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Correction of finite element models using experimental modal data for vibration analysis
Finite Elements in Analysis and Design 14 (1993) 153-162 Elsevier
153
FINEL 321
Correction of finite element models using experimental modal data for vibration analysis Masaaki O k u m a Department of Mechanical Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo, Japan Abstract. A method is presented for correcting finite element models by referring their modal parameters to
experimental modal data. The method is based upon both the sensitivity analysis and the least squares method, and gives a physicallyrealistic solution of appropriate design variables. In this paper, the procedure of the method is first explained, and then two applications are presented. The first application is a correction of a beam structure, as a basic research study. The second is a correction of a practical structure, a "Centre-beam", which is a structural component of an automotive body. The initial finite element models of both cases are successfully corrected to adjust their dynamic characteristics to experimental results. In particular, it is experimentally verified that the corrected finite element model of the "Centre-beam" is able to represent its dynamic characteristics well, even under different boundary conditions. The verification proves the physical validity and the practical availabilityof the corrected model.
Introduction
The finite element method has been widely used as the most practical method for analysing the dynamics of complex mechanical structures, which include most actual structures. T h e r e are two principal problems in applying the finite element method for analysis of actual complex mechanical structures, the first being the difficulty of making reliable finite element models, and the second the computation of solving the equations of motion, for systems which have many degrees of freedom. However, the current development of computers has m a d e it possible to analyse the dynamics of finite element models with many degrees of freedom. This means that the second problem, of computation, is not serious nowadays. So attention has been focused on the first problem, and many researches are reported with regard to it [1-6]. The first problem consists of two items. The first is the means of determining a suitable size of finite elements for analysing the dynamics of objects up to those frequencies desired by analysts [7]. It is not good to use too fine finite element models for analysis up to a comparatively low frequency, because too fine a mesh wastes hours of modelling and computation time. The second item is the means of determining appropriate values of the design variables of all finite elements of models. This p a p e r presents a method with regard to the latter item [8,10]. The modal p a r a m e t e r s of mechanical structures can be extracted accurately from experimentally obtained F R F s by the experimental modal analysis techniques [9]. Methods have been reported for correcting the design variables of finite element models to appropriate values by referring some of their modal p a r a m e t e r s to experimental modal parameters. This p a p e r presents a method for dealing with the problem. The method is based upon the sensitivity analysis [11] and the least squares method. The thicknesses of b e a m finite elements are adopted as the design variables, and the first-order sensitivities of the lowest three natural frequencies and their associated natural 0168-874X/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
154
M. Okuma / Correction of finite element models
modes with respect to the design variables are used on the first application in this paper. The thicknesses of triangular finite elements and the material properties, Young's modulus and density, are adopted as the design variables for the second application. Three reasons can be given why the sensitivities of the material constants are used. (1) Practical objects like the second specimen in this paper have complex shape configurations. Consequently, there are regions where the thickness is not constant nor is the configuration flat, even in the small area modelled by one finite element. (2) The first-order sensitivities are used only, in spite of the strict fact that the sensitivities of the natural frequencies and natural modes have nonlinear relationships with the thicknesses of the finite elements. (3) The values of the sensitivities of the natural frequencies and natural modes are comparatively small. Therefore, if thicknesses only were adopted as the design variables, it would often be necessary to change the thicknesses of finite elements by very large amounts to move the natural frequencies to a required extent, with the result that the thicknesses become unrealistic from the physical viewpoint.
Theory of the method The vector of referred modal parameters, which are obtained by experiment, is denoted as {q0E} here. The vector of their counterparts on a finite element model is denoted as {q0M}. The matrix [Z] denotes the first-order sensitivity matrix, the columns of which consist of the first-order sensitivities of the modal parameters of the finite element model with respect to the design variables. The vector {At} denotes the extent of the modification of the design variables. Then, the following equation is constructed. {(10E} = {(/)M} + [Z]{A/}.
