Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector placed at an intermediate position

Nonlinear dynamic analysis of a Cartesian manipulator carrying an end effector which is placed at different intermediate positions on the span is theoretically investigated with a single mode approach. The governing equation of motion of this system is formulated by using the D’Alembert principle in addition to profuse application of Dirac delta function to indicate the location of the intermediate end effector. Then the governing equation is further reduced to a second-order temporal differential equation of motion by using Galerkin’s method. The method of multiple scales as one of the perturbation techniques is being used to determine the approximate solutions and the stability and bifurcations of the obtained approximate solutions are studied. Numerical results are demonstrated to study the effect of intermediate positions of the end effector placed at various locations on the link with other relevant system parameters for both the primary and secondary resonance conditions. It is worthy of note that the catastrophic failure of the system may take place due to the presence of jump phenomenon. The results are found to be in good agreement with the results determined by the method of multiple scales after solving the temporal equation of motion numerically. In order to determine physically realized solution by the system, basins of attraction are also plotted. The obtained results are very useful in the application of robotic manipulators where the end effector is placed at any arbitrary position on the robot arm.

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