model reduction of large structural systems for active

MODEL REDUCTION OF LARGE STRUCTURAL SYSTEMS FOR ACTIVE VIBRATION CONTROL

By

John Boffa

A thesis submitted in fulfilment of the requirements for the degree of

Master of Engineering

Faculty of Engineering University of Technology, Sydney

Australia

February 2006

CERTIFICATE OF AUTHORSHIP/ORIGINALITY

I certify that the work in this thesis has not previously been submitted for a degree nor has it been submitted as part of requirements for a degree except as fully acknowledged within the text.

I also certify that the thesis has been written by me. Any help that I have received in my research work and the preparation of the thesis itself has been acknowledged. In addition, I certify that all information sources and literature used are indicated in the thesis.

John Boffa (16/02/06)

PUBLICATIONS

Conference Papers:

Paper Title (1): Study on Model Reduction of Large Structural Systems for Active Vibration Control.

Authors: J.Boffa, N.Zhang and B.Samali.

Conference: The 4th Australasian Congress on Applied Mechanics (ACAM 2005).Institute of Materials Engineering Australasia Limited.

Paper Title (2): Experimental Evaluation of Reduced Models of Large Structural Systems for Active Vibration Control.

Authors: J.Boffa and N.Zhang.

Conference: Australian Earthquake Engineering Society Conference. November, (AEES 2005).

Paper Title (3): Active Seismic Response Control of Tall Buildings Based on Reduced Order Model.

Authors: H.Du, J.Boffa and N.Zhang.

Conference: 2006 American Control Conference, Minnesota USA. (http://www.a2c2.org/conferences/acc2006/).

ACKNOWLEDGMENTS

Financial support for this project was supplied by the Faculty of Engineering at the University of Technology, Sydney. This financial support is greatly acknowledged and appreciated.

I am greatly indebted to my supervisor, Associate Professor Dr. Nong Zhang for his support and guidance throughout the whole duration of the project. Other academics that I would also like to thank are Dr. Dikai Liu (my co-supervisor), for his input in the early stages of the project, and Dr. Haiping Du for his advice and co-authorship in a conference paper, towards the end of the project.

Without the ongoing technical assistance and practical advice of Mr. Chris Chapman (scientific officer at UTS), this project would never have been completed, and probably would never have been attempted in the first place. Chris Chapman also wrote the original main Labview program for real-time active control, called “Active Vib Control 4 input v3”, and it was then modified by John Boffa for use on the twenty storey building model. (Labview is a software developed and sold by National Instruments).

Finally, I would like to thank my whole family, in particular my mother, for her continuous moral support and understanding.

TABLE OF CONTENTS

Page

1. INTRODUCTION 1

1.1 LITERATURE SURVEY 1

1.2 THESIS LAYOUT 2

1.3 SUMMARY 2

2. THE STRUCTURAL MODEL 3

2.1 THE STRUCTURAL MODEL – CONFIGURATION #1 3



2.4 BASE PLATE DESIGN 9

2.5 SUMMARY 10

3. THE MATHEMATICAL MODEL 11

3.1 THE FINITE ELEMENT METHOD (USING BEAM ELEMENT) 11

3.2 THE INFLUENCE COEFFICIENT METHOD 14

3.3 THE GUYAN MODEL REDUCTION METHOD 21

3.4 THE DYNAMIC MODEL REDUCTION METHOD 23

3.5 THE MODE-DISPLACEMENT METHOD (Special case of Ma-Hagiwara) 25

3.6 SUMMARY 27

4. OPEN-LOOP ANALYSIS 29

4.1 NUMERICAL SIMULATION #1: (Frequency Response, with =0.01) 29

4.2 NUMERICAL SIMULATION #2: (Frequency Response, with =0.001) 31

4.3 NUMERICAL SIMULATION #3: (Earthquake excitation DSF = 400Hz) 33



4.6 SUMMARY 35

5. CLOSED-LOOP ANALYSIS 37

5.1 STATE SPACE CONTROL 37



5.4 SUMMARY 48

6. EXPERIMENTAL TESTING 49

6.1 FREQUENCY RESPONSE TESTING OF THE LINEAR MOTOR 49

6.2 EXPERIMENTAL MODAL TESTING OF THE STRUCTURAL MODEL 50

6.2.1 EXPERIMENTAL TEST #1 50





6.2.6 MODAL SHAPE TESTING 58

6.2.6.1 EXPERIMENTAL TEST #6 59

6.3 DYNAMIC MODEL REDUCTION METHOD FORCED VIBRATION TEST 61



6.4 COMPARISON OF NATURAL FREQUENCIES FOR THE BUILDING

MODEL (CONFIGURATION #3) 68

6.5 CLOSED-LOOP ANALYSIS USING EXPERIMENTAL PLANT MODEL 69

6.5.1 NUMERICAL SIMULATION #8 (Earthquake excitation DSF = 400Hz) 69

6.6 CLOSED-LOOP ANALYSIS USING EXPERIMENTAL OBSERVERS 70

6.6.1 NUMERICAL SIMULATION #9 (Earthquake excitation DSF = 400 Hz) 70

6.7 REAL-TIME ACTIVE CONTROL TESTING OF THE BUILDING MODEL 71

6.8 SUMMARY 72

7. CONCLUSIONS 73

7.1 OPEN-LOOP NUMERICAL SIMULATIONS 73

7.2 CLOSED-LOOP NUMERICAL SIMULATIONS 74

7.3 EXPERIMENTAL EVALUATION 74

7.4 RECOMMENDATIONS 75

8. REFERENCES 76

9. APPENDIX 78

APPENDIX (A): BILL OF MATERIALS FOR 20 STOREY BUILDING MODEL79

APPENDIX (B): DETAIL DRAWINGS OF 20 STOREY BUILDING MODEL 80

APPENDIX (C): TEST#6, MODAL SHAPE TEST RESULTS 102

APPENDIX (D): TEST #8, 5Hz DMRM FORCED VIBRATION RESULTS 105

APPENDIX (E): OTHER FORCED VIBRATION TEST RESULTS 106

APPENDIX (F): LABVIEW PROGRAMMING FOR REAL-TIME CONTROL 112

ABSTRACT

This thesis studies the applicability of the Dynamic model reduction method that is used

for direct plant order reduction in the active vibration control of large and flexible

structures. A comparison of the performances between the reduced models produced by

the Dynamic model reduction method and those obtained by other common model

reduction methods such as the Guyan method, and the Mode-displacement method have

been carried out. By using a full analytical model of a twenty storey building as the

reference, each three degrees of freedom model was compared by computer simulation.

The open-loop frequency response simulation, open-loop earthquake simulation, and the

closed-loop earthquake simulation were all used to initially evaluate the reduced

models. The accuracy of the frequency responses was assessed with sinusoidal applied

forces, and for the closed-loop dynamic analysis, an active mass damper at the top

storey and a recorded earthquake excitation was used. When compared with the

simulation results of the Guyan method, the Dynamic method has many advantages,

especially in terms of its accuracy at the high frequency range. The Mode-displacement

method produces reduced models that are good for dynamic analysis of open-loop

systems, but it was found to be inconvenient for use in active control. Finally, the

Dynamic model reduction method and Guyan method were compared using

experimental test results. A 2.5m tall building model with 20 floors was used as the

plant, with a linear motor installed at the top storey for the purposes of active-damping.

Although the results of simulations would suggest that both models perform

sufficiently, experimental testing proved that only the Dynamic model performs

adequately for this specific application of active control. The problem associated with

most model reduction methods, such as the Guyan, is that they are based on full-order

models that were derived from the linear elastic theory. The versatility of the Dynamic

model reduction method is such that it provides the option of obtaining system

parameters directly from experiment, not just from theory. The experimental procedure

ensures that the Dynamic model reduction method forms an accurate description of the

real system dynamics. The applicability of this method for obtaining low-order plant

models was demonstrated through real-time active control testing of the model

structure, while it was subject to a sinusoidal excitation. The tests have shown that the

Dynamic model reduction method can be used as an alternative approach for the model

reduction of structural systems for the purpose of active vibration control.

CHAPTER 1: INTRODUCTION__________________________________________________________________________________________________________

1.1 LITERATURE SURVEY

For the active vibration control of complicated mechanical or structural systems a

reduced dynamic model with a very limited number of degrees of freedom and yet

sufficient accuracy is often required. One of the typical applications is the active

vibration control of high rise and flexible building structures subject to earthquake

excitations and wind loads that have mainly low frequency components. In this case,

the dynamic responses of the structural systems concerned contain mainly the

contributions made by a few of their lowest modes of vibration. Consequently, the

vibrations can be effectively controlled based on a reduced-order plant model that

contains only a few of the lowest modes of these structures [1].

The use of a reduced plant model within the controller can minimise the computation

time for determining the feed-back gains required by the actuators and therefore

improve the overall performance of the combined plant-controller system [2,3]. Seto et

al, [1,2] pointed out the importance of having reduced plant models in terms of

meaningful physical parameters such as mass, damping, stiffness parameters and

presented a few successful applications of the active vibration of flexible structures

based on reduced physical low-order plant models. Ma and Hagiwara [4] developed the

Mode-displacement method for obtaining the reduced model of a large structural

system. The resultant models often perform well in structural analysis.

Zhang [5] presented a dynamic model reduction method that produces reduced models

of systems that have a large number degrees of freedom, for dynamic analysis. The

reduced models are formulated from condensed mass, damping and stiffness coefficient

matrices and retain a small number of the lowest modes of the original system. Care

needs to be taken in choosing the reference frequency for taking into account the

dynamic effect of the high modes, and in choosing the master coordinates that are

retained in the reduced models.

Further sources of literature on this subject can be found in: [11] to [22].

1

1.2 THESIS LAYOUT

The three different model reduction techniques that are investigated in this research are

the: (1) Guyan; (2) Dynamic Model Reduction Method (DMRM); and (3) Mode-

displacement (which is a special case of the Ma-Hagiwara method).

Chapter 3 of this thesis contains a description of the three different model reduction

techniques that were used in this research, and chapters 4 and 5 present both simulation

results of the dynamic responses of the open-loop and closed-loop systems under

earthquake inputs. The Pole-placement control technique is discussed next, and then

the vibration testing of the real structural model, including the DMRM test is covered.

The two new reduced models that are formed from the DMRM forced vibration test are

called the “5Hz DMRM model” and the “Static DMRM model”.

The theoretical plant model which was used in all previous simulations is then replaced

with the experiment-based 5Hz DMRM plant model, and all simulations are repeated.

A considerable deviation in behaviour is noticed, when the theoretical models are

compared against this experiment-based plant model.

Finally we compare the only two stable reduced models (the 5Hz DMRM and the Static

DMRM), for real-time active-control testing of the physical structure, while it is subject

to sinusoidal excitation. Significant active-damping is achieved by both of these

reduced plant models when they are used together with the Pole Placement control

technique for real-time control.

1.3 SUMMARY

This thesis studies the applicability of the Dynamic model reduction method that is used

for direct plant order reduction in the active vibration control of large and flexible

structures. A comparison of the performances between the reduced models produced by

the Dynamic model reduction method and those obtained by other common model

reduction methods such as the Guyan method, and the Mode-displacement method have

been carried out. A 2.5m tall building model with 20 floors was used as the plant, with

a linear motor installed at the top storey for the purposes of active-damping.

2

CHAPTER 2: THE STRUCTURAL MODEL__________________________________________________________________________________________________________

2.1 THE STRUCTURAL MODEL - CONFIGURATION #1

The appendix of this thesis contains the detail drawings of the building model

components (they were all drawn by the author of this thesis). The twenty storey

building model was designed so that the distance between its two steel columns could

be adjusted from 50mm to 100mm. The inclusion of this adjustment provided a more

convenient building design that could be tailored to suit the capabilities of the linear

motor and/or shaker equipment. By changing the distance between the columns from

50mm to 100mm the overall building stiffness or flexural rigidity could be modified, as

this distance is directly related to the area moment of inertia of the combined building

column.

The first configuration of the building, let us call it configuration #1, was to have a

50mm distance between its columns. The other parameters of configuration #1 are as

follows: total lumped mass of each floor is 29 kg; the length, width, and height of the

total lumped mass per floor is 354mm, 228mm, and 50mm respectively; the two

columns are made from 100mm × 5mm bright flat steel; the unclamped length of the

columns, i.e. the effective height of the building model, is 2.5m; the distance between

each floor is 76mm; the stationary mass of floor number 20 is 17kg; the active mass

(floor number 21) is approximately 22kg.

The active mass is connected to the top floor by a linear motor. The stationary part of

the linear motor forms the twentieth floor of the building, and it provides the control

force between itself and the active mass (twenty-first floor). For the purposes of

simplification, only one direction of translation was considered throughout this

research.

3

Direction of Motion

FIGURE 1: Twenty Storey Building Model (Sketch) - Configuration #1: (50mm).

4

FIGURE 2: Construction Phase of the 20 Storey Building Model - Configuration #1

As can be seen in Figure 2 and in Figure 3, configuration #1 had four small spacers per

floor of the building that were 16mm wide x 3mm thick. The purpose of these spacers

was to provide alignment during construction and rigidity during vibration for the first

19 floor assemblies of the building. The spacers were positioned at either side of the

columns, and were machined to have a reasonably tight fit onto the column, for

horizontal alignment. The spacers were also designed to reduce the clamping contact

area on the columns, at each floor of the building, so as to more closely replicate the

lump mass assumption of the analytical models.

FIGURE 3: Sketch of Spacers used in Configuration #1

5

FIGURE 4: Close-up of a Floor Assembly of the Building Model (Configuration #1).

Each floor assembly of the building consists of 14 rectangular masses tightly clamped

together. The rectangular masses are 50mm high and 25mm thick. The two

rectangular masses that are placed in between the columns for configuration #1 are only

22mm thick. Therefore the distance between the columns of 50mm is maintained once

the 3mm spacers have been accounted for.


A quick test on configuration #1 of the building, found that the first natural frequency

was lower then the working range of the linear motor. Therefore, in order to slightly

increase the stiffness of the building, and thus increase the natural frequencies, a second

configuration was constructed. Configuration #2 was identical to configuration #1

except that the four spacers per floor of the building were removed. This had the effect

of increasing the clamping contact area on the column, and reducing the effective inter-

floor distance. Therefore, configuration #2 had slightly higher building stiffness and

higher natural frequencies. In configuration #2, the 50mm distance between columns

was maintained by replacing the two 22mm rectangular masses with two 25mm masses,

so that the removal of the 3mm spacers could be compensated for. The only other

6

change was that configuration #2 had an increased active mass quantity of 109kg. With

an active mass of 109kg, the performance of the linear motor was greatly improved.


