The dynamic problem of three-dimensional free surface is numerically studied in this paper. The ALE （Arbitrary Lagrange-Euler） kinematic description is introduced into the control equation system. The ALE descriptio...The dynamic problem of three-dimensional free surface is numerically studied in this paper. The ALE （Arbitrary Lagrange-Euler） kinematic description is introduced into the control equation system. The ALE description method is used to track free surface. Accurate formulations for calculating the normal vector on the free surface are presented. The discrete numerical equations by finite element method are developed by Galerkin weighted residual method. The boundary condition about free-surface tension is represented in the form of weak integration that can be computed by a differential geometry method derived in the present paper. The effect of contact angle is incorporated in the numerical algorithm. Furthermore, the numerical computations are performed and the comparison between computational and analytical results validated the effectiveness of the method. The results of this paper provide a fundamental understandings of the dynamics of liquid free surfaces, in which the surface tension and contact angle boundary conditions are taken into account. Finally, numerical simulation of largescale amplitude sloshing of liquid in a cylindrical container is performed and a numerical analysis of the effect of an annular ring-shaped rigid damping baffle on liquid sloshing oscillations in a cylindrical tank is also carried out.展开更多
This paper is focused on attitude tracking control of a spacecraft that is equipped with flexible appendage and partially filled liquid propellant tank. The large amplitude liquid slosh is included by using a moving p...This paper is focused on attitude tracking control of a spacecraft that is equipped with flexible appendage and partially filled liquid propellant tank. The large amplitude liquid slosh is included by using a moving pulsating ball model that is further improved to estimate the settling location of liquid in microgravity or a zero-g environment. The flexible appendage is modelled as a three-dimensional Bernoulli–Euler beam, and the assumed modal method is employed.A hybrid controller that combines sliding mode control with an adaptive algorithm is designed for spacecraft to perform attitude tracking. The proposed controller has proved to be asymptotically stable. A nonlinear model for the overall coupled system including spacecraft attitude dynamics,liquid slosh, structural vibration and control action is established. Numerical simulation results are presented to show the dynamic behaviors of the coupled system and to verify the effectiveness of the control approach when the spacecraft undergoes the disturbance produced by large amplitude slosh and appendage vibration. Lastly, the designed adaptive algorithm is found to be effective to improve the precision of attitude tracking.展开更多
In this paper, the attitude stability of liquid-filled spacecraft with flexible appendage is investigated. The motion of liquid sloshing is modeled as the spherical pendulum, and the flexible appendage is approached b...In this paper, the attitude stability of liquid-filled spacecraft with flexible appendage is investigated. The motion of liquid sloshing is modeled as the spherical pendulum, and the flexible appendage is approached by a linear shearing beam. Nonlinear dynamic equations of the coupled system are derived from the Hamiltonian. The stability of the coupled system was analyzed by using the energy-Casimir method, and the nonlinear stability theorem of the coupled spacecraft system was also obtained. Through numerical computation, the correctness of the proposed theorem is verified and the boundary curves of the stable region are presented. The increase of the angular velocity and flexible attachment length will weaken the attitude stability, and the change of the filled ratio of liquid fuel tank has a different influence on the stability of the coupled spacecraft, depending on the different conditions. The attitude stability analysis of the coupled spacecraft system in this context is useful for selecting appropriate parameters in the complex spacecraft design.展开更多
The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rat...The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.展开更多
The stability of partly liquid filled spacecraft with flexible attachment was investigated in this paper. Liquid sloshing dynamics was simplified as the spring-mass model, and flexible attachment was modeled as the li...The stability of partly liquid filled spacecraft with flexible attachment was investigated in this paper. Liquid sloshing dynamics was simplified as the spring-mass model, and flexible attachment was modeled as the linear shearing beam. The dynamic equations and Hamiltonian of the coupled spacecraft system were given by analyzing the rigid body, liquid fuel, and flexible appendage. Nonlinear stability conditions of the coupled spacecraft system were derived by computing the variation of Casimir function which was added to the Hamiltonian. The stable region of the parameter space was given and validated by numerical computation. Related results suggest that the change of inertia matrix, the length of flexible attachment, spacecraft spinning rate, and filled ratio of liquid fuel tank have strong influence on the stability of the spacecraft system.展开更多
This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important ...This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important form of mechanical system. It mostly used to stabilize the potential shaping of dynamical system. Free surface movement of liquid inside the container is called sloshing. If there is uncontrolled resonance between the motion of tank and liquid-frequency inside the tank then such sloshing can be a reason of attitude disturbance or structural damage of spacecraft. Equivalent mechanical model of simple pendulum or mass attached with spring for sloshing is used by many researchers. Mass attached with spring is used as an equivalent model of sloshing to derive the mathematical equations in terms of Hamiltonian model. Analytical method of Lyapunov function with Casimir energy function is used to find the stability for spacecraft dynamics. Vertical axial rotation is taken as the major axial steady rotation for the moving rigid body.展开更多
