10NCEE
Tenth U.S. National Conference on Earthquake Engineering
Frontiers of Earthquake Engineering
July 21-25, 2014
Anchorage, Alaska
SUBSTRUCTURE SHAKE TABLE TESTING
WITH FORCE CONTROLLED ACTUATORS
M. Stehman1 and N. Nakata2
ABSTRACT
Real-time hybrid simulation (RTHS) has emerged as a feasible and economical means for
seismic performance assessment of structural systems. It has been successfully applied to study
the system-level response, combining experimental and computational substructures. However,
existing substructure techniques are limited to interface boundaries where the influence of
unbalanced forces are not significant; because existing formulation procedures and experimental
loading are both displacement-based, unbalanced forces are inevitable in conventional RTHS. In
some cases (e.g., testing of extremely rigid specimens, soil-structure boundaries), force
equilibrium becomes more critical than displacement compatibility. To accommodate such
conditions, a force-based approach is essential and has to be developed in RTHS. This study
presents substructure shake table testing as a case study of force-based RTHS. A four-story shear
structure is divided into two substructures. The first story is tested on a shake table while the rest
of the structure is computationally simulated. The interaction between the experimental and
computational substructures is addressed such that the measured acceleration at the top floor in
the experimental substructure is used as the base input to the computational structure while
computed base shear in the computational substructure is fed back to the experimental
substructure through a force-controlled actuator. As such, the overall simulation is performed at
real-time with force-controlled actuators. This study presents the underlying theories of the
substructure shake table test method, centralized actuator control and preliminary numerical
simulations.
1
Graduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, MD 21217
Assistant Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, MD 21217
2
Stehman M and Nakata N. Substructure shake table testing with force controlled actuators. Proceedings of the 10th
National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.
Substructure Shake Table Testing with Force Controlled Actuators
M. Stehman1 and N. Nakata2
ABSTRACT
Real-time hybrid simulation (RTHS) has emerged as a feasible and economical means for seismic
performance assessment of structural systems. It has been successfully applied to study the
system-level response, combining experimental and computational substructures. However,
existing substructure techniques are limited to interface boundaries where the influence of
unbalanced forces are not significant; because existing formulation procedures and experimental
loading are both displacement-based, unbalanced forces are inevitable in conventional RTHS. In
some cases (e.g., testing of extremely rigid specimens, soil-structure boundaries), force
equilibrium becomes more critical than displacement compatibility. To accommodate such
conditions, a force-based approach is essential and has to be developed in RTHS. This study
presents substructure shake table testing as a case study of force-based RTHS. A four-story shear
structure is divided into two substructures. The first story is tested on a shake table and the rest of
the structure is computationally simulated. The interaction between the experimental and
computational substructures is addressed such that the measured acceleration at the top floor in the
experimental substructure is used as the base input to the computational structure while computed
base shear in the computational substructure is fed back to the experimental substructure through a
force-controlled actuator. As such, the overall simulation is performed at real-time with forcecontrolled actuators. This study presents the underlying theories of the substructure shake table
test method, centralized actuator control and preliminary numerical simulations.
Introduction
Substructure shake table testing is a developing experimental technique that combines the fully
dynamic nature of shake table tests with the versatility of real-time hybrid simulation.
Substructure shake table testing allows for the critical components of the test structure to be
evaluated experimentally on a shake table while the remainder of the structure is computationally
analyzed. Since only a portion of the structure is tested experimentally, substructure shake table
testing offers a more economical alternative to its traditional counterpart. While substructure
shake table testing has shown significant promise and potential, it has yet to fully explored.
