ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 21 (2007) 40–57 www.elsevier.com/locate/jnlabr/ymssp Adaptive mode superposition and acceleration technique with application to frequency response function and its sensitivity Zu-Qing Quà Michelin Americas Research & Development Corporation, 515 Michelin Road, Greenville, SC 29605, USA Received 13 December 2005; received in revised form 31 January 2006; accepted 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 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 ÃTel.: +1 864 422 4524; fax: +1 864 422 3219 E-mail address: zuqing.qu@us.michelin.com 0888-3270/$ - see front matter r 2006 Elsevier Ltd All rights reserved doi:10.1016/j.ymssp.2006.02.002 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 41 Nomenclature E F H I K KR M MR n p q Q t X Z K U li /i o omin omax X truncated error vector of the FRFs force or load vector (n  n) receptance matrix (n  n) identity matrix (n  n) stiffness matrix 刻qÞ reduced stiffness matrix defined in Eq (31) q ¯ (n  n) mass matrix 刻qÞ reduced mass matrix defined in Eq (31) q ¯ number of total degrees of freedom of a model design parameter; number of lowest modes to be solved by subspace iteration eigenvalue shifting value 刻qÞ eigenvector matrix of a reduced model q ¯ time response vector (n  n) dynamic stiffness matrix (n  n) eigenvalue matrix (n  n) eigenvector matrix ith eigenvalue ith eigenvector circular frequency of exciting forces or loads under boundary value of the exciting frequencies upper boundary value of the exciting frequencies 刻qÞ eigenvalue matrix of a reduced model q ¯ Superscript l m T FRFs at the low frequency range FRFs at the middle frequency range matrix transpose The dynamic equilibrium of an n-degree-of-freedom undamped system in time domain is generally given by M Xtị ỵ KXtị ẳ Ftị, (1) 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 K o2 MịXoị ẳ Foị, (2) where o is the circular frequency of exciting forces or loads and Z(o) ¼ (KÀo2M) is referred to as dynamic stiffness matrix The frequency response vector X(o) may be expressed as Xoị ẳ K o2 Mị1 Foị (3) The matrix Hoị ẳ K o2 Mị1 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., Hoị ẳ Zoị1 (4) ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 42 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 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 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 The ideas and convergent properties of the m-version, s-version, and ms-version adaptive approaches are presented in Section One scheme for implementing the technique is proposed for each approach A numerical example is provided in Section 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] 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 ; ; ln Þ; ðl1 pl2 p Á Á Á pln ị; U ẳ ẵ/1 ; /2 /n Š, (5) where li and /i are the ith eigenvalue and eigenvector K and U satisfy the following eigenequation and orthogonalities KU ¼ MUK; UT KU ¼ K; UT MU ¼ I; (6) where superscript T denotes matrix transpose I is an identity matrix of n  n From Eq (6) one obtains K À1 ¼ UKÀ1 UT ; M ẳ UT U1 , (7) K o2 Mị1 ¼ UðK À o2 IÞÀ1 UT (8) 2.1 Mode superposition method Introducing Eq (8) into Eq (3) leads to Xoị ẳ UK o2 Iị1 UT F: (9) ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 43 The FRFs may be expanded in the modal space as Xoị ẳ n X /T F r / lr À o2 r r¼1 (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 l1 oị ẳ L X /T F r / lr À o r r¼1 (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-mode truncation scheme is applied If the L1th through L2th modes are selected as the kept modes, the FRFs can be expressed as X m ðoÞ ¼ L2 X /T F r / lr À o2 r r¼L (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 À o2 IÞ in Eq (8) may be expanded as a power series [10], i.e., K o2 Iị1 ẳ S X o2 K1 ịs1 K1 ỵ o2 K1 ịS K o2 Iị1 , (13) s¼1 where S is the level of the mode acceleration and may be any integer larger than zero S ¼ indicates that no power series, mode acceleration, is adopted Substituting Eq (13) into Eq (9), the FRFs can be expressed as " # S X  à s1 Xoị ẳ U o K ị K (14) UT F ỵ U o2 K1 ịS K o2 Iị1 UT F: sẳ1 Using Eq (7), one has s s zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ UKÀs KÀ1 U ¼ U ðKÀ1 UT UÀT UÀ1 UÞ Á Á Á ðKÀ1 UT UT U1 Uị K1 U ẳ K Mị Á Á ðK À1 MÞ K À1 Introducing Eq (15) into the right-hand side of Eq (14) results in S n X o2 S /T F X sÀ1 À1 r o K Mị K F ỵ / Xoị ẳ lr o2 r lr sẳ1 rẳ1 (15) (16) ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 44 If the lowest L modes are selected as the kept modes, the FRFs can be expressed as S L X X o2 S /T F sÀ1 À1 l À1 r X ðoÞ ẳ o K Mị K F ỵ / lr o2 r lr s¼1 r¼1 (17) As defined above, no power series will be applied when S ¼ This means that the first item on the righthand 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 K o2 Mị ẳ K o2 Mị, (18) where K ẳ K À qM; o2 ¼ o2 À q ¯ (19) Usually, the eigenvalue shifting value q is given by o2 ỵ o2 max (20) and should satisfy qalr ðr 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 q% Xoị ẳ K o2 MÞÀ1 F: ¯ (21) When the mode acceleration is applied, the FRFs can be expressed as S n X X o2 À qS /T F À1 r ¯ À1 XðoÞ ẳ o2 K Mịs1 K F ỵ / lr À q lr À o r s¼1 r¼1 (22) 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 L2 S X X o2 À qS /T F sÀ1 ¯ À1 m ¯ r X oị ẳ o K Mị K F ỵ / (23) lr q lr o r r¼L s¼1 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 qHoị qZ oị ẳ qp qp (24) in which p is a design parameter According to the theory of matrix, one has qHðoÞ qZðoÞ À1 ¼ Z À1 ðoÞ Z ðoÞ ¼ HðoÞZ ;p ðoÞHðoÞ, qp qp (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 qZðoÞ À1 ¼ Z À1 ðoÞ Z ðoÞF ¼ HðoÞZ ;p ðoÞXðoÞ qp qp (26) The sensitivity of FRF at the location of xi may be expressed as qxi oị=qp ẳ hi ðoÞZ ;p ðoÞXðoÞ (27) ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 45 in which hi 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 hi ðoÞ and XðoÞ available before evaluate the sensitivity 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 KUp ¼ MUp Kpp , (28) where Up and Kpp are composed of the first p eigenvectors and eigenvalues, respectively If a set of independent initial vectors X 0ị , where q ẳ min2p; p ỵ 8ị suggested by Bathe, are available, the ¯ ¯ q 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., KY iỵ1ị ¼ MX ðiÞ ¯ ¯ q q (29) Y iỵ1ị q as the next estimation of the subspace, then the subspace would If the iterations proceeded use collapse to a subspace of dimension one and only contains the eigenvector corresponding to the lowest eigenvalue (b) To prevent the collapse, the RayleighRitz procedure is adopted, i.e., K iỵ1ị Qiỵ1ị ẳ M iỵ1ị Qiỵ1ị Xiỵ1ị , R R iỵ1ị (30) iỵ1ị where Q and X are the eigenvectors and eigenvalues of the reduced model reduced matrices are given by  T  T K iỵ1ị ẳ Y iỵ1ị KY iỵ1ị ; M iỵ1ị ẳ Y iỵ1ị MY iỵ1ị ¯ ¯ ¯ q q q q R R (K iỵ1ị R and M iỵ1ị ) R The (31) Hence, the (i ỵ 1)th approximation of the rst q eigenvectors X iỵ1ị is q X iỵ1ị ẳ Y iỵ1ị Qiỵ1ị q q (32) X iỵ1ị q iỵ1ị As i increases, the vectors and the values in matrix X will, respectively, converge to the eigenvectors Up and the eigenvalues in matrix Kpp provided that the initial vectors X ð0Þ are not orthogonal to ¯ q one of the required eigenvectors If the eigenvalue shifting technique defined in Eqs (18)–(20) is