The Study of the Tether Motion with Time-Varying Length Using the Absolute Nodal Coordinate Formulation with Multiple Nonlinear Time Scales

Abstract

In this study, the modeling and formulation for tether motion with time-varying length, large rotation, large displacement and large deformation are proposed. A tether or cable is an important element in lift systems, construction machines for transportation and often is used with a time-varying length. In some cases, these systems are large and the tether has a long length, large deformation and large displacement. The dynamic behavior of a tether in extension and retraction using the proposed method is discussed in this paper. In the passage through resonance, significant tether motions with large rotation and large deformation result. In the analysis of this phenomenon, the transient fluctuations of the motion amplitudes are examined and compared with the corresponding steady state motions. The accuracy and the cost of the calculations are also verified by comparison with the experimental results.

Numerical approach to modeling flexible body motion with large deformation, displacement and time-varying length

“…We propose a numerical analysis method for flexible body motion with large deformation, rotation and time-varying length by formulating a variabledomain finite element (VFE) model using the ANCF (Kawaguchi, 2007). We refer to this method as the VFE-ANCF.…”

“…Thus, this VFE method is appropriate for expressing the time-varying length of the flexible body, but the cost of calculation using this method is very high. In order to reduce the cost of calculation, the multiple timescale (MTS) method has been applied (Kawaguchi et al, 2007). However, the timescales used in the MTS method have not been sufficiently considered.…”

Nondimensional analysis of a two-dimensional shear deformable beam in absolute nodal coordinate formulation

“…Then nondimensional global displacements of equation (16) are expressed by equation (17), and the nondimensional global slopes are defined by equation (18)…”

Dynamics of variable-length tethers with application to tethered satellite deployment

“…Vu-Quoc and Li [19] used the finite element method to analyze the nonlinear dynamics of a planar sliding beam with a large deformation and rotation. Zhu and Ni [20] and Zhu and Chen [21] theoretically and Nomenclature r i ; r 0 i position and slope vectors of the ith node of a tether r, r 0 position and slope vectors of any particle on a tether element r 00 spatial derivative of the slope vector of any particle on a tether element v 1 inlet boundary mass flow velocity of a tether v N outlet boundary mass flow velocity of a tether x arc-length coordinate of any point on an un-stretched tether element x 1 arc-length coordinate of the left node of a tether element with inlet boundary mass flow x 2 arc-length coordinate of the right node of a tether element without boundary mass flow t time f distributed external forces on a tether element q generalized coordinates of a tether element N shape functions of a tether element I 3Â3 3 Â 3 identity matrix l un-stretched length of a tether element n normalized coordinate of a tether element _ Ã; € Ã first and second time derivatives of a variable m i , dR i mass and virtual displacement of the ith particle F i applied forces on the ith particle dr virtual displacement of any particle on a tether element dq generalized virtual displacement of a tether element F E , F D distributed elastic forces and distributed internal damping forces on a tether element A cross-sectional area of a tether E Young's modulus of a tether q mass density of a tether J cross-sectional area moment of inertia of a tether e 0 engineering normal strain of a tether element j curvature of a tether element c damping coefficient of a tether element M ele , M v,ele , M q,ele generalized mass matrix, generalized damping matrix, and generalized stiffness matrix of a tether element Q ele generalized forces on a tether element M t , q t generalized mass matrix and generalized coordinates of a tether Q t generalized forces on a tether Lagrangian multiplier associated with the quaternion normalization constraint of the ith rigid body A i b coordinate transformation matrix of the ith rigid body q i,k position vector of the kth joint in the local reference frame of the ith rigid body O À XYZ global reference frame o i À x i y i z i local reference frame of the ith rigid body N b , N t , N C numbers of rigid bodies, tethers, and constraints of a multibody system x generalized coordinates of a multibody system h;h equivalent differential governing equations and nonlinear governing equations of a multibody system at a time instant t 0 , t n initial time instant and the nth time instant in the numerical solution procedure s order of the integration formulation x c , y c Cartesian coordinates for the elevator model L(t) cable length at time t for the elevator model L 0 cable length at the start of movement for the elevator model L end cable length at the end of movement for the elevator model [22] used the absolute nodal coordinate formulation (ANCF) to simulate a planar tether with a time-varying length. However, all of the above studies either simplified the models or restricted the boundary conditions, and it is difficult to directly apply these methods to tethered satellite systems.…”