Nonlinear Modeling of Flexible Manipulators Using Nondimensional Variables

This paper deals with nonlinear modeling of planar one- and two-link, flexible manipulators with rotary joints using finite element method (FEM) based approaches. The equations of motion are derived taking into account the nonlinear strain-displacement relationship and two characteristic velocities, $Ua$ and $Ug$⁠, representing material and geometric properties (also axial and flexural stiffness) respectively, are used to nondimensionalize the equations of motion. The effect of variation of $Ua$ and $Ug$ on the dynamics of a planar flexible manipulator is brought out using numerical simulations. It is shown that above a certain $Ug$ value (approximately $⩾45m∕s$⁠), a linear model (using a linear strain-displacement relationship) and the nonlinear model give approximately the same tip deflection. Likewise, it was found that the effect of $Ua$ is prominent only if $Ug$ is small. The natural frequencies are seen to be varying in a nonlinear manner with $Ua$ and in a linear manner with $Ug$⁠.