Nonlinear Structural Modification and Nonlinear ...

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

Nonlinear Structural Modification and Nonlinear Coupling

1

Taner Kalaycıoğlu1,2, H. Nevzat Özgüven1 Department of Mechanical Engineering, Middle East Technical University, 06800 Ankara, TURKEY 2 MGEO Division, ASELSAN Inc., 06011 Ankara, TURKEY e-mail: [email protected], [email protected]

ABSTRACT Structural modification methods were proved to be very useful for large structures, especially when modification is local. Although there may be inherent nonlinearities in a structural system in various forms such as clearances, friction and cubic stiffness, almost all of the structural modification methods are for linear systems. The method proposed in this work is a structural modification/coupling method developed previously, and extended to systems with nonlinear modification and coupling recently. It is based on expressing nonlinear internal force vector in a nonlinear system as a response level dependent "equivalent stiffness matrix" (the so-called "nonlinearity matrix") multiplied by the displacement vector, by quasi linearizing the nonlinearities using Describing Function Method. Once nonlinear internal force vector is expressed as a matrix multiplication then several structural modification and/or coupling methods can easily be used for nonlinear systems, provided that an iterative solution procedure is employed and convergence is obtained. In this paper, formulations for each of the following cases are given: nonlinear modification of a linear structure with and without adding new degrees of freedom, and elastic coupling of a nonlinear substructure to a main linear structure with linear or nonlinear elements. Case studies for three of those cases are given and an application of the method to a real life engineering problem is demonstrated. 1

INTRODUCTION

Over the last five decades or so, the finite element (FE) method has established itself as the major tool for the dynamic response analysis of engineering structures. However, for the dynamic reanalysis of large engineering structures modified locally, constituting FE model each time is expensive and time consuming, especially when several alternatives are to be studied. Therefore, it will be more practical to predict the dynamic behavior of a modified structure by using dynamic response information of the original structure and dynamic properties of the modifying structure. Various structural modification methods, focusing on the change of dynamic behavior of a structure due to modifications, have been developed in order to reduce the effort involved in the dynamic reanalysis of such systems. Although the structural modification methods based on linearity assumption are available in the literature, a review of which can be found in recent papers of Hang et al. [1-3], these methods cannot be used when there is nonlinearity in the system. During the past two decades, several structural modification/coupling methods have been suggested taking the nonlinear effect into account. Watanabe and Sato [4] used first order describing function in order to linearize the nonlinear stiffness of a beam structure and developed the so-called “Nonlinear Building Block” approach for coupling nonlinear structures with local nonlinearity. Cömert and Özgüven [5] developed a method for calculating the forced response of linear substructures coupled with nonlinear connecting elements. Ferreira and Ewins [6] proposed a new Nonlinear Receptance Coupling Approach for fundamental harmonic analysis based on describing functions. They suggested an approach that is capable of coupling structures with local nonlinear elements whose describing functions are available considering just the fundamental frequency. Then, Ferreira [7] extended the approach and introduced Multi-Harmonic Nonlinear Receptance Coupling Approach. This approach is able to couple linear and nonlinear structures with different types of joints by specifying the multi-harmonic describing functions for all nonlinear joints. Chong and İmregün [8] suggested an iterative algorithm for coupling nonlinear systems with linear ones. Maliha et al. [9] coupled a nonlinear dynamic model of a spur gear pair with linear FE models of shafts carrying them, and with discrete models of bearings and disks. Huang [10] worked on dynamic analysis of assembled structures with nonlinearity.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

In a recent work, [11], the authors of this paper proposed a new approach for dynamic reanalysis of a large linear structure modified locally with a nonlinear substructure. The method suggested is an extension of the method developed by Özgüven [12] for structural modifications of linear systems. In the present study, the proposed approach is further extended for dynamic reanalysis of linear structures coupled with nonlinear substructures by using linear and nonlinear coupling elements. In this paper, applications of the method to modification analysis problems where a linear system is modified with a nonlinear system for the following three cases are presented: structural modification with additional degrees of freedom (DOFs), structural coupling with linear elements and structural coupling with nonlinear elements. Finally, a real life structure modeled with FE method is used in order to show the applicability of the method to large ordered systems. 2

