Design of a single-axis shake table and development of its computational simulation
A Thesis submitted to the faculty of
San Francisco State University
In partial fulfillment of
the requirements for
the Degree
Master of Science
In
Engineering: Structural/Earthquake Engineering
by
Karlel Isaac Dy Cornejo
San Francisco, California
May 2021
Copyright by
Karlel Isaac Dy Cornejo
2021
Certification of Approval
I certify that I have read Design of a single-axis shake table and development of its
computational simulation by Karlel Isaac Dy Cornejo, and that in my opinion this work meets
the criteria for approving a thesis submitted in partial fulfillment of the requirement for the
degree Master of Science in Structural/Earthquake Engineering at San Francisco State
University.
Cheng Chen, Ph.D.
Professor,
Thesis Committee Chair
Jenna Wong Ph.D.
Assistant Professor
Design of a single-axis shake table and development of its computational simulation
Karlel Isaac Dy Cornejo
San Francisco, California
2021
Since its creation, the shake table has proven to be the invaluable to simulating seismic effects in
a controlled state. This study focuses on the design of a single-axis shake table and development
of its computational simulation. Each of the various parts of the shake table’s designs are
detailed while the discussing considerations in the fabrication and assembly process. As part of a
real-time hybrid simulation, a computational simulation is developed as well as to provide an
understanding of the capabilities of the designed shake table. Because of the inherent delay in
real-time hybrid simulations with servo-hydraulic systems, a time-delay compensation method
known as Adaptive Time Series compensation is incorporated into the numerical model. As a
result, accurate actuator control in the simulation is achieved and a better understanding of its
capabilities are gained.
Preface and/or Acknowledgements
The research presented in this study was conducted at Structural Hazards Lab, Department of
Engineering, San Francisco State University, San Francisco, California. During the study, the
director of the department was Dr. Kwok-Siong Teh
I would like to thank my research advisor, Dr. Cheng Chen, for not only his guidance in this study,
but also for encouraging me to pursue a path in Structural Engineering. My appreciation for his
help in my studies and professional career is never-ending.
I would like to thank Esteban Kim for assisting me in the early designs of the shake table. I would
like to also thank my friends and fellow students, Vanessa, Cielo, Sonia, Charlie, Esteban, and
Arturo. This journey would not have been the same without you.
Lastly, I am eternally grateful for my family, my parents without whom any of this would be
possible. This study is dedicated to them.
v
Table of Contents
List of Tables .............................................................................................................................. viii
List of Figures ............................................................................................................................... ix
Chapter 1. Introduction ............................................................................................................... 1
Chapter 2. Shake Table Design.................................................................................................... 3
2.1 Study of Existing Shake Tables ............................................................................................ 3
2.2 Modelling Software .............................................................................................................. 6
2.3 Shake Table Final Design ..................................................................................................... 7
2.3.1 Material Overview ......................................................................................................................................8
2.4 Shake Table Component Breakdown.................................................................................... 9
2.4.1 Base Frame ..................................................................................................................................................9
2.4.2 Base Plates ................................................................................................................................................ 10
2.4.3 Actuator Risers .......................................................................................................................................... 15
2.4.4 Actuator Mount ......................................................................................................................................... 16
2.4.5 Clevis Bracket ........................................................................................................................................... 17
2.4.6 Linear Bearing System .............................................................................................................................. 18
2.4.7 Sliding Table ............................................................................................................................................. 19
2.5 Fabrication .......................................................................................................................... 20
2.6 Test Structure Design .......................................................................................................... 20
Chapter 3. Computational Simulation of Shake Table ........................................................... 22
3.1 Computational Simulation Model Overview ...................................................................... 23
3.2 Displacement Command for Shake Table .......................................................................... 26
3.3 Adaptive Time Series Compensator ................................................................................... 29
3.3.1 Adaptive Time Series Method................................................................................................................... 29
3.3.2 Application to shake table simulation ....................................................................................................... 34
3.4 Controller Subsystem .......................................................................................................... 36
3.4.1 PID Control Theory................................................................................................................................... 36
3.4.2 Application to simulation .......................................................................................................................... 37
3.5 Actuator Subsystem ............................................................................................................ 39
3.6 Specimen Subsystem and Output........................................................................................ 41
3.7 Simulation Outcomes .......................................................................................................... 46
4. Conclusions and Future Work ............................................................................................... 50
4.1 Conclusions ......................................................................................................................... 50
4.2 Recommendations for Future Work.................................................................................... 50
vi
References .................................................................................................................................... 52
vii
List of Tables
Table 1 – Specifications and characteristic properties of the SFSU shake table system........ 7
Table 2 – Component Material List ............................................................................................ 8
Table 3 – Parameter Values Used in Simulation...................................................................... 23
viii
List of Figures
Figure 2.1 – West Michigan University Shake Table ................................................................ 4
Figure 2.2 – UCSD NEES Shake Table....................................................................................... 5
Figure 2.3 – Completed Model of Shake Table .......................................................................... 8
Figure 2.4 – Base Frame Dimensions .......................................................................................... 9
Figure 2.5 – Sliding Table Base Plate Dimensions ................................................................... 11
Figure 2.6 – Actuator Base Plate Dimensions .......................................................................... 12
Figure 2.7 – Sliding Table Base Plate Connection Callouts .................................................... 13
Figure 2.8 – Actuator Base Plate Connection Callouts ........................................................... 14
Figure 2.7 – Actuator Riser Dimensions ................................................................................... 15
Figure 2.8 – Actuator Mount Dimensions ................................................................................ 16
Figure 2.9 – Clevis Bracket Dimensions ................................................................................... 17
Figure 2.10 – One assembled set of the linear bearing system................................................ 18
Figure 2.11 – Sliding Table Dimensions.................................................................................... 19
Figure 3.1 – Simulink Model ...................................................................................................... 25
Figure 3.2 – Block Diagram of Command Subsystem ............................................................. 26
Figure 3.3 – El Centro time histories and scaled displacement signal ................................... 28
Figure 3.4 – Adaptive time series compensator block diagram .............................................. 32
Figure 3.5 – ATS Compensator Block....................................................................................... 33
Figure 3.6 – Input vs. Uncorrected Output Base Displacement ............................................. 35
Figure 3.7 – Input vs. Corrected Input Base Displacement .................................................... 35
ix
Figure 3.8 – Block Diagram of Controller Subsystem ............................................................. 38
Figure 3.9 – PID Controller Block............................................................................................. 39
Figure 3.10 – Actuator and Specimen Block Diagram ............................................................ 40
Figure 3.11 – Block Diagram of Specimen Subsystem: One-Story Structure ....................... 43
Figure 3.12 – One-Story Structure Output............................................................................... 43
Figure 3.13 – Block Diagram of Specimen Subsystem: Three-Story Structure.................... 45
Figure 3.14 – Example Outputs of a Three-Story Structure Specimen ................................. 45
Figure 4.1 – Force Output of Simulation .................................................................................. 46
Figure 4.1 – Output Displacment with 0.1 Hz Structure......................................................... 47
Figure 4.2 – Actuator Force Output with 0.1 Hz Structure .................................................... 48
Figure 4.3 – Output Displacement with 10 Hz Structure ........................................................ 48
Figure 4.4 – Actuator Force Output with 10 Hz Structure ..................................................... 49
x
1
Chapter 1. Introduction
An earthquake’s unpredictable and volatile nature has remained at the forefront of design in
seismic-prone areas. The effects and hazards that earthquakes pose have perpetually plagued
engineers and builders. Throughout the years, engineers have developed techniques to study and
mitigate the effects earthquakes have on buildings and other structures. Before the emergence of
experimental testing facilities, engineers relied on post-earthquake reconnaissance which required
studying buildings that sustained little damage to those that were completely collapsed. It was only
after an earthquake that engineers and builders would know if the constructed buildings of their
designs were successful or not. This trial-and-error approach posed a great danger to those
investigating the damages but more importantly to those that occupied the buildings. The ability
to assess the viability of design rules through experimental test facilities changed the landscape of
design and earthquake engineering. Developments and improvements of these facilities over time
have allowed engineers to test their structural behavioral theories in a controlled and safe
environment and accurately confirm and improve upon theoretical designs prior to constructing
the actual structures (Crewe 1998).
