Shaker Control in the Presence of Nonlinearities

Control systems for shakers are necessary to remove both shaker dynamics as well as coupling effects between the shaker and the structure under test. Little attention has been given to shaker control for nonlinear structures. This paper presents the design, implementation and results of a shaker controller for a horizontal, cantilevered beam subjected to a non-uniform magnetic field. The objective of the shaker control is to ensure the acceleration at the tip of the beam follows a prescribed waveform. The parameters of the nonlinearity were determined using an extended Kalman filter, and the nonlinearity was subsequently cancelled via feedback linearization. NOMENCLATURE C Controller DOF Degree(s) of Freedom EKF Extended Kalman Filter P Plant, or physical system SISO Single input, single output Modal Space C System damping matrix I Identity matrix K System stiffness matrix M System mass matrix MAC Modal Assurance Critera POC Pseudo Orthogonality Criteria TU Mapping matrix between all DOF and reduced DOF U Eigenvectors or mode shapes e Pertaining to experimentally derived parameters u Pertaining to analytically derived parameters xn Entire analytical DOF xa Analytically correlated DOF xd Analytically discarded DOF Ω Natural frequency of mode shapes α Rayleigh damping coefficient for mass weighting β Rayleigh damping coefficient for stiffness weighting ωn Homogeneous natural frequency ζ Damping ratio Enhanced Kalman Filter C Equivalent strength of magnet (Nm) F Force from magnet Fx Force from magnet in direction tangent to arc of beam ∆1 Distance between equilibrium points collinear to Fx ∆2 Distance between end of beam and magnet, perpendicular to ∆1 x Distance towards magnet from initial equilibrium point Feedback Linearization Controller B Transfer function relating force applied and acceleration at the end of the beam Be Estimated transfer function relating force applied and acceleration at the end of the beam Pe Estimated plant transfer function Vl Voltage into the linear plant Vn Voltage into the nonlinear plant d Grouped parameter relating magnet force to equivalent magnet strength x& & Acceleration of tip of beam