Model Updating of FST Lisboa FST06e Prototype Chassis Miguel Andr´ e dos Santos Ponte Duarte [email protected]Instituto Superior T´ ecnico - Universidade de Lisboa, Lisboa, Portugal June 2019 Abstract This work had the objective of implementing a model updating process, based on experimental modal data, to the chassis of the FST06e prototype in order to obtain a numerical model that would represent the real structure as close as possible. From the experimental testing of the chassis, simulating free-free condition, and subsequent modal identification using the Characteristic Response Function (CRF) was possible to obtain the modal properties of the chassis. The model updating process used the experimental results to, through an optimization process, achieve a set of design variables that led to results of eigenvalue simulations closer to the experimental references. To conclude, the initial and updated models were both modified to perform a torsional stiffness simulation and compare each of the results with the experimental result of a torsional stiffness test, thus evaluating the quality of the updating process. Keywords: Chassis, Experimental Modal Analysis, Modal Identification, Characteristic Response Function, Model Updating 1. Introduction Formula student, which is an international com- petition organized by Formula Society of Automo- tive Engineers (FSAE) [4], has the aim of challeng- ing students to design, build and test racing cars according to a specific set of rules. Over the last two decades, the technical charac- teristics of the cars evolved overwhelmingly with the application of state of the art techologies, highly graded materials and increasingly complex designs. These continuous improvements led to faster cars that had to whithdstand higher loads due to their increase in performance. One of the greatest im- provements regards the chassis of the cars, is the use of carbon fiber sandwich monocoques instead of the classic space-frame. To evaluate the structure’s real condition and compare it against the predicted results, from the numerical simulations, it is necessary to perform experimental tests such as vibration tests. In such tests, the structure is excited in a pre-determined frequency range and its response is measured. From the relation between the excitation and the response is possible to obtain a set of curves that are the Frequency Response Functions (FRF). These FRF, that are unique for each combination of excitation and measurement coordinates, contain the modal parameters of the structure, such as the ressonant frequencies, the damping ratio and the modal con- stants. To obtain these parameters precisely and com- pute the mode shapes it is necessary to apply a modal identification method to the experimental re- sults. The choice of the method depends on the ob- jectives of the test and also on how the results were gathered. Nowadays, due to the evolution over the last decades, there is a vast range of methods avail- able. Those methods are classified by the domain of the analysis, whether the identification of the FRFs is performed in a direct or indirect manner, the number of modes that are possible to identify and finally the number of FRF the method allows to use simultaneously [6]. This work was performed using the Characteristic Response Function (CRF), which was developed in IST. The results from the vibration tests can be used individually but can also be used in other tools such as model updating methods. These methods have the objective of correcting or adjusting numer- ical models using experimental results, so that the model yields results closer to those obtained exper- imentally. Once again, there is a vast number of methods that are basically divided between direct or iterative methods [13, 3]. In this work the Model Updating Toolbox provided by Siemens NX[9] 10 was used. The application of these tools in car chassis, were investigated by Sani el al[11], Oktav citeOktav2017 and Schedlinski et al [12], where the authors ob- tained the dynamic characteristics of the chassis 1
Transcript
Model Updating of FST Lisboa FST06e Prototype Chassis
Instituto Superior Tecnico - Universidade de Lisboa, Lisboa, Portugal
June 2019
Abstract
This work had the objective of implementing a model updating process, based on experimentalmodal data, to the chassis of the FST06e prototype in order to obtain a numerical model that wouldrepresent the real structure as close as possible. From the experimental testing of the chassis, simulatingfree-free condition, and subsequent modal identification using the Characteristic Response Function(CRF) was possible to obtain the modal properties of the chassis. The model updating process usedthe experimental results to, through an optimization process, achieve a set of design variables that ledto results of eigenvalue simulations closer to the experimental references. To conclude, the initial andupdated models were both modified to perform a torsional stiffness simulation and compare each ofthe results with the experimental result of a torsional stiffness test, thus evaluating the quality of theupdating process.Keywords: Chassis, Experimental Modal Analysis, Modal Identification, Characteristic ResponseFunction, Model Updating
1. Introduction
Formula student, which is an international com-petition organized by Formula Society of Automo-tive Engineers (FSAE) [4], has the aim of challeng-ing students to design, build and test racing carsaccording to a specific set of rules.