A common example of an experimental test facility is the shake table. Since the first, the
shake table has ushered in many avenues in earthquake response analysis and earthquake hazard
mitigation. Because of its unique ability to subject the test structure to inertia loads representative
of earthquake ground motions, the shake table has proven itself to be an invaluable asset to the
engineering community (Crewe 1998; Sinha 2009). A shake table can be generally broken down
into three main components: (1) the test specimen and the platform it is fixed on; (2) a number of
servo-hydraulic actuators, up to six, that apply load to the platform; and (3) a controller, which
2
sends electrical signals to the servo-hydraulic actuators based on the desired force or displacement
and the actual output value. As the shake table becomes more complex, so does the construction
and operation of the shake table. Each of these components can be varied to accommodate for the
user’s needs. The interaction between these components determines the shake table’s size, power,
and degrees of freedom. (Crewe 1998; Luco 2010).
This study focuses on laying the groundwork for the design, construction, and operation of
a single-axis shake table for San Francisco State University. The paper is split into two main parts:
the first section details the physical design of the shake table and its components while the second,
describes the computational simulation of the designed shake table and its outcomes. The study
concludes with future considerations and possible improvements that can be made to the shake
table design and the computational simulation.
3
Chapter 2. Shake Table Design
When designing a shake table, it is imperative to determine its desired capabilities. Platform size,
actuator power and stroke, and degrees of freedom are the primary aspects to consider when
designing a shake table. These decisions are heavily influenced by the allowable budget and the
available facility space. The simplest form of an earthquake simulator is a single-axis shake table.
Although earthquakes are multidirectional in nature, many institutions, like EUCENTRE and
UCSD’s NEES program, simulate earthquakes with a single-axis table. With its relatively low cost
of construction and operation compared to a multi-axis shake table, facilities can focus most of
their budget to increasing the size and power of their shake table (Ozcelik 2007, Sinha 2009).
Through a study of various shake tables (see Section 2.1), a single-axis shake table was
favored to realize a feasible design with the available resources. The single-axis shake table will
be operated with an already acquired MTS Actuator (See section 2.3 for specifications). Therefore,
the majority of the work is focused on designing a simplistic apparatus to utilize the actuator’s
inherent capabilities. To achieve this, the shake table is designed to exploit its weight and planned
reaction frame to create a self-contained system. This avoids any mounting or consideration for
the foundation of the shake table and allows the shake table to be relocated to virtually any location
that can accommodate the required electricity and hydraulic needs of the servo-hydraulic system.
2.1 Study of Existing Shake Tables
A study of various shake tables was conducted to determine what could be achievable with the
allowed budget and what components were imperative for the SFSU shake table. This section
4
discusses the main take-aways from each of the studied shake tables and how it influences the
design of the SFSU shake table.
Figure 2.1 – West Michigan University Shake Table
The West Michigan University (WMU) Shake Table is a small-scale reaction frame shake
table. It is operated by two Parker-Hannifin Series 2HX actuators, one for the sliding table and one
for the test structure, rated for 22.2 kN (5,000 lbs). It has a maximum test structure load of 444.8
N (100 lbs) and a maximum testing area of 3’ x 3’ (Holtz et al. 2009).
5
Figure 2.2 – UCSD NEES Shake Table
THE UCSD NEES Shake Table is a single degree of freedom system that can shake fullscale structures. The combined maximum force capacity of the actuators is 6.8 MN and can exhibit
a peak acceleration of 4.2g and peak velocity of 1.8m/s. It has a testing area of 7.6m x12.2m and
has safety towers at each corner to contain the damage of the testing structure within the testing
area. It is an example of what can be done with a large space and an enormous budget. As of 2018,
a grant from NSF has allowed for upgrades to a 6-DOF system (Luco et al.2009, Ozcelik et al.
2008).
The study of two shake tables at either end of the budget spectrum has allowed for better
insight to what is desired and achievable for the SFSU shake table. It was determined that a single
degree of freedom shake table would be feasible for the types of research that we expect the shake
6
table to be used for and that a reaction frame would be beneficial for cost as well as ease of
transportation and setup. Although the capabilities of UCSD’s shake table are not obtainable for
the current build, its study has given an outlook of what can be done to SFSU’s shake table in the
future and shows how important budget and space play into designing a shake table.
