PERFORMANCE OF A DYNAMIC VIBRATION ABSORBER USING A ...

International Review of Mechanical Engineering (I.RE.M.E.), Vol. xx, n. x

Manuscript received July 2012, revised July 2012 Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

PERFORMANCE OF A DYNAMIC VIBRATION ABSORBER

USING A MAGNETO-RHEOLOGICAL DAMPER

Mahmoud H.Salem1, M.N.Anany 2, M.El-Habrouk 3, Sohair F. Rezeka4

Abstract – A dynamic vibration absorbers are essentially used to protect equipment from steady-

state harmonic disturbances. This paper studies the performance of a dynamic vibration absorber

using a magneto-rheological (MR) damper. A mathematical model for a two-degree of freedom

system is developed and simulated for different values of the current input of the MR damper. A

Bouc-Wen model for hysteresis of MR damper is implemented. The responses of the primary mass of

the dynamic absorber system subjected to external sinusoidal inputs are evaluated. Experiments

were conducted for different values of absorber masses and input currents and the amplitude of

vibration of the primary mass is recorded at different frequencies. Linear Quadratic Regulator

(LQR) is used to control the force produced by the MR damper in order to minimize the main mass

vertical displacement. The controlled force acquired for the MR damper is transformed to equivalent

current to be applied to the current driver of the damper. The performance of LQR control is

compared to that of the optimum-tuned-passive vibration absorber system using viscous dampers.

The simulation results show that integrating controlled MR dampers in vibration absorber system is

feasible and effective for heavy vibrating systems. LQR control of the MR damper is more effective

than the optimum passive conventional absorber in both suppressing the vibration of the main mass

and reducing the transmitted forces.

Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Dynamic vibration absorber, MR damper, optimum-tuned-absorber, LQR control

NOMENCLATURE

A Parameter to determine the hysteresis loop in Bouc-

Wen-model (dimensionless)

c0b Constant that determine c0 , Ns/m

c1a Constant that determine c1 ,Ns/m

c1 b Constant that determine c1 , Ns/mV

f Frequency of external force , Hz

fd Force of Damper , N

k1 Main Mass stiffness, N/m

k2 Absorber Mass stiffness, N/m

m Mass of Eccentricity, kg

m1 Mass of Main, kg

m2 Mass of Absorber, kg

n Parameter that determine the hysteresis loop Bouc-

Wen model (dimensionless)

x0 Initial displacement of spring k1 associated with

nominal damper force to accumulator, m

x1, x2 Displacement of m1 and m2, m

Abbreviations DVA Dynamic Vibration Absorber

TMD Tuned Mass Damper

LQR Linear Quadratic Regulator

MR Magneto-Rheological

MRDVA Magneto-Rheological Dynamic Vibration Absorber

Greek Symbols Scaling value for the Bouc-Wen model, N/m

a Constant that determine , N/m

b Constant that determine , N/mV

γ, µ Parameters that determine the hysteresis loop in

Bouc-Wen model, m-2

µ' Mass Ratio (dimensionless)

Constant to govern the first- order filter, s-1

I. Introduction

Vibration absorbers have been widely applied to

structures, machines, bridges and even in buildings and

foundations during seismic hazards [1-9]. The main aim

of the absorption system is to isolate the primary mass

from extreme vibrations in order to minimize damage or

failure of the system. To increase the life span of

machines, engineers have used vibration absorber

systems throughout all the aspects of design.

The dynamic vibration absorber (DVA) or tuned-mass

damper (TMD) is considered as a passive vibration

control device. When a mass-spring system, referred to

as a primary system, is subjected to a harmonic excitation

at a constant frequency, its steady-state response can be

Mahmoud H .Salem , M.N. Anany, M. El-Habrouk,and S.F.Rezeka

Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. xx, n. x

suppressed by attaching a secondary mass-spring system

or DVA. This idea was pioneered by Watts [2] in 1883

and Frahm [3] in 1909. However, a DVA consisting of

only a mass and spring has a narrow operation region and

its performance deteriorates significantly when the

exciting frequency varies. The performance robustness

can be improved by using a damped DVA that consists

of a mass, spring, and damper. The key design

parameters of a damped DVA are its tuning parameters

and damping ratio. The first mathematical theory on the

damped DVA was presented in 1928 by Ormondroyd

and Den Hartog [4]. Since then, many efforts have been

made to seek optimum parameters for the damped DVA.

