Three kinematic representations for modeling of highly flexible beams and their applications

Presented here are three kinematic representations of large rotations for accurate modeling of highly flexible beam-like structures undergoing arbitrarily large three-dimensional elastic deformation and/or rigid-body motion. Different methods of modeling torsional deformation result in different beam theories with different mathematical characteristics. Each of these three geometrically exact beam theories fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, exact coordinate transformations, and orthogonal virtual rotations. The derivations are presented in detail, a finite element formulation is included, fully nonlinear governing equations and boundary conditions are presented, and the corresponding form for numerically exact analysis using multiple shooting methods is also derived. These theories are compared in terms of their appropriate application areas, possible singular problems, and easiness for use in modeling and analysis of multibody systems. Nonlinear finite element analysis of a rotating beam and nonlinear multiple shooting analysis of a torsional bar are performed to demonstrate the capability and accuracy of these beam theories.

[1]  D. F. Parker.  An asymptotic analysis of large deflections and rotations of elastic rods , 1979 .

[2]  A. Nayfeh,et al.  Linear and Nonlinear Structural Mechanics , 2002 .

[3]  V. Berdichevskiĭ.  On the energy of an elastic rod , 1981 .

[4]  Charis J. Gantes,et al.  Deployable Structures : Analysis and Design , 2001 .

[5]  David J. Ewins,et al.  Modal Testing: Theory, Practice, And Application , 2000 .

[6]  Robert L. Taylor,et al.  Non-linear dynamics of flexible multibody systems , 2003 .

[7]  P. Frank Pai,et al.  Geometrically Exact Total-Lagrangian Modeling and Nonlinear Finite Element Analysis of Highly Flexible Multibody Systems , 2010 .

[8]  P. Frank Pai,et al.  Experimental Nonlinear Dynamics of a Highly Flexible Spinning Beam Using a 3-D Motion Analysis System , 2010 .

[9]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[10]  Giancarlo Genta,et al.  Dynamics of Rotating Systems , 2005 .

[11]  P. Frank Pai,et al.  Highly Flexible Structures : Modeling, Computation, and Experimentation , 2007 .

[12]  Christopher H. M. Jenkins,et al.  Gossamer spacecraft : membrane and inflatable structures technology for space applications , 2001 .

[13]  W. Olson,et al.  Finite element analysis of DNA supercoiling , 1993 .

[14]  The Role of Saint Venant’s Solutions in Rod and Beam Theories , 1979 .

[15]  J. Hearst,et al.  The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling , 1994 .

[16]  B. Miedema,et al.  NOTES Techniques: present and future , 2008, European Surgery.

[17]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[18]  Dewey H. Hodges,et al.  Nonlinear Composite Beam Theory , 2006 .

[19]  T Schlick,et al.  Modeling superhelical DNA: recent analytical and dynamic approaches. , 1995, Current opinion in structural biology.

[20]  M. Borri,et al.  A large displacement formulation for anisotropic beam analysis , 1986 .

[21]  Bauchau Olivier,et al.  Flexible Multibody Dynamics , 2010 .

[22]  Ahmed A. Shabana,et al.  Dynamics of Multibody Systems , 2020 .