Nonlinear Dynamics of Flexible Cartesian Manipulators

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In this work, the nonlinear dynamic analyses of flexible Cartesian manipulators with elastic, viscoelastic and magnetoelastic beam materials for different applications are carried out. From exhaustive literature review, it is observed that though Cartesian manipulator has been used in many applications such as space exploration, hazardous nuclear power plant, micro surgery, precision manufacturing, and in many other industrial applications, very limited research has been carried out to enhance their productivity by making them lightweight or flexible, as these flexible manipulators cause severe vibration problems. Hence, in this work, an attempt is made to study the flexible Cartesian manipulator and develop some strategy to control the vibration of the presently used elastic manipulators by incorporating viscoelastic and magnetoelastic manipulator. Here, single-link flexible Cartesian manipulator with a payload has been modeled either as an transversely vibrating Euler-Bernoulli beam with roller-support at one end and tip mass at the other end, or as a transversely vibrating cantilever beam with end mass. The roller-supported end is assumed to have periodic motion. In some analyses, to simulate applications like welding, spraying, metal cutting etc. when the endeffector of the manipulator is in contact with the work environment, and has been subjected to forces from the environment, the endeffector of the manipulator has been modeled as a point mass subjected to harmonically varying axial tip force. In this work, 9 different flexible Cartesian manipulator models have been studied. Here, DD AlembertDs principle is used to derive the spatial governing equation of motion for elastic, viscoelastic and magnetoelastic manipulators, which is discretized into their temporal equations of motion by using generalized GalerkinDs method. These nondimensional temporal equations of motion contain many nonlinear terms which include cubic geometric and inertial terms due to large transverse deflection, nonlinear damping terms, and nonlinear parametric excitation terms along with linear stiffness, damping, forcing and parametric excitation terms. Due to the presence of many nonlinear terms, the perturbation methods viz., first and second order method of multiple scales, and method of normal forms are used to reduce the second order temporal equation of motion to a set of first order differential equations, which are then reduced to a set of nonlinear algebraic equation for steady state condition. For different resonance conditions, these reduced equations are solved numerically to find the steady state response of the systems. Taking various physical system parameters such as amplitude and frequency of support motion; mass ratio (ratio of mass of the payload and the beam); static and dynamic amplitude and frequency of axial force and magnetic field; viscous damping; loss factor; material conductivity and relative permeability of the material numerical simulations have been carried out to find the transient and steady state response, their stability and critical bifurcations. Time response, frequency response, instability regions, phase portraits, PoincareDs section, and basins of attraction are used to analyze the system behaviour for different resonance conditions. Initially, study has been carried out to investigate the vibration of harmonically varying roller-supported elastic manipulators with and wi...
Supervisor: Santosh Kr Dwivedy