Study of a Modified Friction Device for the Control of
Civil
Structures
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
By
Mohamed Abdellaoui Maane
French Engineer Degree
Ecole Speciale des Travaux Publics, Paris
Class of 2010
JUL 15 2010
LIBRARIES
ARCHIVE
SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL
ENGINEERING IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF
MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2010
©2010 Mohamed Abdellaoui Maane. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or here
after created.
Signature of Author:
-
Department of Civil and Environmental Engineering
May 14"h, 2010
Certified by:
Jerome J. Connor
Professor of Civil and Environmental Engineering
"7T: jess$unervisor
Accepted by:
Daniele Veneziano
Chairman, Departmental Committee for Graduate Studies
Study of a Modified Friction Device for the Control of Civil
Structures
By
Mohamed Abdellaoui Maane
Submitted to the Department of Civil and Environmental Engineering on May 14, 2010 in Partial
Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and
Environmental Engineering at the Massachusetts Institute of Technology.
Abstract
In structural engineering, vibrations created by transient loads input energy to the structure.
Control devices can be used to dissipate this energy in a civil structure. In this research, a new
semi-active dissipation device is studied: the Modified Friction Device (MFD). The MFD is a
control device that consists of a spring, a viscous element and a braking element similar to car
brakes. Because it is still a conceptual design, the device's frictional behavior has been modeled
using the LuGre friction model that takes the stiction and Stribeck effect into account in the
dynamic behavior. Then, several simulations have been run to evaluate how the device behaves
and how it dissipates energy in a structure under seismic and wind loads. Three interesting
results emerged. Firstly, the LuGre friction model is a very good approximation for this device,
as the comparison between experimental data from a friction device and the model to adapt the
model parameters showed. Secondly, the MFD can display a behavior similar to the MR damper
but with a more mechanically robust and reliable system. Thirdly, several control schemes have
been implemented and the MFD can be used as a force actuator as well as a passive device.
Furthermore, this device meets realistic constraints of control and constructability.
Thesis Supervisor: Jerome J. Connor
Title: Professor of Civil and Environmental Engineering
Acknowledgement
I would like to express my sincere gratitude and appreciation to my tutor, Simon Laflamme, for
sharing his knowledge and research about dampers and motion based design, and for his
constructive comments, advice and guidance that enable the completion of this research. Simon
guided me in my research by emphasizing the important aspects and provided me with the
necessary tools to achieve this research. I would also like to thank him for all the time he spent
explaining me different ways to solve design issues. Finally, I would like to thank him for the
methodology he taught me during my research. It was a remarkable experience.
I would like to thank Professor Jerome J. Connor for his expert guidance, discussion and for his
encouragement and support at all levels. I also would like to thank Professor Connor for giving
me the strong technical knowledge base with which to address motion based design issues.
Professor Connor's approach to teaching and advising is irreplaceable - his rich knowledge and
continuously encouraging manner provide the best environment for growth as an engineering
student.
I would like to thank Dr Douglas Taylor from Taylor Devices Inc. for providing experimental
data about friction devices and for the advice he provided to do this research.
I would like to thank Fernando Pereira-Mosqueira for his advice, support and motivation during
the research and the writing process of this thesis. I would also like to thank him for sharing his
knowledge and his data.
I would like to thank Geoffroy Larrecq for his support and motivation during the research and
the writing process of this thesis.
I would like to thank my parents for their support during this Masters Degree and during my
education in a general manner.
Finally, I would like to thank all the students of Civil Meng for this year and for making MIT
one of the most enjoyable experiences of my life.
Table of Contents
Intro du ctio n ...................................................................................................................................................
9
1. Sem i-active devices.................................................................................................................................11
1.1 Introduction.......................................................................................................................................11
1.2 Variable stiffness device ...................................................................................................................
11
1.3 Friction device using friction pads...............................................................................................
12
1.3.1 Piezoelectric friction device...................................................................................................
12
1.3.1 Lorenz dry friction device......................................................................................................
13
1.4 Rheological dampers.........................................................................................................................14
1.4.1 Electro-rheological dampers....................................................................................................
14
1.4.2 M agneto-rheological dampers...............................................................................................
16
1.4.3 M odels for rheological dampers.............................................................................................
16
2. M FD : M odified Friction device.............................................................................................................22
2.1 General Design..................................................................................................................................22
2.2 Friction element.................................................................................................................................23
2.3 Force generated.................................................................................................................................24
3. Friction m odels........................................................................................................................................25
3.1 Introduction.......................................................................................................................................25
3.2 Friction phenomenon.........................................................................................................................25
3.3 Classical Static m odels......................................................................................................................27
3.3.1 Coulomb friction........................................................................................................................28
3.3.2 Coulomb plus viscous friction...............................................................................................
28
3.3.3 Stiction plus Coulomb plus viscous friction...........................................................................
29
3.3.4 Non-linearity: Stribeck effect.................................................................................................
29
3.4 Dynamic m odels................................................................................................................................30
3.4.1 The Dahl M odel..........................................................................................................................30
3.4.2 The Bristle m odel.......................................................................................................................32
3.4.3 The LuGre model.......................................................................................................................33
3.4 Choice Justification...........................................................................................................................34
4. Fit to experim ental data...........................................................................................................................36
4.1 M ethodology .....................................................................................................................................
36
4.2 Adaptation of ao ................................................................................................................................
39
4.3 Adaptation of ai ...............................................................................................................................
39
4.4 Adaptation of a2 .....................................---
.......
.--------.
. . . . . . . . . . -.. . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.5 Adaptation of F, and vs......................................................................................................................41
5. Com putational implem entation ...............................................................................................................
48
5.1 Earthquake used ................................................................................................................................
48
5.2 M odel building properties and state space formulation ...............................................................
49
5.3 Control rules......................................................................................................................................51
5.3.1 Linear Quadratic Regulator ....................................................................................................
52
5.3.2 Clipped rule ................................................................................................................................
53
5.3.3 Sliding controller........................................................................................................................53
6. Simulations..............................................................................................................................................55
6.1 Passive M FD .....................................................................................................................................
56
6.1.1 Control with one MFD at the first floor .....................................................................................
56
6.1.2 Control with two M FD s at the first and second floor..............................................................
61
6.1.3 Control w ith two M FD s at the first and third floor ...............................................................
62
6.2 M FD controlled with clipped rule:...............................................................................................
63
6.2.1 Control w ith one M FD at the first floor .................................................................................
63
6.2.2 Control with tw o M FD s at the first and second floor.............................................................
68
6.2.3 Control with two M FD s at the first and third floor ...............................................................
69
6.3 MFD controlled with clipped optimal rule and a sliding mode controller:.................70
6.2.1 Control with one M FD at the first floor .................................................................................
6.3.2 Control with two M FD s at the first and second floor..........................
......................
6.3.3 Control with two M FD s at the first and third floor ...............................................................
6.4 D iscussion: ........................................................................................................................................
70
74
75
76
7. Conclusion...............................................................................................................................................78
References ...................................................................................................................................................
79
Table of Figures:
FIGURE 1: VARIABLE STIFFNESS DEVICE. .....................................................................................................................
11
FIGURE 2: CONCEPTUAL BEHAVIOR OF THE VARIABLE STIFFNESS DEVICE. ...............................................................
12
FIGURE 3: THEORETICAL BEHAVIOR OF THE PIEZOELECTRIC FRICTION DEVICE. ........................................................
12
FIGURE 4: PIEZOELECTRIC FRICTION DAM PER. ............................................................................................................
13
FIGURE 5: SCHEME OF THE SEMI-ACTIVE DRY FRICTION DEVICE. .............................................................................
13
FIGURE 6: SCHEME OF THE ER DAMPER DESIGNED IN[8]...................................
FIGURE 7: STRESS AND VELOCITY PROFILE OF THE ER FLUID WHEN THE ER DAMPER IS USED .................................
15
15
FIGURE 8: SCHEM E OF THE M R DAM PER ...................................................................................................................--
16
FIGURE 9: BOUC-W EN MODEL FOR THE M R DAMPER
.....................
........
.......................................................
17
18
FIGURE 10: MODIFIED BOUC-WEN MODEL PROPOSED BY ....................................................
FIGURE 11: FORCE VS. DISPLACEMENT OF THE MR DAMPER MODELED BY THE MODIFIED BOUC-WEN MODEL UNDER A
0.66 Hz SINUSOIDAL EXCITATION OF AMPLITUDE
0.0152 M.
...........................................
.. ...................
20
FIGURE 12: FORCE VS. VELOCITY OF THE MR DAMPER MODELED BY THE MODIFIED BOUC-WEN MODEL UNDER A 0.66
Hz SINUSOIDAL EXCITATION OF AMPLITUDE 0.0 152 M...........
.................................................
......................
21
FIGURE 13: SCHEM E OF TH E M FD ...............................................................................................................................
22
FIGURE 14: DUO-SERVO DRUM BRAKE SCHEME ........................................................................................................
23
FIGURE 15: SIMPLIFICATION OF THE FRICTION PHENOMENA .................................................
26
FIGURE 16: FRICTION BETWEEN A SOLID AND ITS SUPPORT ......................................................................................
26
FIGURE 17: COULOM B FRICTION ..................................................................................................................................
28
FIGURE 18: COULOM B PLUS VISCOUS FRICTION ........................................................................................................
29
FIGURE 19: STICTION PLUS COULOMB PLUS VISCOUS FRICTION .....................
.
......................
........................
29
F IG URE 20: STRIBECK EFFECT .....................................................................................................................................
30
FIGURE 21: FRICTION FORCE IN FUNCTION OF THE DISPLACEMENT FOR THE DAHL MODEL .........................................
31
FIGURE 22: ASPERITIES BETWEEN TWO SURFACES INCONTACT ........................
