Prediction and assignment of latent roots of damped asymmetric systems by structural modifications

ARTICLE IN PRESS Mechanical Systems and Signal Processing 23 (2009) 1920–1930

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Prediction and assignment of latent roots of damped asymmetric systems by structural modifications Huajiang Ouyang  Department of Engineering, University of Liverpool, Brownlow Street, Liverpool L69 3GH, UK

a r t i c l e i n f o

abstract

Article history: Received 21 April 2008 Received in revised form 29 July 2008 Accepted 4 August 2008 Available online 7 August 2008

This paper studies the latent roots of damped asymmetric systems in which the stiffness matrix is asymmetrical. The asymmetric terms are due to ‘external’ loads and are represented by a parameter or parameters. The latent roots of such asymmetric systems are complex and the real parts become positive at some critical values of the parameter(s) (critical points). The work reported in this paper consists of two parts. The first part presents a method for predicting the latent roots of the damped asymmetric system from the receptance of the damped symmetric system. The second part presents an inverse method for assigning latent roots by means of mass, stiffness and damping modifications to the damped asymmetric system again based on the receptance of the unmodified damped symmetric system. The simulated numerical examples of a friction-induced vibration problem show the complexity in assigning stable latent roots for damped asymmetric systems. It is found that it is quite difficult to assign the real parts of latent roots to stabilise the originally unstable asymmetric system and sometimes there is no solution to the modification that is intended to assign certain latent roots. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Asymmetric system Damped Latent roots Prediction Assignment Structural modification Friction-induced vibration Flutter Instability

1. Introduction The mass matrix and stiffness matrix of engineering structures can be assumed to be symmetrical, respectively, positive definite and semi-positive definite in general. The damping matrix of these structures is also symmetrical and at least semi-positive definite. The eigensolutions of such structures are well studied and the vibration of such structures is stable, apart from a few rigid-body modes when the stiffness matrix is only semi-positive definite. There are, however, engineering problems whose stiffness matrices are asymmetrical. Usually asymmetry is produced not by the structure itself, but by some external loads [1] interacting with the structure, such as friction in brakes noise problems [2] or airflow in aeroelastic flutter problems [3]. These external loads acting on a structure are considered to be internal to the system that contains the structure. In this paper, a structure is understood to be a building or a machine while a system is considered to include a structure and the forces acting on the structure. In linear or linearised models of friction-induced vibration, friction introduces asymmetric terms into the stiffness matrix [4–11]. Hoffmann et al. [6] demonstrated that as an off-diagonal stiffness element proportional to the friction coefficient increased two pairs of purely imaginary eigenvalues would coalesce (the two frequencies became identical) at a critical value and become complex with one pair having positive real parts above this critical value in their two-degrees-offreedom undamped system with sliding friction. Hoffman and Gaul [10] went on to show that viscous damping could  Tel.: +44 151794 4815; fax: +44 151794 4848.

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destabilise the friction-induced vibration of a similar two-degrees-of-freedom system. The perturbation method presented by Huang et al. [8] was able to predict complex eigenvalues and the critical friction coefficient from the frequencies and modes of the symmetric system. However the formulas are very complicated. Flutter instability in the form of complex eigenvalues with positive real parts is assumed to signify squeal noise in frictional brakes. The standard approach for brake squeal analysis in automotive industry [2] is computing many complex eigenvalues at gradually varying values of interesting system parameters [12]. Needless to say this is a very expensive process. This paper presents a method for determining the latent roots of the damped asymmetric system based on the receptance of the damped symmetric system, which is easier to obtain. The same method is developed to assign latent roots of the damped asymmetric system by means of mass, stiffness and damping modifications to the asymmetric system when it is unstable. As the eigenvalue problem and the receptance of the symmetric system are usually well understood and much easier to solve and even to measure than the asymmetric system, this receptance-based method has a big advantage. The idea of determining the solutions of a modified system from the receptance of the original system has been used in the context of structural modifications [13–16] and structural vibration control [1,17]. However, assignment of the real parts of latent roots of asymmetric systems proves more difficult to achieve. Shi et al. [18] performed structural design optimisation for suppressing disc brake squeal (believed to be the result of friction-induced vibration), in which 46 1 g masses were evenly distributed on the back of the friction pads and they were able to shift all the unstable latent roots (complex eigenvalues) into the left half of the complex eigenvalue plane. Chu [19] recently presented a robust algorithm for pole assignment of second-order damped systems with a symmetric stiffness matrix through feedback control. Chu also outlined the approach to pole assignment for damped asymmetric systems when there are no degenerate eigenvalues. While the poles in that study were all stable (with negative real parts), this paper deals with unstable latent roots (equivalent to poles in control theory) and re-assigns them by structural modifications to produce a stable system. 2. Asymmetric stiffness matrices The free vibration problem of a structure may be written as 2

ðK þ l MÞu ¼ 0

(1)

where M and K are mass and stiffness matrices. When damping is considered, the free vibration problem becomes 2