(1)
At least in the vicinity of the initial design variables, eqn. (1) applies accurately, in line with Taylor's expansion theorem. Three cases can arise with regard to the dimensions of eqn. (1): the number of equations may (i) exceed, (ii) equal, or (iii) be less than the number of design variables. In case (iii), the simultaneous equations are indeterminable. The singular value decomposition method and the pseudo-inverse method work mathematically, but only produce one feasible solution which is not necessarily physically sensible. The equations of case (ii) are determinable, but, it is not convenient to set up the same number of equations as the number of design variables to be modified. Case (i) is practical, and a physically proper solution will be determined. The number of observed natural frequencies is experimentally limited, in general. The associated natural modes should then be extracted from FRFs at a sufficient number of measurement locations on an object. Equation (1) in the case (i) can be solved by the least squares algebra, with weighting functions as {At} = ([Z]t[W][Z])-I[z]t[w]({~E}
- {~M})"
(2)
The weighting functions should be arranged so that all elements of the vector, {q~E} -- {q0M}, are comparable. Without such weightings, numerical calculation difficulties may occur because the values of referred natural frequencies are in general much larger figures than those of the natural modes. Then the design variables should be improved by applying {/new} = {tinitial} + {At},
(3)
where {tinitial} and {t,ew} are the initial design variables and the improved ones, respectively.
M. Okuma / Correction of finite element models
155
Reading F.E.Model and Modal P a r a m e t e r U~ Sensitivity Analysis to thicknesses CAL. [W] and L.S.M
Sensitivity Analysis to Young's Ratio and Density.
I CAt. tt0J a.d t.S.M I
NO Fig. 1. The procedure of the method presented
The distance between the referred experimental modal parameters and the in counterparts in finite element models is not negligible in general applications. The relationship between the referred modal parameters and the design variables is nonlinear. Therefore, the process mentioned above must be iterated. The procedure is illustrated in Fig. 1.
The first application: a simple beam model
A beam, of length 500 mm and cross-section 10 x 10 mm 2, is the specimen for the first application [7]. The beam is modelled with ten beam finite elements, as shown in Fig. 2. The lengths of the sides of the ten finite elements which are parallel to the direction of bending deformation are set up individually with randomly incorrect values, from 7 mm to 16 mm, as the design variables to be corrected. These incorrect design variables of the initial finite element model are improved by referring to its first three natural frequencies and the values
E. • Finite Element
X lo~ mm ~
E.1
E.2
Z .x IE3 I 500mm
Egl El0 Fig. 2. The finite element model of the beam structure of the first application
M. Okuma / Correction of finite element models
156
Table 1 Correction of design variables Element No.
Wi (mm)
W1 (mm)
1 2 3 4 5 6 7 8 9 10
15.0 14.0 12.0 8.0 11.0 7.0 11.0 12.0 14.0 16.0
11.4 9.51 10.0 10.1 9.65 10.1 10.0 9.46 10.0 9.02
Real widths are all 10 mm. Wi are initial widths, W I are widths corrected by least squares method.
of their associated natural modes at five measurement locations, using the method. The experimental natural frequencies and natural modes for references are obtained by applying an experimental modal analysis technique to FRFs on impact vibration testing. Table 1 shows the result of the improvement of those design variables. All initially incorrect design variables have been improved. Here, the experimental natural frequencies and natural modes are not necessarily perfect, due to some experimental errors. The weighting functions used in the computation are not perfect either. These factors are the reasons why the design variables of the result are not 10 mm exactly. Figure 3 shows the improvement from the viewpoint of observation of FRFs. Figure 4 illustrates the correspondence of the three referred natural modes and the fourth natural mode. The results of this application of the presented method illustrate its fumdamental validity.
The second application: a structural component of an automotive body As an application of the method to actual mechanical structures, the correction of a finite element model of a so-called "Centre-beam" is presented here [9]. The structure is shown in Fig. 5. It consists of two engine-mounting bracket parts, and a structural beam part. The structure is built in at the centre bottom of an automobile engine compartment, in order to
~3.14
0 ~3.14 -2 10 .............
z E
I~i
. . . .