applied to the subspace iteration approach, pffiffiffi the p eigenpairs around the frequency q will be obtained If only one eigenpair is required, q is set as one and ¯ the subspace iteration method becomes the inverse iteration method 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 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 46 errors of the FRFs resulted from Eqs (11) and (12) are given by E l1 oị ẳ n X /T F r /, lr o r rẳLỵ1 L1 X E m oị ẳ rẳ1 (33) n X /T F /T F r r /r ỵ / lr o2 l r o2 r rẳL ỵ1 (34) Dene /rj as the jth element of the vector ur The jth elements elj ðoÞ and em ðoÞ of the error vectors E l1 ðoÞ and j expressed in Eqs (33) and (34) become E m oị elj oị ẳ n X /T F r j , lr À o2 rj rẳLỵ1 em oị ẳ j L1 X rẳ1 (35) n X /T F /T F r r jrj ỵ j lr À o2 l À o2 rj r¼L þ1 r (36) The upper boundaries of the absolute values of these two errors are given by    n  X  /T F   l   r  ej ðoÞp  l À o2 jjrj j, rẳLỵ1 (37) r     L1 À1 T  n X  /T F   m  X  /r F   jj j þ  r  ej ðoÞp  l À o2  rj l o2 jjrj j r rẳL ỵ1 r r¼1 (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 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, madaptive 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 Decompose the stiffness matrix K ẳ LU Use subspace iteration to extract pX2ị eigenpairs Evaluate vectors Rr ¼ /T F/r ðr ¼ 1; 2; ; p À 1Þ r Compute the initial approximation of the FRFs at all frequencies using the p modes: X 0ị ẳ p1 X r¼1 For 5.1 5.2 5.3 Rr lr o2 m ẳ p; p ỵ 1; p ỵ 2; , loop: Select the mth eigenpair or calculate it using inverse iteration together with orthogonalisation process Compute the constant vector Rm ¼ /T F/m m For the frequencies at which the FRFs not converge: ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 47 5.3.1 Evaluate the incremental of the FRFs: DX mị oị ẳ Rm lm À o2 5.3.2 Compute the total of the FRFs: X mị oị ẳ X m1ị oị ỵ DX mị oị 5.3.3 Check the convergence:  ðmÞ  DX ðoÞ ZX ¼  ðmÞ  pe X ðoÞ 5.4 If the FRFs at all the frequencies are convergent, exit this loop Output the results 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 to S may be obtained from these two equations as L X o2 SÀ1 /T F S SÀ1 À1 l À1 r DX oị ẳ o K Mị K F /r , (39) lr lr rẳ1 DX m oịS ẳ o2 K ¯ ¯ À1 ¯ MÞSÀ1 K À1 FÀ L2 X o2 À qSÀ1 /T F r r¼L1 lr À q lr /r The series truncated errors resulted from these two equations are given by n X o2 S /T F l r /, E oị ẳ lr o2 r lr rẳLỵ1 E m oị ẳ L1 À1  X r¼1 o2 À q lr À q S n X o2 À qS /T F /T F r r /r ỵ / lr o2 lr q lr o2 r rẳL ỵ1 (40) (41) (42) From the above two equations, the convergent conditions of Eqs (17) and (23) may be defined as [3] o2 olLỵ1 , max lL1 oo2 ; (43) lL2 ỵ1 4o2 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 l2 ðoÞ and E m ðoÞ are given by  S  T    n X o  /r F   l   jj j, (45) ej oịp lr lr o2  rj rẳLỵ1 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 48  S  T       L1 À1  n X o2 À qS  /T F   m  X o2 q  /r F  jj j ỵ        r ej ðoÞp  l À q  l À o2  rj  l À q  l À o2 jjrj j r r r r rẳL ỵ1 rẳ1 (46) When conditions (43) and (44) are satisfied, the coefficients ðo2 =lr ÞS and ðo2 À q=lr À qÞS 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 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: Decompose the stiffness matrix K ¼ LU Evaluate the natural frequencies and the corresponding mode shapes using subspace iteration Select the L and calculate the constant vectors Rr ¼ /T F/r for r ¼ 1; ; L r Compute the initial approximation of the FRFs at all frequencies X ð0Þ ðoÞ ¼ L X r¼1 For 5.1 5.2 5.3 Rr lr À o2 (a) S ¼ 1; 2; use the MAM: S1ị Calculate Y ẳ MX A (Y ẳ F for S ẳ 1) Sị Solve for X A from equation LUX Sị ẳ Y using