ðK þ lC þ l MÞu ¼ 0

(2) 2

where C is damping matrix. Usually, l in Eq. (1), which is a linear eigenvalue problem, and l in Eq. (2), which is a quadratic eigenvalue problem, are usually both called an eigenvalue. To add confusion, if K is asymmetric, l can be complex. In order to help subsequent discussion, l and u in both equations are called the latent root and latent vector, respectively. This terminology was due to Lancaster [1]. Two typical ways of introducing friction into a system were shown in [2,5,6]. It is clear that both ways introduce asymmetry in the stiffness matrix. As friction coefficient m increases, the asymmetric terms become greater and there can be two scenarios whereby the latent roots gain positive real parts. In the first scenario, a pair of conjugate latent roots cross the vertical imaginary axis, as shown in the simulated examples later in the paper and also illustrated by Huseyin [20] two decades ago. In the second scenario, two conjugate pairs of the latent roots are getting closer and closer and eventually coalesce when the real parts become non-zero (half of them become positive and the other half become negative) [6]. Flutter instability sets in at this point, known as a critical point [20]. This mechanism whereby friction causes flutter instability has been known in the study of brake squeal as mode-coupling [8–10]. Huseyin [20] made extensive studies of stability of various types of low-degrees-of-freedom dynamic systems. The focus in this paper is on damped systems with an asymmetric stiffness matrix. As the major issue in asymmetric systems is the lack of stability, assignment of the real parts to produce stable systems is much more important than assignment of frequencies, which has been studied extensively for symmetric systems. The asymmetry comes from the interaction between the external loads and the structure involved and the problem is selfexcited vibration. As external excitation is usually not a big issue in self-excited vibration problems, assignment of frequencies in these problems are not as important as assignment of the real part to steer an unstable system into the stable territory. This proves to be quite difficult as simulated examples presented later in the paper will show. The main reasons are due to the fact that the real part of a latent is insensitive to the change in mass and stiffness while the imaginary part is insensitive to change in structural damping. As a first attempt to assign latent roots for the sake of system stability, only simple, local mass, stiffness or damping modifications are explored in the paper. More sophisticated techniques of structural modifications can be found in [13,16,21,22]. 3. Prediction of latent roots It is fairly easy to determine the latent roots of a simple system with a small number of degrees-of-freedom, like the one studied in [6,10], which admits close-form solutions. For systems with a fairly large number of degrees-of-freedom,

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however, numerical methods must be used and this is a very costly process. For each value of interesting system parameters, a complex eigenvalue analysis for the whole asymmetric system must be carried out [12]. This paper provides a more efficient method for predicting the latent roots of the asymmetric system. Although friction generates asymmetric terms in the stiffness matrix, it should be recognised that the number of such asymmetric terms is very small in comparison with the total number of degrees-of-freedom in a system. Among the total 180,000 degrees-of-freedom in the Mercedes vented disc brake system model [7,23], there are only about 1600 frictionaffected degrees-of-freedom. While among the total 37,100 degrees-of-freedom in the Mercedes solid disc brake system model [24], there are only about 900 friction-affected degrees-of-freedom. Even in the matrix block that corresponds to the friction-affected degrees-of-freedom, there can be a large number of zeros.

3.1. Basic formulation The equation of motion of damped system with frictional stiffness can be written as Mx€ þ Cx_ þ ðK þ

n X

mi kci Ei Þx ¼ f

(3)

i¼1

where n is the number of friction-affected degrees-of-freedom in the stiffness matrix, mi and kci are the friction coefficient and contact stiffness of the ith frictional stiffness term, and matrix Ei represents the locations of the friction forces, x and f are, respectively, the displacement vector and the force vector (other than friction). It must be pointed out that Eq. (3) can represent a variety of applications [1,20], not limited to friction-induced vibration. Laplace transform of Eq. (3) is ½ðMs2 þ Cs þ KÞ þ

n X

mi kci Ei XðsÞ ¼ FðsÞ

(4)

i¼1

where s is Laplace constant, X and F are Laplace transforms of x and f. Eq. (4) may be re-written as ½ðK þ sC þ s2 MÞ þ

n X

mi kci epi eTqi XðsÞ ¼ FðsÞ

(5)

i¼1

where ej ¼ f 0

0

0

:

:

0

:

0

0

1

0

0

0

:

:

:

0

0

0gT

is a vector whose elements are all zero except its jth element that is 1. pi and qi (piaqi) are the row and column of the ith asymmetric term. Multiplying both sides of Eq. (5) by the receptance matrix H(s) ¼ (K+sC+s2M)1 yields " Iþ

n X

# T i kci HðsÞepi eqi

m

XðsÞ ¼ HðsÞFðsÞ

(6)

i¼1

where I is identity matrix having the same dimension as M. As mentioned before, the number of friction-affected degrees-of-freedom n is much smaller than the total Pn T number of degrees-of-freedom. This means that matrix i¼1 mi kci HðsÞepi eqi have only n columns of non-zero elements. When all the friction-affected degrees-of-freedom are gathered sensibly in forming the stiffness matrix K, only an n  n matrix fully determines the transfer functions of those relevant elements of X. The method is coded in MATLAB for both symbolic and numerical computations and the application of the method is demonstrated below through simulated examples.

