Optimal suppression of parametric vibration in discs under rotating frictional loads

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215: 65 2001Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

H Ouyang and J E MottersheadOptimal suppression of parametric vibration in discs under rotating frictional loads

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Optimal suppression of parametric vibration in discsunder rotating frictional loads

H Ouyang and J E Mottershead*

Department of Engineering, The University of Liverpool, UK

Abstract: This paper investigates the parametric resonances of a stationary disc excited by a rotating

frictional load and in¯ uenced by a series of mass± spring± damper systems with or without friction.

The genetic algorithm is used to ® nd out the right number of mass± spring± damper systems and their

optimal positions in order to reduce and even eliminate the dynamic instability caused by the rotating

friction as a follower force on the disc surface. It is found that, if these mass± spring± damper systemsinvolve no or low friction, they can reduce or suppress the dynamic instability of friction-induced

parametric resonances when correctly located, but they have, at best, no eŒect when the level of

friction is high.

Keywords: disc, parametric resonance, dynamic instability, stabilizer, dry friction, genetic algorithm

NOTATION

a; b inner and outer radii of the disc

c; f; k; m damping, friction force, stiŒness and mass

of the exciter respectively

cj; fj; kj; mj damping, friction force, stiŒness and mass

of the j th stabilizer respectively

D ¯ exural rigidity of the disc

D¤ Kelvin-type damping coe� cient of thedisc

E Young’ s modulus of the disc material

h thickness of the disc

i ¡1p

qkl…qrs† modal coordinate for k…r† nodal circles

and l…s† nodal diameters for the disc

r radial coordinate in the cylindrical

coordinate system

r0; rj initial radial positions of the exciter andthe jth stabilizer respectively

Rkl combination of Bessel functions to

represent the mode shape of the disc in the

radial direction

t time

w de¯ ection of the disc in the cylindrical

coordinate system

¯…¢† Dirac delta function

¯kl Kronecker delta

³ circumferential coordinate in the

cylindrical coordinate system

³j initial angular position of the jth

stabilizer

l characteristic exponent in exp…il½† whichdescribes the dynamic response of the disc

in the time domain

¸ Poisson’ s ratio of the disc material

¹ damping coe� cient of the disc …ˆ D¤=2D†» mass density of the disc

¼ detuning parameter

Âkl mode shape function for the transverse

vibration of the disc corresponding to qkl

!kl natural (circular) frequency

corresponding to qkl~« constant rotating speed of the disc (rad/s)

1 INTRODUCTION

There are many mechanical devices that involve a disc

and an auxiliary system in relative rotation, such as car

disc brakes, computer discs and circular saw blades.

There have also been many investigations into the

dynamic instability of such devices. Early researchincludes a stationary disc excited by a rotating load [1, 2]

or a spinning disc past a stationary load [3]. All these

studies pointed out that the system could become

unstable at some speci® c values of system parameters

(such as mass and stiŒness) and running conditions evenwhen friction between the two contacting components in

The MS was received on 24 June 1999 and was accepted after revisionfor publication on 10 February 2000.* Corresponding author: Department of Engineering, The University ofLiverpool, Brownlow Hill, Liverpool L69 3GH, UK.

65

C09099 ß IMechE 2001 Proc Instn Mech Engrs Vol 215 Part C at University of Liverpool on March 20, 2014pic.sagepub.comDownloaded from



relative rotation is absent. A spinning disc subjected to

stationary frictional load was studied by Ono et al. [4].The friction force modelled as a follower force made the

system more unstable. The transverse vibration of a

stationary disc excited by a rotating mass± spring± dam-

per system or a pad with dry friction can become

unstable under certain circumstances in the subcritical

range [5, 6]. It has been shown that the rotating frictionas a follower force is highly destabilizing. The rotating

mass is similarly destabilizing. A comprehensive survey

of friction-induced instabilities in discs can be found in

reference [7]. From previous experience on the analysis

of the dynamic instabilities of discs, it is very likely thatthe addition of one (or several) rotating mass± spring±

damper system can suppress excessive vibration when

used carefully.

The study and prevention of unstable vibration is

very important in industry. Excessive vibration cangenerate oŒensive noises and degrade the performance

of machinery. It causes major concern about the

reliability and quality of components. These practical

problems can be approximated as a rotating-load-on-

stationary-disc model or a stationary-load-on-rotating-

disc model. One way of eliminating the dynamicinstability as a result of parametric resonances in these

problems is to use stabilizers. The stabilizers are ide-

ally dampers but may contain mass and stiŒness and

even introduce friction in practice. This paper sets out

to ® nd the number of such stabilizers and their opti-mal positions so that best dynamic stability perfor-

mance can be achieved by using the genetic algorithm,

which has not been applied before to the stabilizer

optimization problem. It is found from numerical

simulations that low-friction stabilizers are very eŒec-tive in reducing vibration but high-friction stabilizers

are not.