.....................
........................
32
FIGURE 23: EVOLUTIONARY VARIABLE THAT CORRESPONDS TO BRISTLE DEFLECTION ...........................
33
F IG URE 24: STRIBECK EFFECT .....................................................................................................................................
34
FIGURE 25: VARIATION OF THE STROKE IN TIME DURING THE TEST - DATA PROVIDED BY TAYLOR DEVICES INC .......
37
FIGURE 26: VARIATION OF THE FORCE DEVELOPED BY THE FRICTION DEVICE IN TIME DURING THE TEST - DATA
PROVIDED BY TAYLOR D EVICES INC. ..................................................................................................................
37
FIGURE 27: FORCE VS. DISPLACEMENT - DATA PROVIDED BY TAYLOR DEVICES INC.................................................38
FIGURE 28: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION E0 FOR DIFFERENT VALUES OF El
AND FOR
E2=2000,
FS= 12000 AND VS=0.12........................................................................................................39
FIGURE 29: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION OF El (A) FOR DIFFERENT
VALUES OF VS, (B) FOR DIFFERENT VALUES OF FS, (C) FOR DIFFERENT VALUES OF z2. ............................
40
FIGURE 30: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION OF E2..................................41
FIGURE 31: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION OF Fs AND Vs ------------...............
42
FIGURE 32: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION OF FS AND Vs (SMALLER VALUES
FO R THE TWO VARIA BLES)...................................................................................................................................43
FIGURE 33: VARIATION OF THE SUM OF THE SQUARE OF THE RESIDUALS IN FUNCTION OF FS AND VS (SMALLER VALUES
FO R TH E TWO V ARIA BLES). ..................................................................................................................................
FIGURE 34: M ODEL FITTING A
13
KIPS FRICTION DEVICE..........................................................................................
43
44
FIGURE 35: FORCE VS. DISPLACEMENT OF THE MODEL UNDER A 0.66 Hz SINUSOIDAL EXCITATION OF AMPLITUDE
0 .0 152M . .............................................................................................................................................................
46
FIGURE 36: FORCE VS. DISPLACEMENT OF THE MODEL UNDER A
0.66 Hz SINUSOIDAL
EXCITATION OF AMPLITUDE
0 .0 152M . .............................................................................................................................................................
FIGURE 37: TIME HISTORY OF EL CENTRO EARTHQUAKE
....................................................
FIGURE 38: SHOCK RESPONSE SPECTRA OF EL CENTRO EARTHQUAKE ................
...........................................
46
48
49
FIGURE 39: DISPLACEMENT OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD AT FULL
V O LTA G E. ............................................................................................................................................................
56
FIGURE 40: DISPLACEMENT OF THE THIRD FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD AT FULL
V O LTA G E. ............................................................................................................................................................
57
FIGURE 41: INTER-STOREY DISPLACEMENT OF THE THIRD LEVEL UNCONTROLLED AND CONTROLLED WITH AN MFD AT
FU LL V O LT AG E . ...................................................................................................................................................
58
FIGURE 42: ACCELERATION OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD AT FULL
V O LTA G E . ............................................................................................................................................................
58
FIGURE 43: ACCELERATION OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD AT FULL
V O LTA G E. ............................................................................................................................................................
59
FIGURE 44: INTER-STOREY DISPLACEMENT (A) AND ACCELERATION (B) OF THE FIRST FLOOR CONTROLLED WITH MR
AND M FD AT FULL VO LTAGE. .............................................................................................................................
60
FIGURE 45: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
SEC ON D FLO O R . ...................................................................................................................................................
61
FIGURE 46: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
THE FIRST AN D SECON D FLOO R. ...........................................................................................................................
61
FIGURE 47: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
T H IR D FLO O R. ......................................................................................................................................................
62
FIGURE 48: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
THE FIRST AN D TH IRD FLOOR...............................................................................................................................62
FIGURE 49: INTER-STOREY DISPLACEMENT OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD
(CLIPPED RU LE AN D L Q R)...................................................................................................................................63
FIGURE 50: DISPLACEMENT OF THE THIRD FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD (CLIPPED
OPTIM AL CONTRO L AN D L Q R ). ...........................................................................................................................
64
FIGURE 51: INTER-STOREY DISPLACEMENT OF THE THIRD FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD
(CLIPPED OPTIM AL CONTROL AND
LQ R). ............................................................................................................
65
FIGURE 52: ACCELERATION OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD (CLIPPED
OPTIM AL CONTRO L AND LQ R ). ...........................................................................................................................
FIGURE 53: ACCELERATION OF THE THIRD FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD (CLIPPED
OPTIM AL CONTROL AND LQ R ). ...........................................................................................................................
65
66
FIGURE 54: INTER-STOREY DISPLACEMENT (A) AND ACCELERATION (B) OF THE FIRST FLOOR CONTROLLED WITH MR
AND
MFD
(CLIPPED OPTIMAL CONTROL AND
LQR). ..........
........................................
66
FIGURE 55: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
SECO N D FLOO R . ...................................................................................................................................................
68
FIGURE 56: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
THE FIRST AN D SECON D FLOOR. ...........................................................................................................................
68
FIGURE 57: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
T HIR D FLO O R . ......................................................................................................................................................
69
FIGURE 58: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
THE FIRST AN D TH IRD FLOO R...............................................................................................................................69
FIGURE 59: INTER-STOREY DISPLACEMENT OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD
(CLIPPED OPTIMAL RULE AND A SLIDING MODE CONTROLLER). ...................................................
71
FIGURE 60: ACCELERATION OF THE FIRST FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD (CLIPPED
OPTIMAL RULE AND A SLIDING MODE CONTROLLER). ........
.................................
................. 71
FIGURE 61: DISPLACEMENT OF THE THIRD FLOOR UNCONTROLLED AND CONTROLLED WITH AN MFD (CLIPPED
OPTIMAL RULE AND A SLIDING MODE CONTROLLER). ...........
....
.................
..................................
72
FIGURE 62: INTER-STOREY DISPLACEMENT OF THE THIRD LEVEL UNCONTROLLED AND CONTROLLED WITH AN MFD
(CLIPPED OPTIMAL RULE AND A SLIDING MODE CONTROLLER). ...........................................................................
73
FIGURE 63: DISPLACEMENT OF THE THIRD FLOOR CONTROLLED WITH AN MFD (CLIPPED OPTIMAL RULE AND A
SLIDING MODE CONTROLLER) AND A MFD AT FULL VOLTAGE. ....................
.................................................
73
FIGURE 64: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
SEC O N D FLOO R . ...................................................................................................................................................
74
FIGURE 65: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
THE FIRST AND SECON D FLOOR. ...........................................................................................................................
74
FIGURE 66: DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT THE FIRST AND
T H IR D FLO O R. ......................................................................................................................................................
75
FIGURE 67: INTER-STOREY DISPLACEMENT OF THE FIRST (A) AND THIRD (B) FLOOR CONTROLLED WITH TWO MFDS AT
TH E FIRST AN D TH IRD FLOO R. ..............................................................................................................................
75
Introduction
With the appearance of high rise buildings in the early 20th century, the problem of vibrations in
structures has become an issue in the design. These vibrations can damage the structure and thus
reduce its life span. These vibrations also induce motion in the structure that can be perceived by
the occupants. In order to control these vibrations, several devices have been designed by
structural engineers in recent years.
In the literature related to these systems, one can notice that several of the proposed devices do
not meet realistic constraints in terms of usability and efficiency. During extraordinary events,
earthquakes for instance, a general power failure is likely to occur. Because of this power failure,
it is important to provide energy to the device using independent energy sources to ensure
security. For this reason, devices requiring significant external power are not adapted to real life
constraints.
Semi-active control devices are usually used for controlling civil structures and solve the power
failure problem, because they require low energy. A simple battery can be used to power these
devices. Among semi-active control devices are variable friction dampers and rheological
dampers. Semi-active devices generate force by changing their inner characteristics. For
instance, for a friction device with friction pads, the force is generated by the contact between the
pads and the friction element. In the past few years, a new type of semi active magnetorheological dampers (MR dampers) has been widely studied. A rheological fluid is a fluid whose
stiffness characteristics can be controlled. MR dampers attracted researchers because they can
generate high forces and are easily controllable by a small energy source. However, because MR
dampers are new, they still display some issues that can be problematical for large scale
implementation.
The Modified Friction Device, called MFD in this paper, eliminates some of the disadvantages of
the MR damper such as the sedimentation and the fluid leakage problem [1]. The MFD is
composed of a spring, a viscous element and a braking system proposed by [1]. Therefore, it is
more mechanically robust and reliable than the MR damper as it eliminates the issues previously
described and provides a minimum load created by the stiffness element and the viscous element.
The MFD is still at its conceptual design. In order to capture its behavior without experimental
data and test results, a model has been implemented in MATLAB. The friction model used to
describe the dynamic behavior is the LuGre friction model [2]. It has been designed to optimize
the modeling of friction between two surfaces. This model is one of the most complete friction
models as it takes into account the stiction phenomenon and the Stribeck effect that occurs
during friction.
The strategy of this research was, first, to compare this model with experimental data provided
by Taylor Devices Inc. The experimental data is from a viscous damper that behaves in friction
for small displacement. Thus, by extracting the data for small displacements, the behavior of a
friction device has been obtained. The goal was to adapt the parameters of the LuGre model to fit
a real friction device. Then, the parameters obtained with the fit were extrapolated to obtain a
200 kN force output at full voltage. Finally, in order to compare the MFD with existing devices,
a six storey building has been used to simulate the device. The MFD was compared to a 200 kN
MR. The MR damper was simulated using a modified Bouc-Wen model designed and tested in
[9]. In order to do these simulations, different schemes have been studied. First, the MFD has
been studied as a passive device. Then, a clipped rule controller was implemented to control the
voltage applied to the device. Finally, a clipped rule controller coupled with a sliding controller
was implemented. A linear quadratic regulator has been used to determine the required control
force. A series of studies have been done using one MFD at the first storey, two MFDs at the
first and second storey, and two MFDs at the first and last storey were carried out. This study
was designed to compare the MFD to the MR damper, to compare different control schemes for
the device, and to compare different implementation schemes.
