1 adaptive mode superposition and acceleration technique with application to frequency response function and its sensitivity

Mechanical Systems
and
Signal Processing
Mechanical Systems and Signal Processing 21 (2007) 40–57
Adaptive mode superposition and acceleration technique with
application to frequency response function and its sensitivity
Zu-Qing Qu
Ã
Michelin Americas Research & Development Corporation, 515 Michelin Road, Greenville, SC 29605, USA
Received 13 December 2005; received in revised form 31 January 2006; accepted 5 February 2006
Available online 22 March 2006
Abstract
An adaptive mode superposition and acceleration technique (AMSAT) is proposed and implemented into the
computation of frequency response functions (FRFs) and their sensitivities. Based on the mode superposition and mode
acceleration methods for the FRFs, m-version, s-version, and ms-version adaptive schemes are presented. In these schemes,
the error resulted from the mode truncation and/or series truncation is, at first, estimated at every specific frequency,
respectively. Then, one more mode (called m-version), or one more level of the series (called s-version), or the combination
(called ms-version) is included in the computation of the FRF when its error is greater than the error tolerance. The new
FRF is recalculated and its error is re-evaluated. This procedure is repeated until all the errors fall below the specified
value. Although only the implementation of FRFs and their sensitivities is demonstrated in this paper, the proposed
adaptive technique may be applied to the computation of dynamic responses in time domain and their sensitivities,
sensitivity of eigenpairs, modal energy, etc. One numerical example is included to demonstrate the application of the
proposed adaptive schemes. The results show that the present schemes work very well. The s- and ms-version adaptive
schemes are much more efficient than m-version scheme. Since the intention of this paper is to propose these new
procedures, the damping, particularly the non-classical damping, is not included due to the complexity.
r 2006 Elsevier Ltd. All rights reserved.
Keywords: Numerical methods; Modal analysis; Adaptive technique; Frequency response function; Sensitivity analysis; Mode
superposition; Mode acceleration
1. Introduction
Frequency response function (FRF) is a very important characteristic of a dynamic system because it
includes not only the resonance and antiresonance frequencies of the system but also the amplitudes of the

responses under unit excitations. Due to its good qualities to represent a dynamic model, it has been playing a
very important role in many areas such as finite-element model updating or modification, structural damage
detection or identification, dynamic optimisation, system or parameter identification, vibration and noise
control, etc.
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The dynamic equilibrium of an n-degree-of-freedom undamped system in time domain is generally given by
M
€
XðtÞþKXðtÞ¼FðtÞ, (1)
where M and K are real symmetric mass and stiffness matrices , respectively. X(t) and F(t) are, respectively, the
displacement an d exciting force or load vector s. The corresponding dynamic equilibrium in frequency domain
is given by
ðK À o
2
MÞXðoÞ¼Fð oÞ, (2)
where o is the circular frequency of exciting forces or loads and Z(o) ¼ (KÀo
2
M) is referred to as dynamic
stiffness matrix. The frequency response vector X(o) may be expressed as
XðoÞ¼ðK À o
2
MÞ
À1
FðoÞ. (3)

The matrix HðoÞ¼ðK À o
2
MÞ
À1
is usually called receptance matrix and each element of this matrix
represents a single-input–single-output FRF. Clearly, the receptance matrix is an inverse matrix of the
dynamic stiffness matrix, i.e.,
HðoÞ¼ZðoÞ
À1
. (4)
ARTICLE IN PRESS
Nomenclature
E truncated error vector of the FRFs
F force or load vector
H (n Ân) receptance matrix
I (n Ân) identity matrix
K (n Ân) stiffness matrix
K
R
ð
¯
q Â
¯
qÞ reduced stiffness matrix defined in Eq. (31)
M (n Ân) mass matrix
M
R
ð
¯
q Â

¯
qÞ reduced mass matr ix defined in Eq. (31)
n number of total degrees of freedom of a model
p design parameter; number of lowest modes to be solved by subspace iteration
q eigenvalue shifting value
Q ð
¯
q Â
¯
qÞ eigenve ctor matrix of a reduced model
t time
X response vector
Z (n Ân) dynamic stiffness matrix
K (n Ân) eigenvalue matrix
U (n Ân) eigenvector matrix
l
i
ith eigenvalue
/
i
ith eigenvector
o circular frequency of exciting forces or loads
o
min
under boundary value of the exciting frequencies
o
max
upper boundary value of the exciting frequencies
X ð
¯

q Â
¯
qÞ eigenva lue matrix of a reduced model
Superscript
l FRFs at the low frequency range
m FRFs at the middle frequency range
T matrix transpose
Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 41
The element of the vector X(o) can be single-input–single-output as well as multi-input–single-output FRF.
This depends on the exciting force vector F(o). In the following the more general form of FRF X(o) will be
considered.
Eq. (2) can be looked as a group of linear equations and solved directly and exactly when the system has a
small number of degrees of freedom or only the FRFs at a few frequencies are interested. In this approach, the
decomposition of the system dynamic matrix, forward and back substitution processes are involved at each of
the exciting frequencies. Consequently, it is very computationally expensive when the numbers of the degrees
of freedom and the exciting frequencies are large.
In the mode superposition method (MSM) [1], the FRF is expressed as the summation of the
contributions of all modes in the model. The decomposition, forward and backward substitutions
become unnecessary. However, the eigenvalues and their corresponding eigenvectors should be
available. As we know, it is very difficult and unnecessary to calculate all the eigenpairs, eigenvalues and
the corresponding eigenvectors, of a large model. This means that the mode truncation scheme is
generally utilised and the mode-truncated error is, hence, introduced. To reduce the truncated errors, mode
acceleration method (MAM) [2–4] has been proposed. This approach can improve the accuracy of FRFs
very quickly. However, several problems will be encountered when implementing the MAM to practical
problems: (1) How do we know that the results have the necessary accuracy? (2) How many modes are
necessary to evaluate the FRF accurately? (3) How many items, levels, should be considered in the power
series? (4) Is it possible to use the modes and levels efficiently because their effect on the accuracy changes with
frequency?
An adaptive technique, called adaptive mode superposition and acceleration technique (AM SAT), will be
proposed to solve these problems mentioned above. The MSM and the MAM are to be reviewed concisely in