2 FRICTION-INDUCED PARAMETRIC

RESONANCES OF THE DISC

A circular disc, modelled as a thin plate, excited by arotating mass± spring± damper system initially located at

…r; ³† ˆ …r0; 0†, and acted on by a series of rotating mass±

spring± damper systems initially located at …r; ³† ˆ…rj; ³j†, j ˆ 1; 2; . . . ; is shown in Fig. 1. The friction

between the mass± spring± damper systems and the disc ismodelled as a follower force [5]. The follower force

hypothesis means that the tangential friction force fol-

lows the deformed surface of the disc and changes its

direction when the disc vibrates. As a result, the follower

force has a transverse component when the disc deformsduring vibration. The equation of the transverse motion

of the whole system in a cylindrical coordinate system® xed to the disc centre is

»h@2w

@t2‡ D¤r4 _w ‡ Dr4w

ˆ ¡ 1

r¯…r ¡ r0†¯…³ ¡ ~« t†

£ m@

@t‡ ~«

@

@³

2

‡c@

@t‡ ~«

@

@³‡ k ¡ f

@

r@³w

¡ 1

rjˆ1

¯…r ¡ rj†̄ …³ ¡ ~« t ¡ ³j†

£ mj@

@t‡ ~«

@

@³

2

‡cj@

@t‡ ~«

@

@³‡ kj ¡ fj

@

r@³w

…1†

where the terms in the ® rst square bracket on the right-hand side of equation (1) represent the active force from

the exciter which can cause dynamic instability of the

disc. The terms in the second square bracket represent

the passive mechanisms, referred to as the stabilizers.

Though they could cause instability when wrongly used,

they are intended to stabilize the unstable vibrationcaused by the rotating friction load. Ideally, only a series

of dampers should be used to stabilize the vibrations. In

general, however, a damper may also involve some mass

and some stiŒness represented by a simple spring in this

article. Therefore, if the values of the additional massesand spring constants are ill chosen, or the positions of

these discrete mass± spring± damper systems are not

located in the right positions, they will not suppress the

unstable vibration and could well increase the instabil-

ity. As the dampers are also in contact with the discsurface, a certain amount of friction exists between the

Fig. 1 Circular disc with the exciter and the stabilizer

66 H OUYANG AND J E MOTTERSHEAD

Proc Instn Mech Engrs Vol 215 Part C C09099 ß IMechE 2001 at University of Liverpool on March 20, 2014pic.sagepub.comDownloaded from



dampers and the disc surface. This work ® rst determines

the regions of instability corresponding to diŒerentsystem parameters and running conditions and then

seeks to establish the `optimal positions’ of the stabi-

lizers so as to reduce the likelihood of dynamic

instability. The number of such stabilizers is also dis-

cussed.

The transverse motion of the disc can be expressed inmodal coordinates such that

w…r; ³; t† ˆ1

kˆ0

1

lˆ¡1Âkl…r; ³†qkl…t† …2†

where

Âkl…r; ³† ˆ 1

»hb2Rkl…r†exp…il³† …3†

The modal functions satisfy the orthonormality condi-

tions

b

a

»h ·ÂklÂrsr dr d³ ˆ ¯kr¯ls

b

a

D ·Âklr4Ârsr dr d³ ˆ !2rs¯kr¯ls

…4†

where the overbar denotes complex conjugation.

Substituting equations (2) and (3) into (1) and then

using equation (4) leads to

�qkl ‡ 2¹!2kl _qkl ‡ !2

klqkl

ˆ ¡ 1

»hb2

1

rˆ0

1

sˆ¡1Rrs…r0†Rkl…r0†exp‰i…s ¡ l † ~« tŠ

£ m… �qrs‡i2s ~« _qrs ¡ s2 ~«2qrs† ‡ c… _qrs‡is ~« qrs†

‡ k ¡ isf

r0

qrs

¡ 1

»hb2j

1

rˆ0

1

sˆ¡1Rrs…rj†

£ Rkl…rj†exp‰i…s ¡ l †… ~« t ‡ ³j†Š

£ mj… �qrs‡i2s ~« _qrs ¡ s2 ~«2qrs† ‡ cj… _qrs‡is ~« qrs†

‡ kj ¡isfj

rjqrs …5†

Equation (5) is an in® nite system of Hill’ s equations.