Substructure shake table testing allows for structural components of interest to be
evaluated while the remainder are modeled computationally. Previous researchers used this
technique to test the performance of energy dissipation devices such as dampers [1], bearings [2]
and even support structures for electrical devices [3]. In these works, the computational
1
Graduate Student, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, MD 21217
Assistant Professor, Dept. of Civil Engineering, Johns Hopkins University, Baltimore, MD 21217
2
Stehman M and Nakata N. Substructure shake table testing with force controlled actuators. Proceedings of the 10th
National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014.
substructure is the majority of the test structure and the experimental portion is simply a result of
the response of the computational system. The authors of [4] investigated a more complex
situation where the bottom or middle stories of the structure are tested experimentally. These
tests required both displacement and force dependent boundary conditions: a shake table satisfied
the displacement boundary condition while the force condition was supplied using a
displacement feedback actuator. The authors of [5] expanded on this technique where inertial
masses were used to impose the computational force on the experimental substructure.
Substructure shake table testing has yet to be fully investigated when the experimental
substructure is the same proportion and scale as the computational system. Furthermore, there
has yet to be a study that fully addresses the situations where the boundary conditions are highly
sensitive to acceleration [6, 7] and force [8]. The study presented here investigates a substructure
shake table technique where the lowest stories are tested experimentally and the remainder of the
structure is computationally analyzed. Here a shake table is used to impose to the ground
accelerations to the experimental substructure and a force-controlled actuator is used to impose
the computational forces into the experimental substructure.
This paper presents the problem formulation, framework for the test method, actuator
control and preliminary data from a numerical case study.
Formulation of Substructure Shake Table Testing
In this study, substructure shake table testing is investigated on a linear elastic multi-story singlebay shear building. In the following formulation the spatial partition scheme is such that the
lower stories of the complete structure are tested experimentally and the remaining upper stories
are evaluated computationally. This section presents the equations of motion for the test structure
along with compatibility conditions that ensure accuracy of the RTHS.
Equations of Motion
A schematic of the entire test structure is shown in Fig. 1a and a dynamically equivalent
partitioned structure is shown in Fig. 1b. The response of the n-story entire simulation is viewed
as the reference response for the hybrid structure. Using Newton’s second law of motion the
equation of motion for each floor of the entire structure can be written as:
(1)
where mi is the mass of each floor; xi is the displacement of each floor relative to the
ground; Ri is the restoring force (including stiffness and damping effects) of the i-th floor
resulting from relative motion of the between adjacent floors and
is the acceleration of the
ground. Since the equations of motion were derived for earthquake loading, the ground
acceleration is the only input to the entire structure.
mc _ nc
mn
xn
Rn
Experimental
Substructure
mne +1
Rne +1
x ne +1
mne
Rne
Interface
Floor
xn
Re _ n
e
e
x1
xg
a.) Entire Structure
Figure 1.
mc _1
xc _1
Rc _1
fe
me _ ne
xg _ c
xe _ ne
Computational
Substructure
me _1
m1
R1
xc _ nc
Rc _ nc
xe _1
Re _1
Shake Table
xg _ e
b.) Partitioned Structure
Partitioning scheme for the entire structure using substructure shake table testing.
In the substructured system, the entire structure is partitioned into two structures, namely
an ne -story experimental substructure evaluated on a shake table and a nc -story computational
substructure, where ne and nc are chosen such that ne + nc = n . As shown in Fig 1b, an additional
force, fe , is added to the top floor of the experimental substructure which incorporates the
restoring from the first floor of the computational substructure. Thus the equations of motion for
each floor of the experimental substructure are written as:
(2)
(3)
where xe _ i is the i-th floor displacement of the experimental substructure relative to the
shake table; Re _ i is the i-th floor restoring force and
is the experimental ground
acceleration imposed by the shake table. Eq. 2 holds for the first ne −1 floors of the experimental
floor while Eq. 3 represents the interface floor where information from the computational
substructure is imposed in the experiment. Similarly, the equations of motion for the floors of the
computational system are expressed as:
(4)
where xi is relative to the base of the computational substructure and
is the ground
acceleration input for the computational substructure. In this particular configuration Eq. 2 and 3
are handled experimentally while Eq. 4 is solved numerically with real-time communication
between the two substructures.