Section 2. The FRFs both at the low frequency range and at the middle frequency range are considered. For
convenience, the classical subspace iteration method together with the inverse iteration method is listed in
Section 3. The ideas and convergent properties of the m-version, s-version, and ms-versi on adaptive
approaches are presented in Section 4. One scheme for implementing the technique is proposed for each
approach. A numerical example is provided in Section 5. The advantages and disadvantages of each scheme
are discussed in this section. Although only the FRFs and their sensitivities are utilised to demonstrate the
adaptive techniques, the proposed schemes are valid for many situations where the MSM and MAM are
required [5–9].
2. MSM and MAM
Assume the eigenvalue and eigenvector matrices of the model defined in Eq. (1) are K and U, and
K ¼ diagðl
1
; l
2
; ; l
n
Þ; ðl
1
pl
2
p ÁÁÁpl
n
Þ; U ¼½/
1
; /
2
/
n
, (5)
where l

i
and /
i
are the ith eigenvalue and eigenvector. K and U satisfy the following eigenequation and
orthogonalities
KU ¼ MUK; U
T
KU ¼ K ; U
T
MU ¼ I; (6)
where superscript T denotes matrix transpose. I is an identity matrix of n Ân. From Eq. (6) one obtains
K
À1
¼ UK
À1
U
T
; M ¼ U
ÀT
U
À1
, (7)
ðK À o
2
MÞ
À1
¼ UðK À o
2
IÞ
À1

U
T
. (8)
2.1. Mode superposition method
Introducing Eq. (8) into Eq. (3) leads to
XðoÞ¼UðK Ào
2
IÞ
À1
U
T
F: (9)
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5742
The FRFs may be expanded in the modal space as
XðoÞ¼
X
n
r¼1
/
T
r
F
l
r
À o
2
/
r
. (10)

As aforementioned, the mode truncation scheme is usually necessary for the mode superposition. According
to the value of the exciting frequency, the mode truncation can be divided into middle–high-mode truncation
and low–high-mode truncation [3,4]. The ‘‘low’’, ‘‘middle’’, and ‘‘high’’ frequency ranges are defined in quality
in this paper rather than in quantity. For example, the low frequency range denotes that this frequency range
covers the lowest natural frequency and several higher orders of frequency. The middle frequency range
indicates that this range is away from the lowest frequency. Those frequencies that are far away from the
lowest frequency are denoted by high frequencies. Definitely, these definitions highly depend upon the value
and density of natural frequencies. However, these definitions should not have effect on the procedures to be
proposed.
In the middle–high-mode truncation scheme, both the middle and the high modes of the system are
truncated, i.e., only the modes at the low frequency range are used to calculate the FRFs. If the low L modes
are selected, the FRFs defined in Eq. (10) become
X
l
1
ðoÞ¼
X
L
r¼1
/
T
r
F
l
r
À o
2
/
r
. (11)

When the exciting frequencies lie in the middle frequency range of the system, the number of the kept modes
will be very large if Eq. (11) is utilised to calculate the FRFs. Hence, the low–high-mod e truncation scheme is
applied. If the L
1
th through L
2
th modes are selected as the kept modes, the FRFs can be expressed as
X
m
1
ðoÞ¼
X
L
2
r¼L
1
/
T
r
F
l
r
À o
2
/
r
. (12)
The superscript l and m in Eqs. (11) and (12) denote the FRFs at the low and the middle frequency ranges,
respectively.
2.2. Mode acceleration method

The inverse of matrix ðK Ào
2
IÞ in Eq. (8) may be expanded as a power series [10], i.e.,
ðK À o
2
IÞ
À1
¼
X
S
s¼1
ðo
2
K
À1
Þ
sÀ1
K
À1
þðo
2
K
À1
Þ
S
ðK À o
2
IÞ
À1
, (13)

where S is the level of the mode acceleration and may be any integer larger than zero. S ¼ 0 indicates that no
power series, mode acceleration, is adopted. Substituting Eq. (13) into Eq. (9), the FRFs can be expressed as
XðoÞ¼U
X
S
s¼1
ðo
2
K
À1
Þ
sÀ1
K
À1
"#
U
T
F þ U ðo
2
K
À1
Þ
S
ðK À o
2
IÞ
À1
ÂÃ
U
T

F: (14)
Using Eq. (7), one has
UK
Às
K
À1
U ¼ U ðK
À1
U
T
U
ÀT
U
À1
UÞÁÁÁðK
À1
U
T
U
ÀT
U
À1
UÞ
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflffl{
s
K
À1
U ¼ðK
À1
MÞÁÁÁðK

À1
MÞ
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
s
K
À1
. (15)
Introducing Eq. (15) into the right-hand side of Eq. (14) results in
XðoÞ¼
X
S
s¼1
ðo
2
K
À1
MÞ
sÀ1
K
À1
F þ
X
n
r¼1
o
2
l
r

S

/
T
r
F
l
r
À o
2
/
r
. (16)
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 43
If the lowest L modes are selected as the kept modes, the FRFs can be expressed as
X
l
2
ðoÞ¼
X
S
s¼1
ðo
2
K
À1
MÞ
sÀ1
K
À1
F þ

