Abstract:Summary
As a typical high‐dimensional nonlinear dynamic problem, the difficulties of the dynamic analysis on the spatial flexible damping beam mostly result from the coupling between the spatial motion and the transverse vibration. Considering the coupling effect and the weak structure damping, the dynamic behaviors of the spatial flexible beam are investigated by a complex structure‐preserving method in this paper. Based on the variational principle, the dynamic model of the spatial flexible damping beam is e… Show more
“…Considering the orbit-attitude coupling effect of the large-stiffness slender spatial component shown in Figure 1, a spatial rigid rod model simplified from the spatial flexible damping beam (Hu and Deng, 2018; Hu et al, 2017a) will be investigated by the structure-preserving method (Hu et al, 2013a, 2017b; Hu, 2018) in this article. The purpose of proposing this simplified model is to improve the simulation speed of the numerical analysis on the spatial beam if the stiffness of the beam is large enough, and the flexible vibration can be neglected safely.Figure 1.Spatial rigid rod model.…”
Section: Orbit-attitude Coupling Dynamic Model For Spatial Rigid Rodmentioning
confidence: 99%
“…In the last 30 years, the structure-preserving idea has been generalized from the finite-dimensional Hamiltonian system to the infinite-dimensional Hamiltonian system (Bridges, 1997; Marsden et al, 1998) as well as to the nonconservative dynamic system (Hu, 2018; Hu et al, 2013a, 2013b, 2017b, 2020; Hu and Deng, 2019). With the aforementioned developments on the structure-preserving idea, theoretically, most of the spatial components, regardless of rigid or flexible, can be analyzed by the structure-preserving method.…”
An orbit-attitude coupling dynamic model for the spatial rigid rod that is abstracted from the large-stiffness slender components widely used in spatial structures is established, and the symplectic method is used to estimate the validity of the dynamic model by analyzing the coupling dynamic behaviors of the rod in this work. Based on the Hamiltonian variational principle, the orbit-attitude dynamic model of the spatial rigid rod is proposed, and the canonical form of the model is presented first. Then, the symplectic Runge–Kutta method is developed, and the structure-preserving properties of the canonical form, including the conservation law of energy and conservative property in the phase space, are investigated to illustrate the validity of the numerical results obtained by the symplectic Runge–Kutta method subsequently. Finally, the effects of the nonspherical perturbations of the Earth on the coupling dynamic behaviors are investigated numerically. From the simulation results, it can be concluded that the main orbit-attitude coupling dynamic behaviors of the spatial large-stiffness slender component excited by the nonspherical perturbation can be described by the proposed dynamic model ignoring the deformation as well as the transverse vibration of the slender component, which provides an approach for simplifying rapid dynamic analysis on the spatial large-stiffness slender component. In addition, the validity and the structure-preserving properties of the symplectic Runge–Kutta method for the orbit-attitude coupling dynamic problem of the spatial rigid rod are also illustrated.
