72 R. Schultz Fig. 5.13 Distributions of response accuracy (left), force (center-left), current (center-right) and voltage (right) predicted by the coupled DUTshaker model for 2000 sets of shaker locations understand the spread of values and how much location matters in terms of each parameter. As seen in the distribution plots in Fig. 5.13, there is a range of inputs, with some sets of shaker locations requiring much greater input than other locations. These inverse cumulative distribution function plots show the parameter value on the y-axis for each of the 2000 sets of shaker locations arranged small to large on the x-axis. It should be noted that the shaker locations which is minimum for one parameter will often not be minimum or even small in terms of the other parameter, so there is a balance of requirements. Ultimately, the objective is to choose a set of locations which are acceptable in terms of the shaker limitations (current and force), while resulting in low response error. With model predictions for the response and input force, voltage, and current, a set of shaker locations was chosen for the multi-shaker test, shown in the diagram in Fig. 5.12. Importantly, the shaker locations could be chosen based not only on the response accuracy or input force, but also in terms of the shaker current or voltage. In this case, shaker force and current were the limiting factors of interest so any shaker locations which resulted in predicted forces or currents above limitations were omitted from consideration. Unfortunately, PSDs of responses and inputs could not be reported in this paper, however the overall response was well matched by these six shaker locations and the full levels were achieved within the shaker capabilities, as predicted with this modeling effort. The integration of the shaker electro-mechanical model into the DUT model with FBS allows the required inputs for any set of chosen locations to be assessed against all the important shaker limitations, ensuring the test can be run at full-level without risking damage to the shaker equipment. 5.7 Conclusions The objective of this work was to provide a predictive capability to assess the force, voltage, and current requirements needed to achieve some DUT response from a set of shakers. This capability could then be used in pre-test design to determine where shakers could be located to ensure a multi-shaker vibration test was achievable in the laboratory. As moving shakers around is very time consuming in the laboratory, it is best to determine shaker locations beforehand using models. Introducing shaker electro-mechanical models to the pre-test design process allows the test design to account for all the important limitations, ensuring the shakers are capable of achieving the test goals. A lumped parameter shaker electro-mechanical model was implemented. The unknown parameters in this model were calibrated with a measurement of the shaker by itself. This electro-mechanical model was then coupled to structural dynamics models of various systems, including a simple two DOF model (to ensure the FBS process was properly implemented). Next, the shaker model was connected to an experimentally-derived model of a more complicated DUT which demonstrated that the coupled DUT-shaker model was adequately predictive. Finally, the shaker model was connected to various candidate input locations on a FE model of the DUT to design a multi-shaker test with limitations on the maximum shaker force and current as design factors. References 1. Allen, M.S., Rixen, D., Mayes, R. L.: “Short course on experimental dynamic substructuring, module 2: General theory,” Short Course for the International Modal Analysis Conference, Orland, FL (2014) 2. Avitabile, P.: MECH 5150 (Structural dynamic modeling techniques) course notes, system modeling concepts chapter (2017) 3. Tiwari, N., Puri, A., Saraswat, A.: Lumped parameter modeling and methodology for extraction of model parameters for an electrodynamic shaker. J. Low Freq. Noise Vibr. Act. Control. 36(2), 99–115 (2017) 4. Lang, G.F., Snyder, D.: Understanding the physics of electrodynamic shaker performance. Sound Vib. 35(10), 24–33 (2001)
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