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Formulation of Rayleigh damping and its extensions
Cmpurrrs
Pergamon
0045-7949(94)00611-3
& Smrrurrs
Vol. 57. No. 2. DD. 277-285, 1995 Copyright(’ 1995El&r Science Ltd Printed in Great Britain. All rights reserved M)45-7949195 59.50 + 0.00
FORMULATION OF RAYLEIGH DAMPING AND ITS EXTENSIONS Man Liu and D. G. Corman School
of Mechanical and Offshore Engineering, The Robert Gordon University, Schoolhill, Aberdeen AB9 IFR, Scotland, U.K. (Received
4 March
1994)
Abstract-The objective of this paper is to present a study on the formulation of Rayleigh damping for a damped linear mechanical system. By means of examining the existing theory on Rayleigh damping, extensions of Rayleigh damping are proposed in terms of a double series. The extended forms provide more choices in formulating a damping matrix. Two sets of empirically obtained data are utilized as examples to illustrate the derived formulations.
1. INTRODUCTION
ing force is linearly proportional to the velocity. Rayleigh dissipation function takes the form
Formulation of the governing equations of motion of a system described by mass inertia, stiffness and damping matrices can be obtained easiiy. Our comprehension of damping is, however, considerably less encyclopedic than that of mass and stiffness systems. As such, predicting vibration parameters with respect to damping is difficult in practice. The dynamic response and attenuation of vibration are predominantly controlled by damping. Therefore, using an assumption about damping in the analysis is a justified compromise. Even for experimental purposes, the assumption of damping has to be incorporated so that the obtained data can be processed. There are several damping assumptions available which are based on a single degree of freedom system and extended to multi degree of freedom systems. For a single degree of freedom system, damping theories are well established to include, for example, viscous damping, Coulomb damping and structural damping. A dashpot is frequently idealized as a viscous damping element, in which the damping force is assumed to be proportional to the relative velocity between the two ends of it. Coulomb damping is caused by the relative motion of surfaces sliding against each other, whilst structural damping is introduced, together with others, to model energy dissipation of continuous elements. For a continuous system, energy dissipation within materials is supposed to be caused by thermoelasticity and grain boundary viscosity, which involves microstructure characterization. In dealing with a multi degree of freedom system, all the above damping assumptions are applicable. Amongst these, viscous damping and structural (hysteretic) damping models are most commonly used [I]. The viscous damping model assumes that the damp-
Its
where p is an energy dissipation coefficient, C, are discrete damping factors and u(x, y, z, t) is a displacement function for which the normal finite element discretization is available which produces a set of discretized second-order differential equations derived by using the variational principle. The equations of motion of a linear dynamic system can be written as
Pa{*1 + [WI
+ ~~l~x~= VI>
(2)
where [Ml, [C] and [K] are the mass, damping and stiffness matrices and {x} and {f} are displacement and force vectors, respectively. For hysteretic damping, the damping term [C]{i} is replaced by i[H]{x}. The corresponding equations are
bflcf~ + WI + Wl){x) = u-1.
(3)
This equation is not in the real domain. The interpretation of this equation is that the damping force is proportional to displacement rather than velocity, and is 90” out of phase with the displacement. Accordingly, for harmonic vibration, the equivalent damping force can be written as i[H]{x} = o-‘[H]{i}, 271
{_t} = iw{x}.
(4)
278
Man
Liu and D. G. Gorman
It should be noted that hysteretic damping may only be used in the frequency domain. This hysteretic model is introduced based on experimental observations that damping ratios decrease as the frequency increases. It allows many continuous systems to be modelled adequately. The model will however lead to erroneous results for free vibration analysis, in the time domain, since the eigenvalues will have both positive and negative real parts, and the response will diverge if there is no work done by external forces. Due to the problems of obtaining a system’s damping information, there is no practical way of forming the physical damping matrix by using the finite element method even though damping exists in most mechanical systems or structures. Indeed, so common is damping that we tend to forget that it is not in the natural order of things for oscillation to decay. The damping properties of a system can vary according to the structural design or the material used. In the analysis of the dynamic response of any system the damping information is essential for satisfactory results. The common Rayleigh damping [2] given by
2. RAYLEIGH
DAMPING
The advantage of Rayleigh damping is that no alteration is introduced to the mode shapes and the eigensolution or calculated response is thus greatly simplified. General Rayleigh damping is described by a matrix series given by eqn (5) or eqn (Bl) in the Appendix. A two-parameter model, as shown in eqn (6) derived from this series, has been used on many occasions. Some large FEM codes such as NASTRAN and SAP5 have adopted the two-parameter model for dynamical response analysis. For this two-parameter model, it is specifically easy to obtain the parameters, x0 and 2,. by using a least squares fit. The equivalent decoupled form of eqn (5) is
This equation indicates that the damping ratios expressed in terms of the natural frequencies to odd power (2k - I), i.e. c1)-‘, ro’,c)‘, OJ’.W~, Corresponding to the two parameter model. damping ratios are 2[, = a,,(l,, ’ + %,W,.
is a reasonable approximation for small levels of damping. Using the Rayleigh damping will consequently result in the existence of real modes that can decouple the system’s equations of motion. The simplest case is for proportional damping consisting of only two terms from the above formula,
[Cl = %[W + al WI>
(6)
where c(~ and IX, are arbitrary constant coefficients. Historically, the Rayleigh damping is defined by eqn (6). The common Rayleigh damping described by eqn (5) is proposed by Caughey [2], for which the two-term model eqn (6) is only a special case when p =2.
Fig. I. Schematic
are the
_. the
(8)
From this two-parameter damping model, the relationship between the damping ratio and the natural frequency is schematically demonstrated by Fig. 1. Given more than two values of w, and the corrcsponding values of the damping ratios ;,, the coefficients may be determined. For many practical situations, the two-parameter model is not adequate to approximate the damping information obtained from modal analysis. Increasing the number of terms may help improve the model according to eqn (7). However, the decision is, whether to use expression (7) or eqn (5). The proportional damping may well be described by other polynomials. It is thus proposed to starch for
of the two-parameter
damping
model
Formulation
different forms from the normal Rayleigh and extensions are suggested here.
of Rayleigh
damping,
damping
279
Thus
3. EXTENSIONS OF THE RAYLEIGH DAMPING
3.1. Extended series
Rayleigh damping-negative
indexed
Mathematical extension. Recalling the Rayleigh damping from eqn (5) it is noted that the series of summation is from 0 to p - 1, where p is an integer. In fact if the system does not have any rigid mode, whose associated mode shape does not involve elastic deformation, thus it suffices to say w, # 0, the series can be extended to including negative power indexed terms. As opposed to the normal Rayleigh damping matrix denoted by [C,], this series is denoted by [C_ 1, such that -b-I)
[Cl = [Ml 1 ~k([~l-‘mk k=-I P-1 =
[M] 1 a-k([Ml-‘[KI)-k.
(9)
k=l
The proportionality or orthogonality of this series to the real modal matrix, [@I, can be obtained as
This equation
is based on the assumption
([M]-‘[K])-k
that
= [@I[ w;*“][@]-‘.
(11)
This assumption is elucidated in the Appendix. The extended Rayleigh damping is established by combining eqn (9) and the normal Rayleigh damping, i.e. [Cl = [C- I + [C,],
[Cl = WI
c
k=-I
Q([Ml- WIY
L”l
,go
ak([Ml-
‘[Kl)k
P-l
=
The corresponding
c ‘& ([Mlk=-(p-1,
L”l
orthogonal P-
(@;}r[C]{@j} = c
‘[K])‘.
relationship
(12)
is
’
(a_@,‘”
w+a,oi+C1201+.
.‘+$_,wfp-3
+ C(kWfk)d,,
(13)
k=O
where S,, is the Kronecker delta function, and the coefficient associated with UI~ is $ = c(_~ + a,.
>
. (14)
The normal Rayleigh damping series, which is positively indexed, is expanded to give a negatively indexed series. This gives more choices than the normal model in selecting various terms to form the Rayleigh damping for practical use. The two-parameter model can be simply procured by imposing ak = 0 and leaving two other remaining coefficients to be determined, for instance, k # 0, 1; or k # 1, - 1. A proper selection of the terms in the series has a heavy reliance on empiricism and the linear least squares fit depends categorically on the damping ratios observed in experiment for each individual system. Some examples are given in Section 4. Physical interpretation. In the above subsection, it has been proposed to formulate Rayleigh damping with a negatively indexed series. The same extension may be obtained by another way. According to eqn (5) it is noticed that the generator for [C,] is ([Ml-‘[K]). In fact, the standard eigenproblem of matrix [A] = ([Ml-‘[K]) is equivalent to the general eigenproblem of a dynamical system described by [K]{X} =I[M]{X}. From this expression, another equivalent standard eigenproblem is
wl-‘[w{~~
= ; {XI
where [K] is not singular. As result of this we can use the ([K]-‘[MI) instead to form a Rayleigh matrix, i.e.
p-1 +
EO
+
(15)
generator damping
p-1
[c,,l = LKl
c
I=0
ak([Kl-‘[Ml)k.
This form is similar to eqn (5). Its proportionality orthogonality can be verified according to the cedure outlined in the Appendix. It can be corroborated that apart from the first terms in the series, [C,,,] [eqn (16)] is equivalent [C_] [eqn (9)], since [K] = [M]([M]-‘[K]) ([K]-‘[M])A = ([Ml-‘[K]))“.
(16)
and protwo to and
Man
280
Liu and D. G. Gorman Correspondingly,
Thus
the damping
ratios are composed
of
x cQ([M]-‘[K])“’
‘.
(17) If only p2 = 1 and /J’~= 0, (n # 2) one has
Each term rather than the first and second, in eqn (I 6) may be identified in eqn (9). To sum up, if the normal Rayleigh damping matrix is considered as a result of using the stiffness matrix dominant generator ([Ml-‘[K]), then the proposed negatively indexed damping matrix is achieved by utilizing the mass matrix (or, as it is, the flexibility matrix) dominant generator ([K]-‘[MI). Combining the two renders the extension of the Rayleigh damping [eqn (12)]. For a discrete system, it is as easy to form a flexibility matrix as a stiffness matrix [3]. 3.2. Extended indexed
Rayleigh
damping-rational
fraction
series
Extending the work in the previous subsection, the Rayleigh damping model is adapted to any rational number indexed series. According to eqn (A6) in Appendix A, for any rational number, 7 = k/n, the yth root of a positive definite matrix [A] is
[AlA”” = ([Ml-‘[K])~“’ Accordingly, series is
another
ic,,l = I”l
c
(20)
, q. matrix series is thus
+
B,tcJ
(21)
Hence
pi1
[Cl = i: B,,[C,,l = WI i: n = I
n=I
k + + m,
in the matrix must have
.j = I,2 )
limit a, cuzk”’ I = 0,
k++3Z
limit a, W: = 0. h-+X
ak
(26)
(27)
Here wN is the maximum (highest order) natural frequency, wN = max{w,, co*, , w,, .). Thus
This means for a sufficient large number, k, the coefficients in the Rayleigh damping should be far less than the inverse of the maximum natural frequency to an appropriate power. It is also noted that in order that the damping ratios are positive, [, > 0, not all the coefficients xi are required to be positive. For the two parameter Rayleigh damping model [eqn (S)], x0 + X, W: > 0. This leads to a conclusion that at least one of the coefficients must be positive. Assuming a0 b 0 results in c(~2 -a, wf For each natural frequency, this is satisfied by
I=-@-,,
x
(25)
damping
”
ak([Ml-‘[Kl)”
(19)
i=-tJ-1)
[Cl = /II [Cl1+ Bz[Gl-t
each element
B, it is evident
Rayleigh
where n is an integer, n = 1,2, The general Rayleigh damping collectively formulated as
Consequently, a limit,
= 0,
(18)
= [@I[ W:“:“][@]-‘. extended
limit ak([M]m’[K])‘,“r
For n = 1,
[A]? = [Alk”’ = [P]diag{d~~“}[P]~‘. Thus based on eqn (B5) in Appendix that
Accordingly, as opposed to eqn (7) consisting of the natural frequency to only odd powers, it has terms related to the natural frequency to both odd and even powers. If p and q tend to infinity, the limit may exist if some conditions are satisfied. As with a normal series, the necessary, but not sufficient, condition is that
hz([Ml ‘[Kl)““i. (22)
This double series reduces to the normal Rayleigh damping proposed by Caughey in the case q = 1. Therefore, the integer indexed Rayleigh damping series is extended to both the negative and real number indexed series.
-a,
< a,min
l‘l 1 o?“J
Consequently, -a,min
1
i of”
I
QZr<
+m.
281
Formulation of Rayleigh damping For an underdamped additional constraint
system possessing [, < 1, an is imposed from eqn (8) as
fXO + cl,wf < 20,.
Likewise when [C] is Rayleigh damping, one has {4i}r[C]{$,} = O(i #j). Thus [C,] = 0. Accordingly it is found, as expected, that [C] = [C,].
(31)
{4l>W{4,} [Cal= WI PI
Hence
f 1 y 0,
2
I1
-a,min
(32)
w,
This is the relationship that the coefficients two parameter model must satisfy. 3.3. Extended
Rayleigh
for the
damping-unsymmetrical
systems The above extensions are acquired under the condition that the spatial matrices [Ml, [C] and [K] are symmetrical. For an unsymmetrical system, the Rayleigh damping can also be decoupled, since the following formula holds: [Ml-‘[K]
= [@]diag{oj}[@]-‘.
(33)
The derivation is given in Appendix B [eqn (Bl2)]. Likewise, the extended Rayleigh damping as described by eqns (12) and (22) can be diagonalized, and is applicable to unsymmetrical systems. 3.4. Approximate decoupling Proportionality is not an inherent property for any damping matrix, whether it is experimentally identified or theoretically formulated. For a damping matrix [C] that cannot be decoupled by means of the modal matrix transformation, Rayleigh damping is a reasonable approximation. In fact, a damping matrix [C] can be decomposed into two matrices:
[Cl = [c/l + [C”l
(34)
where [C,] is the Rayleigh damping matrix and [C,] is the matrix that makes up the difference between [C] and [C,]. Determination of [C,] and [C,] is easily achieved by the modal transformation
PIWI.
0 sym
(38)
I
It has been proved that neglecting [C,] is equivalent to neglecting an infinitesimal value corresponding to the second order term of the damping ratios when solving the complex natural frequencies [4]. For calculating the dynamic response of the system, Shahruz [.5] proposed the approximate decoupling of the equations of motion of a damped linear system in order to use the real model superposition principle [3]. It is considered that Rayleigh damping is the most convenient model for predicting system dynamic response [6], in particular for multi degree of freedom systems consisting of a large number of degrees of freedom. Because Rayleigh damping is proportional damping, the system governing equations of motion can be decoupled by the use of a modal transformation. Thus, the modal superposition method can be employed for performing the response analysis. Apart from this advantage, Rayleigh damping also offers an easy way of formulating the damping matrix by using the damping ratios acquired experimentally. The Rayleigh damping matrix can be obtained when the damping ratios and mode shapes are available. According to equation (B7), the damping matrix takes the form
[Cl = P-7
2w.1P-’
= WI [@I[ xp, 1Pmfl.