3.2. Application of the method A simulated example is shown in Fig. 1, which may be considered an extension to the two-degrees-of-freedom system studied by Hoffmann et al. [6]. The system has four masses with m1 having a degree-of-freedom in the x direction, m4 having a degree-of-freedom in the y direction, and m2 and m3 having degrees-of-freedom in both directions. The belt moves at a constant speed. The sliding friction at the slider–belt interfaces is governed by Coulomb friction whose static and kinetic friction coefficients are taken to be the same. This is a simplification and avoids stick-slip vibration from happening. It is important to note that damping is very important in an asymmetric system. In a symmetric system, damping leads to negative real parts in the latent roots and makes a system more stable. In an asymmetric

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k8 m4 k5

c3

k6 k7 k3

k2

k1 m1

k9

k10 k4

m2

m3

c1 c0

c2 kc

n1

kc

f1

n2

f2

Fig. 1. An asymmetric system of friction-induced vibration.

system, however, damping has been found to be destabilising (increasing the real parts of latent roots) in general [10,11]. The mass, damping and stiffness matrices corresponding to displacement vector x ¼ {x1 y4 x2 x3 y2 y3}T are 2

2

3

m1

6 6 6 6 6 6 M¼6 6 6 6 6 4

m4 m2 m3 m2 m3

k11

k13

6 6 6 6 6 k13 6 K¼6 6 6 6 6 4

k22

k25

k25 k33

k34

k34

k44

k35

k35 k55

k46

7 7 7 7 7 7 7; 7 7 7 7 5

2

c1 6 6 6 6 6 c1 6 C¼6 6 6 6 6 4

3

c1 0 c1 þ c2

c2

c2

c2 c0 c3

3

7 7 7 7 7 7 7, 7 7 7 7 5

7 7 7 7 7 7 7 k46 7 7 7 7 5 k66

where k11 ¼ k1 þ k2 ; k34 ¼ k3 ;

k13 ¼ k2 ;

k22 ¼ k6 þ k8 ;

k35 ¼ 0:5ðk7  k5 Þ;

k55 ¼ kc þ k6 þ 0:5ðk5 þ k7 Þ;

k25 ¼ k6 ;

k44 ¼ k3 þ k4 þ 0:5k10 ;

k33 ¼ k2 þ k3 þ 0:5ðk5 þ k7 Þ, k46 ¼ 0:5k10 ,

k66 ¼ kc þ k9 þ 0:5k10

In the example, mi ¼ m(i ¼ 1, 2, 3, 4), ci ¼ c(i ¼ 0, 1, 2, 3), ki ¼ k(i ¼ 1, 2, 3, 4, 6, 7, 8, 9), ki ¼ 0.5k(i ¼ 5, 10). kc ¼ 1.1k. m ¼ 1 kg, c ¼ 0.5 Ns/m and k ¼ 100 N/m. f1, n1, f2 and n2 are, respectively, the friction force and (pre-compression) normal force acting at the slider–belt interfaces. The asymmetric part of the stiffness matrix is given by mkcE where 2

0 60 6 6 60 E¼6 60 6 6 40 0

3

0

0

0

0

0

0

0

0

0

0

0

0

1

0 0

0 0

0 0

0 0

07 7 7 07 7 17 7 7 05

0

0

0

0

0

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Table 1 Latent roots of the undamped and damped symmetric systems, and undamped and damped asymmetric systems. j sj sj sj sj sj sj sj

of of of of of of of

undamped symmetric system damped symmetric system undamped asymmetric system (m ¼ 0.4) undamped asymmetric system (m ¼ 0.5) damped asymmetric system (m ¼ 0.4) damped asymmetric system (m ¼ 0.4171) damped asymmetric system sj (m ¼ 0.5)

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

79.88i 0.0279.88i 710.27i 710.39i 0.00710.27i 0.00710.29i 0.01710.38i

711.64i 0.07711.64i 711.50i 711.45i 0.08711.50i 0.08711.49i 0.09711.45i

7i14.25 0.25714.25i 714.91i 715.21i 0.25714.90i 0.25714.95i 0.25715.21i

716.76i 0.28716.76i 716.25i 715.98i 0.25716.25i 0.25716.21i 0.25715.98i

718.66i 0.20718.66i 718.85i 718.94i 0.13718.81i 0.12718.82i 0.08718.86i

720.13i 0.68720.10i 719.77i 719.63i 0.79719.78i 0.79719.77i 0.83719.70i

Notice that the asymmetric terms are located at (3, 5) (y2 affecting x2 and) and (4, 6) (y3 affecting x3), all through friction. Therefore Eq. (6) becomes 02