Over the last two decades, the technical charac-teristics of the cars evolved overwhelmingly with theapplication of state of the art techologies, highlygraded materials and increasingly complex designs.These continuous improvements led to faster carsthat had to whithdstand higher loads due to theirincrease in performance. One of the greatest im-provements regards the chassis of the cars, is theuse of carbon fiber sandwich monocoques insteadof the classic space-frame.
To evaluate the structure’s real condition andcompare it against the predicted results, from thenumerical simulations, it is necessary to performexperimental tests such as vibration tests. In suchtests, the structure is excited in a pre-determinedfrequency range and its response is measured. Fromthe relation between the excitation and the responseis possible to obtain a set of curves that are theFrequency Response Functions (FRF). These FRF,that are unique for each combination of excitationand measurement coordinates, contain the modalparameters of the structure, such as the ressonantfrequencies, the damping ratio and the modal con-stants.
To obtain these parameters precisely and com-pute the mode shapes it is necessary to apply amodal identification method to the experimental re-sults. The choice of the method depends on the ob-jectives of the test and also on how the results weregathered. Nowadays, due to the evolution over thelast decades, there is a vast range of methods avail-able. Those methods are classified by the domainof the analysis, whether the identification of theFRFs is performed in a direct or indirect manner,the number of modes that are possible to identifyand finally the number of FRF the method allowsto use simultaneously [6]. This work was performedusing the Characteristic Response Function (CRF),which was developed in IST.
The results from the vibration tests can be usedindividually but can also be used in other toolssuch as model updating methods. These methodshave the objective of correcting or adjusting numer-ical models using experimental results, so that themodel yields results closer to those obtained exper-imentally. Once again, there is a vast number ofmethods that are basically divided between director iterative methods [13, 3]. In this work the ModelUpdating Toolbox provided by Siemens NX[9] 10was used.
The application of these tools in car chassis, wereinvestigated by Sani el al[11], Oktav citeOktav2017and Schedlinski et al [12], where the authors ob-tained the dynamic characteristics of the chassis
1
and verify the respective numerical models. Al-though, in motorsports or even Formula Studentthe availability of studies is very limited either be-caused of the limited access secrecy in which thewhole industry is immersed or because these toolsare not applied since they are quite time-consumingand take a well-established framework to be able toreach a conclusion in such complex structures.
The structure under study in this work wasthe chassis of the FST Lisboa prototype FST06e.The chassis is a carbon fiber sandwich monocoque,in which the skins are made of several layers oftwo types of carbon fibers, uni-directional and bi-directional, and the core is made of aluminum hon-eycomb. The design and strcutural evaluation wasperformed in order to comply with the FSAE rules[5].
2. BackgroundThis section briefly introduces the concept of
modal analysis and its general formulation for sys-tems with multiple degrees-of-freedom as well asthe formulations to obtain the modal parametersthrough experimental modal analysis. It is also pre-sented the modal identification method that wasused and applied to the experimental results. Fi-nally, a brief introduction of the tools and methodsthat were used on the model updating process isdone.
2.1. Modal AnalysisThe objective of modal analysis is to compute
the frequencies and mode shapes of a system thatcan be of single or multiple degrees-of-freedom andbe damped or undamped. This is used to under-stand how structures behave under dynamic cir-cumstances.
This work uses the hysteretic damping which is adamping model that describes the phenomenon ofenergy dissipation based on the material’s hystere-sis. This model also portrays the behavior of thedamping mechanism of the structure as frequencyindependent, as usually verified.
As an example, Figure 1 shows a generic systemwith multiple degrees-of-freedom, under forced vi-bration and with hysteretic damping.
Figure 1: Multi-degree-of-freedom model
For such systems, with external forces fi(t)(i =1, 2, ..., N) applied to the masses and forcing themto move, from the equilibrium of forces is possible
to obtain the governing equations, in a matricialform, in the following way:
[M ]{x(t)
}+ i[H]
{x(t)
}+ [K]
{x(t)
}={f(t)
}(1)
where [M ], [H] and [K] are the respective matricesof mass, damping and stiffness. All these matri-ces are square and symmetric, f(t) is the vectorof external forces and x(t), x(t) and x(t) are thecorresponding vectors of acceleration, velocity anddisplacement.