In the process of stage separation of recoverable liquid launch vehicles,because of the large amount of residual fuel in the storage tanks,the influence of liquid sloshing on separation safety must be considered.Consi...In the process of stage separation of recoverable liquid launch vehicles,because of the large amount of residual fuel in the storage tanks,the influence of liquid sloshing on separation safety must be considered.Considering calculation simplicity and operation practicability,the Moving Pulsating Ball Model(MPBM)of large amplitude liquid sloshing is introduced into the calculation of launch vehicle stage separation.Combining the dynamic equation of the model with the energy relationship during"breathing movement",the formula calculating the force of liquid on the rigid body is derived.Compared with the calculations of commercial CFD calculation software,the accuracy of MPBM model is verified.Then,all the external forces and moments are applied to the rigid body of the stages,so that the translational and rotational dynamic equations of the stages are obtained respectively.According to the relative position of the two stages,the geometric shape of the interstage section and the engine of the second stage,the minimum clearance in the separation process can be decided to guarantee that the separation process is safe.展开更多
According to the discrete Hamilton–Pontryagin variational principle,we construct a class of variational integrators in the real vector spaces and extend to the Lie groups for the left-trivialized Lagrangian mechanica...According to the discrete Hamilton–Pontryagin variational principle,we construct a class of variational integrators in the real vector spaces and extend to the Lie groups for the left-trivialized Lagrangian mechanical systems by employing the spectral-collocation method to discretize the corresponding Lagrangian and kinematic constraints.The constructed framework can be transformed easily to the well-known symplectic partitioned Runge–Kutta methods and the higher order symplectic partitioned Lie Group methods by choosing same interpolation nodes and quadrature points.Two numerical experiments about the orbit propagation of Kepler two-body system and the rigid-body flow propagation of a free rigid body are conducted,respectively.The simulating results reveal that the constructed update schemes can possess simultaneously the excellent exponent convergence rates of spectral methods and the attractive long-term structure-preserving properties of geometric numerical algorithms.展开更多
基金The project supported by the National Natural Science Foundation of China （10272022, 10572022） and the Basic Research Foundation of Beijing Institute of Technology （000Y07）. The English text was polished by Yunming Chen.
文摘The dynamic problem of three-dimensional free surface is numerically studied in this paper. The ALE （Arbitrary Lagrange-Euler） kinematic description is introduced into the control equation system. The ALE description method is used to track free surface. Accurate formulations for calculating the normal vector on the free surface are presented. The discrete numerical equations by finite element method are developed by Galerkin weighted residual method. The boundary condition about free-surface tension is represented in the form of weak integration that can be computed by a differential geometry method derived in the present paper. The effect of contact angle is incorporated in the numerical algorithm. Furthermore, the numerical computations are performed and the comparison between computational and analytical results validated the effectiveness of the method. The results of this paper provide a fundamental understandings of the dynamics of liquid free surfaces, in which the surface tension and contact angle boundary conditions are taken into account. Finally, numerical simulation of largescale amplitude sloshing of liquid in a cylindrical container is performed and a numerical analysis of the effect of an annular ring-shaped rigid damping baffle on liquid sloshing oscillations in a cylindrical tank is also carried out.
基金supported by the National Natural Science Foundation of China (Grants 11472041, 11532002)the Doctoral Fund of Ministry of Education of China (Grant 20131101110002)
文摘This paper is focused on attitude tracking control of a spacecraft that is equipped with flexible appendage and partially filled liquid propellant tank. The large amplitude liquid slosh is included by using a moving pulsating ball model that is further improved to estimate the settling location of liquid in microgravity or a zero-g environment. The flexible appendage is modelled as a three-dimensional Bernoulli–Euler beam, and the assumed modal method is employed.A hybrid controller that combines sliding mode control with an adaptive algorithm is designed for spacecraft to perform attitude tracking. The proposed controller has proved to be asymptotically stable. A nonlinear model for the overall coupled system including spacecraft attitude dynamics,liquid slosh, structural vibration and control action is established. Numerical simulation results are presented to show the dynamic behaviors of the coupled system and to verify the effectiveness of the control approach when the spacecraft undergoes the disturbance produced by large amplitude slosh and appendage vibration. Lastly, the designed adaptive algorithm is found to be effective to improve the precision of attitude tracking.
