Integrated Force Shaping and Optimized Mechanical Design in Underactuated Linear Vibratory Feeders

Integrated Inverse Dynamics and Optimized Mechanical Design in Underactuated Linear Vibratory Feeders Under Periodic Excitation

“…Further, since the system is non-minimum phase, a strategy to stabilize the internal dynamics (relating the desired output trajectory and the unactuated coordinates) is proposed through the output redefinition method. In particular, the paper extends the inverse dynamics method for linear vibratory feeders proposed by the authors in [27] and [28] in the simpler case of a sinusoidal trajectory reference, to the more general case of a generic periodic reference trajectory. Conversely to the above-mentioned papers where an algebraic problem was solved to compute the actuation forces, here, a differential-algebraic problem is adopted because of the presence of the internal dynamics that is governed by ordinary differential equations.…”

“…1, employed in manufacturing plants to convey products (see, e.g., [3]). The tray of the feeder is modeled as a flexible beam through N b Euler-Bernoulli planar beam elements, as proposed in [28]. The vertical displacements and the nodal rotations of the tray are denoted as y i and i , respectively (i = 1,…,N b + 1).…”

“…Translations that are orthogonal to the xy plane of motion are not considered. The tray is actuated through N A independent actuators modeled as suspended masses (as usually done in linear feeder [28]), whose absolute translations are x a i = x t + s i cos f and y a i = y i + s i sin f (i = 1,…,N A ). y i is the tray vertical translation in the point of attachment of the actuators and s i is the tray-actuator relative displacement along the direction defined by the angle f = 2 − f , with f being the throw angle, that is a key parameter in the feeder design to ensure the desired flow.…”

“…y i is the tray vertical translation in the point of attachment of the actuators and s i is the tray-actuator relative displacement along the direction defined by the angle f = 2 − f , with f being the throw angle, that is a key parameter in the feeder design to ensure the desired flow. The model of the actuators, whose mass is m a i and k a i is the stiffness of the leaf spring connecting it to the tray, is here omitted for brevity (for more details about the system model, please refer to [3,27,28]).…”