A constrained generalised-\(\alpha\) method for coupling rigid parallel chain kinematics and elastic bodies. (English) Zbl 1311.74015
Summary: A problem arises from combining flexible rotorcraft blades with stiffer mechanical links, which form a parallel kinematic chain. This paper introduces a method for solving index-3 differential algebraic equations for coupled stiff and elastic body systems with closed-loop kinematics. Rigid body dynamics and elastic body mechanics are independently described according to convenient mathematical measures. Holonomic constraint equations couple both the parallel chain kinematics and describe the coupling between the rigid and continuum bodies. Lagrange multipliers enforce the kinetic conditions for both sets of constraints. Additionally, to prevent numerical inaccuracy from inverting stiff mechanical matrices, a scaling factor normalises the dynamic tangential stiffness matrix. Finally, example tests show the verification of the algorithm with respect to existing computational tests and the accuracy of the model for cases relevant to the problem definition.
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| [1] | Agrawal O (1991) General approach to dynamic analysis of rotorcraft. J Aerosp Eng 4(1):91-107 · doi:10.1061/(ASCE)0893-1321(1991)4:1(91) |
| [2] | Armero F, Romero I (2001) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. part 1: low-order methods for two model problems and nonlinear elastodynamics. Comput Methods Appl Mech Eng 190:2603-2649 · Zbl 1008.74035 · doi:10.1016/S0045-7825(00)00256-5 |
| [3] | Arnold M, Brüls O (2007) Convergence of the generalized-\[ \alpha\] α scheme for constrained mechanical systems. Multibody Syst Dyn 18:185-202 · Zbl 1121.70003 · doi:10.1007/s11044-007-9084-0 |
| [4] | Bauchau O (1995) Multi-body formulation for rotorcraft dynamic analysis. Am Helicopter Soc 51st Annu Forum 2:1788-1798 |
| [5] | Bauchau O, Bottasso C, Nikishkov Y (2001) Modeling rotorcraft dynamics with finite element multibody procedures. Math Comput Model 33:1113-1137 · Zbl 1198.93132 · doi:10.1016/S0895-7177(00)00303-4 |
| [6] | Bauchau O, Kang NK (1993) A multibody formulation for helicopter structural dynamic analysis. Am Helicopter Soc 38(22):3-14 · doi:10.4050/JAHS.38.3 |
| [7] | Bauchau OA (2007) DYMORE user’s manual. Georgia Institute of Technology, Atlanta |
| [8] | Bindzi I et al (1992) Dynamic analysis of robotic manipulators with closed kinematic chains. IEEE Reg 10 Conf Tencon 92:639-643 |
| [9] | Blajer W (1999) State and energy stabilization in multibody systems. Mech Res Commun 26(3):261-268 · Zbl 0969.70505 · doi:10.1016/S0093-6413(99)00022-1 |
| [10] | Blajer W (2002) Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems. Multibody Syst Dyn 7:265-284 · Zbl 1007.70006 · doi:10.1023/A:1015285428885 |
| [11] | Bottasso C, Dopico D, Trainelli L (2008) On the optimal scaling of index three differential algebraic equations in multibody dynamics. Multibody Syst Dyn 19:3-20 · Zbl 1136.70001 · doi:10.1007/s11044-007-9051-9 |
| [12] | Bramwell ARS (1976) Helicopter dynamics. Edward Arnold, London |
| [13] | Cesnik CES et al (2001) Dynamic response of active twist rotor blades. J Smart Mater Struct 10:62-76 · doi:10.1088/0964-1726/10/1/306 |
| [14] | Chung J, Hulbert G (1993) A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\[ \alpha\] α method. ASME J Appl Mech 60:371-375 · Zbl 0775.73337 · doi:10.1115/1.2900803 |
| [15] | Gear CW, Leimkuhler B, Gupta GK (1985) Automatic integration of Euler-Lagrange equations with constraints. J Comput Appl Math 12/13:77-90 · Zbl 0576.65072 · doi:10.1016/0377-0427(85)90008-1 |