Therefore, only an approximate solution may be found.

The dynamic behaviour of equation (5) is determined bythe system parameters and operating conditions. The

disc can be unstable at some particular values of the

system parameters and operating conditions in thesubcritical speed range because the rotating friction is

represented as a follower force. The instability thus

caused is also referred to as a parametric resonance since

it is caused by cyclical variation of the system para-

meters and not by an external applied load.

3 METHOD OF MULTIPLE SCALES

When any one of the parameters of the rotating mass±

spring± damper systems after scaling is very small, a

perturbation method may be used, which can reduce the

amount of computation that would be necessary by

using other methods such as the state-space method.

The method of multiple scales [8] is particularly suitablefor solving the above problem, and similar problems

[5, 9].

To use the method of multiple scales, the following

new variables are introduced so that a common small

parameter can be extracted from equation (5) which canthen be simpli® ed:

½ ˆ !crt; kl ˆ !kl

!cr; « ˆ

~«

!cr

…6†

where

!cr ˆ min!kl

l; k ˆ 0; 1; 2; . . . ; l ˆ 1; 2; . . . …7†

Substitution of equations (6) and (7) into equation (5)

yields

d2qkl

d½2‡ 2¹!cr

2kl

dqkl

d½‡ 2

klqkl

ˆ ¡1

rˆ0

1

sˆ¡1Rrs…r0†Rkl…r0†exp‰i…s ¡ l †«½Š

£ m

»hb2

d

d½‡ is«

2

qrs‡c

»hb2!cr

d

d½‡ is« qrs

‡k

»hb2!2cr

¡isf

r0»hb2!2cr

qrs

¡j

1

rˆ0

1

sˆ¡1Rrs…rj†Rkl…rj†exp‰i…s ¡ l †…«½ ‡ ³j†Š

£ mj

»hb2

d

d½‡ is«

2

qrs ‡ cj

»hb2!cr

d

d½‡ is« qrs

‡ kj

»hb2!2cr

¡ isfj

rj»hb2!2cr

qrs …8†

A small perturbation parameter, ", is introduced withthe purpose of scaling the system parameters in equation

OPTIMAL SUPPRESSION OF PARAMETRIC VIBRATION IN DISCS 67




(8) and writing it in an amenable form for a perturbation

approach. It follows that

"m" ˆ m

»hb2; "c" ˆ c

»hb2!cr; "k" ˆ k

»hb2!2cr

;

"f" ˆ f

»hb2!2cr

; "¹" ˆ ¹!cr "m"j ˆ mj

»hb2;

"c"j ˆ cj

»hb2!cr; "k"j ˆ kj

»hb2!2cr

; "f"j ˆ fj

»hb2!2cr

…9†

and equation (8) becomes

d2qkl

d½2‡ 2"¹"

2kl

dqkl

d½‡ 2

klqkl

ˆ ¡1

rˆ0

1

sˆ¡1Rrs…r0†Rkl…r0†exp‰i…s ¡ l †«½Š

£ "m"d

d½‡ is«

2

qrs ‡ "c"d

d½‡ is« qrs

‡ " k" ¡ isf"

r0

qrs

¡j

1

rˆ0

1

sˆ¡1Rrs…rj†Rkl…rj†exp‰i…s ¡ l †…« ½ ‡ ³j†Š

£ "m"jd

d½‡ is«

2

qrs ‡ " c"jd

d½‡ is« qrs

‡ " k"j ¡isf"j

rjqrs …10†

This complicated system of equations can be best han-

dled by the method of multiple scales [8]. Note that,

when the small parameter is introduced in this paper, it

is tacitly assumed that all the right-hand side quantitiesin equation (9) are small. If any one of them is not small,

that system parameter should be left intact so that the

small parameter is not used to scale it down. In that

case, the same quantity will appear in the zero-order

version of equation (10) after the expansion of qkl interms of ". On the other hand, if none of the right-hand

side terms is small, usually a perturbation method like

the method of multiple scales is not valid. In that case,

the method of state space can be used, which results in a

much heavier computing load.