Considering a solution on the form of x(t) =Xeiωt inserted into the homogeneous part of equa-tion 1, yield a generalized eigenvalue problem de-scribed in the following manner:[
[K]− ω2 [M ] + i [H]] {X}
= 0 (2)
where [H] = η k, and η is the damping loss factor.Solving the eigenvalue problem described by
equation 2, results in the matrix ([\λr\) which con-tains N eigenvalues and also ([Φ]) which is a massnormalized matrix containing N eigenvectors.
For each mode r, (λr) is known as the complexeigenvalue and can be written as:
λr = ω2r(1 + iηr) (3)
where ωr is the natural frequency.
2.2. Experimental Modal AnalysisIn real structures, such as the FST06e chassis,
their complexity turns the application of the analyt-ical approach impractical. For such situations, thesolution is to resort to experimental modal analysisand obtain the response function, H(w), via exper-imental testing.
This function is defined as
H(ω) =X(ω)
F (ω)(4)
where X is the structures response and F is theinput. This function is frequency dependent and isdesignated as Frequency Response Funtion (FRF).
The relation between the modal model and theexperimental results may be written as [6]:
[α(ω)] = [Φ][ω2r(1 + ηr)− ω2]−1[Φ]T (5)
where [α(w)] is the receptance matrix from the ex-perimental test.
Considering the residuals, (Rjk), each term of thereceptance matrix can be defined, in an approxi-mated form, as [6]:
αjk(ω) =Xj
Fk
∼= Rjk(ω) +∑m2
r=m1
rAjk
ω2r−ω2+iηrω2
r(6)
2
where N is the total number of modes, r is thevibration mode and rAjk is the modal constant.
The complex residual term, Rjk, usually assumesthe following form:
Rjk(ω) ∼= −1
ω2MRjk
+1
KRjk
(7)
with Mjk and Kjk being the respective mass andstiffness residuals.
The modal constant, rAjk, obeys to a relation-ship, given by equations (8) and (9), that makesit possible to evaluate the whole receptance matrix[α(w)] knowing only one full line or column.
rAjk = φjrφkr (8)
rAjj = φ2jr or rAkk = φ2
kr (9)
where φjr and φkr are the elements of the eigenvec-tor Φr corresponding to coordinates j and k.
In order to identify the modal parameters of thestructure from the experimental data, it is neces-sary to apply a modal identification method. In thiswork, the process of modal identification was doneby resorting to BetaLab [8] software that is based onthe Characteristic Response Function (CRF) [7, 15].
This method was developed on the premise thata specific mode, r, is dominant in the viccinity of itsown natural frequency. Thus, the contribution of allthe other modes can be neglected if well spaced andwith modal constants within the same magnitude.Considering this, it is reasonable to approximatethat effect by a constant, leading to
α(ω) ∼=A
ω2r − ω2 + iηrω2
r
+ Constant (10)
Picking the responses of three, not necessarilyconsecutive, points of a measured FRF ((ωi, αi),(ωk, αk) and (ωl, αl)) and applying some mathemat-ical modelling and intermediate steps, that are de-scrbied in [7, 15], it is possible to achieve the prop-erly called Characteristic Response Function, β(ω)
β(ω) =
(αi−αk)(ω2k−ω
2l )
(αk−αl)(ω2i−ω2
k)− 1
ω2i − ω2
l
(11)
that can also be defined as:
β(ω) =1
ω2r − ω2 + iηrω2
r
(12)
To compute the modal constant, it is necessary topick two points in the viccinity of mode r, (ω1, α1)and (ω2, α2), and apply the following equation[10]:
A =[α1−α2](ω2
r−ω21+iηrω
2r)(ω2
r−ω22+iηrω
2r)
ω21−ω2
2(13)
These modal constants obey to the same relation-ships established in equation (8) and (9).
To obtain a set of consistent modal constantsfrom the experimentally derived, and eventually in-consistent, Silva et al[16] developed a method thatmakes use of a matricial operator that yields a setthat obeys to the consistency relationships.