...........
..........
..........
. ........
....
..
......
.
1. Semi-active devices
1.1 Introduction
The purpose of this part is to introduce the reader to some of the concepts of semi-active friction
devices existing and that have been researched in the past. This list is not exhaustive and only
presents a few features of some of the devices or control systems. First, the concept of a variable
stiffness device using friction will be presented. Then, two friction dampers using friction pads
will be introduced. Then, a friction device that has been installed in trucks will also be discussed.
Finally, a brief presentation of ER and MR dampers will be done.
1.2 Variable stiffness device
The device that is presented in this part has been designed by [4]. The designers of this device
used a commercial hydraulic cylinder filled with nitrogen instead of oil. Using this gas, they
were able to change the stiffness by charging the cylinder at different pressures. [4] related the
pressure on the cylinder to the effective stiffness generated.
The research that has been done by [4] showed that their device behaves similarly to a spring
with an adjustable unstretched length [4]. Because of the intrinsic design of this device, it
behaves like a spring rather than like a damper. Thus, for high velocities, the load transmitted to
the structure through the device is smaller in comparison with other control devices.
~jJz
z
a)
b)
Figure 1: Variable stiffness device. [4]
..................................................
....
..................
......
"I'll",
.................
.. ..
The picture on the right of Figure 1 shows the device as it has been designed. The left part of
Figure 1 illustrates the conceptual behavior of the device. In Figure 1, ko is the structural
stiffness, c is the structural damping and k, is the actuator stiffness.
Figure 2: Conceptual behavior of the variable stiffness device. [4]
The device behaves such as if friction was created between A and B. This friction will cause the
device to be stiffer and will then create more force. The friction between A and B is created by
the gas actuator presented earlier in this section.
1.3 Friction device using friction pads
1.3.1 Piezoelectric friction device
This device has been designed by [5] for the control of single degree of freedom systems. The
device is based on a piezoelectric actuator acting on friction pads. The theoretical behavior of the
device can be summarized as follows:
N
&
MFenia
Ft
0
M
mmiJZ
Nc
Figure 3: Theoretical behavior of the piezoelectric friction device. [5]
........................................................................................
......................
...........
...........
.
The control is done on the normal force N that creates friction. N is created by the piezoelectric
actuator pushing on friction pads. This system enables the device to generate significant force
with a small stroke in the piezoelectric actuator.
Fnion
Pad
Housing
Actuator
Shant
Air
spfingBemng
Figure 4: Piezoelectric friction damper. [5]
By controlling the piezoelectric actuator, it is possible to control the device and then generate
forces depending on the need.
1.3.1 Lorenz dry friction device
This device has been designed in [15] in order to isolate a car engine from the vibrations created
by the shape of a road.
1: fntfin d
2: cod
3: ak
4: loose
5:fictiunpad
9: P NpM
6&imn
1c:Iremar guide
7:Iinear gie
t: 1Eo coin
: positio semn
Figure 5: Scheme of the semi-active dry friction device. [15]
The friction force is generated by the contact between the pads and a cylindrical element. To
bring the pads in contact with the cylindrical element, an electromagnetic actuator is used. The
friction force can be controlled by measuring the air gap (number three in Figure 5). [15] shows
that the air gap is related to the electromagnetic force exerted by a complex relation that has been
established in the same paper. Because the air gap is related to the electromagnetic actuator
force, it is also related to the friction force generated by the friction device. Thus, by monitoring
the width of the air gap it is possible to know the friction force and then change it by adapting the
magnetic field in the actuator.
1.4 Rheological dampers
1.4.1 Electro-rheological dampers
Electro-rheological (ER) fluids are non-conductive fluids where micron sized particles are in
suspension [6]. The strength capacities of these fluids can change tremendously under an
electrical current. Because of this behavior change, these fluids have been considered for the
control of civil structures.
The principle of ER damper is that the change in the stiffness of the fluid under an electrical
current increases the friction force between the fluid and the lateral walls of the device. As the
stiffness characteristics of the fluid can greatly vary, the friction force created can be significant.
The flow of the fluid starts again when the stress applied to the fluid is bigger than the yield
stress.
An ER damper has been designed in [7]. Different configurations can be used for this kind of
dampers. Figure 6 represents the general scheme of ER dampers:
Outer Cylinder Electrode
Inner Cylinder Electrode
Figure 6: Scheme of the ER damper designed in [8]
The stiffness of the ER fluid is controlled by an electrical current. The more the load exerted by
the piston is important, the more stiffness is needed from the fluid to counteract the pressure on
the piston head. This extra-load can be generated by applying more electrical current to the
device.
OUTER CYLINDER OF BYPASS (GROUND)
INNER ROD (ELECTRODE)
ER-MATERIAL IN SOLID STATE
ER-MATERIAL IN FLUID STATE
vx(r)
Y=L
t~(0, stress
____
h/2
r
-xrdemand
Figure 7: Stress and velocity profile of the ER fluid when the ER damper is used [5].
The flow of the fluid can be modeled with the Hagen-Poiseuille theory. Figure 7 shows the
behavior of the fluid under use. Once an electrical current is applied, the stiffness of the fluid
changes. Thus, there is more friction with the lateral walls and then more stress.
1.4.2 Magneto-rheological dampers
Magneto-rheological (MR) fluids exhibit the same behavior as Electro-rheological (ER) fluids.
The fluid becomes stiffer when a magnetic field is applied to it. Thus, the semi solid behavior
created by a magnetic field creates more friction with the lateral walls of the device, which
implies a force that counteracts the motion. By changing the magnetic field, one can control the
yield strength of the fluid and then control the effective force created by the device. The
following figure presents a general scheme of an MR damper:
Wires to
M
Ma
iMRFluid
c
Accumulator
RodPiston
Figure 8: Scheme of the MR damper [9]
1.4.3 Models for rheological dampers
S.R. Hong et al. [8] did a comparison between different damping force models for an electrorheological fluid damper. The study focused on: the Bingham plastic model, the hysteretic
Bingham plastic model, the hysteretic biviscous model, the Bouc-Wen model, and the hydromechanical model. A fit was made to adapt the model to experimental data. Then, the models
were compared to each other and to the experimental data.
This study showed that the Bingham plastic model is not able to capture the behavior of the
hysteresis force in the preyield [8]. The other models capture both postyield and preyield
hysteresis fairly. To evaluate the efficiency of these models, the error between the experimental
data and the model results is evaluated using the relative root mean squared (RMS) force error:
Relative R MS error = [=
mdlt)
moe ]2x
X~k
[E=1
k)-
[j=1
[fexp (tk)]
2
]2p
2
1
y
In the previous equation, fexp is the experimental force and fmodel is the force provided by the
model for the same displacement. This comparison showed that the most accurate model is the
Bouc-Wen model. However, modified models have been studied later in the literature and are
more accurate than the simple Bouc-Wen model.
Spencer et al. [9] studied the MR damper to establish an accurate modeling of this device.
ZO
Bouc-Wen
zx
ko
CO
Figure 9: Bouc-Wen model for the MR damper [9]
In Figure 9, F corresponds to the force in the system, x corresponds to the stroke of the MR
damper, xo corresponds to the rest position of the device, ko and co are respectively the stiffness
coefficient and the damping coefficient associated with the model.
The Bouc-Wen model provides results that are very close to the experimental data. However, it
fails in predicting the non-linear force-velocity relation in the region where acceleration and
velocity have opposite signs and for small velocities. This model is versatile and exhibits a wide
range of hysteretic behavior.
The equilibrium of the bar provides the following equation:
F = co
x
+ ko (x - xO) + az
where z is the evolutionary variable governed by:
z=
-y|xlz |i|"-
-
p*Izi + Ak
In order to better predict the behavior of the MR damper, a modified Bouc-Wen model was
proposed in [9]:
x
y
Bouc-Wen
co
F
Figure 10: Modified Bouc-Wen model proposed by [9]
In Figure 10, a dashpot co has been introduced to simulate the roll-off has been observed during
experimental tests for the zones where acceleration and velocity have opposite signs. ki
represents the accumulator stiffness. y represents the displacement of the first rigid bar while x
represents the force of the second rigid bar. ko and co are the same as for the Bouc-Wen model. xo
is the initial displacement of the spring k1 .
This model has showed better performance to capture the behavior of the MR damper. [9]
compared the error of several models and it appeared that this scheme was more efficient than
the other models.
The equations governing this model have been established. The modified Bouc-Wen model takes
into account an evolutionary variable z.
The equilibrium of forces in both sides of the first rigid bar provides:
c9=
az+ ko(x-y)+ cO (
-9)
The evolutionary variable is governed by the following equation:
z
= -yIk -
p'z 12i|-1 - fl(* - 9)Iz| + A(k -
Solving for y:
1
9co +
ci
,(az + ko (x - y) + co k)
Then, by adding the loads of the upper part of the model to the lower part, the force F obtained
is:
F
=
az+ co(* -9)+
kO(x-y)+ k 1 (x-xo)
Both the ER and MR dampers are semi- active devices. They are interesting because they offer a
performance and adaptability features comparable with active control devices. They eliminate
the problem of energy requirement as they both can be fed with commercial batteries.
The behavior obtained with this model under a 0.66 Hz sinusoidal excitation of 0.0152m
amplitude is:
............
- - -----------
. ......
