Identification of Non-linear Joint Parameters by using Frequency Response Residuals

Marc Böswald1) and Michael Link2)

1)Research Assistant, 2)Professor University of Kassel, Mönchebergstr. 7, D-34109 Kassel, Germany

Nomenclature Scalars:

m mass c viscous damper constant k spring stiffness u displacement u velocity u acceleration t time u complex amplitude of harmonic displacement u∆ complex amplitude of relative displacement between two degrees of freedom f excitation force

f complex amplitude of harmonic excitation force Ω circular frequency of excitation

Rf restoring force of non-linear element j imaginary unit ξ modal viscous damping ratio µ modal mass γ modal stiffness ω circular eigenfrequency α updating parameter of non-linear damper element β updating parameter of non-linear spring element J objective Function

Vectors:

X real normal mode shape

p parameter vector

r residual vector

ν data vector

h vector containing frequency response function data at several frequency points

g partition of the sensitivity matrix

Matrices: [ ]M mass matrix

[ ]C viscous damping matrix

[ ]K stiffness matrix

[ ]G sensitivity matrix

[ ]W weighting matrix

pW regularization matrix

Indices:

A initial underlying linear model eq equivalent value of non-linear quantity im imaginary part of complex quantity nl non-linear re real part of complex quantity a analysis data t test data k linearization point, current design point i summation index for underlying linear modal quantities r summation index for non-linear damper elements s summation index for non-linear spring elements 0 initial values (of updating parameters)

Abstract In this paper, a method is presented, which aims at the extension of the finite element method for the description of the dynamic behavior of structures with non-linear joints. Such non-linear joints may either be structural connections, such as bolted or riveted joints between otherwise linear substructures, or may be non-linear interface conditions, such as friction, clearance, or bilinear stiffness between pre-loaded contacting surfaces. Such types of non-linear joints often exhibit a linear characteristic when the structure vibrates at relatively low amplitudes, whereas non-linear effects are activated in the case of large vibration amplitudes. The approach presented is based on the extension of linear finite element models of overall assembled structures by discrete non-linear elements to take into account the local non-linear effects at the joints. The harmonic balance method is used to derive effective stiffness and damping coefficients of these non-linear joint elements which allows for the frequency domain representation of the overall assembled non-linear model. This non-linear frequency domain model can be considered as the basis for non-linear frequency response analysis to calculate the steady-state forced response of a non-linear model due to harmonic excitation. The FE-model parameters describing the linear stiffness and damping properties of a joint generally suffer from a huge amount of uncertainty like, for example, the calculation of the effective stiffness and damping properties of a simple bolted joint. FE-model parameters describing non-linear stiffness and damping properties of a joint are even more subject to uncertainty. Therefore, the application of a model updating procedure is proposed, which allows for the adjustment of not only linear but also non-linear joint parameters. This is achieved by minimizing the differences between experimental and analytical non-linear frequency response functions. The application of this model updating procedure thus yields a finite element model with an improved prediction capability in the large vibration amplitude regime where non-linear joint effects are activated. The procedure to be presented was developed within the European Research Project CERES and was applied there for non-linear frequency response analysis and non-linear joint parameter identification of a complex large order finite element model of an aero-engine structure.

Introduction The finite element (FE) method is extensively applied in product development as a standard tool for structural analysis. FE models are used there to simulate the response of structures due to static or dynamic loads with the