2.4. Model UpdatingAfter obtaining the experimental and numerical
results, the model updating process contemplatesseveral steps.
First of all, it is necessary to identify which modescorrespond to each other between both sets. Andquantify that correspondency. This is an impor-tant factor in the process, since the success of theupdating is directly related to correspondence be-tween sets.
This work uses the Modal Assurance Criterion(MAC)[2]. This method serves as an indicator anda measure of consistency between two estimates ofmodal vectors [1].
MAC is mathematically defined as:
MAC(A,X) =|∑No
j=1 ΦXjΦ∗Aj|2∑No
j=1 ΦXjΨ∗Xj
∑Noj=1 ΦAjΦ∗
Aj
(14)
where ΦA and ΦX are the work and referencemode shape vectors, No the number of measuredcoordinates and ∗ denotes the complex conjugate.The result of this operation is a scalar between zero(where there is no consistent correlation betweenmodes at all) and one (accounting for consistentcorrelation).
After that, and to evaluate the choice of the pa-rameters to be adjusted on the updating process, ordesign variables, is necessary to calculate the sen-sitivities. The sensitivities translate the suscepti-bility each of the correlated modes eigenvalues andeigenvectors have on the change of each of the de-sign variables.
In the software that was used, Siemens NX 10[9], for the frequencies, the sensitivity ∆λi of theith eigenvalue is presented as:
∆λi ={φi}T ([∆KR]− λi [∆MR]) {φi}
{φi}T [MR] {φi}(15)
while the mode shape sensitivity ∆φi, which is ex-pressed in terms of frequency sensitivity ∆λi, is pre-sented as:
{∆φi} = − ([KR]− λi [MR])−1
([∆KR]−∆λi [MR]− λi [MR]) {φi} (16)
3
In equations (15) and (16), [KR] and [MR] are thestiffness and mass reduced matrices while [∆KR]and [∆MR] are the stiffness and mass reduced ma-trices sensitivities.
The updating process was performed using thegenetic algorithm, implemented in Siemens NX 10[9], and with the following objective function:
f(∆DVj) = min(∑Nt
i=1Ai |εi|+O∑NDV
j=1 Bj |∆DVj |)
(17)
in which, the first summation regards the targeterror, εi, that accounts for the difference betweenthe result of each iteration (frequencies and/ormode shapes) and the reference values. The sum-mation is done over the total number of active tar-gets, NT , and the term Ai represents the weight ofith target.
The second summation, which is done over thetotal number of free design variables NDV , accountsfor the design variables changes, ∆DVj . The termsO and Bj are respectively the overall design variableweight and the weight of the jth design variable.
With this objective function, while the error isminimized in the first term, the second term hasthe task of contributing for a solution that doesnot yield a result far from the initial solution withregard to the design variables.
3. ImplementationThis section describes the procedures and
methodologies used to implement all the numericalanalysis and perform the experimental testing.
3.1. Experimental Testing and Modal IdentificationThe experimental eigenfrequencies of the chassis
were obtained using the setup presented in Figure2. This setup simulates a free-free configurationby suspending the structure with elastic cords sothat the rigid body modes have considerably lowerfrequencies than the flexible modes.
The structure was excited using a shaker that wasconnected to the chassis via a stinger. In the pointof the stinger, was connected a force transducerthat was measuring the input. The vibration inputthat was used was random noise. The input coor-dinate changed throughout the testing, dependingon which coordinate to excite. The test was per-formed in the frequency range of 0 to 260 HZ, witha resolution of 0.5 Hz.
To measure and record the data, an acecelerome-ter was used which was placed in different locationsalong the chassis, depending on the coordinates thatwanted to be measured.
The analog signals from the accelerometer andforce transducer were amplified before being fed tothe digital analyser. There, they were fast Fouriertranformed in order to obtain the FRF.
This softwares acts as a fully interactive platformin which is possible to identify the modal parame-ters of MIMO systems. It allows the processing ofan indefinite number of FRFs. In each FRF, eachmode is treated independently as in a SDOF anal-ysis. It also allows the user to change or tweak theresults that are obtained, regenerating the curves inreal-time for quality jugdement.