2.2 Modelling Software
Autodesk Inventor was the primary program used to design the shake table. It provided a simple
and, more importantly, accurate modeling software to ensure the precision of dimensions and the
fitting of parts without issue. Inventor’s built-in Finite Element Analysis was used to prevent
failure in any component of the shake table. Inventor was also equipped with the ability to
transform the 3-D part models into isometric 2-D plans for manufacturing as seen in Figures 2.4 –
2.11. Fusion 360 was used for rendering and animations of the 3-D model.
7
2.3 Shake Table Final Design
Table 1 lists the general specifications and characteristic of the main components of the shake table
while Figure 2.3 depicts the completed model of the SFSU shake table with all the components
labeled.
Table 1 – Specifications and characteristic properties of the SFSU shake table system
General shake table properties
Overall Size
3’ x 9’ (0.91 m x 2.74 m)
Approximate Overall Weight
2 kips (8.89 kN)
Table Size
3’ x 3’ (0.91 m x 0.91 m)
Maximum Payload
1.3 kips (5.78 kN)
Maximum Displacement
± 3in (±75 mm)
Servo-hydraulic Specifications
Make
MTS Model 244.21 A-05
Force Rating
11 kip (50 kN)
Stroke Length
6 in (150 mm)
Hydraulic Power Unit
MTS SilentFlo Model 505.2
Hydraulic Service Manifold
MTS Model 293.12
Servovalve
MTS Model 252.24G-01
Controller
MTS FlexTest 60 Controller System
Linear Bearing System Specifications
Linear Ball Bearing Pillow Block
PBC Linear Model IPP24G
Shafting
PBC Linear Model NIL24
Shafting Diameter
1.5”
Shafting End Support Block
PBC Linear Model NSB24
Dynamic Load Rating
1600 lbs (7.11 kN)
8
Figure 2.3 – Completed Model of Shake Table
2.3.1 Material Overview
Table 2 lists the different materials intended for the shake table design. It was imperative to use as
few materials as possible to prevent error in construction as well as avoid chemical reactions that
occur between two different types of metals over time.
Table 2 – Component Material List
Material
Shake Table Part
Aluminum 6061
•
•
•
Sliding Table
Pillow Blocks
End Support Blocks
Hardened Steel
•
Linear Shafting
A36 Steel
•
•
•
•
•
Base Frame
Base Plates
Actuator Mount
Actuator Risers
Clevis Bracket
9
2.4 Shake Table Component Breakdown
This section describes the various parts that make up the shake table and details instructions for
the fabrication and assembly of the shake table.
2.4.1 Base Frame
The base frame is comprised of A36 HSS 3” x 3” x 0.375” steel tubes to form the main frame
depicted in Figure 2.4. 8 - 2.5” long A36 L 2.5” x 2.5” x 0.5” angles with 5/8” holes are welded
onto the frame to provide a fixed connection for the base plates to be installed on top 5/8” steel
nuts and bolts. This will allow for relative ease of disassembly for transportation of the shake table.
The approximate total weight of the base frame is 400 lbs. A steel gantry or equivalent equipment
is needed to safely transport and effortlessly move the base frame during assembly and set up.
Figure 2.4 – Base Frame Dimensions
10
2.4.2 Base Plates
The base plates are composed of 1” A36 steel plates. Figures 2.5 and 2.6 detail the necessary
modifications to the steel plates, mainly threaded holes, to facilitate connections for the various
parts that will be installed on the base plates. Figures 2.7 and 2.8 illustrates where the parts will be
installed.
The base plates are split into two to avoid difficulties in fabrication, transportation, and
assembly with one large plate. Each plate will weigh approximately 555 lbs. A standard pallet jack
and steel gantry is recommended for transportation and assembly of the shake table. Aluminum
was considered for the base plate material to greatly reduce the weight of the base plates, but
concerns with cost and steel/aluminum corrosion ultimately led to steel for the base plate material.
11
Figure 2.5 – Sliding Table Base Plate Dimensions
12
Figure 2.6 – Actuator Base Plate Dimensions
13
Figure 2.7 – Sliding Table Base Plate Connection Callouts
14
Figure 2.8 – Actuator Base Plate Connection Callouts
15
2.4.3 Actuator Risers
The actuator risers maintain the actuator’s vertical and horizontal alignment with the sliding table.
This ensures there is minimal damage between the two components and efficient energy transfer.
Each actuator riser, two in total, require 2 - 2.5” long A36 L 2.75” x 2.5” x 0.25” steel angles with
a A36 5” x 5” x 0.25” steel plate welded at the specified location pictured in Figure 2.7. 4 – 5/8”
holes are bored at symmetrical locations specified in view A in Figure 2.7. 1.5” long bolts will be
needed to mount the actuator mount to the base plate. Two triangular A36 2.5” x 1.75” x 0.25”
steel plates are welded onto the steel angle to provide additional support for any out of line forces
during operation.
Figure 2.7 – Actuator Riser Dimensions
16
2.4.4 Actuator Mount
The actuator mount attaches the actuator to the base plate. An A36 7.5” x 9” x 1” thick steel plate’s
side is welded onto an A36 18” x 9” x 0.5” steel plate and is supported by 0.5” thick A36 gusset
plates. 0.63” holes are bored onto the vertical plate to mount onto the actuator while 5/8” holes are
bored through the horizontal plate to attach to the base plate.
Figure 2.8 – Actuator Mount Dimensions
17
2.4.5 Clevis Bracket
The clevis bracket connects the piston rod end of the actuator to the sliding table. Two A36 6.5” x
2.5” x 0.5” steel plates are welded onto an A36 6.5” x 6.5” x 0.5” steel plate as shown in Figure
2.9, View C. The steel plates are punctured to attach to the sliding table and the actuator.
Figure 2.9 – Clevis Bracket Dimensions
18
2.4.6 Linear Bearing System
The linear bearing system is comprised of PBC Linear End Support Blocks, Ball Bearing Pillows,
and Hardened Steel Linear Shafts with a nominal diameter of 1.5”. Figure 2.10 shows one set, of
two, of the linear bearing system. This proprietary product allows for the sliding table to move
with little friction along one axis and is easily interchangeable when a replacement is needed. The
total combined weight of the system is approximately 50 lbs and will have a combined dynamic
load rating of 1600 lbs.
Figure 2.10 – One assembled set of the linear bearing system
19
2.4.7 Sliding Table
A modified 6061 36” x 36” x 1” Aluminum plate will make up the sliding table. Three 0.5” holes
are added to the top of the table to attach onto the clevis bracket. At the bottom of the plate, 4 sets
of 9/32” threaded holes are dilled 0.5” deep for the pillow blocks to attach to via ¼” bolts. The
entire table will have a 5” grid of 9/16-12 UNC threaded holes to provide a mounting area for test
specimens.