intention to gradually replace expensive tests conducted on prototypes. In most cases, FE models are based on a linear elastic material law which does often not reflect the real structural behavior where non-linearity can always be observed, at least to a certain extent. In case of static analysis, non-linear modeling is more and more used in order to achieve a better approximation of real structural behavior. In structural dynamics, non-linear behavior is most often taken into account in case of non-linear time domain analyses which involve the application of numerical integration schemes. Even though available, non-linear frequency domain techniques like non-linear frequency response analysis, can nowadays not be considered a standard tool of commercial FE software. Such tools are still subject of research at universities and yet have not overcome the development stage with application to academic examples. Non-linear behavior may be characterized by a non-linear relationship between element forces and nodal displacements, where the so called restoring force is a non-linear function of the structure’s displacement response. In case of dynamic analyses, restoring forces are a function not only of displacement but also of velocity, i.e. a non-linear restoring force is a function of the system state. Techniques for the characterization of non-linear behavior were developed based on these non-linear restoring force functions which can be derived from experimental data. These techniques are referred to as force-state mapping or restoring force surface method, [1-2], and in the ideal case they can yield an equivalent mechanical model for the actual non-linearity of the structure under investigation. Non-linear frequency response functions (FRFs) can be obtained experimentally from step sine tests with reasonable effort and are a powerful tool for the characterization of non-linear behavior. In most cases, the structure is exposed to a single point harmonic excitation introduced by an electro dynamic shaker whose excitation force amplitude is maintained at a constant level using a feedback loop controller. If additionally a low pass filter is used to filter out higher harmonics from the measured response signals, the non-linear frequency response of the fundamental harmonic can be obtained experimentally. This fundamental non-linear frequency response function deviates from a linear FRF, i.e. it appears distorted, where the distortion characteristic depends on the type of non-linearity activated in the tests. These characteristic distortions generally increase with increasing excitation force level, or respectively, with increasing vibration amplitudes. A non-linear modeling approach should be considered if the deviations from linear behavior are large, such that the response prediction using a linear FE model is inaccurate. However, when modeling non-linear behavior, the assumptions posed on the type of non-linearity and the parameters describing the non-linear element restoring forces are always subject to uncertainty such that the prediction capability of an FE model with such a non-linear supplement is not necessarily improved. The accuracy of the results obtained by a non-linearly extended FE model can be improved by minimizing the differences between experimental non-linear FRFs obtained from step sine tests and analytical non-linear FRFs obtained from simulations with the FE model by adapting the numerical values for the parameters of the non-linear elements. This may raise the question, if the distortion characteristic of a non-linear FRF contains enough information to properly identify the parameters of the non-linear elements which were used to simulate the analytical non-linear FRFs. In [3], this question was discussed in the context of academic examples. The work presented here is also concerned with this question and the progress made on this subject is presented together with recent results obtained from the identification of non-linear joint parameters of large order FE models.

Modeling of Non-linear Behavior Generally, a distinction should be made here between global and local sources of non-linear behavior. Global non-linearities may be characterized by large portions of a structure actually showing non-linear behavior, e.g. when a structure is exposed to excessively high loading or the vibration of a structure stiffened by cables. Such non-linearities shall not be considered here. Local sources of non-linearity may be characterized by very limited areas of a structure showing strong non-linear behavior, whereas the remaining part of the structure is still linear. Such local non-linearities are often encountered at the interfaces between linear components of an overall assembled structure. Examples for such non-linear phenomena are friction in case of shear force transmission between two joined parts, or the bilinear stiffness of pre-loaded joints in case of normal load transmission. In order to adequately represent the non-linear behavior of such joints, the load path through the joint has to be analyzed and modeled as far as necessary. Thereby, the conditions of load application have to be considered together with the mechanisms of load compensation. Modeling the load path thus includes the representation of possible eccentricities frequently existing at bolted flange joints. An example for a load path investigation through a joint is shown in figure 1. There, the 3D geometry of a bolted scalloped flange joint is shown, together with its deformation due to a static tensile load indicated by the cut through the deformed FE model of that joint.

Figure 1: Geometry and deformation of a scalloped flange joint with bolt eccentricity

The FE models used for dynamic analysis are generally not 3D models but are in most cases models with a relatively coarse meshing comprised of shell and beam elements. A static analysis of the load path using a 3D model thus can only provide insight into the deformation characteristic of the joint in order to derive load path modeling strategies applicable to the coarse dynamic models. After the load path through a joint of a dynamic model has been modeled adequately, non-linear elements may be introduced to represent possible joint non-linearities. In the work presented here, this is done by means of non-linear 2-degree-of-freedom elements which are introduced between those degrees of freedom of the joint which have the largest relative displacement due to a typical operational loading condition. These non-linear 2-degree-of-freedom elements represent coupling mechanisms between the otherwise linear substructures of an overall assembled structure and carry non-linear effects such as friction, contact, clearance, etc. An example for modeling the non-linear behavior of a bolted flange joint is shown in figure 2, where non-linear 2-degree-of-freedom elements are used for non-linear coupling of the two joined linear substructures.

