The dynamics and limits on the scaling of a flexible kinetic sculpture

Abstract

Highly¯exible structures may be scaled up by dierent criteria that lead to dierent degrees of structural integrity as the scale increases. This paper ®rst explores the eects of dierent scaling rules, focusing in particular on a double-size version of the Len Lye kinetic sculpture Blade, whose aesthetic performance characteristics must be preserved as the size is increased. Stresses in the sculpture increase, reaching ®rst an economic limit beyond which frequent fatigue failure makes the sculpture too costly to operate. As the scale further increases, a size is reached where the sculpture will collapse. A dimensional parameter for comparing the values of the life to failure for dierent construction materials is also developed. The second part of the paper summarizes the results of dierent analytical approaches to the dynamics of Blade, both with and without the inclusion of an axial acceleration load due to gravity. The results of analyses are found to be in good agreement with experimental data. The paper closes by discussing some of the design implications of implementing the drive system for a double-size version of Blade that has been built and is now on public display. Keywords: kinetic sculpture,¯exible structure dynamics, dynamic scaling of structures NOTATION a blade length (m) A mn coecient used in the series representation of transverse blade de¯ection b blade width (m) B k Y C k constant terms used in the Gram±Schmidt process C m Blade material cost ($akg) D¯exural rigidity of a plate Eh 3 a121 À # 2 (N m) E Young's modulus (N/m 2 ) g gravitational constant (m/s 2 ) h blade thickness (m) I second moment of area (m 4 ) k 1 Y k 2 Y k 3 constants in the blade life and life±cost equations m mass per unit length (kg/m) N in-plane load per unit width (N/m) N f number of blade stress cycles to failure N pf number of blade performances to failure N pfa$ number of blade performances to failure per dollar cost P dimensionless blade life parameter R radius of curvature (m) t time (s) T 3 length of Blade performance in the third-order mode shape (s) w transverse blade displacement (m) x axial blade coordinate (m) y coordinate across the width dimension of the blade (m) stability factor crit stability factor at the critical buckling load ik Kronecker delta loss coecient ! roots of the characteristic frequency equation for a clamped±free beam " frequency parameter # Poisson's ratio & density (kg/m 3 ' normal axial or bending stress (N/m 2 ) 0 characteristic functions of a clamped±free beam 2 characteristic functions of a free±free beam 3 frequency (r/s) characteristic value 3 2 &habaD Subscripts iY kY mY nY pY qY rY s positive integers o original scale sc scaled The MS was