The Finite Element Method Simulation of Active Optimal
Vibration Attenuation in Structures
Submitted to the College of Graduate Studies and Research
in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
Department of Mechanical Engineering
University of Saskatchewan
The Finite Element Method (FEM) based computational mechanics is applied to simulate
the optimal attenuation of vibrations in actively controlled structures. The simulation
results provide the forces to be generated by actuators, as well as the structures response.
Vibrations can be attenuated by applying either open loop or closed loop control
strategies. In open loop control, the control forces for a given initial (or disturbed)
configuration of the structure are determined in terms of time, and can be preprogrammed
in advance. On the other hand, the control forces in closed loop control depend only on
the current state of the system, which should be continuously monitored.
Optimal attenuation is obtained by solving the optimality equations for the problem
derived from the Pontryagin’s principle. These equations together with the initial and
final boundary conditions constitute the two-point-boundary-value (TPBV) problem.
Here the optimal solutions are obtained by applying an analogy (referred to as the beam
analogy) between the optimality equation and the equation for a certain problem of static
beams in bending. The problem of analogous beams is solved by the standard FEM in the
spatial domain, and then the results are converted into the solution of the optimal
vibration control problem in the time domain. The concept of the independent-modalspace-
control (IMSC) is adopted, in which the number of independent actuators control
the same number of vibrations modes.
The steps of the analogy are programmed into an algorithm referred to as the Beam
Analogy Algorithm (BAA). As an illustration of the approach, the BAA is used to
simulate the open loop vibration control of a structure with several sets of actuators.
Some details, such as an efficient meshing of the analogous beams and effective solving
of the target condition are discussed.
Next, the BAA is modified to handle closed loop vibration control problems. The
algorithm determines the optimal feedback gain matrix, which is then used to calculate
the actuator forces required at any current state of the system. The method’s accuracy is