The basic idea of the method of multiple scales is tode® ne diŒerent time-scales as multiples of the integer

powers of ". Thus

T0 ˆ ½; T1 ˆ " ½; T2 ˆ "2½; . . . …11†

and a solution, qkl is sought in the form

qkl ˆ q…0†kl ‡ " q

…1†kl ‡ "2q

…2†kl ‡ ¢ ¢ ¢ …12†

When equations (11) and (12) are introduced into (10)and the resultant equation is separated into the various

orders of smallness, a series of new equations for dif-

ferent orders of " can be derived (for details, see refer-

ences [5] and [9]). Notably among them, the ® rst-order

equation (after dropping the subscript ") is

D20q

…1†kl ‡ 2¹ 2

klD0q…1†kl ‡ 2

klq…1†kl

ˆ ¡2…D0 ‡ ¹ 2kl†D1q

…0†kl ¡

1

rˆ0

1

sˆ¡1Rrs…r0†Rkl…r0†

£ exp i…s ¡ l †«½Š‰m…D0‡is«†2 ‡ c…D0‡is«†

‡ k ¡ isf

r0

q…0†rs

¡j

1

rˆ0

1

sˆ¡1Rrs…rj†Rkl…rj†exp‰i…s ¡ l †…«½ ‡ ³j†Š

£ mj…D0‡is«†2 ‡ cj…D0‡is«† ‡ kj ¡isfj

rjq…0†

rs

…13†

where

D0 ˆ d

dT0; D1 ˆ d

dT1

From previous work [5] it can be seen that single-mode

resonances close to

2l« ˆ 2 kl; l > 0

and combination resonances close to

…s § l †« ˆ rs § kl; s > l; l50

are likely to appear. The formula for only the following

combination resonances are presented in this investiga-tion:

…s ‡ l †« ˆ rs ¡ kl; s > l; l50

In this resonance range,

…s ‡ l †« ˆ rs ¡ kl ‡ "¼; s > l; l50 …14†





and the equation for determining the characteristic

exponent l is

2 rs…l ‡ ¼† ¡ 2i¹ 3rs ¡ R2

rs…r0† ·D¡0rs ¡

j

R2rs…rj† ·D¡j

kl

£ 2 kll ¡ 2i¹ 3kl ¡ R2

kl…r0†D‡0kl ¡

j

R2kl…rj†D‡j

kl

¡ Rkl…r0†Rrs…r0† ·D¡0rs ‡

j

Rkl…rj†Rrs…rj† ·D¡jrs

£ exp‰¡i…s ‡ l †³jŠ

£ Rkl…r0†Rrs…r0†D‡0kl ‡

j

Rkl…rj†Rrs…rj†D‡jkl

£ exp‰i…s ‡ l †³jŠ ˆ 0 …15†

where

D‡0kl ˆ ¡m… kl ‡ l«†2 ‡ i‰c… kl ‡ l«† ¡ lf Š ‡ k

D¡0kl ˆ ¡m… kl ¡ l«†2 ¡ i‰c… kl ¡ l«† ‡ lf Š ‡ k

D‡jkl ˆ ¡mj… kl ‡ l«†2 ‡ i‰cj… kl ‡ l«† ¡ lfjŠ ‡ kj

D¡jkl ˆ ¡mj… kl ¡ l«†2 ¡ i‰cj… kl ¡ l«† ‡ lfjŠ ‡ kj

…16†

As mentioned before, diŒerent values of the system

parameters and operating conditions (parameter) bring

about diŒerent dynamic behaviour in terms of dynamic

stability. It is a common practice to show the areaswhere instability appears on a parameter plane, for

example, on a plane where the rotating speed of the disc

is the ordinate and the friction force is the abscissa.

When the areas of instability (referred to as regions of

instability) on each plane of every pair of system para-meter± operating conditions are obtained, a global pic-

ture of the dynamic stability in¯ uenced by all the system

parameters and operating conditions can be conceived.

The condition where a pair of l have purely real roots

de® nes the transition curves which divide the parameter

plane into regions of stability and regions of instability.The regions of instability have a characteristic wedge-

like appearance [5] in the parameter plane, the areas of

which can be characterized by their width "¼ at a pre-

scribed value of the system parameter (abscissa) con-

cerned. Therefore, the total width of all the regions ofinstability on a parameter plane, denoted by "¼, is a

measure of the degree of dynamic instability of the

whole system. Usually, the regions of instability on a

parameter plane overlap. In that case, "¼ is not the

simple sum of the width of each individual region ofinstability. Rather, it is the overall width of all the

combined regions of instability in that plane. Another

phenomenon should also be noted, namely that, whenthe width of an individual region of instability decreases,

the tip of this wedge-like region tends to move to the

right, resulting in a smaller area, i.e. reduced instability.