X
L
r¼1
o
2
l
r

S
/
T
r
F
l
r
À o
2
/
r
. (17)
As defined above, no power series will be applied when S ¼ 0. This means that the first item on the right-
hand side of Eq. (17) will be zero and the mode superposition expression (11) is resulted. Therefore, the MSM
may be looked as a particular case of the MAM.
Considering the eigenvalue shifting technique, one has
ðK À o
2
MÞ¼ð
¯
K À
¯

o
2
MÞ, (18)
where
¯
K ¼ K À qM;
¯
o
2
¼ o
2
À q. (19)
Usually, the eigenvalue shifting value q is given by
q %
o
2
min
þ o
2
max
2
(20)
and should satisfy qal
r
ðr ¼ 1; 2; ; nÞ. o
min
and o
max
are the under and upper boundary values of the
exciting frequencies. Substituting Eq. (18) into Eq. (3), the FRFs are obtained as

XðoÞ¼ð
¯
K À
¯
o
2
MÞ
À1
F: (21)
When the mode acceleration is applied, the FRFs can be expressed as
XðoÞ¼
X
S
s¼1
ð
¯
o
2
¯
K
À1
MÞ
sÀ1
¯
K
À1
F þ
X
n
r¼1

o
2
À q
l
r
À q

S
/
T
r
F
l
r
À o
2
/
r
. (22)
Assume that the L
1
th through the L
2
th modes are selected as the kept modes when the low–high-mode
truncation scheme is applied. The FRFs at the middle frequency range of the system are given by
X
m
2
ðoÞ¼
X

S
s¼1
ð
¯
o
2
¯
K
À1
MÞ
sÀ1
¯
K
À1
F þ
X
L
2
r¼L
1
o
2
À q
l
r
À q

S
/
T

r
F
l
r
À o
2
/
r
. (23)
Similarly, Eq. (12) is a particular case of Eq. (23).
2.3. Sensitivity of FRF
Using Eq. (4), the sensitivity matrix of the FRF matrix may be expressed as
qHðoÞ
qp
¼
qZ
À1
ðoÞ
qp
(24)
in which p is a design parameter. According to the theory of matrix, one has
qHðoÞ
qp
¼ Z
À1
ðoÞ
qZðoÞ
qp
Z
À1

ðoÞ¼HðoÞZ
;p
ðoÞHðoÞ, (25)
where Z
;p
ðoÞ is the sensitivity matrix of the dynamic stiffness matrix with respect to the design parameter.
Similar expressions were used by Lin and Lim [9] for the case that the design parameter is a mass or stiffness at
a certain coordinate location. The sensitivities of FRF vector XðoÞ is given by
qXðoÞ
qp
¼ Z
À1
ðoÞ
qZðoÞ
qp
Z
À1
ðoÞF ¼ HðoÞZ
;p
ðoÞXðoÞ. (26)
The sensitivity of FRF at the location of x
i
may be expressed as
qx
i
ðoÞ=qp ¼ h
i
ðoÞZ
;p
ðoÞXðoÞ (27)

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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5744
in which h
i
ðoÞ is the ith row or column vector of the receptance matrix. The sensitivity matrix is generally
highly sparse and dependant upon the design parameter. If p is a local parameter, the matrix Z
;p
ðoÞ is locally
populated. Therefore, it is unnecessary to make all the elements of h
i
ðoÞ and XðoÞ available before evaluate
the sensitivity.
3. Subspace iteration and inverse iteration methods
As stated above, the eigenpairs of the system should be available before the MSM and MAM are applied to
compute the FRFs. Subspace iteration method, the Lanczos method, conjugate gradient method,
condensation method, and Ritz vector method are the frequently used approaches to extract partial
eigenpairs from large degrees of freedom systems [11]. The basic subspace iteration [12] will be listed concisely
in the following.
The basic subspace iteration is a combination of the simultaneous inverse iteration an d the Rayleigh–Ritz
procedure. Its objective is to solve for the lowest p eigenpairs satisfying the gen eralised eigenvalue Eq. (6). If
the first p eigenvalues and their corresponding eigenvectors are considered, the eigenproblem (6) can be
rewritten as
KU
p
¼ MU
p
K
pp
, (28)
where U

p
and K
pp
are composed of the first p eigenvectors and eigenvalues, respectively.
If a set of independent initial vectors X
ð0Þ
¯
q
, where
¯
q ¼ minð2p; p þ 8Þ suggested by Bathe, are available, the
basic subspace iteration method obtains the new set of approximate eigenve ctors by the following two steps:
(a) A new subspace is obtained by using the simultaneous inverse iteration method, i.e.,
KY
ðiþ1Þ
¯
q
¼ MX
ðiÞ
¯
q
. (29)
If the iterations proceeded use Y
ðiþ1Þ
¯
q
as the next estimation of the subspace, then the subspace would
collapse to a subspace of dimension one and only contains the eigenvector corresponding to the lowest
eigenvalue.
(b) To prevent the collapse, the Rayleigh–Ritz procedu re is adopted, i.e.,

K
ðiþ1Þ
R
Q
ðiþ1Þ
¼ M
ðiþ1Þ
R
Q
ðiþ1Þ
X
ðiþ1Þ
, (30)
where Q
ðiþ1Þ
and X
ðiþ1Þ
are the eigenvectors and eigenvalues of the reduced model ( K
ðiþ1Þ
R
and M
ðiþ1Þ
R
). The
reduced matrices are given by
K
ðiþ1Þ
R
¼ Y
ðiþ1Þ

