DETERMINATION OF STATIC STIFFNESS OF MECHANICAL STRUCTURES FROM OPERATIONAL MODAL ANALYSIS

DETERMINATION OF STATIC STIFFNESS OF MECHANICAL STRUCTURES FROM OPERATIONAL MODAL ANALYSIS A. Melnikov, K. Soal, J. Bienert Mr, Technische Hochschule Ingolstadt, Germany, anton.melnikov@tum.de Mr, Stellenbosch University, South Africa, keithsoal@gmail.com Prof, Technische Hochschule Ingolstadt, Germany, joerg.bienert@thi.de ABSTRACT The experimental determination of static stiffness is an important task in structural design. The two current methods include clamping the structure, applying pre-defined loads, and measuring the displacements, or performing EMA and interpolating the frequency response function to Hz. Both methods require high experimental effort in laboratory setup. This paper presents a new idea, whereby the structure is measured during normal operating conditions, and OMA is used to reconstruct the stiffness matrix. The challenge of unscaled eigenvectors in operational modal analysis is overcome using mass modification scaling, whereby the structure is measured at a baseline and mass modified condition. Three different models were investigated: a basic discrete model, a laboratory ladder frame and a car body. Investigations were conducted into the effects of the position and magnitude of the mass modification, the number of assigned degrees of freedom of the system, the mode shape error effect and the effect of modal truncation. Key findings include significantly reduced reconstruction errors when the mass modification is a scalar multiple of the mass matrix. It was also found that the accuracy of the reconstructed stiffness matrix is strongly dependent on the uncertainty in the eigenvectors, as well as how the model is truncated. A finite element model of the ladder frame was used to test different modification strategies and to compare to results from operational modal analysis. A key finding which was not revealed by the discrete model was the importance of including the RBM in the stiffness matrix reconstruction. Finally experimental modal analysis and operational modal analysis were conducted on an Audi TT. The reconstructed stiffness matrices, bending and torsion static stiffness are then compared and discussed. Keywords: operational modal analysis, oma, measurement, static stiffness, car, body in white, biw, audi

. INTRODUCTION The dynamic behavior of most mechanical structures is fundamentally linked to their performance. The creation of mathematical models which can explain these physical responses is therefore key in the design and optimization process. The automotive industry provides such an example, where the vehicle chassis plays an integral role in the vehicle dynamics - handling, steering and road holding ability and in Noise Vibration Harshness (NVH) improvement. The chassis is also responsible for housing and linking the main elements - engine, wheels and drive train, and is a crucial safety barrier for the vehicle occupants. Two important metrics are the chassis bending and torsional stiffness, which can be calculated either analytically or experimentally. Analytical methods are limited by complex geometries, materials and joints, often requiring thorough experimental validation and updating. The two current experimental techniques include clamping the structure, applying pre-defined loads, and measuring the displacements, or performing experimental modal analysis and interpolating the frequency response function to Hz. Both methods require high experimental effort in laboratory setup. This paper presents a new idea, whereby the structure is measured during normal operating conditions, and Operational Modal Analysis (OMA) is used to reconstruct the stiffness matrix. The challenge of unscaled eigenvectors in OMA is overcome using mass modification scaling, whereby the structure is measured at a baseline and mass modified condition. The mathematics of the reconstruction technique are presented first, followed by a discrete model simulation investigating the error effects of various mass modification configurations. The technique is then applied to a laboratory ladder frame structure. A finite element model of the ladder frame was also created and is compared to the experimental results. Finally the technique is applied to a vehicle chassis Body in White (BIW).. FUNDAMENTAL IDEA.. Basic equations As a starting point the discrete Multiple Degree of Freedom (MDOF) ordinary differential equation in the notation Mẍ + Bẋ + Kx = f(t) () is used, whereby M represents the mass matrix, B the damping matrix and K the stiffness matrix. In most engineering applications, for example in large steel structures, the damping part can be suppressed and the system of equations becomes much easier leading to the eigenvalue problem KΦ = MΦΛ. () The matrix Λ is a diagonal matrix and contains the eigenvalues corresponding with the position of the eigenvectors in the matrix Φ. Alternatively the state space notation can be used to include general viscous damping into the equation by including of the momentum balance. This leads to the following eigenvalue problem with twice the number of Degrees of Freedom (DOFs) A Φ = A Φ Λ () which can be solved like the undamped case in Equation. For reasons of simplicity the notation of undamped systems will be used in the following discussion, which can be expanded to damped systems based on the state space formulation. The mass normalization of the eigenvectors is an important procedure to access the complete set of properties of the modal matrix. The calculation step can be done in the following way Φ = Ψ[Ψ T MΨ] /. ()