Den Hartog [5], first tackled the optimum solution of a

damped DVA that is attached to a classical primary

system, i.e., a system free of damping. His study utilized

the feature of ‘‘fixed-point’’ frequencies, i.e., frequencies

at which the response amplitude of the primary mass is

independent of the absorber damping. Based on the

‘‘fixed-points’’ theory, Den Hartog found the optimum

tuning parameter and defined the optimality for the

optimum absorber damping. Based on this optimality,

Brock [6] derived an analytical solution for the optimum

damping ratio of the damped DVA.

Magneto-Rheological fluids (or simply “MR” fluids)

belong to the class of controllable fluids. The essential

characteristic of MR fluids is their ability to reversibly

change from free-flowing, linear viscous liquids to semi-

solids having controllable yield strength in milliseconds

when exposed to a magnetic field. This feature provides

simple, quiet, rapid response interfaces between

electronic controls and mechanical systems. MR fluid

dampers are relatively new semi-active devices that

utilize MR fluids to provide controllable damping forces

[10-13]. In order to characterize the performance of the

MR damper, several models were proposed by many

investigators [10-12]. Spencer et al. [10] proposed a

modified Bouc–Wen model to describe the MR damper

behavior. This model can accurately capture both the

force–displacement and the force–velocity hysteresis

loops, which involves as many as 14 parameters [9]. MR

damper as described by Bouc-Wen model was applied in

many research areas [14-19].

The main objective of this paper is to study the dynamic

vibration absorber response upon using a magneto-

rheological damper. The feasibility of the absorber is

investigated analytically and experimentally.

II. Mathematical Modeling

Figure 1 shows a schematic diagram of Magneto-

Rheological Damper Dynamic Vibration Absorber

(MRDVA). m1 is the primary mass and 𝑚2 is the

absorber mass. 𝑘1 is the supporting stiffness and 𝑘2 is the

absorber stiffness and an absorber damper of a damping

coefficient 𝑐2.

The equations of motion of the two masses may be

written as:

m1 x1̈ = meω2sin ωt − k1x1 − k2(x1 − x2) − fd (1)

m2 x2̈ = k2(x1 − x2) + fd (2)

Fig.1.A schematic diagram of MRDVA

where x1 and x2 are the displacements of the primary and

absorber masses, respectively. fd is the magneto-

rheological damping force. The amplitude of the exciting

force due to eccentricity is Fo = meω2, where m is the

rotating unbalanced mass and e is the radial eccentricity.

Equations 1 and 2 can be rearranged in the following

form,

x1̈ = − (k1+k2

m1) x1 + (

k2

m1) x2 − (

fd

m1) + (

meω2

m1) sinωt

(3)

x2̈ = − (k2

m2) x2 + (

k2

m2) x1 + (

fd

m2) (4)

MR Damper: To evaluate the potential of the MR damper

in vibration control applications and to take full

advantage of their unique features, a mathematical model

needs to be developed that can accurately describe the

behavior of the MR dampers. The mathematical model

proposed by Ai and Liao [13] is adopted in this study.

The hysteretic behavior in the damper was described by

the Bouc-Wen model. The mechanical realization of the

MR damper is shown in Fig.2.