In order to make one final change in the configuration of the building, it was decided to

shift the stiffness and natural frequencies even higher. This decision was made on the

basis that the new shaking equipment that was soon to arrive in the lab, had a higher

frequency working range than the current building configuration. Therefore, the third

and final building configuration (configuration #3) had a distance of 100mm between

its two columns, but was otherwise identical to configuration #2. As a result of this

change, the stiffness and natural frequencies of configuration #3 were considerably

higher than all previous arrangements. Configuration #3 has an active mass of 109kg.

FIGURE 5: Twenty Storey Building Model (Sketch) - Configuration #3: (100mm)

7

FIGURE 6: Twenty Storey Building Model - Configuration # 3: (100mm).

8

2.4 BASE PLATE DESIGN

The pictorial drawings below show that the base plate design was configured to allow

for clamping of the building at 50mm or 100mm distances between columns.

FIGURE 7: Base Plate Design of the Twenty Storey Building Model.

FIGURE 8: Base Plate Design of the Twenty Storey Building Model

9

2.5 SUMMARY

The twenty storey building model was designed so that the distance between its two

steel columns could be adjusted from 50mm to 100mm. The inclusion of this

adjustment provided a more convenient building design that could be tailored to suit the

capabilities of the linear motor and/or shaker equipment. By changing the distance

between the columns from 50mm to 100mm the overall building stiffness or flexural

rigidity can be modified, as this distance is directly related to the area moment of inertia

of the combined building column. The distance between the two columns directly

affects the natural frequencies of the whole building model.

10

CHAPTER 3: THE MATHEMATICAL MODEL__________________________________________________________________________________________________________

The Finite Element Method (FEM) was used to create the full (high) order mathematical

model of the building (plant) with two degrees of freedom per floor. The beam element

was used, with one rotation and one translation at each end. Therefore, for the twenty

storey building, with twenty lump masses, the FEM model had 40 degrees of freedom

(dof /DOF) in total. The influence coefficient method (for continuous structures) was

also employed in this study, to obtain very similar results as the FEM, with the

advantage that it uses only half of the degrees of freedom that the FEM requires. The

influence coefficient method only uses one translation (one direction of motion) at each

floor of the building. By using the influence coefficient method, we can easily obtain a

“semi-reduced” model, which can then be used to create our three reduced models.

3.1 THE FINITE ELEMENT METHOD (USING BEAM ELEMENT)

The finite element method is a numerical procedure that can be used to obtain solutions

to a large class of engineering problems, not just vibration analysis. The finite element

method uses integral formulations to create a system of algebraic equations, so that a

solution for each element can be approximated by a continuous function. The global

solution is then obtained by assembling all individual (element) solutions, and results in

a seamless or continuous approximation of the entire (global) structure.

A simple beam element consists of two nodes, one node at each end. At each node

there are two degrees of freedom (dof), one transverse translation and one rotation

(slope). Unlike the frame element, the simple beam element does not contain any axial

translations. Therefore there are four nodal values in total associated with a simple

beam element, as in Figure 9.

Since there are four parameters in Figure 9, we will use a third order polynomial in

Equation (1), with four unknown coefficients to represent the displacement of the

simple beam.

432

23

1 cxcxcxcv (1)

11

yUi1

Uj1

xi

Ui2

j

Uj2

L

FIGURE 9: The Simple Beam Element

The end conditions of the beam have the following values, when substituting in the

nodal variables.

For node i: when “v” is the displacement in the y direction at x = 0,

14)0( iUcv

For node i: when dv/dx is the slope at x = 0,

23)0(

iUcdx

dv

For node j: when “v” is the displacement in the y direction at x = L;

1432

23

1)( jUcLcLcLcLv

For node j: when dv/dx is the slope at x = L;

2322

1 23)(jUcLcLc

dx

Ldv

We now have four equations with four unknowns. Solving for c1, c2, c3, and c4, and

rearranging yields Equation (2) as follows:

122

221123

222113 21312iijijijiji UxUxUU

LUU

LxUU

LUU

Lv

The following equations describe the bending moment and shear force in the beam

element:

3

3

)( EIxV

Therefore the bea

dx

vd (3)

m element stiffness equations and matrix can be derived from the

above when taking into consideration the positive shear force and positive bending

moment sign conventions in Figure 10.

2

2

)(dx

vdEIxm

12

Vi

FIG ent. URE 10: Internal Force Sign Convention: Positive Shear, Positive Bending Mom

212133 jjiii Ldx

3

612612)0(LUULUU

EIvdEIV (4)

22

122

132

2

2646)0(jjiii ULLUULLU

L

EI

dx

vdEIm (5)

212133

3

612612)(jjiij LUULUU

L

EI

dx

LvdEIV (6)

22

122

132

2

4626)(jjiij ULLUULLU

L

EI

dx

LvdEIm (7)

In matrix form the above equations can be expressed as follows:

13 612612 jj ULLLV

2

2

1

22

22

4626

2646612612

j

i

i

j

i

i

U

U

U

LLLL

LLLL

LL

EI

m

m

V

(8)

and therefore the stiffness matrix for a simple beam element can be written as

22

3 612612 LLL

22

4626

2646612612

]

LLLL

LLLL

LL

EIK (9)

The mass matrix for a simple beam elem

from reference [6]. Please refer to [6] for a derivation, and for symbolic definitions.

[

ent is presented in Equation 10, and was taken

22 313422135422156

LLLL

LL

LM (1

22 422313221561354420LLLL

LL0)

Vj

mi mj

i j

L

13

When assembling the global stiffness and mass matrices from the twenty simple beam

elements of the building model, forty-two

total. After taking into consideration that the ground floor of the building in rigidly

fixed in displacement and rotation, the total degrees of freedom becomes forty.

coefficient method for the formation of a semi-reduced mathematical

odel of the twenty storey building, is based on the deflection of a cantilever beam

cted in Figure 11.

FIGURE 11: Deflection of Cantilever Beam Subject to Point Load at End

degrees of freedom are initially created in

Therefore the global FEM model, using the simple beam element produces one

translation and one rotation at each lump mass of the building. Alternatively, the FEM

frame element can be used, and this adds another degree of freedom to each lump mass

in the form of an axial compression. If the FEM frame element is used, a global model

consisting of sixty degrees of freedom is created for our twenty storey building model.

The frame element was used in this analysis with little or no change in result from the

simple beam element and will therefore not be discussed here. The influence

coefficient method was used as an alternative method of modelling to the FEM beam

element, and resulted in an instant halving of the dof, without creating any change in

performance.

3.2 THE INFLUENCE COEFFICIENT METHOD

The influence

m

subject to a point load (one end free one end fixed), as depi

P

Z

P

y

X

yM

M

Z

14

Please refer to reference [7], for the beam deflection and slope Equations (11) and (12).

232 36

36

XXZEI

PXZX

EI

Py (11)

EI

PZyM 3

3

EI

PZM 2

2

(12)

ariable, as in Figure 12.

the point of application of the load,

which is no longer at the end of the beam.

From the above equations let us now derive the more general situation when Z becomes

v

FIGURE 12: Deflection of Cantilever Beam Subject to Point Load

Therefore, the two Equations (13) now apply to

EI

PZyM 3

3

EI

PZM 2

2

(13)

However applies from the point of application of the load, and all along until the

nd of the beam.

cenario (1) occurs when: Z2 X2 ZT . Therefore,

M

e

The following 2 scenarios are associated with Figure 12.

S

Z

P

ZT=Z+Z2

Z2

X X2

y

M

yM

15

236

XXZEI

Py

2223

6XZXZZ

EI

Py TT

222223

6XZZXZZZ

EI

Py

222222

6XZZXZZ

EIy

P

222222

6XZZZXZ

EI

Py

222222

6ZXZZXZ

EI

Py

222222

6ZXZZXZ

EI

Py

22222

222 .22

6ZXZXZZZXZ

EI

Py

322

22222

222222

23 22426

ZXZXZZXZXZZXZZEI

Py Z

32222

23 326

ZXZXZZEI

Py

3

322

322

2

3

33

223

22

62.

Z

ZX

Z

ZXZ

Z

Z

EI

ZPy

and finally,

32222

3

21

231

3.

Z

ZX

Z

ZX

EI

ZPy (14)

16

Scenario (2) occurs when 0 X2 Z2. Therefore,

dx

dyXZyy M .22

MM XZyy .22

EIEI 23PZ

XZPZ

y .2

22

3

ZEI

PZZX

EI

PZy

.2.

33

3

3

22

3

ZZX

EI

PZ

EI

PZy

21.3.

33 22

33

and finally,

Z

ZX

EI

PZy

231.

322

3

(15)

ow if we consider yi as being the deflection of the ith floor of the building, then Fii is

the inertial force at the ith floor. The same can be said about the jth floor in Figure 13.

: Floor(i) and Floor (j) of the Twenty Model

N

FIGURE 13 Storey Building

iyi

yj

FiFloor (i)

iFjFloor ( j)

17

The inertial forces at the ith and jth floors in Figure 13 are as follows:

iii

i ymF and (16)

nd for sinusoidal motion,

jjij ymF

a

tSinAy jj (17)

jj2 (18)

(19)

floor therefore becomes:

tSinAy

j jyy 2

The inertial force at the jth

jjij ymF 2 (20)

jjij ymF .. 2 and i ii

i ymF .. 2 (21)

The superpo

escribed by yi. Where i,j is the deflection at the ith floor that is caused by a unit

.

IGURE 14: Deflection at ith floor due to a unit inertial force at the jth floor

sition of deflections for the ith floor of the building, ie for 1 i 20, can be

d

inertial force at the jth floor, as depicted in Figure 14

i,jFloor (i)

F

Unit Inertial ForceFloor ( j)

18

For the twenty storey building, the superposition of deflections for the ith floor, caused

y inertial forces at every other floor becom s Equation (22) as follows:

herefore the deflection superposition at each floor of the building (i = 1 to 20), is

iven by the following representati equations and is labelled Equation (23).

is more convenient to represent the Influence Coefficient in matrix form [ ].

(24)

i

i

i

i

i

j

j

iijiii

j

j

i

F

F

F

F

F

y

y

y

y

y

20

19

2

1

20,2019,20,202,201,20

20,1919,19,192,191,19

20,19,,2,1,

20,219,2,22,21,2

20,119,1,12,11,1

20

19

2

1

(25)

)22(.................. 20,2019,1918,18,3,32,21,1 ii

ii

ii

jiiji

ii

ii

ii FFFFFFF

20,120iF

20,2019,1918,18,3,32,21,1 .................. ii

ii

ii

jiiji

ii

ii

ii FFFFFFFy

b e

y

T

g on of twenty

19,11918,118,13,132,121,111 .................. iij

ij

iii FFFFFFy

(23)

20,202019,201918,2018,203,2032,2021,20120 .................. iiij

ij

iii FFFFFFFy

It

20,2019,20,202,201,20

20,1919,19,192,191,19

20,19,,2,1,

20,219,2,22,21,2

20,119,1,12,11,1

j

j

iijiii

j

j

Therefore,

i

19

202 y20

192

19`

2

22

2

12

1

20,2019,20,202,201,20

20,1919,19,192,191,19

20,19,,2,1,

20,219,2,22,21,2

20,119,1,12,11,1

20

19

1

m

ym

ym

ym

ym

y

y

y

y

ii

j

j

iijiii

j

j

i (26)

202

20

192

19`

2

22

2

12

1

20

19

2

1

.

ym

ym

ym

ym

ym

y

y

y

y

y

iii (27)

19

2

1

19

2

1

20

19

2

1

..

000000

0000

00000000

.

y

y

y

y

m

m

m

m

y

y

y

y

y

iii (28)

2y

2

202000 ym

2... YMY (29)

(30) 211 ... YMY

21 ... YMIY (31)

0.. 21 YMY (32)

0.. 2111 YMMYM (33)

20

0. 2 (34) .11 YIYM

0.211 YIM

Where = 2 is an eig

atrix: [A] = [M-1. -1] (which is equivalent to the matrix [A] = [M-1K]). Thus the

inverse of the influence co

] = [ ]-1 for describing the analytical model of the building. This results in a

mathematical model with twenty dof,

EM model. The performance of the influence coefficient method is very similar to

that of the full order FEM model, with the obvious advantage of using half of the dof.

Therefore the influence coefficient m

odel, and will be applied to the model reduction techniques that are detailed below.

.3 THE GUYAN MODEL REDUCTION METHOD

x.

his static transformation ignores the dynamic effect of the applied loads and creates an

or this reason, the Guyan

odel reduction method is only accurate in the low frequency range, and this will be

(35)

envalue with a corresponding eigenvector {Y} of the square

m

efficient matrix can be effectively used as a stiffness matrix

[K

which amounts to half of the dof used by the full

F

ethod will now be referred to as the “full” order

m

3

The Guyan method is historically the most fundamental model reduction technique that

has been commonly used since its inception in 1965, by Robert J. Guyan [8]. The

reduced stiffness matrix formed by the Guyan method is very accurate, because all of

the elements of the original full stiffness matrix contribute in its formation, and none of

the structural complexity is lost. However, the reduced mass matrix that is formed by

the Guyan method is not very accurate, when compared to other reduced models. The

reason for this inaccuracy is that the Guyan method uses a static transformation

between the eliminated and retained coordinates for obtaining the reduced mass matri

T

increasing error as the frequency of excitation is increased. F

m

demonstrated later in the simulation results.

From the original full order model, we can describe the equation of motion as follows:

FKXXCXM (36)

Where M represents the full order mass matrix, C represents the full order damping

matrix, and K is the full order stiffness matrix. The displacement vector is represented

21

by (X), and its derivatives are velocity X and acceleration X respectively. The

damping in our steel structural model is negligible and is omitted in Equation (37) for

simplicity. When damping becomes significant however, refer to [8] for its inclusion in

Equation (37) and in the rest of the Guyan reduced modelling procedure.

FKXXM (37)

of the reduced model

s of the three

west vibration modes. Therefore, in our example of the twenty storey building

model, we selec

election applies to all of the reduced models. Let us now re-arrange the equation of

S

C

S

C

F

F

X

X

KK

KK

.

2221

1211

nd therefore, we define the Guyan “static” transformation as Equation (41).

(41)

In our example, the reduced Guyan model is of order nC = 3, and therefore the dynamics

is governed by the following Equation (38):

FXKXM (38)

Where M

CCCCC

C is a 3x3 reduced mass matrix and KC is a 3x3 reduced stiffness matrix. As

with all of the model reduction techniques, care needs to be taken when selecting the

three master coordinates because they must not occur at any of the node

lo

ted the master coordinates at floor numbers 7, 13, and 20, and this

s

motion of the full order system in partitioned form as follows.

