Shape Control of Beams by Piezoelectric Actuators

The shape control of beams by piezoelectric actuators is addressed analytically. Solutions are presented for a beamsubjectedto differentboundaryconditions.Thesolutionsshowhowandhowmuchthepiezoelectricactuators can ine uence the shape of a beam. Several case studies arealso presented to show the applications of the analytical solutions in the various analyses relevant to shape control of beams by piezoelectric actuators. The limitation of the actuation forces produced by piezoelectric actuators makes it dife cult to realize global and local precise shape control. In this application, the actuation forces produced by piezoelectric actuators are used to change the shapes of e exible structures. Some work has been reported on the shape control of beams, plates, and shells during the past few years. Some numerical methods, includ- ing the Ritz method, 3,4 e nite difference method, 5 and e nite element method 6i 10 have been used to model and analyze the problems rel- evant to shape control of e exible structures by piezoelectric actua- tors. However, due to the approximation of the numerical methods, the reported results can only give rough information of the global shape changes of the structures induced by piezoelectric actuators. Theycannot showthe detailed localshapechange information,such as information on the slope or curvature change of a structure at one point. The detailed local shape change information induced by piezoelectricactuators,whichshowshowthepiezoelectricactuators can change the shape of a structure or what types of shape change the piezoelectric actuators can make to a structure, is important for the design and analysis of such a piezoelectric smart structure. To obtain this information, one needs to analytically solve the problem of shape control. In general, it is very dife cult and complicated to solveanalyticallytheproblemforatwo-dimensionalplate;however, a one-dimensional beam is another case, and an analytical solution can be possibly obtained with acceptable complexity. The conclu- sions for a two-dimensional plate can be drawn from those for a one-dimensional beam. Thispaperwillpresenttheanalyticalsolutionsofthedee ectionof a beam induced by both piezoelectric actuators and external forces. In accordance with the solutions, the detailed local shape change information induced by piezoelectric actuators will be addressed. Using the solutions, one can analytically perform various analyses relevant to the shape control of beams by piezoelectric actuators. Case studies will also be carried out to show the applications of the solution.