¯
q

T
KY
ðiþ1Þ
¯
q
; M
ðiþ1Þ
R
¼ Y
ðiþ1Þ
¯
q

T
MY
ðiþ1Þ
¯
q
. (31)
Hence, the (i þ 1)th approximation of the first
¯
q eigenvectors X
ðiþ1Þ
¯
q
is
X

ðiþ1Þ
¯
q
¼ Y
ðiþ1Þ
¯
q
Q
ðiþ1Þ
. (32)
As i increases, the vectors X
ðiþ1Þ
¯
q
and the values in matrix X
ðiþ1Þ
will, respectively, converge to the
eigenvectors U
p
and the eigenvalues in matrix K
pp
provided that the initial vectors X
ð0Þ
¯
q
are not orthogonal to
one of the required eigenvectors.
If the eigenvalue shifting technique defined in Eqs. (18)–(20) is applied to the subspace iteration approach,
the p eigenpairs around the frequency
ffiffiffi

q
p
will be obtained. If only one eigenpair is required,
¯
q is set as one and
the su bspace iteration method becomes the inverse iteration method.
4. Adaptive mode superposition and acceleration technique
4.1. m-version
The FRFs are expressed as the summation of all the contributions of each mode as shown in Eq. (10). It will
be replaced by Eqs. (11) and (12) when the mode truncation schemes are used. The corresponding truncated
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 45
errors of the FRFs resulted from Eqs. (11) and (12) are given by
E
l
1
ðoÞ¼
X
n
r¼Lþ1
/
T
r
F
l
r
À o
2
/
r

, (33)
E
m
1
ðoÞ¼
X
L
1
À1
r¼1
/
T
r
F
l
r
À o
2
/
r
þ
X
n
r¼L
2
þ1
/
T
r
F

l
r
À o
2
/
r
. (34)
Define /
rj
as the jth element of the vector u
r
. The jth elements e
l
j
ðoÞ and e
m
j
ðoÞ of the error vectors E
l
1
ðoÞ and
E
m
1
ðoÞ exp ressed in Eqs. (33) and (34) become
e
l
j
ðoÞ¼
X

n
r¼Lþ1
/
T
r
F
l
r
À o
2
j
rj
, (35)
e
m
j
ðoÞ¼
X
L
1
À1
r¼1
/
T
r
F
l
r
À o
2

j
rj
þ
X
n
r¼L
2
þ1
/
T
r
F
l
r
À o
2
j
rj
. (36)
The upper boundaries of the absolute values of these two errors are given by
e
l
j
ðoÞ







p
X
n
r¼Lþ1
/
T
r
F
l
r
À o
2








jj
rj
j, (37)
e
m
j
ðoÞ







p
X
L
1
À1
r¼1
/
T
r
F
l
r
À o
2








jj
rj
jþ
X
n

r¼L
2
þ1
/
T
r
F
l
r
À o
2








jj
rj
j. (38)
Clearly, the total values of all items on the right-hand side of Eqs. (37) and (38) become smaller
and smaller when the number of the modes included in the mode superposition increases. This means
that the values of the upper boundary reduce with the increase of the number of modes included.
Therefore, the FRF obtained from Eq. (11) or (12) converges to the exact values when the number of modes
increases.
However, we do not know how many modes are enough to compute the FRFs with the prescribed accuracy.
Also, the error is a function of the exciting frequencies as shown in Eqs. (35) and (36). For the same prescr ibed
accuracy of the FRFs, different numbers of modes might be required at different frequencies. Hence, m-

adaptive technique becomes necessary.
In the m-version adaptive technique, one or several more modes are included in the mode superposition
only at the frequencies that the corresponding FRs have higher errors than the prescribed. The following
scheme shows the main logic of this technique.
1. Decompose the stiffness matrix K ¼ LU.
2. Use subspace iteration to extract pðX2Þ eigenpairs.
3. Evaluate vectors R
r
¼ /
T
r
F/
r
ðr ¼ 1; 2; ; p À 1Þ.
4. Compute the initial approximation of the FRFs at all frequencies using the p À1 modes:
X
ð0Þ
¼
X
pÀ1
r¼1
1
l
r
À o
2
R
r
.
5. For m ¼ p; p þ1; p þ 2; , do loop:

5.1. Select the mth eigenpair or calculate it using inverse iteration together with orthogo nalisation process.
5.2. Compute the constant vector R
m
¼ /
T
m
F/
m
.
5.3. For the frequencies at which the FRFs do not converge:
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5746
5.3.1. Evaluate the incremental of the FRFs:
DX
ðmÞ
ðoÞ¼
1
l
m
À o
2
R
m
.
5.3.2. Compute the total of the FRFs:
X
ðmÞ
ðoÞ¼X
ðmÀ1Þ
ðoÞþDX

ðmÞ
ðoÞ.
5.3.3. Check the convergence:
Z
X
¼
DX
ðmÞ
ðoÞ




X
ðmÞ
ðoÞ




pe.
5.4. If the FRFs at all the frequencies are convergent, exit this loop.
6. Output the results.
7. Similarly, the m-adaptive scheme for the FRFs at the middle frequency range may be obtained. For that
case, the subspace iteration and inverse iteration methods should be used with the eigenvalue shifting
technique.
4.2. s-version
The idea of the s-adaptive, series-adaptive, technique comes from Eq. (17) or (23). The incremental of the
FRFs from the level of S À 1toS may be obtained from these two equations as
DX

l
2
ðoÞ
S
¼ðo
2
K
À1
MÞ
SÀ1
K
À1
F À
X
L
r¼1
o
2
l
r

SÀ1
/
T
r
F
l
r
/
r

, (39)
DX
m
2
ðoÞ
S
¼ð
¯
o
2
¯
K
À1
MÞ
SÀ1
¯
K
À1
F À
X
L
2
r¼L
1
o
2
À q
l
r
À q