(39)
Alternatively, the damping matrix can be obtained by using the proposed Rayleigh damping matrix characterized by eqn (22). 4. EXAMPLES
Example 1
Pl’([c/l+ [C,l)[@l = Pl’[ClPl. Because
[@]r[M][@] = [I], it simplifies
(35)
to
[Cdl+ [Co1= Pl-‘PITKl[@lP-’ = ~~l~~lr~~l’~~l~~l~~lr’[~l.
(36)
Since the transformed Rayleigh damping matrix [@]‘[C,][@] consists of non-zero diagonal terms and zeros elsewhere, and [@]‘[C,][@] contains all nonzero off diagonal terms, one has
[Cdl= [w[@lr[ {~,)TCl{d4~I[@nm
(37)
This example illustrates the formulation of Rayleigh damping by using the fractional index series. As opposed to the two parameter (Rayleigh damping) model described by [C] = cr,[M] + E, [K], two terms are selected, they are [C] = c(, 6[M]([M]-‘[K])‘.16 and GI,[K], corresponding to (k = 1, n = 6) and (k = 1, n = I), respectively. The coefficients for the two-parameter model are identified by using the linear least-square fitting method. Given values of the natural frequencies and the corresponding values of the damping ratios, s0 and a, for the two-parameter model can be determined according to eqn (8), by using a straight line fit taking the y axis to be 21,~~ and the x axis to be
Man Liu and D. G. Gorman
282 Table
Mode
1. Damping
Angular frequency
from the two parameter Two-parameter Damping ratios
Damping ratios
0.507 0.541 0.764 1.356 2.718 3.924 4.184 5.082 5.349 6.965 7.664 7.881
2 3 4 5 6 I 8 9 10 11 12
ratios
13.605 12.765 9.123 5.345 3.118 2.612 2.561 2.471 2.463 2.524 2.589 2.612
14.900 5.000 6.800 3.400 1.800 7.000 1.300 2.500 3.000 2.300 3.900 1.100
t [C] = c+,[M]([M]-‘[K])“~
+ a, [K], the standard
Table
Mode
1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23
2. Damping
Identified modal Natural frequency (Hz) 108.9 293.1 360.0 380.4 414.0 483.8 553.1 591.1 732.2 872.5 902.1 1017.3 1028.6 1034.2 1045.6 1067.1 1093.2 1131.9 1187.3 1253.8 1310.0 1337.2 1432.5
t [Cl = ~~,~t~I([~l-‘[~l)’
ratios
5.65 3.91 1.87 2.06 2.70 2.55 2.62 2.00 I .66 1.45 1.20 0.40 0.23 0.58 0.60 1.17 0.73 1.07 0.70 0.82 0.81 0.68 0.60
a, = 0.0684 LX,= 0.0022
9.602 9.202 7.353 5.122 3.459 2.936 2.867 2.697 2.662 2.548 2.534 2.533
model? Standard deviation 1.49 x IO’
MI6” 0.0606 c(, = 0.0013
model. The standard deviation from the proposed two-term model may not be the minimum one.
Example
2
This example show the formulation of the Rayleigh damping matrix for a structural pipework used in modal analysis with thin coupling hoses [8]. Both the two-parameter model ([Cl = IZQ[M]+ c(,[K]) and the extended two-term model ([Cl = x,,.~[M]([M]~‘[K])“~ + tl, [K]‘14 + a, [K]) are utilized. The coefficients for both models are identified by using the least-squares fitting method. From the following table it is shown that, as with the first example, using a fractional power term will improve the fitting, and its standard deviation is smaller than the normal two-parameter model.
Two-parameter Damping ratios 9.24 3.41 2.77 2.62 2.40 2.04 1.78 1.66 1.32 1.98 1.04 0.91 0.90 0.89 0.88 0.86 0.83 0.80 0.75 0.70 0.66 0.65 0.59
4 + a, [K], the standard deviation
Two-term Damping ratios
is o2 = 2XA[‘.
from the two parameter
parameters Damping ratios
model Standard deviation 1.97 X 10-z
deviation
WT. For the extended two term model, its coefficients can be identified by using the linearized least-squares fit. It should be noted, however, that the relationship between the damping ratios and the natural frequencies are not linear, no matter whether the two-parameter model or the extended two-term model is used. For the two-parameter model, this relationship is expressed by eqn (8) and is illustrated by Fig. 1. It is misleading that the relationship between the damping ratios and the natural frequencies for the two-parameter model was illustrated as a linear one by Bathe and Wilson (Fig. 8.4 in Ref. [7]). From Table 1 extracted from Ref. [8] it is found that using a fractional power term will improve the fitting, and the associated standard deviation is smaller than that of the normal two-parameter
model and proposed model
model and the extended model Standard deviation 33.6397
2; = 1007.435 ct; = -7.966 x IO-’
02 = 2ZA5?
model
Two-term Damping ratios 5.37 3.10 2.72 2.63 2.48 2.22 1.99 1.89 1.54 1.25 1.20 1.00 0.98 0.97 0.95 0.92 0.88 0.82 0.74 0.64 0.56 0.53 0.40
modelt Standard deviation 7.8264
@114 56.9 I8 a; = -7.714 X 10-4
Formulation
of Rayleigh
In this example, the natural frequencies in Hertz are used directly (see Table 2), and the obtained coefficients t(.; are not the same as those found by using the angular frequencies, CX?.To convert from c(.J to CL,.,the associated scaling factors are (27r)G-‘), i.e. ~L.J = (27~)~-‘cr,., (y = l/4, 1). This is derived according to the relationship between the natural frequencies and the angular frequencies (rad SK’), (Hz) o, = 2r$. According to the above extended two-term model, the damping matrix is [C] = c+[M]([M]-‘[K])‘!~ + LX, [K]. Thus, the corresponding damping ratios are obtained
2&o, = 51,i4W/‘*+ tc,wf.
(40)
21,(276) = @~(2r!f)‘~~ + ‘xi(27rf;)‘.
(41)
Equivalently,
According to this equation, if the natural frequencies in Hz are used in fitting, the coefficients obtained are LY;’ = (271)27_‘c$, (y = l/4, 1). From the result shown in this table, it is evident that the proposed model renders a smaller standard deviation than the normal two parameter model. It is also noted that the second coefficient, CX;or aI, for both models is negative. Theoretical discussions given in Section 3.2 indicate that not all coefficients are required to be positive to produce a positively definite damping matrix. Summing up, this section has assimilated some points relating to the discussions in this paper. These two examples have shown the advantage of the extended Rayleigh damping model. The disadvantage of the extended model is in the computational aspects. The damping matrix created by using such terms to fractional powers requires decomposition of the matrix [Ml-‘[K]. The consequence is that the damping matrix does not have the same banded property that the mass and the stiffness matrices possess. 5. CONCLUDING
damping REFERENCES
L. Cronin, Eigenvalue and eigenvector determination of nonclassically damped dynamic systems.
1. D.
Comput. Struct. 36, 133-138 (1990). 2. T. Caughey, Classical normal modes in damped linear systems. J. appl. Mech. 27, 269-271 (1960). Elements of Vibration Analysis. 3. L. Meirovitch, McGraw-Hill, New York (1986). 4. M. Liu and J. Wilson, On the damping ratio of multi degree of freedom systems. Commun. appl. numer. Meth. 8, 265-271 (1992). Approximate decoupling of the 5. S. M. Shahruz, equations of motion of damped linear systems. J. Sound Vibr. 136, 51-64 (1990). 6. D. J. Ewins, Modal Tesfing: Theory and Practice. Research Studies Press, Taunton, Somerset (1984). 7. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1976). interaction effects on vi8. M. Liu, Fluid-structural brations of pipework. Ph.D. Thesis, The Robert Gordon University, October (1993). 9. M. Liu and J. Wilson. Criterion for decounhne. dvnamic equations of motion of linear gyroscopic systems: AIAA J. 30, 2989-2991 (1992).