(39)

Where XC is the master coordinates, and XS is the slave coordinates, and because the

forces are only applied to the master coordinates, FS = 0. Therefore,

02221

1211

SC

CSC

XKXK

FXKXK (40)

a

CS XKKX 211

22

22

Substituting Equation (41) back into Equation (40) yields the following:

CCC FXKCC

CCC

FXKKKK

FXKKKXK

211

221211

211

221211

(42)

Therefore, the reduced Guyan stiffn

C (43)

lease refer to reference [8] for a full derivation of the reduced mass matrix, which is as

follows:

(For static FC)

ess matrix becomes Equation (43):

211

221211 KKKKK

P

ST

ST

SSC TMTMTTMMM 22211211 (44)

where:

UCTION METHOD (DMRM)

erence [5]. This

ection provides only a brief summary. A continuous structure can be represented with

good accuracy by a dis

is finite. For simplicity, it is assumed that the structure considered is free of damping.

otion of the structure is therefore described by the

211

22 KKTS (45)

3.4 THE DYNAMIC MODEL RED

The details of the dynamic reduction procedure can be found in ref

s

crete model of order n as long as the frequency range of interest

The governing equation of m

following differential Equation (46):

MX KX F , (46)

where M and K represent respectively the mass and stiffness m

represent respectively the acceleration, displacement, and the excitation force vector of

If one selects degrees of freedom of the original system to be retained in

the condensed model, the motion of the structure at the chosen master coordinates can

en be described by Equation (47).

(47)

atrices, and F,X X

n . norder c

th

M X K X F , c c c c c

23

where Mc and Kc are respectively the corresponding reduced mass and stiffness

matrices, and ,X Xc c and Fc represent respectively the acceleration, the displacement

and the excitation force vector of order nc at the chosen master coordinates.

From Equation (47), for a simple harmonic input with unit amplitudes applied to the

retained master coordinates, the dynamic response matrix of the condensed system is

determined as Equation (48), where “I” represents the identity matrix.

IMIM 2

Xc determined from the condensed model is the

same as that determined from e orig nal model.

XK1 . (48) cccc

In order for the condensed model to best approximate the original one, two essential

requirements are introduced: (a) the condensed model retains nc number of natural

frequencies and the corresponding modes at the chosen master coordinates of interest

from the original model and (b) for the same unique harmonic forces applied at the

master coordinates, the response matrix

th i

To meet the first requirement, the system matrix M Kc c1 is determined as:

c c cB M K , (49)

vibration

testing. Using Equation (48) and the obtained matrices and , the mass matrix of

the reduced model can be determined as,

Consequently, the stiffness matrix is determined as,

ed either by solving the eigenvalue

problem of the original model or from modal testing. After the condensed model is

tes du rces can be

computed from Equation (48) and hence, ic re ponse at tho

1 1

where is the eigenvalue matrix and is the corresponding modal matrix. To meet the

second requirement, the response matrix Xc must be determined from the original

structural system which has a large number of dof, or alternatively from

B Xc c

M X I M Kc c c c1 2 1 1( ) . (50)

K M Bc c c . (51)

It should be noted that the nc number of chosen natural frequencies, the corresponding

modes and the response matrix Xc can be obtain

obtained, the responses at the master coordina e to the applied fo

the dynam s s se eliminated

coordinates can also be obtained in terms of the computed responses at the master

coordinates.

24

As damping always exists in the actual structural systems and is difficult to be modelled

accurately, modal damping is therefore used for the reduced models. The level of the

modal damping is determined b

n, the damping matrix of the low-

order model is

,

y experience or by experimental modal testing on the

systems. Assuming that the damping ratios are know

determined as the following:

C diagc i i[ ( )]2 (52)

where 2

1

i i is a diagonal matrix, i and i ci n, , , ...,1 2 , are the ith modal damping

ratio and natural frequency respectively.

Theoretically, there is no particular restriction on choosing the natural frequencies and

the corresponding modes to be kept in the condensed model. However, since most

applied loads have frequencies corresponding to the lower range, it is suggested that a

umber of lowest modes of the original model be retained in the condensed model

Ma and Hagiwara presented the mode superposition method that is to obtain the m

frequency response of a coupled acoustic-structural system by using a few of the

requency range of i

requency responses analysis of complicated

n

unless one has a particular interest in higher modes. Only the lower frequencies are

used to produce low-order models.

3.5 THE MODE-DISPLACEMENT METHOD (Special case of Ma-Hagiwara)

odal

vibration modes that are in the f nterest. The approach was

developed because of the need for the f

acoustic-structure coupled systems that require not only truncating the higher modes

but also the lower ones. A brief presentation of the Ma-Hagiwara method is given here,

and further details can be found in reference [4].

It is assumed that the frequencies of applied load are within the range of [ a b, ]

( a b ), and m and n are the num

s

bers of the lowest and highest m

s, where m satisfies

odes of coupled

ystem m a and n satisfies n b . Then, the accurate solution of

n

h

c

( ) ( ), (53)

the considered system in Equation 53, can be written as:

X t Q X ti ii n0

25

Where the first term in the above equation is the approximated solution and X th ( ) is

the residual error, which represents the contributions of the truncated modes i iQ (i=1,

..., n0-1, nc+1, ..., n).

The approximated solution is represented as the following,

X t K j C M Fj

jQh c c

c i i c

i i i c ci

i n

nc 2 2 2i( ) ( ) { ( )}2 1

2 220

(54)

where

QF

m ji

T

i i i i( )2 22 (55) i i

is the frequency of applied load; c is the reference frequency for compensation;

i ik mi is the ith undamped natural frequency; i is the ith natural modal

ing in the

full order model is determined as

, (56)

where

coefficient and F F ei ij t

0 is the applied load at ith coordinate. The damp

system is considered by adding the modal damping into the system model, and the

damping matrix of the

C Diag i iT[ ( )]2

and i i n, , , ...,1 22 i i is a diagonal matrix, i , are the ith modal damping ratio

ral frequency respectively, and is the modal shape matrix of the full order

system. The error of the responses determi ed by quatio (54) nd co

true ones is

and natu

n E n a mpared to the

X tj

j jQh

i i i c cii

i i i c ci

i ni

c

( ) 2 21

2 212 2

Qjc i c

n

ic i c

n( ) ( )2 21 2 22 20

rameters a

From the Ma-Hagiwara approximation, when

(57)

where all the pa re the same as those in Equation (54). The first term will

disappear when n0=1, which means that no low modes are truncated in the

approximated solution.

c n, 0 1 , th give

emen

1

is s the Mode-

displac t approximation, i.e.,

X t Qd i

n

i

c

( ) ; (58) i

26

and when c n0 10, , it gives the Mode-acceleration approximation, i.e.,

X t K F Qai

ii

n

i

c

( ) ( )1 2

1

. (59)

i

c 1

(60)

The errors are respectively:

X t Qn

( ) ; d ii n

X t Qn

( ) ( )2 . a ii n

i

c 1

(61)

ents of mat

i

As the elem rix i iQ vary with different signs, it is difficult to judge which

rror is the largest one from the summations in Equations (57, 60& 61). If the above

three variations of the Ma-Hagiwara model reduction method are to be used, it becom

necessary to investigate their effectiveness in terms of their accuracy in describing the

tructural systems’ dynamic behaviour. Which model reduction method is the most

lly dependent on the applications and damping levels in the

onsidered systems. There is currently no quantified error analysis of the resulting

reduced models av

building is known, it was proven by numerical simulations that the Mode-displacement

as the best performance, out of all of the Ma Hagiwara methods. Therefore, only the

Mode-displacement met here onwards, as it best represen

Ma-Hagiwara technique for our particular application. As the errors given in equations

7, 60 & 61) and the error of low-order models obtained by the Dynamic model

e

es

s

effective and suitable is usua

c

ailable. However, since the full-order model of our twenty storey

h

hod will be used from ts the

(5

reduction method cannot be quantified, in particular, when damping exists in the

considered system, the true comparison must be obtained through detailed numerical

simulations based on the 20 dof structural system.

3.6 SUMMARY

Several theoretical approaches to the mathematical modelling of the plant were

discussed in this chapter. The Finite Element Method (FEM) was initially presented for

27

the creation of the full-order mathematical model of the building, with two degrees of

freedom per floor. The influence coefficient method (for continuous structures) was

then presented, with the advantage that it uses only half of the degrees of freedom as

the FEM. The influence coefficient method creates a “semi-reduced” mathematical

model that yields very similar performance characteristics to the FEM.

By using the influence coefficient method as the new “full-order” model, the three

different model reduction techniques were derived, based on the procedures outlined in

this chapter. The three theoretical model reduction techniques that were detailed in this

hapter are the: (1) Guyan; (2) Dynamic Model Reduction Method (DMRM); and the

t (which is a special case of the Ma-Hagiwara method when the

ference frequency is set to infinity).

c

(3) Mode-displacemen

re

28

CHAPTER 4: OPEN-LOOP ANALYSIS________________________________________________________________________________________________________

.1 NUMERICAL SIMULATION # 1: (FREQUENCY RESPONSE, WITH =0.01)

his simulation was based on configuration #1 of the building model, which was

riginally modelled by a 20 degree of freedom (20 dof) lumped mass system. Only

teral (bending) vibration in one direction is considered, and the stiffness and mass

atrices of the original model were obtained by using the influence coefficient method.

he modal parameters such as natural frequencies and corresponding modal shapes of

e structure were then determined from the stiffness and mass matrices. The damping

as simulated by using a constant damping ratio of 0.01, which is commonly used for

teel structures. For investigating the accuracy of the low order models obtained by the

uyan, Dynamic model, and Mode-displacement methods, the low order size of 3 dof

inal structure and

ed

irectly from the original full order model. Figure 15 below shows the frequency

__

4

T

o

la

m

T

th

w

s

G

was chosen. The 3 dof models retain the first three modes of the orig

use master coordinates at floor numbers 7, 13 and 20. The frequency responses of these

low-order models are computed and compared with the true ones that were obtain

d

responses at the master (physical) coordinates computed from the full-order model and

the low-order models obtained by the three different model reduction methods of the

structure within the frequency range of zero to 50Hz. The same simple harmonics with

unit amplitudes were applied to all of the three chosen coordinates.

0 5 10 15 20 25 30 35 40 45 5010

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

The frequency responses at 1st master coordinate (damping ratio=0.01)

resp

onse

am

plitu

des

(m)

frequencies of applied loads (Hz)

FIGURE 15: Frequency Response at 1st Master Coordinate (Floor #7, = 0.01)

29

0 5 10 15 20 25 30 35 40 45 5010

-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

The frequency responses at 2nd master coordinate (damping ratio=0.01)


resp

onse

am

plitu

des

(m)

FIGURE 16: Frequency Response at 2nd Master Coordinate (Floor #13, = 0.01)

0 5 10 15 20 25 30 35 40 45 5010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

The frequency responses at 3rd master coordinate (damping ratio=0.01)


resp

onse

am

plitu

des

(m)

rdFIGURE 17: Frequency Response at 3 Master Coordinate (Floor #20, = 0.01)

:THE ABOVE FIGURE COLOUR CODES

BLACK = Full-order model; BLUE = Guyan; Green = Dynamic (DMRM); Red = Mode-displacement.

The peaks in Figures 15 to 17 would suggest that the Mode-displacement method

interprets damping in a different manner to the other methods despite that the same

level of damping was used. This effect is lessened when = 0.001, in Figures 18 to 20.

30

4.2 NUMERICAL SIMULATION # 2: (FREQUENCY RESPONSE, WITH =0.001)

Numerical simulation #2 is identical to simulation #1, except that the damping ratio has

been changed to 0.001.

0 5 10 15 20 25 30 35 40 45 5010

10-3

-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

The frequency responses at 1st master coordinate (damping ratio=0.001)

resp

onse

am

plitu

des

(m)


FIGURE 18: Frequency Response at 1st Master Coordinate (Floor #7, = 0.001)

0 5 10 15 20 25 30 35 40 45 5010

-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

The frequency responses at 2nd master coordinate (damping ratio=0.001)


resp

onse

am

plitu

des

(m)

FIGURE 19: Frequency Response at 2nd Master Coordinate (Floor #13, = 0.001)

31

0 5 10 15 20 25 30 35 40 45 5010

-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

The frequency responses at 3rd master coordinate (damping ratio=0.001)


resp

onse

am

plitu

des

(m)

FIGURE 20: Frequency Response at 3rd Master Coordinate (Floor #20, = 0.001)

he peaks in the above graphs represent the three lowest natural frequencies of the plant

models (2.3, 14.1, and 39.3Hz), and they are compared against the full order 20 degree

of freedom system (the black line). As can be seen in the graphs, the reduced model

based on the DMRM (green line) retains the chosen three natural frequencies and

corresponding modal shapes accurately from the original system. Therefore the

predicted dynamic responses based on the DMRM have very good accuracy at a

particular frequency range that often covers the frequency range of excitation forces

applied to the system. In contrast to this, the reduced model obtained by the Guyan

method (the blue line) does not retain all of the chosen vibration modes precisely and

consequently has poor accuracy in the higher frequency range (third natural frequency).

The results also show that the performance of the low order model obtained by the

Mode-displacement method (the red line) is good at all frequencies, but that it is quite

sensitive to the damping level in mainly the high frequency range. This phenomenon

rdinate, when they

are produced by different damping ratio levels, of = 0.01 and = 0.001. The deviation

in the peaks of the red line from the black line is evident in simulation #1, but it

disappears in simulation #2, ie when the damping ratio is significantly reduced.

T

can be noticed by comparing two responses at the same master coo

32

4.3 NUMERICAL SIMULATION #3:(EARTHQUAKE EXCITATION DSF =400Hz )

For the simulations of the responses of open-loop system, the recorded El Centro

earthquake data was used. The original data had many dominate low frequency

components and was sampled at 50Hz (DSF= 50Hz). The original data sampling

frequency was scaled up by a factor of 8 (to a 400Hz sampling frequency) in order to

shift the dominant frequency components to a higher range. In doing so, the higher

modes of the reduced plant models were also excited under the modified earthquake

input. The 400Hz sampled earthquake input has major dominant frequencies between 9

and 17Hz, with some minor dominant frequencies occurring between 17 to 47Hz. At

this sampling frequency, the total frequency content of the earthquake ranges from zero

to approximately 130Hz. The three degrees of freedom (3dof) reduced models were

used as the plant model in each simulation, and the accelerations of the top storey of the

building were plotted (below). These three reduced models were compared against a

presentative of the

ll-scale model due to its inclusion of twice as many modes of vibration. As can be

6dof Mode-displacement model, as it was considered to be more re

fu

seen in Figure 21, the 6dof response has greater acceleration amplitudes because it

includes the extra modes of vibration. For this reason, all of the 3dof responses appear

to underestimate the true response of the full-structure as estimated by the 6dof model.