“…Considering the orbit-attitude coupling effect of the large-stiffness slender spatial component shown in Figure 1, a spatial rigid rod model simplified from the spatial flexible damping beam (Hu and Deng, 2018; Hu et al, 2017a) will be investigated by the structure-preserving method (Hu et al, 2013a, 2017b; Hu, 2018) in this article. The purpose of proposing this simplified model is to improve the simulation speed of the numerical analysis on the spatial beam if the stiffness of the beam is large enough, and the flexible vibration can be neglected safely.Figure 1.Spatial rigid rod model.…”
Section: Orbit-attitude Coupling Dynamic Model For Spatial Rigid Rodmentioning
confidence: 99%
“…In the last 30 years, the structure-preserving idea has been generalized from the finite-dimensional Hamiltonian system to the infinite-dimensional Hamiltonian system (Bridges, 1997; Marsden et al, 1998) as well as to the nonconservative dynamic system (Hu, 2018; Hu et al, 2013a, 2013b, 2017b, 2020; Hu and Deng, 2019). With the aforementioned developments on the structure-preserving idea, theoretically, most of the spatial components, regardless of rigid or flexible, can be analyzed by the structure-preserving method.…”
An orbit-attitude coupling dynamic model for the spatial rigid rod that is abstracted from the large-stiffness slender components widely used in spatial structures is established, and the symplectic method is used to estimate the validity of the dynamic model by analyzing the coupling dynamic behaviors of the rod in this work. Based on the Hamiltonian variational principle, the orbit-attitude dynamic model of the spatial rigid rod is proposed, and the canonical form of the model is presented first. Then, the symplectic Runge–Kutta method is developed, and the structure-preserving properties of the canonical form, including the conservation law of energy and conservative property in the phase space, are investigated to illustrate the validity of the numerical results obtained by the symplectic Runge–Kutta method subsequently. Finally, the effects of the nonspherical perturbations of the Earth on the coupling dynamic behaviors are investigated numerically. From the simulation results, it can be concluded that the main orbit-attitude coupling dynamic behaviors of the spatial large-stiffness slender component excited by the nonspherical perturbation can be described by the proposed dynamic model ignoring the deformation as well as the transverse vibration of the slender component, which provides an approach for simplifying rapid dynamic analysis on the spatial large-stiffness slender component. In addition, the validity and the structure-preserving properties of the symplectic Runge–Kutta method for the orbit-attitude coupling dynamic problem of the spatial rigid rod are also illustrated.
“…e effects of higher order terms on the orbit and attitude motions were analyzed. Hu et al [16] established the coupled dynamic model of space flexible beams by FFRF and analyzed the effect of weak damping on the vibration of beams according to six different initial values. Afterwards, considering the effect of aspheric perturbation, Hu and Deng [17] established the coupled dynamic model of flexible spatial beams by using FFRF and analyzed the effect of nonsphere perturbation of the Earth and weak damping of beams on the transverse vibration and the stable fixed point of beams.…”
In this paper, a dynamic model for long-term on-orbit operation of the tethered solar power satellite (Tethered-SPS) is established. Because the Tethered-SPS is a super-large and super-flexible structure, the coupling among the orbit, attitude, and structure vibration of the system should be considered in comparison with the traditional spacecraft dynamic models. Based on the absolute nodal coordinate formulation (ANCF), the dynamic equation of the Tethered-SPS is established by Hamiltonian variational principle, and the dual equations of the Hamiltonian system are established by introducing generalized momentum through Legendre transformation. The symplectic Runge–Kutta method is used for numerical simulation, and the validation of the modeling method is verified by a numerical example. The effects of orbital altitude, initial attitude angle, length of solar panel, and orbital eccentricity on the orbit and attitude of the Tethered-SPS are analyzed. The numerical simulation results show that the effect of orbital altitude and length of solar panel on the orbital error of midpoint of beam is small. However, the initial attitude angle has a significant effect on the orbital error of midpoint of solar panel. The effect of the length of solar panel on the attitude angle of the system is not significant, but orbital altitude, orbital eccentricity, and initial attitude angle of the system severely affect the attitude angle of the system. Then, the stability of the system is affected.
“…(1) reduces to the classical SMK equation which was considered for exact travelling wave solutions and Cls in [49][50][51]. Moreover, one can nd more details on the construction of analytical, exact, numerical solutions, and other information for classical NLPDEs, in [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69].…”
Abstract:In this work, Lie symmetry analysis for the time fractional simpli ed modi ed Kawahara (SMK) equation with Riemann-Liouville (RL) derivative, is analyzed. We transform the time fractional SMK equation to nonlinear ordinary di erential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in the Erdelyi-Kober (EK) sense. We solve the reduced fractional ODE using a power series technique. Using Ibragimov's nonlocal conservation method to time fractional partial di erential equations, we compute conservation laws (Cls) for the time fractional SMK equation. Some gures of the obtained explicit solution are presented.
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