In this case Φ represents for the modal matrix of mass-normalized eigenvectors and Ψ the matrix of arbitrarily scaled eigenvectors. Based on Equation and including the mass normalization of the eigenvectors from Equation the mass and stiffness matrices can be reconstructed from the formulation Φ T MΦ = I Φ T KΦ = Λ by inverting the modal matrix Φ and multiplying it from right and left, it leads directly to the mass and stiffness matrices M = (Φ T ) IΦ K = (Φ T ) ΛΦ (). The inversion of the modal matrix becomes more complicated when the number of DOFs and the number of modes are different. In this case the use of the pseudo inverse is necessary. The general formulation uses the Singular Value Decomposition (SVD) to obtain the pseudo inverse in the following form Φ = USV H Φ + = V S + U H... Mode shape rescaling and matrix reconstruction There is no direct method to obtain the normalized shape vectors from OMA, however there are a few approaches of how to approximate it. These methods are mostly based on the sensitivity of eigenfrequencies and mode shapes described by DRESIG in [] with the following conditions: eigenvector remains within modification x i and small mass change M M. A Sensitivity-based method to estimate the scaling factors was presented by PARLOO in [] and was validated on a full scale bridge in subsequent years []. The frequency shifts between the original and mass-modified system were used for re-scaling (mass-normalizing) of the mode shapes. Later the formulation of PARLOO was established as [] ω α = ω ˆx T () M ˆx and the formulation of BRINCKER, with only small differences was establishes as α = ω ω ω ˆxT M ˆx. For easier use the BRINCKER formulation can be used in matrix form as A = [(Λ Λ m )(diag{ψ T MΨ}Λ m ) ] () with the diagonal matrix A containing the scaling factors α. The rescaling can be done by multiplying the factors by the eigenvectors to obtain an approximation of the modal matrix Φ Φ = ΨA where it must be remembered that the quality depends on the fulfillment of Equations 8 and 9. The system matrices can then be reconstructed by M r = ( Φ T ) + I Φ + K r = ( Φ T ) + Λ Φ + using two OMA data sets - an original or baseline and a mass modified system. () (7) (8) (9) () () ()

. DISCRETE MODEL For testing purposes a simple discrete model with DOFs was created as shown in Figure. m k k k k m m m m Figure : Simple undamped degrees of freedom system with one Rigid Body Mode/s (RBM) The model consists of equal masses connected by springs with equal stiffness and can move only in one direction. Analyzing the model leads to one RBM, which was chosen to show the functionality of the method independent of RBM. The complete set of mode shapes is shown in Figure. Mode shape by Hz Mode shape by. Hz Mode shape by.9 Hz Mode shape by 8. Hz Mode shape by 9.79 Hz Figure : Complete set of mode shapes for basic discrete model... Results for different system setups There are a number of ways to setup the system, which have an effect on the quality of the results. The errors obtained by applying a single % mass modification at one degree of freedom and varying its position are shown in the Figure. The highest error is present when the mass modification is located at relative error.7..... Relative error estimation mass error stiffness error. mass modification position Figure : Global relative error for mass and stiffness. the center node. This becomes obvious when considering Figure since the center node is located at a vibration nodal point for modes and, and therefore the mass modification does not contribute to the frequency shift of these modes.

The variation of the mass magnitude from % to % while keeping the mass located at the first node leads to the errors shown in Figure. The errors are seen to decrease with reduced mass. This is as relative error....... Relative error estimation mass error stiffness error..8... mass modification Figure : Global relative error for mass and stiffness for different mass modifications a result of the reduction of the change in the mode shape. The effect of the number of applied mass modifications is shown in Figure where a % mass modification is applied to all nodes. The stiffness Reconstructed stiffness Absolute error 8e- e- e- - - e- Figure : Reconstructed stiffness matrix in N m and absolute error in N m. matrix approximation is seen to be very accurate with an essentially numerically zero error. It was also observed that the reconstructed mass matrix had a constant offset of approximately %, the reasons for which are unknown. That offset could be avoided by connecting the system to a fixed environment by an additional weak spring at the first node. The errors as a function of the stiffness of the weak spring are shown in the Figure. The error for the mass matrix becomes very small in relation to the RBM by. Relative error estimation. relative error.. mass error stiffness error.8. weak springs boundary stiffness in N/m 8 Figure : Global relative error for mass and stiffness for different weak spring stiffness. applying a small stiffness, which could be an indication of numerical ill conditioning or a failure of the Equation at Hz frequency... Mode error effect Mode shapes obtained from experimental data contain errors and are not exactly orthogonal. To model this phenomenon the eigenvectors of the unconstrained discrete model were mistuned with a % error