Figure 2: Non-linear modeling approach for bolted flange joint using rigid elements

in conjunction with non-linear 2-degree-of-freedom elements

The load path representation of the bolted flange joint shown in figure 2 allows that a part of the loads to be transmitted through the joint is passed through the bolt. A superposition of linear and non-linear spring/damper elements is introduced at the bottom of the flange in order to carry the non-linear effects possibly existing at the joint. These elements connect the x-direction degrees of freedom of the nodes at the bottom of the flange and may represent, for example, different joint stiffnesses in tension and compression conditions by means of a bilinear restoring force function. The location at the bottom of the flange was chosen since large relative displacements occur there when the structure is exposed to typical loadings, see figure 1. When disregarding the non-linear elements, this approach for load path modeling can also be used for modeling the effective linear stiffness of the joint in a physically meaningful way. The stiffness of the linear spring introduced at the bottom of the flange has to be identified from linear test data coming from a standard modal survey test (eigenfrequencies and mode shapes). In addition, the joint modeling approach shown in figure 2 can be extended to, or respectively, applied to different approaches of flange modeling, e.g. when offset beam elements are used to represent the flanges instead of shell elements as shown in figure 2, see [4]. After non-linear elements have been introduced, most commercial FE codes provide numerical integration schemes to analyze the non-linear response due any type of time-varying loading. This time domain analysis would in principle also be used to calculate non-linear frequency response curves by calculating the steady state

rigid connections shell elements

bolt centre line

x

y

F F

superposition of linear and non-linear elements

responses as the result of a series of non-linear analyses using harmonic excitation at different excitation frequencies. This procedure, however, may be inefficient if not impossible in case of weakly damped structures where the decaying time of the initial transient response may be considerable.

The Harmonic Balance Method The Harmonic Balance Method, see [5], allows for analyzing the steady-state fundamental harmonic non-linear structural response due to harmonic excitation in the frequency domain. The transformation to the frequency domain is done by calculating equivalent stiffness and damping parameters for the non-linear elements which would normally prevent such a transformation. The frequency domain representation of the non-linear equation of motion is then the basis for non-linear frequency response analysis. The theory behind the Harmonic Balance Method shall be discussed in the following on a single degree of freedom system, whose non-linear equation of motion shall be written in a very general form:

( ) ( , , ) ( )Rmu t f u u t f t+ = , (1)

where ( , , )Rf u u t is a non-linear restoring force function (e.g. due to friction, clearance, bilinear stiffness etc.). In case of a viscously damped linear single degree of freedom oscillator this restoring force function would contain the contribution of a viscous damper and a linear spring:

, ( , , ) ( ) ( )R linf u u t cu t ku t= + . (2)

The fundamental assumption behind the Harmonic Balance Method is that the response of a non-linear system due to harmonic excitation can be approximated by a harmonic function in the frequency of excitation, i.e. the total non-linear response of the system is dominated by the fundamental harmonic response:

( )( )( )( )2

ˆ( ) sinˆ ˆ( ) sin ( ) cos

ˆ( ) sin

u t u tf t f t u t u t

u t u tϕ

≈ Ω= Ω + → ≈Ω Ω ≈ −Ω Ω

. (3)

If this assumption is sufficiently fulfilled, the non-linear restoring force function ( , , )Rf u u t can be decomposed into a Fourier series. Truncating this Fourier series after the fundamental terms yields the following approximation for the restoring force function:

( ) ( )0 1 1( , , ) sin cosRf u u t a a t b t= + Ω + Ω +… , (4)

where 0a , 1a and 1b are the Fourier coefficients which can be calculated from the following integrals ranging over one period of vibration:

2

00

1 ( , , ) ( )2 Ra f u u t d t

π

π= Ω∫ , (5a)

2

10

1 ( , , ) cos( ) ( )Ra f u u t t d tπ

π= Ω Ω∫ , (5b)

2

10

1 ( , , ) sin( ) ( )Rb f u u t t d tπ

π= Ω Ω∫ . (5c)

The aim of applying the Harmonic Balance Method is approximating non-linear restoring force functions such as friction, contact, or clearance, by equivalent spring and damper forces:

( ) ( )( ) ( )

ˆ ˆ( , , ) ( ) ( ) sin cosR eq eq eq eq

u t u t

f u u t k u t c u t k u t c u t≈ + = Ω + Ω Ω . (6)

A comparison of the coefficients of equation (4) and equation (6) yields two equations from which the equivalent stiffness and damping parameters can be identified:

1ˆ( )ˆeqbk uu

= , (7a)

1ˆ( )ˆeqac uu

=Ω

. (7b)

