APPROXIMATE SOLUTIONS FOR VIBRATIONS OF DEPLOYING APPENDAGES

The existing investigations of the dynamics of deploying appendages are numerical in nature. This paper presents approximate analytical solutions that may be used as "benchmarks" to check numerical solutions of more general equations of motion and for quick estimates in the early design phase. For beam-type appendages, the square of the length varies linearly with time. For tethered systems, results for the common exponential deployment (or retrieval) are presented. The solutions provide new qualitative results that, during extension (or retraction), the amplitudes of transverse vibration of beam-type and tether-type appendages are proportional to L1/2 and L-1/4, respectively, where L is the instantaneous length of the appendage. Thus, when the length is increasing during deployment, the vibratory amplitude of a beam increases whereas that of a tether decreases. The converse is true during retraction. The approximate solutions agree reasonably well with computer-intensive numerical solutions of more general equations.