3.2. Finite Element Modelling
The finite element model of the chassis was builtfrom the outer surface of the CAD model, in orderto match with the coordinates in which the experi-mental testing was performed. This model was usedin the initial modal analysis simulation, to comparethe natural frequencies and mode shapes with theexperimental results and proceed to the model up-dating process.
In order to match the FE model with the realstructure, the firewall component was modelledwith a crack as close as possible to the real one.
Table 1 presents the densities of all materials be-fore and after being multiplied by the weight ratio.
The properties for the isotropic materials, steeland aluminum, are defined as the typical mechani-cal properties while for the orthotropic are as pre-sented in the material’s specsheets.
Figure 3 presents the complete meshed model ofthe chassis.
Figure 3: Complete meshed model
3.3. Model UpdatingThe model updating process was applied after ob-
taining the experimental results and over the nu-merical model of the chassis that was built. Thisprocess emcompasses several steps.
From the complete model, which initially con-sidered the material properties presented in Table2 and 3, the natural frequencies and mode shapeswere obtained. Those results were compared to theexperimental ones in order to indentify the discrep-ancies and how far the numerical model was fromthe reality.
Since the updating process requires the numericalmodel to have the same number of DoF... as wereexperimentally measured, the initial FEM model
was reduced using the set of experimental coordi-nates. With this, the stiffness and mass matriceswere condensed but maintained as much as pos-sible the dynamic characteristics of the structure.Addicionally in this step the future design vari-ables of the updating process were chosen. Con-sidering the whole model and the knowledge aboutthe properties, the longitudinal Youngs modulus ofthe unidirectional fiber, E1u, and the longitudinaland transversal Youngs modulus of the bidirectionalfiber, E1b and E2b respectively, were chosen.
The modal analysis results of the reduced modelwere then correlated with the experimental results,using the MAC correlation, in order to identifywhich modes presented correspondence between thesets and to define the pairs of modes that were goingto be used in the updating process.
The updating process made use of those modesto optimize their correlation in terms of frequenciesby changing the design variables. This process wasrepeated in several iterations that contemplated dif-ferent target values for each of the correlated pairs.
Once the results of the optimization, concerninguniquely the correlated pairs, were within the fre-quency error values that were desired, the reducedmodel and then the complete model were updatedwith the new values of the design variables and therespective simulations were performed once again.The effect of the updating process on the completefrequency range can only be verified after updatingand simulating the reduced and complete models.
Finally, it is possible to compare the results withthe experimental set and evaluate the success of themodel updating process.
3.4. Torsional Stiffness Model
The development of the torsional stiffness modelaims to evaluate whether the changes on the ma-terial properties, after the updating, would lead topositive results in a static situation in the same waythat happened in the dynamic evaluation (modalanalysis).
The numerical results, before and after the up-dating, were compared against the experimental re-sult that were previously obtained by the team FSTLisboa. Each simulation considered 5 load cases(100, 250, 500, 750 and 1000 N) to evaluate thelinearity.
Figure 4 presents the numerical model after beingmodified in order to simulate the torsional stiffnesstest as close as possible to the experimental setup.
To compute the torsional stiffness in Nm/deg,from the numerical simulation, the following equa-tion is used:
Kt =2× F × d
arctan(∆z1+∆z22×d )
(18)
5
Figure 4: Numerical model for torsional stiffness simulation
where F is the magnitude of the applied force inNewton, d is distance between the force applicationpoint and the longitudinal axis of the car, ∆z1 and∆z2 are the measured vertical displacements in theforce application point.
4. ResultsThis section presents the results of the several
steps of this work. Starts with the results from themodal analysis of the FEM model with the initialmaterial properties and it is followed by the resultsfrom the experimental testing. Then the results andrespective comparisons of the model updating pro-cess are presented. Finally, the torsional stiffnesssimulation results are presented.
4.1. Modal Analysis - Initial ResultsThe modal analysis to the chassis, with the ini-
tial mechanical properties, yielded 21 modes. Therespective frequencies are presented in Table 4.
In Table 4, the 14 modes highlighted in grey haveconsiderably higher amplitudes on the firewall re-gion in comparison with the remaining structure.Thus, it is reasonable to consider them local modes.The local modes were not considered in furtheranalysis since this work is focused on the global be-havior of the structure.