Figure 2.11 – Sliding Table Dimensions
20
2.5 Fabrication
Because of the requirement of space and specific equipment to fabricate these parts, especially the
base frame and base plates, Fabrication cannot be done on campus. Therefore, Maxx Metals in San
Carlos, attention Francisco, has already been contacted to procure the materials and fabricate the
pieces per plan. (See Appendix B for material and fabrication quotes). The parts that they will be
fabricating are the base frame, base plates, actuator risers, actuator mount, and clevis bracket. The
unmodified Aluminum sliding table has already been acquired and Maxx Metals has been
contacted to bore the holes to the specifications in Figure 2.11. All these parts are on hold because
of the pandemic shutdown. When ready, payment can be sent for the work to begin. The parts will
be delivered to the campus. Because of the unavoidable heavy components, additional equipment
is required to move the large pieces to the shake table facility. The linear bearing system parts have
also been acquired from PBC Linear and are currently in campus storage.
2.6 Test Structure Design
Any test structure to be tested on the shake table is mainly limited by size and weight. The base of
the structure must be less than 2.75’x3’. This is to avoid the clevis bracket and securely mount the
base to the sliding table. It is not recommended to design a base greater than the sliding table as
the structure base may cause unexpected damage to the actuator and its surroundings. There is
essentially no height limit as the height is restricted by the height of the facility the shake table is
housed in. It is recommended, that for higher structures, a protective system should for the MTS
actuator to prevent any falling debris in the case of failure in the test structure. For the future
improvements, a larger sliding table would necessitate in increasing the length of the linear bearing
system, the shake table frame, and base plates itself.
21
The weight of the test structure is restrained by the linear bearing system. The dynamic
load rating of the linear bearing system is 1600 lbs and with the sliding table, which weighs approx.
300 lbs, the maximum test structure weight is 1300 lbs. The servo-hydraulic system has no issue
shaking a combined weight of 1600 lbs. Any desire to increase the maximum weight limit would
require a more expensive linear bearing system and a reassessment of the shake table. Multiple
tests were conducted simulating a 1300 lb. single-story test structure with a 2% damping ratio
throughout a frequency range of 0.1 -10 Hz under a scaled El Centro ground motion. The results
are detailed in Section 3.7
22
Chapter 3. Computational Simulation of Shake Table
Because of the complexity in the interactions between a shake table’s various components, it is
difficult to achieve the desired properties without some form of preliminary test. A numerical
model, or computational simulation, can be created to determine a shake table’s capabilities before
actual construction and operation. This provides a safe and controlled manner for verifying the
shake table’s limits (Williams 2001). This verified and true method has allowed many institutions
to confirm their designs and make necessary changes to achieve their desired goal. Furthermore,
incorporating the computational simulation to a shake table substructure test has allowed for
accurate reproduction of seismic effects (Mahin et al. 1975). First proposed by Hakuno et al.
(1969) and further developed through many studies that were encapsulated by Iemura (1985),
Takanashi et al. (1987), and Mahin et al. (1975), the real-time hybrid simulation, or real-time
hybrid experimental method, has become backbone of earthquake research (Horiuchi et al. 1999).
As with the design of a shake table, the development of its computational simulation is just as
important.
For the SFSU shake table, a computational simulation is created to assess the capabilities
of the designed shake table and provide a basis for the real-time hybrid simulation. The
simulation’s results provide insight to what is achievable with the shake table design. The
computational simulation of the shake table focuses on three main components of the shake table
system: (1) the controller system (FlexTest 60), (2) the actuator and servo-hydraulic system and
(3) the test specimen and table. Using MATLAB/Simulink, the FlexTest 60 controller system and
the servo-hydraulic actuator are simulated using a modified model by MTS while a state space
model is added to simulate the platform and a simplified test specimen. A time-delay compensation
23
method known as Adaptive Time Series is also incorporated into the model to account for inherent
delays during substructure testing. This chapter discusses the background and development of
these methods in the numerical model and details the outcomes from running the simulation with
parameters listed in Table 3.
3.1 Computational Simulation Model Overview
This section begins with the path a signal takes through the model detailing the subsystems of the
computational model along the way. The model’s results are discussed to provide insight on the
capabilities of the shake table. Table 3 presents the parameters used in the simulation while Figure
3.1 depicts the entire shake table model in Simulink.
Table 3 – Parameter Values Used in Simulation
Parameter
Value
Controller
Proportional Gain
8.0
Integral Gain
0.0
Derivative Gain
0.0
Type
Displacement-Based
Actuator
Maximum Piston Displacement
Total Internal Volume
Piston Cross Section Area
Supply Pressure
3 in (75 mm)
11 gal
3.90 in2 (25.16 cm2)
3000 psi
Test Specimen
Mass
25 kg
Natural Frequency
2 Hz
Damping Ratio
2%
Ground Motion
Event
El Centro, 1940
24
Max Displacement (Raw)
0.2133 m
Max Acceleration (Raw)
0.341 s2
Event Duration
m
31.2 sec
25
Figure 3.1 – Simulink Model
26
3.2 Displacement Command for Shake Table
The cmd block in the Simulink model is responsible for generating the reference signal in the
Simulink model. As depicted in Figure 3.2, a reference signal can be generated from the cyclic
signal generator or a ground motion time history, both of which are set up through MATLAB code.
When the latter is selected, it is imperative to know what type of time history is loaded into the
simulation. The model allows the input of all three different time history types: acceleration,
velocity, and displacement. A user-defined switch is incorporated to easily transform the loaded
time history into a displacement-based signal via the input selector.
Figure 3.2 – Block Diagram of Command Subsystem
In the simulation, an acceleration time history of the El Centro earthquake was inputted as
GM.mat. Figure 3.3 depicts the resulting velocity and displacement time histories via integration
blocks within the cmd subsystem and shows how the displacement signal is scaled to ensure the
27
actuator’s capabilities are utilized, but not exceeded. This is done by scaling the resulting
displacement time history via a gain block, where the gain is
𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺 =
𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
so that the maximum signal displacement does not exceed the maximum actuator
displacement while maintaining the nature of the ground motion’s displacement. In the simulation,
this value is set at 60 mm to be on the safer side.