0 1 2 3 4 5-3

-2

-1

0

1

2

3Open-Loop 6DOF Mode-Disp Response

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-3

-2

-1

0

1

2

3Open-Loop 3DOF DMRM Response

TIME (Sec)

AC

CE

L. (

m/s

2 )

Open-Loop 3DOF Guyan Response Open-Loop 3DOF Mode-Disp Response

0 1 2 3 4 5-3

-2

-1

0

1

2

3

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-3

-2

-1

0

1

2

3

TIME (Sec)

AC

CE

L. (

m/s

2 )

FIGURE 21: Open-Loop Response with Ground Earthquake Excitation (DSF= 400Hz)

33

In Figure 21, when comparing all of the 3 dof models against the 6 dof model, the

Mode-displacement appears to perform most accurately and the Guyan model performs

with the least accuracy.

From numerical simulation #3 and onwards, the Runge-Kutta method for numerical

integration was used. Its details can be found in most vibration literature.


(Numerical simulation #4 is identical to simulation #3, except that the data sample

frequency of the earthquake has been doubled to DSF = 800Hz). When the data sample

frequency of the earthquake is doubled to 800 Hz in Figure 22, the deviation of the 3dof

models from the 6dof model increases further, with the Mode-displacement still

performing most accurately. The reason for the increase in deviation of the 3dof models

from the 6dof model is that the higher modes that occur in the 6 dof model are not

included in the 3dof models. However, these high modes that are excited by a DSF of

800Hz are not generally excited by an average earthquake, and can be ignored.

0 0.5 1 1.5 2 2.5-2

-1

0

1


1

TIME (Sec)

AC

CE

L. (

m/s

2 )

0

-1

-2


0 0.5 1 1.5TIME (Sec)

AC

CE

L. (

m/s

2 )

2 2.5

2

0 0.5 1 1.5 2 2.5-2

-1

0

1

Open-Loop 3DOF Guyan Response

TIME (Sec)

AC

CE

L. (

m/s

2 )

2

0 0.5 1 1.5 2 2.5-2

-1

0

1

Open-Loop 3DOF Mode-Disp Response

TIME (Sec)

AC

CE

L. (

m/s

2 )


34


Numerical simulation #5 is identical to simulation #3, except that the data sample

equency of the earthquake has been reduced to DSF = 100Hz.

ls from the 6

of model decreases. When the earthquake input was modified by using a data sample

he reason for this is that only the low modes of vibration are excited at this low

fr

As the data sample frequency is decreased the deviation of the 3 dof mode

d

frequency of 100Hz, as in Figure 23, all four models yield almost identical responses.

T

frequency range, and these low modes are all included in the 3dof reduced models.

However it is unrealistic to expect that an earthquake would act at such a low frequency

range. For the most accurate representation of a real earthquake acting on a real twenty

storey building, numerical simulation #3 should be used only.

0 5 10 15 20-4

-2

0

2


(m

/s2 )

TIME (Sec)

AC

CE

L.

-2

-4

0

2


(m

/s2 )

0 5 10TIME (Sec)

AC

CE

L.

15 20

4

0 5 10 15 20-4

-2

0

2

Open-Loop 3DOF Guyan Response

TIME (Sec)

AC

CE

L. (

m/s

2 )

4

0 5 10 15 20-4

-2

0

2

Open-Loop 3DOF Mode-Disp Response

TIME (Sec)

AC

CE

L. (

m/s

2 )


4.6 SUMMARY

The frequency responses were determined by applying harmonic excitation forces to the

The reduced open-loop reduced plant models which have three degrees of freedom.

35

model based on the DMRM retains the chosen three natural frequencies and

corresponding modal shapes accurately from the original system. Therefore the

e of excitation forces

pplied to the system. In contrast to this, the reduced model obtained by the Guyan

erations of the top storey

of the building were plotted. These three reduced models were compared against a 6dof

Mode-displacement model, as it was considered to be more representative of the full-

scale model due to its inclusion of twice as many modes of vibration. The 6dof response

has greater acceleration amplitudes because it includes the extra modes of vibration.

For this reason, all of the 3dof responses appear to underestimate the true response of

the full-structure as estimated by the 6dof model.

Of all the 3dof models, the Mode-displacement appears to perform most accurately at an

earthquake DSF of 400Hz, and the Guyan model performs with the least accuracy.

When the data sample frequency of the earthquake is increased, the deviation of the

3dof models from the 6dof model increases further, and as the data sample frequency is

decreased the deviation decreases as well. When the earthquake input was modified by

using a sample frequency of 100Hz, all four models gave almost identical responses.

The reason for this is that only the low modes of vibration are excited at this low

frequency range, and these modes are all included in the 3dof reduced models.

low frequency

nge. For the most accurate representation of a real earthquake, acting on a real twenty

l simulation #3 should be used only.

predicted dynamic responses based on the DMRM have very good accuracy at a

particular frequency range that often covers the frequency rang

a

method does not retain all of the chosen vibration modes precisely and consequently has

poor accuracy in the higher frequency range (third natural frequency). There are no

significant discrepancies between the DMRM and the Mode-displacement model

reduction methods, as both of these models perform closely to the full-scale model.

For the earthquake simulations of the responses of open-loop system, the recorded El

Centro earthquake data was used. The three degrees of freedom (3dof) reduced models

were used as the plant model in each simulation, and the accel

However it is unrealistic to expect that an earthquake would act at such a

ra

storey building, numerica

It should be noted here that throughout chapter 4, a 6dof plant model was used to

represent the full-order 20dof model. This was done for computational reasons only.

36

CHAPTER 5: CLOSED-LOOP ANALYSIS__________________________________________________________________________________________________________

5.1 STATE SPACE CONTROL

Although it is widely acknowledged that the Linear Quadratic Regulator (LQR) is a

more common control method for structural control, the Pole Placement control

technique was used here because of its simplicity. The Pole Placement control was

configured so that parameters such as the desired closed-loop damping ratios (active-

damping ratios) and the desired closed-loop natural frequency of the active mass could

be adjusted. By increasing these parameters, more control force is produced. It should

is much more

structural control.

ace variables of Figure 24 are made clearer when we define the absolute

be noted however, that in a real system, an unlimited control force is not usually

available. Please refer to [9], for a complete study of Pole Placement control, as the

following is only a brief explanation.

Pole Placement control is only one form of state space control. For state space control

we must first start with the definition of our state variables. Due to the availability of

very accurate linear transducers in the lab, it was decided that for this research, we

would limit our modelling to displacement feedback only. It is acknowledged

however, that the use of accelerometers and thus acceleration feedback

common for

As far as our control modelling is concerned, we only need to deal with the 3 degree of

freedom (reduced) models, as they now represent our “plant” or twenty storey building.

However, by adding an active mass to the structure, at the top storey of the building, we

increase our 3 dof plant models to 4 dof, as represented in Figure 24. Each “un” in

Figure 24 represents a relative displacement (state space variable), as defined below.

u1 = Displacement of lump mass 1, relative to the ground. u2 = Displacement of lump mass 2, relative to the ground. u3 = Displacement of lump mass 3, relative to the ground. u4 = Displacement of lump mass 4, relative to lump mass 3.

The state sp

displacements of each lump mass, including the ground mass, as in Figure 25. Each

37

absolute displacement denoted by “Xn”, is relative to some arbitrary point in space that

he main advantage of using relative displacements as our state variables, as opposed to

not evident when using absolute displacements as our state variables, and the control

has no motion at all.

FIGURE 24: Simplified block diagram of reduced plant model in state space

FIGURE 25: Definition of absolute displacements

T

absolute displacements, is that for earthquake ground excitations we are conveniently

provided with a ground movement input vector. The ground movement input vector is

m1

m2

m3

m4

k1

k2

k3

X1

X2

X3

X4

X0

m1

m2

m3

k

m4

1

k2

k3

u1

u2

u3

u4

38

modelling becomes restricted. Let us now express the state variable nu as follows,

where u is the relative acceleration, and is the absolute acceleration. nXn

)( 0344

344

033

022

011

XuXu

XXu

XXu

XXu

XXu

(62)

Similarly, for accelerations,

XXu

XXu

XXu

XXu

344

033

022

011

)( 0344 XuXu

(63)

The lateral stiffness (in one horizontal direction only) of each column segment can now

be defined as “kn”. Let us assume that X4>X3>X2>X1>X0, therefore the free body

diagrams (FBD) of the each lump mass can be represented by the block diagrams below

(in Figures 26 to 29), using state variables “un” as follows:

FIGURE 26: Free body diagram of lump mass #1 : (FBD.1)

m1

k1(u1)

k2(u2-u1)


m2

k2(u2-u1)

k3(u3-u2)

39

In FBD.3 and FBD.4 below, it is assumed that the direction of the control force (FC) is

consistent with that which is required to produce the assumption X4>X3 that was

mentioned above. However, it is only necessary to ensure that the control force in

FBD.3 and FBD.4 are equal but have opposite directions, in accordance with Newton’s

third law of motion.


m3

k3(u3-u2)

FC

m4


FC

By applying Newton’s second law of motion to FBD.1, we have the following

XMF

011 XumF

0111 XmumF (64)

By defining the direction to the right hand side as positive, we have the following:

221211 ukukukF

0111221211 Xmumukukuk

40

012212111 Xmukukkum (65)

Similarly, for FBD.2, FBD.3, and FBD.4, we have the following equations respectively:

02332321222 Xmukukkukum (66)

(67)

(68)

ented in matrix form as follows:

03332333 XmFukukum C

044434 XmFumum C

Equations (66) to (68) can be pres

04

03

02

01

4

3

2

1

33

3322

221

4

3

2

1

44

3

2

1

00

000000000

00000000000

Xm

Xm

Xm

Xm

F

F

u

u

u

u

kk

kkkk

kkk

u

u

u

u

mm

m

m

m

C

C

(Equation 69)

From the three different model reduction methods that were discussed previously, we

lready have the 3dof mass and 3dof stiffness matrices, which can be represented by the

following:

a

33

3322

221

3

3

2

1

3

0

0;

000000

kk

kkkk

kkk

K

m

m

m

M dofdof (70)

should be noted here that the reduced mass and stiffness matrices in Equation (69),

with

ee body diagrams. In reality however, the 3dof reduced matrices that were created for

the open-loop analysis, were not produced by free body diagrams, but from the three

ifferent model reduction techniques that were detailed previously. Therefore the actual

It

were produced from a 3dof representative lumped mass model that was analysed

fr

d

41

3dof matrices are of a distributed form, as in Equation (71), and are not in a lumped

mass form as described in Equation

2322213

333231

2322 ;sss

sssK

ppp

pp dof (71)

For closed-loop analysis, we need to add an extra degree of freedom to our 3dof reduced

(open-loop) models. By adding the active m

atrices, we effectively add a final row and column to Equation (71). These final rows

and columns are depicted in

000000

;

0000

4

44

4 dofdof K

mm

(72)

We are therefore left with a 4dof control model as described in Equation (73). With a

put vector and a ground movement disturbance vector both occurring on

e right hand side of the equation, as in Equation (69).

04

03

02

4

3

2

333231

232221

4

3

2

44

333231

23 0

000000

0000

Xm

Xm

Xm

F

F

u

u

u

sss

sss

u

u

u

mm

ppp

p

C

C

(73)

The matrix Equation (73) can be represented by the following symbolic Equation (74).

here [M] is the mass matrix, [K] is the stiffness matrix, {F} is the control force input

(69).

131211

21

131211

3

sss

p

ppp

M dof

333231

ass (mass number 4) to our 3dof reduced

m

Equation (72), and were taken from Equation (69).

00

M

control force in

th

0111312111

2221

131211 000 Xmusssu

pp

ppp

W

vector, and {Z} is the ground movement disturbance vector. {U} is the relative

displacement state vector, and therefore U is the relative acceleration vector.

ZFUKUM (74)

42

For Pole Placement Control, we must now convert Equation (74) into state-space form.

In true state space form, the state variables include not just the four displacements, but

lso the four velocities. Let us now redefine the state space variable (relative

displacement, and relative velocity) as the following vector: {X}.

Where: (75)

a

U

UXand

U

UX ,

The degrees of freedom of our model has now doubled from 4dof to 8dof, and our

equation of motion in state space form becomes:

ZDFBXAX (76)

[A] is our state space system matrix with size 8x8. [B] is t

matrix, and [D] is the ground movement disturbance coefficient matrix. The vectors

{F} and {Z} were described previously in Equations (73) and (74). The state space

matrix [A] is expressed in Equation (77), where [I] is the 4x4 identity matrix, [0]

(77)

Therefore the two coefficient matrices in Equation (76) can be described by the 8x4

atrix as follows:

According to Ogata [9], the velocity feedback variables should not simply be derived

rom the differentiation of the displacement feedback variables. The reason for this is

that the differentiation of a

noise generally fluctuates more rapidly then the command signal. But by using state

he control force coefficient

system

is the 4x4 zero matrix, and [C] is the 4x4 damping matrix.

CMKM

I

A11

0

m

1

]0[M

DB (78)

f

signal always decreases the signal-to-noise ratio because

43

observation, which is the estimation of unmeasurable states based on the measurement

of the output and control variables, we can create a full state reconstruction. The

estimated full state consists of all four estimated relative displacements and all four

estimated relative velocities, and will be denoted by X~ .

Where: U

UXandU

UX ~

~~;~~~

e two Equations (81) and (82). We

need to be careful here to note that the matrix described by the letter “K” does not

represent stiffness anymore, bu

onsistent with the terminology used in OGATA [9], and most other control literature.

(79)

For Pole Placement state-space control using full state reconstruction, we can redefine

our single equation of motion as consisting of th

t now denotes the 1x8 state feedback gain matrix, and is

c

Therefore, the multiplication term denoted by: XK~ , is now representative of our

negative feedback control force signal, and is of the scalar form. Therefore the matrix

[T] is now required, to transform this scalar value into the vector form described by {F}

previously.

