Eigensensitivity Analysis in Variable Geometry Trusses

Abstract

T HE adaptive structures of variable geometry are light mechanical systems capable of modifying their geometry and mechanical properties to adapt to different operating conditions. There are different kinds of mechanical variable geometry systems [1], amongwhich stand out the variable geometry trusses (VGTs) [2– 5]. These structures are a subset of the former, generally comprising a large number of biarticulated bars to form a complex truss. Some of these bars are active elements; that is, their length can vary in a controlled way to enable the actuation of the VGT. One of the most important problems to be solved in VGTs is vibration, essentially due to two factors. On the one hand, they are generally slimline structures, making them easily excitable at low frequencieswith large amplitudes, whichmight interferewith correct VGT operation, thereby harming its accuracy, and even resulting in collisions with environmental obstacles [5–7]. On the other hand, as VGTs may have highly different configurations throughout their operation, the dynamic properties (natural frequencies and vibration modes) also change to a large extent. There have been numerous contributions onvibratory dynamics of VGTs. There are a few works, like those of Keane and Bright [8], Keane [9], andNair andKeane [10], which opt for optimum redesign of the structure geometry using evolutionary methods to determine structural geometries presenting a better dynamic response; the most widely accepted procedure for improving and/or controlling dynamic properties is the inclusion of active, semiactive, or passive elements to control its dynamic response. In this sense, Bilbao et al. [11] carried out a detailed revision of the state of the art and proposed a new methodology for the optimal location of damping elements. This Note further contributes to vibratory dynamics of VGTs, describing a tool developed to efficiently estimate the dynamic properties of VGTs (frequencies and modes) throughout their movement, by responding to two needs. First, the information that frequencies and modes supply is often sufficient to foresee the dynamic behavior without having to perform costly direct dynamic analyses, as verified in [12]. Second, should said analyses be necessary, they can be approached from the modal superposition viewpoint, for which a tool that efficiently estimates natural frequencies and vibration modes is necessary. A linear estimation is proposed tomodel the variation of these dynamic properties throughout the VGTevolution, so that there is no need to calculate them per position but only for a fraction of these positions, whichmay result in considerable computational saving. Thus, a methodology estimating the value of natural frequencies and vibration modes in a new position from their value in the immediately prior position is employed. This is done by differentiating natural frequencies and vibration modes with respect to nodal coordinates at the starting position and developing a first-order series around the starting position to extrapolate the values of natural frequencies and vibration modes for a new position.