• Experimental Data : Corrected Model : Initial Model
11
c
500
1000
1500
Frequency (Hz)
2000
Fig. 3. Observation of improved FRFs of the beam model
M. Okuma / Correction of finite element models • Corrected Modes by Proposed Method ......
157
: Initial Modes ......
: Experiment
: Experiment
1st
3rd
~
~
~
'
~
4th 7
Fig. 4. Observation of improved natural modes of the b e a m model
support the engine. It is considered important for a finite element model of the structure to represent the dynamics accurately up to about 200 Hz. The specimen is modelled as the finite element model shown in Fig. 6, with due regard to the appropriate size of triangular finite elements adopted [7]. The finite element model is structured with 221 nodal points and 461 triangular finite elements. The front engine mounting bracket, the smaller bracket part, is neglected in the modelling, because it is supposed not to affect the global dynamics of the structure. The thicknesses of the finite elements are classified into 40 groups, as shown in Fig. 7. An equal thickness is set on the finite elements in each group, and the thicknesses of these 40 groups are adopted as the design variables. The referred natural frequencies are the first and the second natural frequencies under free-free boundary conditions and the referred values of the associated natural modes are
Fig. 5. T h e object, " C e n t r e - b e a m " , of the second application
M. Okuma / Correction oJfinite element models
158
56
t~£~
19£
7.
t/;\ Fig. 6. The finite element model of the centre b e a m
those at the 46 measurement locations denoted by black solid circles in Fig. 8. The first natural mode is the first-order torsional mode. The second natural mode is the first bending mode. These referred modal parameters are identified from measured FRFs using multi-sine wave excitation testing. The number of the referred values is 94. The number of adopted design variables is 42, being the thicknesses of the forty groups plus the material density and Young's modulus. Consequently, the least squares method is applicable.
Engine Mount Bracket ~ ~~ .....
4
~ ~._~'8.
!~ ~
,~. ~
Upper Plate
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Fig. 7. Classification of finite elements into 40 groups
M. Okuma / Correction of finite element models Driving point 2 (at C l a m p e d Condition)
Mass ~ added Point 1 \C
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Fig. 8. Schematic view of location of driving points and measurement points on experimental tests
.
Experiment ............ Updated ................. initial
i0 s
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tJ'
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~:
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i
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200
......
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I I
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300
400
Frequency (Hz)
Fig. 9. FRFs of the initial and the corrected models, together with the experimental pattern
Table 2 shows a comparison of the referred natural frequencies from experiment and the initial finite element model. The solid lines and the dotted lines in Figs. 9 and 10 denote experimentally obtained FRFs and the counterpart FRFs calculated with the initial finite element model, respectively. These comparisons show the big difference of dynamic characteristics between the initial finite element model and the actual structure. It will often be difficult to improve inaccurate finite-element models by hand for structures as in this application. The FRFs of the model corrected by the method presented in this paper are shown by the broken lines in Figs. 9 and 10. Table 3 shows the natural frequencies from experiment, the initial model and the corrected model, together. Two natural frequencies in the range up to 200 Hz, from experiment and the corrected finite element model, correspond well. The design variables, the thicknesses and material constants, are shown in Table 4. The correction seems
Table 2 Natural frequencies of experiment and the initial model Mode
Experiment
FIE Model
MAC
1 2
133.1 (Hz) 151.3 (Hz)
216.8 (Hz) 240.6 (Hz)
0.9227 0.8893
a
M A C
-
Modal Assurance Criterion
a
M. Okuma / Correction of finite element models"
160
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Fig. 10. F R F s of the initial and the corrected models, together with the experimental pattern
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............ : F.E.M at clamped condition