200
150-
max
100.
z
*
.
.
.
.
.
---
75% V
max
V-max
50 -
0-
-ax
-25% V
-- --
...... I.............................
~-50-
-100-
-150
-5.02
0.005
0
-0.015 -0.01 -0.005
displacement (m)
0.01
0.015
0.02
Figure 11: Force vs. displacement of the MR damper modeled by the modified Bouc-Wen model under a0.66 Hz
sinusoidal excitation of amplitude 0.0152 m.
............
200
150
- -25% V
-
......
50%V
100 .
---
-0
75%
max
. ----
max
Vmax
-
-
.60/
-100-
II
-
-
-
-- - - - -
-100
,
-W
-150
-2 .015
.001
0.005
0.005
0
velocity (mIS)
0.01
0.015
Figure 12: Force vs. velocity of the MR damper modeled by the modified Bouc-Wen model under a 0.66 Hz
sinusoidal excitation of amplitude 0.0152 m.
2. MFD: Modified Friction device
2.1 General Design
The Modified Friction Device proposed by [1] consists of a friction element, a viscous element
and a spring.
The values of the spring and viscous element have been designed such that under 0 volt the force
applied by the device is 20kN. As the goal was to implement a device that can generate a force of
200kN, the friction element can generate a maximum force of 180kN. This force added to the
20kN provided by the spring and the viscous element provides a load of 200kN. This friction
element is commanded by a 12-volt battery. Thus, the control of the friction device is done by
changing the voltage between OV and 12V.
dashpot element
. F
variable friction
element
x
Figure 13: Scheme of the MFD [1]
In Figure 13, F represents the applied load, x the stroke of the device, which is the relative
displacement of the device.
The unique feature of the MFD is that the friction element is a braking system. Braking systems
have been widely studied because of their importance in cars, trucks and industrial applications.
The railroad industry, for instance, uses large scale friction elements that can produce significant
forces. Thus, while generating a significant force, the MFD presents the advantage that it is more
............
. ..
..
....
..
reliable and more mechanically robust than the other devices that it can be compared to such as
the ER damper or the MR damper.
2.2 Friction element
The braking system used for the MFD is a Duo-Servo Drum brake described in [1]. It is
composed of an external drum, two internal shoes that will act as friction shoes, a hydraulic
actuator that can be controlled to vary the friction force and a link between the two shoes. The
shoes are anchored to the drum at a single point.
0W
b
a
a
-0&.....
Scheme
loads in the braking system
Figure 14: Duo-Servo drum brake scheme [1]
In Figure 14, W represents the load exerted by the hydraulic actuator. Rn is the reaction exerted
by the rigid link. F is the friction force created by the contact between the drum and the shoes. N
is the force normal to the friction force. The radius of the brake is noted r; b and a represent the
distance between respectively the top and the bottom to the center of the scheme.
The way this device works is as follows:
e
The actuator is required by a control system to exert a force on the shoes.
" The load is transmitted from the shoe to the link that creates a reaction.
" The shoes are then put in contact with the external drum and create a friction force.
" The drum rolls on its support.
This braking system has the advantage of amplifying the force more easily because of its
intrinsic design [1]. The rotational design and the fact that the two shoes are linked to each other
with the link and the actuator make the system to be self-energizing. Thus, the MFD produces
forces more easily than a linear braking system. The actuator used in [1] will develop a load of
2kN. The study made by [1] during the design showed that with the MFD characteristics, the
force amplification factor of the device is 58.5.
Again the complete study of this element has been made in [1] and the results concerning the
characteristics of the device are:
*
p.= 0.46 where t is the friction coefficient of the elements.
* b, =0. 1m and t=0.025m where bw and tw are respectively the width and the thickness of
both shoes.
*
The drum radius is r-0.2m.
*
The actuator is actually two 2kN-actuators separated by a rigid link and operating under a
0 volt-12 volts current provided by commercial batteries. These two actuators are
mounted at 0.6 m up from the center of the drum and the shoes finishes at 0.3 m down
from the center of the system.
2.3 Force generated
The force generated is:
Fmfd
=
Ffriction + kx + ci
where Ffriction is the load created by the braking element, k and c are the characteristics of the
spring and viscous element.
The MFD is still at the conceptual design. As the device has not been built yet, experimental data
providing the exact behavior of the device has not been yet generated. Thus, in order to evaluate
the performance of this device, it is important to model it accurately using a friction model that
will capture the MFD behavior as well as possible.
3. Friction models
3.1 Introduction
Friction is the phenomenon that appears when two physical surfaces are in contact. Friction can
appear in different cases. For instance, for magneto rheological dampers, it results from the
contact between the rheological fluid and the lateral walls of the device. Friction also appears
when two solid elements in contact are sliding at different velocities.
This phenomenon is very important in all the engineering fields as it often occurs, for instance, in
machines, car transmissions, valves and brakes. Because this phenomenon is non-linear, it is
important to capture the real frictional behavior as accurately as possible. This will result in a
reduction of the errors consequential to the non-linear behavior.
In civil engineering, friction is used to dissipate energy. Some control devices use friction to
function. Because friction can generate high level forces, it can be used for large scale control.
3.2 Friction phenomenon
Friction is a tangential force between two surfaces in contact that acts in opposition to the
displacement. It depends on the properties of the materials in contact, the contact geometry, the
velocity of the two surfaces and the type of contact involved (e.g., lubricated, dry).
As it is shown in Figure 15, the mechanism that is involved in a dry contact can be simplified as
the contact between the micro-asperities of the material.
FFigure 15: Simplification of the friction phenomena [10].
This contact will make each of the asperities carry a load, the summation of which will
equilibrate the normal load N. The deformation of these asperities is elastic until the tangential
load exceeds the shear strength of the material used [10]. Then, the deformation becomes plastic.
V
Figure 16: Friction between a solid and its support [10].
Friction phenomena also depend on the normal force exerted between two solids. Figure
16shows the case of a solid sliding on a support. A force Fe applied to the solid creates a friction
force F. F depends on the normal force N and follows the classical law for friction
F=yN
where p is the friction coefficient. The relation between friction force and normal force can be
used to design controllable friction devices. For instance, an actuator can be used to raise or
lower the normal force and then provide the friction force that is needed.
There are two main types of friction. The first is static friction. This type of friction does not
involve a motion of the two surfaces in contact relatively to each other. The second is dynamic
friction and involves a motion of the surfaces relatively to each other. The classical law for
friction is applied differently in each friction case by changing the friction coefficient p. Thus,
we have pstatic and pdynamic. In the case of dry sliding, the friction coefficient does not depend on
the normal load N. To manipulate this coefficient, it is possible to use an interface between the
bodies [11].
When there is lubrication, it is more difficult to assess the frictional behavior as the properties of
the lubricant affect the friction phenomenon. For low velocities and a low pressure distribution,
friction depends on shear forces in the fluid because of the hydrodynamic effects in the lubricant.
These shear forces are related, for instance, to the viscosity of the fluid used and shear velocity.
For high velocities and a high pressure distribution, the lubricant reacts as a solid. High pressure
in the contact area turns the liquid to an amorphous solid phase. Shear forces in the lubricant, in
this case, are not related to the shear velocity. As a result, the difficulty in evaluating the friction
force generated increases [11]. For solid lubricants, the shear strength of the lubricants decreases
with increasing velocity. Thus, the friction coefficient p also decreases with increasing velocities.
However, when the lubricant layer is thick enough to separate totally the two bodies, the friction
coefficient increases as hydrodynamic effects due to the lubrication become significant. This
phenomenon is called the Stribeck effect and is difficult to model. Therefore, it is difficult to
model properly friction phenomena. Several friction models have been designed for both static
and dynamic friction and capture approximately all the parameters of friction.
3.3 Classical Static models
The classical static friction models are numerous. Some of these models account for several
aspects of the friction and others are limited to the minimum aspects.
Friction acts in opposition with the displacement. It depends of the contact area and the velocity
v. The classical formula that describes it is
Ffriction = Fsign(v)
3.3.1 Coulomb friction
Coulomb friction is defined by F= p N where N is the normal force and p the friction coefficient.
Figure 17: Coulomb friction [11].
Figure 17 illustrates Coulomb friction. The value of the force for v=O can be anywhere between
-F and +F.
3.3.2 Coulomb plus viscous friction
Coulomb plus viscous friction corresponds to the case where there is lubrication between the two
elements that create friction. Viscous friction is associated to the viscous force - F, - that is
created. Viscous friction is described by
F = Fv
Viscous friction is usually coupled with Coulomb friction. The resulting friction force - Ffiaon is
Ffriction = F%|vlasign(v)
This formulation provides a better approximation to the experimental data [12], [13].
Figure 18 shows the profile of force vs. velocity for this type of friction.
Figure 18: Coulomb plus viscous friction [11].
3.3.3 Stiction plus Coulomb plus viscous friction
In static friction, stiction is shorter than in dynamic friction. In [14], the authors introduced
stiction in a static model by assuming it as a force that counteracts the external forces and that is
higher than the Coulomb force. The resulting friction force is
Ffriction = IFsign(F)
ifv = 0 and IFe| < IF|
if v = 0 and IFe| > |F|
where Fe represents the external force applied to the solid and Fs the stiction force.
When the velocity is null, the friction force depends on the external force. Thus, stiction can take
any value between - Fs and + Fs.
Figure 19 illustrates the force vs. velocity profile of this type of friction.
Figure 19: Stiction plus Coulomb plus viscous friction [11].
3.3.4 Non-linearity: Stribeck effect
This last model is the result of the observations and research made by Stribeck. The friction
decreases continuously in terms of velocity dependence [16]. The proposed description for this
phenomenon is
F(v)
Fe
Fssign(Fe)
Friction =
if v = 0
if v = 0 and IFe| < |Fs|
otherwise
F (v) is commonly a function of the form
F(v) = F + (Fs- F)evs
+ Fv
where F is the Coulomb friction force and F is the stiction force.
