IMECE2010-40380 : Resonance Modeling and Control in Structures by means of Magnetorheological...

1 Copyright © 2010 by ASME

Proceedings IMECE2010 2010 ASME International Mechanical Engineering Congress and Exposition

November 12-18, 2010, British Columbia, Vancouver, CANADA

IMECE2010-40380

RESONANCE MODELLING AND CONTROL IN STRUCTURES BY MEANS OF MAGNETORHEOLOGICAL DAMPERS

Mario F. Letelier Department of Mechanical Engineering

University of Santiago of Chile [email protected]

Dennis A. Siginer Department of Mechanical Engineering

The Petroleum Institute [email protected]

Juan S. Stockle Department of Mechanical Engineering


Cristian Herrera Department of Mechanical Engineering


ABSTRACT

Many structures such as automobile bodies, bridges and

buildings are subjected to external forces due to the nature of

the environment in which they exist. When the external

excitation frequency is similar to the structure's natural

frequency a resonance effect is generated, increasing the energy

and the amplitude of the oscillation, often causing catastrophic

situations.

This paper presents a method to decrease such resonant

oscillations using magnetorheological dampers.

INTRODUCTION

The phenomenon of resonance appears very often in nature,

in biological, chemical, mechnical systems, etc. Some of these

systems are subjected to different kinds of external loads with

oscillation frequencies often hard to predict on time when there

are high frequencies loads. One example of this is a car driver

who finds an uneven road that causes decreased vehicle control

[1].

This paper presents a method to dissipate the effect of

resonance by the application of magnetic fields arranged

around the edges of a circular tube that transports a fluid whose

properties change when a magnetic field is applied.

Assuming the pressure gradient as a function of parameters

that model the resonance behavior, the physical problem can be

studied. The purpose is to find the form that the flow stress

characterized by the magnetic field must take to decrease the

resonant effect on the flow.

ANALYSIS

Using the momentum equation along the axial coordinate

for a circular tube

� �� + �

� + �� = − �

�� (1)

where � is the density of the fluid, � is the sheer stress, � is the

axial velocity, and − �� is the pressure gradient,

The flow is assumed to behave as a Bingham plastic,

represented by the following equation:

� = �� + � �− �� (2)

where � is the sheer stress, �� is the yield stress, and � is the

viscosity. The flow term �� is the unknown that must be

determined in this work, which is in charge of reducing the

resonance.

Substituing (2) in (1) and writing − �� = � we have

� �� − �∇�� = � − ��

� (3)

Taking the following factor scale:

� = ��∗

https://www.researchgate.net/publication/238073569_Phenomenological_Model_of_a_Magnetorheological_Damper?el=1_x_8&enrichId=rgreq-4b9684fd-dbd4-4fd9-b619-5f2861ee9c72&enrichSource=Y292ZXJQYWdlOzI1NzM0MTU1MTtBUzoxODgxNTM3NzIxMjYyMTBAMTQyMTg3MDc2MzE1Ng==


� = ��∗ (4) � = !�∗

where �� represents a reference velocity, �� is a reference time,

and ! is the radius of the circular tube.

Substituting (4) in (3) and dividing by "��

�� we have

��∗��∗ − ��!� �∇��∗ = �� − ��!��

��∗ (5)

Taking the following considerations:

��!� � = 1 → �� = �!�� (6)

y

��"�� = �∗ → where �� = "��

�� (7)

��'"�� = ��∗ → where �� = '"��

�� (8)

we get the momentum equation in its nondimensional form:

��∗��∗ − ∇�w∗ = ϕ∗ − τ�∗r∗ (9)

In what follows, the asterisks are omitted in nondimensional

equations.

Incorporating the following model for the pressure gradient

function that simulates the resonance:

�(�) = ��./01�234(5�) (10)

we can get a variety of behaviors by varying parameters ��, 8, 5, 4, where ��, 4 represent the amplitude, 5 the

frequency, and 8 is a direct relation of the resonance period.

This function is shown graphically in the following plot.