The model is governed by the following equations:

𝑦 ̇ = 1

c0 +c1 [ z+ c0 x+ k0 (x–y

z =- x – y z z n – 1 – µ x – yz n + A x – y

𝑓𝑑= c1 y+ k3 ( x– x0) (5)



Fig.2. A mechanical model of the MR damper (c2).[13]

where z is a variable that accounts for the history

dependence of the applied response. The model

parameters depend on the voltage v to the driver as

follows:

= (u) = a+ b u,

c1= c1 (u) = c1a+c1bu,

c0 = c0 (u) = c0a+ c0bu (6)

Where u is given as the output of the first order filter:

u= - (u-) (7)

In this model, the accumulator stiffness is represented

by k3, and the viscous damping observed at large

velocities by co. A dashpot, represented by c1, is included

in the model to introduce the nonlinear roll-off that was

observed at low velocities, ko is present to control the

stiffness at large velocities, and xo is the initial

displacement of spring k3 associated with the nominal

damper force due to the accumulator. In this model, there

are 14 parameters, (coa, cob, ko, c1a, c1b, k3, xo, αa, αb, γ, µ,

Α, n, η) to characterize the MR damper. The optimized

values for the 14 parameters are determined by fitting the

model to the experimental data obtained in the

experiments [14]. The resulting parameters are given in

Table 1.

Using Equations 1 to 7, a Simulink model is constructed

for the purpose of obtaining MRDVA responses.

III. Experimental Test Rig

Figure 3 illustrates the experimental test rig used in this

study. It consists of two groups of masses, m1 and m2,

four groups of springs of total of twelve springs, and the

MR damper. The main mass in m1 is an Aluminum

square base with dimensions of 500x500x20 mm. A set

of four identical springs supports m1. A set of another

four identical springs and copper sleeves, locking any

overshooting from m1 during excitation, are mounted on

guiding shafts that are fixed to a heavy steel structure

supporting the whole model. Vertically mounted in the

middle of m1 two steel shafts, each is 580 mm long with

20 mm diameter. The shafts carry m2 over two identical

springs. Another two identical springs and two cupper

sleeves fit above m2 on the same shafts limiting any

overshoot from m2 during excitation and locking it. The

exciting unit is located under the Aluminum base and its

mass is counted with m1. The total mass of m1 is 48 kg

and m2 is 6 to 13 kg.

Excitation System: The vibration of the model is induced

from the rotation of two identical spur gears. They are

Table 1. Parameters for the MR damper model. value parameter Value parameter

12441 N/m a 784 Ns/m c0a

38430

N/mV

b 1803

Ns/mV

c0b

13632 m-2 3610 N/m k0

2059020 m-2 µ 14649

Ns/m

c1a

58 A 34622

Ns/mV

c1 b

840 N/m k3 .0245 m x0

2 n 190 s-1

Fig.3. Experimental test rig

Shaft

Bearing

Copper

Sleeves

Primary

mass

m1

Absorber

mass

m2 Absorber

Spring

k2

c2 ( MR Damper)

Main

Spring

k1



made from a non metallic material. Each gear is 124

mm outer diameter and has tooth of module 2 mm,

carved through the 20 mm face of the gears. Each gear

carries a steel rotating unbalance mass of 214 g, fixed at

a radius of 42 mm from the gear centre. The two

unbalance masses are mounted opposing each other on

the two gears symmetrically. This configuration allows

summing their centrifugal forces in the vertical direction

only while they cancel each other in the horizontal

direction, giving a maximum force in the direction of the

springs (vertical direction). Each gear is mounted on two

steel shafts and secured by a 6 mm square key; each shaft

is 32 mm in diameter. Snap rings are mounted on both

sides of the gears faces to prevent their slippage along the

shaft. The rotating shafts carrying the two gears are made

of steel to reduce their elastic deflection caused by the

rotating unbalance force. The two shafts are attached to

the lower surface of the Aluminum base by means of four

ball bearings, housed in FSB (P204) seats, bolted beneath

it. A high speed unidirectional flexible shaft, with chuck

at one end, drives the gear assembly. The flexible shaft is

driven by a single phase, 0.3675 kW electrical motor,

with maximum rotation speed of 2850 R.P.M. at 50/60

Hz. An SV008iC5 LG, variable frequency inverter, links

between the electric motor and the electricity source. It

allows varying the frequency of the input electric current

from 0 to 400 Hz, which in turn varies the speed of the

motor. Frequency can be fine adjusted via the inverter

and consequently the electric motor speed.

