Nonlinear three-dimensional beam theory for flexible multibody dynamics

Abstract

In flexible multibody systems, it is convenient to approximate many structural components as beams or shells. Classical beam theories, such as Euler–Bernoulli beam theory, often form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the beam’s section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions. Kinematically, the problem is decomposed into an arbitrarily large rigid-section motion and a warping field. The sectional strains associated with the rigid-section motion and the warping field are assumed to remain small. As a consequence of this kinematic decomposition, the governing equations of the problem fall into two distinct categories: the equations describing geometrically exact beams and those describing local deformations. The governing equations for geometrically exact beams are nonlinear, one-dimensional equations, whereas a linear, two-dimensional analysis provides the detailed distribution of three-dimensional stress and strain fields. Within the stated assumptions, the solutions presented here are the exact solution of three-dimensional elasticity for beams undergoing arbitrarily large motions.

Appendices

Appendix A: Hamiltonian matrices

Matrix , of size 2n×2n, is said to be Hamiltonian if it satisfies the following property

(73)

where the skew-symmetric matrix is defined as

(74)

Note the following properties of matrix :

(75a)

(75b)

In view of these properties, the definition (73) of Hamiltonian matrices can be recast as . Because matrix is skew-symmetric, the following property holds for any vector $\underline{\mathcal{U}}$:

(76)

Property (73) implies that the most general form of a Hamiltonian matrix is

(77)

where matrices , , and are of size n×n, and matrices and are symmetric. Clearly, the transpose of a Hamiltonian matrix is also Hamiltonian.

1.1 A.1 Eigenvalues of Hamiltonian matrices

Let λ and μ be two eigenvalues of a Hamiltonian matrix and the associated eigenvectors are denoted $\underline{\mathcal{U}}_{\,\lambda}$ and $\underline{\mathcal{U}}_{\,\mu }$, respectively, i.e., and . Pre-multiplying the first equation by and the second by leads to and , respectively. Because matrix is symmetric, the right-hand sides of these two equations are identical, and subtraction yields . If λ+μ≠0, the following symplectic orthogonality results:

(78)

Let λ be an eigenvalue of a Hamiltonian matrix associated with eigenvector $\underline{\mathcal{U}}_{\,+\lambda}$, i.e., . It then follows that and properties (73) and (75a) then imply . Clearly, if $\underline{\mathcal{U}}_{\,+\lambda}$ is an eigenvector of associated with eigenvalue λ, is an eigenvector of associated with eigenvalue −λ. Because the spectra of eigenvalues of matrices and are identical, the eigenvalues of Hamiltonian matrices are symmetric about the imaginary axis, i.e., occur in pairs of opposite sign, ±λ. In summary,

(79a)

(79b)

where Eqs. (79a) and (79b) express identical properties for eigenvalues +λ and −λ, respectively. Vectors $\underline{\mathcal{U}}_{\,+\lambda }$ and can also be interpreted as the right and left eigenvectors of matrix , both associated with eigenvalue +λ.

It is convenient to define matrix , whose columns store the right eigenvectors associated with eigenvalues −λ and +λ. These vectors enjoy the following properties

(80)

Indeed, property (76) implies and vectors $\underline{\mathcal{U}}_{\,+\lambda}$ and $\underline{\mathcal{U}}_{\,-\lambda}$ can be normalized to enforce . The following pseudo-orthogonality statement in the space of Hamiltonian matrix then follows

(81)

Appendix B: Symplectic matrices

Matrix , of size 2n×2n, is said to be symplectic if it satisfies the following property

(82)

where the skew-symmetric matrix is defined by Eq. (74). The inverse of a symplectic matrix is expressed easily as

(83)

Indeed, using definition (82) leads to .

Recasting Eq. (82) as and using Eq. (83) to express the inverse of the symplectic matrix leads to , and finally,

(84)

This means that the transpose of a symplectic matrix is itself symplectic. It is proved easily that if matrix is symplectic, its inverse is also symplectic. Furthermore, the product of two symplectic matrices is also symplectic. Note that matrices and are symplectic.

A close connection exists between symplectic and Hamiltonian matrices. First, their definitions are closely related: and , for Hamiltonian and symplectic matrices, respectively. Next, consider matrix , defined through the following transformation, . Using Eq. (83) leads to , which implies that matrix is symmetric if matrix is itself symmetric. Consequently, if matrix is Hamiltonian, so is matrix . It follows that the transformation of a Hamiltonian matrix by a symplectic matrix yields a Hamiltonian matrix, i.e.,

(85)

Consider the following partition of a symplectic matrix:

(86)

where matrices , , , and are of size n×n. The definition of a symplectic matrix then implies

(87a)

(87b)

(87c)

The last two properties imply that products and form symmetric matrices.