Publications
Evaluation of a six-DOF electrodynamic shaker system
The paper describes the preliminary evaluation of a 6 degree of freedom electrodynamic shaker system. The 8 by 8 inch (20.3 cm) table is driven by 12 electrodynamic shakers producing motion in all 6 rigid body modes. A small electrodynamic shaker system suitable for small component testing is described. The principal purpose of the system is to demonstrate the technology. The shaker is driven by 12 electrodynamic shakers each with a force capability of about 50 lbs (220 N). The system was developed through an informal cooperative agreement between Sandia National Laboratories, Team Corp. and Spectral Dynamics Corporation. Sandia provided the laboratory space and some development funds. Team provided the mechanical system, and Spectral Dynamics provided the control system. Spectral Dynamics was chosen to provide the control system partly because of their experience in MIMO control and partly because Sandia already had part of the system in house. The shaker system was conceived and manufactured by TEAM Corp. Figure 1 shows the overall system. The vibration table, electrodynamic shakers, hydraulic pumps, and amplifiers are all housed in a single cabinet. Figure 2 is a drawing showing how the electrodynamic shakers are coupled to the table. The shakers are coupled to the table through a hydraulic spherical pad bearing providing 5 degrees of freedom and one stiff degree of freedom. The pad bearing must be preloaded with a static force as they are unable to provide any tension forces. The horizontal bearings are preloaded with steel springs. The drawing shows a spring providing the vertical preload. This was changed in the final design. The vertical preload is provided by multiple strands of an O-ring material as shown in Figure 4. Four shakers provide excitation in each of the three orthogonal axes. The specifications of the shaker are outlined in Table 1. Four shakers provide inputs in each of the three orthogonal directions. By choosing the phase relationships between the shakers all six rigid body modes (three translation, and three rotations) can be excited. The system is over determined. There are more shakers than degrees of freedom. This provided an interesting control problem. The problem was approached using the input-output transformation matrices provided in the Spectral control system. Twelve accelerometers were selected for the control accelerometers (a tri-axial accelerometer at each corner of the table (see Figure 5). Figure 6 shows the nomenclature used to identify the shakers and control accelerometers. A fifth tri-axial accelerometer was placed at the center of the table, but it was not used for control. Thus we had 12 control accelerometers and 12 shakers to control a 6-dof shaker. The 12 control channels were reduced to a 6-dof control using a simple input transformation matrix. The control was defined by a 6x6 spectral density matrix. The six outputs in the control variable coordinates were transformed to twelve physical drive signals using another simple output transformation matrix. It was assumed that the accelerometers and shakers were well matched such that the transformation matrices were independent of frequency and could be deduced from rigid body considerations. The input/output transformations are shown in Equations 1 and 2.