Where:

11

00

T (80)

XKTBZDXAX P PP

~ (81)

XXCKeXKBXAX OO

~~~~ (82)

Equations (81) and (82), we assume that the true state {X} is approximated by the In

estimated state X~ of the observer mathematical model, which is subscripted with the

letter “O”. The observer model “O” was derived from the model reduction procedures

outlined earlier in this thesis, and can alternatively be derived from experiment. The

observer model should not be confused with the plant model which is subscripted with

44

the letter “P”, and represents the real (physical) plant. The term [Ke] serves as a

weighting matrix and can also be described as the observer gain matrix, which is

different from the state feedback gain matrix described previously. The last term on the

right hand side of Equation (82) is a correction term that involves the difference

etween the measured output {X} and the estimated output X~b . This correction term

serves the purpose of eliminating the error between the real plant and the observer

tical) model. The coefficient matrix [C] is

easured and observed states, to ensure that only the relevant quantities are compared.

(mathema multiplied by the difference in the

m

We cannot compare velocities because the measured state does not produce velocities, it

only produces four displacements (with four zero velocity terms). Therefore the

estimated velocities must be deleted from the comparison, and this is achieved by the

following matrix [C]. Where [I] is the 4x4 identity matrix and [0] is the 4x4 zero

matrix.

0I

of the observed-state feedback control system that was

ch for the comparison of the different model reduction

techniques (observer models). The overall Pole Placement control technique was kept

performan

compared. Please refer to Equations (81) and (82) when using this diagram. The state

edback gain matrix [K], and the observer gain matrix [Ke] in the Figure 30, were

designed separately for each reduced m

For simplicity, the closed-loop poles were configured so that the desired closed-loop

natural frequencies were equal to the open-loop natural frequencies of each reduced

model. The closed-loop damping ratio (ac

roughout. For the simulations that follow, it was found that the Mode-displacement

00C (83)

Figure 30 is a block diagram

used throughout this resear

the same, so that the ce of the different reduced observer models could be

fe

odel, but by the same Pole Placement process.

tive damping ratio) was kept constant

th

method produces reduced models that are good for dynamic analysis of open-loop

systems but that it is inconvenient for use in the active vibration control of closed-loop

systems. The unstable closed-loop responses of the Mode-displacement method have

not been presented here, but it should be noted that the Mode-displacement does work

well when the complete plant and controller system remains in principal co-ordinates.

45

FIGURE 30: Observed-State Feedback Control System, for same size Plant and Observer

ERICAL SIM

2

5.2 NUM ULATION#6 (EARTHQUAKE EXCITATION DSF =100Hz)

In the simulation of closed-loop system responses under earthquake input, we have

added an active mass damper to the top level of the building model and therefore

increased the reduced models by one degree of freedom. The same 6dof (now 7dof)

mode-displacement model was used to replace the actual (physical) plant in all the

closed-loop simulations. The previous 3dof (now 4dof) models of the plant were used

within the controller (observer) for the purpose of estimating any unmeasurable states.

The configuration #1 of the building model was used here again, and all system

parameters were kept unchanged in these simulations, so that the performance of the

4dof observers could be isolated. The simulations presented in Figure 31 use an

earthquake data sample frequency (DSF) of 100Hz because the Guyan model performs

best at this frequency. The graphs show that the DMRM performs slightly better then

the Guyan method, during closed-loop control. Both the DMRM and Guyan models

dampen-out the response effectively as they estimate states in an accurate manner. The

data sample frequency of the earthquake was then altered again so that the effect of

higher frequency contents of excitation could be examined on the closed-loop system

performance of the 4dof models. The numerical simulation #7 presented in Figure 3

Ke

AO

BO

C

-K

B

AP

BBP ++

++++ +_

X

X~

PLANT

OBSERVER

46

uses an earthquake data sample frequency of 400Hz, and clearly shows that the DMRM

erfor n method, during closed-loop control. The

esponse m

stimates unmeasurable states in a m

p ms significantly better then the Guya

D mpMRM da ens-out the r ore effectively then the Guyan, because it

e ore accurate manner.

0 5 10 15 20-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Modified El Centro Earthquake

TIME (Sec)

AC

s)

4

CE

L. (m

/2

0 5 10 15 20-4

-2

0

2

Open-Loop Response of 7DOF Mode-Disp Plant(Theoretical Plant)

ME (Sec)

AC

s2 )C

EL.

(m

/

TI

Closed-Loop Res(Theoretical O

0 5 10 15 20-1

-0.5

0

0.5

1

1.5

ponse of 4DOF DMRM Observer bserver using Theoretical Plant)

AC

CE

L. (m

/s)

1.5

TIME (Sec)

2

0 5 10 15 20-1

-0.5

0

0.5

1

Closed-Loop Response of 4DOF GUYAN Observer (Theoretical Observer using Theoretical Plant)

AC

CE

L. (m

/s2 )

TIME (Sec)

FIGURE 31: Closed-Loop Response with Ground Earthquake Excitation (DSF= 100Hz)

5.3 NUMERICAL SIMULATION#7 (EARTHQUAKE EXCITATION DSF =400Hz)

0 1 2 3 4 5-0.3

-0.2

-0.1

0

0.1

0.2

0.3


TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-3

-2

-1

0

1

2

3

Open-Loop Response of 7DOF Mode-Disp Plant(Theoretical Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-2

-1

0

1

2

Closed-Loop Response of 4DOF DMRM Observer(Theoretical Observer using Theoretical Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-2

-1

0

1

2

Closed-Loop Response of 4DOF GUYAN Observer(Theoretical Observer using Theoretical Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )


47

efinition of the state space

variables, the relative displacements of each of the master co-ordinates of the building

was used, ie: master co-ordinate 1 (Floor 7); master co-ordinate 2 (floor 13); master

coordinate 3 (floor 20); and master coordinate 4 (the active mass).

All system parameters were kept unchanged in these simulations, so that the

performance of the 4dof theoretical observers could be isolated. The graphs show that

the theoretical DMRM performs slightly better then the Guyan method, during closed-

loop control simulations, when a DSF of 100Hz is used.

The data sample frequency (DSF) of the earthquake was then raised to 400Hz so that

the effect of higher frequency contents of excitation could be examined on the closed-

loop system performance. The theoretical DMRM model performs significantly better

then the Guyan method during closed-loop control at this chosen DSF. The DMRM

ates

nmeasurable states in a more accurate manner.

We also attempted to use the 4dof reduced observer model obtained by the Mode-

displacement method in the simulation of closed-loop system responses under

earthquake inputs, but unstable responses were obtained when the control input was

applied to the 7dof plant model. The Mode-displacement method works exceptionally

well only when the complete plant-controller system remains in principal co-ordinates,

but this is inapplicable to a real (physical) plant system. Current research is being

undertaken to rectify this problem.

It should be noted here, that all of the closed-loop simulations contained in this chapter

were based on theoretically attained controller and plant models. These performance

results are drastically affected when the theoretical plant model is replaced by an

experiment-based plant model, as presented in chapter 6.

5.4 SUMMARY

The state-space control technique was presented in this chapter, and was used

throughout this research because of its simplicity. For the d

dampens-out the response more effectively then the Guyan, because it estim

u

48

CHAPTER 6: EXPERIMENTAL TESTING________________________________________________________________________________________________________

onto the motor, its

erformance characteristics were enhanced. Below is a frequency response and phase

(with a 1.8Hz offset allowed

r). Therefore according to this result, the motor can perform well when activated by a

__

6.1 FREQUENCY RESPONSE TESTING OF THE LINEAR MOTOR

In order to determine the capabilities of the linear motor for the purposes of active

control, it was necessary to acquire its frequency response characteristics. In the

process of doing this, it was found that by placing extra active mass

p

response plot of the linear motor while the force output of the motor is measured against

command force (input). Approximately 109kg of active mass was rigidly attached to

the motor, and a 10% per 10V gain value was used, as this configuration was found to

be optimal. The plot reveals a -3dB point value of 34.8Hz

fo

command force signal that is of a frequency of 34.8Hz or less. However, further tests

also revealed that the actual lower limit of the motor’s performance was approximately

2Hz. Below this frequency, the response characteristics start to become nonlinear.

FIGURE 33: Frequency & Phase Response of Linear Motor; under Force Control Mode

49

6.2 EXPERIMENTAL MODAL TESTING OF THE STRUCTURAL MODEL

order to verify all of the previous numerical simulations, it was necessary to conduct

ing that

as required was modal testing, which in its most common form is known as the

BUILDING CONFIGURATION: Configuration #2; with unrestrained active mass.

VIBRATION DIRECTION: Only one lateral direction of vibration was

considered in this test, as depicted in Figure 34.

IGURE 34: Direction of Motion for Experimental Test #1

In

testing on the real (physical) twenty storey building model. The first type of test

w

measurement or acquisition of natural frequencies and modal shapes. The modal testing

methodologies have been omitted here, because they can be found in many other

sources of literature such as Wang [10]. A quick test on configuration #1 of the twenty

storey building model found that the first natural frequency of the structure was well

below that which was predicted by all analytical models. The source of this discrepancy

was unknown at the time, and due to the limitations of the linear motor (actuator) and

shaking equipment, it was decided that the natural frequencies of the building model

should be shifted upwards. In order to achieve this goal, the structure was modified

according to configuration #2, and then finally to configuration #3.

6.2.1 EXPERIMENTAL TEST #1

F

m1

m2

m3

m4

k1

k2

k3

Active Mass Unrestrained

DIRECTION OF VIBRATION

50

TEST DESCRIPTION: Impact (Hammer) Test

TEST OBJECTIVE: To determine the lowest five natural frequencies

of the twenty storey building model, according to

the vibration direction depicted in Figure 34.

TEST EQUIPMENT: Two accelerometers, an impact hammer, a power

supply and a dynamic signal analyser were used in

this test.

TEST RESULTS: fn1 (Test 1)= 2.875 Hz

fn2 (Test 1)= 9.625 Hz

fn3 (Test 1)= 19.625 Hz

fn4 (Test 1)= 29.750 Hz

fn5 (Test 1)= 45.375 Hz

FIGURE 35: Natural Frequency Test Results for Test #1

51

CONCLUSION: The first five natural frequencies of the building for

test #1 are given in the results section above. The

mass

is restrained to the top of the building model and

6.2.2 EXPERIMENTAL TEST #

IGURATION: the active mass restrained.

VIBRATION DIRECTION: ction of vibration was

icted in Figure 36.

FIGURE 36: Direction of Motion for Experimental Test #2



of the twenty storey building model according to

the vibration direction depicted in Figure 36, while

ass is restrained to the top floor.

smaller peaks that occur below 2.875 Hz are from

the other modes of vibration, or are possibly from

the fixed active mass mode (when the active

cannot slide).

2

BUILDING CONF Configuration #2; with

Only one lateral dire

considered in this test, as dep

the active m

m1

m2

m3

m4

k1

k2

k3


Restrained Active Mass

52

TEST EQUIPMENT:

TEST RESULTS:

6.688 Hz

fn4 (Test 2)= 26.375 Hz

5 (Test 2)= 36.625 Hz

Two accelerometers, an impact hammer, a power


this test.

fn1 (Test 2)= 1.438 Hz

fn2 (Test 2)= 7.438 Hz

fn3 (Test 2)= 1

fn



BUILDING CONFIGURATION: Configuration #2; with the active mass restrained.

IBRATION DIRECTION: Only one horizontal direction of vibration (into the

throughout this test, as

depicted in Figure 38.

V

page) was considered

53






the active mass is restrained to the top floor.

eters, an impact hammer, a power


this test.

f (Test 3)= 1.625 Hz

n2

f (Test 3)= 27.875 Hz

m4


e)


m3

k3

m2

(Into the pagk2

m1

k1

TEST EQUIPMENT: Two accelerom

TEST RESULTS: n1

f (Test 3)= 10.250 Hz

n3

fn4 (Test 3)= 51.550 Hz

fn5 (Test 3)= 80.000 Hz

54



BUILDING CONFIGURATION: Configuration #2; with the active mass restrained.

(twisting) direction of

vibration was considered throughout this test, as


VIBRATION DIRECTION: Only one rotational

depicted in Figure 40.

m1

m2

m3

k

m4


(Twisting)


3

k2

k1

55





the active mass is restrained to the top floor.

TEST EQUIPMENT: Two accelerometers, an impact hammer, a power

dynamic signal analyser were used in

this test.

(Test 4)= 3.813 Hz

supply and a

TEST RESULTS: fn1


CONCLUSION: The only conclusive natural frequency that could

be measured for the rotational direction of

vibration was the highest peak in the graph, which

occurred at 3.813 Hz. It was very difficult to

isolate the rotational direction without exciting

other modes of vibration at the same time. The

minor peaks in the graph are more than likely due

modes of vibration which were

recorded earlier, as their values seem to coincide.

to the other

56


BUILDING CONFIG Configuration #3; with unrestrained active mass.

Only one lateral direction of vibration was

considered in this test, as depicted in Figure 4

URATION:

VIBRATION DIRECTION:

2.



EST OBJECTIVE: To determine the lowest three natural frequencies