THEORY

The structural modification method proposed by Özgüven [12] more than two decades ago can be used for modified linear systems with or without additional DOFs. The method is for dynamic reanalysis of systems where there is a modification in the mass, stiffness and/or damping of the system. The frequency response functions (FRFs) of a modified system are calculated from those of the original system and the dynamic stiffness matrix representing the modifications in the system by using the following equations [12]: −1

 H *  =  I +  H  Z    H   11  [ ]  11   11    11 

(1)

T

 H *  =  H *  =  H  [ I ] − Z   H *    11   11    12   21   21  

(2)

H *  =  H  − H  Z  H *   22   22   21   11   12 

(3)

where [H] and [H*] are receptance matrices of the original and modified systems, respectively. [Z] represents the dynamic stiffness matrix of structural modifications, and superscripts 1 and 2 denote the coordinates on which a modification is applied and the remaining coordinates, respectively. The method has been extended in the same work for modifications which require additional DOFs, in other words, for modifications which also causes coupling of another system to the original one. The resultant equations for such cases are as given below [12]: −1 H*   [ ] [ ]  H [0]  Hba   ba  =  I 0  +  bb  .[ Z]   H*  [ 0] [ 0]  [0] [ I ]   [0]    ca    

H*   bb  H*   cb 

(4)

H*  [ I ] [ 0] H  0  −1 H  [ 0]  bc   bb   bb  =  .[ Z]    + *  [ ] [ ] [ ] 0 0   I    [ 0] [ I ]   0 Hcc  

H*  = H  −  H  aa   aa   ab 

|

H*  [ 0][ Z]  ba    *  Hca 

 H*  H*  H*  = H  [ 0] [ I ] −[ Z]  bb    ab   ac   ab  H*   cb  

|

H*   bc  H*   cc 

|

(5)

(6)

(7)

where the subscript a represents the coordinates that belong to the original system only, the subscript b denotes connection coordinates, and the subscript c represents coordinates that belong to modifying structure only. The method is most useful when such modifications are on limited number of coordinates, that is, when modification is local. Then the order of the matrix to be inverted will be reduced considerably, irrespective of the total size of the original structure.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

The method can be extended to modification of a linear system where modifying system has nonlinearity. Nonlinear internal forces can be included in the analysis by considering an additional equivalent stiffness matrix in the calculations which is a function of unknown response amplitudes [11]. Then, the dynamic stiffness matrix for the modifying system showing nonlinear behavior will take the form

[ Z ( X )] = [ ∆K ] − ω 2 [ ∆M ] + jω [ ∆C ] + j [ ∆D] + [ ∆( X )]

(8)

|

where [∆K], [∆M], [∆C] and [∆D] represent stiffness, mass, viscous and structural damping matrices of the modifying structure, respectively, and {X} is the amplitude vector of harmonic response of the system. The formulation for [∆(X)], named as “nonlinearity matrix”, was first introduced by Budak and Özgüven [13, 14] for certain types of nonlinearities, and later extended by Tanrıkulu et al. [15] for any type of nonlinearity by using Describing Function Method. The elements of nonlinearity matrix are given [15] as n

∆kk = ∑ vkm

|

(9)

m =1

∆kk = −vkm ,

( k ≠ m)

(10)

where subscripts k and m represent two engagement coordinates of a nonlinear element. Here, due to the nonlinearity matrix, [Z(X)] will be response level dependent, and therefore solution can only be obtained by employing an iterative solution procedure. The details of the above formulation can be found in reference [11]. 2.1

Formulation for Nonlinear Structural Modification without Additional DOFs

For nonlinear structural modifications without additional DOFs, equations (1) to (3) are used where dynamic stiffness matrix for the modifying system is expressed as follows:

 Z11 ( X )  = ∆K11  − ω 2 ∆M11  + jω  ∆C11  + j ∆D11  +  ∆11 ( X ) 

|

(11)

Here, the subscript 1 refers to the coordinates of the system where there is a modification. To be able to use equations (1) to (3), renumbering of the coordinates of the original system may be necessary. It should be also noted that although there is a matrix inversion in the formulation, the size of the matrix to be inverted is equal to the total DOFs of the modifying system (size of the matrix [Z11(X)]). Therefore, the method is most useful when local modifications are applied to large ordered systems. 2.2

Formulation for Nonlinear Structural Modification with Additional DOFs

When the modification is such that modifying nonlinear system does not only change the system properties of the original system at some coordinates, but also couples another system to the original system, then the total DOF of the modified system will be increased. Such a structural modification case is referred to as structural modification with additional DOFs. In such applications, equations (4) to (7) are used for the receptance of modified system. [Z(X)] in these equations will be as shown in equation (8). 2.3

Formulation for Nonlinear Structural Coupling with Linear Elements

The same formulation given in section 2.2 can also be used for analyzing coupled systems by treating the problem as an equivalent structural modification problem as shown in Fig. 1. That is, for each connection node on the original system a massless node is added to the coupled subsystem. Firstly, the stiffness matrix of the coupled subsystem, [∆K], is expanded. For example, if p number of massless nodes are added to the coupled subsystem where the DOF per node is q, p×q number of rows and columns are added to the stiffness matrix of the coupled subsystem. Then, the stiffness values of the linear elastic coupling elements are inserted in proper locations of added rows and columns of [∆K]. The mass, nonlinearity, viscous and structural damping matrices of the coupled

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

subsystem should also be expanded in the same way. However, just zeros will be inserted in these rows and columns. Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, subsystem, the problem can be taken as a nonlinear structural modification problem as defined in section 2.2.

Fig. 1 Structural coupling problem with linear elastic elements

2.4

Formulation for Nonlinear Structural Coupling with Nonlinear Elements

Again, the same formulation given in section 2.2 can be used for analyzing coupled systems by treating the problem as an equivalent structural modification problem as shown in Fig. 2. That is, for each connection node on the original system a massless node is added to to the coupled subsystem as in the previous case. The only difference will be the nonlinear character of the connection elements. Therefore, again; stiffness, mass, nonlinearity, viscous and structural damping matrices of the coupled subsystem are expanded by adding p×q number of rows and columns, where p is the number of massless nodes added to the coupled subsystem and q is the DOF per node.

Fig. 2 Structural coupling problem with nonlinear elements

However this time, the added rows and columns of the nonlinearity matrix [∆( [ X)] will be filled with proper elements representing the nonlinear connection elements. If there are linear stiffness counterparts of the connecting elements, these values will also be properly inserted into the expanded rows and columns columns of the stiffness matrix of the coupled subsystem. Lastly, by defining additional massless nodes as new rigid connection nodes of the coupled subsystem, the problem can be taken as a nonlinear structural modification problem as defined in section 2.2. 3

CASE STUDIES

In this section applications of the proposed method firstly to discrete systems and then to real engineering structure modeled with FE method will be given. The first three case studies illustrate modification and coupling analyses of two discrete subsystems in three main categories, namely, nonlinear structural modification with additional DOFs, nonlinear structural coupling with linear elements and nonlinear structural coupling with nonlinear elements. An application for structural modification without additional DOF is not given in this paper, since it was well investigated in our recent paper [11]. In the modification last case study, a real life engineering problem with nonlinear structural modification will be considered.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

3.1

Nonlinear Structural Modification Modification with Additional DOFs

In this case study, nonlinear modification of a linear discrete system is considered. As can be seen from Fig. 3, the modifying nonlinear system adds a new DOF to the original linear system.