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Fig. 11. Predicted F R F s u n d e r clampedfree boundary condition
Table 3 Natural frequencies of the corrected model Mode
Exp. (Hz)
Initial (Hz)
U p d a t e d (Hz)
1 2 3 4 5
133.10 151.30 236.09 258.30 265.79
216.8 240.6 435.9 596.3 750.6
135.7 149.6 275.7 380.3 462.6
M. Okuma / Correction of finite element models
161
Table 4 Modified design variables of the finite element model of the centre-beam Initial value
Updated value
Density (kg/mm 3) Young's modulus (N/mm)
7.86 × 10 -5 206,00
1.24 × 10-4 130,000
Group
Initital value (mm)
Updated value (mm)
Group
Initial value (mm)
Updated value (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 3.00 3.00 3.00 2.00 2.00 2.00 1.39 1.39
1.54 1.87 2.99 1.24 2.68 2.41 1.29 2.19 2.36 1.54 1.84 1.99 4.11 1.98 3.31 1.81 1.36 2.08 1.22 1.50
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1.39 1.39 1.39 1.39 1.39 4.00 4.00 2.00 2.00 2.00 2.00 1.39 1.39 1.39 1.39 4.00 4.00 4.00 4.00 4.00
2.18 2.50 4.66 2.81 1.63 0.76 2.57 4.48 1.94 1.53 1.71 3.14 0.80 1.72 2.00 4.01 4.37 3.34 3.48 3.66
to be successful, from the result. On the other hand, a trial of correcting the initial finite element model without adopting the material properties among the design variables was unsuccessful, in that the referred modal parameters of the model could not converge to the
le-3
....
~ ~8-4L~ i\~ :4
1~-~1
:Experiment ............ :F.E.Model at Mass Added C o n d i t i o n
r i
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114¥+ • Z!
5131X+ /
114Y+ : Z1
Fig. 12. Predicted FRFs after attaching two weights on the object, under free-free boundary conditions
162
M. Okuma / Correction of finite element models
values of the experimentally obtained modal parameters because some thickness values became negative before convergence. In order to verify whether the corrected finite element model is physically proper, the dynamics of the specimen under different boundary conditions should be observed. The finite element model can be observed to be proper if it represents the dynamics accurately even under different boundary conditions. As the first verification, the dynamics is calculated for the model under the boundary condition that the right end of the specimen is clamped, as shown in Fig. 8. One of the F R F s is shown in Fig. 11. The F R F s calculated with the model correspond very well with their experimental counterparts. As a second verification, the F R F s of the specimen after attacking two weights under the free-free boundary condition are calculated using the model, and compared with their experimental counterparts. The two weights, of 730 g and 700 g, were connected directly at the points denoted " M a s s added points" in Fig. 8. The result is shown in Fig. 12. The dynamic properties up to about 200 Hz taken from the calculation and experiment correspond very well.
Conclusion
This p a p e r presents a method for correcting finite element models by referring their modal parameters to the experimentally obtained counterparts. The method allows the physically realistic determination of appropriate values of the design variables. The first application as basic research, is a correction of a b e a m structure. The second is a correction of a practical structure, a " C e n t r e - b e a m " structural component of an automotive body. The initial finite element models of those specimens are successfully corrected with regard to the dynamic characteristics. On the second application, it was found that the adoption of material constants is effective for practical applications. It was also verified experimentally, in order to prove the physical validity of the model, that the corrected finite element model of a " C e n t r e - b e a m " represents its dynamic characteristics well even under different boundary conditions.
References
[1] M. BARUCH,A/AA J. 20 (11), p. 1623, 1982. [2] B.J. DOBSON,Proc. 2nd 1MAC, p. 1231, 1984. [3] B. CAESARand J. PETER, A/AA J. 25 (11) p. 1494, 1987. [4] J.C. WEI et al., Proc. 7th IMAC, p. 1231, 1989. [5] J.C. O'CALLAHANand C.M. CHOU, Int. £ AnaL Exp. Modal AnaL 4 (1), p. 8, 1989. [6] Q. ZHANGet al., Int. J. Anal Exp. Modal Anal 4 (2), p. 39, 1989. [7] K. SOHN,M. OKUMAand A. NAGAMATSU,Trans. JSME, 57 (544C) p. 3741, 1991 (in Japanese). [8] K. SOHN,M. OKUMAand A. NAGAMATSU,Trans. JSME 57 (537C), p. 1591, 1991 (in Japanese). [9] A. NAGAMATSU,Modal Analysis, Baifukan, 1985 (in Japanese). [10] K. SOHN,M. OKUMAand A. NAGAMATSU,Tram. JSME 58 (554C), October 1992 (in Japanese). [11] R.L. Fox, and M.P. KAeOOR,AIAA J. 6 (12), p. 2426, 1968.