This formula illustrates the non-linearity of the friction profile showed in Figure 20.
F
Figure 20: Stribeck effect [11].
The issue in the use of such a model is to detect when the velocity is null, which corresponds to
the limit between a sticking and a sliding friction [11], [16].
The static models presented in this section capture part of the friction characteristics. However,
because of the new applications of friction in engineering, more precision was needed. Dynamic
friction models have been designed to provide more accuracy in modeling friction. Furthermore,
numerical simulations are difficult to achieve with static models as sticking is difficult to
distinguish from sliding.
3.4 Dynamic models
3.4.1 The Dahl Model
The Dahl model is based on the stress-strain curve that is defined by
dF
-
dx
= U-(1 -
F
sign(v))"
Fc
In this equation, v is the velocity, a is the stiffness coefficient, Fe is the Coulomb force and
x determines the shape of the stress-strain curve [17]. The curve obtained with this equation is
presented in Figure 21.
810PO
Figure 21: Friction force in function of the displacement for the Dahl model [11].
The friction force, in this case, is a function of the displacement. Because it depends on the
displacement and not on the velocity, the model is called rate independent, which is one of the
main characteristics of this model. As it has been said earlier, the stiction phenomenon and the
Stribeck effect are rate dependent [11], [16], [17]. Thus, this model cannot model these two
phenomena.
A time domain model of the Dahl model can be obtained by
dF
-
dt
_
dFdx
--
dxdt
=dF
-v
dx
F
=o-1
sgv)a
Fc
This model is the most general dynamic friction model as it is a generalization of the Coulomb
friction phenomenon. It models neither the Stribeck effect nor the stiction phenomenon.
Usually the coefficient x is taken to be 1. In this case, the time domain equation becomes
F
dF
- = U-(1 - sign(v))v
dt
F
Introducing the variable z such as F= a z, the model can be written
dz
cIvI
dt
F
F =
oz
3.4.2 The Bristle model
The most interesting aspect of this model is that it tries to capture the frictional behavior at the
microscopic scale. This friction model has been developed in [18].
The irregularities of the surfaces that are in contact are the areas where friction happens. These
irregularities are thought as bonds between flexible bristles. As the surfaces move relatively to
each other, the bristles act like springs when in contact with the bonds. This creates the friction
force generated by the contact.
Figure 22: Asperities between two surfaces in contact [10].
The friction force is then the sum of the forces created by each bristle [18]. If we note (O the
stiffness of every bristle,
xt
the position of the bristle and bt the position of the bound, the friction
force created is
F =
iocN (xt - be)
In this equation, N is the number of bristles. It represents the number of contacts between both
surfaces.
This model is difficult to use for simulation because it is complicated to know the number of
microscopic contacts between the surfaces. Because it is complex, the Bristle model is not
efficient when N increases.
3.4.3 The LuGre model
The LuGre model is inspired from both the Dahl model and the Bristle interpretation of friction.
This model has been established in [19]. The concept is the same as the Dahl model except that
the LuGre model aims to capture both stiction and the Stribeck effect. It is related to the Bristle
model in the sense that it introduces an evolutionary variable that, for the LuGre model,
corresponds to the bristle deflection (Figure 23).
V
No
Figure 23: Evolutionary variable that corresponds to bristle deflection [10].
The model can be described by
dz
dt
|v|
g (v)
dz
F = aoz + a1(v) -+ f (v)
In these equations, z represents the bristle deflection and v represents the velocity of the sliding
motion. The second equation corresponds to the equation of motion of the system. It has been
written using the same principle as the Bristle model. In this equation, Co corresponds to the
stiffness of the bristles, I corresponds to the damping and f(v) is a function that describes the
viscous damping. The viscous damping is linearly proportional to the velocity and can be written
f
(v) = 02 V.
The function g (v) is a function that describes the Stribeck effect. The Stribeck effect has already
been described in section 3.3 related to static models.
F
Figure 24: Stribeck effect [2].
A reasonable function to describe the rate-dependent relationship of the Stribeck effect is
g(v) = ao + a1 e vs
When the velocity is null,
g(v) = ao + a1
which corresponds to the stiction force. When the velocity is high,
g(v) = ao
which corresponds to Coulomb friction. Thus, the description for this model is
dz
lvi
-t= V - Uo g(v)'
V z
dt
g(v) = ao + a1 e
F = a0z +
,)
1 i+ au
2v.
3.4 Choice Justification
In order to study the Modified Friction Device, we needed a friction model that provides as much
accuracy as possible. The Dahl model, even if designed for simulation purposes, does not capture
stiction and the Stribeck effect [16], [17]. One of the purposes of this research is to show that the
Modified Friction Device is efficient for real life applications. The MFD is at the conceptual
design phase and we do not have experimental data about the device. The LuGre model captures
both stiction and the Stribeck effect. For this reason, the LuGre model is the best candidate to be
used in this research.
4. Fit to experimental data
4.1 Methodology
In order to adapt the model to a real friction device, Taylor Devices Inc. provided experimental
data from a viscous device behaving in pure friction for small displacements. Thus, by extracting
the data from the force vs. displacement profile for small displacements, it was possible to obtain
the behavior of a real friction device.
The dynamic of the LuGre model has been studied in section 3. As shown in section 3 of this
paper, the dynamic of the LuGre model is described by
dz
dt
|v|
g (V)
-V1
2
g (v) = ao + ale Is
F = cz + a1 i + a2 v
In these equations, Yo represents the stiffness of the device, a1 represents the damping of the
device and 72 represents the viscous damping of the device. o corresponds to the coulomb
friction and ao+ai captures the stiction. By calling Fc the coulomb friction and Fs the stiction, F,
=o and Fs = Fe + cc1 . The equations guiding our system are
dz
-t= V dt
UO
0
|v|
V z
g (v)
g (v) = F + (Fs - F)je vs
F = cz + a 12 + a 2 v
Thus, by changing the parameters of the model a fit of the real behavior can be obtained. Then,
the 200 kN Modified Friction Device model can be obtained by extrapolating the results of the
fit.
The force and stroke variation in time of the real device used to adapt the model is provided in
Figure 25 and Figure 26.
0.2
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
5
10
15
20
25
30
35
40
45
time (s)
50
Figure 25: Variation of the stroke in time during the test - Data provided by Taylor Devices Inc.
-5
-10
-15
5
10
15
20
25
30
time (s)
35
40
45
50
Figure 26: Variation of the force developed by the friction device in time during the test - Data
provided by Taylor Devices Inc.
............................
.
The force vs. displacement profile has also been generated based on the experimental data.
15
10-
0.
0-
UL
-10i
-1b.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
displacement(in)
Figure 27: Force vs. displacement - Data provided by Taylor Devices Inc.
Figure 27 shows the behavior of this device. The fit aimed to adapt the model to a real device. As
the LuGre model takes into account several parameters, for example, the coulomb friction, the
stiction and the Stribeck effect, one can change these parameters to adapt the real behavior.
As it has already been explained in the part related to the friction models, the LuGre model
introduces 5 parameters: aO, al, a2, Fs and v. Fc is fixed to set the force created by the device.
To do this fit, the least square method was used. The goal was to obtain the same behavior for
the force vs. displacement profile. Thus, it has been necessary to reorganize both the
experimental data and the data generated by the model. This has been done using MATLAB. The
reorganization consisted in a comparison of the force generated by the model and the force from
the experimental data for a same displacement.
The final parameters have been selected in order to minimize the sum of the squares of the
residuals. The residuals correspond to the difference between the experimental data yi and the
LuGre force corresponding to the same displacementf (co,, a,
r = yi -
f(o, U1, a2 , Fs, vs)
2 ,F,
v).
..
Then the sum of the square of the residuals is calculated for all the displacements:
n
S=
r
t=1
This study enabled to assess the effect of each of the parameters. As this study has 5 variables,
some plots justifying the choice of these parameters are presented with explanations.
4.2 Adaptation of Yo
The following figures show the behavior of the sum of the square of the residuals when ao varies.
A
a
1
3
A
a
~6
E
H
9
lso
Figure 28: Variation of the sum of the square of the residuals in function aO for different values of al and
for a2=2000, Fs=12000 and vs=O. 12.
The study showed that the profile for the variation of the sum of the square of the residuals in
function of ao was the same for different set of values for 02, Fs and vs. The more Go is small the
more the sum is big and then the more the model is wrong. Thus, a value of 9,000,000 has been
chosen for YO.
4.3 Adaptation of a,
The following figures show the behavior of the sum of the square of the residuals when ai varies.
...
. .....
. .....
(a)
(b)
24
x10..
050
sigma 2
sigma1
(c)
Figure 29: Variation of the sum of the square of the residuals in function of ai (a) for different values of vs, (b) for
different values of F,, (c) for different values of a2 .
The previous figure shows that c01 does not impact the variation of the sum of the square of the
residuals. This sum is constant for ai in the three configurations presented. These configurations
are the same if we vary the parameters. This is due to the fact that the loading is not harmonic.
Thus, an arbitrary value of 100,000 has been chosen for
40
1.
4.4 Adaptation of 0 2
Figure 30 shows the behavior of the sum of the square of the residuals when y2 varies. Several
plots have been have been generated and the variation of the sum in function of 0 2 is the same for
all the parameters.
6
5
4
co 3-
2-
1
0
0
0.2
0.4
0.6
0.8
1
sigma2
1.2
1.4
1.6
1.8
2
x 104
Figure 30: Variation of the sum of the square of the residuals in function of a 2.