Fig. 3 Structural modification with additional DOFs

In the same figure, figure, there exists a cubic stiffness element between coordinates 4 and 5 showing hardening behavior. Parameters of this nonlinear element and the properties of both subsystems are given as follows: m1 = m2 = m3 = 1kg and m4 = m5 = 0.5 kg , k1 = k 2 = k 3 = k4 = k 5 = k6 = 1000 N/m,


)NL = K0 x + β x 3 where K0 = 0 N/m and β = 2x106 N/m3 n( x,x

|

|

(12)

Structural damping with a loss factor of 0.0015 is assumed in the analysis for all linear elastic elements. Frequency response of the modified system at the point where a harmonic force of magnitude 4 N is applied is shown in Fig. 4.

Fig. 4 Frequency response of m3 after modification

The results show that the nonlinearity in the modifying system becomes effective in all four modes of the modified system which reveals the importance of including nonlinearity in this specific case. Although the nonlinearity changes the frequency responses around resonances considerably by causing a jump, which is a typical response behavior due to cubic stiffness element, no convergence problem problem is observed in the solution when the proposed method is used. Note that, since the proposed method is an FRF based method, only the FRFs related with the DOFs we are interested and with the connection DOFs are to be included in the calculation. Moreover, Moreover, the size of the matrix to be inverted is 2 by 2 in this example, which is the order of the modifying system (and therefore it would still be 2, even though the size of the original system were much

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

higher). These are the important features of the method which makes it more advantageous for large ordered systems with local modification. 3.2

Nonlinear Structural Coupling with Linear Elements

As the second case study, nonlinear structural coupling analysis of the same linear and nonlinear discrete system coupled with a linear elastic element, as shown in Fig. 5, is considered.

Fig. 5 Structural coupling of linear and nonlinear systems with a linear element

Here, the stiffness of the linear elastic element is taken as 200 N/m. Assuming structural damping with a loss factor of 0.0015 again for all linear elastic elements, frequency response of the modified system at the point where a harmonic force of magnitude 4 N is applied, is obtained as shown in Fig. 6.

Fig. 6 Frequency response of m3 after coupling

It can be seen from the figure that nonlinearity is more effective on 2nd, 3rd and 5th modes of the system compared to the other two modes. Note that, since the proposed method is based on FRFs, in addition to the FRFs related with the connection DOFs, it is sufficient to include only the FRFs related with the required DOFs in the calculations. Moreover, the size of the matrix to be inverted during calculations is again 2 by 2 in this example (which is the size of the coupled subsystem). This saves considerable computational time especially in large ordered original systems as long as the modification is of small order. This feature of the method makes it very desirable in parametric studies, for instance, while investigating the effects of the linear elastic coupling element stiffness on system response. In Fig. 7, the effect of different stiffness values of the linear elastic coupling element on the system response around 3rd resonance is examined in detail. It can be seen from the figure that increasing values of the linear elastic coupling element stiffness will not only shift the 3rd natural frequency to higher frequencies, but also will increase the effect of nonlinearity on this mode.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

Fig. 7 Frequency responses around 3rd resonance for different linear elastic coupling elements (a) kLC = 200 N/m, (b) kLC = 400 N/m, (c) kLC = 600 N/m 3.3

Nonlinear Structural Coupling with Nonlinear Elements

The third case study discusses nonlinear structural coupling analysis of the same linear and nonlinear discrete systems with nonlinear elements. The linear coupling element used in the previous case study is kept the same, but an additional nonlinear coupling element is considered as shown in Fig. 8.

Fig. 8 Structural coupling of linear and nonlinear systems with a linear and a nonlinear element

As the nonlinear coupling element, a linear spring having a stiffness of magnitude 200 N/m and 0.02 m clearance on each side is inserted between two coupling coordinates. Assuming structural damping with a loss factor of 0.0015 again for all elastic elements, frequency response of the modified system at the point where the harmonic force of magnitude 4 N is applied is obtained as shown in Fig. 9.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

Fig. 9 Frequency response of m3 after coupling with a linear and a nonlinear element

When Figures 6 and 9 are compared with each other, it can be observed that nonlinear coupling element affects 1st and 3rd modes of the system more than it does the other modes. Again, the advantage of using only the FRFs related with the required and connection DOFs, which is the FRF related with the 3rd mass in this example, and inverting a matrix only in the size equal to the DOF of the modifying system in this method, the method can be used in design analyses where, for instance, the effects of using different nonlinear coupling elements on the system response are investigated.