The base excitation of the model is provided from the

above excitation system; the gears assembly with the

motors and the current inverter. The meshed gears

provide a harmonic vibration to the model in the form of

“A sin ωt”. When the inverter is set to certain frequency,

the motor is allowed to rotate at constant angular

velocity, providing a steady state condition for

measurements. Although the inverter provides a wide

range of operating frequencies, the main concern in this

work is in the range from 0 to 20 Hz.

The experimental rig specifications are:

Primary Mass m1= 44 kg, Absorber Mass m2=variable

Supporting Stiffness k1 = 55666N/m ,

Absorber Stiffness k2= 21012 N/m

Rotating Unbalance m= 0.244 kg, and

Radial Eccentricity e= 0.05 m

The data measured in the experimental test rig are;

the acceleration of the primary mass m1 and secondary

mass m2. To obtain these parameters, two accelerometers

were available. The first accelerometer was 752A12 from

ENDEVCO with sensitivity 10 mV/ms-2 (for m1). The

second one was 752A13 from ENDEVCO with

sensitivity 100 mV/ms-2 (for m2). These transducers were

attached to BRUEL & KJAER hardware. Displacement

and velocity can be taken from accelerometer by double

integration respectively. The hardware consists of a

number of channels to which transducers can be attached.

It resolves electric signals from transducers and

manipulates them. The signals are processed and

transferred to the computer via Local Area Network link

(LAN). The LAN allows for high speed data

transmission (100 Mbps). The computer utilized was an

Hp notebook with Intel Pentium M processor 1.6 GHz

and 600 MHz CENTRINO. BRUEL & KJAER software

was installed on this notebook. This software is the

manifestation for the data gathered from the transducers.

It is where users can select from different functions,

different data adjustments, different time windows, and

different displays. The software is user-friendly. It is an

arrangement of icons in task groups. Icons, when

pressed, reveal the graphical displays of the user selected

functions.

IV. Uncontrolled System Results and

Discussions

Undamped System: The experimental measurements and

the simulation results are obtained and compared firstly

with no damping in the system upon using different

values for absorber mass m2. The results are reported in

figures 4 to 7.

= 5.5kg 2h mamping witdo n ,1xFig.4. Displacement

.5kg8= 2amping with mdo n ,1x. Displacement 5Fig.

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3x 10

-3

Am

plit

ude X

1 (

m)

Frequency (Hz)

X1 Amplitude m2=5.5kg Experimental and Simulation

Experimental m2=5.5kg

Simulation m2=5.5kg

Experimental

Single Degree of Freedom

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

3

3.5x 10

-3

Am

plit

ude X

1 (

m)

Frequency (Hz)

X1 Amplitude m2=8.5kg Experimental and Simulation


Simulation m2=8.5kg

Experimental




Figures 4,5 and 6 show both the experimental and

theoretical variation of the primary mass displacement

amplitude x1 versus the operating frequency for an

absorber mass of 5.5, 8.5 and 12 kg, respectively and for

single degree of freedom test cases. For the above

primary mass–spring system, the natural frequency

determined experimentally is fn= 6.2 Hz as seen in the

figures and fn= 5.8 Hz by calculating the natural

frequency mathematically. The theoretical results show

that the absorber can isolate the primary mass completely

from the base excitation. Meanwhile, the experiments

render minimum amplitude of primary mass oscillations

between 3.8 and 5.5 mm.

kg12= 2amping with mdo n ,1x. Displacement 6Fig.