APPENDIX
Al
Transformation
A. ROOT MATRIX
of similitude
The nth root operation of matrices is defined here according to advanced matrix theories. Given a matrix [A], its kth root, [A]“!+is defined as [B], if [B]’ = [A].
(Al)
Accordingly, when [A] has a matrix of similitude which is a positively definite diagonal matrix (and [A] must be then positive definite)
PI = diag{d,},
WV
namely,
[Al = V’IPI[f’-’ then matrix
REMARKS
In this paper, Rayleigh damping is re-examined and two more extended Rayleigh damping forms are established. The index of the damping series has been expanded from a positive integer set, to a real number set. It has been attempted to add a new dimension to the understanding of proportional damping. The corresponding damping matrix is characterized by a double series. The proposed expressions have enhanced the Rayleigh damping and provide alternatives for formulating a proportional damping matrix. On the basis of the simplicity and power of Rayleigh damping, the extended fractional indexed Rayleigh damping models as proposed in Section 3, were fitted to available damping data obtained in the experimental modal analysis. Identification of Rayleigh damping using both the two parameter model and the extended model is thus successfully demonstrated.
283
[B] is found
(A3)
to be
[B] = [P][D]‘/‘[P]-‘[P]diag{dj!‘}[P]-‘,
(A4)
[P] is a transformation matrix of similitude. The eigenvalues of matrix [A] are not affected by this transformation. Thus the diagonal [D] is the spectrum matrix of [A], containing all the eigenvalues. Therefore [A]‘:” = [B] = [P]diag{dj;k)[P]-‘. Hence
for any rotational
number,
y = k/n,
[B]
[A];’ = [Apfl = [B]’ = [B][B]
= [P]diag{d:‘“}[P]-‘. The inverse can be achieved
(A5)
(A6)
easily as
([A];‘)-’ = ([P]diag{dt!“}[P]-‘))’ = ([PI-‘)-’
= [P]diag{d;”
diag-‘{df,“}([P])-’ “}[P]-‘.
(A7)
Man Liu and D. G. Gorman
284 Thus in general,
Consequently ([A]‘) = ([P]diag{d;}[P]-‘),
y can be both positive
[M]~‘[K]
(Ag) hence
and negative.
([Ml-‘[K])”
A2. Orthogonal transformation If [PI-’ = [P]r, the above transformation is called the orthogonal transformation since [P] [PIT = [I]. When [A] is a symmetrical and positive definite matrix, there is bound to be a orthogonal matrix [D] into which [A] may be decomposed, i.e.
[Al = [~IPIW~
= [@I[ WY I[@]“.
Based on eqn (B6) and in accordance and (B4). it arrives at
(JW
with eqns (Bl), (B3)
(A9)
= [@17[M][@] 1 q diag{w,‘“) k=”
where [P] is the eigenvector matrix, and [D] contains the eigenvalues of [A]. For an unsymmetrical matrix [A], it can be expressed as
(AlO)
PI = [Ql[~l[f’l’~ where [P] eigenvector
W)
= [@I[ w; ][@I~‘.
and [Q] are, respectively, the matrices and [Q][P]‘= [I].
right
and
left
Thus the damping relation is
matrix is diagonalized.
The orthogonal
APPENDIX B. RAYLEIGH DAMPING
The normal form:
Rayleigh
damping
matrix
is in the following
where 6,, is the Kronecker
function,
I i =j i 0 i #j’
6,, = p-1
[Cl = Pfl 1 ~,w-‘m”~ 1-0
@I)
where [Ml, [C] and [K] are, respectively, the inertial mass, the viscous damping, and stiffness matrices. When p = 2, one has
[Cl = %[W + a, [Kl.
(W
This is the basic form of the Rayleigh damping matrix. Rayleigh damping is known as the proportional damping. As this name stands for, the Rayleigh damping matrix can be diagonalized by the real mode matrix [@I extracted from solving the eigenvalues and eigenvectors of its associated undamped system ((M, K)), indicating on this occasion that the damping matrix shares the same eigenvectors with ((M, K)). Thus Rayleigh damping does not at all introduce any change in the vibration patterns. Moreover, the resulted governing equations of motion can be decoupled completely. Each of the decoupled equations can be analyzed independently and the normal mode superposition principle is applicable. Accordingly, the system’s eigensolutions can be sought from its undamped version of the model. Once the natural frequencies and mode shapes have been extracted, the vibratory characteristics will be completed by adding the damping information. It is therefore Rayleigh damping that most researchers would relish to use. Bl.
The significance of the orthogonality which results in decoupling of the governing equations of motion implies that the real mode superposition principle is applicable for system analysis. In addition, the spatial parameters are associated with the modal parameters by these orthogonal relations. B2. Unsymmerrical systems When discussing a general damped vibration system, spatial matrices are not necessary to be symmetrical. In this case, the left and right eigenvector matrices, [Y] and [@I, are involved and two eigensolutions are required to yield the complete information of vibratory characteristics. As will be shown now, the Rayleigh damping matrix [Bl] can be diagonalized by the left and right eigenmatrices. Similar to symmetrical systems, the orthogonality relations in the matrix form are [Y]‘[M][@]
[Ml-’ =[@][‘f’]‘,
= I,
[Ml-‘[K]
[@]‘[K][@]
= diag{cu:).
(B3)
It can be easily shown that the real modes are also orthogonal in terms of the Rayleigh damping matrix. From eqn (B3) we have [Ml-‘[@][@lr,
[K] = [@]-‘I w; ][@I-‘.
(B4)
(Y]‘[K][@]
= diag{of}.
(BIO)
[K] = [Y]-‘[
w; ][@I-‘.
PI 1)
Consequently = [@I[ w; ][@I-‘.
(Bl2)
This turns out to be identical in form to the one for symmetrical systems, eqn (B5). Using eqns (B6), (BIO) and (Bl I) yields p-1
w~clPl= [@]r[M][@]
= I,
Equivalently,
Symmetrical systems
For a symmetrical dynamical system, its mass, damping and stiffness are symmetrical and non-negative definite. The corresponding undamped version of the system concerned possesses eigenvalue and normalized eigenvector matrices. The matrix form of the real mode orthogonal relation is
(B9)
[w[~l (
= [Y]r[M][@]
c %Pl[W:” 1Pr’ [@I I=” > c s(~diag(w,‘“} I=”
Formulation This has demonstrated diagonalized.
that
the damping
matrix
can
of Rayleigh be
damping
285
see if it is Rayleigh damping. The criterion for a symmetrical system to possess Rayleigh damping is derived by Caughey as
B3. Criterion of Rayleigh damping
It has been shown that if a damping matrix is formulated by Rayleigh damping, it can be decoupled by the real mode matrices extracted from its associated undamped system. Given a damping matrix, however, there is no need to work out proper coefficients to expand it into a series in order to
[CIW-WI
= vwf-‘[a
0314)
But this condition is only the necessary condition for unsymmetrical systems. Further conditions are required to ensure that the damped system possesses the real modes [9].