1

B6 B6 0 B6 B6 0 B6 B6 B6 0 B6 B4 0 @ 0

0

0

0

0

1

0

0

0

0 0

1 0

0 1

0 0

0

0

0

1

0

0

0

0

0

2

3

0

6 60 07 7 6 7 60 07 7 þ mkc 6 6 60 07 7 6 7 60 05 4 0 1

0

0

0

h13

0 0

0 0

0 0

h23 h33

0

0

0

h43

0

0

0

h53

0

0

0

h63

9 318 X1 > > > > > > 7C> > h24 7C> > Y4 > > > > 7C> > > > < = C X h34 7 2 7C ¼ HF 7C h44 7C> > X3 > > > 7C> > > > > C 7 h54 5A> > Y2 > > > > > :Y > ; h64 3 h14

(7)

which can be re-written as 2

mkc h13 mkc h23 mkc h33 0 0 1 mkc h43 0 0 0 1 þ mkc h53 0 0 0 0 mkc h63

1 6 60 6 60 6 6 60 6 60 4

0 1 0

0 0 1

0 0 0

38

9

mkc h14 > > X1 > > > 7> > > mkc h24 Y4 > 7> > > > > > 7> > > < = 7 mkc h34 7 X2 ¼ HF 7 mkc h44 X3 > 7> > > > 7> > > 7> > mkc h54 Y2 > > > 5> > > > > : 1 þ mkc h64 Y3 ;

(8)

where hij (i, j ¼ 1,2,y,6) are the elements of the receptance matrix H of the symmetric system. It can be observed that the coefficient matrix of Eq. (8) has least number of non-zero elements in the matrix block of rows 5–6 and columns 5–6, coinciding with the locations of y2 and y3 in the displacement vector x. This is actually no coincidence and reflects the nature of the matrix formulation and presents an advantage of the present method. From Eq. (8), one may observe that "

1 þ mkc h35

mkc h36

mkc h45 1 þ mkc h46

#(

Y2 Y3

) ¼

8 6 9 P > > > > h F > > 5j j > > < = j¼1 6 P > > > > > h6j F j > > > : ;

(9)

j¼1

Note that the receptance matrix H is symmetrical and from Eqs. (8) and (9) h53, h54, h63 and h64 are replaced with h35, h45, h36 and h46. This should help reveal the relationship between the locations of the friction-affected degrees-of-freedom and those receptance matrix elements involved. The latent roots of a system are the poles of its transfer function or the determinant equation of Eq. (9), that is " #! mkc h45 1 þ mkc h35 ¼0 (10) det mkc h36 1 þ mkc h46 It can be seen that only four (n  n) related receptance matrix elements are involved in Eq. (10). The latent roots of the undamped symmetric system, damped symmetric system, and the damped asymmetric systems at m ¼ 0.4 and 0.5 are obtained using the conventional approach (MATLAB eig function) to an eigenvalue problem and are listed in Table 1. A MATLAB numerical-symbolic program combined with Maple is coded for the present method. The present method yields identical results (to four decimal places) to those obtained by MATLAB eig function. and the latent roots at this The critical friction coefficient is found to be mcr ¼ 0.4171 by the conventional approach pffiffiffiffiffiffiffi critical friction coefficient that will cross the vertical imaginary axis are 710.29i, where i ¼ 1. If damping is ignored, it is found that mcr ¼ 0.5484, which indicates that viscous damping as given in Fig. 1 is actually destabilising. It was found that damping is stabilising in a damped asymmetric system when it is proportional [10,11].

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4. Inverse method for assignment of latent roots It must be clear now that adding damping is not a desirable way of improving the stability of asymmetric systems. If damping is used for this purpose, the resulting system damping must be proportional. To achieve proportional damping, damping must be added to all degrees-of-freedom generally. This would defeat the purpose of structural modifications, which are applied to a small number of degrees-of-freedom. With damping being unreliable, mass and stiffness modifications must be used. However, the real part of a latent root is much less sensitive to mass and/or stiffness change than the imaginary parts. This nature makes assignment of stable latent roots of an unstable asymmetric system by structural modifications more difficult than assignment of natural frequencies of a symmetric system. This section explores the attachment of mass and spring elements to assign latent roots based on the receptance of the original damped symmetric system. For simplicity, connecting a mass to an existing degree-of-freedom with a spring is not considered in the paper as that increases the number of degree-of-freedom. It is useful to point out that the symmetric system, whose receptance is to be used, includes the contact stiffness appearing in the relevant diagonal elements of the stiffness matrix but excludes the asymmetric terms that are also related to the contact stiffness. It is already known from the previous section that at m ¼ 0.4171 there are a pairs of latent roots at the flutter boundary. All the subsequent modifications are intended to re-assign this pair of latent roots that become unstable at m ¼ 0.4171 so that not only this pair of latent roots gains a negative real part but also the real parts of all other pairs of latent roots remain negative (hence a stable system). It will be seen that that is not always achievable. It should also be explained that at the flutter boundary, usually there are only one pair of distinct unstable latent roots. Therefore the assignment of only one pair of latent roots is investigated in the paper. 4.1. Point mass modifications Suppose it is desired that this pair of latent roots become 0.10710.00i. As a latent root of an asymmetric system is generally complex with a real part and an imaginary part, quite often a solution does not exist unless there are twice as many added (mass or spring) elements as the number of latent roots to be assigned. Three sets of locations of adding mass element(s) are considered: (1) Dm1 and Dm4 to x1 and y4; (2) Dm2 to x2 and y2; and (3) Dm3 to x3 and y3. Eq. (6) for this modified system becomes, respectively " # n X ðK þ sC þ s2 MÞ þ mi kci epi eTqi þ s2 ðDm1 e1 eT1 þ Dm4 e2 eT2 Þ XðsÞ ¼ FðsÞ (11) i¼1