4 EFFECT OF THE STABILIZERS

Without the stabilizers, the regions of instability of thedisc excited by the rotating exciter can be quite large,

depending on the values of the system parameters and

operating conditions. Adding additional dampers alone

can reduce instability. Here, it is intended to investigate

how additional mass± spring± damper stabilizers aŒect

the dynamic instability and how instability can be

reduced by optimal positioning of these stabilizers. Thegenetic algorithm (GA) is suitable for this purpose.

The GA is a numerical search technique simulating

the process of natural evolution. A feasible solution is

represented by a binary string, which is analogous to achromosome in a biological system [10]. The genetic

algorithm works with a population of such strings.

Within an evolution cycle, there are three basic opera-

tions on these strings: reproduction, crossover and

mutation. A speci® c example will be used to demon-strate how it is used.

The surface of the annular disc is conceptually

discretized into 4 £ 32 small cells by a uniformly dis-

tributed mesh of diameters and circles, as illustrated in

Fig. 2. Another way of describing the conceptually dis-

cretized disc is that it consists of four concentric circles

(circles 1 to 4 from the inner radius to the outer radius)of cells. As there are 128 ˆ 27 cells altogether, each cell

can be represented by a 7-digit binary string. Suppose

three stabilizers are used, which are located at the cen-

tres of three of these cells. The three 7-digit strings,

which represent the three cells the stabilizers occupy, are

Fig. 2 Annular disc conceptually discretized into four circles

of cells





The intention is to study the regions of instability of the

disc under the in¯ uence of the exciter and the stabilizers

and then determine the optimal positions of the stabi-

lizers so as to have the smallest regions of instability, i.e.to minimize "¼. Here, a very brief description is given

of how the GA is used for this purpose.

Initially, choose randomly a number of 7-digit binary

strings and form a set of 21-digit binary strings in the

case of three stabilizers. Calculate "¼ for all possiblecombination resonances from equation (15) according

to the positions that every 21-digit binary string repre-

sents and add them together to get "¼. Those strings

resulting in small "¼ are deemed ® t for subsequent

reproduction:

1. Reproduction. This is a process where the ® t strings

are incorporated into the mating pool, in which the

number of appearances of the ® t strings is deter-

mined by the weighted roulette wheel rule [10].

2. Crossover. The strings deemed ® t for reproductionin the mating pool are randomly chosen to form

couples. Each couple exchanges part of their strings

to produce oŒspring.

3. Mutation. Mutation modi® es a small fraction of a

string, ¯ ipping 0 and 1 in that part of the string. Theauthors’ experience shows that, without mutation,

the search would usually converge to a local optimal

solution in a smaller part of the solution space,

which is spanned by the strings produced in previous

evolution cycles.

In a typical application there is a cost function which

characterizes the performance or the crucial attribute

(for example, the weight of an aircraft, the shortest route

to a destination and so on) of a system. The goal of

optimization is to minimize the cost function, "¼ inthis investigation. Those strings leading to small "¼are considered ® t and retained for reproduction. The ® t

parent strings tend to produce ® t and even ® tter oŒ-

spring strings. However, without mutation, the ® t parent

strings can only produce the ® ttest oŒspring stringwithin the solution space determined by the parent

strings. In so doing, only a local optimum is ensured

since the ® rst batch of parent strings is randomly chosen

and usually does not span the space that happens to

cover the global optimum. Therefore, mutation is

necessary to achieve a global optimum. On the other

hand, mutation and crossover may produce less ® t oŒ-

spring strings occasionally. An `orthodox’ genetic algo-rithm would not do anything about the problem since,

in the long run, un® t strings are sieved out and dis-

carded. However, computation with un® t strings takes

longer and slows down convergence to the global opti-

mum. As a result, most GA users tend to revise the pureGA to suit their need.

In this investigation, a record of poor strings pro-

duced in a previous cycle is kept and, if the poor strings

are reproduced in the next few cycles, they are discarded

and additional crossover and/or mutation operations

are performed to produce ® t strings. This record isconstantly updated. This is one of the many ideas

embraced in the so-called tabu search method [11].

Thus, a computation cycle (generation) consists of

reproduction, crossover and mutation. The reproduc-

tion is really a process of selection of ® t strings. Here,both the weighted roulette rule of an ’orthodox’ GA and

an empirical rule based on the knowledge gained about

the in¯ uence of the stabilizers are used in the selection.

The latter may be considered as a type of tabu search.