4.2. Experimental Results and Modal IdentificationFrom the experimental tesitng a set of 83 FRFs
Comparing the frequencies with the ones fromthe numerical solution, it is immediately percept-able that with similar frequency ranges, the num-ber of modes is considerably different. It is worthmentioning that the firewall was not instrumentedin the experimental testing.
Regarding the mode shapes, their computationdid not yield results according to what would be ex-pected. Figure 6 shows the first experimental modeof the chassis, as an example.
Figure 6: 1st mode of vibration, 78.652Hz
It is clearly visible that only on the left sideof the chassis presents visible amplitudes, yieldingshapes that do not present the symmetries (or anti-symmetries) that were expected.
The possible causes for this lack of quality onthe mode shapes might have been due to erroneousmeasurements or problems on the identification andsubsequent post-processing and computation of themode shapes.
Figure 7 presents the MAC correlation betweenthe initial FE model and experimental results. Itnoticeable that there is low correlation values, with
6
the highest around 0.3. Additionally, some of thehighest correlated modes are local modes, from theinitial model, and are very dispersed. Some of thesemodes present frequencies that are cosiderably dis-tanced on the studied spectrum.
Figure 7: MAC matrix between experimental results andinitial FE model
From these results, it is fair to consider that thenumerical model does not characterize correctly thedynamic behavior of the real structure.
4.3. Model Updating ResultsThis section contemplates the reduced model re-
sults along with the summarized results from thefive iterations that were performed. Finally, is pre-sented a brief discussion of the results.
From the reduced model eigenvalue solution, 7modes were obtained. Table 6 presents them.
Even with the reduction on the number of Dof,the frequency results present considerable discrep-ancies with the experimental results in Table 5.
Figure 8 presents the MAC correlation of the re-duced model results with the experimental results.Once again, the values are considerably low and areseveral reduced model modes (work) correlated withthe experimental ones (reference)
Despite the MAC values being lower than the de-sired, due to the lack of quality of the experimental
Figure 8: MAC matrix results between experimental modalanalysis results and reduced analysis
mode shapes, the pairs that presented a value of atleast 0.2 were considered.
Table 7 summarizes the paired modes as well asthe relative errors between them.
Table 7: Initial error between reference and work frequen-cies and mode shapes
Figure 9 presents the correlated experimental andreduced model mode shapes, side-by-side.
(a) 1st experimental modeand 1st reduced model mode
(b) 2nd experimental modeand 4th reduced model mode
(c) 5th experimental modeand 6th reduced model mode
Figure 9: Experimentally identified and reduced modelmode shapes side-by-side
Figure 10 presents the sensitivity matrix. Thismatrix represents the sensitivity that each corre-lated frequency or mode shape has on changes ineach of the design variables.
The first three collumns refer to the frequencieswhile the last three refer to the mode shapes. Itcan be seen that the mode shapes present almostnull sensitivity to any change. This was expected
7
Figure 10: Sensitivity Matrix of the Design Variable Con-sidering the Reference Modes
since any change will affect the structure globally,changing the amplitudes but not the proper shapes.
Regarding the frequencies, the highlight is on thesecond frequency that is highly sensible to the longi-tudinal Young’s modulus of the unidirectional fiber.All the plies of these fiber are aligned with the lon-gitudinal axis of the chassis and since the secondmode is a longitudinal bending mode, any changeon the stiffness specifically on that direction wouldcause variations on the frequency.
Table 8 presents the target weights, regarding thefrequencies, that were used in each iteration. It-eration 5 is not presented since the target weightreffering to the frequencies were equal to iteration4. In iteration 5 were added the mode shapes astargets, but with a weight of 1.00. The objectivewas to evaluate whether the inclusion of the modeshapes would improve the results.
The target weights represent the importancegiven for each one of them. The first target hasthe heighest weight since is the one in which theconfidence of the correlation is higher, despite theMAC values. This is justified since represents thefirst mode of both sets, leading to a smaller proba-bility of changes on the mode order. The other twotargets, as seen in Figure 8, correlate modes thatbeside havinga bigger frequency error, have farthestmode numbers,
Starting with the design variables, it can be seenthat the first design variables presented the biggestchange from the initial value, 1.0. This was ex-pected as shown in the sensitivity matrix in Figure10. The value on the other collumn below each iter-ation represents the mechanical property, in GPa.