SÀ1
/
T
r
F
l
r
/
r
. (40)
The series truncated errors resulted from these two equations are given by
E
l
2
ðoÞ¼
X
n
r¼Lþ1
o
2
l
r

S
/
T
r
F
l

r
À o
2
/
r
, (41)
E
m
2
ðoÞ¼
X
L
1
À1
r¼1
o
2
À q
l
r
À q

S
/
T
r
F
l
r
À o

2
/
r
þ
X
n
r¼L
2
þ1
o
2
À q
l
r
À q

S
/
T
r
F
l
r
À o
2
/
r
. (42)
From the above two equations, the convergent conditions of Eqs. (17) and (23) may be defined as [3]
o

2
max
ol
Lþ1
, (43)
l
L
1
À1
oo
2
min
; l
L
2
þ1
4o
2
max
. (44)
Eq. (43) indicates that the eigenvalue of the truncated mode should be higher than the square of the highest
exciting frequency. Eq. (44) shows that the frequencies corresponding to the truncated modes should lie
outside of the exciting frequency range.
Similarly, the upper boundaries of the jth element of error vectors E
l
2
ðoÞ and E
m
2
ðoÞ are given by

e
l
j
ðoÞ






p
X
n
r¼Lþ1
o
2
l
r

S
/
T
r
F
l
r
À o
2









jj
rj
j, (45)
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 47
e
m
j
ðoÞ






p
X
L
1
À1
r¼1
o
2
À q

l
r
À q








S
/
T
r
F
l
r
À o
2








jj
rj

jþ
X
n
r¼L
2
þ1
o
2
À q
l
r
À q








S
/
T
r
F
l
r
À o
2









jj
rj
j. (46)
When conditions (43) and (44) are satisfied, the coefficie nts ðo
2
=l
r
Þ
S
and ðo
2
À q=l
r
À qÞ
S
in Eqs. (45) and
(46) decrease with the increase of the level S. This makes the whole rth items at the right-hand side of Eqs. (45)
and (46) smaller and smaller.
From the discussion above, the accuracy of the FRFs may be improved by increasing the level of the MAM.
However, we do not know how many levels of the MAM are necessary for the purpose of accuracy. Research
also showed that the MAM has different effect on the accuracy of the FRFs at different frequencies [3].
Consequently, the s-adaptive technique is necessary. One more level of the MAM is only included at the
frequencies that the corresponding FRs have higher errors than the error tolerance. Its main logic is shown in

the following scheme:
1. Decompose the stiffness matrix K ¼ LU.
2. Evaluate the natural frequencies and the corresponding mode shapes using subspace iteration.
3. Select the L and calculate the constant vectors R
r
¼ /
T
r
F/
r
for r ¼ 1; ; L.
4. Compute the initial approximation of the FRFs at all frequencies
X
ð0Þ
ðoÞ¼
X
L
r¼1
1
l
r
À o
2
R
r
. (a)
5. For S ¼ 1; 2; use the MAM:
5.1. Calculate Y ¼ MX
ðSÀ1Þ
A

(Y ¼ F for S ¼ 1).
5.2. Solve for X
ðSÞ
A
from equation LUX
ðSÞ
A
¼ Y using the forward and backward substitutions.
5.3. For the frequencies at which the FRFs do not converge:
5.3.1. Evaluate the incremental of the FRFs
DX
ðSÞ
ðoÞ¼o
2SÀ2
X
ðSÞ
A
À
X
L
r¼1
o
2SÀ2
l
S
r
R
r
.
5.3.2. Compute the total of the FRFs

X
ðSÞ
ðoÞ¼X
ðSÀ1Þ
ðoÞþDX
ðSÞ
ðoÞ.
5.3.3. Check the convergence
Z
X
¼
DX
ðSÞ
ðoÞ




X
ðSÞ
ðoÞ




pe.
5.4. If the FRFs at all the frequencies are convergent, exit this loop.
6. Output the FRFs and other results.
The s-adaptive scheme for the FRFs at the middle frequency range may be similarly obtained. For that
case, the subspace iteration and inverse iteration methods should be used with the eigenvalue shifting

technique.
4.3. ms-version
From the error Eqs. (45) and (46), we know that the accuracy of the FRFs increa se with the increase
of the level of the MAM. However, the practical accuracy might become worse if the level is too high
because of the numerical truncated error in computation. Hence, we cannot improve the accuracy of
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5748
FRFs and their sensitivities by increasing the levels of the MAM infinitely. Fortunately, as shown in Eqs. (45)
and (46), the errors may also be reduced by increasing the number of modes considered. Consequently, the m-
version adaptive scheme will be utilised to these approximate FRs which errors are higher than the prescribed
value after the maximum level of the MAM is used . This adaptive technique is called ms-version. The main
logic is
1. Decompose the stiffness matrix K ¼ LU.
2. Evaluate the natural frequencies and the corresponding mode shapes using subspace iteration.
3. Select the L and S
max
; calculate the constant vectors R
r
¼ /
T
r
F/
r
for r ¼ 1; ; L.
4. Compute the initial approximation of the FRFs using Eq. (a) at all the frequencies.
5. For S ¼ 1; 2; ; S
max
use the MAM.
Steps 5.1 through 5.3 are the same as those in the s-version adaptive scheme. If the FRFs at all the
frequencies are convergent, exit this loop and go to step 7.