Both the theoretical and experimental values of the first

and second natural frequencies f1 and f2 are listed in

Table 2. The results indicate that the increase in the

absorber mass yields a decrease in the value of the

bandwidth of operation. It can be noticed that the values

resulted from the experiments are slightly higher for the

first natural frequency and lower for the second one. This

can be attributed to the errors in the determination of the

values of the stiffness of the springs. Since the system is

considered as undamped, the amplitudes of the vibrations

reach infinity in the simulated results. Due to the

presence of the internal damping in the guides of the

primary mass and absorber mass, the experimental results

show how the small amount of damping affects the

response of the primary mass(x1) and reduces the

vibration amplitude.

Figures 7a presents the theoretical results of the tuned

frequency for various absorber masses. The primary mass

displacement x1 was zero at a frequency of 9.8 Hz, 8.6

Hz, 7.9 Hz, 7.1 Hz, 6.7 Hz and 6.1 Hz for absorber mass

of 5.5 kg, 7.2 kg 8.9 kg, 11.5 kg, 12 kg and 14 kg,

respectively. Figure 7b shows the tuned frequency at

which the mass exhibits zero displacement where it

isolates the system completely versus the various masses.

Also it can be seen at the natural frequency of oscillation

of the one degree of freedom at (6.2 Hz), the optimum

absorber mass is about 14 kg.

Table 2. Absorber mass and corresponding natural

frequencies. Absorber

massm2,

kg

First Natural Frequency f1,

Hz

Second Natural Frequency

f2, Hz

Simulation Experiments Simulation Experiments

5.5 5.4 5.5 12.4 11.2

8.5 5.2 5.0 9.7 8.6

12 4.8 5.2 8.2 7.8

Fig.7a. Displacement x1 without damping for different

absorber masses

Fig.7b.Tuned Frequency versus absorber mass m2

Damped System: The MR damper was added to produce

the damping effect during the operation of the

experiments. To ensure the homogeneity of an MR fluid,

a trial run was first performed for approximately 10

cycles of oscillations. This ensures that the fluid has been

dispersed before the actual data acquisition commences.

Figures (8 to 12) illustrate the variation of the

displacement amplitudes of vibration x1 with the

operating frequency for constant values of the input

current I = 0A, 0.25A, 0.75A and 1A for different values

of the absorber masses m2= 5.5, 7.2, 8.5, 11.5 and 12 kg.

0 2 4 6 8 10 120

0.5

1

1.5

2

2.5

x 10-3

Am

plit

ude X

1 (

m)

Frequency (Hz)

X1 Amplitude m2=12kg Experimental and Simulink

Experimental m2=12kg

Simulation m2=12kg

Experimental


0 2 4 6 8 10 120

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Am

plit

ude X

1 (

m)

Frequency (Hz)

X1 Amplitude Simulation at Different Masses with No damping

Mass=5.5kg

Mass=7.2kg

Mass=8.9kg

Mass=11.5kg

Mass=12kg

Mass=14kg

4 5 6 7 8 9 10 11 12 13 14 15

6

6.5

7

7.5

8

8.5

9

9.5

10

X: 13.67

Y: 6.2

Tuned F

requency

Mass (kg)

Tuned Frequency at different Masses



Fig.8. Displacement x1, with MR damping at current input 0A

Fig.9. Displacement x1, with MR damping at current input 0.25A

Fig .10. Displacement x1, with MR damping at current input 0.5A

Fig .11. Displacement x1, with MR damping at current input 0.75A

Fig.12.Displacement x1, with MR damping at current input 1A

The results show that the response of the primary mass

is never zero in the damped case. The maximum value of

the displacement amplitude of vibration x1 of the primary

mass occurs at absorber mass m2 = 12.5kg and applied

current I = 1 A, as shown in figure 12. It is clear from the

figures that the two masses m1 and m2 are virtually

clamped and the system behaves as a single degree of

freedom. This indicates that the damping ratio resulted

upon using the MR damper with the system under

consideration is infinity.