TEST EQUIPMENT:

TEST RESULTS:

m3

m4Active Mass Unr

T

the vibration direction depicted in Figure 42.

Two accelerometers, an impact hammer, a power


this test.

fn1 (Test 5)= 3.18 Hz

fn2 (Test 5)= 12.10 Hz

fn3 (Test 5)= 24.20 Hz

m1

m2

k1

k2

k3

estrained


57

6.2.6 MODAL SHAPE TESTING

For all harmonic excitation testing and for the DMRM forced vibration test that will be

resented later, it was necessary to remove the linear motor from the building model

ent stat

this required modification was that he

uilding, there is a very large passive damping force between the active mass and the

twentieth floor. For forced vibration testing, this passive damping force is significant

and greatly affects the test results. By simply restraining the active mass and leaving

the motor on the building, a misleading result is also achieved because the active mass

would then form part of the twentieth floor. In active-control mode, the active mass

does not form part of the twentieth floor, it forms the control force, and therefore this

option was not viable. The equivalent static mass (of 17.1 kg) depicted in Figure 43 is

only representative of that part of the motor that is fixed to the twentieth floor of the

building, and therefore the active mass was ignored for this equivalence calculation.

FIGURE 43: Photo of the 17.1 kg Equivalen L

p

and replace it with an equival ic mass, (as depicted in Figure 43). The reason for

when the motor is left to idle on the top of t

b

t inear Motor Static Mass

58

The m dal shapes of the building moo del configuration #3 are presented here, because

e final active-control testing utilizes these results. For a more detailed presentation of


TEST DESCRIPTION: Harmonic Excitation Test

TEST OBJECTIVE: To determine the lowest three reduced modal

shapes of the twenty storey building model,

according to the vibration direction depicted in

th

these test results please refer to the appendix of this thesis.

6.2.6.1 EXPERIMENTAL TEST #6

BUILDING CONFIGURATION: Configuration #3; with the linear motor removed,

and replaced by the equivalent linear motor static

mass.



Figure 44.

m1

m2

m3

17.1 kg

k1

k2

k3

Equivalent Static Motor Mass


59

TEST EQUIPMENT: Three Eddie-current sensors with a power supply,

a dynamic signal analyser, and an electromagnetic

shaker were used in this test.

FIGURE 45: Electromagnetic Shaker with Stinger attachment for Test #6 and Test #8.

TEST RESULTS: NORMALISED REDUCED MODAL SHAPES (Test 6)

FREQUENCY: 24.20 Hz 12.10 Hz 3.18 Hz

FLOOR #20: 1.000000 1.000000 1.000000

FLOOR #13: -0.936997 -0.368681 0.694248

FLOOR #07: 0.518767 -0.965475 0.329552 Table 1: Normalised Reduced Mo Test Results. dal Shape

r active-control models are 3dof.

CONCLUSIONS: The normalised reduced modal shapes for test #6

are given in the above results table. Only the

reduced (three floor values) were required here,

because ou 60

6.3 DYNAMIC MODEL REDUCTION METHOD FORCED VIBRATION TEST

nty storey building model reveThe modal test results of the real twe aled that all forms of

nalytical (theoretical) modelling had failed for this application of active control. The

percentage error in natural frequency and modal shapes produced by all of the

theoretical models was extremely high, and it became evident that these theoretical

models could not be used for real-time active control purposes. (This conclusion will be

proved later in this thesis, when the theoretical models are compared against a real

experiment-based plant model). The problem associated with the Guyan method, and

with most other model reduction methods, is that they assume that the system behaves

strictly according to linear elastic theory. The versatility of the Dynamic model

reduction method is such that it provides the option of obtaining system parameters

from experiment, not just from theory. The following experimental procedures ensure

that the Dynamic model reduction method forms an accurate description of the real

system dynamics.





TEST DE ON: atic DMRM For ed Vibration Test

TE E: o determin ass and sti ices

dynamics.

T:

a

mass.


SCRIPTI The St c

ST OBJECTIV T e accurate m ffness matr

that represent the real system

TEST EQUIPMEN Three Eddie-current sensors with a power supply,

a dynamic signal analyser, an assortment of

weights, and a weight pulley system were used in

this test.

61

FIGURE 46: Direction of Motion for Experimental Test #7 and Test #8.

m1

m2

m3

k

17.1 kgEquivalent Static Motor Mass

1

k2

k3


TEST METHOD: The displacements were measured at all three master

es of the building while a range of static loads

were applied separately to each master co-ordinate at a

of wh

elements of the static response matrix XC, (that was

mentioned previously under section 3.4 of this thesis).

The sy

frequencies and modal shapes presented in test results #6.

fore our

Static DMRM mass matrix can be determined as follows:

M C

applied

coordinat

time. From this, linear graphs were obtained, the slopes

ich describe the displacement per unit force

stem matrix BC is determined from the natural

So that: ,11CCC KMB and there

1211 )( IKMX CCC . Where the frequency of the

load is equal to zero in this case (ie 0 ), and

our matrices become:

C (Sta

Mc (Sta

tic) = 1CX , and

tic) =

K

1CB CX

62

TEST RESULTS (Test 7):

RESULTS FOR FORCES APPLIED TO MASTER CO-ORDINATE #1, FLOOR #7

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

0.0022

25 50 75 100 125 150 175 200

SP

LA

CE

ME

NT

(m

etre

s)D

I

FORCE (N)

Master Co-ordinates, versus Applied Force. FIGURE 47: Graph of Displacement at the

RESULTS FOR FORCES APPLIED TOR CO-ORDINATE #2, FLOOR #13MASTE

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

0.0022

0.0024

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

FORCE (N)

DIS

PL

AC

EM

EN

T (

met

res)

0.0023

-o inFIGURE 48: Graph of Displacement at the Master Co rd ates, versus Applied Force.

63

RESULTS FOR FORCES APPLIED TOMASTER CO-ORDINATE #3, FLOOR #20

0.0012

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

0.0019

0.002

0.0021

0.0022

0.0023

0.0024

10 15 20 25 30 35 40 45 50 55

FORCE (N)

DIS

PL

AC

EM

EN

T (

met

res)

FIGURE 49: Graph of Displacement at the Master Co-ordinates, versus Applied Force.

HE ABOVE GRAPH COLOUR CODE: GREEN = Master Co-ordinate #1, Floor #7

PINK = Master Co-ordinate #2, Floor #13

CYAN = Master Co-ordinate #3, Floor #20

From the results presented in experiment #6 and re-arranging, we have:

(84)

(85)

T

0000.10000.10000.19370.03687.06942.0

5188.09655.03296.0

231200000.578000022.399

64

and from the results section of experiment #7, we have

(86) 4101851.01064.00423.01059.00774.00333.00451.00355.00242.0

)( xStaticX C

6102631.04102.01058.04048.09824.06470.0

1034.06760.01647.1)( xStaticKC (87)

418.382936.0032.16790.5110.145757.235852.32524.782.131

)(StaticM C (88)

ONCLUSION: The results of the Static DMRM forced vibration

49. From the

slopes of these three graphs, the response matrix

XC was determined, and the reduced static mass

and stiffness matrices were calculated according to

th

this test. When the static loads are applied to the

first master co-ordinate (floor #7), the resultant

graph (Figure 47) contains some characteristics of

hysteresis. This problem was addressed by using a

problem can be rectified by conducting separate

measurements for the loading and unloading of the

static force (weights).

C

test are presented in Figures 47 to

e procedure detailed in the methods section of

line of best fit. For future testing however, this

65




mass.

IBRATION DIRECTION: Only one lateral direction of vibration was


TEST DESCRIPTION: The 5Hz DMRM Forced Vibration Test

TEST OBJECTIVE: To determine accurate mass and stiffness matrices

that represent the real system dynamics.

EST EQUIPMENT: Three Eddie-current sensors with a power supply,

a dynamic signal analyser, a load cell, and an

electromagn

EST METHOD: Instead of a static load, this time a sinusoidal load

at a frequency of 5Hz was applied separately to

V

T

etic shaker were used in this test.

T

each master coordinate of the building at a time.

The dynamic load was measured continuously by

a load cell attached to the stinger, as in Figure 45.

All displacements were measured concurrently

with the load, and this produced a dynamic

response matrix XC, which contains the

displacement per unit load elements. A frequency

of 5Hz was chosen here, because it is the lowest

frequency that the shaking equipment is rated at,

and because 5Hz occurs in between the first and

second natural frequencies of the building model.

(It was important to choose an excitation

frequency that was considerably away from any

66

natural frequency of the building, in order to avoid

resonance).

frequencies and modal shapes presented in

test results #6. So that:

. Where the

frequency of the applied load is equal to 5 Hz in

The system matrix BC is determined from the

natural

,11CCC KMB

and therefore our 5 Hz DMRM mass matrix can be

determined as follows: 1211 )( IKMXM CCCC

this case 5..2.ie , and our matrices become: 121 )()5( IBXHzM CCC

CCCC BIBXHzK .)()5( 121

Sinusoidal graphs were produced by the siTEST RESULTS gnal

analyser according to the method outlined above,

00493.00258.000233.00000.0

).1()5( HzX C

and the signal amplitudes of these graphs were

used to calculate XC as follows. (Please refer to

the appendix of this thesis for a more detailed

presentation of these test results).

4101139.0968.0532.

x (89) 00903.00548.0

339.5241.0

2028.

5730.322295.11114.782568.92145.1001.112

)5( HzM C (90)

67

107602.30404.1

x

2396.22733.30502.19709.87444.69381.59116.9

)5( HzKC5 (91)

CONCLUSION:

tion test are presented in

model which is based on

experiment, and not on the th previous

models such as the Guyan method were based on.

ental model, because it

ations modes of the

structure more accurately, while at the same time,

preserving the lower modes reasonably well.

6.4 COMPARISON OF NATUR

MODEL (CONFIGURATION #3)

In the table below, the percentage err

presented, based on the true system rom experiment.

his error, which increases as the vibration mode increases, will become evident in the

ake simulations that follow. In Table 2 and in all of the simulations that follow,

everything was re-calculated to reflect the changes according to the new building design

(configuration #3). It should be noted here that the theore cal m

exceptional accuracy when it was applied to a previous single-pole building design,

ter-floor shear associated with that design. Table 2 shows the

iation in natural frequency that this shear effect produced in the new design. VIBRATION MODE: 1st MODE 2nd MODE 3rd MODE

The reduced mass and stiffness matrices for the

5Hz DMRM forced vibra

the above results. These matrices can now be used

to form our real plant

eory that the

The 5Hz experimental model is expected to be

more representative of the true system dynamics

then the static experim

describes the higher vibr

AL FREQUENCIES FOR THE BUILDING

or in natural frequency of our theoretical models is

dynamics which was determined f

T

earthqu

ti odelling worked with

because there was no in

drastic dev

LINEAR ELASTIC THEORY: 4.26 Hz 26.63 Hz 74.4 Hz

EXPERIMENTAL TEST #5: 3.18 24.20 Hz Hz 12.10 Hz

% ERROR 34% 120% 207%

Table 2: Comparison of Natural Frequencies obtained from Theoretical and Experimental Procedures.

68

6.5 CLOSED-LOOP ANALYSIS USING EXPERIMENTAL PLANT MODEL 6.5.1 NUMERICAL SIMULATION #8 (EARTHQU

Figure 50, the Mode-displacement plant from simulation #7 is replaced with a 5Hz

t model, which is

outlined in #8. It is only when th

compared against a real plant mod

become obviously unstable. The sa

400Hz). The reason for this instabi

modal shapes of all of the theore

(physical) building model, and sec

deviate significantly from the real m

the theoretical models are based on t

closely followed in reality (especially

AKE EXCITATION DSF =400Hz)

In

DMRM real plan based on the experimental procedure that was

e theoretical observer models (in Figure 50) are

el such as this, that we notice the active control

me situation occurs for any chosen DSF (not just

lity is twofold: Firstly the natural frequencies and

tical models deviate considerably from the real

ondly, their forced response characteristics also

odel. These discrepancies have occurred because

he linear elastic spring theory, which is not always

for our twin-column flat steel building design).

0 1 2 3 4 5-0.3

-0.2

-0.1

0

0.1

0.2

0.3


s)

0 1 2 3 4 5

0.5

TIME (Sec)

AC

CE

L (m

/2

-1

-0.5

0

1

Open-Loop Response of 3DOF 5Hz DMRM Plant (Experimental Plant)

AC

C/s

2 )E

L. (

m

TIME (Sec)

0 0.2 0.4 0.6 0.8 1 1.2-3

-2

-1

0

1

2

3

Closed-Loop Response of 4DOF DMRM Observer (Theoretical Observer using Experimental Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 0.2 0.4 0.6 0.8 1 1.2-3

-2

-1

0

1

2

3

Closed-Loop Response of 4DOF GUYAN Observer (Theoretical Observer using Experimental Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )


To the ased Guy l is repl h a

Stati 51, which erived fro ecific e

procedure ou st #7. In addition to ry- MRM m

replaced by a 5Hz DMRM experiment-based model, which was derived from test #8.

solve this problem of instability, theory-b an mode aced wit

c DMRM model in Figure was d m the sp xperimental

tlined in te this, the theo based D odel is also

69

6.6 CLOSED-LOOP ANALYSIS USING EXPERIMENTAL OBSERVER 6.6.1 NUMERICAL SIMULATION #9 (EARTHQUAKE EXCITATION DSF =400Hz)

0 1 2 3 4 5-0.3

-0.2

-0.1

0

0.1

0.2

0.3


TIME (Sec)

AC

CE

L (m

/s2 )

0 1 2 3 4 5-1

-0.5

0

0.5

1

Open-Loop Response of 3DOF 5Hz DMRM Plant (Experimental Plant)

TIME (Sec)

AC

CE

L. (

m/s

2 )

0 1 2 3 4 5-1

-0.5

0

0.5

1

Closed-Loop Response of 4DOF 5Hz DMRM Observer (Experimental Observer using Experimental Plant)

L. (

m/s

2 )

TIME (Sec)

AC

CE

-0.5

-1

0

0.5

1

Closed-Loop Response of 4DOF Static DMRM Observer(Experimental Observer using Experimental Plant)

L. (

m/s

2 )