" ðK þ sC þ s2 MÞ þ

n X

#

mi kci epi eTqi þ s2 Dm2 ðe3 eT3 þ e5 eT5 Þ XðsÞ ¼ FðsÞ

(12)

i¼1

" 2

ðK þ sC þ s MÞ þ

n X

# T i kci epi eqi

m

2

þs

Dm3 ðe4 eT4

þ

e6 eT6 Þ

XðsÞ ¼ FðsÞ

(13)

i¼1

Multiplying both sides of Eqs. (11)–(13) by 02 1 þ s2 Dm1 h11 s2 Dm4 h12 B6 2 B6 s Dm1 h21 1 þ s2 Dm4 h22 6 detB B6 s2 Dm1 h51 s2 Dm4 h52 @4 2 s Dm1 h61 s2 Dm4 h62 02

1 þ s2 Dm2 h33

B6 det@4 s2 Dm2 h53 s2 Dm2 h63 02

1 þ s2 Dm3 h44

B6 det@4 s2 Dm3 h54 s2 Dm3 h64

H and further manipulation yields 31 mkc h13 mkc h14 7C 7C mkc h23 mkc h24 7C ¼ 0 7C 1 þ mkc h53 mkc h54 5A mkc h63 1 þ mkc h64

s2 Dm2 h35 þ mkc h33 1 þ s2 Dm2 h55 þ mkc h53 s2 Dm2 h65 þ mkc h63

(14)

31

mkc h34 7C mkc h54 5A ¼ 0 1 þ mkc h64

(15)

31

mkc h43 s2 Dm3 h46 þ mkc h44 7C 1 þ mkc h53 s2 Dm3 h56 þ mkc h54 5A ¼ 0 2 mkc h63 1 þ s Dm3 h66 þ mkc h64

(16)

For mass modification (1), two pairs of solutions are found to be (0.082, 0.103) and (0.672, 0.244). The new latent roots are given in Table 2. Indeed, a pair of latent roots of 0.10711.00i results in both modifications. However, another pair of latent roots emerge in either modification whose real part is positive and hence is undesirable, even though these look like a plausible modification. Experience from the author reveals that it is more likely that a modification that intends to stabilise a pair of latent roots might make a different, originally stable pair of latent roots unstable, as seen in the above modifications. This is in fact a common feature also observed in other types of structural modifications to assign complex

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Table 2 Latent roots of the mass-modified system to assign 0.10711.00i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dm1 ¼ 0.082, Dm4 ¼ 0.103 sj at Dm1 ¼ 0.672, Dm4 ¼ 0.244

0.03710.56i 0.10711.00i

0.10711.00i 0.02711.07i

0.26715.21i 0.26716.08i

0.25716.24i 0.60717.97i

0.15718.55i 0.03718.10i

0.78719.92i 1.06726.19i

Table 3 Latent roots after mass modification of adding Dm3 ¼ 0.6112 to assign 0.01710.80i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj

0.01710.80i

0.09711.43i

0.48717.39i

0.15718.64i

0.72725.49i

0.83726.95i

Table 4 New latent roots of the stiffness modified system to assign 0.01710.80i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dk2 ¼ 69.90, Dk3 ¼ 102.89 sj at Dk2 ¼ 68.23, Dk3 ¼ 389.86

0.01710.80i 0.01710.80i

0.09711.44i 0.09711.44i

0.28716.18i 0.28716.19i

0.15718.78i 0.16718.77i

0.28719.02i 0.71720.40i

0.69720.45i 711.17

latent roots. Of course there is not such an issue in the structural modification of a symmetric system. Incidentally, the process of calculating these solutions for mass modification (1) is given in detail as a sample calculation in Appendix A. For mass modification (2), Eq. (15) does not yield a solution, which means that this particular mass modification is incapable of assigning a pair of patent roots of 0.10710.00i (but this does not exclude its capability of assigning a different pair of latent roots). For modification (3), latent roots with a real part of 0.1 cannot be assigned to produce a stable system. If the ambition is dropped, however, it is possible to produce a stable system. The new latent roots of the system modified by Dm3 ¼ 0.6112 to assign 0.01710.80i are listed in Table 3. It can be observed that mass modification (1) requires more computations or more measured receptance elements than mass modifications (2) and (3). It is found that the latent roots of the lowest frequency are more difficult to be re-assigned. Finally, there are other locations where mass modifications may be applied but they are not presented in this paper. 4.2. Ground-connected spring modifications Two sets of locations of adding ground-connecting springs are explored: (1) at x3 and y3 and (2) at y2 and y3. For these two sets of modifications, the equations to solve are 02 31 1 þ Dk2 h44 mkc h43 mkc h44 þ Dk3 h46 B6 7C 1 þ mkc h53 mkc h54 þ Dk3 h56 det@4 Dk2 h54 (17) 5A ¼ 0 Dk2 h64 mkc h63 1 þ mkc h64 þ Dk3 h66 " det

