method of multiple scales
Springer Online Journal Archives 1860-2000
Abstract A critical problem in designing large structures for space applications, such as space stations and parabolic antennas, is the limitation of testing these structures and their substructures on earth. These structures will exhibit very high flexibilities due to the small loads expected to be encountered in orbit. It has been reported in the literature that the gravitational sag effect under dead weight is of extreme importance during ground tests of space-station structural components [1–4]. An investigation of a horizontal, pinned-pinned beam with complete axial restraint and undergoing large-amplitude oscillations about the statically deflected position is presented here. This paper presents a solution for the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity. The governing equation of motion used for the analysis is the Bernoulli-Euler type modified to include the effects of mid-plane stretching and gravity. Boundary conditions are simply supported such that at both ends there is no bending moment and no transverse and axial displacements. These boundary conditions give rise to an initial tension in the statically deflected shape. The displacement function consists of an assumed space mode using a simple sine function and unknown amplitude which is a function of time. This assumption provides for satisfaction of the boundary conditions and leads to an ordinary differential equation which is nonlinear, containing both quadratic and cubic functions of the amplitude. The perturbation method of multiple scales is used to provide an approximate solution for the fundamental frequency-amplitude relationship. Since the beam is initially deflected the small-amplitude fundamental natural frequency always increases relative to the free vibration situation provided in zero gravity. The nonlinear equation provides for interactions between frequency and amplitude in that both hardening and softening effects arise. The coefficient of the quadratic term in the nonlinear equation arises from the static (dead load) portion of the deflection. This quadratic term, depending upon its magnitude, introduces a softening effect that overcomes the hardening term (due to initial axial tension developed by deflection) for large slenderness ratios. For very large slender, immovable-end beams, the fundamental natural frequency is greater than that of beams without axial constraints undergoing small amplitude oscillations. This phenomenon is attributed to the stiffening effect of the statically-induced axial tension. However, the stiffening effect of axial tension in beams with slenderness ratios greater than approximately 392 undergoing large-amplitude symmetric-mode oscillations is overpowered by the presence of gravitational loading.
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