基金supported by the National Natural Science Foundation of China (Grants 11472041, 11532002)the Doctoral Fund of Ministry of Education of China (Grant 20131101110002)
文摘In this paper, the attitude stability of liquid-filled spacecraft with flexible appendage is investigated. The motion of liquid sloshing is modeled as the spherical pendulum, and the flexible appendage is approached by a linear shearing beam. Nonlinear dynamic equations of the coupled system are derived from the Hamiltonian. The stability of the coupled system was analyzed by using the energy-Casimir method, and the nonlinear stability theorem of the coupled spacecraft system was also obtained. Through numerical computation, the correctness of the proposed theorem is verified and the boundary curves of the stable region are presented. The increase of the angular velocity and flexible attachment length will weaken the attitude stability, and the change of the filled ratio of liquid fuel tank has a different influence on the stability of the coupled spacecraft, depending on the different conditions. The attitude stability analysis of the coupled spacecraft system in this context is useful for selecting appropriate parameters in the complex spacecraft design.
基金the National Natural Science Foundation of China (Nos. 11472041,11532002,11772049,and 11802320)。
文摘The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.
基金supported by the National Natural Science Foundation of China (11472041, 11532002)the Innovation Fund Designated for Graduate Students of Beijing Institute of Technology (2015CX10003)the Research Fund for the Doctoral Program of Higher Education of China (20131101110002)
文摘The stability of partly liquid filled spacecraft with flexible attachment was investigated in this paper. Liquid sloshing dynamics was simplified as the spring-mass model, and flexible attachment was modeled as the linear shearing beam. The dynamic equations and Hamiltonian of the coupled spacecraft system were given by analyzing the rigid body, liquid fuel, and flexible appendage. Nonlinear stability conditions of the coupled spacecraft system were derived by computing the variation of Casimir function which was added to the Hamiltonian. The stable region of the parameter space was given and validated by numerical computation. Related results suggest that the change of inertia matrix, the length of flexible attachment, spacecraft spinning rate, and filled ratio of liquid fuel tank have strong influence on the stability of the spacecraft system.
基金supported by Higher Education Commis- sion of Pakistan,National Natural Science Foundation of China(11072030)Ph.D.Programs Foundation of Ministry of Education of China(20080070011)+1 种基金Scientific Research Foundation of Ministry of Education of China for Returned Scholars(20080732040)Program of Beijing Municipal Key Discipline Construction
文摘This paper presents a canonical Hamiltonian model of liquid sloshing for the container coupled with spacecraft. Elliptical shape of rigid body is considered as spacecraft structure. Hamiltonian system is an important form of mechanical system. It mostly used to stabilize the potential shaping of dynamical system. Free surface movement of liquid inside the container is called sloshing. If there is uncontrolled resonance between the motion of tank and liquid-frequency inside the tank then such sloshing can be a reason of attitude disturbance or structural damage of spacecraft. Equivalent mechanical model of simple pendulum or mass attached with spring for sloshing is used by many researchers. Mass attached with spring is used as an equivalent model of sloshing to derive the mathematical equations in terms of Hamiltonian model. Analytical method of Lyapunov function with Casimir energy function is used to find the stability for spacecraft dynamics. Vertical axial rotation is taken as the major axial steady rotation for the moving rigid body.
基金supported by the National Natural Science Foundation of China(Nos.12132002,12202044)。
文摘In the process of stage separation of recoverable liquid launch vehicles,because of the large amount of residual fuel in the storage tanks,the influence of liquid sloshing on separation safety must be considered.Considering calculation simplicity and operation practicability,the Moving Pulsating Ball Model(MPBM)of large amplitude liquid sloshing is introduced into the calculation of launch vehicle stage separation.Combining the dynamic equation of the model with the energy relationship during"breathing movement",the formula calculating the force of liquid on the rigid body is derived.Compared with the calculations of commercial CFD calculation software,the accuracy of MPBM model is verified.Then,all the external forces and moments are applied to the rigid body of the stages,so that the translational and rotational dynamic equations of the stages are obtained respectively.According to the relative position of the two stages,the geometric shape of the interstage section and the engine of the second stage,the minimum clearance in the separation process can be decided to guarantee that the separation process is safe.
基金This work was supported by the National Natural Science Foundation of China(Grants 11772049,and 12132002).
文摘According to the discrete Hamilton–Pontryagin variational principle,we construct a class of variational integrators in the real vector spaces and extend to the Lie groups for the left-trivialized Lagrangian mechanical systems by employing the spectral-collocation method to discretize the corresponding Lagrangian and kinematic constraints.The constructed framework can be transformed easily to the well-known symplectic partitioned Runge–Kutta methods and the higher order symplectic partitioned Lie Group methods by choosing same interpolation nodes and quadrature points.Two numerical experiments about the orbit propagation of Kepler two-body system and the rigid-body flow propagation of a free rigid body are conducted,respectively.The simulating results reveal that the constructed update schemes can possess simultaneously the excellent exponent convergence rates of spectral methods and the attractive long-term structure-preserving properties of geometric numerical algorithms.
