Passive damping of large space structures
The complex stiffness matrix and the mass matrix of a uniaxial bar subjected to constrained layer damping over its entire length are derived exactly by solving the differential equations of motion of the three-layered structure. The stiffness and mass matrices of a bar with segmented damping treatments are obtained by assembling the corresponding matrices for each segment and eliminating the internal nodes using a reduction procedure similar to static condensation. The natural frequencies, mode shapes, and loss factor of a pinconnected truss containing several damped members are computed by three different methods: truss finite element (TFE) method (exact), equivalent beam element (EBE) method, and scaled beam element (SEE) method, each method being more efficient than the preceding one. A 10-bay plane truss is considered as an example to illustrate each method. The EBE method yields very good results, but the savings in computation is not significant. The SBE method reduces the computational effort drastically and gives reasonably approximate results. I. Introduction T HE space structures of future space stations and other such facilities would be typically latticed, lightweight, and flexible. During regular operation of these space stations, they would be subjected to excitation, causing undesirable low-frequency vibrations. The control of these vibrations is vital for the successful operation of the space structure. In addition to active controls, passive damping techniques can be employed to minimize the mass of components of the active control systems. Constrained layer damping is one of the efficient passive vibration control techniques.1 In this paper we have developed a series of analytical and numerical techniques for the analysis of a large space structure (pin-connected truss) subjected to constrained layer damping. Actually, these techniques are equally applicable to any type of passive damping treatment. Figure 1 depicts the hierarchy of models that can be used in analyzing a passively damped large space structure. First, a closed-form expression is derived for computing the complex stiffness matrix and mass matrix of a uniaxial bar subjected to damping treatment along the entire length (Fig. la). The problem of segmented treatment on a uniaxial bar is solved by considering the bar as an assemblage of fully treated bars (Fig. Ib). The stiffness and mass matrices of the various uniaxial members are assembled to form the global stiffness and mass matrices of the truss structure (Fig. Ic). The loss factor can be computed using the modal strain energy method.2 If the truss has a large number of members—which is typical of large space structures—then an equivalent beam or plate model can be derived. There are several approaches for deriving the equivalent continuum model for an undamped space structure, e.g., Sun et al., 3 Noor et al.,4 and Lee.5 All of these methods are based on the assumption that the large space structure has a repeating unit cell. In this paper we have modified the method described by Lee5 to derive the complex stiffness and mass matrix of an equivalent beam element, which can then be used in the analysis of large structures. Two models, equivalent beam element (EBE) method (Fig. Id) and scaled beam element (SBE)