the forward and backward substitutions A For the frequencies at which the FRFs not converge: 5.3.1 Evaluate the incremental of the FRFs DX Sị oị ẳ o2S2 X Sị A L X o2S2 rẳ1 lS r Rr 5.3.2 Compute the total of the FRFs X ðSÞ oị ẳ X S1ị oị ỵ DX Sị oị 5.3.3 Check the convergence  Sị  DX oị ZX ẳ  ðSÞ  pe X ðoÞ 5.4 If the FRFs at all the frequencies are convergent, exit this loop 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 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 49 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 mversion 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 Decompose the stiffness matrix K ¼ LU Evaluate the natural frequencies and the corresponding mode shapes using subspace iteration Select the L and Smax; calculate the constant vectors Rr ¼ /T F/r for r ¼ 1; ; L r Compute the initial approximation of the FRFs using Eq (a) at all the frequencies 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 For m ¼ L ỵ 1; L ỵ 2; ; 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  Smax o DX mị oị ẳ Rm lm o2 lm 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 Numerical example and discussions A two-dimensional plane frame as shown in Fig 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  1011 N=m2 , mass density r ¼ 7800 kg=m3 , area of cross-section A ¼ 2:4  10À4 m2 , and area moment of inertia I ¼ 8:0  10À9 m4 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 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 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 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 L1 ¼ L2 ¼ 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 and ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 50 Fig Schematic of a two-dimensional plane frame Table Former 15 natural frequencies of the plane frame (rad/s) Mode Freq Mode Freq Mode Freq 242.059 804.888 931.090 1182.69 1207.22 10 1264.52 1307.22 1325.26 1333.37 1383.99 11 12 13 14 15 1421.37 1449.77 1457.41 1476.97 1499.34 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 reduced by onetenth, 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 tolerance is reduced by one-tenth again ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 10-4 10-5 Exact 0.001 0.0001 0.00001 10-5 Exact 0.001 0.0001 0.00001 10-6 10-7 FRF FRF 10-6 10-7 10-8 10-8 10-9 10-10 10-9 10-10 51 10-11 200 (a) 400 600 Frequency (rad/s) 800 1000 10-12 1300 (b) 1320 1340 1360 1380 1400 Frequency (rad/s) Fig 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 100 140 120 80 80 Mode Mode 100 60 40 60 40 20 20 0 200 400 (a) 600 800 1000 200 (b) Frequency (rad/s) 400 600 800 1000 1380 1400 Frequency (rad/s) Fig Number of modes used at low frequency range: (a) e ¼ 0:001 and (b) e ¼ 0:00001 140 100 120 80 Mode Mode 100 60 40 80 60 40 20 1300 (a) 20 1320 1340 1360 Frequency (rad/s) 1380 1300 1400 (b) 1320 1340 1360 Frequency (rad/s) Fig Number of modes used at middle frequency range: (a) e ¼ 0:001 and (b) e ¼ 0:00001 When the error tolerance is 0.001, the maximum and minimum numbers of modes used in the mode superposition are, respectively, 95 and 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 Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 52 the FRFs, the total number becomes 9595 (95  101) This means that some computational work may be saved when the proposed m-version adaptive scheme is implemented This advantage is also shown clearly 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 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 Consequently, the MSM is sometimes not a good approach to 10-5 Exact Approximate 10-6 FRF 10-7 10-8 10-9 10-10 10-11 10-12 1300 1320 1340 