Plane motion dynamics of a space-based crane-manipulator of the arm type taking into account the links’ bending

The paper considers the plane motion dynamics of a three-link crane-manipulator consisting of two rods elastic in bending and a massive solid body attached to them at their end (load capture). All links are interconnected by hinge joints with the given (controlled) relative rotation angles. The first link is also connected to the mobile base. The system mathematical model was developed to calculate its non-stationary oscillations and dynamic loading of links at arbitrary in time and large relative angles of the links rotation as the solid bodies. The rods bending deformations (elastic transverse displacements and rotation angles of the axes) were considered to be insignificant (linear). Each rod ending was represented by the Ritz method using two given shape functions with the unknown coefficients, which were taken as the generalized coordinates. In this case, the “cantilever” bending forms of each rod with the given transverse displacement and angle of rotation at their ends were used. A simplified version was considered: each rod bending was represented by only one given form of its bending by a single moment applied at the end. System oscillation equations were obtained based on the principle of probable displacements in the generalized coordinates (four equations or two in a simplified form). Since rotation angles at the ends of the elastic rods were added to the given rotation angles of the non-deformable rods, total angles were left under the signs of sines and cosines without linearization for convenience of calculations and increasing accuracy. As a result, such equations of non-stationary oscillations remained non-linear. They were integrated by numerical methods using standard programs in one of the computer algebra systems. Examples are provided with the comparative results estimates obtained in the two-term and one-term approximations of the function in regard to the bending form of the system bending rod joints.

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