The effects of the input current applied to the MR

damper on the displacement amplitudes of vibration x1

are examined in Figures (13 to 17) for constant values of

the absorber masses m2= 5.5, 7.2, 8.5, 11.5 and 12.5 kg.

The resonant frequencies are decreased from 5.2 Hz to

4.7 Hz as the absorber mass is increased from 5.5 kg to

12.5 kg. The increase of the input current has an adverse

effect on the amplitude of vibration specially with the

increase of absorber mass. This means that increase of

the damping force does not always suppress the

vibrations. Figure 18 summarizes the results of the

performance of the dynamic absorber with MR damper

integrated to a light vibrating system.

Fig .13. Displacement x1 ,with damping (m2 = 5.5 kg)

2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10

-3 X1 Amplitude at I=0A

Frequency (Hz)

Am

plit

ude X

1 (

m)






2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10

-3 X1 Amplitude at I=0.25A

Frequency (Hz)

Am

plit

ude X

1 (

m)






2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)






2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)






2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9x 10

-3

Frequency (Hz)

Am

plit

ude X

1 (

m)

X1 Amplitude at I=1A






2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10

-3 X1 Amplitude at m=5.5kg

Frequency (Hz)

Am

plit

ude X

1 (

m)

I=0A

I=0.25A

I=0.5A

I=0.75A

I=1A



Fig.14. Displacement x1 ,with damping (m2 = 7.2 kg)



Fig.17.Displacement x1 ,with damping (m2 = 12.5 kg)

It can be realized from Figure 18 that the change of input

current in this system does not show a great deal of

change in the amplitude x1 when small absorber masses

are adopted. It can be concluded that the MR damper is

too large for the proposed system since the increase of

the damping force resulted from the increase of the input

current degrades the absorber performance. From the

parameters of MR damper listed in Table 1, the damper

produces a minimum force of 100N at zero current to a

maximum force of 3000N at I=2.75A. Therefore, a larger

system is required for further study of the MR damper

performance.

Fig.18. Experimental maximum displacement amplitudes of the

primary mass versus input current to damper for different values of the

absorber mass.

V. Controlled System Results and

Discussions

The new system specifications are: Primary mass m1 =3183𝑘𝑔, supporting stiffness k1 = 4.5MN/m, and

absorber stiffness k2= 0.6283 MN/m.

Optimal-Tuned Passive Absorber: An approach is

examined that uses an approximation of the optimal

damper force. Using the heavy system data, a tuned

vibration absorber with a constant damping coefficient

can be set by the following optimal conditions [16].

c = 2m2ωn√3μ′

8(1+μ′)3 (8)

The absorber tuned to

ωa

ωn=

1

1+μ′ (9)

where μ′ is the ratio of the masses of the absorber and

main mass, ωa is the natural frequency of the absorber

mass and absorber spring, and ωn is the natural

frequency of the main mass and foundation spring.

2 3 4 5 6 7 8 90

1

2

3

4

5

6

7x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)

I=0A

I=0.25A

I=0.5A

I=0.75A

I=1A

2 3 4 5 6 7 8 90

1

2

3

4

5

6x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)

I=0A

I=0.25A

I=0.5A

I=0.75A

I=1A

2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)

I=0A

I=0.25A

I=0.5A

I=0.75A

I=1A

2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9x 10


Frequency (Hz)

Am

plit

ude X

1 (

m)

I=0A

I=0.25A

I=0.5A

I=0.75A

I=1A



The absorber mass to main mass ratio μ′ was set to 0.2,

and the absorber mass m2=636 kg. The main mass

foundation natural frequency is 6 Hz.

