Eigenvalue inverse formulation for optimising vibratory behaviour of truss and continuous structures

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1 Computers and Structures 8 () Eigenvalue inverse formulation for optimising vibratory behaviour of truss and continuous structures H. Bahai *, K. Farahani, M.S. Djoudi Department of Systems Engineering, Brunel University, Uxbridge, UK Received 1 November ; accepted 4 July Abstract This paper presents formulations for inverse optimisation of vibration behaviour of finite element models of both truss and continuous structures. The proposed algorithms determine the required modifications on truss and continuous structures to achieve specified natural frequencies. The modification can be carried out globally or locally on the structures stiffness and matrices and the formulation can also be used to add new structural members to achieve the desired response. Numerical examples and finite element implementation of the developed method are provided to demonstrate the feasibility of the formulations. Ó Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. Keywords: Inverse problem; Structural modifications; Desired frequencies; Structural vibration; Eigenvalues; Pin-jointed structures; Cross-sectional area 1. Introduction The conventional approach for optimisation of structural vibration behaviour is to conduct a series of modifications on the structureõs mathematical model. Each series requires the analysis of the modified model, which is usually only slightly different from the model previously analysed. This complete reanalysis of the system is often an expensive and time consuming task, as all the system equations are subjected to the same solution procedure repeatedly. To eliminate the need to reanalyse the whole structure, some effort has in the past been put towards developing new formulations to compute the required parameters changes to yield the desired natural frequencies. Early researchers in this area [1,] based their formulation on ReyleighÕs work by utilising the first order terms of TaylorÕs series expansion. Chen and Garba [3] used the iterative method to modify structural systems. Later Baldwin and Hutton [4] * Corresponding author. address: hamid.bahai@brunel.ac.uk (H. Bahai). presented a detailed review of structural modification techniques and they classified them into three categories of the techniques based on small modification, localised modification and those based on modal approximation. Further research on structural modification was carried out by Tsuei and Yee [5 7] who presented a method of shifting the desired eigenfrequencies using the forced response of the system. The method is based on modification of either the mass or stiffness matrix by treating the modification of the system matrices as an external forced response. This external forced response is formulated in terms of the modification parameters. Thus creating a modified eigenvalue problem. More recently Zhang and Kim [8] investigated the use of mass matrix modification to achieve desired natural frequencies. McMillan and Keane [9] investigated a method of shifting eigenfrequencies of a rectangular plate by adding concentrated mass elements. Sivan and Ram [1 1] extended further the research on structural modification by studying the construction of mass spring system with prescribed natural frequencies, they obtained stiffness and mass matrices using the orthogonality principles. However, the resulting stiffness or mass matrix may not be physically implemented. In //$ - see front matter Ó Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S ()49-3

