Improved method for calculating booster to launch pad interface transient forces

Abstract

This paper presents a methodology for the calculation of the transient interface forces between a launch vehicle and its launch pad. The method presented is closed form and, thus, is mathematically exact. The method uses variable constraint equations to couple the launch vehicle to the pad. The resulting interface forces are monitored each time step, and as the launch vehicle separates from the pad, the constraint equations are modified accordingly. The only assumption in the method is that the forces acting on the booster and pad, including the subject interface forces, vary linearly between integration time steps. This is the same assumption used in standard linear transient loads analyses. In addition, the nature of the constraint equations allows a great reduction in the number of coupled equations to be solved. Solving the resulting small system of equations yields the interface forces, which are then used to solve for the booster and pad response by the standard linear transient solution method. Use of constraint equations also frees the analyst from calculating coupled boosterpad systems modes, which are a necessity in some methodologies currently in use. Nomenclature} A,A' ,B,B' = coefficients of integrationD = all known variables to the modal displacement solution F t F' = coefficients of integration / = physical interface force {J = physical force G,G ' = coefficients of integration / = identity matrix K = discrete physical stiffness matrix P = generalized force q = dummy variable V = all known variables to the modal velocity solution x = discrete physical displacement (3 = modal damping coefficient A = displacement difference d = interface displacement mismatch £ = modal displacement degree of freedom </ > = eigenvectors co = eigenvalues 0 = null matrix Subscripts b = booster / = modal degree of freedom j = physical degree of freedom k = modal degree of freedom n = time step p = launch pad Superscripts r = residual mode t = untruncated modes, i.e., modes retained in the analysis T = matrix transposeO Introduction NE of the prime drivers in booster-payload system liftoff response is the time history of the pad forces. As the