It can be seen from equations (7a) and (7b) that the equivalent stiffness and damping parameters are dependent on the amplitude u of the harmonic displacement response ( )u t . The dependence on the factor time could be eliminated by the application of the Harmonic Balance Method. Since the amplitude of the displacement response u is generally unknown at the beginning of the response analysis, the determination of the equivalent parameters

eqk and eqc has to be performed iteratively. Furthermore, the Fourier coefficients 1a and 1b , which are needed to calculate the equivalent parameters, are generally dependent on the parameters describing the non-linear restoring force functions ( , , )Rf u u t , e.g. the different stiffness in the tension and the compression regime in case of bilinear spring. When using a complex approach for the excitation and for the displacement response in equation (3), with the complex excitation force amplitude ˆ ˆ ˆ

re imf f j f= + and the complex response amplitude ˆ ˆ ˆre imu u j u= + , and

when additionally using Euler’s equation cos( ) sin( )j te t j tΩ = Ω + Ω with the imaginary unit 1j = − , equation (3) can be brought to the following form:

2

ˆ( )

ˆ ˆ( ) ( )

ˆ( )

j t

j t j t

j t

u t ue

f t fe u t j ue

u t ue

Ω

Ω Ω

Ω

≈ ℜ= ℜ → ≈ℜ Ω

≈ℜ −Ω

. (8)

Introducing the complex approach of equation (8) into the equation of motion (1) together with the equivalent stiffness and damping parameters yields the non-linear equation of motion in the frequency domain:

( ) ( )( )2 ˆˆ ˆ ˆeq eqm j c u k u u f−Ω + Ω + = . (9)

Equation (9) is still a non-linear equation, because the equivalent parameters eqc and eqk are dependent on the

magnitude of the complex displacement response amplitude u . This means, that the application of the Harmonic Balance Method does not yield linear equation of motion, but rather provides a means for transformation into the frequency domain where a non-linear equation of motion has to be solved afterwards. The extension of the Harmonic Balance Method to 2-degree-of-freedom elements for coupling of linear substructures is achieved by introducing the element matrices of a non-linear spring element and a non-linear damper element:

( ) ( ) 1 1ˆ ˆ

1 1eq eqK u k u− ∆ = ∆ −

, (10)

( ) ( ) 1 1ˆ ˆ

1 1eq eqC u c u− ∆ = ∆ −

. (11)

These 2x2 matrices can be assembled with the overall linear stiffness and damping matrices of an overall assembled structure. Thereby, the equivalent stiffness and damping parameters eqk and eqc are dependent on

the magnitude of the complex relative displacement response amplitude 2 1ˆ ˆ ˆu u u∆ = − between the two degrees

of freedom 1u and 2u . If several identical non-linear elements are introduced into a structure at various positions, the actual relative displacement amplitude will be different at the different positions of the non-linear elements. As a consequence, the equivalent stiffness and damping values of the nominal identical non-linear elements will be different, too. The Harmonic Balance Method shall now be discussed on a simple cantilever beam structure. It is composed of two nominal identical beam components connected by a piecewise linear rotational spring and has a large mass attached to the beam tip, see figure 3. The non-linear restoring force function of the piecewise rotational spring is sketched in figure 3 where the softening stiffness character of this type of spring can be observed.

Figure 3: Cantilever beam with piecewise linear rotational spring

The linear frequency response and the non-linear frequency response for 50 N excitation force applied to the large mass were analyzed using the Harmonic Balance response software HBResp. Figure 4 shows the frequency response obtained at the excitation degree of freedom of the cantilever beam model. It can be seen that the piecewise linear spring produces a softening stiffness non-linearity which is indicated by the shift of the resonance peak towards lower frequencies. The frequency response curve for increasing frequency increment (i.e. stepping from a low frequency towards a higher frequency, red dashed curve in figure 3) differs from that for decreasing frequency increment (i.e. stepping from a high frequency towards a lower frequency, blue dash-dotted curve in figure 3). In the frequency range where these curve do not coincide indicates a region with non-unique stable solution branches, i.e. there are two possible stable responses to a given excitation.