Fig. 1 Graphic representation of the pressure gradient that

models the resonance phenomenon.

The resonance effect can be decreased postulating a function 9(�, �) formed by the pressure gradient and the yield stress.

9(�, �) = �(�) − :(�)� (11)

:(�) in equation (11) represents the yield stress in a

nonadimensional form of Bingham constitutive model, which

takes a form determined by the change in the magnetic field.

This is the function to be determined to stop the increase of the

pressure gradient's amplitude that is caused by the forces acting

on the system.

Applying the operator ( )∫1

0

2 rdr to equation (11)

2 ;(9(�, �)�)<� = �(�) − 2:(�)=

� (12)

and assuming

2 ;(9(�, �)�)<� = >234(5�) (13)=

�

we can solve for the yield stress term that was sought:

:(�) = �(�) − > sin(5�)2 (14)

As already mentioned, the function > 234 (5�) has the purpose

of decreasing the effect of the pressure gradient, keeping the

same frequency value, but including a smaller amplitude

modified by parameter >. Its graphic representation is shown in

Figure 2.

Fig. 2. > 234 (5�) function

The solution of Eq. (9) is found using a superposition method.

Deleting the yield stress term in (9) gives

�� − ∇�w = ϕ(t) (15)


The solution of this equation can be divided into a particular

solution and a homogeneous solution. The particular solution is

found by expanding it as a series:

�C(�, �) = D=(�)(1 − �) + D�(1 − ��) + DE(� − �) + ⋯ (16)

Replacing (16) in (15) and arranging in terms of powers of the

radius we can get the time functions D=(�), D�(�), DE(�) … ., where the odd values, i.e., D=(�), DE(�), DI(�) …. are null and

the even values D�(�), DJ(�), DK(�) …. agree with the following

relation:

4D� + <D�<� + <DJ<� + <DK<� + ⋯ = �(�) (17)

DJ = 14�

<D�<� (18)

DK = 16�

<DJ<� (19)

DL = 18�

<DK<� (20)

Replacing equations (18), (19), (20) in Eq. (17) we get an

equation in terms of a single unknown:

4 MD� + 12�

<D�<� + 12�4�

<�D�<�� + 12�4�6�

<ED�<�E + ⋯ N = �(�) (21)

Eq. (21) is written in terms of D�, proposing the following

solution:

D� = /01�(OP2(5�)(!� + !=� + !�� + ⋯ ) + (22) + sin(5�) (Q� + Q=� + Q��+. . . ))

The degree of the polynomials in Eq. (22) was determined by

comparing solutions, i.e., first polynomials of the 8th

degree

were taken, finding a solution for the velocity, then a solution

was found for polynomials of the 9th

degree, and when the

results were compared it was estimated that for the range of

parameters used in this paper the solutions are equivalent.

Replacing Eq. (22) in Eq. (21) and giving values to the

parameters of function �(�) we can determine the polynomial's

coefficients by means of a system of equations arranged in

terms of t, and replacing D�(�) in Eqs. (18), (19), (20),… we

solve for the values of DJ(�), DK(�)DL(�), getting the

expression for the velocity �C(�, �).

Once the solution of equation (15) is obtained, the stress term

that appears in Eq. (9) is included, and then replacing (16) in

(9) and arranging in terms of powers of r, the following set of

equations is obtained.

D=(�) = −:(�) (23)

4D� + RSTR� + RSU

R� + RSVR� + RSW

R� + RSXR� + ⋯ = �(�) (24)

DE = 13�

<D=<� (25)

DJ = 14�

<D�<� (26)

DI = 15�

<DE<� (27)

DK = 16�

<DJ<� (28)

DY = 17�

<DI<� (29)

DL = 18�

<DK<� (30)

Replacing equations (23), (25), (26), (27), (28), (29), (30) in Eq

(24) we get again an equation in terms of D�:

4 MD� + 12�

<D�<� + 12�4�

<�D�<�� + 12�4�6�

<ED�<�E + ⋯ N = �(�)

(31)

+ M<:(�)<� + 1

3�<�:(�)

<�� + 13�5I

<E:(�)<�E +N

This equation can be rearranged separating the right side terms.