The outcome is a reduction in the number of compu-tation cycles, though each cycle becomes slightly more

complicated than when the knowledge about the speci® c

problem is not applied. Incidentally, the initial popula-

tion of the strings is also checked for ® tness. There are

other versions of the GA. When the knowledge about a

problem can be used to guide the search, a large savingcan be achieved. It is reported that, in one particular

application, the computation is reduced by 97 per cent

[12] using the GA combining speci® c knowledge about

the subject matter. Another example shows that the GA

can handle very complex optimization problems [13].The advantage of the GA over purely random search

techniques lies in its guidance from the evolution prin-

ciples. As ® tter oŒspring from the genetic algorithm

evolve, "¼ is reduced. In the end, a global minimum

of "¼ is found. If the GA is left on its own withoutintervention, the cost value would frequently ¯ uctuate,

though the general trend would be towards con-

vergence. Fluctuation of the cost value means extra

search eŒort.

…r; ³† ˆ 0:0875m;3

16p 0:0745 m;

1

2p …0:1135 m; p†

7-digit basic strings 0010001 0100100 1000111

The combined string 0010001010001001000111

linked to form a single 21-digit binary string. One

example of such a combined string is shown below:





5 NUMERICAL EXAMPLES AND ANALYSIS

To see how the additional stabilizers aŒect the dynamic

instability of the disc, four case studies are presented

here for a disc clamped at the inner radius and free at the

outer radius. The principal dimensions and properties of

the disc are a ˆ 0:067 m, b ˆ 0:12 m, h ˆ 0:01 m,

E ˆ 120GPa, ¸ ˆ 0:25, » ˆ 7000 kg=m3 and ¹ ˆ 0, and

for the exciter "m ˆ 0:24, "c ˆ 5 £ 10¡5, "k ˆ 0:7 and

"f ˆ 5 £ 10¡6 m (which amounts to a friction force of

about 200 N). Note that the unit of the friction force

parameter is the unit of length because of the way this

parameter is scaled in equation (9). The natural fre-

quencies (rad/s) and normalized frequencies of the disc

from the classical thin plate theory are given in Table 1.

It should be stated that the fundamental modes of the

disc clamped at the inner radius and free at the outerradius have zero nodal circles. The ® rst mode with

one nodal circle (without a nodal diameter) corresponds

to the eleventh distinct frequency of the disc as

93 642 rad/s, which is just over 89 638 rad/s for the

mode with nine nodal diameters and no nodal circles.Additionally, modes with nodal circles alone will not

cause parametric resonances in the disc [5, 9]. In other

words, parametric resonances of a disc under a rotating

follower force always involve at least one nodal dia-

meter mode.In order that the dynamic instability for an exciter

without stabilizers can be compared with subsequent

cases when both the exciter and stabilizers are present,

the regions of instability of the former are shown in

Figs 3a and b (all the regions of instability are the

wedge-like areas enclosed by the curves shown in therelevant ® gures) for r0 ˆ 0:1 and 0.1135m, and "¼ is

0.3046 rad/s and 0.7136 rad/s respectively. It is clear

that, the further away from the disc centre, the more

destabilizing the friction force is. This is because the

modes of the disc have greater de¯ ection at greaterradial position when the disc is clamped at the inner

radius but free at the outer radius. For other boundary

conditions, the conclusion might be quite diŒerent.

Case 1. The data for the frictionless stabilizer(s) are

"mj ˆ 0:1, "cj ˆ 10¡5, "kj ˆ 0:01, "fj ˆ 0, r0 ˆ 0:1 m.The question will arise as to what happens when the

stabilizers bring in friction. In practice, friction between

the stabilizers and the disc surface is unavoidable owing

to sliding contact. It seems more realistic to include

friction in the stabilizers, which is studied in cases 2 to 4.

Case 2. The friction for the stabilizer(s) "fj ˆ 10¡6 m.

Other data remain the same as in case 1.

Case 3. The friction for the stabilizer(s) "fj ˆ 1:5£10¡6 m. Other data remain the same as in case 1.

Case 4. The friction for the stabilizer(s) "fj ˆ 2 £ 10¡6

(m). Other data remain the same as in case 1.

Before presenting the results, some terms to be used to

describe them should be clari® ed. A solution is a set of

positions on the disc surface that the stabilizers occupy.

An optimal solution means that the best performance

(explained later) is achieved where the stabilizer or sta-

bilizers occupy an optimal set of positions. Because ofdisc symmetry about the diameter passing through the

exciter, the optimal solutions are always in pairs and are

symmetric to the angular position of the exciter. This

symmetry is exploited in choosing the ® tter oŒspringstrings by the genetic algorithm to reduce computing

eŒort.