LQR Control Method: Equations 3 and 4 can be put in

state-space form as:

�̇�(𝑡) = 𝐴 𝑋 + 𝐵 𝑢𝑑 + 𝑑(𝑡), (10)

𝑌(𝑡) = 𝐶 𝑋(𝑡)

where X(t) = [ 𝑥1 �̇�1 𝑥2 �̇�2] , ud = fd (the controlled

damping force of the MR dampers), and d(t) is the

exciting disturbing force. The Linear Quadratic

Regulator (LQR) control algorithm is employed to

evaluate the effectiveness of the semi-active vibration

absorber system with MR dampers [15]. The

performance index is chosen as:

𝐽 = ∫ {[𝑋𝑇(𝑡)𝑄 𝑋(𝑡) + 𝑢𝑑(𝑡)𝑇 𝑅 𝑢𝑑(𝑡)] 𝑑𝑡}∞

0 (11)

where Q and R are symmetric semi-positive definite and

positive definite matrices and defined as:

𝑄 = [

1𝑒7 00 1𝑒7

0 0 0 0

0 00 0

1 0 0 1

] 𝑎𝑛𝑑 𝑅 = [1]

The control law that minimizes Equation (11) is given

by:

ud (t) = - K X(t) (12)

Where K = R-1BTS, and S is determined by solving

Riccatti equation: S B R-1 B S - SA - ATS = Q

For the new system specifications, the state feedback

gains are:

K =

[-7468.5563 ; -1824.5806 ; 7.9495e08 ; 1152.2516].

The controlled force acquired from the MR damper

cannot be commanded; only the voltage v applied to the

current driver for the MR damper can be directly

changed. To induce the MR damper to generate

approximately the desired optimal control force fdc, the

command signal v is selected as follows; When the MR

damper is providing the desired optimal force (i.e., fd =

fdc), the voltage applied to the damper should remain at

the present value. If the magnitude of the force produced

by the damper fd is smaller than the magnitude of the

desired optimal force fdc and the two forces have the

same sign, the voltage applied to the current driver is

increased to the maximum level so as to increase the

force produced by the damper to match the desired

control force. If fd ≥ fdc , then the voltage is set to zero.

Results of Controlled System: The effects of varying the

amplitude of the exciting force (Fo) from 100 N to 1000

N on the main mass at different frequencies have been

studied for both optimally-passive-tuned absorber and for

controlled MR damper. Figures 19 to 24 show that

optimally-passive-tuned absorber render higher values

for the main mass displacement than that resulted from

the Linear Quadratic control MR damper. As the

frequency is increased to 12 Hz the amplitude of the

transmitted displacement to the main mass converge for

both dampers. The figures also illustrate that the

maximum amplitude of the main mass is attained at f= 6

Hz. As the exciting force increases from 100 N to 1000

N, the amplitude of the main mass increases from

0.07mm to 0.706mm, respectively, in case of passive

damper at f=6Hz. The corresponding values for the MR

dampers are 0.037mm and 0.432mm, respectively.

Figures (25) and (26) present the response of the

displacement of the main mass and the damping force for

the two modes of the simulation for Fo = 1000 N and f=6

Hz. The maximum damping force due to applying

passive damper is 514 N, and that produces by controlled

MR damper is 125 N.

100N of mplitudeforce aat 1x. Displacement 9.1Fig

200N of mplitudeforce aat 1xFig.20. Displacement

2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

8x 10

-5

Am

plit

ude X

1 (

m)

Frequency (Hz)

X1 Amplitude vs Frequency ( Amplitude=100 N )

Passive

LQR

2 4 6 8 10 12 14 160

0.5

1

1.5x 10

-4

Am

plit

ude X

1 (

m)

Frequency (Hz)


Passive

LQR



400N of mplitudeforce a at 1xFig .21. Displacement



Fig.24. Displacement x1 at force amplitude of 1000N

Fig.25. Vertical displacement vs. time at f=6 Hz

Fig.26. The damping force of MR damper vs. time at f

=6Hz

2 4 6 8 10 12 14 160

1

2

x 10-4

Am

plit

ude X

1 (

m)

Frequency (Hz)


Passive

LQR

2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-4

Am

plit

ude X

1 (

m)