6. For m ¼ L þ1; L þ2; ; do loop.
Steps 6.1 through 6.4 are similar to the steps 5.1 through 5.4 in the m-version adaptive scheme except the
incremental of the FRFs
DX
ðmÞ
ðoÞ¼
o
2
l
m

S
max
1
l
m
À o
2
R
m
.
7. Output the FRFs and other results.
The ms-adaptive scheme for the FRFs at the middle frequency range may be obtained similarly. For that
case, the subspace iteration and inverse iteration methods should be used with the eigenvalue shifting
technique.
5. Numerical exampl e and discussions
A two-dimensional plane frame as shown in Fig. 1 is considered in the following. The frame has 10 layers
with 1.0 m height and 4.0 m width for each layer. The properties of each beam in the frame are the following:
modulus of elasticity E ¼ 2:0 Â10
11

N=m
2
, mass density r ¼ 7800 kg=m
3
, area of cross-section
A ¼ 2:4 Â 10
À4
m
2
, and area moment of inertia I ¼ 8:0 Â10
À9
m
4
. The frame is discretised into 134 elements
and 55 nodes using the finite-element method. The number of the total degrees of freedom is 160. The lowest
15 natural frequencies are listed in Table 1. The modulus of the diagonal element through node A, which is
highlighted in Fig. 1, is selected as the design parameter. The input and output are all assumed to be at node A
in the horizontal direction.
5.1. m-version
The approximations of the FRFs at the low and middle frequency ranges, 0–1000 and 1300–1400 rad/s,
are plotted in Fig. 2. In these figures, the exact values are also included for comparison. In these two figures
and all others followed, 101 frequency steps are implemented to compute the FRFs and their sensitivities. This
means that the frequency step sizes used for the low and middle frequency range are, respectively, 10
and 1 rad/s.
For the approximate FRFs at the low frequency range, the first two modes, i.e., L ¼ 2, are first included
in the mode superposition. L
1
¼ L
2
¼ 8 are originally selected for the FRFs at the middle frequency

range. Definitely, these modes are not enough to compute the FRFs accurately. Hence, m-version
adaptive scheme is used to increase the number of the modes according to the error distribution with
respect to the excited frequencies. The numbers of modes used in the mode superposition are shown in
Figs. 3 and 4.
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 49
The approximate FRFs at the low frequency range with e ¼ 0:001 are close to the exact at most
of the frequencies. The error is a little big at some frequencies between 600 and 1000 rad/s as shown in
Fig. 2(a). However, this error tolerance is absolutely too big for the FRFs at the middle frequency
range. The differences between the approximate FRFs with e ¼ 0:001 and the exact, shown in Fig. 2(b),
are significant. One pseudo-antiresonance frequency is resulted from the approximate FRFs around
1350 rad/s. To increase the accuracy of the computed FRFs, the error tolerance is reduc ed by one-
tenth, i.e. e ¼ 0:0001. The corresponding FRFs at the low frequency range are very close to the exact
except some errors around the antiresonance frequencies. Unfortunately, the errors are still very
big for the approximate FRFs at the middle frequency range. Hence, the error toleran ce is reduced by
one-tenth again.
ARTICLE IN PRESS
Fig. 1. Schematic of a two-dimensional plane frame.
Table 1
Former 15 natural frequencies of the plane frame (rad/s)
Mode Freq. Mode Freq. Mode Freq.
1 242.059 6 1264.52 11 1421.37
2 804.888 7 1307.22 12 1449.77
3 931.090 8 1325.26 13 1457.41
4 1182.69 9 1333.37 14 1476.97
5 1207.22 10 1383.99 15 1499.34
Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5750
When the error tolerance is 0.001, the maximum and minimum numbers of modes used in the mode
superposition are, respectively, 95 and 3 for the FRFs at the low frequency range as shown in Fig. 3(a). The
summation of the numbers of modes at all the frequencies in Fig. 3(a) is 3975. If the 95 modes are used for all

ARTICLE IN PRESS
0 200 400 600 800 1000
0
20
40
60
80
100
Mode
Frequency (rad/s)
0 200 400 600 800 1000
0
20
40
60
80
100
120
140
Mode
Frequency (rad/s)
(a)
(b)
Fig. 3. Number of modes used at low frequency range: (a) e ¼ 0:001 and (b) e ¼ 0:00001.
1300 1320 1340 1360 1380 1400
0
20
40
60
80

100
Mode
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
0
20
40
60
80
100
120
140
Mode
Frequency (rad/s)
(a)
(b)
Fig. 4. Number of modes used at middle frequency range: (a) e ¼ 0:001 and (b) e ¼ 0:00001.
0 200 400 600 800 1000
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5

10
-4
Exact
0.001
0.0001
0.00001
FRF
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
10
-12
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
Exact
0.001
0.0001
0.00001
FRF

Frequency (rad/s)
(a)
(b)
Fig. 2. FRFs resulted from m-adaptive scheme with e ¼ 0:001; 0:0001 and 0:00001: (a) low frequency range; and (b) middle frequency
range.
Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 51
the FRFs, the total number becomes 9595 (95 Â101). This means that some computational work may be
saved when the proposed m-version adaptive scheme is implemented. This advantage is also shown clear ly in
Fig. 4(a).
The convergent rate of the mode superposition is very slow. As shown in Fig. 2, there is no
much improvement of the accuracy when the error tolerance is reduced from 0.0001 to 0.00001. Compared
to the results for e ¼ 0:001, the summations of the numbers of modes used in mode superposition
become 11,992 and 11,194 for e ¼ 0:0001 and 0.00001, respectively. This means about 119 and 111 modes,
74% and 69% of the total modes, should averagely be included in the mode superposition to obtain the
FRFs with e ¼ 0:0001 and 0.00001, respectively. Unfortunately, the accuracy of the FRFs is still very low. To
show this clearly, the former 120 modes, 75% of the number of total modes, are included to compute
the FRFs at the middle frequency range. These results are plotted in Fig. 5. The 120th natural
frequency is 13,202 rad/s which is about nine times bigger than the highest frequency, 1400 rad/s,
in this frequency range to be considered. Clearly, both the number of modes and the frequency range,
0–13,202 rad/s, are much greater than those recommended in the engineering analysis. Unfortunately,
the errors are still significant. Conseque ntly, the MSM is sometimes not a good approach to
ARTICLE IN PRESS
1300 1320 1340 1360 1380 1400
10
-12
10
-11
10
-10
10

