Comprehensive Application of Implicit Trapezoidal Rule with Numerical Jacobian in Rotor Dynamics

doi:10.3901/JME.2025.01.162

Abstract

Abstract: In recent years, geometrically exact beam theory is widely used in rotor structural dynamics to model composite blade undergoing large deflections. The relationship between movements and elastic deformations based on this theory shows strong nonlinearity and the generalized forces term is complex and huge. Therefore, it is difficult to derive the Jacobian matrix and determinant of the blade dynamics equation in analytical form, which brings great challenges to the implicit integration method. A trapezoidal integration utilizing numerical Jacobian matrix is proposed to solve the dynamic equations of helicopter rotor blade, in which the Jacobian matrix and stiffness matrix are numerically computed by central difference method with Romberg extrapolation scheme. Accuracy of the methods is evaluated on dynamic problems from one cantilevered beam and the other fully articulated rotor blade. Numerical stability and efficiency are also investigated by changing update policies of Jacobian matrix and mesh configurations of typical rotor blade. Numerical results show that the integration method with numerical Jacobian performs well on predicting natural frequencies and transient responses in rotor dynamics.

Key words: rotor dynamics, numerical integration, Jacobian matrix, extrapolation method, quasi-Newton method

CLC Number:

V214

WU Jie, YU Zhihao. Comprehensive Application of Implicit Trapezoidal Rule with Numerical Jacobian in Rotor Dynamics[J]. Journal of Mechanical Engineering, 2025, 61(1): 162-171.

