A nonlinear stress-strain relation

Orthwein, William C.

doi:10.1016/0020-7683(68)90044-9

“…where Q 1 and Q 2 are the shear forces; N is the axial force; M 1 and M 2 are the bending moments; and T is the twisting moment. Using the third-order Maclaurin series expansion of nonlinear stress-strain relations in Orthwein (1968) results in the constitutive equations:…”

Section: Kineticmentioning

confidence: 99%

Boundary tracking control of flexible beams for transferring motions

Duc

¹

2020

International Journal of Systems Science

2

0

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Although flexible beams for transmitting both translational and rotational large motions are used in practice such as ocean drill pipes, their control has not been considered. This paper develops boundary feedback controllers to stabilize these beams at their reference configurations. Exact nonlinear partial differential equations governing motion of the beams in three-dimensional space are derived and used in the control design. The designed controllers guarantee globally practically asymptotically stability of the beam mo-tions at the reference states (i.e., positions and rotations of a straight beam moving axially with a desired velocity and rotating around its axial axis with a desired velocity). In the control design and analysis of well-posedness and stability, we utilize different transformations between the earth-fixed and body-fixed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed for a class of evolution systems in Hilbert space. Simulation results are also included to illustrate the effectiveness of the proposed control design.

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“…where Q 1 and Q 2 are the shear forces; N is the axial force; M 1 and M 2 are the bending moments; and T is the twisting moment. Using the third-order Maclaurin series expansion of nonlinear stress-strain relations in [43] results in the constitutive equations:…”

Section: Kineticmentioning

confidence: 99%

Stochastic stabilization of slender beams in space: Modeling and boundary control

Duc

¹

,

Lucey

²

2018

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This paper considers the problem of modeling and boundary feedback stabilization of extensible and shearable slender beams with large deformations and large rotations in space under both deterministic and stochastic loads induced by flows. Fully nonlinear equations of motion of the beams are first derived. Boundary feedback controllers are then designed for global practical exponential p-stabilization of the beams based on the Lyapunov direct method. A new Lyapunov-type theorem is developed to study well-posedness and stability of stochastic evolution systems (SESs) in Hilbert space.

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“…where Q 1 and Q 2 are the shear forces; N is the axial force; M 1 and M 2 are the bending moments; and T is the twisting moment. Using the third-order Maclaurin series expansion of nonlinear stress-strain relations in [34] results in the constitutive equations:…”

Section: Kinematicsmentioning

confidence: 99%

“…where ϵ 11 is a positive constant to be chosen, and we have used Dr 0 = col(0, 0, 1) and the second equation of (34). Using the third equation of (34) and integration by parts, we can calculate LU 12 as follows:…”

Section: Control Designmentioning

confidence: 99%

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Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback

Duc

¹

2018

Journal of Sound and Vibration

12

0

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Boundary feedback controllers are designed to stabilize Timoshenko beams with large translational and rotational motions in space under external disturbances. The exact nonlinear partial differential equations governing motion of the beams are derived and used in the control design. The designed controllers guarantee globally practically asymptotically (and locally practically exponentially) stability of the beam motions at the reference state. The control design, well-posedness and stability analysis are based on various relationships between the earth-fixed and body-fixed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed to study well-posedness and stability for a class of evolution systems in Hilbert space. Simulation results are included to illustrate the effectiveness of the proposed control design.

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A nonlinear stress-strain relation

Cited by 14 publications

References 6 publications

Boundary tracking control of flexible beams for transferring motions

Boundary tracking control of flexible beams for transferring motions

Stochastic stabilization of slender beams in space: Modeling and boundary control

Stabilization of exact nonlinear Timoshenko beams in space by boundary feedback

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