This figure shows that the more
c2
is big the more the difference between the model and the
experimental data is important. For this reason, a small value has been chosen for 02. For the
MFD, 02=1,000.
4.5 Adaptation of F, and v,
Finally, for Fs and vs, which represent respectively the stiction and the Stribeck effect, it
appeared that there was an exact value for which the model is a good fit of the experimental data.
The following figures shows the behavior of the sum of the square of the residuals when F, and
.....
. .. ...............
...
vs vary.
8000..
6000
U)
4000
2000
0.2
2.5
0.15
x 104
1
0.05
Figure 31: Variation of the sum of the square of the residuals in function of F, and v,.
This figure shows that there is a general tendency: the more F, and v, are big, the more the model
is different from the experimental data. There is a double curvature but the figure does not show
it so a second figure with small values of the two variables has been generated.
42
....................................
......
..............
....
1200
O
1000
--
800
--
600
-
400200
x 104
-
0.15
1.5
Fs
0.2
0.1
1
0.05
Figure 32: Variation of the sum of the square of the residuals in function of F, and v, (smaller values for the two
variables).
Figure 32 shows that there are two local minima.
Fs
1 0.05
S
Figure 33: Variation of the sum of the square of the residuals in function of F, and v, (smaller values for the two
variables).
In Figure 33, the areas where the minimum can be located are the red line and the red rectangle.
Using the matrices generated during the study, the minimum was found to be for Fs=14,000 and
vs=0. 12.
The Best fit has been obtained with the next values:
Table 1: Parameters for the model fitting of the 13 kips friction device.
0O
9000000 kN/m
a0
100000 kN.s/m
02
1000 kN.s/m
Fs
14000 kN.s/m
vs
0.12 m/s
2
I
With these parameters the fit was:
10|-
-10 1-
.0.15
-0.1
-0.05
0
0.06
0.1
Displacement (in)
Figure 34: Model fitting a 13 kips friction device.
0.15
By extrapolation, this model was adapted to have the 200 kN Modified Friction Device.
Table 2: Parameters for the 200 kN Modified Friction Device.
(7o
50,000,000 kN/m
G1
1000OkN.s/m
(32
6.5 kN.s/m
Fs
1.1697 Fc
a
2
p3
1
vs
0.025 m/s
Fcmax
160 kN
K
200000 N/m
C
100000 N.s/m
190
100
These parameters have been adapted to obtain a behavior for the friction element similar to the
experimental data from the 13 kips device. K and C are the values of the spring and the dashpot
of the MFD.
The model of the MFD, i.e. with these parameters, has been simulated for a sinusoidal load of
amplitude 0.5 inch (0.0152m) and a frequency of 0.667 Hz.
200
-
150- ------------
-.- no voltage
--- 2% Vmax
.50% V
100-
50
V
---------
max
0-
-50
.................
-100
-----------------
-150
r
I
I
I
0
0.005
-0.02 -0.015 -0.01 -0.005
displacement (m)
I
I
0.01
0.015
0.02
Figure 35: Force vs. displacement of the model under a 0.66 Hz sinusoidal excitation of amplitude
0.0152m.
200
-.- no voltage
150
100
50
--
---
25% Vmax
..
50% Vmax
75% Vmax
Vmax
060-100-150
-2w(1
5
-0.01
-0.005
0
0.005
0.01
0.015
velocity (mis)
Figure 36: Force vs. displacement of the model under a 0.66 Hz sinusoidal excitation of amplitude
0.0152m.
Figure 35 and 36 illustrate the behavior of the MFD under a sinusoidal excitation of frequency
0.66 Hz and amplitude 0.0152 m.
5. Computational implementation
5.1 Earthquake used
The simulations have been done under the earthquake El Centro that occurred in Imperial Valley
in 1940. The time history of the North-South component of this earthquake is shown in Figure
37.
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
5
10
15
20
TIME (SEC)
Figure 37: Time history of El Centro earthquake [wI].
25
30
0
C
-j
0.1
0.1
10
1
NATURAL FREQU ENCY(Hz)
Figure 38: Shock response spectra of El Centro earthquake [wI].
5.2 Model building properties and state space formulation
The building used for the simulation of the MFD and the comparison with a 200 kN MR damper
has been extracted from a building in Boston. The stiffness and mass matrix of the three degree
of freedom model used are the stiffness and masses of the three first floors of the building [20].
For the purpose of the simulation, they have been scaled to obtain a maximum deflection of
0.005m when submitted to El Centro.
1 162240
M=4
0
0
0
305830
0
1 3.1005
-1.1420
2.2584
-1.1164
K =- -1.1420
2
0
6.2009
C = -2.2841
0
-2.2841
4.5168
-2.2327
01
kg
0
222760]
01
-1.1164
3.8247]
x 108 N/in
0
-2.2327
7.6494-
X
106 N. s/m
In order to simulate the building using MATLAB, the state space formulation is the most
convenient way to describe the system.
This formulation is derived from the equations of motion of the system. For a multi-degree of
freedom system (n degrees of freedom and r actuators), the equation of motion is:
MX+CX+KX
Eff-MEgXg.
In this equation, M is the mass matrix of the system, C is the damping matrix and K is the
stiffness matrix. These three matrices are n x n matrices. f is the control force and Xg is the
acceleration of the ground .Ef is the matrix that informs of the position of the actuators in the
system. Ef is a n x r matrix. Eg is the matrix that informs about the impact of the earthquake on
the structure. The earthquake creates a d'Alembert force at each floor. Thus, Eg is a n x1 matrix
with only ones.
This equation can be reduced to the following function by multiplying it by M1 and by passing
all the terms except the X to the left side of the equation.
X = -M-1C9 - M-1KX - M-1Eff - Egig.
Using the following system:
X =X
t9 -M-1C
- M-1KX - M-'Eff - Eg)g
Noting Y the state of the system such that:
Y =[ |.1
The previous system can be written:
sOxa
Y
matricesare
-M-1
In x n
K - M-1 C J-M-1
Ej
The state space matrices are :
A=
I nx n
Onyn
I-M-1 K - M-1 C
- E9g
-
9
Bf
=
B'q
[ M-1 Ef]
Bf Onxr
- Eg
Using this formulation, the structure has been implemented in MATLAB. The LuGre model
previously discussed enabled to simulate the device under El Centro earthquake.
5.3 Control rules
A Linear Quadratic Regulator (LQR) will be implemented to provide the control force that has to
be applied to the structure. The force provided by the LQR will be used to find the electric
current that has to be applied to the MFD to generate this force.
In order to control the voltage required in the MFD, several control approaches are possible. In
this research, we focused on two control rules. A clipped control rule has been used to compare
the performance of the MFD with the MR damper because this control rule is easy to implement
for both devices. Another control rule has been implemented in [1] and couples a clipped control
in the hysteresis i.e. for small velocities and a sliding controller outside the hysteresis. This
control rule has been designed because of the simplicity of the clipped rule for dynamics that are
complex to invert [1].
In order to compare the MFD with the MR damper, it is important to apply the same control rule.
Applying two different control rules during the comparison can alter the results as a control rule
can perform better than another one. The hysteretic behavior of the MR damper is difficult to
know which makes it difficult to know when the device works in the hysteresis. The clipped
scheme is simple to apply and enables to control the device properly without taking this issue
into account.
This will allow investigating various strategy of implementation of the MFD: passive and active
control.
5.3.1 Linear Quadratic Regulator
In order to define the required force, the Linear Quadratic Regulator can be used.
The Linear Quadratic Regulator (LQR) is a control scheme that aims to optimize the control of a
dynamic system and, at the same time, minimize the amount of energy to do this control [14].
The classical state space formulation is of the form:
X = AX + Bu.
In this formula, A and B represent the state space matrix, X represents the state vector and u the
input forces such as control forces, wind loads and earthquakes.
A quadratic cost function is defined [14]:
I = 1f[XtQ X + uR u]dt
20
where
Q is the matrix that commands the weighting for the control of the
state and R is the
matrix that commands the weighting for the price. These two matrices have to be chosen by the
designer. The operator "t" means transposed.
The solution of this problem, which is the control force that minimizes the state and the energy
needed for this reduction is:
F = -KX.
In the previous formula, K is the gain matrix.
The gain matrix is obtained by:
K = R-1BtP.
In the previous formula, P is the solution of the Riccati equation:
-PA - AtP + PBR-BtP - Q = 0
MATLAB has been used for the simulations of this research. This software has a function that
calculates directly the gain. The inputs for this function are the state matrices.
5.3.2 Clipped rule
The clipped control rule is based on providing no voltage or full voltage depending on the law
the designers specify. The rule used for this research is the following [1]:
V= Vmax, when IFreqI >
IFdamperl
and sign ( velocity) = sign ( Freq)
0, Otherwise
In this equation Freq is the required force provided by the LQR and Fdamper is the force
effectively generated by the damper.
5.3.3 Sliding controller
The sliding control is a control theory that alters the dynamics of a system by applying high
frequency changing forces [w3]. Thus, by introducing a disturbance in the system, a control
force for instance, we can tend to a target system.
This controller is used to control the MFD out of the hysteresis and aims to control the applied
voltage in order to provide the required force [1]. Thus, the goal is to generate the same force as
the force required. Then, the surface is defined as s= Fact-Freq where Freq is the required force
provided by the LQR and Fact is the force generated by the actuator. The goal is to drive this
difference to be null.
The Lyapunov function for scalars can be written:
1
2
V=-s2
2
To minimize this function we calculate the derivative of this function:
V=sx5
V = S[Fact - Freq]-
This study was done in [1] . As a result, to ensure the negative definiteness of the Lyapunov
function, the control voltage can be chosen such that:
(-o-22 - kmfdk
Vrea = Vact + -sign(k)
-
fCmfdX-- + Freq - ES)
where c represents the system uncertainty and ri represents the delay of the voltage (Cf. chapter
3).
