Nonlinear analysis of pretwisted rods using 'principal curvature transformation'. I - Theoretical derivation

Three-dimensional nonlinear vibrations of composite beams — I. Equations of motion

“…Comparing equations (74) and (75) with the equations (A13) and (A14) used by Rosen et al [7] for the nonlinear curvatures, we find that each of the latter is missing a third-order term. Comparing equations (74) and (75) with the equations (A13) and (A14) used by Rosen et al [7] for the nonlinear curvatures, we find that each of the latter is missing a third-order term.…”

“…In the development of nonlinear equations of motion for flexible beams, three successive Euler-like rotations tire conmlonly used to relate the deformed and undeformed states [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. [3tit different sequences of Euler rotations will result in different equations of motion [11.…”

“…They also identified several nonlinear terms. Rosen et al [3,7] derived a set of equations that is more accurate than those of Hedges and Dowell [2] by including some nonlinear terms with order higher than two. The restllts show that there tire systematic differences between theory and experiment when tip deflections become large because the equations of motion include only quadratic nonlinearities and are expanded about the undcformcd rather than the deformed position 15i.…”

Nonlinear Dynamics of a Flexible Multirod (Multibeam) System

“…This matrix can be derived using various levels of accuracy. An accurate expression of the elastic rotation matrix for slender rods was presented by Rosen et al (1987). In many practical cases slender rods undergo moderate elastic rotations (where products of the rotations are small compared to unit) while the elastic strains remain small.…”

“…Using Eq. (7) instead of the exact expressions results in simplifications in the strain expressions and thus simplifies the expressions associated with the rod elasticity (for more details see Rosen et al, 1987). These simplifications lead to a more efficient model for common cases of moderate elastic deformations.…”