Mechanical Systemsand Signal Processing Mechanical Systems and Signal Processing 21 2007 40–57 Adaptive mode superposition and acceleration technique with application to frequency respon
Trang 1Mechanical 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 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
r2006 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
www.elsevier.com/locate/jnlabr/ymssp
0888-3270/$ - see front matter r 2006 Elsevier Ltd All rights reserved.
doi:10.1016/j.ymssp.2006.02.002
Tel.: +1 864 422 4524; fax: +1 864 422 3219.
E-mail address: zuqing.qu@us.michelin.com.
Trang 2The dynamic equilibrium of an n-degree-of-freedom undamped system in time domain is generally given by
where M and K are real symmetric mass and stiffness matrices, respectively X(t) and F(t) are, respectively, the displacement and exciting force or load vectors The corresponding dynamic equilibrium in frequency domain
is given by
where o is the circular frequency of exciting forces or loads and Z(o) Ử (Ko2M) is referred to as dynamic stiffness matrix The frequency response vector X(o) may be expressed as
The matrix Hđoỡ Ử đK o2Mỡ1 is usually called receptance matrix and each element of this matrix represents a single-inputỜsingle-output FRF Clearly, the receptance matrix is an inverse matrix of the dynamic stiffness matrix, i.e.,
Nomenclature
E truncated error vector of the FRFs
F force or load vector
H (n n) receptance matrix
I (n n) identity matrix
K (n n) stiffness matrix
KR đốq ốqỡ reduced stiffness matrix defined in Eq (31)
M (n n) mass matrix
MR đốq ốqỡ reduced mass matrix 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 đốq ốqỡ eigenvector matrix of a reduced model
X response vector
Z (n n) dynamic stiffness matrix
K (n n) eigenvalue matrix
U (n n) eigenvector matrix
li ith eigenvalue
/i ith eigenvector
o circular frequency of exciting forces or loads
omin under boundary value of the exciting frequencies
omax upper boundary value of the exciting frequencies
X đốq ốqỡ eigenvalue matrix of a reduced model
Superscript
l FRFs at the low frequency range
m FRFs at the middle frequency range
T matrix transpose
Trang 3The 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 (AMSAT), 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-version 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 diagðl1; l2; ; lnÞ; ðl1pl2p plnÞ; U ¼ ½/1; /2 /n, (5) where li and /i are the ith eigenvalue and eigenvector K and U satisfy the following eigenequation and orthogonalities
where superscript T denotes matrix transpose I is an identity matrix of n n From Eq (6) one obtains
2.1 Mode superposition method
Introducing Eq (8) into Eq (3) leads to
Trang 4The FRFs may be expanded in the modal space as
XðoÞ ¼Xn
r¼1
/TrF
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
Xl1ðoÞ ¼XL
r¼1
/TrF
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-mode truncation scheme is applied If the L1th through L2th modes are selected as the kept modes, the FRFs can be expressed as
Xm1ðoÞ ¼XL 2
r¼L 1
/TrF
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 o2I Þ in Eq (8) may be expanded as a power series[10], i.e.,
ðK o2I Þ1¼XS
s¼1
ðo2K1Þs1K1þ ðo2K1ÞSðK o2I Þ1, (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
XðoÞ ¼ U XS
s¼1
ðo2K1Þs1K1
UTF þ U ðo 2K1ÞSðK o2I Þ1
Using Eq (7), one has
UKsK1U ¼ U ðK1UTUTU1UÞ ðK1UTUTU1UÞ
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{s
K1U ¼ ðK1MÞ ðK1MÞ
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{s
K1 (15) Introducing Eq (15) into the right-hand side of Eq (14) results in
XðoÞ ¼XS
ðo2K1MÞs1K1F þXn
o2
lr
S
/TrF
Trang 5If the lowest L modes are selected as the kept modes, the FRFs can be expressed as
Xl2ðoÞ ¼XS
s¼1
ðo2K1MÞs1K1F þXL
r¼1
o2
lr
S
/TrF
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
where
¯
Usually, the eigenvalue shifting value q is given by
q o
2
minþo2max
and should satisfy qalrðr ¼ 1; 2; ; nÞ omin and omax are the under and upper boundary values of the exciting frequencies Substituting Eq (18) into Eq (3), the FRFs are obtained as
When the mode acceleration is applied, the FRFs can be expressed as
XðoÞ ¼XS
s¼1
ðo¯2K¯1MÞs1K¯1F þXn
r¼1
o2q
lrq
/TrF
Assume that the L1th through the L2th 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
Xm2ðoÞ ¼XS
s¼1
ðo¯2K¯1MÞs1K¯1F þXL 2
r¼L 1
o2q
lrq
/TrF
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
qZ1ðoÞ
in which p is a design parameter According to the theory of matrix, one has
qHðoÞ
qp ¼Z
1ðoÞqZðoÞ
where Z;pðoÞ 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 ¼Z
1ðoÞqZðoÞ
qp Z
The sensitivity of FRF at the location of ximay be expressed as
Trang 6in which hiðoÞ 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;pðoÞ is locally populated Therefore, it is unnecessary to make all the elements of hiðoÞ 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 and the Rayleigh–Ritz procedure Its objective is to solve for the lowest p eigenpairs satisfying the generalised eigenvalue Eq (6) If the first p eigenvalues and their corresponding eigenvectors are considered, the eigenproblem (6) can be rewritten as
