When performing modal analyses of active flexible multibody systems, both controller effects and flexible body dynamics should be included in a multidisciplinary system model. Control system software, such as MATLAB and Simulink, usually supports both controller design and control system simulation, in which the mechanical system can be modeled with rigid bodies, lumped masses, inertias, springs, dampers or analytical equations. This will cause the flexible body dynamics to be predicted by very simplified models. In active flexible multibody dynamics software systems, such as FEDEM, feedback type controllers will typically calculate loads applied to the mechanical model based on feedback measurements of the system. This approach works well in a time domain analysis when the controller drives the mechanical model with applied loads based on the given controller algorithms. However, a major problem occurs in modal analyses of the closed-loop system. In a free vibration analysis, all loads are set to zero, which decouples the controller and mechanical model. As a result, the mechanical model becomes singular in all controlled degrees of freedom. A common approach by mechanical engineers when performing modal analyses of active flexible multibody systems is to introduce additional boundary conditions for the system degrees of freedom affected by controllers. This causes the flexibility in the different joints of the mechanism to be omitted since the joints are made rigid at relevant positions. Another common, though inaccurate, solution to this problem is to represent the controller effects by virtual springs, dampers and inertias in the mechanical model. Nonetheless, this approach is only applicable for simple control systems in which the mechanical engineer knows how to transform the controller into an equivalent mechanical model. Additionally, when designing and optimizing active flexible multibody systems, the engineer also has to update two system models simultaneously: one for the modal analysis and one for the time domain dynamic simulation.
This thesis presents a method for performing modal analyses of active flexible multibody systems in a finite element environment based on the generalized eigenvalue problem. The mechanical equivalent properties of position, velocity or acceleration feedback proportionalintegral- derivative (PID) controllers are derived, and it is shown how these properties can be included into a system model appropriate for modal analysis. Controllers containing noncollocated sensors and actuators are also covered. Since the controller parameters may not be explicitly defined for the engineer working in a finite element environment, a method for deriving the controller gains for PID controllers using perturbations is also presented. Two versions of the modal analysis method are presented: one simplified n-dimensional undamped version and one complete 3n first-order version which include all system properties. The derived theory is verified throughout this work through examples. The presented method is of relevance to mechatronic products involving vibrational issues in disciplines such as robotics, aerospace and aviation, military, etc.