Pergamon
0045-7949(94)00611-3
& Smrrurrs
Vol. 57. No. 2. DD. 277-285, 1995 Copyright(’ 1995El&r Science Ltd Printed in Great Britain. All rights reserved M)45-7949195 59.50 + 0.00
FORMULATION OF RAYLEIGH DAMPING AND ITS EXTENSIONS Man Liu and D. G. Corman School
of Mechanical and Offshore Engineering, The Robert Gordon University, Schoolhill, Aberdeen AB9 IFR, Scotland, U.K. (Received
4 March
1994)
Abstract-The objective of this paper is to present a study on the formulation of Rayleigh damping for a damped linear mechanical system. By means of examining the existing theory on Rayleigh damping, extensions of Rayleigh damping are proposed in terms of a double series. The extended forms provide more choices in formulating a damping matrix. Two sets of empirically obtained data are utilized as examples to illustrate the derived formulations.
1. INTRODUCTION
ing force is linearly proportional to the velocity. Rayleigh dissipation function takes the form
Formulation of the governing equations of motion of a system described by mass inertia, stiffness and damping matrices can be obtained easiiy. Our comprehension of damping is, however, considerably less encyclopedic than that of mass and stiffness systems. As such, predicting vibration parameters with respect to damping is difficult in practice. The dynamic response and attenuation of vibration are predominantly controlled by damping. Therefore, using an assumption about damping in the analysis is a justified compromise. Even for experimental purposes, the assumption of damping has to be incorporated so that the obtained data can be processed. There are several damping assumptions available which are based on a single degree of freedom system and extended to multi degree of freedom systems. For a single degree of freedom system, damping theories are well established to include, for example, viscous damping, Coulomb damping and structural damping. A dashpot is frequently idealized as a viscous damping element, in which the damping force is assumed to be proportional to the relative velocity between the two ends of it. Coulomb damping is caused by the relative motion of surfaces sliding against each other, whilst structural damping is introduced, together with others, to model energy dissipation of continuous elements. For a continuous system, energy dissipation within materials is supposed to be caused by thermoelasticity and grain boundary viscosity, which involves microstructure characterization. In dealing with a multi degree of freedom system, all the above damping assumptions are applicable. Amongst these, viscous damping and structural (hysteretic) damping models are most commonly used [I]. The viscous damping model assumes that the damp-
Its
where p is an energy dissipation coefficient, C, are discrete damping factors and u(x, y, z, t) is a displacement function for which the normal finite element discretization is available which produces a set of discretized second-order differential equations derived by using the variational principle. The equations of motion of a linear dynamic system can be written as
Pa{*1 + [WI
+ ~~l~x~= VI>
(2)
where [Ml, [C] and [K] are the mass, damping and stiffness matrices and {x} and {f} are displacement and force vectors, respectively. For hysteretic damping, the damping term [C]{i} is replaced by i[H]{x}. The corresponding equations are
bflcf~ + WI + Wl){x) = u-1.
(3)
This equation is not in the real domain. The interpretation of this equation is that the damping force is proportional to displacement rather than velocity, and is 90” out of phase with the displacement. Accordingly, for harmonic vibration, the equivalent damping force can be written as i[H]{x} = o-‘[H]{i}, 271
{_t} = iw{x}.
(4)
278
Man
Liu and D. G. Gorman
It should be noted that hysteretic damping may only be used in the frequency domain. This hysteretic model is introduced based on experimental observations that damping ratios decrease as the frequency increases. It allows many continuous systems to be modelled adequately. The model will however lead to erroneous results for free vibration analysis, in the time domain, since the eigenvalues will have both positive and negative real parts, and the response will diverge if there is no work done by external forces. Due to the problems of obtaining a system’s damping information, there is no practical way of forming the physical damping matrix by using the finite element method even though damping exists in most mechanical systems or structures. Indeed, so common is damping that we tend to forget that it is not in the natural order of things for oscillation to decay. The damping properties of a system can vary according to the structural design or the material used. In the analysis of the dynamic response of any system the damping information is essential for satisfactory results. The common Rayleigh damping [2] given by
2. RAYLEIGH
DAMPING
The advantage of Rayleigh damping is that no alteration is introduced to the mode shapes and the eigensolution or calculated response is thus greatly simplified. General Rayleigh damping is described by a matrix series given by eqn (5) or eqn (Bl) in the Appendix. A two-parameter model, as shown in eqn (6) derived from this series, has been used on many occasions. Some large FEM codes such as NASTRAN and SAP5 have adopted the two-parameter model for dynamical response analysis. For this two-parameter model, it is specifically easy to obtain the parameters, x0 and 2,. by using a least squares fit. The equivalent decoupled form of eqn (5) is
This equation indicates that the damping ratios expressed in terms of the natural frequencies to odd power (2k - I), i.e. c1)-‘, ro’,c)‘, OJ’.W~, Corresponding to the two parameter model. damping ratios are 2[, = a,,(l,, ’ + %,W,.
is a reasonable approximation for small levels of damping. Using the Rayleigh damping will consequently result in the existence of real modes that can decouple the system’s equations of motion. The simplest case is for proportional damping consisting of only two terms from the above formula,
[Cl = %[W + al WI>
(6)
where c(~ and IX, are arbitrary constant coefficients. Historically, the Rayleigh damping is defined by eqn (6). The common Rayleigh damping described by eqn (5) is proposed by Caughey [2], for which the two-term model eqn (6) is only a special case when p =2.
Fig. I. Schematic
are the
_. the
(8)
From this two-parameter damping model, the relationship between the damping ratio and the natural frequency is schematically demonstrated by Fig. 1. Given more than two values of w, and the corrcsponding values of the damping ratios ;,, the coefficients may be determined. For many practical situations, the two-parameter model is not adequate to approximate the damping information obtained from modal analysis. Increasing the number of terms may help improve the model according to eqn (7). However, the decision is, whether to use expression (7) or eqn (5). The proportional damping may well be described by other polynomials. It is thus proposed to starch for
of the two-parameter
damping
model
Formulation
different forms from the normal Rayleigh and extensions are suggested here.
of Rayleigh
damping,
damping
279
Thus
3. EXTENSIONS OF THE RAYLEIGH DAMPING
3.1. Extended series
Rayleigh damping-negative
indexed
Mathematical extension. Recalling the Rayleigh damping from eqn (5) it is noted that the series of summation is from 0 to p - 1, where p is an integer. In fact if the system does not have any rigid mode, whose associated mode shape does not involve elastic deformation, thus it suffices to say w, # 0, the series can be extended to including negative power indexed terms. As opposed to the normal Rayleigh damping matrix denoted by [C,], this series is denoted by [C_ 1, such that -b-I)
[Cl = [Ml 1 ~k([~l-‘mk k=-I P-1 =
[M] 1 a-k([Ml-‘[KI)-k.
(9)
k=l
The proportionality or orthogonality of this series to the real modal matrix, [@I, can be obtained as
This equation
is based on the assumption
([M]-‘[K])-k
that
= [@I[ w;*“][@]-‘.
(11)
This assumption is elucidated in the Appendix. The extended Rayleigh damping is established by combining eqn (9) and the normal Rayleigh damping, i.e. [Cl = [C- I + [C,],
[Cl = WI
c
k=-I
Q([Ml- WIY
L”l
,go
ak([Ml-
‘[Kl)k
P-l
=
The corresponding
c ‘& ([Mlk=-(p-1,
L”l
orthogonal P-
(@;}r[C]{@j} = c
‘[K])‘.
relationship
(12)
is
’
(a_@,‘”
w+a,oi+C1201+.