where Up and Kpp are composed of the first p eigenvectors and eigenvalues, respectively
If a set of independent initial vectors Xð0Þ¯q , where ¯q ¼ minð2p; p þ 8Þ suggested by Bathe, are available, the basic subspace iteration method obtains the new set of approximate eigenvectors by the following two steps: (a) A new subspace is obtained by using the simultaneous inverse iteration method, i.e.,
If the iterations proceeded use Yðiþ1Þ¯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 procedure is adopted, i.e.,
where Qðiþ1Þand Xðiþ1Þare the eigenvectors and eigenvalues of the reduced model (Kðiþ1ÞR and Mðiþ1ÞR ) The reduced matrices are given by
Kðiþ1ÞR ¼Yðiþ1Þ¯q T
K Yðiþ1Þ¯q ; Mðiþ1ÞR ¼Yðiþ1Þ¯q T
Hence, the (i þ 1)th approximation of the first ¯q eigenvectors Xðiþ1Þ¯q is
As i increases, the vectors Xðiþ1Þ¯q and the values in matrix Xðiþ1Þ will, respectively, converge to the eigenvectors Up and the eigenvalues in matrix Kpp provided that the initial vectors Xð0Þ¯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 frequencypffiffiffiq
will be obtained If only one eigenpair is required, ¯q is set as one and the subspace 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
Trang 7errors of the FRFs resulted from Eqs (11) and (12) are given by
El1ðoÞ ¼ Xn
r¼Lþ1
/TrF
Em1ðoÞ ¼LX1 1
r¼1
/TrF
lro2 /rþ Xn
r¼L 2 þ1
/TrF
Define /rjas the jth element of the vector ur The jth elements eljðoÞ and emj ðoÞ of the error vectors El1ðoÞ and
Em1ðoÞ expressed in Eqs (33) and (34) become
eljðoÞ ¼ Xn
r¼Lþ1
/TrF
emj ðoÞ ¼LX1 1
r¼1
/TrF
lro2 jrjþ Xn
r¼L 2 þ1
/TrF
The upper boundaries of the absolute values of these two errors are given by
eljðoÞ
n
r¼Lþ1
/TrF
lro2
emj ðoÞ
L 1 1
r¼1
/TrF
lro2
jjrjj þ Xn
r¼L 2 þ1
/TrF
lro2
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 prescribed 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 Rr¼/TrF/rðr ¼ 1; 2; ; p 1Þ
4 Compute the initial approximation of the FRFs at all frequencies using the p 1 modes:
Xð0Þ¼Xp1
r¼1
1
lro2 Rr
5 For m ¼ p; p þ 1; p þ 2; ., do loop:
5.1 Select the mth eigenpair or calculate it using inverse iteration together with orthogonalisation process 5.2 Compute the constant vector Rm¼/TmF/m
5.3 For the frequencies at which the FRFs do not converge:
Trang 85.3.1 Evaluate the incremental of the FRFs:
DXðmÞðoÞ ¼ 1
lmo2 Rm 5.3.2 Compute the total of the FRFs:
XðmÞðoÞ ¼ Xðm1ÞðoÞ þ DXðmÞðoÞ
5.3.3 Check the convergence:
ZX ¼ DXðmÞðoÞ
XðmÞðoÞ 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 1 to S may be obtained from these two equations as
DXl2ðoÞS¼ ðo2K1MÞS1K1F XL
r¼1
o2
lr
S1
/TrF
lr
DXm2ðoÞS¼ ðo¯2K¯1MÞS1K¯1F XL 2
r¼L 1
o2q
lrq
/TrF
lr
The series truncated errors resulted from these two equations are given by
El2ðoÞ ¼ Xn
r¼Lþ1
o2
lr
S
/TrF
Em2ðoÞ ¼LX1 1
r¼1
o2q
lrq
/TrF
lro2 /rþ Xn
r¼L 2 þ1
o2q
lrq
/TrF
From the above two equations, the convergent conditions of Eqs (17) and (23) may be defined as [3]
lL11oo2
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 El2ðoÞ and Em2ðoÞ are given by
eljðoÞ
n
o2
lr
S
/TrF
lro2
Trang 9emj ðoÞ
L 1 1
r¼1
o2q
lrq
S /TrF
lro2
jjrjj þ Xn
r¼L 2 þ1
o2q
lrq
S /TrF
lro2
When conditions (43) and (44) are satisfied, the coefficients ðo2=lrÞS and ðo2q=lrqÞS 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 Rr¼/TrF/r for r ¼ 1; ; L
4 Compute the initial approximation of the FRFs at all frequencies
Xð0ÞðoÞ ¼XL
r¼1
1
5 For S ¼ 1; 2; use the MAM:
5.1 Calculate Y ¼ MXðS1ÞA (Y ¼ F for S ¼ 1)
5.2 Solve for XðSÞA from equation LUXðSÞA ¼Y 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
DXðSÞðoÞ ¼ o2S2XðSÞA XL
r¼1
o2S2
lSr Rr. 5.3.2 Compute the total of the FRFs
XðSÞðoÞ ¼ XðS1ÞðoÞ þ DXðSÞðoÞ
5.3.3 Check the convergence
ZX¼ DXðSÞðoÞ
XðSÞðoÞ 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 increase 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
Trang 10FRFs 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 Smax; calculate the constant vectors Rr¼/TrF/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; ; Smax 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 þ 1; L þ 2; ; 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
DXðmÞðoÞ ¼ o
2
lm
S max
1
lmo2 Rm
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 example and discussions
A two-dimensional plane frame as shown inFig 1is 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 1011N=m2, mass density r ¼ 7800 kg=m3, area of cross-section
A ¼ 2:4 104m2, and area moment of inertia I ¼ 8:0 109m4 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 inFig 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 inFig 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 L1¼L2¼8 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