28
Figure 3.3 – El Centro time histories and scaled displacement signal
29
3.3 Adaptive Time Series Compensator
3.3.1 Adaptive Time Series Method
In real-time hybrid simulations, the inherent dynamics of a servo-hydraulic system can cause a
time-delay in the signal from the actuator response to the command displacements. This creates a
desync between the actuator target displacement xt and measured displacement xm. Chae et al.
(2012) cites the works of Horiuchi et al.; Carrion, Phillips and Spencer; Zhao et al.; Darby et al.;
Wallace et al.; and Chen and Ricles to explain the developments in time-delay compensation.
Although initial developments in time delay compensation were based on a constant actuator
delay, the nonlinearity in the servo-hydraulic and experimental substructure systems has raised the
possibility of the nonlinearity in the actuator delay. Thus, nonlinear or adaptive delay
compensation methods became the focus of researchers. Chae et al. (2012) notes that these
nonlinear methods relied on calibration prior to performing a real-time hybrid simulation. As Chae
et al. (2012) describes, the calibration requires manual tuning of the adaptive gains which mostly
becomes a trial-and-error process. To prevent this, Chae et al. (2012) developed an adaptive delay
compensation method that updates the coefficients of the system at each time step of the simulation
using the least squares method. This section details the work of Chae et al. to provide background
to the ATS compensator incorporated in the computational simulation.
The Adaptive Time Series Method expands upon Equation (3-1a) expressed in the discrete
time domain where uc is the compensated displacement command, xt is the target displacement, A
is the amplitude error factor, and the k is a time index.
where
𝑢𝑢𝑘𝑘𝑐𝑐 = 𝑎𝑎0 𝑥𝑥𝑘𝑘𝑡𝑡 + 𝑎𝑎1 𝑥𝑥̇ 𝑘𝑘𝑡𝑡 + ⋯ + 𝑎𝑎𝑛𝑛
𝑑𝑑𝑛𝑛 𝑥𝑥𝑘𝑘𝑡𝑡
𝑑𝑑𝑡𝑡 𝑛𝑛
(3-1a)
30
𝑎𝑎𝑗𝑗 =
𝜏𝜏𝑗𝑗
𝐴𝐴𝑗𝑗 !
(3-1b)
, 𝑗𝑗 = 0, 1, … , 𝑛𝑛
Equation (3-1a) and (3-1b) are based on a developed model-based feedforward
compensator and a constant time delay and amplitude error. This nonlinearity is accounted for by
varying aj adaptive in accordance with the response of the actuator.
In the ATS compensator, the coefficients aj at time tk are obtained from the relationship
between the input and measured actuator displacements via the least squares method, where the
minimized objective function Jk, defined as
𝑐𝑐
𝑒𝑒𝑒𝑒𝑒𝑒 2
𝐽𝐽𝑘𝑘 = ∑𝑞𝑞𝑖𝑖=1(𝑢𝑢𝑘𝑘−𝑖𝑖
− 𝑢𝑢𝑘𝑘−𝑖𝑖
)
(3-2)
𝑐𝑐
In Equation (3-2), 𝑢𝑢𝑘𝑘−𝑖𝑖
is the compensated input actuator displacement at time 𝑡𝑡𝑘𝑘−𝑖𝑖 and
𝑒𝑒𝑒𝑒𝑒𝑒
𝑢𝑢𝑘𝑘−𝑖𝑖
is the estimated compensated input actuator displacement at time 𝑡𝑡𝑘𝑘−𝑖𝑖 on the basis of Equation
𝑚𝑚
(3-1a) using the measured actuator displacement 𝑥𝑥𝑘𝑘−𝑖𝑖
at time 𝑡𝑡𝑘𝑘−𝑖𝑖 and its time derivatives
𝑒𝑒𝑒𝑒𝑒𝑒
𝑚𝑚
𝑚𝑚
𝑢𝑢𝑘𝑘−𝑖𝑖
= 𝑎𝑎0𝑘𝑘 𝑥𝑥𝑘𝑘−𝑖𝑖
+ 𝑎𝑎1𝑘𝑘 𝑥𝑥̇ 𝑘𝑘−𝑖𝑖
+ ⋯ + 𝑎𝑎𝑛𝑛𝑛𝑛
𝑚𝑚
𝑑𝑑𝑛𝑛 𝑥𝑥𝑘𝑘−1
𝑑𝑑𝑡𝑡 𝑛𝑛
(3-3)
The values of the coefficients ajk in Equation (3-3) are found using with
𝐀𝐀 = (𝐗𝐗 m T 𝐗𝐗 m )−1 𝐗𝐗 m T 𝐔𝐔𝐜𝐜
(3-4)
𝑑𝑑𝑛𝑛
𝑚𝑚
𝑚𝑚
𝑚𝑚
Where A = [a0k a1k … ank]T, Xm = [𝐱𝐱 𝑚𝑚 𝐱𝐱̇ 𝑚𝑚 … … 𝑑𝑑𝑡𝑡 𝑛𝑛 (𝐱𝐱 𝑚𝑚 )], 𝐱𝐱 𝑚𝑚 = [𝑥𝑥𝑘𝑘−1
𝑥𝑥𝑘𝑘−2
… 𝑥𝑥𝑘𝑘−𝑞𝑞
]T, and
𝑐𝑐
𝑐𝑐
𝑐𝑐
U𝐜𝐜 = [𝑢𝑢𝑘𝑘−1
𝑢𝑢𝑘𝑘−2
… 𝑢𝑢𝑘𝑘−𝑞𝑞
]T . From here, the compensated input actuator displacement at time tk is
calculate using equation (1a)
𝑢𝑢𝑘𝑘𝑐𝑐 = 𝑎𝑎0𝑘𝑘 𝑥𝑥𝑘𝑘𝑡𝑡 + 𝑎𝑎1𝑘𝑘 𝑥𝑥̇ 𝑘𝑘𝑡𝑡 + ⋯ + 𝑎𝑎𝑛𝑛𝑛𝑛
𝑑𝑑𝑛𝑛 𝑥𝑥𝑘𝑘𝑡𝑡
𝑑𝑑𝑡𝑡 𝑛𝑛
(3-5)
The coefficients in Equation (3-5) can be used to equate the amplitude error factor and time
delay where
31
𝐴𝐴𝑘𝑘 ≅
1
𝑎𝑎0𝑘𝑘
, 𝜏𝜏𝑘𝑘 ≅
𝑎𝑎1𝑘𝑘
(3-6)
𝑎𝑎0𝑘𝑘
The use of higher order terms in Equation (3-5) can achieve good actuator displacement
tracking. In reality, higher-order terms can greatly affect the accuracy of the higher order time
derivatives of the target displacement xt due to inherent noise within the integration algorithm. The
SFSU computational simulation uses a second-order system where
𝑢𝑢𝑘𝑘𝑐𝑐 = 𝑎𝑎0𝑘𝑘 𝑥𝑥𝑘𝑘𝑡𝑡 + 𝑎𝑎1𝑘𝑘 𝑥𝑥̇ 𝑘𝑘𝑡𝑡 + 𝑎𝑎2 𝑥𝑥̈ 𝑘𝑘𝑡𝑡
(3-7)
and the target velocity and acceleration are approximated using the finite difference method
𝑥𝑥̇ 𝑘𝑘𝑡𝑡 =
𝑡𝑡
𝑥𝑥𝑘𝑘𝑡𝑡 − 𝑥𝑥𝑘𝑘−1
∆𝑡𝑡
, 𝑥𝑥̈ 𝑘𝑘𝑡𝑡 =
𝑡𝑡
𝑡𝑡
𝑥𝑥𝑘𝑘𝑡𝑡 − 2𝑥𝑥𝑘𝑘−1
− 𝑥𝑥𝑘𝑘−2
∆𝑡𝑡 2
,
(3-8)
This same method can be used to calculate the measured actuator velocity 𝐱𝐱̇ 𝑚𝑚 and
