Increased Controllability in Component Testing using Structural Modifications
Paper i proceeding, 2008
In ordinary component testing for durability, the component's in situ mechanical environment is mapped to what can be achieved in dynamic test rigs under controlled conditions in the testing laboratory. In order for such tests to be meaningful, the response history obtained in the laboratory test needs to closely resemble the targeted response history observed in field. Such responses are more often than not strain and acceleration responses. The reconstruction of a target output time history may prove impossible however, if the controllability in the test system setup is too low. In a typical situation, if it is found that the system is not controllable, the rig's excitation setup may be altered. Such alterations include increasing the number of shakers or repositioning of shakers. Unfortunately, the natural approach to modify the excitation setup is not always readily available for dynamic test rigs, as these typically consist of a number of hydraulic actuators of considerable size. However, it is evident that another possibility to increase the controllability is by changing the system eigenvectors, and thus the system itself. We use a state-space representation of the test system, ẋ = Ax+Bu, y = Cx, in which y is the response and u is the drive signals to the test rig. In mathematical terms test system excitation alterations mean that the B-matrix of the state-space equation is modified, and test system alterations mean that the A-matrix, and thus its eigenvectors, are modified. Using a controllability metric for a certain state φi from a certain input uj, i.e. column vector of B, bj, as proposed by Hamdan & Nayfeh, it is easy to see the relevance of such modifications. Their controllability metric is namely the angles between the invariant subspaces of A (normally the eigenvectors) and the columns of B, calculated as: To be able to study system alterations in an efficient way, a parameterized structural modification of the component is proposed, in which a passive structural component - typically a spring-mass system - is added to the component to be tested. In this article, a two-step method for such modifications is introduced. In the first step we aim at finding a system which is at least marginally controllable, whereafter we maximize the controllability of the resulting system with respect to a given parameter set to obtain a set of optimal structural modifications. These are then elected as starting positions when optimizing the passive component parameters with respect to the minimization of input force and output error of the combined system.