-9
10
-8
10
-7
10
-6
10
-5
Exact
Approximate
FRF
Frequency (rad/s)
Fig. 5. FRFs computed using the former 120 modes.
0 200 400 600 800 1000
10
-21
10
-20
10
-19
10
-18
10
-17
Exact
A
B
Sensitivity
Frequency (rad/s)

1300 1320 1340 1360 1380 1400
10
-21
10
-20
10
-19
10
-18
10
-17
10
-16
10
-15
10
-14
Sensitivity
Frequency (rad/s)
(a) (b)
Exact
A
B
Fig. 6. Sensitivities of FRFs resulted from m-adaptive scheme with e ¼ 0:00001: (a) low frequency range; and (b) middle frequency range.
Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5752
compute the FRFs especially for those around the antiresonance frequencies because it is impractical
to compute more than one-half of the total modes for a large model. This is one reason that the MAM is
necessary for FRFs.
ARTICLE IN PRESS
0 200 400 600 800 1000

10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Exact
0.02
0.01
FRF
Frequency (rad/s)
1300 1320 1340 1360 1380 140
0
10
-10
10
-9
10
-8
10
-7
10
-6
10

-5
FRF
Frequency (rad/s)
(a) (b)
Exact
0.02
0.01
Fig. 7. FRFs resulted from s-adaptive scheme with e ¼ 0:02 and 0:01: (a) low frequency range; and (b) middle frequency range.
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
8
Level
Frequency (rad/s)
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
8

Level
Frequency (rad/s)
(a) (b)
Fig. 8. Number of levels of the MAM at low frequency range: (a) e ¼ 0:02; (b) e ¼ 0:01.
1300 1320 1340 1360 1380 1400
0
1
2
3
4
5
6
7
8
Level
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
0
1
2
3
4
5
6
7
8
Level
Frequency (rad/s)
(a) (b)
Fig. 9. Number of levels of the MAM at middle frequency range: (a) e ¼ 0:02 and (b) e ¼ 0:01.

Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 53
The sensitivities of FRFs computed from the m-version ada ptive scheme with error tolerance e ¼ 0:00001,
denoted by case A, and by the lowest 120 modes, denoted by case B, are plotted in Fig. 6. For compari son
purpose, the exact values are also included. Clearly, both cases give large errors at many frequencies
considered.
5.2. s-version
The FRFs at the same frequency ranges as those used above are considered again. According to the
convergent conditions in Eqs. (43) an d (44) and the natural frequencies listed in Table 1, L ¼ 3, L
1
¼ 7 and
L
2
¼ 10 are selec ted for the s-version adaptive scheme. The approximate FRFs resulted from the s-version
scheme is shown in Fig. 7. The corresponding levels of the MAM are plotted in Figs. 8 and 9. When the error
tolerance is 0.02, the difference between the approximate and the exact FRFs is already very small. The
convergence of the s-version scheme is much faster than the m-version. Only three or four modes are used in
the MAM while the accuracy of the FRFs are much higher than those resulted from the mode superposition
using the former 120 modes.
Besides the accuracy, the computational work is also a very important factor for a numerical method. The
major computational work of the m- and s-version is listed in the following:
 Matrix decomposition: The LU decomposition of the stiffness matrix K or the shifted stiffness matrix
¯
K is
usually very expensive because it is proportional to the cubic of the number of degrees of freedom (n).
According to Ref. [13], the number of manipulation, including multiplication and addition, is about 2n
3
/3;
As shown in the m- and s-version schemes, the decomposi tion of the stiffne ss or shifted stiffness matrix is
only performed once for both schemes.
 Subspace iteration or inverse iteration: The number of operations in the subspace iteration is problem

dependent. To simplify it, we use two assumptions: (1) assume
¯
q ¼ 1 in Eqs. (29), (31), and (32). The
subspace iteration becomes the invers e iteration under this assumption, and the eigenvalue analysis of the
reduced model in the subspace, shown in Eq. (30), will not be considered. Similarly, the manipulation of the
orthogonalisation in the inverse iteration is also ignored. Hence, the number of operation is about 6n
2
for
each of the iterations. (2) Twenty iterations are average required in the subspace iteration or the inverse
iteration to make the mode(s) convergent. (Actual ly, the number of iterations is usually much higher than
20 for the modes which eigenvalues are far away from the value q if no more eigenvalue shift is applied.)
Consequently, one mode takes at least 120n
2
manipulations.
 Power series: The major computational work in the power series is listed in Steps 5.1 and 5.2 in the s-version
adaptive scheme. The number of multiplication an d addition is about 4n
2
.
According to these assumptions, about 15,227n
2
operations are required for the m-version scheme with
e ¼ 0:00001 at the low and the middle frequency range. However, only 487n
2
and 611n
2
operations are used in
the s-version scheme with e ¼ 0:02. Compared to the m-version adaptive scheme, the computational work in
the s-version adaptive scheme can be ignored.
As shown in Figs. 8 and 9, the levels of the MAM change with the frequencies. The minimum levels for the
four cases are all one while the maximum are 5, 8, 6, and 7, respectively. Compared with the traditional MAM