0 1 2 3 4 5TIME (Sec)

AC

CE


The test frequencies of zero and 5Hz were both chosen specifically, because they fall

well below the third natural frequency of the structure, and are close to, but lower than

the dominant frequency of excitation. The third natural frequency is significant here,

because it is the highest mode of vibration that we are trying to control. As can be seen

in Figure 51, both experiment based DMRM models perform well as observers, when

compared against a real (physical) plant model.

It has been proven in the above earthquake simulations that the theoretical observer

models cannot be used with our real plant model, because their closed-loop response is

nstable. We have also proved by earthquake simulation that both of our experiment-u

based observer models work effectively well when compared against a real plant model.

However, to truly measure the performance of these observer models, we must now

conduct real-time active control testing on the physical twenty storey building model.

70

6.7 REAL-TIME ACTIVE CONTROL TESTING OF THE BUILDING MODEL

igures 52 to 55 prove that both the Static DMRM and the 5Hz DMRM experiment-

based reduced models effectively mitigate oscillations during active vibration control

testing. These tests were conducted on the real 2.5 metre high model structure

(configuration #3). All of these graphs compare the active-damping mode against the

passive-damping mode of the system when a sinusoidal force is applied to 18th floor. At

a frequency of excitation of 12.1Hz, the 5Hz model performs in a very similar manner

to the Static model. But when excited at 24.2Hz, the 5Hz model performs much better

than the Static one, because it describes the higher vibration modes more accurately.

The exact same Pole Placement control technique was used here as in all of the

numerical simulations. However, in this case, the application was real-time, and the

programming had to be converted into the Labview language, as illustrated in

APPENDIX (F). C. Chapman wrote the main Labview program called “Active Vib

Control 4 input v3” and it was then modified by J. Boffa. The two sub-programs called

“Control Calc 3” and “4” were written by Boffa for the purpose of estimating all

F

unmeasurable states, and for full-state reconstruction.

Top Floor Response of Structure with 5Hz DMRM Observer, and 12.1Hz Sinusoidal Excitation.

-0.6

-0.4

-0.2

0

0.2

0.4

0.60 0.5 1 1.5 2 2.5 3 3.5 4

Time (Seconds)

Dis

pla

cem

ent

(mm

)

Passive DampingActive Damping

Top Floor Response of Structure with Static DMRM Observer, and 12.1Hz Sinusoidal Excitation

-0.6

-0.4-0.2

0

0.20.4

0.60 0.5 1 1.5 2 2.5 3 3.5

Time (Seconds)

Dis

pla

cem

ent

(mm

)

4


FIGURE 52: 12.1Hz Sinusoidal Experimental Test Results of 5Hz DMRM observer

FIGURE 53: 12.1Hz Sinusoidal Experimental Test Results of Static DMRM observer

Top Floor Response of Structure with 5Hz DMRM Observer, and 24.2Hz Sinusoidal Excitation

-0.6

-0.4-0.2

0

0.20.4

0.60 0.5 1 1.5 2 2.5 3 3.5 4

Time (Seconds)

Dis

pla

cem

ent

(mm

)


Top Floor Response of Structure with Static DMRM Observer, and 24.2Hz Sinusoidal Excitation

-0.6

-0.4-0.2

0

0.20.4

0.60 0.5 1 1.5 2 2.5 3 3.5

Time (Seconds)

Dis

pla

cem

ent

(mm

)

4


FIFURE 54: 24.2Hz Sinusoidal Experimental Test Results of 5Hz DMRM observer

FIGURE 55: 24.2 Hz Sinusoidal Experimental Test Results of Static DMRM observer

71

6.8 SUMMARY

The modal test results of the real twenty storey building model revealed that all forms of

theoretical modelling had failed for this application of active control. The percentage

error in natural frequency and modal shapes produced by all of the theoretical models

was extremely high, and it became evident that these theoretical models could not be

used for real-time active control purposes. This conclusion was proven when the

theoretical observer models were compared against a real experiment-based plant

model, in which unstable closed-loop responses were produced. It should be noted here

that all theoretical modelling worked with exceptional accuracy when they were applied

a previous single-pole building design. This is because there was no inter-floor shear

orce is applied to 18 floor. At

a frequency of excitation of 12.1Hz, the 5Hz model performs in a very similar manner

to the Static model. But when excited at 24.2Hz, the 5Hz model performs much better

than the Static one, because it describes the higher vibration modes more accurately.

to

associated with that design.

It was proven in this chapter that both the Static DMRM and the 5Hz DMRM

experiment-based reduced models effectively mitigate oscillations during active

vibration testing. These tests were conducted on the real 2.5 metre high model structure

(configuration #3). All of these graphs compare the active-damping mode against the

passive-damping mode of the system when a sinusoidal f th

72

CHAPTER 7: CONCLUSIONS________________________________________________________________________________________________________

d consequently has poor accuracy in the higher frequency

nge (third natural frequency). There are no significant discrepancies between the

ear

underestimate the true response of the full-structure as estimated by the 6dof model.

f all the 3dof models, the Mode-displacement appears to perform most accurately

hen an earthquake DSF of 400Hz is used, and the Guyan model performs with the

ast accuracy. When the data sample frequency of the earthquake is increased, the

eviation of the 3dof models from the 6dof model increases further, and as the data

mple frequency is decreased the deviation decreases as well. When the earthquake

put was modified by using a sample frequency of 100Hz, all four models gave almost

entical responses. The reason for this is that only the low modes of vibration are

cited at this low frequency range, and these modes are all included in the 3dof

duced models. However it is unrealistic to expect that an earthquake would act at

ch a low frequency range. For the most accurate representation of a real earthquake,

ting on a real twenty storey building, numerical simulation #3 should be used only.

__

7.1 OPEN-LOOP NUMERICAL SIMULATIONS

For the numerical simulations presented in chapter 4, the reduced model based on the

DMRM retains the chosen three natural frequencies and corresponding modal shapes

accurately from the original system. Therefore the predicted dynamic responses based

on the DMRM have very good accuracy at a particular frequency range that often

covers the frequency range of excitation forces applied to the system. In contrast to

this, the reduced model obtained by the Guyan method does not retain all of the chosen

vibration modes precisely an

ra

DMRM and the Mode-displacement model reduction methods, as both of these models

perform closely to the full-scale model.

For the earthquake simulations of the responses of open-loop system, the three reduced

models were compared against a 6dof Mode-displacement model, as it was considered

to be more representative of the full-scale model due to its inclusion of twice as many

modes of vibration. The 6dof response has greater acceleration amplitudes because it

includes the extra modes of vibration. For this reason, all of the 3dof responses app

to

O

w

le

d

sa

in

id

ex

re

su

ac

73

7.2 CLOSED-LOOP NUMERICAL SIMULATIONS

se simulations, so that the

erformance of the 4dof theoretical observers (reduced models) could be isolated. The

e also attempted to use the 4dof reduced observer model obtained by the Mode-

ss

to stiffness ratio at each level of the building is greatly increased (as in our example).

All system parameters were kept unchanged in the

p

graphs show that the theoretical DMRM performs slightly better then the Guyan

method, during closed-loop control simulations, when a DSF of 100Hz is used.

The data sample frequency of the earthquake was then raised to 400Hz so that the effect

of higher frequency contents of excitation could be examined on the closed-loop system

performance. The theoretical DMRM model performs significantly better then the

Guyan method during closed-loop control at this chosen DSF. The DMRM dampens-

out the response more effectively then the Guyan, because it estimates unmeasurable

states in a more accurate manner.

W

displacement method in the simulation of closed-loop system responses under

earthquake inputs, but unstable responses were obtained when the control input was

applied to the plant model. The Mode-displacement method works exceptionally well

only when the complete plant-controller system remains in principal co-ordinates, but

this is inapplicable to a real (physical) plant system. Current research is being

undertaken to rectify this problem.

7.3 EXPERIMENTAL EVALUATION

The applicability of the Dynamic model reduction method to the active vibration control

of large structural systems has been demonstrated from the presented test results. The

versatility of the Dynamic model reduction method is such that it provides the option of

obtaining system parameters from experiment, not just from theory. The problem with

theory-based model reduction techniques is that they rely on theoretical techniques

alone. These theory-based techniques resulted in a drastic deviation in performance

from the real structural model, as determined from physical testing on it, and produced

unstable observers for active control. The theory-based techniques could be drastically

improved by including the inter-floor shear effect that becomes prevalent when the ma

74

The experimental procedure outlined in this thesis ensures that the Dynamic model

duction method forms an accurate description of the real system dynamics, and can be

mpted;

ey are being directed towards better control algorithms.

) For the experimental test #8, an improvement in the results can easily be achieved

) As mentioned in the conclusion section of test #7, hysteresis can be avoided by

e loading and the unloading of the static

force (weights).

re

performed at any convenient frequency including zero. Care needs to be taken when

choosing this test frequency, as it should be as close as possible to, but lower than, the

predicted dominant frequency of the excitation force. Further attempts at improving

the active-damping effect of the real structural system are currently being atte

th

7.4 RECOMMENDATIONS

It is recommended that the following list of suggestions be undertaken for the benefit of

future research in this area.

(1

by using a similar technique to that of test #7. Instead of just recording one

sinusoidal force and the resultant displacements, several tests at the same frequency

should be conducted. Each test should contain a different force magnitude, so that

linear graphs can be produced as in Figures 47 to 49. The slopes of these graphs

will provide a more accurate XC matrix, then that obtained by the previous single

test.

(2

conducting separate measurements for th

(3) Future research could possibly be directed towards improving the theoretical

mathematical models of the building in order to correct their unacceptable error,

and/or the development and optimisation of various control algorithms. The

theory-based modelling could be drastically improved by including the inter-floor

shear effect that becomes prevalent when the mass to stiffness ratio at each level of

the building is greatly increased (as in our current building model design).

75

CHAPTER 8: REFERENCES__________________________________________________________________________________________________________

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Flexible Structures Using Unobservability and Uncontrollability and Its Application in

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] Kajiwara, I., Nakamatsu, A. and Inagaki, T., ‘Reduced Modeling of Structure of

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81,

[5]

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[6] Moaveni, S., ‘Finite Element Analysis Theory and Application with ANSYS’,

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[8]

Aero

[9]

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Zhang, N., ‘Dynamic Condensation of Mass and Stiffness Matrices’, Journal of

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2003.

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Guyan, R. J., ‘Reduction of Stiffness and Mass Matrices’ American Institute of

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hou, K., Doyle, J. C., and Glover, K., ‘Robust and Optimal Control’, Prentice

d IRS techniques’, Journal of Sound and Vibration., 186(2), pp.311-323,

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I-based Synthesis’, Automatica,

ol. 32, pp.1007-1014, 1996.

AIAA Journal, Vol.6, No.7, pp.1313-1319, 1968.

s’, IEEE Transactions on

with Spillover using

feedback’, IEEE Cont. Sys. Mag., vol. 17, pp.19-35, 1997.

[12] Moore, B. C., ‘Principal component analysis in linear systems: controllability,

observability, and model reduction’, IEEE Trans. Auto Contr. V

[13] Balas, M. J., ‘Feedback control of flexible systems’, IEEE Trans. Automat. Contr.,

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[14] Z

Hall, New Jersy, 1995.

[15] Du, H., and Shi, X., ‘Low-order H controller design using LMI and genetic

algorithm’, in Proc. Amer. Contr. Conf., pp.511–512, 2002.

[16] Friswell, M. I., Garvey, S. D. and Penny, J. E. T., ‘Model reduction using Dynamic

and iterate

1

[17] Gahinet, P., ‘Explicit Controller Formulas for LM

v

[18] Craig, R. R., and Bampton, M. C. ‘Coupling of substructures for dynamic

analysis’,

[19] Meirovitch, L., ‘Dynamics and Control of Structures’, Wiley, New York, 1990.

[20] Balas, M. J., ‘Feedback Control of Flexible System

Automatic Control, Vol. AC-23, pp.673-679, 1978.

[21] Chait, Y., and Radcliffe, C. J., ‘Control of Flexible Structures

an Augmented Observer’, Journal of Guidance, Control and Dynamics, Vol.12, No.2,

pp.155-161, 1989.

[22] Liu, W. and Hou, Z., ‘Model Reduction in Structural Vibration Control and its

