Dynamic Characteristic Formulations for Jointed Space Structures

New mathematical formulations are described for estimating the dynamic characteristics of deployable space structures. The two main effects of the structure joints are transitions in natural frequencies and energy dissipation. These effects are formulated individually by using a nonlinear spring model and friction- and impact-damping models. The total effects of the joints are obtained by integrating the models using energy loss factors. The formulations are quite efficient compared to numerical analyses because the dynamic characteristics can be obtained mathematically for each cycle without time-consuming calculations. Analyses and experimental evaluations show the dynamic characteristics and demonstrate the validity of the formulations. Nomenclature A = cross-sectional area in the joint a = amplitude in the contact condition (maximum deflection of the beam) b = joint width (clearance width) C = decrement gradient E = Young’s modulus of the joint EFL = quarter-cycle energy dissipation caused by friction EFLg = quarter-cycle friction loss caused by gravity E I = internal energy of the beam ERL = quarter-cycle energy dissipation caused by impact e = restitution coefficient of the joint f = natural frequency of the beam fb = shearing stress by friction (friction stress) on bottom surface in the joint fu = shearing stress by friction (friction stress) on upper surface in the joint f0 = natural frequency in the noncontact condition f1 = natural frequency in the contact condition h = joint height h0 = offset between the centroid axis and the neutral axis (zero-strain axis) of the joint (Fig. 6) I = moment of inertia of area of the joint k = equivalent stiffness of the beam k0 = stiffness in the noncontact condition k1 = stiffness in the contact condition L = beam length l = effective joint length (clearance length) M = bending moment in the joint M0 = bending moment at x = 0 m = mass of the beam N = shearing force in the joint n = vertical stress (constricted stress) in the joint P = external force on the beam Q = vertical force at x = l q = vertical stress in the joint R = vertical force at x = 0 r = vertical stress in the joint T = reactive force in the joint t = time