Another positive outcome is that in all iterationsthe value of E1u is always the highest, which is inagreement with the nature of the materials. Andalso that the other two design variables/mechanicalproperties present similar values.
Table 10 presents the individual and overall errorsof each target in each one of the five iterations.
From these results, and putting aside the fifth it-eration by now, iteration 4 yielded the best results.This is mainly justified by the individual error ofthe first target, to which was given the greatestimportance through the target weights. The othertargets, in which the confidence was lower, presentmuch higher errors and their evolution throughoutthe iteration varies considerably. The overall erroriteration 1 and 2 was considerably prejudiced by theindividual error of the first and second target whilein iteration 3 and 4 was mainly due to the target 2.
Concerning iteration 5, can only be comparedagainst iteration 4 since they share the same tar-get weigths for the frequencies. It can be seen thatdespite the big evolution on the first target errorin both models, the other two showed smaller im-provements or even evolved negatively. Regardingthe design variables/mechanical properties, the sim-ilarity between E1b and E2b is not as much as initeration 4.
Figure 11 shows the correlation between the ex-perimental results and iteration 4 and 5.
(a) Iteration 4 (b) Iteration 5
Figure 11: MAC matrices of updated complete model andexperimental results
From Figure 11, it can be concluded that the in-
8
Table 10: Iteration 1 to 5 individual and overall error comparison
clusion of the mode shapes as targets did not im-proved the correlation of the sets. The dispersionand correlation of local modes with the experimen-tal ones was maintained as well as the low MACvalues.
Table 11 presents the frequency results of itera-tion 4 and 5, side-by-side. In light grey are high-lighted the local mode shapes of both sets while inthe darker shade are indicated which modes werecorrelated with the experimental results, i.e., pairsof modes with a MAC value higher than 0.2.
Comparing the results of the correlated modeswith the experimental results (Table 5), can be seenthat in iteration 5 two of the frequencies are closerto the reference in comparison with iteration 4. Al-though, their respective MAC value decreased.
From this, one can conclude that despite the ap-proximation of the frequencies, the uncertainty onthe correlation increased. This corroborates the ini-tial decision of excluding the mode shapes due totheir lack of quality amd possible negative effect onthe results.
4.4. Torsional Stiffness ResultsThe results from the torsional stiffness simula-
tions, before and after the updating process, werecompared against the experimental value which was
2950 Nmdeg .
Table 12 summarizes the results. Note that thetorsional stiffness value is the mean value from allthe load cases in each simulation. The error collumnis relative to the experimental value that sets thereference.
It is immediately noticeable that the result fromthe initial model, with the initial mechanical prop-erties, considerably overestimates the torsional stiff-ness of the structure.
Considering uniquely the error, iteration 3 wouldbe the best. Although, considering also the resultsfrom the updating process, it is reasonable to reckonthat the results from iteration 4 and 5 can be suit-able candidates for the best approximation.
As an overview, acknowledging the eventual un-derestimation of the experimental test and overesti-mation of the numerical simulation, the results fromiteration 4 and 5 can be considered the ones thatbest represent the structure since the real value oftorsional stiffness might be between the ones thatwere obtained. It is important to understand thatthe results of the torsional stiffness analysis arehighly influenced by the model updating processand so, it is necessary to interpret these results withsome caution and reservations.
5. Conclusions
The main goal of this work was to implementa model updating process, based on experimentalmodal data, so that the numerical model of theFST06e chassis would present results as close aspossible to the real structure.
The experimental testing and consequent modalidentification process allowed the obtention of 6 res-onant frequencies of the structure. Although, thequality of the mode shapes was considerably be-low what was expected and desirable. This mighthave been caused by erroneous measurements or
9
problems on the identification and subsequent post-processing of the results. However, it was decidedto proceed with the study and consider the possibleeffects on the final results.
The comparison of the results prior to the updat-ing showed considerable differences that were at-tenuated with the application of the model updat-ing process. From iteration 1 to 4, it was possibleto verify a convergence towards the reference. Al-though, very low MAC correlations were obtained.