Compatibility Conditions for Dynamic Equivalence
In order for the substructure system to have completely equivalent dynamic properties to that of
the entire structure, certain conditions have to be satisfied at every instance in time during the
RTHS. Aside from an accurate choice of parameters for the substructure system ( me, mc , story
stiffness and damping) there are three main compatibility conditions of interest in this type of
substructure shake table testing.
The experimental setup has to satisfy two compatibility conditions. The first condition is
that the ground acceleration produced by the shake table matches the reference ground
acceleration, that is
. Since shake tables use hydraulic actuators to produce motion, this
compatibility requires high performance acceleration tracking of shake tables. While this is a
challenging issue, there has been significant progress in the enhancement of acceleration tracking
of shake tables (namely [6, 9]).
The second experimental compatibility condition requires the correct force to be applied
at the top floor of the experimental substructure, namely fe = Rc _1 . Since hydraulic actuators are
normally used to apply experimental forces, this condition requires accurate force tracking
performance of hydraulic actuators. Force feedback control of actuators has been addressed in
the past with great success recently ([8, 10]).
The computational process requires that the input ground acceleration to the
computational substructure is equal to the absolute acceleration of the top floor of the
experimental substructure, which is to say that
. This condition assures that the
computational substructure is responding to the correct base excitation. With modern
measurement systems (accelerometers) and real time data acquisition this condition is
straightforward to satisfy.
With the compatibility conditions formally stated, it is apparent that the implementation
of the substructure shake table testing method presented here is fundamentally different from
traditional RTHS. Namely the experimental conditions do not rely on accurate displacements but
rely on accelerations and forces, which are far more sensitive. The computational conditions are
now accelerations compared to forces in traditional RTHS. Since the RTHS is highly sensitive to
errors in the compatibility conditions, extreme care must be taken when the system is
implemented.
Actuator Control for Substructure Shake Table Testing
Since substructure shake table testing relies heavily on hydraulic actuators for the performance
of the experimental substructure, the problem of actuator control must be properly addressed.
This study utilizes a centralized control scheme to handle coupling between the actuators. In this
technique, the shake table has a single independent controller while the force actuator has two
controllers. The block diagram of the substructure shake table test method, including the actuator
control systems is shown in Fig 2.
Computational
Substructure
xg _ c
Rc _1
- ef
Centralized Force
Controller
Cf
uf
CD
xg
Conv
xg
- ed
PID
Experimental
Substructure
fe
xe _ ne + xg _ e
ud
xg _ e
Independent Shake Table Control loop
Figure 2. Block diagram of substructure shake table test method including actuator control
system.
Fig. 2, introduces additional variables where: the Conv block converts the ground
acceleration to an equivalent ground displacement, xg is the reference ground displacement; ed
and e f are the tracking errors between the shake table and force controlled actuators
respectively; C f and CD are the controllers for the force-controlled actuator; the shake table is
assumed to have a PID controller and ud and u f are the voltages sent to each of the actuator’s
servo-valves.
Since both the shake table actuator and the external force actuator are connected through
the experimental substructure, there will be coupling between both actuators. However, since
shake tables are typically controlled through displacement feedback, the effect of the force
actuator is negligible and the shake table can be controlled independently. The same cannot be
said for the force-controlled actuator [10]. Therefore, in this control technique, the force actuator
has two controllers: one for reference tracking and disturbance rejection, C f , and one to
eliminate the effect of the shake table dynamics on the force controlled actuator, CD .
In terms of actuator control, the experimental substructure can be viewed as a two-input
two-output system:
⎧⎪ x
g_e
⎨
⎪⎩ fe
⎫⎪ ⎡ H x u
g d
⎬ =⎢
⎢
⎪⎭ ⎣ H feud
⎧⎪ u ⎫⎪
H xgu f ⎤⎧⎪ ud ⎫⎪
d
⎥⎨
⎬ = H EXP ⎨
⎬
⎥
uf ⎪
H feu f ⎪⎩ u f ⎪⎭
⎪
⎩
⎭
⎦
(5)
and the controllers are determined based on the relations in Eq. 5. The shake table PID controller
can be tuned solely from the relation H xgud . Once the shake table controller is fixed, the force
feedback controller, C f , can be designed solely using a loop shaping, [8], approach on H feu f . If
the shake table is completely independent of the force-controlled actuator then H xgu f = 0 and a
straightforward choice of CD will decouple the force-controlled actuator from the shake table.
