulder, USA, 4-6 August 19992 Flexible Multibody Systems –Statics, kinematics and dynamics of multibody systems –Appropriate representation of 3D rotations –Geometric nonlinearities with finite rotations and displacements –Flexible joints and members: –Continuous elements –Discrete elements (springs, flexible joints,...) –Rigid joints and members: –Rigid bodies and joints –Constraint dynamics by implicit time step integration

3. V USNCCM, Boulder, USA, 4-6 August 19993 Introduction Gear modeling: –Usually: parallel straight gears –Several papers on particular cases (e.g. Vibrations in a planetary system with stiffness...) –Industrial applications require treating all cases: - planetary : 행성 톱니 바퀴식의

4. V USNCCM, Boulder, USA, 4-6 August 19994 Introduction Work objectives : –Develop a formulation for gears consistent with the multibody systems formulation –General formulation for any kind of gears –Account for all gearing forces –Possibility of modeling gear trains, planetary systems, etc –Include flexibility and backlash - backlash : 톱니 바퀴 사이의 틈, 반동 / 반발, 역회전

5. V USNCCM, Boulder, USA, 4-6 August 19995 Constraint Dynamics Formulation Equations of motion: –System with holonomic constraints: unconstrained Lagrangian generalized displacements and velocities non conservative forces holonomic constraints set holonomic constraint forces: scale and penalty factors Nonlinear DAE’s solved by implicit time-stepping

6. V USNCCM, Boulder, USA, 4-6 August 19996 Gear Pair Kinematics Basic definitions: –Joint formed by two wheels with centers A and B centers position normal frames at each wheel Convention ( ): first vector normal to wheel, second one initially pointing to contact point C Current wheel frames:

7. V USNCCM, Boulder, USA, 4-6 August 19997 Gear Pair Kinematics normal frames at the support Convention( ): first vector normal to each wheel, and second vector always pointing towards contact point Distance between centers and relative orientation of both wheels is kept constant by an external support to assure correct engagement.

8. V USNCCM, Boulder, USA, 4-6 August 19998 Gear Pair Kinematics Convention (tooth frame ): First vector parallel to tooth base, second vector along tooth vertical line Cone and helix angles: at wheel A

9. V USNCCM, Boulder, USA, 4-6 August 19999 Gear Pair Kinematics –Relation between { } and with { } is computed in terms of kinematic variables at wheel A –At wheel B: –Relation between support frames at each wheel:

10. V USNCCM, Boulder, USA, 4-6 August 199910 Gear Pair Kinematics Expression of at the reference configuration –Position of contact point : after solving: –Third vector : –Wheel B: –It can be shown:

11. V USNCCM, Boulder, USA, 4-6 August 199911 Gear Pair Kinematics Joint degrees of freedom Generalized coordinates: wheels center position wheels rotations angular displacements in frame measure of deformation and backlash

12. V USNCCM, Boulder, USA, 4-6 August 199912 Gear Pair Kinematics Relation between current wheel and support frames: with: Twelve physical dofs: 6 dofs at wheel A + 6 dofs at wheel B + 1 dof generalized mesh deformation - 1 dof relative rotation constraint Three constraints to be imposed (Grüber): #constr = #gen.coord. - #phys.dofs = 15 - 12 = 3 then, three Lagrange multipliers: Constraints ?

13. V USNCCM, Boulder, USA, 4-6 August 199913 1.Kinematics relation between angular displacements: with normal module of gear teeth pressure angle in normal plane teeth numbers of each wheel The Lagrange multiplier, times the scale factor, is the normal contact force. 2.Hoop contact: with position of ideal contact point in terms of kinematics variables at wheel A and B 3.Unicity of unit vectors computed in terms of variables at wheels A and B: Constraint Equations

14. V USNCCM, Boulder, USA, 4-6 August 199914 Constraint Equations Variation of vectors Variation of contact point position

15. V USNCCM, Boulder, USA, 4-6 August 199915 Constraint Forces Variation of constraints

16. V USNCCM, Boulder, USA, 4-6 August 199916 Constraint Forces and Stiffness Matrix Forces of constraint Stiffness contribution Second order derivatives of constraint assure exact computation of tangent behavior; i.e. vibration eigenfrequencies.

17. V USNCCM, Boulder, USA, 4-6 August 199917 Mesh Deformation and Backlash Teeth contact: Rigid wheels with elastic teeth Nonlinear stiffness along instant pressure line Contact damping Mesh stiffness variation Backlash Mesh forces from potential mesh stiffness loaded transmission error circumf.backlash

18. V USNCCM, Boulder, USA, 4-6 August 199918 Mesh Deformation and Backlash Mesh stiffness: function of the normalized pitch displacement accounts for variations of instantaneous stiffness (i.e. # teeth in contact) Load transmission error: first harmonic of hoop displacement amplitude due to errors with respect to theoretical gear profiles

19. V USNCCM, Boulder, USA, 4-6 August 199919 Radial Component of Contact Forces Non holonomic nature: –Pressure line orientation changes according to sign of transmitted torque –The radial component acts by trying to always separate both wheels –The holonomic constraint only considers the axial and hoop components of the contact force Computation of radial contact force:

20. V USNCCM, Boulder, USA, 4-6 August 199920 Radial Component of Contact Forces Contribution to internal forces: Contribution to tangent stiffness:

21. V USNCCM, Boulder, USA, 4-6 August 199921 Gear-Rack Pair Generalized coordinates:

22. V USNCCM, Boulder, USA, 4-6 August 199922 Conical Straight Bevel Pair Radius Normal modulus Pressure angle Teeth number Cone angle Helix angle

23. V USNCCM, Boulder, USA, 4-6 August 199923 Three Wheels System Diameter Normal modulus Pressure angle Teeth number Cone and helix angles spring

24. V USNCCM, Boulder, USA, 4-6 August 199924 Dynamics of a Geared Shaft System Diameter Normal modulus Pressure angle Teeth number Cone and helix angles Mesh stiffness Mesh damping Load transmission error

25. V USNCCM, Boulder, USA, 4-6 August 199925 Dynamics of a Geared Shaft System Case A: Rigid supports without clearanceCase B: Flexible supports without clearance Supports stiffness and damping :

26. V USNCCM, Boulder, USA, 4-6 August 199926 Dynamics of a Geared Shaft System Case C: Rigid supports with clearance Clearance : Case D: Flexible supports with clearance Supports stiffness and damping : Clearance :

27. V USNCCM, Boulder, USA, 4-6 August 199927 Vibration Analysis of a Gear Pair Normal modulus Pressure angle Teeth number Mass & inertia wheel A Mass & inertia wheel B Cone angles Helix angles Mesh stiffness

28. V USNCCM, Boulder, USA, 4-6 August 199928 Vibration Analysis of a Gear Pair Supports stiffness : Frequencies [Hz] : Program Reference

29. V USNCCM, Boulder, USA, 4-6 August 199929 Conclusions !We developed a general formulation for the analysis of multibody systems with gears !The formulation accounts for: Bmesh stiffness fluctuation Bbacklash, friction Berrors with respect to theoretical gear !Inclusion of second order derivatives of constraints allows to compute tangent vibration frequencies