The optimal location for a single stabilizer is ® rst

sought and knowledge about its in¯ uence on the stabi-

lity of the disc is gained and can be used to help select

better solutions for the two- and three-stabilizer pro-blems. The initial population of the combined strings is

12 in some problems and 16 in others. For problems

with only one stabilizer, optimal solutions are obtained

after 12± 20 cycles of computation, depending on how ® t

the initial, randomly selected strings are. Since there are

62 feasible solutions for single-stabilizer problems, usingthe GA may not be worthwhile. Actually, the merit of

the GA can only be seen when the number of feasible

solutions is large. For problems with three stabilizers,

there are 249 984 feasible solutions. Convergence to aglobal optimum is achieved after generations of about

70± 120, when the record of poor strings is consulted and

previous knowledge about the stability of the disc is

used. For example, it is already known that, the further

away from the centre of the disc a friction force lies, the

more unstable the disc is made to be. However, itbecomes complicated when mass, spring stiŒness and

damping are all involved in the stabilizer, as these have

diŒerent eŒects on the stability of the disc.

There can be more than one pair of optimal solutions

for a speci® c group of system parameters and operating

conditions. The best performance is either the smallest

"¼ or a complete elimination of instability up to the

exciter friction level of " f ˆ 5 £ 10¡6 m. The latter

always occurs in the cases of more than one pair of

optimal solutions. The numbers of optimal solutionsand the corresponding "¼ for diŒerent numbers of

stabilizers are summarized in Table 2. Intuitively, opti-

mal positions may seem to be at circle 4. This pre-

sumption is partly true when the friction level of the

stabilizer(s) is very low. Even in those cases, there may

be optimal positions other than at the outer radius of thedisc. When the friction level of the stabilizers is high, the

Table 1 Natural frequencies (rad/s) and normalized fre-

quencies

…k; l† (0,0) …0; §1† …0; §2† …0; §3† …0; §4†

!kl 14 271 14 633 16 072 19 367 25 103 kl 2.27 2.33 2.56 3.09 4.00





optimal positions are at circle 1, where "¼ changes

very slightly compared with "¼ without the stabi-

lizers. The optimal positions for one stabilizer at dif-

ferent friction levels are given in Table 3.

Numerical results show that the numbers of optimalsolutions and the positions of them depend on the fric-

tion level of the stabilizers (when the friction level of the

exciter is ® xed at " f ˆ 5 £ 10¡6 m). When using one

frictionless stabilizer, there are three pairs of optimal

solutions, two of them on circle 4 and the third on circle3. Actually, one stabilizer on circle 4 is su� cient for this

purpose. At a low friction level of "fj ˆ 10¡6 m, optimal

solutions include only positions on circle 4. Case 3 yields

similar results as in case 2. However, as the friction level

of the stabilizers increases further, their stabilizing eŒect

diminishes. Since the further away from the disc centre

the more destabilizing the friction, at the higher friction

level of "fj ˆ 2 £ 10¡6 m in cases 4, the optimal positions

for the stabilizers are on circle 1 (close to the clamped

inner radius of the disc) and they are no longer eŒectivein curbing the vibration of the disc where they are close

to the inner radius.

Whenever the optimal positions are thought to be

located on circle 1 or 4, the search for optimal positions

of the stabilizers can be limited there. Then the searchspace is considerably smaller and the convergence is

much faster.

From Table 3 it can be seen that, at a low friction

level, it would nearly always be best to put stabilizers on

the outer radius of the disc, as in case 2. When the level

Fig. 3 Regions of instability without stabilizers: (a) r0 ˆ 0:1 m, "¼ ˆ 0:3046 rad/s; (b) r0 ˆ 0:1135m,

"¼ ˆ 0:7136 rad/s





of friction is not very low, the situation becomes com-

plicated. This is because the friction of the stabilizers,

when located at the outer radius, has a larger destabi-

lizing eŒect than inside the disc surface, and the aggre-gate results of the stabilizing eŒect of dampers of the

stabilizers and the destabilizing eŒect of friction of the

stabilizers determine where the best places are to put

those stabilizers when friction is present. The implica-

tion is that the search for optimal positions of the sta-bilizers has to be conducted on the whole disc surface.