2 398 H. Bahai et al. / Computers and Structures 8 () Ref. [1] Sivan and Ram developed a new algorithm based on JosephÕs work [13] which involves the solution of the inverse eigenvalue problem. In the last few years the work on the inverse problem done by Gladwell [14] started to be taken seriously by engineers and researchers interested in this field of engineering. The work is applied to both discrete and continuous systems. In this paper an efficient formulation between the geometric and material properties of structures and their eigenvalues is established for both truss structures and those made up of continuous elements. The formulation allows the shifting of the natural frequencies and solves for the required modification on chosen geometric and material properties.. Inverse formulation for truss structures The first formulation establishes a relationship between the eigenfrequencies of finite element truss elements and the structural parameters of the system. For a discrete system such as mass and spring systems with only few degrees of freedom the formulation which accounts for such relationship is easily obtained and hence the change of stiffness or mass required for shifting the eigenvalues can be readily evaluated. However, for systems with a large number of degrees of freedom and continuous systems special algorithms have to be developed. For the new system to be constructed, the modification carried out on the structural properties of the system must have a physical meaning and be realisable. For example in the case of truss structures the cross-sectional area of the bars can be modified to shift the eigenfrequencies. This would result in stiffness as well as mass modification. In the following section a formulation giving the cross-sectional area modification as a function of the required eigenfrequency is developed. For a pin-jointed truss structure both the stiffness and mass modifications can be expressed as functions of the area modification of any member in the structure. DK ¼ DA½K Š DM ¼ DA½M Š ð1þ where DK and DM are the variations or modifications on the system stiffness and mass matrices respectively, DA is the change in the area of the modified member and ½K Š and ½M Š are the matrices containing the coefficients of the stiffness and mass participation of the member to be modified respectively, i.e. they are the stiffness and mass matrices of the modified member where the area is taken as unity. The equation of motion for the free vibration of a dynamic system is given by: ðk k MÞd ¼ ðþ where K is the stiffness matrix of the system, M is the mass matrix, d is the displacement vector and k is the eigenvalue of the original system. If a modification DA is carried out on any member of the structure, this would result in modifications in both stiffness and mass matrices of the structure and hence the equation of motion becomes: ðk þ DK k d M k d DMÞd ¼ ð3þ where k d is the new eigenvalue of the modified structure. Eq. (3) can be transformed to modal co-ordinates by putting d ¼ Uu where U is the mass normalised modal matrix. Hence, ðk þ DK k d M k d DMÞUu ¼ ðku þ DKU k d MU k d DMUÞu ¼ ð4þ ð5þ If we pre-multiply the above equation by U T and use the orthogonality characteristic of U with respect to K and M we obtain the following equation: ðx þ U T DKU k d I k d U T DMUÞu ¼ ðþ where X is the diagonal eigenvalue matrix and I is the unity matrix. Eq. () can be written as ðx k d IÞu ¼ ðu T DKU k d U T DMUÞu or u ¼ ðx k d IÞ 1 ðu T DKU k d U T DMUÞu By pre-multiplying both sides by U we obtain Uu ¼ UðX k d IÞ 1 ðu T DKU k d U T DMUÞu Rearranging Eq. (9) gives: Uu ¼ UðX k d IÞ 1 U T ðdk k d DMÞUu ð7þ ð8þ ð9þ ð1þ By substituting for DK, DM and Uu by DA½K Š, DA½M Š and d respectively we obtain: d ¼ DAUðX k d IÞ 1 U T ðk k d M Þd This can be written as fdg ¼ DA½FŠ½GŠd ¼ DA½RŠfdg where ½F Š¼UðX k d IÞ 1 U T ; and ½RŠ ¼½FŠ½GŠ ½GŠ ¼½K k d M Š ð11þ ð1þ Eq. (1) can be written in the form of simultaneous equations as:

3 H. Bahai et al. / Computers and Structures 8 () d 1 ¼ DAðR 11 d 1 þ R 1 d þþr 1n d n Þ d ¼ DAðR 1 d 1 þ R d þþr n d n Þ... d n ¼ DAðR n1 d 1 þ R n d þþr nn d n Þ After rearranging Eq. (13), we obtain, ðda 1 þ R 11 Þd 1 þ R 1 d þþr 1n d n ¼ R 1 d 1 þðda 1 þ R Þd þþr n d n ¼... R n1 d 1 þ R n d þþðda 1 þ R nn Þd n ¼ ð13þ ð14þ Eq. (14) can be simplified in a matrix form as: DA þ R 1;1 R 1;... R 1;n d 1 R ;1 DA 1 þ R ;... R >< ;n d >= >: >; R n;1 R n;... DA 1 þ R n;n d n ¼ ð15þ where the terms R i;j are function of the eigenvalue k d. The characteristic equation of the modified system for the eigenvalue k d is given by: DA 1 þ R 1;1 R 1;... R 1;n R ;1 DA 1 þ R ;... R ;n ¼ R n;1 R n;... DA 1 þ R n;n ð1þ Eq. (1) are for global modification where all the bars are to be modified at the same time and in this case n is equal to the total number of unconstrained degrees of freedom. However, if this is not the case then only the terms corresponding to the nodes associated with the modified bars are retained. For example if only one bar is to be modified, then Eq. (1) would be reduced to four equations for plane truss and to six equation for space truss, where the associated nodes are not constrained. A solution for the above problem can be obtained by solving the characteristic Eq. (1) and obtaining DA..1. Algorithm For a given truss structure: (1) Obtain the stiffness and mass matrices K and M. () Run a modal analysis to obtain the natural eigenvalues ½XŠ and the eigenvectors fh i g. (3) Compute the mass normalised modal matrix ½UŠ. (4) Obtain the desired eigenvalue k d from the desired frequency f d. (5) Compute the matrix F ¼ UðX k d IÞ 1 U T. () Input the number of the bar to be modified. (a) Compute the stiffness and mass matrices K and M, taking A for the member as unity. (b) Compute the matrix G ¼ K k d M. (c) Carry out the matrix multiplication ½RŠ ¼½F Š½GŠ. (d) Compute DA 1 from the characteristic equation jrj ¼. (e) Determine DA which represents the necessary variation of the cross-sectional area of the bar considered. (7) Repeat step to consider another member. The proposed method also allows for the addition of new members to the structure to achieve the required frequency. In this case if the new bar is to be added between two existing nodes then only the other geometric and material properties as well as the nodes numbers associated with the new bar need to be entered in step for K and M to be computed. However, if the new bar is to be connected to a new node, then of course that node has to be considered in step 1 with any corresponding boundary conditions. If more than one member is to be altered simultaneously, then in step above, all the numbers of the members to be modified must be entered. Also in step (a), K and M would refer to the assembled stiffness and mass matrices for the considered members. 3. Formulation for continuum elements For the case of the finite element models made up of continuum elements we start again from the eigenvalue problem of the systems free vibration: ðk kmþu ¼ ð17þ The objective here is to obtain a first order approximate solution for the required modification in design variables to obtain a certain shift in eigenvalues of the system. The system design variables could include geometrical or material properties of the structure such as thickness, nodal coordinates or the YoungÕs modulus of elasticity. To obtain the sensitivity of an eigenvalue of the structure with respect to design variable, Eq. (17) is differentiated with respect to the design variable b: ðk km k MÞU þðk kmþu ¼ ð18þ where ð Þ¼o=ob. Considering the orthogonality property of eigenvectors with respect to mass matrix, and multiplying (18) by U T, we get: k ¼ U T ðk km ÞU ð19þ As a linear or first order TaylorÕs expansion series, it can be assumed that for a given structure, a change in a single design variable b will affect the eignevalue of the system:

4 4 H. Bahai et al. / Computers and Structures 8 () k d ¼ k þ k Dt ðþ where k may be obtained from (19). In order to apply the method to multi-variate design optimisation problems, Eq. (19) is rewritten as: k i;j ¼ U T i ðk ;j k i M ;j ÞU i ð1þ where the the subscript ði; jþ denotes the derivative of the ith eigen parameter with respect to the jth design variable. In general, the number of design variables may be more than the number of eigenvalues to be shifted. To ensure equality of number of equations and variables, we can adjust the number of the eigenvalue shifts to match the number of design variables and set spurious eigenvalue variations to zero. Therefore, using the first order Taylor series expansion for multi-variable functions, we can write: Dk i ¼ Xj¼n k i;j Db j i ¼ 1;...; n ðþ j¼1 Rewriting () in matrix form, we get: % Variation of bars cross section bar 1 bar bar 3 bar 7 bar 8 bar 9 bar 1 bar 11 bar % Variation of first frequency Fig.. Variation of first frequency with required modification on bars area. fdkg ¼½LŠfDbg ð3þ where fdkg and fdbg are the vectors containing eigenvalues shifts and change in the design variables, respectively, and the matrix ½LŠ is defined as: L ij ¼ ok i ð4þ ob j The linear system of Eq. (3) is solved for fdbg in terms of the required fdkg. The last example will show the accuracy of this method for continuous finite element structures. 4. Numerical examples 4.1. Twelve bar truss m m m Fig. 1. Twelve bar truss structure m.5m 1.5m The first example is a 1 bar truss cantilever as shown in Fig. 1. This example is used to illustrate the modifications required on the cross-sectional area of the bars to shift the lowest frequency. The addition of new bars is also considered in this example. The material properties and the cross-sectional area of the bars are as follows: E ¼ 1 11 N/m, q ¼ 78 kg/m 3 and A ¼ m for all bars. The lowest natural frequency of the structure has been increased by Df ¼ 5% through steps of.5% and for each step the required change in the cross-sectional area of each bar is obtained. These are shown in Fig.. It can be seen that while an increase in the cross-sectional area of some bars, for example 1,, 3 and 7, is necessary to achieve the desired frequency, other bars require a reduction in their areas. This is due to the fact that the cross-sectional area affects both the mass and stiffness matrices of the structure. It is also noticed that the fixed frequency may not be achieved by varying the area of some bars, for example in this case, a shift in the frequency by % cannot be obtained by modifying the cross-sectional area of bars 1, 3 and 9 only. Therefore, if no restraints are imposed on the selection of the members to be modified, the most ÔsensitiveÕ members of the structure are selected. To further illustrate the effectiveness of the developed formulation, two practical applications are carried out on the above truss structure. In the first case it was desired to shift the first frequency from to 45. Hz by restricting alteration to the cross-sectional area of bar 3. The results obtained from the developed method showed that an increase in the cross-sectional area of bar C 1 from to 1: m is required. The results showing the lowest five natural frequencies