acceleration 𝒙𝒙̈ 𝑚𝑚 used in Equation (3-3). It is important to note that the measured actuator
displacement contains sensor noise. To account for this, a low pass filter is introduced to the model
to remove high-frequency noise from the measure displacement 𝐱𝐱 𝑚𝑚 to achieve a better estimate of
𝐱𝐱̇ 𝑚𝑚 and 𝒙𝒙̈ 𝑚𝑚 using the finite difference method. A low pass filter is also added to the compensated
𝑐𝑐
actuator input displacement 𝑢𝑢𝑘𝑘−𝑖𝑖
to reach a synchronized set of data due to the time delay that the
low pass filter adds. Figure 3.4 illustrates this process in a generalized Simulink block diagram
while Figure 3.5 depicts the actual block diagram within the computational simulation.
32
Figure 3.4 – Adaptive time series compensator block diagram
33
Figure 3.5 – ATS Compensator Block
34
3.3.2 Application to shake table simulation
After the signal is scaled within the displacement command block, it is adjusted within the ATS
Compensator. The Adaptive Time Series Compensator is responsible for introducing a
compensated displacement to adjust the output displacement to minimize the time delay. As
explained before, this is done by continuously updating the coefficients of the system transfer
function using online real-time linear regression analysis. To effectively account for the
nonlinearity of the hydraulic actuator, the ATS compensator does not utilize user-defined adaptive
gains compared to other time-delay compensation methods. This results in less time in calibrating
the ATS compensator and more accuracy in actuator control.
As seen in Figure 3.6, the simulation without the ATS Compensator has a time delay of
approximately 100 msec whereas when the ATS compensator is incorporated in the model, seen
in Figure 3.7, the time delay is practically zero. Figure 3.7 also illustrates the compensated
displacement introduced by the ATS compensator.
35
Figure 3.6 – Input vs. Uncorrected Output Base Displacement
Figure 3.7 – Input vs. Corrected Input Base Displacement
36
3.4 Controller Subsystem
3.4.1 PID Control Theory
The MTS FlexTest 60 Controller System utilizes a PIDF system with a forward loop filter. This
control loop can be manually or automatically tuned. The shake table numerical model currently
requires manual tuning therefore, an understanding of PIDF control is needed.
A PIDF controller is a type of feedback controller in which the controller determines the
input signal based on the original signal and the output signal, or the feedback signal (Visioli
2006). Figure 3.1 shows the typical components of a feedback control loop. It is comprised of three
types of control actions: a proportional action, an integral action, and a filtered derivative action.
The proportional action is responsible for the current control error and can be
expressed as
𝑢𝑢(𝑡𝑡) = 𝐾𝐾𝑝𝑝 𝑒𝑒(𝑡𝑡) = 𝐾𝐾𝑝𝑝 (𝑟𝑟(𝑡𝑡) − 𝑦𝑦(𝑡𝑡)),
(3-9)
where 𝐾𝐾𝑝𝑝 is the proportional gain, 𝑒𝑒(𝑡𝑡) is the control error, 𝑟𝑟(𝑡𝑡) is the reference signal, and 𝑦𝑦(𝑡𝑡)
is the process, or controlled, variable. Because of its simplicity in increasing or decreasing the
control variable, u, when the control error is large, the transfer function for a pure-proportional
controller is
𝐶𝐶(𝑠𝑠) = 𝐾𝐾𝑝𝑝 .
(3-10)
Although viable, the primary disadvantage with using a pure proportional controller is the
steady-state error that it creates. Because of this, a bias term 𝑢𝑢𝑏𝑏 . is introduced to Equation 3-9
𝑢𝑢(𝑡𝑡) = 𝐾𝐾𝑝𝑝 𝑒𝑒(𝑡𝑡) + 𝑢𝑢𝑏𝑏 ,
(3-11)
where constant is assigned to 𝑢𝑢𝑏𝑏 to reduce the steady-state error to zero. A more efficient way to
mitigate the steady-state error is with the integral action.
37
The integral action, also known as the automatic reset, is proportional to the integral of the
control error and is related the control error’s past values. It is expressed as
𝑡𝑡
𝑢𝑢(𝑡𝑡) = 𝐾𝐾𝑖𝑖 ∫0 𝑒𝑒(𝜏𝜏)𝑑𝑑𝑑𝑑,
where 𝐾𝐾𝑖𝑖 is the integral gain. This corresponds to a transfer function of
𝐶𝐶(𝑠𝑠) =
𝐾𝐾𝑖𝑖
𝑠𝑠
1
= 𝐾𝐾𝑝𝑝 (1 + 𝑇𝑇 𝑠𝑠).
𝑖𝑖
(3-12)
(3-13)
This transfer function allows the integral action to automatically apply the correct value of
𝑢𝑢𝑏𝑏 thus reducing the steady-state error to zero at each time-step.