| [16] | Ghorashi M (2009) Dynamics of elastic nonlinear rotating composite beams with embedded actuators. Ph.D. thesis, Carleton University, Ottawa |
| [17] | Gransden D (2011) Aeroservoelasticity of articulated rotor hubs. Ph.D. thesis, Carleton University, Ottawa · Zbl 0775.73337 |
| [18] | Hodges DH, Dowell EH (1974) Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. Tech. Rep. D-7818, NASA, Washington |
| [19] | Johnson W (1980) Helicopter theory. Princeton University Press, Princeton |
| [20] | Johnson W (2000) CAMRAD II: comprehensive analytical model of rotorcraft aerodynamics and dynamics. Johnson Aeronautics, Palo Alto |
| [21] | Khude N, et al. (2007) A discussion of low order numerical integration formulas for rigid and flexible multibody dynamics. In: Proceedings of the 6th ASME international conference on multibody system, nonlinear dynanics and control IEDTC07, pp 1-12 |
| [22] | Kövecses J, Piedboeuf JC, Lange C (2003) Dynamics modeling and simulation of constrained robotic systems. IEEE/ASME Trans Mechatron 8(2):165-177 · doi:10.1109/TMECH.2003.812827 |
| [23] | Kuhl D, Crisfield MA (1999) Energy-conserving and decaying algorithms in non-linear structural dynamics. Int J Numer Methods Eng 45:569-599 · Zbl 0946.74078 · doi:10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A |
| [24] | Laulusa A, Bauchau O (2008) Review of classical approaches for constraint enforcement in multibody systems. J Comput Nonlinear Dyn 3(1):11,004.1-11,004.8 · doi:10.1115/1.2803257 |
| [25] | Lin SK (1990) Dynamics of the manipulator with closed chains. IEEE Trans Robot Autom 6(4):496-501 · doi:10.1109/70.59361 |
| [26] | Luh JYS, Zheng YF (1985) Computation of input generalized forces for robots with closed kinematic chain mechanisms. IEEE J Robot Autom 1(2):95-103 · doi:10.1109/JRA.1985.1087008 |
| [27] | Lunk C, Simeon B (2006) Solving constrained mechanical systems by the family of Newmark and \[\alpha\] α-methods. J Appl Math Mech 86(10):772-784 · Zbl 1116.70008 |
| [28] | Murray JJ, Lovell GH (1989) Dynamic modelling of closed chain robotic manipulators and implications for trajectory control. IEEE Trans Robot Autom 5(4):522-528 |
| [29] | Negrut D et al (2000) Method in the context of index 3 differential algebraic equations of multibody dynamics. Int J Numer Methods Eng 00:1-6:1-25 |
| [30] | Simeon B (2006) On lagrange multipliers in flexible multibody dynamics. Comput Methods Appl Mech Eng 195:6993-7005 · Zbl 1120.74517 · doi:10.1016/j.cma.2005.04.015 |
| [31] | Simo JC, Tarnow N (1992) The discrete energy-momentum method: conserving algorithms for nonlinear elastodynamics. J Appl Math Phys 43:757-792 · Zbl 0758.73001 · doi:10.1007/BF00913408 |
| [32] | Simo JC, Tarnow N (1994) A new energy and momentum conserving algorithm for the nonlinear dynamics of shells. Int J Numer Methods Eng 37(15):2527-2549 · Zbl 0808.73072 · doi:10.1002/nme.1620371503 |
| [33] | Simo JC, Tarnow N, Doblaré M (1995) Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int J Numer Methods Eng 38:1431-1473 · Zbl 0860.73025 · doi:10.1002/nme.1620380903 |
| [34] | Truong KV (2005) Dynamics studies of the ERATO blade, based on finite element analysis. 31st Eur Rotorcr Forum, pp 94.1-94.8 · Zbl 0969.70505 |
| [35] | Zheng YF, Luh JYS (1986) Joint torques for control of two coordinated moving robots. Tech. rep., IEEE Int. Conf. on Robotics and Autom., San Francisco |
| [36] | Zheng ZC, Ren G, Cheng YM (1999) Aeroelastic responce of a couple rotor/fuselage system in hovering and forward flight. Arch Appl Mech 69:68-82 · Zbl 0949.74019 · doi:10.1007/s004190050205 |
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