5 H. Bahai et al. / Computers and Structures 8 () Table 1 The five lowest natural frequencies (Hz) of original and modified structure Original structure Modified structure Area of bar 3 increased to 1:7 1 4 m for both original and modified structure are given in Table 1. In the second case it was decided to shift the second frequency of the original structure was shifted from to 11. Hz by restricting the modification to the cross-sectional area of bar. The solution obtained showed that an increase in the cross-sectional area of bar from to 13: m is required. The results showing the lowest five natural frequencies for both original and modified structure for this case are also given in Table Adding new structural members Area of bar increased to 13: m As mentioned above the method can also deal with the possibility of adding new bars to an existing structure to achieve a desired frequency. In this example bars 13 and 14 are added separately, in order to shift the lowest natural frequency of the truss. Fig. 3 shows that to shift the frequency by 5% only a small sectional area for bar 13 is required and this cannot be achieved by adding bar 14 alone. This last example shows how easily the method can be used to modify the frequencies of existing structures Space truss structure The second example consists of the tower shown in Fig. 4. The dimensions and material properties are shown in the same figure. The cross-sectional areas for each bar is given by: A ¼ m for C 1 and C bars (corner columns in bottom and top levels respectively) A ¼ 1:5 1 4 m for B 1 bars (horizontal members in bottom level) A ¼ :8 1 4 m for B bars (horizontal members in top level) A ¼ :8 1 4 m for T 1 bars (diagonal members in bottom level) A ¼ :4 1 4 m for T bars (diagonal members in top level) The sensitivity of the lowest natural frequency to any modification on the cross-sectional area of different bars is first investigated. Fig. 5 shows the percentage variation of the first natural frequency with the required percentage variation on the cross-sectional area of the bars. It is seen that the first frequency is most sensitive to bars C 1 and C and one can only specify the desired shifting of the first natural frequency to obtain the necessary required modification on the cross-sectional area of either bars C 1 or C. 1.m B C E=.1x1 11 N/m r =78kg/m 3 1 T Member 13 Member 14 Cross sectional area 9 3 B 1 T 1 C % Variation of frequency 3m Fig. 3. Variation of first frequency with the area of added bars. Fig. 4. Space truss structure.