The derivative action is based on the predicted future values of the control error. This
ideally can be expressed as
𝑢𝑢(𝑡𝑡) = 𝐾𝐾𝑑𝑑
𝑑𝑑𝑑𝑑(𝑡𝑡)
𝑑𝑑𝑑𝑑
where 𝐾𝐾𝑑𝑑 is the derivative gain. Its transfer function is
,
𝐶𝐶(𝑠𝑠) = 𝐾𝐾𝑑𝑑 𝑠𝑠.
(3-14)
(3-15)
There are many combinations of these actions but this simulation will utilize the parallel
form of a PID controller. In the parallel form,
𝐶𝐶𝑝𝑝 (𝑠𝑠) = 𝐾𝐾𝑝𝑝 +
𝐾𝐾𝑖𝑖
𝑠𝑠
+ 𝐾𝐾𝑑𝑑 𝑠𝑠
(3-16)
The three actions are completely separated and allow for the integral and derivative actions
to be turned off by setting their gains to zero.
3.4.2 Application to simulation
The compensated signal enters the controller subsystem as the command signal. The controller
subsystem communicates with the actuator via feedback loops to guarantee that the output signal
matches the desired input. The MTS Simulink model employs three feedback signals, i.e,
38
displacement (fbk in inport 2), force (ffbk in inport 3), and the differential pressure (dpfbk in inport
4) as shown in Figure 3.8. Depending on the users’ choice, the system can easily be switched
between displacement control and force control using the feedback selector. In this study, the
servo-hydraulic actuator for the shake table is under displacement control, therefore the
displacement feedback is used.
For every iteration, the command signal is adjusted by the ATS compensator to compensate
for the inherent time delay. This command signal is further adjusted within the PID controller
influenced by the displacement feedback and the differential pressure feedback, which is
responsible for the stability of the output as illustrated in Figure 3.9. The resulting signal passes
through the feedforward filter and saturation block into the actuator system. The compensated
command signal is illustrated in Figure 3.7.
Figure 3.8 – Block Diagram of Controller Subsystem
39
Figure 3.9 – PID Controller Block
3.5 Actuator Subsystem
The actuator subsystem is comprised of two blocks: the actuator block and the flow
subsystem block, as seen in Figure 3.8. Their parameters, specified in Table 3 above, are modified
to match the specifications of the servo-hydraulic system of the shake table. The signal from the
controller block is fed into the actuator system as the valve command. The valve command along
with, the supply pressure, return pressure, displacement feedback, and velocity feedback provide
the actuator subsystem with necessary information to output force, flow, and differential pressure.
The flow signal provides the flow subsystem with information on how to supply pressure into the
actuator. This in turn, results in a force and differential pressure output. The force and differential
pressure are looped back into the command system to complete the feedback loops, while the same
force signal is sent into the specimen block to shake the specimen.
40
Figure 3.10 – Actuator and Specimen Block Diagram
41
3.6 Specimen Subsystem and Output
The specimen subsystem employs the use of discrete state space to model the base sliding table as
well as a simple one-story structure. As seen in Figure 3.11, the force signal from the actuator
block is first converted into Newtons before being applied to the state space model. The specimen
subsystem is modeled by
(3-17a)
𝑥𝑥𝑛𝑛+1 = 𝐴𝐴𝑥𝑥𝑛𝑛 + 𝐵𝐵𝐵𝐵𝑛𝑛
(3-17b)
𝑦𝑦𝑛𝑛 = 𝐶𝐶𝐶𝐶𝑛𝑛 + 𝐷𝐷𝐷𝐷𝑛𝑛
where 𝑥𝑥𝑛𝑛 represents the states, 𝑢𝑢𝑛𝑛 , the inputs, and 𝑦𝑦𝑛𝑛 , the outputs. While A, B, C, and D, are the
state-space matrices. For a one-story structure specimen in this study, the interaction between the
sliding table at which the displacement is applied is represented by the matrix 𝐴𝐴
0
⎡ 𝐾𝐾
⎢− 𝑀𝑀
𝐴𝐴 = ⎢ 1
0
⎢ 𝐾𝐾
⎣ 𝑀𝑀2
1
𝐶𝐶
− 𝑀𝑀
0
𝐶𝐶
1
𝑀𝑀2
0
𝐾𝐾
𝑀𝑀1
0
𝐾𝐾
− 𝑀𝑀
2
0
⎤
⎥
1 ⎥
𝐶𝐶 ⎥
− 𝑀𝑀 ⎦
2
𝐶𝐶
𝑀𝑀1
(3-18)
where K is the stiffness, C is the damping, 𝑀𝑀1 is the mass of the sliding table, and 𝑀𝑀2 is the mass
of the structure. The stiffness and damping coefficients are automatically calculated using the
following,
𝐾𝐾 = 𝑀𝑀2 (2𝜋𝜋𝜔𝜔𝑛𝑛 )2, 𝐶𝐶 = 2𝑀𝑀2 ζ(2𝜋𝜋𝜔𝜔𝑛𝑛 )
(3-19)
where �� is the natural frequency of the structure and ζ is the damping ratio. �� and ζ are user
defined. The 𝐵𝐵 matrix represents the input matrix of the system. Since the only input for the system
is the actuator force applied to the sliding table 𝑀𝑀1 ,
42
0
⎡1⎤
𝐵𝐵 = ⎢𝑀𝑀1 ⎥
⎢0⎥
⎣0⎦
(3-20)
The C matrix determines the main outputs of the system in the y equation. For
1
𝐶𝐶 = �0
0
0 0 0
1 0 0�,
0 1 0
(3-21)
The results of this C matrix are the base displacement, base velocity, and the absolute roof
displacement. Although any output can be obtained from the state space model, the base
displacement and the base velocity are required outputs for all modeled structures. As seen in
Figure 3.10, the base velocity loops back into the actuator subsystem while the base displacement
loops back into the ATS compensator as the measured actuator displacement.
The D matrix is rarely used but it is responsible for applying any additional inputs for the
y output. In this simulation,
0
𝐷𝐷 = �0�
0
(3-22)
With the parameters listed in Table 3, the resulting responses of the structure are plotted in
Figure 3.12. In comparison to the El Centro displacement, the resulting base displacement
embodies the action of the earthquake. The state space model allows for the ease of modelling
simple single degree of freedom structures. The acceleration, velocity and displacement of the
structure can be easily derived and modeled in the simulation.