Using the previous assumption the decoupling controller can be formed as:
CD = −
H feud
H feu f
⋅ (PID)
(6)
This choice of the decoupling controller will reduce the effect of shake table on the forcecontrolled actuator (complete decoupling is achieved only if H xgu f = 0 ). This control technique,
assumes that the shake table is uninfluenced by the force actuator. However in reality some
coupling may exist, but as long as the coupling is very small relative to the other relationships in
Eq. 5 this control technique is still valid.
Once the controllers are defined, the system is ready to run substructure shake table
testing. However due to the influence of actuator delays on the stability of RTHS, additional
measures may be needed to remove the delay from the force controlled actuator. Actuator delay
compensation techniques have been well established and the appropriate compensation algorithm
should be chosen based on the specific constraints of the individual actuator. It should be
mentioned that delay compensation is not needed for the shake table since the ground motion is
pre-defined and the experimental system drives the RTHS.
Numerical Case Study
In this section, a numerical case study is performed to investigate the capabilities of substructure
shake table testing including centralized actuator control. Matlab Simulink is used to simulate
both the substructure shake table test and the reference entire structure. In this study, the
response of a 4-story linear-shear structure subjected to ground motion is investigated. A
substructure shake table test of the 4-story structure is completed to investigate the capabilities of
the test method. For the substructured system, the first floor is experimental while the upper
three floors are computational (refer to Fig. 1 with n = 4 , ne = 1 and nc = 3 ).
Realistic parameter values are selected that are compatible with the size of the shake table
at Johns Hopkins. The parameters and models for the shake table and force controlled actuators
are selected in accordance with [8]. For simplicity, each story of the structure has the same
N
Ns
physical properties: m = 70 kg , k = 5⋅10 4
and c = 187
. With these parameter choices
m
m
the first vibration mode of the entire structure is 1.48 Hz with a damping ratio of 1.75%.
Before the results of substructure table testing are discussed, the performance of the
experimental substructure is investigated. The performance of the experimental setup depends on
a few criteria, namely: the ability of each actuator to accurately track its reference signal with
little effects of coupling and the time delay of the force-controlled actuator. The actuator
controllers were designed based on the methodology from the previous section. Results from step
input simulations are used to evaluate the effectiveness of the centralized control strategy. First a
step displacement is sent to the shake table while the force actuator has a zero force reference,
then the shake table has a constant displacement reference while the force actuator receives a
step force input. The results from this simulation are shown in Fig. 3.
Figure 3. Performance of experimental setup during step input tests: a.) shake table
displacement; b.) force from second actuator.
The results from this simulation show that the choice of controllers yields very good
performance from the experimental setup. The shake table is able to follow the reference
displacement with no influence from the force actuator Fig. 3a. The force-controlled actuator is
also able to track the reference step force with only a small influence from the shake table. The
force actuator has a 4N response to the shake table step, which quickly dies out. It is worth
noting the same simulation was investigated without the decoupling controller and in that case
the force actuator had a 425N response to the shake table step. Thus the decoupling controller
reduces the interaction between the shake table and force actuator by approximately 99%.
While both actuators are able to successfully track their reference inputs independently,
the force actuator has a relatively large time delay of about 12.5ms. This time delay is too large
for implementation in RTHS. To reduce this delay, the reference force is passed through a delay
compensator block before being sent to the force actuator. The delay compensation algorithm
used here is an inverse based compensation method known as feed-forward control. Where the
reference signal is sent through a pseudo-inverse model of the closed loop force actuator. With
the addition of the delay compensation algorithm, the force actuator time delay is brought down
to 4ms, which is suitable for implementation in the substructure shake table test.