153
FINEL 321
Correction of finite element models using experimental modal data for vibration analysis Masaaki O k u m a Department of Mechanical Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo, Japan Abstract. A method is presented for correcting finite element models by referring their modal parameters to
experimental modal data. The method is based upon both the sensitivity analysis and the least squares method, and gives a physicallyrealistic solution of appropriate design variables. In this paper, the procedure of the method is first explained, and then two applications are presented. The first application is a correction of a beam structure, as a basic research study. The second is a correction of a practical structure, a "Centre-beam", which is a structural component of an automotive body. The initial finite element models of both cases are successfully corrected to adjust their dynamic characteristics to experimental results. In particular, it is experimentally verified that the corrected finite element model of the "Centre-beam" is able to represent its dynamic characteristics well, even under different boundary conditions. The verification proves the physical validity and the practical availabilityof the corrected model.
Introduction
The finite element method has been widely used as the most practical method for analysing the dynamics of complex mechanical structures, which include most actual structures. T h e r e are two principal problems in applying the finite element method for analysis of actual complex mechanical structures, the first being the difficulty of making reliable finite element models, and the second the computation of solving the equations of motion, for systems which have many degrees of freedom. However, the current development of computers has m a d e it possible to analyse the dynamics of finite element models with many degrees of freedom. This means that the second problem, of computation, is not serious nowadays. So attention has been focused on the first problem, and many researches are reported with regard to it [1-6]. The first problem consists of two items. The first is the means of determining a suitable size of finite elements for analysing the dynamics of objects up to those frequencies desired by analysts [7]. It is not good to use too fine finite element models for analysis up to a comparatively low frequency, because too fine a mesh wastes hours of modelling and computation time. The second item is the means of determining appropriate values of the design variables of all finite elements of models. This p a p e r presents a method with regard to the latter item [8,10]. The modal p a r a m e t e r s of mechanical structures can be extracted accurately from experimentally obtained F R F s by the experimental modal analysis techniques [9]. Methods have been reported for correcting the design variables of finite element models to appropriate values by referring some of their modal p a r a m e t e r s to experimental modal parameters. This p a p e r presents a method for dealing with the problem. The method is based upon the sensitivity analysis [11] and the least squares method. The thicknesses of b e a m finite elements are adopted as the design variables, and the first-order sensitivities of the lowest three natural frequencies and their associated natural 0168-874X/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
154
M. Okuma / Correction of finite element models
modes with respect to the design variables are used on the first application in this paper. The thicknesses of triangular finite elements and the material properties, Young's modulus and density, are adopted as the design variables for the second application. Three reasons can be given why the sensitivities of the material constants are used. (1) Practical objects like the second specimen in this paper have complex shape configurations. Consequently, there are regions where the thickness is not constant nor is the configuration flat, even in the small area modelled by one finite element. (2) The first-order sensitivities are used only, in spite of the strict fact that the sensitivities of the natural frequencies and natural modes have nonlinear relationships with the thicknesses of the finite elements. (3) The values of the sensitivities of the natural frequencies and natural modes are comparatively small. Therefore, if thicknesses only were adopted as the design variables, it would often be necessary to change the thicknesses of finite elements by very large amounts to move the natural frequencies to a required extent, with the result that the thicknesses become unrealistic from the physical viewpoint.
Theory of the method The vector of referred modal parameters, which are obtained by experiment, is denoted as {q0E} here. The vector of their counterparts on a finite element model is denoted as {q0M}. The matrix [Z] denotes the first-order sensitivity matrix, the columns of which consist of the first-order sensitivities of the modal parameters of the finite element model with respect to the design variables. The vector {At} denotes the extent of the modification of the design variables. Then, the following equation is constructed. {(10E} = {(/)M} + [Z]{A/}.