1 þ mkc h53 þ Dk2 h55

mkc h63 þ Dk2 h65

mkc h54 þ Dk3 h56 1 þ mkc h64 þ Dk3 h66

#! ¼0

(18)

It turns out after repeated attempts that stiffness modifications using ground-connected springs are not as effective in assigning the real parts of unstable latent roots as mass modifications. Neither stiffness modification (1) nor (2) is capable of assigning 0.10710.00i without producing another pair of unstable latent roots. If a smaller decay rate is tolerable, a stiffness modification can still work, as demonstrated by the examples below. Suppose that a pair of latent roots of 0.01710.80i is desired. Stiffness modification (1) permits two solutions (69.90, 102.89) and (68.23, 389.86). The new latent roots are given in Table 4. The second solution is apparently unrealistic, considering the total stiffness of y3 of the original asymmetric system is only 260 N/m. It would create a total negative stiffness at y3. As a result, there would appear a real positive latent root, indicating divergence. It should also be mentioned that stiffness modification (2) is incapable of assigning this pair of latent roots, without producing a new pair of unstable latent roots. These results are not presented. Next, assignment of a pair of latent roots of 0.007710.228i is attempted by stiffness modification (2). Two pairs of solutions (68.64, 82.26) and (28.91, 99.58) are found. The new latent roots are given in Table 5. Both pairs successfully

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Table 5 New latent roots of the stiffness modified system to assign 0.07710.228i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dk2 ¼ 68.64, Dk3 ¼ 82.26 sj at Dk2 ¼ 28.91, Dk3 ¼ 99.58

0.007710.228i 0.007710.228i

0.050712.254i 0.066711.871i

0.250715.018i 0.249712.614i

0.259718.506i 0.251715.105i

0.930719.913i 0.050719.378i

0.004720.027i 0.878719.760i

Table 6 New latent roots of contact stiffness-modified system to assign 0.1711.0i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dkc1 ¼ 35.17, Dkc2 ¼ 67.39 sj at Dkc1 ¼ 64.37, Dkc2 ¼ 158.69

0.01710.05i 0.1678.24i

0.10711.00i 0.12710.40i

0.25713.30i 0.10711.00i

0.26715.60i 0.25715.49i

0.15718.12i 0.14717.62i

0.74719.90i 0.73719.96i

produce a stable system. As the spring constants in the second pair are smaller than those in the first pair, the new frequencies achieved by the second solution are overall smaller than those of the first modification. 4.3. Contact spring modifications The examples in Section 4.2 clearly illustrate the limitation of applying ground-connected springs in assigning stable latent roots. Alternatively, any two-degrees-of-freedom of the original asymmetric system may be connected by a spring. However, one such spring adds four extra terms to the total stiffness matrix and the equation for determining the right spring constant so as to assign desired latent roots would be more complicated than Eqs. (17) and (18). Although that may produce a feasible modification, the workload would be much higher than another possibility to be explored in this subsection, that is, modification of the contact stiffness. Denote the stiffness modifications to the two contact springs in Fig. 1 by Dkc1 (left) and Dkc2 (right). Eq. (6) now becomes ½I þ HðsÞfDkc1 e5 eT5 þ Dkc2 e6 eT6 þ mðkc þ Dkc1 Þe3 eT5 þ mðkc þ Dkc2 Þe4 eT6 gXðsÞ ¼ HðsÞFðsÞ The latent roots satisfy the following equation " #! 1 þ Dkc1 h55 þ mðkc þ Dkc1 Þh53 Dkc2 h56 þ mðkc þ Dkc2 Þh54 det ¼0 Dkc1 h65 þ mðkc þ Dkc1 Þh63 1 þ Dkc2 h66 þ mðkc þ Dkc2 Þh64

(19)

(20)

