Structural modifications to reduce the LOS-error in large space structures

This paper deals with the study of the dynamic behavior of large space structures due to changes in the stiffness of the members. The stiffness of the members was modified to satisfy an optimum design criterion, to satisfy static displacements associated with the line-of-sight (LOS) and to satisfy frequency constraints. The response of the LOS to different initial displacement conditions was investigated for various modified designs. Introduction There are different stringent requirements on Large Space Structures (LSS) because of their use, environment and the methods usedkto deploy them. LSS have to be a minimum weight design and also satisfy all the imposed performance requirements. One of the important requirement is the Line-ofSight (LOS) pointing. The error associated with this has to be within a specified value for proper operation of the vehicle. The LOS error may be defined as the movement of a node point or a linear combination of movements at different locations in the structure due to a dynamic disturbance. The LOS error depends on the stiffness of the structure, the distribution of nonstructural mass and the control system used to minimize the error. The correct procedure to obtain an optimum design would be to include all the design variables in the formulation of the problem and to derive a proper algorithm to minimize the specified objective functions and satisfy all the constraints. In order to solve this problem it would be necessary to deal with two disciplines of LSS design such as structural design and control theory. This is a difficult problem to solve because it would be necessary to take into consideration the interaction of the two disciplines. In this paper a simple procedure to investigate the effect of structural modifications on the passive control of the LOS error is used. The structure is modified to satisfy constraints on static displacements associated with the LOS and the frequency distribution. Different designs are then compared for their dynamic behavior. The structure used in this investigation is a tetrahedral truss discussed in Ref. 1. This structure was devised by Draper Labs as one of the simplest nonplaner geometries capable of representing a Large Space Structure. A short description of the methods used to modify and analyze the structure is given in the next section. *Aerospace Engineer, Member AIAA **Professor, Associate Fellow AIAA Static Analysis The relation between the static and the displacement vector Iu3 for n degrees of freedom is given by [ K l f u}={P} load vector {PI a structure with where [K] is the nxn symmetric stiffness matrix of the structure. The displacement u at a node point j in the structure is given by j where m is the number of elements and Qij is the flexibility coefficient given by Here {uIi and {sjli are the displacements associated with the ith element due to the applied load vector {PI and the virtual load vector {sJ}. In Eq 3 [kIi is the stiffness matrix of the ith element and Ai is the ith design variable, which in the case of bar elements will be the cross-sectional area. The solution to Eq 1 gives the displacements in the structure, and Eq 2 and 3 can be used to determine the displacement at a specified node point. Dynamic Analysis The equations of motion for a structure are given by where [MI is a nxn mass matrix which includes the contribution from the structural and the nonstructural mass of the structure, [El is a nxn damping matrix and [Dl is the nxp applied load distribution matrix relating the applied load vector C~(t)l and the coordinate system. For the free undamped motion Eq 4 becomes Since the free oscillations are simple harmonic, the displacement u(t) can be written as This paper is declared a worl of the U.S. Government and therefore is in the public domaln. where w is the circular frequency and {U} is the vector defining the amplitudes of motion. Substituting Eq 6 in Eq 5 and cancelling eht gives the homogeneous set of equations, Eq 15 is known as the state input equation and the state output equation is given by t Multiplying this equation by {u) and rearranging gives where {y) is a qxl output vector and [C] is a qx2n output matrix. The complete solution to Eq 15 is given by2 where wj is the jth circular frequency associated with the jth eigenvector {Uj). Using the coordinate transformation, where $(t) is the state transition matrix and is given by where [$] is the modal matrix whose columns are eigenvectors {UIj (normalized with respect to the mass matrix [MI), Eq 4 can be transformed into n uncoupled system of differential equations as In Eq 19 11 and fare the vectors and $, B, A are the matrices. For discrete time steps Eq 19 can be suitably modified and evaluated at each time interval. In the case where there are no externally applied loads or the input vector If} is zero, Eq 19 becomes where [MI= [CI %Inxn identify matrix [El = [+25w Jnxn diagonal damping matrix (12) The numerical solution to Eqs 19 or 21 can be obtained by using available subroutines. Algorithm to Minimize Displacements The optimization problem for minimizing the displacements-in a structure for a given weight of the structure W can be stated as: 2 [K]= [*w ,lnxn matrix of eigenvalues (13) In Eq 12 {5) is the vector of modal damping factors. The second order Eq 10 can be reduced to a first order equation by using the state variable vector {XI given by minimize the displacement function 'G' subject to The state space representation of Eq 10 using Eq 14 can be written as where pi is the density of the material, Ai is the design variable and 9.i is the volume parameter associated with the ith element. Using Eq 2 the function G may be written as where [A] is 2nx2n plant matrix, [B] is 2nxp input matrix and {f) is an px2n control input vector. The plant matrix and the input matrix are of the form m G= 1 a single displacement i=l Ai [A] = -0-1-1-[-2;-2J can be written as 4,5 ' 1 m Q G= I: R. 1 3 a linear combination j=l J ~ = ~ Ai of the displacements (24) where v+l and v are the iteration numbers and r is the parameter that controls the step size. The value of the Lagrange parameter X which is required to use in Eq 32 can be calculated by solving Eqs 28 through 30 for Ai substituting in Eq 22 and solving for A. This gives 2 a linear combination of the squares of the displacements (25) where Rj is a set of specified parameters and pl is the number of parameters. Using Eqs 22 and 23 through 25, the Lagrangian can be written as for a single displacement, where X is the Lagrange multiplier. Differentiating Eq 26 with respect to the design variables Ai and setting the resulting equations to zero, the optimality conditions can be written as for a linear combination of displacements, Using Eqs 23 through 25 the optimality conditions for the three objective functions can be written as single displacement for a linear combination of the square of the displacements. The algorithm to minimize the displacements then consists of modifying the design variables by using Eq 32 until the optimality criterion is satisfied. linear combination of the displacements Algorithm to Satisfy Frequency Constraints The optimization problem for the minimum weight of the structure with constraints on the fundamental frequency and the frequency distribution of the higher vibration modes can be defined as: linear combination of the square of the displacements minimize the weight of the structure PI Qij 1=X 1 2R.G j=l j b:pili subj acted to m Qij In Eq 30 Gj = I q. Egs 28 through 30 can be i=l written as 2 2 =(w -a w )<1 j=l,.. ., j j l P1 where the value of Bi depends on the nature of the objective function. Using the optimality criterion a recurrence relation to modify the design variables and gj are the actual and the desired valz!?Efglhe constraints. Gl is the s ecified value S of the fundamental frequency and ajml is the desired value of the square of the jth frequency. Using Eqs 36 and 38 the Lagrangian can be written as: Differentiating this equation with respect to the design variable Ai and setting the resulting equations to zero gives6 Eq 45 was derived by assuming that all the elements would satisfy the optimality criterion. However, the cross sectional areas of some of the elements would be governed by the minimum size requirement. In that case these elements are not included in the summation in Eq 45. One of the design requirements can be that the fundamental frequency or some other frequency of the structure should be equal to a specified value. This would require the scaling of all the design variables. The scaling factor 6j is given by6