Fig. 10 Frequency responses around 3rd resonance for different nonlinear coupling elements (a) kNLC = 200 N/m, δ = 0.02 m, (b) kNLC = 200 N/m, δ = 0.04 m, (c) kNLC = 400 N/m, δ = 0.02 m

In Fig. 10, the effect of different stiffness values of nonlinear coupling element on the system response around 3rd resonance is examined in detail. It can be seen from Fig. 10 that typical response distortion due to clearance type of nonlinearity is observed as an abrupt change in the frequency response at the point of transition where the response amplitude reaches to the value of clearance. As an expected result, the displacement value where this abrupt change occurs differs depending on the

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

value of the clearance (compare Fig. 10 (a) and (b)). On the other hand, for nonlinear spring elements having different stiffness values but the same clearance, this abrupt change occurs at the same displacement value but the frequency responses after that point show different behaviors due to different additional linear spring stiffnesses of the nonlinear coupling elements after the response amplitude reaches to the value of clearance (compare Fig. 10 (a) and (c)). 3.4

Nonlinear Structural Modification - A Real Life Engineering Problem

As the last case study, a real life engineering problem, a shaft and mirror plate assembly usually used in land platforms for optical purposes, is considered (Fig. 11).

Fig. 11 The shaft and mirror plate assembly

The mirror plate is a costly part due to its well machined reflective surface, so once it is designed it is not desired to be modified further in the design optimization of the assembly depending on the vibration characteristics of the platform it is mounted. Therefore, when it is used in a platform, it may be necessary to modify the shaft and/or bearings, in order to minimize the vibration of the mirror plate so that its reflection performance is increased. In order to make a more precise analysis, nonlinearity introduced by the bearings is included in the dynamic analysis, which can easily be handled by the nonlinear structural modification analysis method suggested here. Solid elements are used in the FE model of the mirror plate with 3 DOFs per node yielding 2655 total DOFs (Fig. 12). The shaft is also modeled by using solid elements with 3 DOFs per node resulting in 186 total DOFs. The FE model of the shaft is also shown in the same figure.

Fig. 12 The FE model of (a) the mirror plate and (b) the shaft

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

Material properties of the mirror plate made of an aluminum alloy and of the shaft made of a structural steel are given in Table 1. Table 1. Material properties of the mirror plate and the shaft Mirror plate Shaft 200 GPa 71 GPa Young’s Modulus 0.3 0.33 Poisson’s Ratio 3 2770 kg/m 7850 kg/m3 Density

Here, mirror plate is taken as the original structure since it is not desired to be changed during the design phase of the assembly. Shaft and bearing assembly on the other hand is taken as the nonlinear modifying structure where ball bearings at the two ends of the shaft, shown in Fig. 11, are modeled as grounded nonlinear springs in horizontal and vertical directions. The nonlinear behavior of the ball bearings can be taken to be cubic in nature [16]. The nonlinear parameters of the ball bearings are taken as follows:


)NL = K0 x + β x3 where K0 = 2x10 2 N/m and β = 5x107 N/m3 n( x,x

|||

|

(12)

In the analysis, firstly the receptances of the mirror plate are calculated for connection points and for any other point we might be interested in (i.e., points of which response is required or a force is applied to, by using the standard modal analysis). Then, the structural modification method is employed and the receptances of the required points on the modified structure are calculated. As the response of corner points on the mirror have the major importance since they are more prone to effect the reflection performance, the FRF related with a point near one of the corners of the mirror plate is calculated when a harmonic force of magnitude 2 N is applied to the same point. The calculated frequency response is shown in Fig. 13 with the linear FRF of the assembly without considering bearing nonlinearity. Using structural modification method, it is very easy and fast to recalculate the response for any design change in the shaft and/or bearings.