1360 Frequency (rad/s) 1380 1400 Fig FRFs computed using the former 120 modes 10-14 10-17 Exact A B 10-16 Sensitivity Sensitivity 10-18 Exact A B 10-15 10-19 10-17 10-18 10-19 10-20 10-20 10-21 (a) 200 400 600 Frequency (rad/s) 800 10-21 1300 1000 (b) 1320 1340 1360 1380 1400 Frequency (rad/s) Fig Sensitivities of FRFs resulted from m-adaptive scheme with e ¼ 0:00001: (a) low frequency range; and (b) middle frequency range ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 53 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 10-4 10-5 Exact 0.02 0.01 Exact 0.02 0.01 10-6 10-6 FRF FRF 10-5 10-7 10-7 10-8 10-8 10-9 10-9 200 400 600 800 Frequency (rad/s) (a) 10-10 1300 1000 1320 1340 1360 1380 1400 Frequency (rad/s) (b) Fig FRFs resulted from s-adaptive scheme with e ¼ 0:02 and 0:01: (a) low frequency range; and (b) middle frequency range 6 5 Level Level 4 3 2 1 0 200 400 600 800 1000 Frequency (rad/s) (a) 200 400 600 800 1000 1380 1400 Frequency (rad/s) (b) Fig Number of levels of the MAM at low frequency range: (a) e ¼ 0:02; (b) e ¼ 0:01 6 5 Level Level 4 3 2 1 1300 (a) 1320 1340 1360 Frequency (rad/s) 1380 1300 1400 (b) 1320 1340 1360 Frequency (rad/s) Fig Number of levels of the MAM at middle frequency range: (a) e ¼ 0:02 and (b) e ¼ 0:01 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 54 The sensitivities of FRFs computed from the m-version adaptive 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 For comparison 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) and (44) and the natural frequencies listed in Table 1, L ¼ 3, L1 ¼ and L2 ¼ 10 are selected for the s-version adaptive scheme The approximate FRFs resulted from the s-version scheme is shown in Fig The corresponding levels of the MAM are plotted in Figs and 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 2n3/3; As shown in the m- and s-version schemes, the decomposition of the stiffness 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 in Eqs (29), (31), and (32) The ¯ subspace iteration becomes the inverse 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 6n2 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 (Actually, 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 120n2 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 and addition is about 4n2 According to these assumptions, about 15,227n2 operations are required for the m-version scheme with e ¼ 0:00001 at the low and the middle frequency range However, only 487n2 and 611n2 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 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 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 3 Level Level 55 2 0 200 400 600 800 1320 (b) Frequency (rad/s) (a) 1300 1000 1340 1360 1380 1400 Frequency (rad/s) Fig 10 Number of levels of the MAM with e ¼ 0:005: (a) low frequency range; and (b) middle frequency range 3 Mode Mode 2 0 200 400 600 800 Frequency (rad/s) (a) 1300 1000 1320 1340 1360 1380 1400 Frequency (rad/s) (b) Fig 11 Number of modes of the MAM with e ¼ 0:005: (a) low frequency range; and (b) middle frequency range 10-14 10-17 Exact A B C Exact A B C 10-15 Sensitivity Sensitivity 10-18 10-19 10-16 10-17 10-18 10-20 10-19 10-21 (a) 200 400 600 Frequency (rad/s) 800 10-20 1300 1000 (b) 1320 1340 1360 1380 1400 Frequency (rad/s) Fig 12 Sensitivities of FRFs resulted from s- and ms-adaptive scheme: (a) low frequency range; and (b) middle frequency range ARTICLE IN PRESS 56 Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 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 not converge if we just use s-version scheme In the present computation the number of maximum level is set as 