(1)
At least in the vicinity of the initial design variables, eqn. (1) applies accurately, in line with Taylor's expansion theorem. Three cases can arise with regard to the dimensions of eqn. (1): the number of equations may (i) exceed, (ii) equal, or (iii) be less than the number of design variables. In case (iii), the simultaneous equations are indeterminable. The singular value decomposition method and the pseudo-inverse method work mathematically, but only produce one feasible solution which is not necessarily physically sensible. The equations of case (ii) are determinable, but, it is not convenient to set up the same number of equations as the number of design variables to be modified. Case (i) is practical, and a physically proper solution will be determined. The number of observed natural frequencies is experimentally limited, in general. The associated natural modes should then be extracted from FRFs at a sufficient number of measurement locations on an object. Equation (1) in the case (i) can be solved by the least squares algebra, with weighting functions as {At} = ([Z]t[W][Z])-I[z]t[w]({~E}
- {~M})"
(2)
The weighting functions should be arranged so that all elements of the vector, {q~E} -- {q0M}, are comparable. Without such weightings, numerical calculation difficulties may occur because the values of referred natural frequencies are in general much larger figures than those of the natural modes. Then the design variables should be improved by applying {/new} = {tinitial} + {At},
(3)
where {tinitial} and {t,ew} are the initial design variables and the improved ones, respectively.
M. Okuma / Correction of finite element models
155
Reading F.E.Model and Modal P a r a m e t e r U~ Sensitivity Analysis to thicknesses CAL. [W] and L.S.M
Sensitivity Analysis to Young's Ratio and Density.
I CAt. tt0J a.d t.S.M I
NO Fig. 1. The procedure of the method presented
The distance between the referred experimental modal parameters and the in counterparts in finite element models is not negligible in general applications. The relationship between the referred modal parameters and the design variables is nonlinear. Therefore, the process mentioned above must be iterated. The procedure is illustrated in Fig. 1.
The first application: a simple beam model
A beam, of length 500 mm and cross-section 10 x 10 mm 2, is the specimen for the first application [7]. The beam is modelled with ten beam finite elements, as shown in Fig. 2. The lengths of the sides of the ten finite elements which are parallel to the direction of bending deformation are set up individually with randomly incorrect values, from 7 mm to 16 mm, as the design variables to be corrected. These incorrect design variables of the initial finite element model are improved by referring to its first three natural frequencies and the values
E. • Finite Element
X lo~ mm ~
E.1
E.2
Z .x IE3 I 500mm
Egl El0 Fig. 2. The finite element model of the beam structure of the first application
M. Okuma / Correction of finite element models
156
Table 1 Correction of design variables Element No.
Wi (mm)
W1 (mm)
1 2 3 4 5 6 7 8 9 10
15.0 14.0 12.0 8.0 11.0 7.0 11.0 12.0 14.0 16.0
11.4 9.51 10.0 10.1 9.65 10.1 10.0 9.46 10.0 9.02
Real widths are all 10 mm. Wi are initial widths, W I are widths corrected by least squares method.
of their associated natural modes at five measurement locations, using the method. The experimental natural frequencies and natural modes for references are obtained by applying an experimental modal analysis technique to FRFs on impact vibration testing. Table 1 shows the result of the improvement of those design variables. All initially incorrect design variables have been improved. Here, the experimental natural frequencies and natural modes are not necessarily perfect, due to some experimental errors. The weighting functions used in the computation are not perfect either. These factors are the reasons why the design variables of the result are not 10 mm exactly. Figure 3 shows the improvement from the viewpoint of observation of FRFs. Figure 4 illustrates the correspondence of the three referred natural modes and the fourth natural mode. The results of this application of the presented method illustrate its fumdamental validity.