Frequency (Hz)


Passive

LQR

2 4 6 8 10 12 14 160

1

2

3

4

5

6x 10

-4

Am

plit

ude X

1 (

m)

Frequency (Hz)


Passive

LQR

2 4 6 8 10 12 14 160

1

2

3

4

5

6

7

8x 10

-4

Am

plit

ude X

1 (

m)

Frequency (Hz)


Passive

LQR

18.5 19 19.5 20-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 10

-3

Am

plit

ude X

1 (

m)

Time (s)

X1 Amplitude 1000 at Freq=6Hz

Passive

LQR

18.5 19 19.5 20-600

-400

-200

0

200

400

600

Dam

pin

g F

orc

e (

N)

Time (s)

Force Amplitude 1000 at Freq=6Hz

Passive

LQR



VI. Conclusions

A semi-active vibration absorber system with MR

dampers has been investigated experimentally and

analytically. The governing equations of the two degree-

of-freedom vibration absorber model integrated with MR

dampers were developed. The linear quadratic control

was adopted to illustrate the feasibility and effectiveness

of controlled MR dampers on vibration absorber systems.

The dynamic responses of the main mass with secondary

MR absorber system implementing the linear quadratic

control to external sinusoidal forces were evaluated and

compared with those using conventional optimum-tuned-

passive absorber system using viscous dampers. It was

found that, the MR damper, controlled by the proposed

linear quadratic algorithm, can significantly attenuate the

vibrations of the foundation main mass subjected to large

exciting forces. MR dampers in vibration absorber

systems are feasible and effective for heavily vibrating

systems. The performance of LQR control is compared to

that of the optimum-tuned-passive vibration absorber

system using viscous dampers. The simulation results

show that integrating controlled MR dampers in vibration

absorber system achieves a significant decrease in main

mass displacement in heavy vibrating systems. LQR

control of the MR damper is more effective than the

optimum passive conventional absorber in both

suppressing the vibration of the main mass and reducing

the transmitted forces.

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Authors’ information

1- Mahmoud H Salem received the B.Sc.

degree from Arab Academy for Science & Technology & Maritime Transport, Alex, Egypt

in 2009 in Mechanical Engineering. He is a

Graduate Teaching Assistant of Mechanical Engineering in Arab Academy for Science &

Technology & Maritime Transport, Alex, Egypt

.His professional interests include mechanical vibrations intelligent control systems for autonomous vehicles, modeling and simulation of dynamic

systems and mechatronics.



2- Mohammed N. Anany received his B.Sc. and M.Sc. degrees from the Arab Academy for Science &

Technology AAST, Egypt, in 1998 and 2003,

respectively. He obtained his PhD from Univesity of Stuttgart, Germany in 2010. He is working for the

time being as a lecturer at the AAST. He is interested

in modeling and simulation of mechanical systems.

3- Mohamed El-Habrouk received the B.Sc.

and M.Sc. degrees from Alexandria Univ., Egypt in 1992 and 1995, respectively and the

PhD degrees in Electrical Engineering from

Brunel University, London, UK in 1998. He

is a lecturer of Electrical Engineering in

Alexandria University, Egypt. His

professional interests include autonomous Robotics, Power Electronics, Automatic

Control Applications, Microcontrollers, Modeling and Simulation of

dynamic systems and mechatronics.

4- Sohair F. Rezeka received the B.Sc. degree from Alexandria Univ., Egypt in 1976 and the M.Sc. and

the PhD degrees in Mechanical Engineering from

Wayne State University, Detroit, Michigan in 1980, and 1984, respectively. She is a professor of

Mechanical Engineering in Alex. Univ., Egypt and

now she is on leave at Arab Academy for Science and Technology, Alex., Egypt. Her professional interests include intelligent

control systems for autonomous vehicles and HVAC systems, fault

diagnostics and identification, modeling and simulation of dynamic systems and mechatronics.

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