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 and 7, respectively The corresponding modes included in the MAM are and 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 much 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 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 situations 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 antiresonance frequencies Because more than half of the total modes are required in this method, the eigenvalue analysis of the model 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 number 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 and 10 Hence, it is unnecessary to worry about the numerical truncation in the computers The convergence with respect to the mode is much more faster than the m-version 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 currently 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-classical 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 ARTICLE IN PRESS Z.-Q Qu / Mechanical Systems and Signal Processing 21 (2007) 40–57 57 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 References [1] A.K Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice-Hall, Englewood-Cliffs, NJ, 1995 [2] M.A Akgun, A new family of mode-superposition methods for response calculations, Journal of Sound and Vibration 167 (2) (1993) 289–302 [3] Z.-Q Qu, Hybrid expansion method for frequency responses and their sensitivities, Part I: undamped systems, Journal of Sound and Vibration 231 (1) (2000) 175–193 [4] Z.-Q Qu, Accurate methods for frequency responses and their sensitivities of proportionally damped systems, Computers and Structures 79 (1) (2001) 87–96 [5] C Huang, Z.-S Liu, S.-H Chen, An accurate modal method for computing response to periodic excitation, Computers and Structures 63 (3) (1997) 625–631 [6] J.M Dickens, J.M Nakagawa, M.J Wittbrodt, A critique of mode acceleration and modal truncation augmentation methods for modal response analysis, Computers and Structures 62 (6) (1997) 985–998 [7] Y.-Q Zhao, S.-H Chen, S Chai, Q.-W Qu, An improved modal truncation method for responses to harmonic excitation, Computers and Structures 80 (1) (2002) 99–103 [8] V.S.C Rao, S.R Chaudhuri, V.K Gupta, Mode acceleration approach to seismic response of multiply supported secondary systems, Earthquake Engineering and Structural Dynamics 31 (10) (2002) 1603–1621 [9] R.M Lin, M.K Lim, Derivation of structural design sensitivities from vibration test data, Journal of Sound and Vibration 201 (5) (1997) 613–631 [10] Z.-Q Qu, Model Order Reduction Techniques with Applications in Finite Element Analysis, Springer, London, 2004 [11] A.F Bertolini, Review of eigensolution procedures for linear dynamic finite element analysis, Applied Mechanics Reviews 51 (2) (1998) 155–172 [12] K.J Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996 [13] W.H Press, Numerical Recipes in Fortran: The Art of Parallel Scientific Computing, Cambridge University Press, New York, 1992  ... 0.000 01 10-5 Exact 0.0 01 0.00 01 0.000 01 10-6 10 -7 FRF FRF 10 -6 10 -7 10 -8 10 -8 10 -9 10 -10 10 -9 10 -10 51 10 -11 200 (a) 400 600 Frequency (rad/s) 800 10 00 10 -12 13 00 (b) 13 20 13 40 13 60 13 80 14 00 Frequency. .. approach to 10 -5 Exact Approximate 10 -6 FRF 10 -7 10 -8 10 -9 10 -10 10 -11 10 -12 13 00 13 20 13 40 13 60 Frequency (rad/s) 13 80 14 00 Fig FRFs computed using the former 12 0 modes 10 -14 10 -17 Exact A B 10 -16 Sensitivity. .. 10 -16 Sensitivity Sensitivity 10 -18 Exact A B 10 -15 10 -19 10 -17 10 -18 10 -19 10 -20 10 -20 10 - 21 (a) 200 400 600 Frequency (rad/s) 800 10 - 21 1300 10 00 (b) 13 20 13 40 13 60 13 80 14 00 Frequency (rad/s)