Nonlinear analysis of pretwisted rods using 'principal curvature transformation'. II - Numerical results

Rosen, Alon; Loewy, Robert G.; Mathew, Mathew B.

doi:10.2514/3.9669

Cited by 19 publications

(4 citation statements)

References 5 publications

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“…1), then the position vector of any point after deformation is given by e z + ye yl + ze z (2) where ^ is the warping displacement of the cross section, which is to be assumed small compared to v and w. Substitution of Eqs. (lb) and (Ic) into Eq.…”

Section: The Kinetic Energy Of a Deformed Rodmentioning

confidence: 99%

Nonlinear dynamics of slender rods

1987

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The nonlinear equations of motion of a rod undergoing small strains and moderate elastic rotations are derived. The derivation uses a variational approach and the equations of motion are obtained by applying Lagrange's equations. The results of a recently developed "principal curvature transformation" are employed to account for the structural aspects. Generalized coordinates are also used to establish an efficient analysis method. By a direct perturbation of the nonlinear equations of motion, the relations governing the small vibrations of a rod superimposed on finite static deformations are obtained. The theory is validated by comparing its results with experimental results available in the literature. Good agreement is obtained. In addition, the influences of nonlinear effects on such vibrations are pointed out and discussed.

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Section: The Kinetic Energy Of a Deformed Rodmentioning

confidence: 99%

Nonlinear dynamics of slender rods

1987

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“…The tip twist angle θ Tip vs the inclination angle γ v , for three different cases of end load P TiP is shown in Figure 13(d). The proposed results are compared with corresponding numerical results (model D) in Rosen et al (1987a), and with the experimental results in Dowell and Traybar (1975). This comparison shows that the present results are in good agreement with the previously published results, especially with the experimental results.…”

Section: Cantilever Beam Subjected To a Vertical End Load At The Free...mentioning

confidence: 98%

Utilizing Tait-Bryan Angles for Large Displacement Corotational Finite Element Static Analysis of Spatial Beams

Elerian,

Shebl,

Elkaranshawy

2024

Lat. Am. j. solids struct.

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In this work, a corotational finite element formulation is suggested for spatial beams with geometrically nonlinear behavior subjected to static loads. We returned to the three successive rotation angle procedure, mainly the Tait-Bryan angles. By carefully defining the trigonometric rules for all rotation angles, the singularity problem, that had limited the use of these angles, is avoided. Three different types of coordinate systems are used: a fixed global coordinate system that stays fixed throughout the analysis, a fixed local coordinate system that is fixed and precisely attached to each element, and a corotational local frame for each element that moves and rotates together with the element throughout the analysis. The rigid body motion can easily be separated from the overall deformation since the deformation is always tiny relative to the corotational frame. An incremental-iterative method is used for the solution based upon the Newton-Raphson method. Different examples are solved to demonstrate the practicality, correctness, and accuracy of the proposed method. The solutions converge at a relatively quick rate.

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“…In the development of nonlinear equations of motion for a flexible beam posed in threedimensional space, two or three successive Euler-like rotations are commonly used to obtain the exact transformation matrix that relates the deformed and undeformed states and accounts for the geometric nonlinearities [9,12,[21][22][23][24][25][26][27][28][29][30][31], And, Taylor series expansions coupled with an ordering scheme are usually used to obtain nonlinear polynomial equations of motion. Alkire [25] showed that the transformation matrix and curvatures obtained by using Euler-like rotations are unique, but the form varies according to the choice of the twist variable.…”

Section: Introductionmentioning

confidence: 99%

“…Alkire [25] showed that the transformation matrix and curvatures obtained by using Euler-like rotations are unique, but the form varies according to the choice of the twist variable. Hence, different sequences of Euler rotations will result in different forms for the equations of motion, and they can be asymmetric because there are implicit asymmetric terms included in the twist variable [21,28,29].…”

Section: Introductionmentioning

confidence: 99%

A nonlinear composite beam theory

Pai

Nayfeh

1992

Nonlinear Dyn

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Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the yon Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the yon Karman type nonlinear theory.

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Nonlinear analysis of pretwisted rods using 'principal curvature transformation'. II - Numerical results

Cited by 19 publications

References 5 publications

Nonlinear dynamics of slender rods

Nonlinear dynamics of slender rods

Utilizing Tait-Bryan Angles for Large Displacement Corotational Finite Element Static Analysis of Spatial Beams

A nonlinear composite beam theory

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