Because,
- E=
e is chosen positive to ensure the negativity of V.
6. Simulations
Different simulations have been conducted in order to evaluate the performance of the MFD.
First, the MFD has been studied at full capacity to evaluate the performance of this device used
in a passive mode.
Then, a control using a Linear Quadratic Regulator (LQR) has been implemented. The
performance of a controlled device depends on the control strategy. In order to compare the
performance of the devices and in order to eliminate the weight of the control in the comparison,
the same control strategy has to be implemented. Therefore, all the controls are realized using the
LQR.
The required force provided by the LQR is converted to an electric current to control the device.
As it has been said in the previous section, two different voltage controllers have been
implemented. In order to compare the MFD with an MR damper that has the same capacity, the
clipped rule has been implemented to control the voltage applied to both devices. The hysteresis
of the MR damper is complex and difficult to invert. An easy way to avoid this problem is to use
the clipped rule as it easy to implement and adapted to singular systems. Then, a clipped rule
coupled with a sliding controller was implemented to control the voltage applied to the MFD.
The passive MFD is compared to an MFD controlled with this controller.
Different implementation schemes have been studied. First, a single MFD was installed at the
first storey. Then, two MFDs were installed at the first and second storey. Finally, two MFDs
were installed at the first and last storey.
During extraordinary events such as earthquakes, it is fundamental to prevent the structure from
damage. Thus, a good control objective is the inter-storey displacement. In this study, the interstorey displacement of the first floor means between the ground and the first floor. The interstorey displacement of the second floor means between the first floor and the second floor. The
inter-storey displacement of the third floor means between the second floor and the third floor.
.................................................
......
........
6.1 Passive MFD
6.1.1 Control with one MFD at the first floor
The first step was to study the MFD at full voltage. This option is simple to implement in a real
building. The MFD will be at full capacity during extraordinary event. For this, sensors able to
feel abnormal ground motion will require the MFD to switch on and generate the 200 kN force
that it has been designed to generate.
x 103
-6 '
0
5
10
15
20
25
30
35
40
time (s)
Figure 39: Displacement of the first floor uncontrolled and controlled with an MFD at full voltage.
Figure 38 shows that the MFD reduces significantly the displacement. The peak displacement of
this first floor is reduced by almost 37% in comparison with the uncontrolled case. The
displacement of this floor corresponds to the inter-storey displacement as it is the first floor.
................
::..:::::::::::::::
...................................................
....
".............
......
....
....
"........
.....
x10 3
----- -uncontrolled
MFD at full-voltage
0.
0
5
10
15
20
25
time (s)
30
35
40
Figure 40: Displacement of the third floor uncontrolled and controlled with an MFD at full voltage.
The displacement of the third floor has slightly been reduced by the action of the passive 200 kN
MFD installed at the first floor. However, this reduction is very small and does not provide
significant information about the real performance of the MFD. For these tests, the device has
been placed at the first storey. In a real building control, a scheme with several MFDs should be
implemented. For instance, an efficient control can be implemented using a MFD every two
storey. However, for earthquakes, the important is the control of the inter-storey displacement
and the MFD reduces the inter-storey displacement between the second and third floor by
18.67%.
6 x 10-3
4
E
e2-
E
0
'A
.2
0
-8-
0
5
10
15
25
20
time (s)
30
35
40
Figure 41: Inter-Storey displacement of the third level uncontrolled and controlled with an MFD at full
voltage.
5
10
15
25
20
time (s)
30
35
40
Figure 42: Acceleration of the first floor uncontrolled and controlled with an MFD at full voltage.
...............
.............
.....uncontrolled
MFD at full-voltage
43.
E 2....
i'
-3
0
5
10
15
20
25
time (s)
30
35
40
Figure 43: Acceleration of the first floor uncontrolled and controlled with an MFD at full voltage.
As Figure 40 shows, the acceleration of the first floor is slightly reduced but not significantly.
The peak acceleration is the same as without any control. The fact that the biggest part of the
force generated by the MFD is generated by the friction element creates sudden changes in the
force sign which does not help in reducing the acceleration. As in this case, the MFD is used to
control a building under an earthquake, the control is focused on the control of the inter-storey
displacement that can result in damages for the building. However, to control wind loads, an
adapted MFD can be designed by increasing the characteristics of the stiffness element and the
viscous element. This way, the force-displacement and the force-acceleration profiles will be
more rounded and a smoother transition will be possible when the force changes sign.
......
..........................................
110~'
FAVID
FAR
2
a
25
3
35
4
45
5
55
2:5
3
wiho
Ps]I1
35
4
4
5
55
I
ismej.sO
Figure 44: Inter-storey displacement (a) and acceleration (b) of the first floor controlled with MR and MFD
at full voltage.
Figure 42 provides a comparison between a 200 kN MR damper and the MFD. This figure shows
both the displacement and the acceleration of the first floor. As it can be seen in the figure, the
reduction's magnitude is similar and both devices provide the same performance. The tests
showed that the MFD performs better in reducing the peak displacement but by less than 1%.
Thus, it can be said that both devices perform the same way.
Table 3: Inter-storey Displacement reduction by the MFD in comparison with a MR damper.
Measure of the performance
MR
MFD
negligible
Table 4: Inter-storey displacement reduction by one MFD at the first floor.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
26.98%
18,67%
To conclude about the MFD at full voltage, it reduces the uncontrolled displacement by
approximately 26.98 % and behaves as well as the MR used as full voltage. Using the MFD as a
passive device is a feasible option as it provides a good control of the structure.
Although the third floor displacement is not reduced significantly, the inter-storey displacement
is reduced by 18.67%.
6.1.2 Control with two MFDs at the first and second floor
In order to improve the control of the displacement, a different scheme can be implemented. Two
MFDs have been implemented respectively at the first and second floor.
210
-cdfi1rlledIedthw Iu MFDs
to 11
10
_0
S
hkmis
Iu
10
ime *}
Figure 45: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and second floor.
X10,
210
10
110
thwe P.1
lihmeis
Figure 46: Inter-storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
second floor.
Table 5: Percentage of reduction of the peak inter-storey displacement with 2 MFDs at the first two floors.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
44.62%
55.03%
The results obtained show that this scheme is far more efficient than the control with a single
MFD at the first floor. Both the displacement of the first floor and the third floor are greatly
reduced. The first floor's peak displacement is reduced by 44.62% in comparison with the
uncontrolled case, while the displacement of the third floor is reduced by 55.03%.
These results show that the device designed in [1] can be very efficient for earthquake vibration
mitigation and efficient implementation schemes can be put in place.
6.1.3 Control with two MFDs at the first and third floor
I
- e
zle
10
hIwo INu
10
Figure 47: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and third floor.
.Xle
10
IM"hn
M
lime I11i
Figure 48: Inter-storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
third floor.
.
....
_ .Mr.
......
...............
Table 6: Percentage of reduction of the peak inter-storey displacement with 2 MFDs at the first and third floors.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
29.61%
16.41%
This scheme is more efficient than the previous one for the control of the displacement of the third
floor. On the other hand, the reduction of the displacement of the first floor is smaller.
6.2 MFD controlled with clipped rule:
6.2.1 Control with one MFD at the first floor
In this section, the MFD is studied as a semi-active device. The control rule used in this case has
been exposed in section 5.3. The clipped rule is used to decide whether or not to switch on the
MFD with regards to a control force that is calculated using the Linear Quadratic Regulator.
x 10~
6r
time (s)
Figure 49: Inter-storey displacement of the first floor uncontrolled and controlled with an MFD (clipped
rule and LQR).
The MFD performs a reduction of approximately 13.76% with regards to the uncontrolled case.
It appears that the MFD performs better at full voltage as a passive device than with control such
as the clipped rule.
3
6X 10-
5
-
uncontrolled
MFD clipped rule
0-
0
5
10
15
20
25
30
35
40
time (s)
Figure 50: Displacement of the third floor uncontrolled and controlled with an MFD (clipped optimal
control and LQR).
The displacement at the top floor is slightly reduced but not significantly because the building is
controlled with a MFD acting at the first floor. Even if the displacement is not reduced, it
appears that the inter-storey displacement between the second and the third floor has been
reduced by 17.18% (Figure 49).
E
0
-2
-
0
10
6
16
20
time (s)
26
36
30
40
Figure 51: Inter-storey displacement of the third floor uncontrolled and controlled with an MFD
(clipped optimal control and LQR).
6
-
4
--uncontrolled
-MFD
clipped rule
3
-1
-
I
6
I
10
I
15
I
20
I
25
I
30
I
35
40
time (s)
Figure 52: Acceleration of the first floor uncontrolled and controlled with an MFD (clipped
optimal control and LQR).
..
....
.. ....
- - ..uncontrolled
MFD clipped rule
4-
-3
-
-
C
0
-1
0
5
10
15
20
25
30
35
40
time (S)
Figure 53: Acceleration of the third floor uncontrolled and controlled with an MFD (clipped
optimal control and LQR).
As it was the case for passive control, the acceleration is not well controlled by the MFD. As it
has already been said, an adaptation of the MFD can be done to obtain a smoother behavior and
ensure a good control of the acceleration.
z4
two C51
Imm *
Figure 54: Inter-storey displacement (a) and acceleration (b) of the first floor controlled with MR and MFD (clipped
optimal control and LQR).
In order to compare the MFD to a 200 kN MR damper, the same control scheme has been
applied to the MR damper. Thus, a clipped rule control where the required force is provided by a
linear quadratic regulator has been implemented. By using the same control scheme, it is possible
to compare the performance of both devices as the performance of the control strategy is the
same and does not affect the comparison.
As it is shown in Table 7, the MFD performs better for reducing the inter-storey displacement
than the MR under the same control strategy.