of random sign for every entry. Using these eigenvectors for the reconstruction led to the stiffness matrix shown in the Figure 7. The structure of the original stiffness matrix is still recognizable, but there are Reconstructed stiffness Absolute error - Figure 7: Reconstructed stiffness matrix in N m and absolute error in N m. locally high errors for some entries. The reconstruction procedure is therefore sensitive to the mode shape errors... Modal truncation To investigate the influence of modal truncation one of the five modes were rejected from the reconstruction. The stiffness matrices with the errors are shown in the following Figures 8 and 9. The truncation of Reconstructed stiffness - - Absolute error Figure 8: Truncation of the mode : reconstructed stiffness matrix in N m and absolute error in N m. Reconstructed stiffness Absolute error 8 - Figure 9: Truncation of the mode : reconstructed stiffness matrix in N m and absolute error in N m. mode, which is the RBM had no effect on the reconstructed matrix, leading to the conclusion that this mode is superfluous in this basic model. The truncation of mode, which is the highest mode completely destroys the result. The errors between mode and mode truncation lie between these extremes, also described in []. The higher modes are therefore of greater importance for the quality of the reconstructed stiffness matrix.

. LABORATORY LADDER FRAME Investigations of a ladder frame shown in the Figure, were conducted as a first laboratory testing step. The frame was attached by weak springs to a fixed environment and the measurements were conducted using accelerometers at locations. The geometry and measurement point positions are described in []. Figure : Ladder frame experimental set-up... Finite Element Method (FEM) evaluation The ladder frame was modeled with FEM and modes were identified with the natural frequencies presented in the Table. The stiffness matrix shown in the Figure was extracted from the FEM Table : FEM calculated frequencies. Mode Frequenzy (Hz) Mode Frequency (Hz) 7 8 9..7..9.. 7.. 8.8.9 7 8 9 8. 88.. 77.7 8. 8. 8.7 8. 9.. model, and was used as the reference stiffness matrix. Four different mass modification strategies were FEM stiffness matrix - Figure : Extracted stiffness matrix from the FEM in N mm. analyzed using the FEM model. Modification with g at the Degree of Freedom (DOF), symmetrical modification at DOF and with g and g and a modification with g at all six points were applied. The results are presented in Table, were it can be seen that cases and are not in

the appropriate range. The reasons are in the validation of the requirements set by the Equations 8 and 9. The results of case show the smallest error, since the mass modification matrix is close to a scalar multiple of the original mass matrix. M = a M, a R, a > ().. OMA with single mass modification OMA was conducted for the single mass modification with g at the DOF and the reconstructed stiffness matrix is shown in Figure. The structure of the two diagonal blocks shows correlation with the FEM calculated reference, however the off-diagonal blocks are different and the values are higher than expected. A reason for the differences between the measurements and the FEM model is expected to be as a result of the single point mass modification of large mass magnitude together with the weakness of the OMA and FEM analysis. The corner joints in the FEM were modeled as mass points without stiffness and damping properties and are therefore also expected to introduce errors. The results of the OMA reconstruction are presented and compared to the FEM reconstructions in Table. OMA stiffness matrix absolute error to FEM 7 - Figure : Reconstructed OMA stiffness matrix K r in N N mm and absolute error in mm (FEM reference). Table : Ladder frame static stiffnesses gained by different mass modification approaches and technology. Case Bend. N mm B. error % Tors. N m rad T. error % FEM reference 9.79-9.9 - FEM g 8..8 8. 7.8 FEM g.9 8.8 7.8.8 FEM g 9..7 7..99 FEM g 9.7.8 98.7.9 OMA g.7.. 7.8

. ADVANCED MODEL - VEHICLE BODY The complexity of the structural system was then increased to that of an Audi TT Coupe Mk BIW as shown in Figure. The doors, hood and tailgate were dismounted to avoid complications of a multibody system with joints. The body was mounted on inflated rubber tubes to provide RBM. Measurements were conducted using accelerometers at points as shown in Figure. The measurement points,,, 7, 8, 9 were focused on for comparison to the ladder frame and for obtaining the bending and torsional stiffness. 9 7 8 Figure : Audi TT Coupé BIW with measurement setup... Experimental Modal Analysis (EMA) Experimental modal analysis was first conducted on the vehicle body. modes were identified and the natural frequencies are listed in Table. The reconstructed mass and stiffness matrices using all the identified modes are shown in Figure. The structure of these matrices is as expected with a dominant main diagonal in the mass matrix and two clusterings in the stiffness matrix due to the dominant stiffness connection between nodes - and 7-9. Table : EMA eigenvalues. Mode Frequency (Hz) Mode Frequency (Hz) Mode Frequency (Hz) 8. 7..8 9. 78.9 8..8 8..8 7. 8.7.. 8.7 8..98 8.8.7 7.78 7 87.8 8.77 8 9.879 9. 9.9 8..97 The calculated results of the static bending stiffness, static torsion stiffness and the mass matrix trace are presented in Figure, the calculation procedures are detailed described in []. It can be seen that including higher modes is important for convergence. The static bending stiffness converged at modes

