State-Space Dynamic Substructuring with the Transmission Simulator Method

Abstract

Dynamic substructuring is a technique to simplify the analysis of complex structures. The vibrational problems of the constituent substructures are analysed and solved individually and their solutions are then assembled to form the global solution. In experimental dynamic substructuring, at least one of the constituent substructures is identified experimentally. The coupling interfaces are commonly simplified in such syntheses, which can result in poor prediction quality in many applications. The transmission simulator was introduced to address this problem. Transmission simulators are well-modelled parts attached to the interface of the substructures to be coupled. This allows for distributed interfaces and a relaxation of the coupling conditions by using the transmission simulator's analytical modes as a basis for the coupling equations, at the cost of adding a decoupling step to the substructuring problem. In this paper, the transmission simulator method is translated to the state-space substructuring domain. The methodology is applied to the Society for Experimental Mechanics' substructuring focus group's Ampair A600 test bed in form of experimental-analytical substructuring. The Ampair wind turbine's hub is used as the transmission simulator and is modelled with finite elements while the three blades, individually attached to the real hub, are experimentally identified. The three experimental blade hub systems are then coupled and two finite element hubs decoupled from the system, using the derived method. Finally, this system is compared to a directly measured hub with three blades by means of frequency response functions and modal properties.

On the use of Lagrange Multiplier State-Space Substructuring in dynamic substructuring analysis

“…To decouple a substructure we may use the Inverse Coupling approach [20], [17]. This decoupling procedure follows the same methodology as coupling.…”

“…This decoupling procedure follows the same methodology as coupling. However, the state-space model of the substructure to be decoupled must be established in negative form [13], [20]. Then from the obtained state-space model, the inputs and outputs of the substructure to be characterized must be retained while the others must be eliminated.…”

“…matrices [M], [K] and [V] are the mass, stiffness and damping matrices respectively. The computation of the negative form of the state-space model given by expression ( 21) can be performed by multiplying the matrices [M], [K] and [V] by −1 [13], [20], as follows:…”

“…The computation of the exact state-space model of the components and of the assembled system was performed by using its mass, stiffness and damping matrices in accordance with equations ( 19) and (20). The mass and stiffness matrices were established by using the Lagrange equations [24], while for simplicity the damping matrix of each substructure was constructed from the respective stiffness one by replacing the stiffness terms with the damping ones.…”

“…When the state-space models to be coupled are transformed into this coupling form, their state-space matrices present a particular structure, for which the first block row of the state equation is representative of a second order differential equation, making it possible to perform coupling by directly summing those block rows [10], [13]. This coupling procedure is possible, because the applied similarity transformation guarantees that the contribution of the interface inputs to the response of the internal states is null.…”