DEICON, Inc., Dayton, OH
(937) 885-4134
deicon@erinet.com
www.deicon.com

The recent trend for testing products in multi–directions and to higher and higher frequencies has created the need to develop excitation systems with high natural frequencies, i.e. having a stiff construction with small moving mass(es). A tri–axial shaker, i.e., a 3–degree of freedom (DOF) excitation mechanism with closed–chain kinematics, is developed that can be used to test different products in multi–directions and to higher and higher frequencies.

Figure 1 Two views of the solid model of the triaxial shaker with the frame

Closed loop control allows this device to be used for testing parts under realistic combined loads that include not only vertical but also two axial horizontal loads. The 6-degree of freedom version of such device can, in addition to 3 axial forces, also load a part with 3 moments. The use of this device can be extended to active vibration control applications such as active seats for off–road vehicles.

Closed–chain mechanisms can be built at relatively low cost, because most parts are similar and are mechanically less complicated than most open chain configurations. One of the concerns about the use of machines with closed–chain kinematics has been the computation time required performing their accurate, model–based control (requiring the real–time computation of the machine dynamics). The actuator coordinates (joint space) have no simple expression in the Cartesian coordinates thus a coordinate transformation is needed. With the latest generation of high-speed processors this can be done with an inexpensive single processor system.

Figure 2 The 3-D motion of the center of the moving paltform

A tri–axial shaker with closed chain kinematics has been virtually prototyped. The closed–chain nature of the mechanism provides a high degree of rigidity to this multi–DOF shaker. The shaker is made up of a moving platform manipulated by 3 articulated legs each having a universal-prismatic-universal joint configuration. To manipulate the moving platform in x, y, and z directions, the 3 legs of the mechanism are simultaneously adjusted. The mechanism is framed using a tubular structure, which provide a high level of stiffness; see Figure 1. Finite element analysis of the frame indicated that the first natural frequency of the frame is 210 Hz.

Figure 2 shows the 3–dimensional motion of the center Figure 3 Block diagram model of the closed loop controlled shaker of the moving platform. The motion is generated by

1. transforming the desired motions of the platform in x, y, and z directions to the leg space using the inverse kinematics of the device, and
2. commanding the leg motions, evaluated in step 1, to the legs (actuators).

Figure 3 Block diagram model of the closed loop controlled shaker

Simultaneous motion of all three legs will generate the original, desired motion of the platform. In the case of Figure 2, the desired motions are 3 sinusoids in x, y, and z directions.

The work volume of the design shown in Figure 1, is approximately 10 inches in diameter and 12 inches in depth. The design can easily be re–scaled to provide a larger or smaller work volume. Despite the fact that only small displacements are required for shaking applications, the shaker is designed to provide a rather large horizontal displacement. This is to accommodate large workpiece heights without using an extra degree of freedom to move the workpeice.

Closed–loop Control

Figure 3 shows the block diagram model of the closed–loop controlled shaker exciting a workpiece. The force interactions between the moving platform and the workpiece are measured in 3 directions with a multi–DOF force sensor mounted on the moving platform and in contact with the workpeice being tested. The measured forces are compared with the desired (reference) forces and the errors are used to drive the controllers, which manipulates the shaker.

Note that the control force, as well as the interaction force between the moving platform and the part are in the Cartesian space. These forces are resolved in the leg space and added to the damping force of each leg. This combined forces is called the leg force. The leg forces are transformed back to the Cartesian space and used to cause the motion of the platform, which in turn describes the interaction force between the platform and the part being tested.

Figure 4 depicts the result of the simulation of the block diagram of Figure 3. The X and Y-direction reference (desired) forces are 1-25 and 1-15 Hz chirp signals, respectively. Clear from Figure 4, the shaker can follow the desired force trajectories in the 3 x, y, and z directions.

Figure 4 The response of the workpiece/platform interaction forces (solid line) to the chirp reference input (dotted line); X-direction (top) and Y-direction (bottom)