Fig. 13 The direct point FRF related with a point near one of the corners of the mirror plate for F = 2 N

Furthermore, the FRF values for the resulting nonlinear system will be a function of the amplitude of the applied harmonic force, which will require the recalculation of the FRFs for each forcing amplitude level even though nothing is changed in either of the subsystems. In such analyses the method suggested here, again, provides drastic computational advantage, since the FRFs of the linear part of the structure (which is usually the major part of the system) are calculated once and then used to find the FRFs for the nonlinear overall system. In this later phase of the computations, which requires iterative solution, only the FRFs related with the points we are interested in (in addition to those of the modifying structure) are used, rather than all DOFs (which would be the case if the coupled nonlinear system were to be analyzed with standard approaches). In this case study, FRF related with a point near one of the corners of the mirror plate is calculated at two more different forcing levels.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

The results are shown in Figures 14 and 15. The effect of forcing level on the FRF related with a point near one of the corners of the mirror plate after modification can easily be observed by comparing Figures 13 to 15.

Fig. 14 The direct point FRF related with a point near one of the corners of the mirror plate for F = 4 N

Fig. 15 The direct point FRF related with a point near one of the corners of the mirror plate for F = 6 N 4

DISCUSSION AND CONCLUSIONS

The noble structural modification method originally developed for linear systems [12] almost two decades ago was recently extended for dynamic reanalysis of linear structures modified locally with a nonlinear substructure [11]. In this paper, the same approach is further applied for dynamic reanalysis of linear structures coupled with nonlinear substructures by using linear and/or nonlinear coupling elements. In this approach, the frequency responses of the modified structure are calculated from those of the original structure and the system matrices of the modifying nonlinear structure (which can be in the form of a coupled nonlinear substructure). The formulation used in this approach is given for each of the four nonlinear modification or coupling cases investigated. Case studies for three of those cases are presented in this paper. Finally, an application of the method to a real life engineering problem is demonstrated.

Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

The method is based on the computation of the FRFs of a modified system from those of the original system and the dynamic stiffness matrix representing the modifications in the system. Due to the nonlinear behavior of the modifying system, the dynamic stiffness matrix turns out to be response level dependant and therefore the solution requires an iterative approach. The formulation is for rigid connection of the nodes of the original and modifying systems. For the cases where a nonlinear subsystem is coupled to a linear system with elastic elements (linear or nonlinear), the problem is treated as an equivalent structural modification problem where to each free end of a connecting elastic element a massless node is added and that node is rigidly coupled to the main system. With numerical case studies, applications of the method are demonstrated. Firstly, a discrete linear system modified with a discrete nonlinear system is considered. The same original system with different types of modification is used in the first three case studies. The effects of modifications and/or coupling elements are demonstrated. The iterative numerical solution was found to be successful as far as convergence to a solution is concerned. It should be noted that since the modified system is a nonlinear system, the calculated FRFs are valid only for the level of the force applied, and different FRFs are obtained when the amplitude of the external harmonic force is changed. As the last case study, a real life engineering problem is considered in order to show the applicability of the method to real structural systems modeled with FE method. In this problem, structural modification analysis of a mirror plate modified with a shaft-bearing assembly, where bearings at the two ends of the shaft are taken as hardening stiffnesses, is studied. It should be noted that since the proposed method is an FRF based method, only the FRFs of the original system related with the DOFs we are interested in, in addition to the ones at the connection DOFs are to be included in the calculations. Although the formulation includes a matrix inversion, the size of the matrix to be inverted is equal to the DOF of the modifying system, and therefore the method is most advantages when the modification is local. Especially in the design of large main structures which may need to be modified locally, the method is very useful and makes it possible for the designer to try various possible design changes or to make a parametric study with minimum computational cost. 5

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Proceedings of the SEM IMAC XXX Conference Jan. 30 - Feb. 2, 2012, Jacksonville, FL USA Copyright 2011 Society for Experimental Mechanics, Inc.

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