[3,4], 48%, 59%, 50%, and 49% of the total levels may be saved. This makes the present scheme is more
computationally efficient than the traditional MAM.
5.3. ms-version
From the error Eqs. (41) and (42) and the results above, we know that the accuracy of the FRFs improve
with the increase of the level of the MAM. However, the practical accuracy might become worse if the level is
too high because of the numerical truncated error in computation. Hence, the accuracy of the FRFs may not
be increased infinitely. To solve this problem, the ms-version adaptive scheme is presented. This is a
combination of the s- and m-version. In this scheme the s-version adaptive scheme is first used. Then, the
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Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–5754
ARTICLE IN PRESS
0 200 400 600 800 1000
0
1
2
3
4
5
Level
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
0
1
2
3
4
5
Level
Frequency (rad/s)
(a)

(b)
Fig. 10. Number of levels of the MAM with e ¼ 0:005: (a) low frequency range; and (b) middle frequency range.
0 200 400 600 800 1000
0
1
2
3
4
5
Mode
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
0
1
2
3
4
5
Mode
Frequency (rad/s)
(a) (b)
Fig. 11. Number of modes of the MAM with e ¼ 0:005: (a) low frequency range; and (b) middle frequency range.
0 200 400 600 800 1000
10
-21
10
-20
10
-19
10

-18
10
-17
Exact
A
B
C
Sensitivity
Frequency (rad/s)
1300 1320 1340 1360 1380 1400
10
-20
10
-19
10
-18
10
-17
10
-16
10
-15
10
-14
Exact
A
B
C
Sensitivity
Frequency (rad/s)

(a)
(b)
Fig. 12. Sensitivities of FRFs resulted from s- and ms-adaptive scheme: (a) low frequency range; and (b) middle frequency range.
Z Q. Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 55
m-version adaptive scheme will be used to the FRFs which errors are still higher than the error tolerance after
a prescribed level of the MAM has been applied.
The FRFs at the low and middle frequency ranges with e ¼ 0:005 are considered. As stated above, the
results do not converge if we just use s-version scheme. In the present computation the number of maxi mum
level is set as 5 and other parameters are the same as those used above. The resulted FRFs are very close to the
exact. The levels and modes used in this scheme are plotted in Figs. 10 and 11, respectively. As shown in Figs.
8(b) and 9(b), the maximum levels of the MAM are 8 and 7, respectively. The correspondi ng modes included
in the MAM are 3 and 4. When the maximum level is set as 5, only one mode is required at some frequencies
while the FRFs have very high accuracy. This means that the convergence of the mode in the ms-version
scheme is mu ch faster than that in the m-version scheme. The reason can be explained clearly from the error
Eqs. (33), (34), (41) and (42).
The sensitivities of the FRFs computed from the s- and ms-version adaptive schemes are given in Fig. 12.
Cases A–C denote the three cases considered above. Clearly, the computed sensitivities are very close the
exact.
6. Conclusions
An adaptive mode superposition and acceleration technique was proposed. Three adaptive schemes, m-, s-,
and ms-version, were presented. Although only the implementation of the new technique into FRFs and their
sensitivities was demonstrated, the technique may be utilised in many situ ations where MSM and MAM are
required. Compared to the traditional MAM, a lot of computation work can be saved by using the adaptive
scheme. Furthermore, the proposed schemes are easier to be implemented into a practical problem.
For some model, the MSM is not a good approach to perform the FRFs analysis especially for the FRFs
around the a ntiresonance frequencies. Because more than half of the total modes are requir ed in this method,
the eigenvalue analysis of the mod el becomes very computationally expensive and sometimes impossible when
the model has a large number of degrees of freedom.
m-version adaptive scheme may save some computational work. One criterion to determine the numb er of
modes was presented in this scheme. Compared with the s-version scheme, there is no requirement for the

minimum modes. However, the convergence is very slow because this scheme is based on the MSM. In the
present numerical example, a very small error tolerance is required to obtain the accurate FRFs. Further
researches show that e ¼ 0:005 is usually enough.
The s-version adaptive scheme is based on the MAM. Also, one criterion to determine the number of levels
was presented. The results showed that only several levels of the MAM are enough to compute the FRFs and
their sensitivities accurately. The convergence of the s-version is much faster than the m-version. Compared
with the m-version scheme, the computational effort in the s-version can be ignored sometimes. Consequently,
the s-version scheme is much more efficient than the m-version scheme. One disadvantage of this scheme is
that the accuracy could not be increased infinitely due to the digit truncation in the computation even though
the MAM is convergent theoretically. Another disadvantage is that a minimum number of the modes are
required to guarantee the convergence.
ms-version is also based on the MAM. Hence, it has all the advantages of the s-version scheme. Because it
also includes the idea of the m-version scheme, the convergence is usually guaranteed. In this scheme, we can
set the maximum level of the MAM as any number between 5 and 10. Hence, it is unnecessary to worry about
the numerical truncation in the computers. The convergen ce with respect to the mode is much more faster than
the m-ve rsion. Similarly, a minimum number of the modes are required to guarantee the convergence.
On one hand, because the convergence of m-version adaptive scheme for the FRFs and their sensitivities is
very slow, the error tolerance cu rrently selected for this special example is very small. The error tolerance for
the FRFs in the s-version adaptive scheme, on the other hand, is relatively bigger. Therefore, new error
estimator becomes necessary for some special problems.
The proposed procedures can be extended to a model with damping, particularly non-cla ssical damping.
However, if the damping is non-classical, the complex modal space and state space formulations generally
need to be used which makes the equations in this paper much more complex. One major intention of this
paper is to propose these new procedures and give some demonstration of good results. Therefore, the
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damping is not included although it is believed that these procedures would be more powerful with the
inclusion of damping. These issues and the related researches are under way.
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