MTS Systems Seismic Shake Table Configurations
Inherent Dynamic Cross-coupling per Shake Table Configuration
(click graphic for larger view)

Exploring Inherent Cross-coupling Among Various Seismic Shake Table Configurations

Numerous considerations factor into the selection of a shake table configuration - or actuator geometry - for seismic testing. Typical considerations affected by a table’s actuator geometry include laboratory floor space and foundation requirements, ease of access for mounting specimens, and the ability to join tables together. However, one critical aspect of table configuration that is seldom considered is the dynamic cross-coupling that occurs between degrees-of-freedom (DOFs) and the limitations this imposes on test fidelity. Recently, Brad Thoen, MTS senior staff engineer, embarked on a study to explore the factors that affect cross-coupling and analyze, compare and rank five common shake table configurations from the standpoint of inherent dynamic cross-coupling.

What is Cross-coupling?

6DOF test systems employ complex, synchronized actuation arrangements in which multiple actuators apply varying forces to the same portion of the test article simultaneously: these actuators are cross-coupled. Shake tables that exhibit inherently low levels of cross-coupling have a natural tendency to move in the desired DOF without causing unwanted motion in t other DOFs, while tables that exhibit inherently high levels of cross-coupling will tend to move in directions other than the one desired under specific conditions.

There are two types of cross-coupling: static cross-coupling, which has no impact on testing due to coordinate transforms; and dynamic cross-coupling, which can generate errors due to the fact that the motion being applied in any given DOF can be unduly influenced by the motions being applied in the other DOFs. These errors can hamper table control and compromise test fidelity.

While cross-coupling is determined largely by a shake table’s actuator geometry, it is also exacerbated by numerous other factors, including aspects of the test like displacement and frequency, specimen scale and center of gravity (CG), a specimen’s ability to tolerate multiple test runs (iterations), and the availability of advanced non-iterative table controls.

Study Methodology

To determine the degree of cross-coupling inherent among various shake table actuator geometries, Thoen first created five virtual shake tables, representing three orthogonal configurations (Balanced, Unbalanced and Pinwheel) and two non-orthogonal configurations (Vee and Stewart).

He then worked to harmonize (to the best extent possible) a variety of performance characteristics across all five virtual tables in all six degrees of freedom; these characteristics included natural frequency, maximum displacement, maximum velocity, maximum force, and closed-loop bandwidth.

Next, Thoen proceeded to compute the cross-coupling for each virtual table in response to five specific use cases (shown below), spanning variations in shake table position, specimen center of gravity and specimen dynamics.

Use CasePosition offset
(X Y Z)
CG offset
(X Y Z)
1: Home position, minimal
(0 0 0)
(0 0 0)
2: Home position, significant
(0 0 0)
(0 0 1)
3: X-only position offset
(0.2 0 0)
(0 0 0)
4: X, Y, and Z position
(0.2 0.2 0.1)
(0 0 0)
5: Lightly-damped resonant specimen (100% of table mass, 5 Hz,
    5% damping, 1m CG)

Finally, Thoen reviewed results generated by the five actuator geometries subjected to the five use cases, comparing plots of velocity frequency response functions (FRFs) from the on-diagonal DOF output to each off-diagonal DOF output.

Results & Conclusions

Thoen’s study and analysis showed that all five shake table configurations (actuator geometries) analyzed exhibited dynamic cross-coupling to some degree, with the Vee configuration having the least amount and the Stewart configuration having the most. The study also revealed how other test factors influence the impact of a table’s cross-coupling: small displacement waveforms prompted less cross-coupling impact on test fidelity than large displacement waveforms; full-scale model testing at < 10 Hz prompted less cross-coupling impact on test fidelity than scaled model testing at > 10 Hz; and specimens with low CG prompted less cross-coupling impact on test fidelity than specimens with high CG.

For those in the process of selecting of a shake table configuration for seismic testing, Thoen’s study offers two practical recommendations:
1.   Advanced control techniques exist that mitigate cross-coupling effects,
      but at a cost of complexity or requiring multiple test runs; owners of
      existing tables have no choice but to rely on such methods.
2.   Buyers of new shake tables should strongly consider a table
      configuration (actuator geometry) that exhibits inherently low


MTS Systems

14000 Technology Drive
Eden Prairie, MN USA

Tel: 952.937.4000
Tel: 800.328.2255
Fax: 952.937.4515