Equivalent Beam Element Research Articles

Spectral Element Approach for the Homogeneous Continuum Representation of a Periodic Lattice Structure

Motivated by the apparent accuracy of the spectral element in comparison with the conventional finite element, a new dynamic continuum modeling method is developed by using the spectral element to represent a large periodic lattice beam as an equivalent continuum beam model. As the first step, the transfer matrix for a representative lattice cell of the periodic lattice beam is numerically derived in terms of the continuum degrees of freedom introduced in this paper. The global dynamic stiffness matrix is then obtained by assembling the spectrally formulated dynamic stiffness matrices for the structural elements within the lattice cell. As the second step, the transfer matrix for an equivalent continuum beam element is analytically derived in terms of the unknown equivalent continuum structural properties. Lastly, the two transfer matrices are forced to be equal to each other to determine the equivalent continuum structural properties of the continuum beam model. In this paper, the equivalent continuum structural properties and vibration characteristics of an equivalent continuum beam model by the present method are compared with those by the other existing continuum methods to show that the present continuum model gives vary reliable bending vibration characteristics of the original lattice structures compared to others.

Dynamic continuum modeling of truss-type space structures using spectral elements

A new continuum modeling approach is developed by utilizing the spectral element method to represent a large periodic truss-type beam as an equivalent continuum extended Timoshenko beam model. First, the transfer matrix for a representative truss cell of the periodic truss-type beam is numerically derived by condensing the global dynamic stiffness matrix by use of the continuum degrees of freedom introduced. The global dynamic stiffness matrix is obtained by assembling the spectrally formulated dynamic stiffness matrices for the structural elements within the truss cell. Next, the transfer matrix for an equivalent continuum beam element is analytically derived so that the unknown equivalent structural properties are contained in the transfer matrix. The two transfer matrices are then forced to be equal to each other to determine the equivalent structural properties of the continuum beam model. The equivalent structural properties and vibration characteristics of a continuum beam model by the present continuum method are compared with those by other continuum methods to prove that the present continuum method can give a very reliable continuum model.

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)