43
Figure 3.11 – Block Diagram of Specimen Subsystem: One-Story Structure
Figure 3.12 – One-Story Structure Output
44
An understanding of state-space equations and multi-degree of freedom systems leads to
modelling structure specimens with multiple floors. For a simple three-story structure with varying
masses, stiffnesses, and damping, the state space matrices are slightly more complicated, shown
below, and additional outputs are setup within the structure subsystem as seen in Figure 3.13
0
⎡ 𝐾𝐾1
⎢−
⎢ 𝑀𝑀1
⎢ 0
⎢ 𝐾𝐾1
⎢ 𝑀𝑀
𝐴𝐴 = ⎢ 2
0
⎢
⎢ 0
⎢
⎢ 0
⎢ 0
⎣
1
𝐶𝐶1
−
𝑀𝑀1
0
𝐶𝐶1
𝑀𝑀2
0
0
0
0
0
𝐾𝐾1
𝑀𝑀1
0
𝐾𝐾1 + 𝐾𝐾2
−
𝑀𝑀2
0
𝐾𝐾2
𝑀𝑀3
0
0
1
⎡0
⎢
𝐶𝐶 = ⎢0
⎢0
⎣0
0
𝐶𝐶1
𝑀𝑀1
1
𝐶𝐶1 + 𝐶𝐶2
−
𝑀𝑀2
0
𝐶𝐶2
𝑀𝑀3
0
0
0
1
0
0
0
0
0
0
0
𝐾𝐾2
𝑀𝑀2
0
𝐾𝐾2 + 𝐾𝐾3
−
𝑀𝑀3
0
𝐾𝐾3
𝑀𝑀3
0
𝐶𝐶2
𝑀𝑀2
1
𝐶𝐶2 + 𝐶𝐶3
−
𝑀𝑀3
0
𝐶𝐶3
𝑀𝑀3
0
0
0
⎡1⎤
⎢ ⎥
⎢𝑀𝑀1 ⎥
⎢0⎥
𝐵𝐵 = ⎢ 0 ⎥
⎢0⎥
⎢0⎥
⎢0⎥
⎣0⎦
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
⎡0⎤
⎢ ⎥
𝐷𝐷 = ⎢0⎥
⎢0⎥
⎣0⎦
0
0
0
0
0
0
0
0
0
1
0
0
0⎤
⎥
0⎥
0⎥
0⎦
0
0
0
𝐾𝐾3
𝑀𝑀3
0
𝐾𝐾3
−
𝑀𝑀3
0
⎤
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
𝐶𝐶3 ⎥
⎥
𝑀𝑀3 ⎥
1 ⎥
𝐶𝐶3
− ⎥
𝑀𝑀3 ⎦
45
Figure 3.13 – Block Diagram of Specimen Subsystem: Three-Story Structure
For this simple three-story structure, the results obtained are the base displacement, base
velocity, and the displacements of the three masses representing the three floors. Figure 3.14
illustrates the displacements of the three floors with mass, stiffness, and damping equal to
parameters set in Table 3.
Figure 3.14 – Example Outputs of a Three-Story Structure Specimen
46
3.7 Simulation Outcomes
There were very little issues simulating the El Centro ground motion. One of the issues was in
manually tuning the PID controller which could only be done via trial-and-error. Therefore, for
each new ground motion that will be simulated with the model, the PID controller must be
manually tuned. On the other hand, the automation of the Adaptive Time Series Compensation has
allowed a greater error range of PID tuning as it compensates for and time-delay and amplitude
errors. With more tests and simulations, the ATS compensator can be fine-tuned by adjusting the
ranges of the coefficients for the shake table.
One of the more important findings from the simulation was the capabilities of the actuator.
With the parameters of the actuator inputted in correspondence with the actual servo-hydraulic
actuator, the acutator’s exerted forces during the simulation were very small. Figure 4.1 illustrates
that with a 25-kg (55 lb) massed single DOF structure, the actuator only utilizes 0.3% of its
capacity.
Figure 4.1 – Force Output of Simulation
47
Since the modeled test structure was light compared to the expected capacity of the shake table, a
few more simulations were conducted with a heavier mass and different natural frequencies. Since
the weight of the test structure is limited by linear bearing system, the maximum payload that be
tested on the current design of the shake table is 590 kg (1300 lbs). Figure 4.2 and Figure 4.3 show
that with max payload and a low natural frequency, the actuator can achieve the El Centro ground
motion with less force. As for a structure with a natural frequency of 10 Hz, Figure 4.4 and Figure
4.5 illustrate that the actuator exerts more force but still to only about 4% of the actuator force
capacity.
Figure 4.1 – Output Displacment with 0.1 Hz Structure
48
Figure 4.2 – Actuator Force Output with 0.1 Hz Structure
Figure 4.3 – Output Displacement with 10 Hz Structure
49
Figure 4.4 – Actuator Force Output with 10 Hz Structure
50
4. Conclusions and Future Work
4.1 Conclusions
This study discussed the proposed design of the SFSU shake table with a limited budget in mind
and detailed each part and component that would complete the shake table. Fabrication of the shake
table with Maxx Metals and recommendations in assembly were discussed. It was also noted that
with some of the heavy parts of the shake table, acquiring necessary equipment to transfer, move,
and lift those parts is imperative and is a major factor in the project’s budget. A computational
simulation was also developed to gain an understanding of the shake table’s capabilities as well as
contribute to the real-time hybrid simulation of the shake table system. The inclusion of the ATS
compensator will prove useful for achieving accurate control.
As a result of the simulations, the current shake table design was shown to not utilized the
servo-hydraulic actuator’s capacities. With the simulations with the current shake table design has
shown that the actuator is not being utilized to its full potential. Because of this, the shake table
may be redesigned to increase its payload capacity and sliding table size. This is dependent on the
facility the shake table will be housed in and the allowed budget for the project. However, with
the current design, the intention to provide the department with a research and educational tool is
can still be achieved.
4.2 Recommendations for Future Work
Future work for the shake table and computational simulation would include:
•
Automatic PID tuning. With the intention of running various earthquake ground motion
and randomized ground motions, it becomes inefficient to manually tune the PID
51
controller for each different test. The emergence of automatic PID tuning will present an
opportunity to significantly reduce shake table testing preparation time.
•
Data acquisition methods. In simulations, the collection and acquisition of results and
outputs is uncomplicated whereas for an experimental substructure, data acquisition
requires additional equipment and device to record the response of the test
structure/specimen. Along with deciding the type of data acquisition devices to acquire
results, further work is needed to deficiently acquire the data to loop back into the
simulation and to analyze.
52
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