On one hand we get

4 MD� + 12�

<D�<� + 12�4�

<�D�<�� + 12�4�6�

<ED�<�E + ⋯ N = �(�) (32)

whose answer is already known because it does not incorporate

any new term, so its solution is the same as that of equation

(21)

The other part of the equation is written in terms of the yield

stress:

4 MD� + 12�

<D�<� + 12�4�

<�D�<�� + 12�4�6�

<ED�<�E + ⋯ N = (<:(�)<� +

(33) 13�

<�:(�)<�� + 1

3�5I<E:(�)

<�E + ⋯ )

Again this solution is divided into two because the term N(t)

contains the terms �(t) and >234(5�), so there will be two

equations of the following form:

4 [D� + =�U

RSUR� + =

�UJURUSUR�U + =

�UJUKURVSUR�V + ⋯ \ = =

� (R](�)R� +

(34) 13�

<��(�)<�� + 1

3�5I<E�(�)

<�E + ⋯ )


4 [D� + =�U

RSUR� + =

�UJURUSUR�U + =

�UJUKURVSUR�V … \ = =

� (Acos(5�) +

(35) 234(5�)a)

Taking the values of the constants that go with the sine and

cosine terms in Eq. (35) as

D = M5> − 5E>3�5� + 5I>

3�5�7�9� − 5Y>3�5�7�9�11�13�N (36)

a = (− 5�>3� + 5J>

3�5�7� − 5K>3�5�7�9�11� +

(37) 5L>3�5�7�9�11�13�15�)

we propose the following solution for Eq. (34):

D�= = /01�bOP2(5�)(!� + !=� + !�� + ⋯ ) + 234(5�)(Q� +Q=� + Q�� + ⋯ )c (38)

Which is solved in the same way as Eq. (21). Now the

remaining terms of :(�), associated with the >234(5�) term

are solved proposing a solution of the type

D�� = ! 234(5�) + Q cos(5�) (39)

The same above considerations are applied, the pressure

gradient variables as well as the flow term are assigned values,

and in this way, replacing Eqs. (34) and (35) we get the

coefficients of the D�=(�) and D��(�) terms and a second

profile of the velocities �� is determined, adding these

solutions to term D�(�) found earlier.

RESULTS

The pressure gradient function that describes the

resonance, and function F, which includes the flow term in

charge of decreasing the resonance effect, are presented. Each

graph is shown with the corresponding velocity profile.

Fig. 3. Pressure gradient versus time for ∅� = 2, 8 = 2, 5 = 3, 4 = 4

Fig. 4. Velocity profile versus radius for ∅� = 2, 8 = 2, 5 = 3, 4 = 4

Fig. 5. F versus time for ∅� = 2, 8 = 2, 5 = 3, 4 = 4





Fig. 25. Pressure gradient versus time for ∅� = 2, 8 = 0.5, 5 = 3, 4 = 4

Fig. 26. Velocity profile versus radius for ∅� = 2, 8 = 0.5, 5 = 3, 4 = 4

Fig. 27. F versus time for ∅� = 2, 8 = 0.5, 5 = 3, 4 = 4





CONCLUSIONS

Has been found the form of the yield stress, for to decrease

the effect of the resonance produced by external forces.

This work can be used for to configurate the values of

yield stress in new design of devices that want to null the effect

of resonance.

REFERENCES

[1] Spencer Jr., B.F., Dyke, S.J., Sain, M.K., and Carlson, J.D.,

(1998). “Phenomenological Model for Magnetorheological

Dampers, “Journal of Engineering Mechanics, ASCE, Vol. 123,

No. 3, pp 9.

[2] S.J. Dyke, B. F. Spencer, M. K. Sain, and J.D.

Carlson.Modeling and control of magnetorheological dampers

for seismic response reduction. Smart Materials and Structures,

5:565 575, 1996.

IMECE2010-40380 : Resonance Modeling and Control in Structures by means of Magnetorheological...

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