The second application: a structural component of an automotive body As an application of the method to actual mechanical structures, the correction of a finite element model of a so-called "Centre-beam" is presented here [9]. The structure is shown in Fig. 5. It consists of two engine-mounting bracket parts, and a structural beam part. The structure is built in at the centre bottom of an automobile engine compartment, in order to
~3.14
0 ~3.14 -2 10 .............
z E
I~i
. . . .
• Experimental Data : Corrected Model : Initial Model
11
c
500
1000
1500
Frequency (Hz)
2000
Fig. 3. Observation of improved FRFs of the beam model
M. Okuma / Correction of finite element models • Corrected Modes by Proposed Method ......
157
: Initial Modes ......
: Experiment
: Experiment
1st
3rd
~
~
~
'
~
4th 7
Fig. 4. Observation of improved natural modes of the b e a m model
support the engine. It is considered important for a finite element model of the structure to represent the dynamics accurately up to about 200 Hz. The specimen is modelled as the finite element model shown in Fig. 6, with due regard to the appropriate size of triangular finite elements adopted [7]. The finite element model is structured with 221 nodal points and 461 triangular finite elements. The front engine mounting bracket, the smaller bracket part, is neglected in the modelling, because it is supposed not to affect the global dynamics of the structure. The thicknesses of the finite elements are classified into 40 groups, as shown in Fig. 7. An equal thickness is set on the finite elements in each group, and the thicknesses of these 40 groups are adopted as the design variables. The referred natural frequencies are the first and the second natural frequencies under free-free boundary conditions and the referred values of the associated natural modes are
Fig. 5. T h e object, " C e n t r e - b e a m " , of the second application
M. Okuma / Correction oJfinite element models
158
56
t~£~
19£
7.
t/;\ Fig. 6. The finite element model of the centre b e a m
those at the 46 measurement locations denoted by black solid circles in Fig. 8. The first natural mode is the first-order torsional mode. The second natural mode is the first bending mode. These referred modal parameters are identified from measured FRFs using multi-sine wave excitation testing. The number of the referred values is 94. The number of adopted design variables is 42, being the thicknesses of the forty groups plus the material density and Young's modulus. Consequently, the least squares method is applicable.
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Fig. 7. Classification of finite elements into 40 groups
M. Okuma / Correction of finite element models Driving point 2 (at C l a m p e d Condition)
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Fig. 8. Schematic view of location of driving points and measurement points on experimental tests
.
Experiment ............ Updated ................. initial
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200
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Frequency (Hz)
Fig. 9. FRFs of the initial and the corrected models, together with the experimental pattern
Table 2 shows a comparison of the referred natural frequencies from experiment and the initial finite element model. The solid lines and the dotted lines in Figs. 9 and 10 denote experimentally obtained FRFs and the counterpart FRFs calculated with the initial finite element model, respectively. These comparisons show the big difference of dynamic characteristics between the initial finite element model and the actual structure. It will often be difficult to improve inaccurate finite-element models by hand for structures as in this application. The FRFs of the model corrected by the method presented in this paper are shown by the broken lines in Figs. 9 and 10. Table 3 shows the natural frequencies from experiment, the initial model and the corrected model, together. Two natural frequencies in the range up to 200 Hz, from experiment and the corrected finite element model, correspond well. The design variables, the thicknesses and material constants, are shown in Table 4. The correction seems
Table 2 Natural frequencies of experiment and the initial model Mode
Experiment
FIE Model
MAC
1 2
133.1 (Hz) 151.3 (Hz)
216.8 (Hz) 240.6 (Hz)
0.9227 0.8893
a
M A C
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Modal Assurance Criterion
a
M. Okuma / Correction of finite element models"
160
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Table 3 Natural frequencies of the corrected model Mode
Exp. (Hz)
Initial (Hz)
U p d a t e d (Hz)
1 2 3 4 5
133.10 151.30 236.09 258.30 265.79
216.8 240.6 435.9 596.3 750.6
135.7 149.6 275.7 380.3 462.6
M. Okuma / Correction of finite element models
161
Table 4 Modified design variables of the finite element model of the centre-beam Initial value
Updated value
Density (kg/mm 3) Young's modulus (N/mm)
7.86 × 10 -5 206,00
1.24 × 10-4 130,000
Group
Initital value (mm)
Updated value (mm)
Group
Initial value (mm)
Updated value (mm)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 3.00 3.00 3.00 2.00 2.00 2.00 1.39 1.39
1.54 1.87 2.99 1.24 2.68 2.41 1.29 2.19 2.36 1.54 1.84 1.99 4.11 1.98 3.31 1.81 1.36 2.08 1.22 1.50
21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1.39 1.39 1.39 1.39 1.39 4.00 4.00 2.00 2.00 2.00 2.00 1.39 1.39 1.39 1.39 4.00 4.00 4.00 4.00 4.00
2.18 2.50 4.66 2.81 1.63 0.76 2.57 4.48 1.94 1.53 1.71 3.14 0.80 1.72 2.00 4.01 4.37 3.34 3.48 3.66
to be successful, from the result. On the other hand, a trial of correcting the initial finite element model without adopting the material properties among the design variables was unsuccessful, in that the referred modal parameters of the model could not converge to the
le-3
....