.‘+$_,wfp-3
+ C(kWfk)d,,
(13)
k=O
where S,, is the Kronecker delta function, and the coefficient associated with UI~ is $ = c(_~ + a,.
>
. (14)
The normal Rayleigh damping series, which is positively indexed, is expanded to give a negatively indexed series. This gives more choices than the normal model in selecting various terms to form the Rayleigh damping for practical use. The two-parameter model can be simply procured by imposing ak = 0 and leaving two other remaining coefficients to be determined, for instance, k # 0, 1; or k # 1, - 1. A proper selection of the terms in the series has a heavy reliance on empiricism and the linear least squares fit depends categorically on the damping ratios observed in experiment for each individual system. Some examples are given in Section 4. Physical interpretation. In the above subsection, it has been proposed to formulate Rayleigh damping with a negatively indexed series. The same extension may be obtained by another way. According to eqn (5) it is noticed that the generator for [C,] is ([Ml-‘[K]). In fact, the standard eigenproblem of matrix [A] = ([Ml-‘[K]) is equivalent to the general eigenproblem of a dynamical system described by [K]{X} =I[M]{X}. From this expression, another equivalent standard eigenproblem is
wl-‘[w{~~
= ; {XI
where [K] is not singular. As result of this we can use the ([K]-‘[MI) instead to form a Rayleigh matrix, i.e.
p-1 +
EO
+
(15)
generator damping
p-1
[c,,l = LKl
c
I=0
ak([Kl-‘[Ml)k.
This form is similar to eqn (5). Its proportionality orthogonality can be verified according to the cedure outlined in the Appendix. It can be corroborated that apart from the first terms in the series, [C,,,] [eqn (16)] is equivalent [C_] [eqn (9)], since [K] = [M]([M]-‘[K]) ([K]-‘[M])A = ([Ml-‘[K]))“.
(16)
and protwo to and
Man
280
Liu and D. G. Gorman Correspondingly,
Thus
the damping
ratios are composed
of
x cQ([M]-‘[K])“’
‘.
(17) If only p2 = 1 and /J’~= 0, (n # 2) one has
Each term rather than the first and second, in eqn (I 6) may be identified in eqn (9). To sum up, if the normal Rayleigh damping matrix is considered as a result of using the stiffness matrix dominant generator ([Ml-‘[K]), then the proposed negatively indexed damping matrix is achieved by utilizing the mass matrix (or, as it is, the flexibility matrix) dominant generator ([K]-‘[MI). Combining the two renders the extension of the Rayleigh damping [eqn (12)]. For a discrete system, it is as easy to form a flexibility matrix as a stiffness matrix [3]. 3.2. Extended indexed
Rayleigh
damping-rational
fraction
series
Extending the work in the previous subsection, the Rayleigh damping model is adapted to any rational number indexed series. According to eqn (A6) in Appendix A, for any rational number, 7 = k/n, the yth root of a positive definite matrix [A] is
[AlA”” = ([Ml-‘[K])~“’ Accordingly, series is
another
ic,,l = I”l
c
(20)
, q. matrix series is thus
+
B,tcJ
(21)
Hence
pi1
[Cl = i: B,,[C,,l = WI i: n = I
n=I
k + + m,
in the matrix must have
.j = I,2 )
limit a, cuzk”’ I = 0,
k++3Z
limit a, W: = 0. h-+X
ak
(26)
(27)
Here wN is the maximum (highest order) natural frequency, wN = max{w,, co*, , w,, .). Thus
This means for a sufficient large number, k, the coefficients in the Rayleigh damping should be far less than the inverse of the maximum natural frequency to an appropriate power. It is also noted that in order that the damping ratios are positive, [, > 0, not all the coefficients xi are required to be positive. For the two parameter Rayleigh damping model [eqn (S)], x0 + X, W: > 0. This leads to a conclusion that at least one of the coefficients must be positive. Assuming a0 b 0 results in c(~2 -a, wf For each natural frequency, this is satisfied by
I=-@-,,
x
(25)
damping
”
ak([Ml-‘[Kl)”
(19)
i=-tJ-1)
[Cl = /II [Cl1+ Bz[Gl-t
each element
B, it is evident
Rayleigh
where n is an integer, n = 1,2, The general Rayleigh damping collectively formulated as
Consequently, a limit,
= 0,
(18)
= [@I[ W:“:“][@]-‘. extended
limit ak([M]m’[K])‘,“r
For n = 1,
[A]? = [Alk”’ = [P]diag{d~~“}[P]~‘. Thus based on eqn (B5) in Appendix that
Accordingly, as opposed to eqn (7) consisting of the natural frequency to only odd powers, it has terms related to the natural frequency to both odd and even powers. If p and q tend to infinity, the limit may exist if some conditions are satisfied. As with a normal series, the necessary, but not sufficient, condition is that
hz([Ml ‘[Kl)““i. (22)
This double series reduces to the normal Rayleigh damping proposed by Caughey in the case q = 1. Therefore, the integer indexed Rayleigh damping series is extended to both the negative and real number indexed series.
-a,
< a,min
l‘l 1 o?“J
Consequently, -a,min
1
i of”
I
QZr<
+m.
281
Formulation of Rayleigh damping For an underdamped additional constraint
system possessing [, < 1, an is imposed from eqn (8) as
fXO + cl,wf < 20,.
Likewise when [C] is Rayleigh damping, one has {4i}r[C]{$,} = O(i #j). Thus [C,] = 0. Accordingly it is found, as expected, that [C] = [C,].
(31)
{4l>W{4,} [Cal= WI PI
Hence
f 1 y 0,
2
I1
-a,min
(32)
w,
This is the relationship that the coefficients two parameter model must satisfy. 3.3. Extended
Rayleigh
for the
damping-unsymmetrical
systems The above extensions are acquired under the condition that the spatial matrices [Ml, [C] and [K] are symmetrical. For an unsymmetrical system, the Rayleigh damping can also be decoupled, since the following formula holds: [Ml-‘[K]
= [@]diag{oj}[@]-‘.
(33)
The derivation is given in Appendix B [eqn (Bl2)]. Likewise, the extended Rayleigh damping as described by eqns (12) and (22) can be diagonalized, and is applicable to unsymmetrical systems. 3.4. Approximate decoupling Proportionality is not an inherent property for any damping matrix, whether it is experimentally identified or theoretically formulated. For a damping matrix [C] that cannot be decoupled by means of the modal matrix transformation, Rayleigh damping is a reasonable approximation. In fact, a damping matrix [C] can be decomposed into two matrices:
[Cl = [c/l + [C”l
(34)
where [C,] is the Rayleigh damping matrix and [C,] is the matrix that makes up the difference between [C] and [C,]. Determination of [C,] and [C,] is easily achieved by the modal transformation
PIWI.
0 sym
(38)
I
It has been proved that neglecting [C,] is equivalent to neglecting an infinitesimal value corresponding to the second order term of the damping ratios when solving the complex natural frequencies [4]. For calculating the dynamic response of the system, Shahruz [.5] proposed the approximate decoupling of the equations of motion of a damped linear system in order to use the real model superposition principle [3]. It is considered that Rayleigh damping is the most convenient model for predicting system dynamic response [6], in particular for multi degree of freedom systems consisting of a large number of degrees of freedom. Because Rayleigh damping is proportional damping, the system governing equations of motion can be decoupled by the use of a modal transformation. Thus, the modal superposition method can be employed for performing the response analysis. Apart from this advantage, Rayleigh damping also offers an easy way of formulating the damping matrix by using the damping ratios acquired experimentally. The Rayleigh damping matrix can be obtained when the damping ratios and mode shapes are available. According to equation (B7), the damping matrix takes the form
[Cl = P-7
2w.1P-’
= WI [@I[ xp, 1Pmfl.