It can be noted that, if the stabilizers are frictionless,

the regions of instability are reduced wherever they are

except close to the clamped inner radius (though they

may not achieve best performance). Figure 4 shows sucha case … "¼ ˆ 0:2217 rad/s) when a single frictionless

stabilizer resides at …r1; ³1† ˆ …0:1135 m, 0). On the other

hand, if the stabilizers introduce friction and even if the

friction level is low, the regions of instability may be

increased when the stabilizers are wrongly located.Figure 5 illustrates one such example … "¼ ˆ

Fig. 4 Regions of instability with one frictionless stabilizer: r0 ˆ 0:1 m, "¼ ˆ 0:2217 rad/s

Table 2 Optimal solutions

Friction level (m) One stabilizer Two stabilizers Three stabilizers"fj ˆ 0 Three pairs of Many pairs of Many pairs of

optimal solutions optimal solutions optimal solutions"¼ ˆ 0 "¼ ˆ 0 "¼ ˆ 0

"fj ˆ 10¡6 Two pairs of Many pairs of Many pairs ofoptimal solutions optimal solutions optimal solutions

"¼ ˆ 0 "¼ ˆ 0 "¼ ˆ 0

"fj ˆ 1:5 £ 10¡6 One pair of Many pairs of Many pairs ofoptimal solutions optimal solutions optimal solutions

"¼ ˆ 0 "¼ ˆ 0 "¼ ˆ 0

"fj ˆ 2 £ 10¡6 One pair of One pair of One pair ofoptimal solutions optimal solutions optimal solutions

"¼ ˆ 0:3087rad/s "¼ ˆ 0:3100rad/s "¼ ˆ 0:3173rad/s

Table 3 Optimal positions for a single stabilizer

Optimal positions "fj ˆ 0 (m) "fj ˆ 10¡6 (m) "fj ˆ 1:5 £ 10¡6 (m)

…rj; ³j† (0.1135m, § 18 p† …0:1135m; § 1

8 p† (0.1135m § 18 p†

(0.1005m, § 18 p† (0.1135m, § 3

16 p†

(0.1135m, § 316 p†





0:5921 rad/s) when two stabilizers with friction of

"fj ˆ 10¡6 m reside at …rj; ³j† ˆ …0:1135 m, 14 p† and

(0.1135 m, 916 p†.

Finally, it should be stressed that the determination of

the regions of instability for each placement of stabi-

lizers takes time. Without using the GA, it would be very

time consuming to ® nd an optimal solution and it is

necessary to make do with a reasonably good solution (alocal minimum, for example). When the number of cells

increases, the advantage of the GA becomes even more

obvious.

From the above analysis it can be seen that adding

stabilizers on to the disc surface as a way of reducing the

instability should be done carefully. The level of friction

between the stabilizers and the disc will aŒect where theyshould be located and if they are an eŒective measure. If

wrongly added, they can destabilize rather than stabilize

the disc when friction is present.

The present work can be extended to friction-inducedparametric resonances of a disc with a negative friction±

velocity slope [14]. Other possible extensions are ® nding

optimal solutions of both the optimal positions and

optimal parameter values of the stabilizers, and even

controlling the dynamic behaviour of the disc. Ofcourse, both optimal positions of the stabilizers and

optimal values of the system parameters can be included

in the optimization process.

6 CONCLUSIONS

In this paper, the formulation of the multiple scales

method is presented for analysing the parametric

resonances of a disc excited by an exciter ofmass± spring± damper with friction and a series of mass±

spring± damper stabilizers with friction or without fric-

tion. The genetic algorithm is used to ® nd out the

optimal positions of the stabilizers in order to reduce oreven suppress the instability at diŒerent friction levels:

1. At low levels of friction the stabilizers are very

eŒective in reducing or even eliminating the dynamic

instability. One stabilizer at the outer radius of the

disc is usually su� cient.2. At higher friction levels the stabilizers are useless.

They increase the regions of instability when put

anywhere except close to the inner radius of the disc.

3. The optimal positions of the stabilizers are at the

outer radius on many occasions but can be else-where. When it is thought that the optimal positions

are at the outer radius, the search should be

conducted on the outer radius and the convergence

will be much faster.

ACKNOWLEDGEMENTS

This investigation is supported by the Engineering andPhysical Sciences Research Council (grants L00322 and

L91061), BBA Friction Limited and LucasVarity plc.

The authors are grateful to Dr Xiaojian Liu of the

University of Portsmouth for helpful discussion on the

genetic algorithm.

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Fig. 5 Region of instability with two stabilizers: "f1 ˆ "f2 ˆ 10¡6 m, r0 ˆ 0:1 m, "¼ ˆ 0:5921 rad/s





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Optimal suppression of parametric vibration in discs under rotating frictional loads