References

[1] MA Hui，LU Yang，WU Zhiyuan，et al. A new dynamic model of rotor–blade systems[J]. Journal of Sound and Vibration，2015，357(24)：168-194.
[2] GUNJIT B，INDERJIT C. University of Maryland advanced rotorcraft code (UMARC)，theory manual， volume I [M]. Maryland：University of Maryland，1994.
[3] FLOROS M W. Elastically tailored composite rotor blades for stall alleviation and vibration reduction[D]. Pittsburgh：The Pennsylvania State University，2000.
[4] GHORASHI M. Nonlinear analysis of the dynamics of articulated composite rotor blades[J]. Nonlinear Dynamics，2012，67：227-249.
[5] WU Jie，DU Xianbin，MA Yijiang. Research of precise time integration method and its derived formats on helicopter rotor dynamics[J]. International Journal of Computational Methods，2020，17(8)：1950059.
[6] LI Kuinian，DARBY A P. A high precision direct integration scheme for nonlinear dynamic systems[J]. Journal of Computational and Nonlinear Dynamics，2009，4(4)：041008.
[7] BAUCHAU O A，THERON N J. Energy decaying scheme for nonlinear beam models[J]. Computer Methods in Applied Mechanics and Engineering，1996，134(1-2)：37-56.
[8] BAUCHCHAU O A，THERON N J. Energy decaying scheme for nonlinear elastic multi-body systems[J]. Computers & Structures，1996，59(2)：317-331.
[9] RUTKOWSKI M J，RUZICKA G C，ORMISTON R A，et al. Comprehensive aeromechanics analysis of complex rotorcraft using 2GCHAS[J]. Journal of the American Helicopter Society，1995，40(4)：3-17.
[10] HARSHA S P. Nonlinear dynamic analysis of a high-speed rotor supported by rolling element bearings[J]. Journal of Sound and Vibration，2006，290(1-2)：65-100.
[11] CELAYA E A，JUAN J A. BDF-α: A multistep method with numerical damping control[J]. Universal Journal of Computational Mathematics，2013，1(3)：96-108.
[12] FANG Luning，KISSEL A，ZHANG Ruochun. On the use of half-implicit numerical integration in multibody dynamics[J]. Journal of Computational and Nonlinear Dynamics，2023，18(1)：014501.
[13] CAO Jianming. Transient analysis of flexible rotor with nonlinear bearings, dampers and external forces[D]. Charlottesville：University of Virginia，2012.
[14] CHANDRA D，GANGULI S R. Modified newton, rank-1 Broyden update and rank-2 BFGS update methods in helicopter trim: a comparative study[J]. Aerospace Science and Technology，2012，23(1)：187-200.
[15] 虞志浩，杨卫东，张呈林. 基于Broyden法的旋翼多体系统气动弹性分析[J]. 航空学报，2012，33(12)：2171-2182. YU Zhihao，YANG Weidong，ZHANG Chenglin. Aeroelasticity analysis of rotor multibody system based on Broyden method[J]. Acta Aeronautica et Astronautica Sinica，2012，33(12)：2171-2182.
[16] TANG Di，BAO Shiyi，LUO Lijia，et al. Study on the aeroelastic responses of a wind turbine using a coupled multibody-FVW method[J]. Energy，2017，141(15)：2300-2313.
[17] MARTIN B，BERNHARD S. Coupled simulation of multibody and finite element systems: an efficient and robust semi-implicit coupling approach[J]. Archive of Applied Mechanics，2012，82：723-741.
[18] HODGES D H. Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams[J]. AIAA Journal，2003，41(6)：1308-1309.
[19] IM B，CHO H，KEE Y J，et al. Geometrically exact beam analysis based on the exponential map finite rotations[J]. International Journal of Aeronautics and Space Sciences，2020，21：153–162.
[20] SHANG Li'na，XIA Pinqi，HODGES D H. Geometrically exact nonlinear analysis of pre-twisted composite rotor blades[J]. Chinese Journal of Aeronautics，2018，31(2)：300-309.
[21] WANG Lin，LIU Xiongwei，RENEVIER N，et al. Nonlinear aeroelastic modeling for wind turbine blades based on blade element momentum theory and geometrically exact beam theory[J]. Energy，2014，76：487-501.
[22] REZAEI M M，ZOHOOR H，HADDADPOUR H. Aeroelastic modeling and dynamic analysis of a wind turbine rotor by considering geometric nonlinearities[J]. Journal of Sound and Vibration，2018，432：653-679.
[23] CELI R，FRIEDMANN P. Rotor blade aeroelasticity in forward flight with an implicit aerodynamic formulation[J]. AIAA Journal，1988，26(12)：1425-1433.
[24] SIAMI A，NITZSCHE F，LEIBBRANDT R，et al. Dynamic response analysis of the helicopter blades with non-uniform structural properties[C]. AIAA SciTech，2021 Forum，2021：0562.
[25] WU Jie，MA Yijiang，WANG Zhidong，et al. Structurally coupled characteristics of rotor blade using new rigid-flexible dynamic model based on geometrically exact formulation[J]. Chinese Journal of Aeronautics，2022，35(6)：186-197.
[26] BABILIO E，LENCI S. On the notion of curvature and its mechanical meaning in a geometrically exact plane beam theory[J]. Int. J. Mech. Sci.，2017，128：277–293.
[27] DAVIDE I，LORENZO D. A fully consistent linearized model for vibration analysis of rotating beams in the framework of geometrically exact theory[J]. J Sound Vib，2017，370：351–371.
[28] JOHNSON W. CAMRAD II: Comprehensive Analytical Method of Rotorcraft Aerodynamics and Dynamics [M]. Johnson Aeronautics，Palo alto，CA，USA，2012.
[29] WANG Qi，YU Wenbin. Geometric nonlinear analysis of composite beams using Wiener-Milenkovic parameters[J]. Journal of Renewable and Sustainable Energy，2017，9：033306.
[30] DUTKA J. Richardson extrapolation and Romberg integration[J]. Historia Mathematica，1984，11(1)：3-21.
[31] ASCHER U M，LINDA R P. Computer methods for ordinary differential equations and differential-algebraic equations[J]. SIAM，Philadelphia，Pennsylvania，1998.
[32] CHENG Tao. Structural dynamics modeling of helicopter blades for computational aeroelasticity[D]. MIT thesis，2002.
[33] THOMSON W T，DAHLEH M D. Theory of vibrations with applications [M]. 5th ed. London：Pearson press，1998.
[34] BAUCHAU O A，KANG N K. A multibody formulation for helicopter structural dynamic analysis[J]. Journal of the American Helicopter Society，1993，38(2)：3-14.
[35] HEFFERNAN R M，GAUBERT M. Structural and aerodynamic loads and performance measurements of an SA349/2 helicopter with an advanced geometry rotor [M]. California：NASA-TM-88370, 1986.
[36] ERIC K. A faster Broyden method, BIT numerical mathematics[J]. SIAM，1991，31(2)：369-372.
[37] PRESS W H，TEUKOLSKY S A，et al. Numerical recipes：the art of scientific computing[M]. 3rd ed. Cambridge：Cambridge University Press，2007.