(39)
Alternatively, the damping matrix can be obtained by using the proposed Rayleigh damping matrix characterized by eqn (22). 4. EXAMPLES
Example 1
Pl’([c/l+ [C,l)[@l = Pl’[ClPl. Because
[@]r[M][@] = [I], it simplifies
(35)
to
[Cdl+ [Co1= Pl-‘PITKl[@lP-’ = ~~l~~lr~~l’~~l~~l~~lr’[~l.
(36)
Since the transformed Rayleigh damping matrix [@]‘[C,][@] consists of non-zero diagonal terms and zeros elsewhere, and [@]‘[C,][@] contains all nonzero off diagonal terms, one has
[Cdl= [w[@lr[ {~,)TCl{d4~I[@nm
(37)
This example illustrates the formulation of Rayleigh damping by using the fractional index series. As opposed to the two parameter (Rayleigh damping) model described by [C] = cr,[M] + E, [K], two terms are selected, they are [C] = c(, 6[M]([M]-‘[K])‘.16 and GI,[K], corresponding to (k = 1, n = 6) and (k = 1, n = I), respectively. The coefficients for the two-parameter model are identified by using the linear least-square fitting method. Given values of the natural frequencies and the corresponding values of the damping ratios, s0 and a, for the two-parameter model can be determined according to eqn (8), by using a straight line fit taking the y axis to be 21,~~ and the x axis to be
Man Liu and D. G. Gorman
282 Table
Mode
1. Damping
Angular frequency
from the two parameter Two-parameter Damping ratios
Damping ratios
0.507 0.541 0.764 1.356 2.718 3.924 4.184 5.082 5.349 6.965 7.664 7.881
2 3 4 5 6 I 8 9 10 11 12
ratios
13.605 12.765 9.123 5.345 3.118 2.612 2.561 2.471 2.463 2.524 2.589 2.612
14.900 5.000 6.800 3.400 1.800 7.000 1.300 2.500 3.000 2.300 3.900 1.100
t [C] = c+,[M]([M]-‘[K])“~
+ a, [K], the standard
Table
Mode
1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23
2. Damping
Identified modal Natural frequency (Hz) 108.9 293.1 360.0 380.4 414.0 483.8 553.1 591.1 732.2 872.5 902.1 1017.3 1028.6 1034.2 1045.6 1067.1 1093.2 1131.9 1187.3 1253.8 1310.0 1337.2 1432.5
t [Cl = ~~,~t~I([~l-‘[~l)’
ratios
5.65 3.91 1.87 2.06 2.70 2.55 2.62 2.00 I .66 1.45 1.20 0.40 0.23 0.58 0.60 1.17 0.73 1.07 0.70 0.82 0.81 0.68 0.60
a, = 0.0684 LX,= 0.0022
9.602 9.202 7.353 5.122 3.459 2.936 2.867 2.697 2.662 2.548 2.534 2.533
model? Standard deviation 1.49 x IO’
MI6” 0.0606 c(, = 0.0013
model. The standard deviation from the proposed two-term model may not be the minimum one.
Example
2
This example show the formulation of the Rayleigh damping matrix for a structural pipework used in modal analysis with thin coupling hoses [8]. Both the two-parameter model ([Cl = IZQ[M]+ c(,[K]) and the extended two-term model ([Cl = x,,.~[M]([M]~‘[K])“~ + tl, [K]‘14 + a, [K]) are utilized. The coefficients for both models are identified by using the least-squares fitting method. From the following table it is shown that, as with the first example, using a fractional power term will improve the fitting, and its standard deviation is smaller than the normal two-parameter model.
Two-parameter Damping ratios 9.24 3.41 2.77 2.62 2.40 2.04 1.78 1.66 1.32 1.98 1.04 0.91 0.90 0.89 0.88 0.86 0.83 0.80 0.75 0.70 0.66 0.65 0.59
4 + a, [K], the standard deviation
Two-term Damping ratios
is o2 = 2XA[‘.
from the two parameter
parameters Damping ratios
model Standard deviation 1.97 X 10-z
deviation
WT. For the extended two term model, its coefficients can be identified by using the linearized least-squares fit. It should be noted, however, that the relationship between the damping ratios and the natural frequencies are not linear, no matter whether the two-parameter model or the extended two-term model is used. For the two-parameter model, this relationship is expressed by eqn (8) and is illustrated by Fig. 1. It is misleading that the relationship between the damping ratios and the natural frequencies for the two-parameter model was illustrated as a linear one by Bathe and Wilson (Fig. 8.4 in Ref. [7]). From Table 1 extracted from Ref. [8] it is found that using a fractional power term will improve the fitting, and the associated standard deviation is smaller than that of the normal two-parameter
model and proposed model
model and the extended model Standard deviation 33.6397
2; = 1007.435 ct; = -7.966 x IO-’
02 = 2ZA5?
model
Two-term Damping ratios 5.37 3.10 2.72 2.63 2.48 2.22 1.99 1.89 1.54 1.25 1.20 1.00 0.98 0.97 0.95 0.92 0.88 0.82 0.74 0.64 0.56 0.53 0.40
modelt Standard deviation 7.8264
@114 56.9 I8 a; = -7.714 X 10-4
Formulation
of Rayleigh
In this example, the natural frequencies in Hertz are used directly (see Table 2), and the obtained coefficients t(.; are not the same as those found by using the angular frequencies, CX?.To convert from c(.J to CL,.,the associated scaling factors are (27r)G-‘), i.e. ~L.J = (27~)~-‘cr,., (y = l/4, 1). This is derived according to the relationship between the natural frequencies and the angular frequencies (rad SK’), (Hz) o, = 2r$. According to the above extended two-term model, the damping matrix is [C] = c+[M]([M]-‘[K])‘!~ + LX, [K]. Thus, the corresponding damping ratios are obtained
2&o, = 51,i4W/‘*+ tc,wf.
(40)
21,(276) = @~(2r!f)‘~~ + ‘xi(27rf;)‘.
(41)
Equivalently,
According to this equation, if the natural frequencies in Hz are used in fitting, the coefficients obtained are LY;’ = (271)27_‘c$, (y = l/4, 1). From the result shown in this table, it is evident that the proposed model renders a smaller standard deviation than the normal two parameter model. It is also noted that the second coefficient, CX;or aI, for both models is negative. Theoretical discussions given in Section 3.2 indicate that not all coefficients are required to be positive to produce a positively definite damping matrix. Summing up, this section has assimilated some points relating to the discussions in this paper. These two examples have shown the advantage of the extended Rayleigh damping model. The disadvantage of the extended model is in the computational aspects. The damping matrix created by using such terms to fractional powers requires decomposition of the matrix [Ml-‘[K]. The consequence is that the damping matrix does not have the same banded property that the mass and the stiffness matrices possess. 5. CONCLUDING
damping REFERENCES
L. Cronin, Eigenvalue and eigenvector determination of nonclassically damped dynamic systems.
1. D.
Comput. Struct. 36, 133-138 (1990). 2. T. Caughey, Classical normal modes in damped linear systems. J. appl. Mech. 27, 269-271 (1960). Elements of Vibration Analysis. 3. L. Meirovitch, McGraw-Hill, New York (1986). 4. M. Liu and J. Wilson, On the damping ratio of multi degree of freedom systems. Commun. appl. numer. Meth. 8, 265-271 (1992). Approximate decoupling of the 5. S. M. Shahruz, equations of motion of damped linear systems. J. Sound Vibr. 136, 51-64 (1990). 6. D. J. Ewins, Modal Tesfing: Theory and Practice. Research Studies Press, Taunton, Somerset (1984). 7. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis. Prentice-Hall, Englewood Cliffs, NJ (1976). interaction effects on vi8. M. Liu, Fluid-structural brations of pipework. Ph.D. Thesis, The Robert Gordon University, October (1993). 9. M. Liu and J. Wilson. Criterion for decounhne. dvnamic equations of motion of linear gyroscopic systems: AIAA J. 30, 2989-2991 (1992).