6 4 H. Bahai et al. / Computers and Structures 8 () % variation of bars area C1 bars C bars B1 bars B bars T1 bars T bars %Error in Frequency change % Error of the method % Thickness change App%DF Exct%DF %Err.DF % Variation of first frequency Fig. 5. Variation of first frequency with the required modification on the cross-sectional area of bars Plane stress problem The third example is a model made up of 1 plane stress continuous elements with thickness.3 m (Fig. ). The dimensions and mechanical properties are shown in the figure. The objective of this case study is to investigate the sensitivity of the thickness of various elements with a shift in the first natural frequency. The first frequency of the structure is f 1 ¼ 99:18 Hz and therefore 1 1 ¼ :. Using ANSYS program, Fig. 7. Error of frequency shift in continuous structure. the first eigenfrequency of the structure was extracted. Table shows the sensitivity of this frequency with respect to the thickness of various elements. It can be seen that element 1 is the most sensitive element in the model. From (1) we have: k d ¼ þ 9894Dt ð5þ and therefore the updated frequency will be: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 9894Dt f d ¼ ðþ p Fig. 7 shows the difference between exact and approximate change of the first frequency against the change in the thickness of element number one, and the percentage error in the frequency change. This figure shows that for 3.% frequency shift, there is a 5% thickness change in element one, and the error is about 9%. 5. Conclusions and discussion Fig.. Plane stress continuous structure. An inverse eigenvalue formulation has been developed in order to determine the required geometrical and material modifications for pre-defined natural frequency values of structures. The method has been validated by applying it to three case studies. The results are compared against exact solutions and it is shown that the error are negligible for small changes of natural frequencies. The proposed algorithm gives an acceptable level of accuracy for all the cases studied. In the case of the truss structure the natural frequencies can be shifted by up to % with a maximum error of 5%. The method Table First eigenvalue sensitivity Element dk=dt )18 87 ) ) ) )19 83 )

7 H. Bahai et al. / Computers and Structures 8 () was extended to handle structures modelled with continuous finite elements. For this case, a study of a plain stress problem was conducted, where the thickness of a finite element was modified to optimise the first frequency of the structure. It was shown that for 3.% frequency shift, there is a 5% thickness change in the element, with an error of about 9%. The great advantage of the formulation proposed above is that the sensitivity analysis is conducted only on small parts of the global stiffness and mass matrices, requiring minimal computational processing time and memory. It is, therefore, not necessary to assemble the stiffness and mass matrices in order to conduct the sensitivity analysis. This is a very important feature of the proposed technique which allows it to be used as an addon tool to most commercial FEA codes which do not readily make the assembled stiffness and mass matrices easily accessible to the user. Moreover, since the algorithm always works with the components of stiffness and mass matrix of individual elements it is much more computationally economical. The other important feature of the proposed technique is that other natural frequencies of the system can be kept constant whilst small modifications are made on the required frequencies. References [1] Belle V. Higher order sensitivities in structural design. AIAA J 198;(February):8 8. [] Vanhonacker P. Differential and difference sensitivities of natural frequencies and mode shapes of natural structures. AIAA J 198;18:1511. [3] Chen JA, Garba JA. Analytical model improvement using modal test results. AIAA J 198;18. [4] Baldwin J, Hutton S. Natural modes of modified structures. AIAA J 1985;3(11): [5] Tsuei YG, Yee E. A method for modifying dynamic properties of undamped mechanical systems. J Dyn Syst, Meas Control 1989;111(September):43 8. [] Yee E, Tsuei YG. Method for shifting natural frequencies of damped mechanical systems. AIAA J 1991;9(11): [7] Yee E, Tsuei YG. Modification of stiffness for shifting natural frequencies of damped mechanical systems. In: Modal analysis, modelling, diagnostics and control analytical and experimental, DE-Vol p. 11. [8] Zhang Q, Kim K-K. A review of mass matrices for eigenproblems. J Comput Struct 1993;4: [9] McMillan AJ, Keane AJ. Shifting reasonances from a frequency band by applying concentrated masses to a thin rectangular plate. J Sound Vibr 199;19():549. [1] Sivan D, Ram YM. Mass and stiffness modification to achieve desired natural frequencies. Commun Numer Meth Eng 199;1: [11] Ram YM. Enlarging a spectral gap by structural modification. J Sound Vibr 1994;17():5 34. [1] Sivan D, Ram YM. Optimal construction of mass-spring system with prescribed model and spectral data. J Sound Vibr 1997;1(3): [13] Joseph KT. Inverse eigenvalue problem in structural design. AIAA J 199;3(1):89. [14] Gladwell GML. Inverse vibration problems for finite element models. Inverse Probl 1997;9(4):41 34.

An inverse strategy for relocation of eigenfrequencies in structural design. Part I: first order approximate solutions

Journal of Sound and Vibration 274 (2004) 481 505 JOURNAL OF SOUND AND VIBRATION www.elsevier.com/locate/jsvi An inverse strategy for relocation of eigenfrequencies in structural design. Part I: first