To evaluate the substructure shake table test method, a simulation is performed with the
entire RTHS system implemented. The ground motion record used for this evaluation is the 1995
Kobe ground motion, with the peak ground acceleration scaled to 0.2 g. A plot of the
acceleration tracking performance of the shake table is shown in Fig 4.
Figure 4. Shake table acceleration during Kobe simulation: a.) entire record; b.) zoomed-in
view.
As shown in Fig. 4, the shake table reproduces the reference ground acceleration within a
reasonable degree of accuracy. The shake table exhibits a small time delay however shows little
to no influence from the force controlled actuator. The performance of the force-controlled
actuator during the simulation is shown in Fig 5.
Figure 5. Experimental force during Kobe simulation: a.) entire record; b.) zoomed-in view.
The force-controlled actuator is shown to accurately replicate the desired force from the
computational substructure through out the simulation. As shown in Fig. 5b, a large amount of
the actuator time delay is removed by using the corrected magenta line as the command to the
actuator. Although the measured force still lags behind the true reference, it is acceptable and the
simulation was stable. It is also worth noting that there is no influence from the shake table
dynamics and the addition of the decoupling controller was successful.
Figs. 4 and 5 indicate that the centralized control strategy allows for a stable simulation
and acceptable performance from both the shake table and the force actuator. Next the accuracy
of the structural performance is discussed. To evaluate the effectiveness of the substructure shake
table test, the substructured response is compared to a simulation of the entire 4-story structure.
To ensure a fair comparison of results, the input ground motion for the entire structure is the
produced acceleration of the shake table during the substructured simulation. A comparison of
the 4th floor absolute acceleration from both simulations is shown in Fig. 6.
Figure 6. 4th floor absolute acceleration during Kobe simulations a.) time histories; b.) Fourier
Transform of time histories.
A comparison of the top floor accelerations from both substructured and entire simulation
shows that the substructure shake table test was able to accurately reproduce the response of the
reference entire structure, Fig. 6. Fig. 6a indicates that the substructured system has almost
identical vibration characteristics as the entire structure. However during the free vibration
portion of the simulation, the response of the substructured system decays quicker than the entire
structure. This observation indicates that the substructure system has slightly more damping than
the entire structure. These observations are again confirmed through a frequency domain
comparison, Fig. 6b. Here both responses have almost identical characteristics except at the first
natural frequency of the structure, where the substructured response has smaller magnitude due
to the larger damping ratio.
Overall the substructure shake table simulation performed exceptionally and was able to
recreate the response of the entire structure within a reasonable tolerance. The RMS error
between the top floor accelerations of both simulations was only 13%. While the results
presented in this study are limited to a single simulation, the simulation data suggests that the
substructure shake table test method presented in this paper could serve as a viable alternative to
full-scale shake table tests.
Conclusions
This paper presented a new concept of substructure shake table testing where the lower stories
are tested experimentally on a shake table and the upper stories are computationally analyzed.
The equations of motion were formulated and compatibility requirements were discussed to
ensure dynamic equivalence between the substructured system and the entire structure. The
nature of this method requires accurate acceleration control of shake tables and force control
techniques for the experimental actuator. A centralized control strategy was developed that
allows for independent control of both the shake table and force-controlled actuator.
A numerical case study was carried out to investigate the performance of the control
strategy as well as the ability of the substructure shake table test method to reproduce the desired
response. The results showed the proposed control strategy was effective in allowing
independent control of the shake table and force actuator. Also, the substructure shake table test
method was able to accurately reproduce the seismic response of the reference entire structure.
This preliminary study has shown that substructure shake table testing has substantial
promise as an experimental testing technique. However, further simulations and experimental
investigations are needed to fully validate substructure shake table testing.
Acknowledgments
This work is supported by the National Science Foundation under an award entitled: “Career:
Advanced Acceleration Control Methods and Substructure Techniques for Shaking Table Tests”.
Grant No. CMMI-0954958.
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