Iteration 5 had the objective of evaluatingwhether the inclusion of mode shapes, even know-ing their lack of quality, would improve the resultsobtained in iteration 4. The overall results did notimproved, proving the initial idea that the inclusionof the mode shapes as targets would not benefit theprocess.
The torsional stiffness simulation had the objec-tive of performing a relative evaluation of the evo-lution of the results after the whole process. Itwas possible to conclude that the updating processyielded results closer to the reference. Nevertheless,knowing that these results are highly influenced bythe outcome of the model updating process thatyield the design variables, it is necessary to havesome reservations on the interpretation of them, de-spite a clear improvement.
As an overview to the whole work and respec-tive results, it is not possible to consider themto be sucessfull. Despite the improvements eitheron the structural dynamic behavior and also onthe torsional stiffness results, they all derived fromthe model updating process that was conditioned,since the begining, by the flawed experimental modeshapes.
To obtain more reliable results would be neces-sary to repeat the experimental test in order toobtain good mode shapes and also develop a FEmodel that included the information about the lo-cal overlaps of plies and also the damaged areas ofthe structure.
References
[1] R. J. Allemang. The Modal Assurance Crite-rion (MAC): Twenty Years of Use and Abuse.Proceedings of SPIE - The International Soci-ety for Optical Engineering, 4753(1):397–405,2002.
[2] R. J. Allemang and D. L. Brown. A cor-relation coefficient for modal vector analysis.In First International Modal Analysis Confer-ence, pages 110–116, 1982.
[3] M. Friswell. Inverse Problems in StructuralDynamics. In Second International Conferenceon Multidisciplinary Design Optimization andApplications, Gijon, Spain, 2008.
[4] FSAE. FSAE Online.
[5] FSAE. FSAE Competition Rules, 2016.
[6] N. M. M. Maia and J. M. M. Silva. Theoreticaland Experimental Modal Analysis. ResearchStudies Press LTD, second edition, 1997.
[7] N. M. M. Maia, J. M. M. Silva, and A. M. R.Ribeiro. A New Concept in Modal Analysis:The Characteristic Response Function. TheInternational Journal of Analytical and Exper-imental Modal Analysis, 9(3):191–202, 1994.
[8] D. Montalvao. BETALab 11.23.3.
[9] NX. Siemens, plano, texas, usa, 2019.
[10] A. M. R. Ribeiro. Desenvolvimento deTecnicas de Analise Dinamica Aplicaveis aModificacao Estrutural. Phd thesis, InstitutoSuperior Tecnico, 1999.
[11] M. S. M. Sani, M. M. Rahman, M. M. Noor,K. Kadirgama, and M. H. N. Izham. Identi-fication of dynamics modal parameter for carchassis. In IOP Conference Series: MaterialsScience and Engineering, volume 17, 2011.
[12] C. Schedlinski, F. Wagner, K. Bohnert,J. Frappier, A. Irrgang, R. Lehmann, andA. Muller. Experimental Modal Analysis andComputational Model Updating of a Car Bodyin White. In Proceedings of ISMA2004 - Inter-national Conference on Noise and VibrationEngineering, pages 1925–1938, Leuven, Bel-gium, 2004.
[13] S. Sehgal and H. Kumar. Structural DynamicModel Updating Techniques: A State of theArt Review. Archives of Computational Meth-ods in Engineering, 23(3):515–533, 2016.
[14] D. C. C. M. Silva. A Modal-Based ContributionTo Damage Location in Laminated CompositePlates. Phd thesis, Instituto Superior Tecnico,2010.
[15] J. Silva, N. Maia, and A. Ribeiro. StructuralDynamic Identification with Modal ConstantConsistency Using the Characteristic ResponseFunction (CRF) Concept. Machine Vibration,5:83–88, 1996.
[16] J. M. M. Silva, N. M. M. Maia, and A. M. R.Ribeiro. Modal Constants Consistency: Appli-cation Of A New Method For Solving Overde-termined Non-Linear Equations. In 12th In-ternational Modal Analysis Conference (IMACXII) Proceedings, number 1, pages 1533–1536,Honolulu, Hawaii, 1994.