Stiffness matrix Mass matrix e+7 e+ -e+ Figure : Reconstructed EMA stiffness matrix K r in N m and mass matrix M r in kg with full modal result set. to approximately kn mm. This is within the expected range from literature [ 8]. The static torsion stiffness did not show clear convergence increasing to a final value of.9 kn deg. This far exceeds the value published by Audi of 8. kn deg by 8%. The trace of the mass matrix converges at modes to 9kg which is in the expected range. Stiffness in kn/deg Mass in kg Stiffness in kn/mm 7 Static bending stiffness Modes included Static torsion stiffness 8 Modes included Mass matrix trace Modes included Figure : Static bending, static torsion stiffness and mass matrix trace as a function of the number of included modes... OMA with Mass Modification Operational modal analysis was then performed with random (in space and time) multi-point ( person) tapping excitation. Mass modifications of..kg =.8%.% of the total mass were made at DOFs ( 7 8 9), DOFs ( 7) and DOF () during consecutive tests in order to scale the eigenvectors. It was found that due to the complexity (not in phase but in identifiable shape or symmetry) of the resulting mode shapes, that it was very difficult to track and link the modal shift after mass modification. Especially since the modes were closely spaced and did not all move by the same amounts i.e. the order of the modes was changed. For this reason a reduced mode set of six identifiable modes was used. The resulting reconstructed mass and stiffness matrices after modal scaling are shown in Figure The mass matrix has a less dominant main diagonal, while the stiffness matrix retains most of the expected matrix structure. The static stiffness results from both EMA and OMA are presented in Table. OMA with -mass scaling provided more accurate results than -mass and -mass scaling as expected

Stiffness matrix Mass matrix e+7 e+7 -e+7 - Figure : Reconstructed OMA stiffness matrix K r in N m and mass matrix M r in kg with full modal result set. from the discrete model. If it is however not possible to place masses at each measurement location for practical reasons, certain points may provide better results, as shown by the -mass vs. -mass comparison. The large error in the torsion stiffness for EMA with full resolution is unclear. It was however seen that including higher modes was important to achieve convergence. The importance of including the RBM in the reconstruction can be seen in the poor results of case. Case Table : Static stiffness results from EMA and OMA. Bend. kn mm knm Tors. mm T. error % EMA full res...9 8. EMA -modes 7.7. 8. EMA full no RBM..9. OMA -masses 7.9.9 9. OMA -masses... OMA -mass.7. 9.. CONCLUSION The determination of static stiffness from OMA reconstruction was tested and benchmarked. A basic discrete model was used to prove the concept and investigate the effects of the mass modification position, number of applied mass modifications and the size of the masses. It was found that reconstruction errors were smallest when the mass modification is a scalar multiple of the mass matrix and the masses are made as small as possible but as large as necessary to exceed measurement error. The reconstruction errors were found to be strongly dependant on the uncertainty in the eigenvectors, as well as by how the model was truncated - with the inclusion of higher modes significantly reducing the error. A ladder frame structure was used to test the idea in the laboratory and a FE model was built to investigate different modification strategies and to compare to results from OMA. Large errors resulted from the OMA reconstruction mainly due to the single mass modification and subsequently poor mode shape scaling. The revelation of the matrix structure could however still be an important factor for understanding and investigating natural phenomena. Finally EMA and OMA were conducted on an Audi TT BIW, and the reconstructed stiffness matrices were used to calculate the bending and torsion static stiffness. Results showed convergence when increasing the number of modes included in the reconstruction. Accurate bending stiffness values were calculated from OMA reconstruction with -mass modification. The most accurate torsional stiffness values were calculated from EMA with a truncated mode set. Importantly it was also found that the RBM played an insignificant role in the ladder frame stiffness reconstruction but a significant role in the vehicle body reconstruction. The technique has shown much potential, but requires further research to quantify and improve the reconstruction error effects. This method has been registered as patent DE 8 A.. [9].

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