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Fig. 12. Predicted FRFs after attaching two weights on the object, under free-free boundary conditions
162
M. Okuma / Correction of finite element models
values of the experimentally obtained modal parameters because some thickness values became negative before convergence. In order to verify whether the corrected finite element model is physically proper, the dynamics of the specimen under different boundary conditions should be observed. The finite element model can be observed to be proper if it represents the dynamics accurately even under different boundary conditions. As the first verification, the dynamics is calculated for the model under the boundary condition that the right end of the specimen is clamped, as shown in Fig. 8. One of the F R F s is shown in Fig. 11. The F R F s calculated with the model correspond very well with their experimental counterparts. As a second verification, the F R F s of the specimen after attacking two weights under the free-free boundary condition are calculated using the model, and compared with their experimental counterparts. The two weights, of 730 g and 700 g, were connected directly at the points denoted " M a s s added points" in Fig. 8. The result is shown in Fig. 12. The dynamic properties up to about 200 Hz taken from the calculation and experiment correspond very well.
Conclusion
This p a p e r presents a method for correcting finite element models by referring their modal parameters to the experimentally obtained counterparts. The method allows the physically realistic determination of appropriate values of the design variables. The first application as basic research, is a correction of a b e a m structure. The second is a correction of a practical structure, a " C e n t r e - b e a m " structural component of an automotive body. The initial finite element models of those specimens are successfully corrected with regard to the dynamic characteristics. On the second application, it was found that the adoption of material constants is effective for practical applications. It was also verified experimentally, in order to prove the physical validity of the model, that the corrected finite element model of a " C e n t r e - b e a m " represents its dynamic characteristics well even under different boundary conditions.
References
[1] M. BARUCH,A/AA J. 20 (11), p. 1623, 1982. [2] B.J. DOBSON,Proc. 2nd 1MAC, p. 1231, 1984. [3] B. CAESARand J. PETER, A/AA J. 25 (11) p. 1494, 1987. [4] J.C. WEI et al., Proc. 7th IMAC, p. 1231, 1989. [5] J.C. O'CALLAHANand C.M. CHOU, Int. £ AnaL Exp. Modal AnaL 4 (1), p. 8, 1989. [6] Q. ZHANGet al., Int. J. Anal Exp. Modal Anal 4 (2), p. 39, 1989. [7] K. SOHN,M. OKUMAand A. NAGAMATSU,Trans. JSME, 57 (544C) p. 3741, 1991 (in Japanese). [8] K. SOHN,M. OKUMAand A. NAGAMATSU,Trans. JSME 57 (537C), p. 1591, 1991 (in Japanese). [9] A. NAGAMATSU,Modal Analysis, Baifukan, 1985 (in Japanese). [10] K. SOHN,M. OKUMAand A. NAGAMATSU,Tram. JSME 58 (554C), October 1992 (in Japanese). [11] R.L. Fox, and M.P. KAeOOR,AIAA J. 6 (12), p. 2426, 1968.