APPENDIX
Al
Transformation
A. ROOT MATRIX
of similitude
The nth root operation of matrices is defined here according to advanced matrix theories. Given a matrix [A], its kth root, [A]“!+is defined as [B], if [B]’ = [A].
(Al)
Accordingly, when [A] has a matrix of similitude which is a positively definite diagonal matrix (and [A] must be then positive definite)
PI = diag{d,},
WV
namely,
[Al = V’IPI[f’-’ then matrix
REMARKS
In this paper, Rayleigh damping is re-examined and two more extended Rayleigh damping forms are established. The index of the damping series has been expanded from a positive integer set, to a real number set. It has been attempted to add a new dimension to the understanding of proportional damping. The corresponding damping matrix is characterized by a double series. The proposed expressions have enhanced the Rayleigh damping and provide alternatives for formulating a proportional damping matrix. On the basis of the simplicity and power of Rayleigh damping, the extended fractional indexed Rayleigh damping models as proposed in Section 3, were fitted to available damping data obtained in the experimental modal analysis. Identification of Rayleigh damping using both the two parameter model and the extended model is thus successfully demonstrated.
283
[B] is found
(A3)
to be
[B] = [P][D]‘/‘[P]-‘[P]diag{dj!‘}[P]-‘,
(A4)
[P] is a transformation matrix of similitude. The eigenvalues of matrix [A] are not affected by this transformation. Thus the diagonal [D] is the spectrum matrix of [A], containing all the eigenvalues. Therefore [A]‘:” = [B] = [P]diag{dj;k)[P]-‘. Hence
for any rotational
number,
y = k/n,
[B]
[A];’ = [Apfl = [B]’ = [B][B]
= [P]diag{d:‘“}[P]-‘. The inverse can be achieved
(A5)
(A6)
easily as
([A];‘)-’ = ([P]diag{dt!“}[P]-‘))’ = ([PI-‘)-’
= [P]diag{d;”
diag-‘{df,“}([P])-’ “}[P]-‘.
(A7)
Man Liu and D. G. Gorman
284 Thus in general,
Consequently ([A]‘) = ([P]diag{d;}[P]-‘),
y can be both positive
[M]~‘[K]
(Ag) hence
and negative.
([Ml-‘[K])”
A2. Orthogonal transformation If [PI-’ = [P]r, the above transformation is called the orthogonal transformation since [P] [PIT = [I]. When [A] is a symmetrical and positive definite matrix, there is bound to be a orthogonal matrix [D] into which [A] may be decomposed, i.e.
[Al = [~IPIW~
= [@I[ WY I[@]“.
Based on eqn (B6) and in accordance and (B4). it arrives at
(JW
with eqns (Bl), (B3)
(A9)
= [@17[M][@] 1 q diag{w,‘“) k=”
where [P] is the eigenvector matrix, and [D] contains the eigenvalues of [A]. For an unsymmetrical matrix [A], it can be expressed as
(AlO)
PI = [Ql[~l[f’l’~ where [P] eigenvector
W)
= [@I[ w; ][@I~‘.
and [Q] are, respectively, the matrices and [Q][P]‘= [I].
right
and
left
Thus the damping relation is
matrix is diagonalized.
The orthogonal
APPENDIX B. RAYLEIGH DAMPING
The normal form:
Rayleigh
damping
matrix
is in the following
where 6,, is the Kronecker
function,
I i =j i 0 i #j’
6,, = p-1
[Cl = Pfl 1 ~,w-‘m”~ 1-0
@I)
where [Ml, [C] and [K] are, respectively, the inertial mass, the viscous damping, and stiffness matrices. When p = 2, one has
[Cl = %[W + a, [Kl.
(W
This is the basic form of the Rayleigh damping matrix. Rayleigh damping is known as the proportional damping. As this name stands for, the Rayleigh damping matrix can be diagonalized by the real mode matrix [@I extracted from solving the eigenvalues and eigenvectors of its associated undamped system ((M, K)), indicating on this occasion that the damping matrix shares the same eigenvectors with ((M, K)). Thus Rayleigh damping does not at all introduce any change in the vibration patterns. Moreover, the resulted governing equations of motion can be decoupled completely. Each of the decoupled equations can be analyzed independently and the normal mode superposition principle is applicable. Accordingly, the system’s eigensolutions can be sought from its undamped version of the model. Once the natural frequencies and mode shapes have been extracted, the vibratory characteristics will be completed by adding the damping information. It is therefore Rayleigh damping that most researchers would relish to use. Bl.
The significance of the orthogonality which results in decoupling of the governing equations of motion implies that the real mode superposition principle is applicable for system analysis. In addition, the spatial parameters are associated with the modal parameters by these orthogonal relations. B2. Unsymmerrical systems When discussing a general damped vibration system, spatial matrices are not necessary to be symmetrical. In this case, the left and right eigenvector matrices, [Y] and [@I, are involved and two eigensolutions are required to yield the complete information of vibratory characteristics. As will be shown now, the Rayleigh damping matrix [Bl] can be diagonalized by the left and right eigenmatrices. Similar to symmetrical systems, the orthogonality relations in the matrix form are [Y]‘[M][@]
[Ml-’ =[@][‘f’]‘,
= I,
[Ml-‘[K]
[@]‘[K][@]
= diag{cu:).
(B3)
It can be easily shown that the real modes are also orthogonal in terms of the Rayleigh damping matrix. From eqn (B3) we have [Ml-‘[@][@lr,
[K] = [@]-‘I w; ][@I-‘.
(B4)
(Y]‘[K][@]
= diag{of}.
(BIO)
[K] = [Y]-‘[
w; ][@I-‘.
PI 1)
Consequently = [@I[ w; ][@I-‘.
(Bl2)
This turns out to be identical in form to the one for symmetrical systems, eqn (B5). Using eqns (B6), (BIO) and (Bl I) yields p-1
w~clPl= [@]r[M][@]
= I,
Equivalently,
Symmetrical systems
For a symmetrical dynamical system, its mass, damping and stiffness are symmetrical and non-negative definite. The corresponding undamped version of the system concerned possesses eigenvalue and normalized eigenvector matrices. The matrix form of the real mode orthogonal relation is
(B9)
[w[~l (
= [Y]r[M][@]
c %Pl[W:” 1Pr’ [@I I=” > c s(~diag(w,‘“} I=”
Formulation This has demonstrated diagonalized.
that
the damping
matrix
can
of Rayleigh be
damping
285
see if it is Rayleigh damping. The criterion for a symmetrical system to possess Rayleigh damping is derived by Caughey as
B3. Criterion of Rayleigh damping
It has been shown that if a damping matrix is formulated by Rayleigh damping, it can be decoupled by the real mode matrices extracted from its associated undamped system. Given a damping matrix, however, there is no need to work out proper coefficients to expand it into a series in order to
[CIW-WI
= vwf-‘[a
0314)
But this condition is only the necessary condition for unsymmetrical systems. Further conditions are required to ensure that the damped system possesses the real modes [9].