If 0.1710.3i are to be assigned, two solutions are found by solving Eq. (20) as (80.49, 102.52) and (64.82, 201.96). Considering the original contact spring constant kc ¼ 110 N/m, the second solution pair is not permissible. The first solution pair leads to a very small contact stiffness at y3 and can hardly be feasible in reality. In order to avoid a large negative stiffness modification that is not permissible, the assigned frequency should be raised. If instead 0.1711.0i are to be assigned, two solutions found from Eq. (20) are (35.17, 67.39) and (64.37, 158.69). The new latent roots for both modifications are given in Table 6. Even though the second solution pair is still not permissible, both pairs of solutions have improved in their feasibility. Both modifications produce stable systems. As the second modification contains a large negative stiffness, a pair of low latent roots emerge. Note that even though Dkc2 ¼ 158.69 is a very big (negative) modification, it is accommodated by the total stiffness of the original stiffness of y3, which is K(6,6) ¼ 260 N/m, and the system becomes stable even after this drastic modification. If an even higher frequency is attempted, for example, 0.1712.0i, it is discovered that no real solution can be found. 4.4. Mass–stiffness modifications A fourth possibility is to apply mass and stiffness modifications simultaneously. One such modification is studied here. If Dm3 is added to x3 and Dk3 to y3, then the equation to solve is 02 31 Dk3 h46 þ mkc h44 1 þ s2 Dm3 h44 mkc h43 B6 7C 1 þ mkc h53 Dk3 h56 þ mkc h54 (21) det@4 s2 Dm3 h54 5A ¼ 0 s2 Dm3 h64 mkc h63 1 þ Dk3 h66 þ mkc h64 Suppose a pair of latent roots of 0.01710.80i is to be assigned. Two pairs of solutions are found as (0.614, 12.449) and (0.590, 290.932). The new latent roots are given in Table 7. As the second solution pair leads to a negative stiffness at y3, divergence in the form of a positive real latent root is expected in the second solution.

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Table 7 New latent roots of mass and stiffness modified system to assign 0.1710.80i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dm2 ¼ 0.614, Dk3 ¼ 12.449 sj at Dm2 ¼ 0.590, Dk3 ¼ 290.93

0.01710.80i 0.01710.80i

0.09711.43i 0.09711.43i

0.26716.43i 0.47717.35i

0.48717.46i 0.16718.64i

0.15718.63i 0.88725.96i

0.91726.62i 75.90

Table 8 New latent roots of damping modified system to assign 0.006710.290i. j

1, 2

3, 4

5, 6

7, 8

9, 10

11, 12

sj at Dc2 ¼ 0.275, Dc3 ¼ 0.524 sj at Dc2 ¼ 0.308, Dc3 ¼ 0.249

0.006710.290i 0.006710.290i

0.037711.492i 0.032711.491i

0.270714.950i 0.240714.948i

0.493716.204i 0.138716.214i

0.020718.816i 0.008718.816i

0.799719.776i 0.798719.777i

4.5. Damping modifications Lastly, a damping modification is explored. Suppose two ground-connected viscous dampers Dc2 and Dc3 are applied to y2 and y3, respectively. The equation to solve is " #! 1 þ sDc2 h55 þ mkc h53 sDc3 h56 þ mkc h54 ¼0 (22) det sDc2 h65 þ mkc h63 1 þ sDc3 h66 þ mkc h64 If assignment of a pair of latent roots of 0.1710.28i is attempted, it is found that Eq. (22) does not yield any real solution. Actually adding any positive damping to both locations makes the system even more unstable. Applying negative damping to y3 or both can make a less ambitious assignment. Such cases are shown in Table 8. It should be pointed out that even though 0.006710.2900i can be assigned by this damping modification, 0.006710.2901i cannot be assigned by this damping modification (no real solution). Moreover, a further addition of negative damping will produce a new pair of unstable latent roots, even though it makes the latent roots of the lowest frequency more stable. So the effectiveness of the damping modification to assign stable latent roots is very limited. If proportional damping may be added, stable latent roots can be easily assigned. For example, if C ¼ M, the real parts of all latent roots at m ¼ 0.4171 are negative. The new critical friction coefficient would be raised to 0.6283. 4.6. Short discussion It can be seen that the real part of a latent root, which determines the stability of lack of it, is rather insensitive to changes of mass and stiffness. So mass and stiffness modifications to assign stable latent roots in a damped asymmetric system has limited success, in comparison with these kinds of modifications to assign frequencies in a symmetric system. Even though the real part is sensitive to damping changes, proportional damping will be required to stabilise an unstable asymmetric system. This would mean apply damping to nearly all degrees-of-freedom, which is hardly feasible in reality. So damping cannot be relied upon to stabilise an unstable asymmetric system. Even though the mass and stiffness modifications explored in this paper do not seem very successful in assign stable latent roots, it is believed that they would be more successful when dealing with real systems with large numbers-of-degrees-of-freedom as the number of frictionaffected degrees-of-freedom in those large systems is only a small percentage of the total number of degrees-of-freedom. This positive sign may be sensed from the structural optimisation work reported in [18]. It seems that a sensible alternative is to use active control [1,19]. From the point of view of vibration control, structural modifications performed in the present paper belong to passive control. One disadvantage of passive control is that a modification to assign some latent roots will normally also change other unassigned latent roots. Thus it is usually impossible to assign some latent roots while keeping other latent roots unchanged by means of passive control, unless modifications at a number of coordinates of the system are made, which means a big increase of computing workload. Even then, there is no guarantee that a solution may be found. Mottershead et al. [25] recently showed that a single input of state-feedback control was capable of assigning all latent roots of damped symmetric systems. It is believed that active control would also be capable of assigning all latent roots of damped asymmetric systems. This will be investigated shortly. 5. Conclusions This paper presents a method for predicting the latent roots of damped asymmetric systems, which have an asymmetric stiffness matrix, based on the receptance of the damped symmetric systems. The method is developed in the inverse analysis for assigning latent roots by means of mass, stiffness and damping modifications. The following findings are made