Application’, 16th ASCE Engineering Mechanics Conference, Seattle, 2003.

77

CHAPTER 9: APPENDIX__________________________________________________________________________________________________________

78

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FF

: RH

S S

mal

l (17

0mm

long

) #0

189

0303

M

8 x3

5mm

Soc

ket H

ead

Cap

Scr

ew IS

O M

etric

16

N/A

N

/A16

onl

y fo

r to

p cl

amp

asse

mbl

y #?

????

????

10

mm

Fla

t Mild

Ste

el (

500m

m x

500

mm

sqr

) 1

?N

/A1

OF

F: B

ase

Pla

te

#???

????

? 10

0mm

x 2

5mm

Fla

t Hig

h T

ensi

le A

lum

iniu

m

1?

N/A

1 O

FF

: 100

mm

Top

Cen

tre

Cla

mp

APPENDIX (A): BILL OF MATERIALS FOR 20 STOREY BUILDING MODEL

79

APPENDIX (B): DETAIL DRAWINGS OF 20 STOREY BUILDING MODEL

80

DY

NA

MIC

RE

SP

ON

SE

MA

TR

IX A

T

EX

CIT

AT

ION

=

NA

TU

RA

LFR

EQ

UE

NC

Y (

3.18

Hz)

18/0

5/20

05

Li

near

Mot

or

rem

oved

from

bu

ildin

g fo

r th

is

test

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

Cal

ibra

te E

ddie

=0.

1250

0000

0m

m/V

olt

Flo

or 2

0 1.

0000

0000

F

orce

Am

plitu

de

Not

Mea

sure

d

Flo

or13

0.69

4248

19

10

0mm

"SP

AN

"

Flo

or 7

0.

3295

5240

E

quiv

alen

t of

Mot

or S

tatic

M

ass

adde

d to

bu

ildin

g

B

uild

ing

Con

figur

atio

n #3

G

RA

PH

(D)&

(E):

E

XC

ITA

TIO

N

@M

.C.3

(F

loor

#20

)

N

orm

alis

e

P to

P (

volta

ge

read

ing)

Max

Min

P to

P

ER

RO

R:

x ca

libra

tepe

r to

p flo

or

CH

.1

3.95

0000

-10.

6500

00-1

4.60

0000

3.95

0000

0.

0000

000.

4937

500.

3303

1358

9

C

H.2

8.

3200

00-9

.980

000

-18.

3000

008.

3200

00

0.00

0000

1.04

0000

0.69

5749

129

CH

.3

11.9

5833

311

.958

333

0.00

0000

11.9

5833

3 0.

0000

001.

4947

921.

0000

00

G

RA

PH

(F)&

(G):

EX

CIT

AT

ION

@

M.C

.3 (

Flo

or #

20)

Nor

mal

ise

P to

P (

volta

ge

read

ing)

Max

Min

P to

P

ER

RO

R:

x ca

libra

tepe

r to

p flo

or

CH

.1

3.74

0000

-10.

7400

00-1

4.48

0000

3.74

0000

0.

0000

000.

4675

000.

3287

9120

9

C

H.2

7.

8800

00-1

0.18

0000

-18.

0600

007.

8800

00

0.00

0000

0.98

5000

0.69

2747

253

CH

.3

11.3

7500

011

.375

000

0.00

0000

11.3

7500

0 0.

0000

001.

4218

751.

0000

00

APPENDIX (C): TEST #6, MODAL SHAPE TEST RESULTS

102

AT

E

XC

ITA

TIO

N =

z)

18/0

5/20

05

build

ing

for

this

test

D

YN

AM

IC R

ES

PO

NS

E M

AT

RIX

NA

TU

RA

LFR

EQ

UE

NC

Y (

12.1

0H

Lin

ear

Mot

or

rem

oved

from

EX

CIT

AT

ION

@M

.C.

20)

Cal

ibra

te

Edd

ie=

0.m

/Vol

t3

(Flo

or #

1250

0000

0m

Flo

or 2

0F

o

1.00

0000

00rc

e A

mpl

itude

Not

Mea

sure

dF

loor

13

-0.3

6868

064

10

0mm

"SP

AN

"

Flo

or 7

-0

.965

474

alen

t of M

oto

Sta

tic M

ass

adde

d to

bui

ldin

g

72

Equ

ivr

Con

fi#3

Bui

ldin

ggu

ratio

n

GR

AP

H(J

)&(K

)&(L

):

EX

TIO

N

@M

.C.3

(F

loor

#20

) C

ITA

Nor

mal

ise

(vol

tage g)

Max

Min

P to

PE

RR

OR

:x

calib

rate

oor

P to

P

read

in

per

top

fl

CH

.1

000

-8.7

4000

0-1

6.57

0000

7.83

0000

0.00

0000

0.97

8750

-0.9

6547

4723

7.83

0

CH

.2

000

--1

2.0.

000

-2.

990

12.6

2000

05.

6100

0099

0000

0000

0.37

375

0.36

8680

641

CH

.3

8.11

0000

--2

1.84

8.0.

000

013

.730

000

0000

1100

0000

01.

0137

51.

0000

00

103

DN

AM

IC R

ES

PO

NS

E M

AT

RIX

AT

18

/05/

2005

Line

ar M

otor

u

YE

XC

ITA

TIO

N =

N

AT

UR

ALF

RE

QU

EN

CY

(24

.20H

z)

rem

oved

from

bfo

r th

is te

st

ildin

g

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

Cal

ibra

teE

ddie

=0.

1250

0000

0m

m/V

olt

Flo

or 2

0 1.

0000

0000

For

ce A

mpl

itude

Not

s

red

Mea

u

Flo

or 1

3 -0

.936

9973

2

100m

m"S

PA

N"

Flo

or 7

0.

5187

6676

tor

to

ldin

g

E

quiv

alen

t of M

Sta

tic M

ass

add

bui

o ed

B

uild

ing

Con

figur

atio

n #3

GR

AP

H(O

)&(P

)&(Q

):

EX

CIT

AT

ION

@

M.C

.3 (

Flo

or #

20)

Nor

mal

ise

P to

P (

volta

ge

read

ing)

Max

MP

to

ER

RO

R:

x ca

libr

op fl

oor

inP

ate

per

t

CH

.1

3.-1

0.69

-0

0 00

037

756

87

0000

0000

14.5

6000

3.87

000

0.00

00.

4850

0.51

8766

CH

.2-1

0.65

0000

-17.

6400

006.

9900

00

0.00

0000

0.87

3750

-0.9

3699

7319

6.

9900

00

CH

.37.

4600

00-1

4.09

0000

-21.

5500

000

000

320

000

7.

4600

00.

000

0.9

500

1.0

0

15

8/

05/2

00 o

f Edd

ie-c

urre

Ref

er to

pag

e 12

nt s

enso

rfo

r 8m

m P

robe

an

d 1m

of c

able

Z

ero

to 2

0 V

olts

=2.

5mm

ddie

-Cur

rent

CA

LIB

RA

TE

all

Cha

nnel

s =

0.12

5 m

m/V

olt

E

104

AT

Hz)

B

uild

ing

Con

figur

atio

n#3

DY

NA

MIC

RE

SP

ON

SE

MA

TR

IXIT

AT

ION

= H

IGH

FR

EQ

UE

NC

Y (

5.0

18/0

5/20

05

Li

near

Mot

or

rem

oved

from

bui

ldin

g fo

r th

is te

st

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

EX

CIT

AT

ION

@

M.C

.2(F

loor

#13

)

E

XC

ITA

TIO

N

@M

.C.1

(Flo

or#7

)

Cha

rge

Am

p m

ultip

y fa

ctor

cha

nges

for

each

floo

r

Cal

ibra

te E

ddie

=0.

1250

0000

0 m

m/V

olt

Flo

or 2

0 0.

0000

1139

Flo

or 2

00.

0000

0903

Flo

or 2

0 0.

0000

0548

Flo

or 1

3 96

80.

0000

0493

Flo

or 1

3 0.

0000

0258

10

0mm

"S

PA

N"

0.00

000

Flo

or 1

3F

loor

7

0.00

0005

32F

loor

70.

0000

0233

Flo

or 7

0.

0000

0000

Sta

tic M

ass

adde

d to

bu

ildin

g

E

quiv

alen

t of M

otor

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

Nor

mal

ise

Nor

mal

ise

P to

P (

vlta

ge

read

ing)

Max

inP

to P

E

RR

OR

:pe

r to

p flo

rpe

r F

orce

o

Mx

calib

rate

o

CH

.1

2.55

0000

-11.

3400

00-1

3.89

0000

2.55

0000

0.

0000

000.

4670

3296

75.

3195

9E-0

6 (m

/N)

0.31

8750

CH

.2

4.64

0000

-11.

8000

00-1

6.44

0000

4.64

0000

0.

0000

000.

8498

1685

9.67

957E

-06

(m/N

) 0.

5800

00C

H.3

5.

4600

00-1

5.1

0

0.00

000.

6825

001.

0000

1.13

902E

-0/N

) 50

000

-20.

6100

05.

4600

0000

005

(mC

H.4

0.

5992

000.

2987

00-0

.300

500

0.59

9200

0.

0000

0059

.920

000

(N)

EX

CIT

AT

ION

@M

.C.2

(F

loor

#13

)

Nor

mal

ise

Nor

mal

ise

P to

P (

v reol

tage

adin

g)M

axM

inP

R:

per

For

ce

P to

E

RR

Ox

calib

rate

per

top

floor

CH

.1

3.41

0000

-11.

0000

00-1

4.41

0000

3.41

0000

0.

0000

000.

4262

500.

2583

3333

32.

3330

6E-0

6 (m

/N)

CH

.2

000

8000

000

00

004.

9261

1E-0

/N)

7.2

00-9

.6-1

6.88

07.

2000

00.

0000

0.90

0000

0.54

5454

545

6 (m

CH

.3

13.2

0000

0-6

.640

000

013

.200

000

9.03

120E

-0(m

/N)

-19.

8400

00.

0000

001.

6500

001.

0000

006

CH

.4

827

0503

0024

0000

00

()

0.1

000.

-0.1

30.

1827

0.00

0018

2.70

0000

N

E

XC

ITA

TIO

N @

M.C

.1 (

Flo

or #

7)

N

orm

alis

e N

orm

alis

e P

to P

(vo

ltare

adin

g)M

axM

inP

R:

per

For

ce

geP

to

ER

RO

x ca

libra

tepe

r to

p flo

or

CH

.1

0.00

0000

0.00

0000

0.00

0000

0.00

000

00.

0000

000

0 (m

/N)

0 0.

000

00C

H.2

5.

5000

00-1

0.11

0000

00.

6875

000.

4708

9041

12.

5807

1E-0

6 (m

/N)

-15.

6100

005.

5000

00

0.00

000

CH

.3

11.6

8000

0-6

.890

000

-18.

5700

0011

.680

01.

4600

001.

0000

005.

4804

8E-0

6 (m

/N)

000.

000

00C

H.4

0.

2664

000.

1486

00-0

.117

800

0.26

60

266.

4000

00(N

)

40

0 0.

000

00

APPENDIX (D): TEST #8, 5Hz DMRM FORCED VIBRATION TEST RESULTS

105

DY

NA

MIC

RE

SP

ON

SE

MA

TR

IX A

T

18/0

5/20

05to

r

EX

CIT

AT

ION

= L

OW

FR

EQ

UE

NC

(6.0

0Hz)Y

Li

near

Mo

rem

oved

from

bu

ildin

g fo

r th

is te

st

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

N .2 3)

E

XC

(

mul

tipy

fact

or c

hang

es fo

r ch

floo

r

Cal

ibra

te E

ddie

=0.

1250

0000

0 m

m/V

olt

EX

CIT

AT

IO@

M.C

(Flo

or #

1

ITA

TIO

N

@M

.C.1

Flo

or#7

)

Cha

rge

Am

p

eaF

loor

20

90

0.00

0005

5F

loor

20.

0000

0558

Flo

or 2

0 0.

0000

0379

Flo

or 1

3 0.

0000

0590

Flo

or 1

30.

0000

0262

Flo

or 1

3 0.

0000

0135

10

0mm

"S

PA

N"

Flo

or 7

0.

0000

0358

Flo

or 7

0.00

0001

22F

loor

7

0.00

0000

82E

quiv

alen

t ota

tic M

ass

adde

d to

f Mot

or

Sbu

ildin

g

EX

CIT

AT

ION

@M

.C.3

(F

l)

N

oor

#20

orm

alis

e N

orm

alis

e

g)x

xp

P to

P (

volta

ge

read

inM

aM

inP

to P

E

RR

OR

:ca

libra

teer

top

floor

per

For

ce

CH

.1

00

0.64

0371

3.58

032E

-06

(m/N

) 2.

7600

0-1

1.18

000

-13.

9400

002.

7600

000.

0000

000.

3450

0023

CH

.4.

5500

0-1

1.83

000

-16.

3800

004

0.00

0000

0.56

8750

1.05

5684

455

5.90

235E

-06

(m/N

) 2

00

.550

000

CH

.4.

3100

0-1

5.77

000

-20.

0800

004

0.00

0000

0.53

8750

1.00

0000

5.59

101E

-06

(m/N

) 3

00

.310

000

CH

.40.

9636

00.

4910

0-0

.472

600

0.96

3600

0.

0000

0096

.360

000

00

(N)

EX

CIT

AT

ION

@M

.C.2

(F

l)

N

oor

#13

orm

alis

e N

orm

alis

e

g)x

xP

to P

(vo

ltage

re

adin

Ma

Min

P to

P

ER

RO

R:

calib

rate

per

top

floor

per

For

ce

CH

.1

00

0.21

8825

421.

2217

0E-0

6 (m

/N)

2.72

000

-11.

4400

0-1

4.16

0000

2.72

0000

0.00

0000

0.34

0000

2C

H.

5.83

000

-10.

3700

0-1

6.20

0000

50.

0000

000.

7287

500.

4690

2654

92.

6185

8E-0

6 (m

/N)

20

0.8

3000

0C

H.3

12.4

3000

-6.9

3000

-19.

3600

000

012

.430

000.

0000

001.

5537

500

5.(m

) 1.

0000

058

300E

-06

/NC

H.4

0.

2783

000.

0859

00-0

.19

0.2

027

2400

7830

0.0

0000

08.

3000

00(N

)

EX

CIT

AT

ION

@M

.C.1

(F

)

Nor

mal

ise

Nor

mal

ilo

or #

7se

g)x

xP

to P

(vo

ltage

re

adin

Ma

Min

P to

P

ER

RO

R:

calib

rate

per

top

floor

per

For

ce

CH

.1

00

0.21

5662

658.

1750

1E-0

7 (m

/N)

1.79

000

-11.

7900

0-1

3.58

0000

1.79

0000

0.00

0000

0.22

3750

1C

H.2

2.

9500

00-1

1.46

0000

-14.

4100

002.

9500

00

0.00

0000

0.36

8750

0.35

5421

687

1.34

728E

-06

(m/N

) C

H.3

8.

3000

00-8

.460

000

-16.

7600

008.

3000

00

0.00

0000

1.03

7500

1.00

0000

3.79

065E

-06

(m/N

) C

H.4

0.

2737

000.

1544

00-0

.119

300

0.27

3700

0.

0000

0027

3.70

0000

(N)

APPENDIX (E): OTHER FORCED VIBRATION TEST RESULTS

106

DY

NA

MIC

RE

SP

ON

SE

MA

TR

IX

18/0

5/20

05

Li

near

Mot

or

ing

f

A

T E

XC

ITA

TIO

N =

HIG

H

FR

EQ

UE

NC

Y (

8.0H

z)

rem

oved

from

bui

ldor

this

test

E

XC

ITA

TIO

N @

M.C

.3 (

Flo

or #

EX

CIT

AT

ION

@

M.C

E

XC

ITA

TIO

N

C f e

Cal

ibra

te

0.12

5000

000

mm

/Vol

t 20

)

.2 (

Flo

or #

13)

@M

.C.1

(Flo

or#7

)

harg

e A

mp

mul

tipy

acto

r ch

ange

s fo

r ac

h flo

or

Edd

ie=

Flo

or 2

0 0.

0000

0125

0.00

0003

380.

0000

0282

F

loor

20

Flo

or 2

0 F

loor

.000

00

100m

m

"SP

AN

" 1

3 0

0357

Flo

or 1

30.

0000

0093

Flo

or 1

3 .0

0000

032

F

loo

.000

00

Equ

ival

ent o

f Mot

or

Sr

7 0

0276

Flo

or 7

0.00

0000

33F

loor

7

.000

0016

7ta

tic M

ass

adde

d to

bu

ildin

g

EX

CIT

AT

ION

@M

.C.3

(F

loor

#20

)

Nor

mal

ise

Nor

mal

ise

P to

P (

volta

ge

rM

axM

inP

to P

E

RR

OR

:x

calib

rate

rp

eadi

ng)

per

top

floo

er F

orce

CH

.1

2.82

0000

-11.

2100

0-1

4.03

000

26

(m/N

) 0

0.8

2000

00.

0000

000.

3525

002.

2031

252.

7610

2E-0

CH

.23.

6500

03

00-1

2.28

0000

-15.

9300

003.

6500

00.0

0000

00.

4562

502.

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000

.350

000

0.00

0000

0.04

3750

1.00

0000

9.77

435E

-08

(m/N

) C

H.4

0.4

0-0

447

600

.226

900

.220

700

0.44

7600

0.00

0000

47.6

0000

0(N

)

EX

CIT

AT

ION

@M

.C.2

(F

loor

#13

) N

orm

alis

e N

orm

alis

eP

to P

M

axM

inP

to P

E

RR

OR

:x

calib

rate

oor

(vol

tage

read

ing)

per

top

flpe

r F

orce

CH

.14.

7-1

0-1

54

9000

0.3

1000

0.1

0000

0.7

9000

00.

0000

000.

5987

500.

8919

9255

17.

4843

8E-0

7(m

/N)

CH

.23.

7-1

1-1

53

3000

0.3

9000

0.1

2000

0.7

3000

00.

0000

000.

4662

500.

6945

9962

85.

8281

3E-0

7(m

/N)

CH

.35.

3-1

0-1

65

7000

0.7

4000

0.1

1000

0.3

7000

00.

0000

000.

6712

501.

0000

008.

3906

3E-0

7(m

/N)

CH

.40.

80

-000

000

.326

300

.473

700

0.80

0000

0.00

0000

800.

0000

00(N

)

EX

CIT

AT

ION

@M

.C.1

(F

loor

#7)

N

orm

alis

e N

orm

alis

eP

to P

(v

olta

gere

Max

Min

P

adin

g)

to P

E

RR

OR

:x

calib

rate

per

top

floor

per

For

ce

CH

.10.

8-1

2-1

20

2000

0.1

3000

0.9

5000

0.8

2000

0-0

.000

000

0.10

2500

0.15

5303

031.

4547

3E-0

7(m

/N)

CH

.24.

4-1

0-1

54

3000

0.6

8000

0.1

1000

0.4

3000

00.

0000

000.

5537

500.

8390

1515

27.

8590

7E-0

7(m

/N)

CH

.35.

2-1

0-1

55

8000

0.1

8000

0.4

6000

0.2

8000

00.

0000

000.

6600

001.

0000

009.

3670

2E-0

7(m

/N)

CH

.4

0.70

4600

0.39

3600

-0.3

1100

00.

7046

000.

0000

0070

4.60

0000

(N)

111

APPENDIX (F): LABVIEW PROGRAMMING FOR REAL-TIME ACTIVE CONTROL

Program Name: "Active Vib Control 4 input v3" Page Description: Block Diagram of Main Program

112

Program Name: "Active Vib Control 4 input v3"Page Description: Front Panel of Main Program

113

Program Name: "Control Calc X" Page Description: Block Diagram of Control Calc X, Page 1. (Pole Placement Control using Runge-Kutta Method)

114

Program Name: "Control Calc X" Page Description: Block Diagram of Control Calc X, Page 2. (Pole Placement Control using Runge-Kutta Method)

115

Program Name: "Control Calc 3" (Using the 5Hz DMRM Model) Page Description: Front Panel of Control Calc 3 (Pole Placement Control using Runge-Kutta Method)

116

Program Name: "Control Calc 4" (Using the Static DMRM Model) Page Description: Front Panel of Control Calc 4 (Pole Placement Control using Runge-Kutta Method)

117

model reduction of large structural systems for active

Documents