Table 7: Inter-storey displacement reduction by the MFD in comparison with an MR damper.
Measure of the performance
MR
MFD
5.33%
Table 8: Inter-storey displacement reduction by one MFD at the first floor.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
13.76%
17.18%
To summarize the results, the MFD performs a reduction of 5.33% in comparison with the MR
damper.
For the inter-storey displacement, the reduction is not as significant as for the passive case. The
displacement of the first floor is much bigger with this scheme than with the passive damper.
The inter-storey displacement between second and third floor is approximately the same under
both controls.
...............
.................
6.2.2 Control with two MFDs at the first and second floor
6
lime ~
*"wm
fIsj.
Figure 55: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and second floor.
uncmdirulled
FIFD clipped is
uncaIlrthIledl
IFD clipped mWi
1
2-
.11-
I4
lhme 10.1
Figure 56: Inter-Storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
second floor.
Table 9: Percentage of reduction of the peak inter-storey displacement.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
48.95%
58.15%
.....................
.
The use of two MFDs, one at the first and one at the second floor, enables a better control of the
displacement of the first and the third floor. With only one MFD at the first floor, the
displacement of the top floor was poorly reduced. With two devices at the first two floors, the
peak inter-storey displacement is reduced by 58.15% at the top floor. The clipped rule used for
two devices also enables a better control than the passive MFD. Both the inter-storey
displacement of the first and the third storey are better reduced than with the passive MFD.
6.2.3 Control with two MFDs at the first and third floor
.X10e
go
thn e N
thn P)
Figure 57: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and third floor.
r IS
II-
I0uF
Figure 58: Inter-storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
third floor.
Table 10: Percentage of reduction of the peak inter-storey displacement.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
31.95%
17.09%
This scheme performs slightly better for the control of the third floor than the one with passive
dampers. However, the control of the first floor is not as efficient as in section 6.2.2. While two
MFDs at the first two floors perform a reduction of 48.95% of the displacement of the first floor,
the scheme studied in this section only performs a reduction of 31.95%. In comparison with the
previous scheme, the peak inter-storey displacement is also poorly reduced.
6.3 MFD controlled with clipped optimal rule and a sliding mode
controller:
6.2.1 Control with one MFD at the first floor
In this section, the passive MFD is compared to the MFD controlled by a clipped rule for small
velocities (hysteretic behavior) and by a sliding mode controller outside of the hysteretic
behavior.
The displacement reduction by this controlled MFD is approximately the same as the MFD
controlled with the clipped optimal rule alone. The uncontrolled displacement is reduced by
13.33% thanks to the MFD studied in this section.
.......................................
x 10-
.61i
I
0
5
I
10
1
15
1
1
20
25
time (s)
I
1
30
35
40
Figure 59: Inter-storey displacement of the first floor uncontrolled and controlled with an MFD (clipped optimal rule
and a sliding mode controller).
-
uncontrolled
MFD controlled
0
*" -
.
U
0
-.L
Id
-2
.30
Ii
~~*
I
I
I
5
10
15
I
I
I
20
25
time (s)
I
30
35
Figure 60: Acceleration of the First floor uncontrolled and controlled with an MFD (clipped optimal rule and a
sliding mode controller).
::::.::.::: .......
M
:
..
.. ...............................
.. ..........
....
As it was the case in 6.1 and 6.2, the acceleration is not well controlled as the forces directions
changes suddenly. To achieve a proper control of the acceleration, a smoother profile can be
achieved by increasing the participation of the stiffness and viscous element in the force
generated by the device.
x10
5
uncontrolled
MFD controlled
0
C6
0
5
10
15
20
25
time (s)
30
35
40
Figure 61: Displacement of the third floor uncontrolled and controlled with an MFD (clipped
optimal rule and a sliding mode controller).
The displacement of the third floor is slightly reduced. The small magnitude of this reduction is
due to the fact that there is only one MFD acting at the first floor. A better control can be
achieved by putting in place more MFDs. However, the inter-storey displacement is reduced by
17.83% (Figure 62).
..........
...........
........
.............
....
...
...
.............
..
................
..
x 10-3
E
*d 2
E
-2
0
.6
-8
time (s)
Figure 62: Inter-storey displacement of the third level uncontrolled and controlled with an MFD
(clipped optimal rule and a sliding mode controller).
10.
. x 10,3
D C-0- id
WD fuN valtage
-
FD-controlled
WFDfuUvoltage
4
0
2
4
6
U
1
wr
12
14
is
I
20
25
(s)
3.5
4
Ume(s)
4.5
5
5.5
a
Figure 63: Displacement of the third floor controlled with an MFD (clipped optimal rule and a sliding mode
controller) and a MFD at full voltage.
The comparison of the controlled MFD with the passive MFD at full voltage showed clearly that
the passive MFD performs better. Even if the controlled MFD performs a reduction of 13.33% in
comparison with the uncontrolled case, the passive MFD performs better as it performs a
reduction of 13.65% in comparison with the controlled MFD.
Table 11: Comparison between the MFD controlled and the passive MFD for the control of the
first floor.
Measure of the performance
MFD controlled
Uncontrolled
MFD full voltage
13.33%
-13.65%
6.3.2 Control with two MFDs at the first and second floor
W10
thm J*
-um"m
II
Figure 64: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and second floor.
tweu j~
Figure 65: Inter-storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
second floor.
Table 12: Percentage of reduction of the peak inter-storey displacement.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
39.08%
39.90%
6.3.3 Control with two MFDs at the first and third floor
5z10
unctadfreedl
camdrolledvelh
110
0
-owsPN
6
110
15
Ib FDs
a
fim $1
Figure 66: Displacement of the first (a) and third (b) floor controlled with two MFDs at the first and third floor.
w
.
unasdrlled
--
MFID contnd ind
5V.
10
6
110
ithII PA
115
2
110c
Figure 67: Inter-storey displacement of the first (a) and third (b) floor controlled with two MFDs at the first and
third floor.
Table 13: Percentage of reduction of the peak inter-storey displacement.
Measure of the performance
Uncontrolled 1st floor
Uncontrolled 3rd floor
MFD
26.87%
19.01%
6.4 Discussion:
Table 14: Summary of the results: reduction in the inter-storey displacement.
Measure of the performance
Uncontrolled 1st
Uncontrolled 3rd
floor
floor
MFD passive v=12V
26.98%
18,67%
With one MFD at
MFD controlled with clipped rule
13.76%
17.18%
the first floor
MFD controlled with clipped rule
13.33%
17.83%
MFD passive v=12V
44.62%
55.03%
MFD controlled with clipped rule
MFD controlled with clipped rule
48.95%
58.15%
39.08%
39.90%
MFD passive v=12V
29.61%
16.41%
MFD controlled with clipped rule
MFD controlled with clipped rule
31.95%
17.09%
26.87%
19.01%
and a sliding controller
With two MFDs at
the first and second
storey
and a sliding controller
With two MFDs at
the first and third
storey
and a sliding controller
The best control has been obtained with two MFDs at the first and second storey controlled by
the clipped rule.
Several schemes and controls have been studied in this research. The MFD constantly switched
"on"9, i.e. used as passive device, showed really interesting results in controlling the inter-storey
displacements during an earthquake.
The controlled cases showed good performance but can certainly be improved by enhancing the
control strategy.
The best control is done by using two devices at the first and second floor. This way, the interstorey displacement is almost divided by two for the clipped rule control.
It appears that the clipped rule combined with the sliding controller is not efficient in comparison
with the passive MFD and the clipped rule controller. A more sophisticated controller has to be
designed to perform better than the passive MFD and the MFD controller with the clipped rule.
When one MFD is used to control the building, the passive MFD is the most efficient scheme.
When the building is controlled with two MFDs, the best control is done by the MFD controlled
with the clipped rule.
The MFD used in this study did not showed tremendous abilities to control the acceleration. In
order to realize a good control of the acceleration with the MFD, a smoother Force vs. Velocity
profile has to be designed. [1] shows that with such a profile, the MFD can control the
acceleration efficiently. A smoother profile can be obtained by a reducing the ratio of the force
created by the variable friction element to the force created by the stiffness and viscous elements
or by controlling the voltage to perform a smooth transition when the force changes sign.
A study to evaluate the rise of temperature in the device was done. In the case of these
simulations, the displacement of the building is really small, of the order of 10~3m. The work
done by the device is not significant and the rise of temperature cannot be assessed properly.
Simulations on a benchmark building can be interesting to do this study.
[20] shows that using this device can be economically efficient for a building in Boston, MA.
This study aimed to evaluate how efficient the replacement of the viscous damper of that tower
by MFDs can be. By using MFDs following an optimal scheme, the price of the installation
would have been reduced by 44% in comparison with viscous dampers installation.
7. Conclusion
The device designed in [1] shows good performance in dissipating earthquake vibration. The
peak inter-storey displacement can be reduced tremendously if a proper control strategy is
implemented. In this research, several strategies have been studied. Some of them are more
adapted than others but all showed acceptable results as the worst reduction in the peak interstorey displacement was 13.33%.
This device is very interesting because it is composed of simple element. The friction element,
which is similar to a car brake, can be built with reliable commercial elements. The MFD is
cheap in comparison with other devices. Furthermore, braking systems have been studied which
make them reliable devices.
The MFD, even if it is at the conceptual design stage, showed promising performances in
controlling a building under an earthquake. Furthermore, it is adapted to real life projects, needs
and constraints.
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Websites:
[wl] http://www.vibrationdata.com/elcentro.htm
[w2] http://en.wikipedia.org/wiki/Linear-quadratic-regulator
[w3]
http://scholar.lib.vt.edu/theses/available/etd5440202339731121/unrestricted/CHAP4_DOC.pdf