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from simulated examples: (1) It is more difficult to assign latent roots, in particular the real parts to produce a stable system, of damped asymmetric systems than to assign frequencies of symmetric systems, (2) three possibilities of a feasible solution, an impermissible solution and no real solution of a modification may result, (3) a stable pair of latent roots can often be assigned but new pair(s) of latent roots may emerge that are unstable, (4) different types of modifications (mass, normal stiffness and contact stiffness) and different locations of modifications may have very different outcomes of latent root assignment, (5) damping modifications are capable of assigning very small negative real parts and very sensitive to the frequency to be assigned, and (6) some modifications can assign desired latent roots and produce a stable system. Modifications in the form of adding an extra mass with a spring have not been studied in this paper and will be explored in near future. Stiffness modifications by linking two-degrees-of-freedom with a spring are not studied in the paper either as they bring about more workload (involve more receptance elements) than the modifications being explored in the paper. On the positive side, critical points of damped asymmetric systems can be assigned by the present method after some modifications to the formulas presented in this paper. The method presented in this paper can be used with only measured receptances without knowing mass, stiffness and damping matrices of the system.

Acknowledgements This work is partly funded by the State Key Laboratory for Structural Analysis of Industrial Equipment of China in Dalian in the form of an international travel grant to the first author. His discussion with Professor John E. Mottershead on pole assignment in control is useful. Appendix A As an example, the solution of the first mass modification of adding Dm1 to x1 and Dm4 to y4 is given below. For this particular modification, Eq. (14) gives ADm1 Dm4 þ BDm1 þ C Dm4 þ D þ ðEDm1 Dm4 þ F Dm1 þ GDm4 þ HÞi ¼ 0

(A.1)

where A ¼ 11.6700, B ¼ 1.9722, C ¼ 4.6256, D ¼ 0.5400, E ¼ 0.2033, F ¼ 0.1802, G ¼ 0.5283 and H ¼ 0.0413. These symbols are only valid in this Appendix A. This equation is equivalent to ADm1 Dm4 þ BDm1 þ C Dm4 þ D ¼ 0 EDm1 Dm4 þ F Dm1 þ GDm4 þ H ¼ 0

(A.2)

It is easy to derive solutions to the above bilinear equations. At first, the following quadratic equation should be solved to get two solutions of Dm4 ðAG  ACF  CEÞðDm4 Þ2 þ ðAH þ BG  ADF  BCF  DEÞDm4 þ BðH  DFÞ ¼ 0

(A.3)

and they are then substituted into the linear Eq. (A.4) to get two solutions of Dm1

Dm1 ¼ 

C Dm4 þ D ADm4 þ B

For this mass modification, it is found from equations (A.3) and (A.4) that

(A.4) (

Dm1 ¼ 0:082 and Dm4 ¼ 0:103

(

Dm1 ¼ 0:672 . Both Dm4 ¼ 0:244

pairs of solutions assign a pair of complex conjugate latent roots of 0.10711.00i. But they make another pair of complex conjugate latent roots gain a positive real part. For equations more complicated than Eq. (A.2), the method of Gro¨bner bases can be used, as done in [16,26]. References [1] D.J. Inman, Vibration with Control, Wiley, New York, 2006. [2] H. Ouyang, W. Nack, Y. Yuan, F. Chen, Numerical analysis of automotive disc brake squeal: a review, International Journal of Vehicle Noise and Vibration 1 (2005) 207–230. [3] D.H. Hodge, G.A. Pierce, Introduction to Structural Dynamics and Aeroelasticity, Cambridge University Press, Cambridge, 2002. [4] M.R. North, Disc brake squeal—a theoretical model, Technical Report 1972/5, Motor Industry Research Association, Warwickshire, England, 1972. [5] W. Nack, Friction induced vibration: brake moan, SAE Paper 951095, SAE, 1995. [6] N. Hoffmann, M. Fischer, R. Allgaier, L. Gaul, A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations, Mechanics Research Communications 29 (2002) 197–205. [7] Q. Cao, H. Ouyang, M.I. Friswell, J.E. Mottershead, Linear eigenvalue analysis of the disc-brake squeal problem, International Journal for Numerical Methods in Engineering 61 (2004) 1546–1563. [8] J. Huang, C.M. Krousgrill, A.K. Bajaj, An efficient approach to estimate critical value of friction coefficient in brake squeal analysis, Journal of Applied Mechanics 74 (2007) 534–541. [9] D. Hochlenert, G. Spelsberg-Korspeter, P. Hagedorn, Friction induced vibrations in moving continua and their application to brake squeal, Journal of Applied Mechanics 74 (2007) 542–549.

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