Figure 4: Linear and non-linear frequency response of the cantilever beam with piecewise linear spring

The resonance frequency of the fundamental bending mode of the underlying linear cantilever beam model is at 114.2 Hz. The results obtained from the HBResp analysis shall be proved by linear and non-linear transient analyses (MSC.Nastran Sol 109 and 129) using harmonic excitation at four different frequencies (112 Hz, 113.4 Hz, 114 Hz and 115.5 Hz). The steady-state response amplitudes obtained form the linear transient analyses are indicated by circles in figure 4 (Newmark implicit integration scheme, time step for direct integration 1e-4 sec). It can be seen that these amplitudes match those obtained form the linear HBResp analysis (this result is used as a check for the consistency of the proportional damping model used here in both, frequency domain and time

F=50N piecewise linear rotational spring

M

ϕ∆hinge

domain analysis). The steady-state response amplitudes obtained form non-linear transient analysis (Newmark implicit integration scheme with Gauß-Newton iteration at each time step to achieve convergence, time step for direct integration 1e-4 sec) are indicated by the blue stars in figure 4. It can be seen that the HBResp results match the non-linear transient response analysis results. This means that a steady-state response of a non-linear structure can efficiently be calculated directly in the frequency domain using the Harmonic Balance method without the need to compute the steady-state response of numerous non-linear transient analyses using harmonic excitation. It should be emphasized that the harmonic balance model is not an equivalent model, because the same non-linear stiffness properties were used both, non-linear transient analysis and Harmonic Balance frequency domain analysis. It should be noted that is was not possible to calculate a unique converged steady-state transient response solution at 112 Hz using MSC.Nastran. One possible reason for this failure may be that the solution is oscillating between the two possible solutions available at 112 Hz. However, the non-linear transient analysis may be stabilized by applying artificial numerical damping (see MSC.Nastran manuals), but in this case the steady-state solution due to harmonic excitation of such a lightly damped structure would be calculated erroneously (because the steady-state solution is sensitive to damping) and should not be used for comparison with the HBResp results. Figure 5 shows typical response plots of a non-linear transient analysis (drive point response and relative response between the degrees of freedom of the non-linear element).

Figure 5: Driving point response and relative response at non-linear element degrees of freedom

for harmonic excitation at 114 Hz

One interesting feature of this non-linear cantilever beam model is that even though there is no non-linear damping in the model, the response amplitudes of the non-linear model when excited near the resonance of the underlying linear model are lower than those predicted by the underlying linear model. In addition, if only the frequency response function for increasing frequency increment is considered (red dashed curve in figure 4), it can be seen that the response magnitudes are much less than those obtained with the linear model. This response curves are often measured if such a structure is tested by a step sine tests starting from a low frequency and stepping towards a higher frequency only. The result would then be interpreted as non-linear damping even though there is no damping non-linearity! When using the Harmonic Balance Method as it is done here, higher harmonic terms of the Fourier series (4) are disregarded in the calculation of the equivalent stiffness and damping parameters. The same applies to the constant term of the Fourier series. Regarding the higher harmonic terms, it has to be noted, that the Harmonic Balance Method as discussed here, yields the steady-state displacement amplitude of a monofrequent harmonic response function for each frequency point of interest. This monofrequent harmonic response function can be considered as a “best-fit” in the sense of a least-squares approximation of the general non-linear multi-harmonic displacement response. Furthermore, it has to be noted that the contribution of the higher harmonic response portions to the total general non-linear response will only be significant when the structure is excited in the vicinity of a resonance. Consequently, when the structure is excited at a frequency far away from a resonance, the general non-linear response is approximately monofrequent and harmonic. The constant term 0a of the Fourier series can completely be disregarded for non-linear elements having odd

restoring force functions (i.e. when ( ) ( )R Rf u f u− = − ), because in such cases the Fourier coefficient 0a will

vanish anyway. When considering general types of restoring force functions with ( ) ( )R Rf u f u− ≠ − , the Fourier

coefficient 0a will not vanish. In this case, the structure will vibrate with an offset about its position of static equilibrium, i.e. the non-linear system is becoming pre-loaded when approaching resonance. This constant part of the displacement response is neglected in the Harmonic Balance Method and is in most cases filtered out during vibration measurements as part of the signal conditioning, or respectively, cannot be measured experimentally when using piezo-electric accelerometers. The effect of vibration with an offset about the position of static equilibrium can be seen in figure 6. There, the time histories of the non-linear displacement response and the non-linear velocity response are shown. Further considerations with respect to the application of the Harmonic Balance Method and the accuracy of the solution to be obtained are given in [6].

0 50 100 150 200 250 300-4

-2

0

2

4

6

8

10Time History of Displacement Response

Time [s]

Dis

pl. [

m]

0 50 100 150 200 250 300-6

-4

-2

0

2

4

6Time History of Velocity Response

Time [s]

Vel

o. [m

/s]

-4 -2 0 2 4 6 8 10-7

-6

-5

-4

-3

-2

-1

0

1

2

3Restoring Force

Displacement [m]

Forc

e [N

]

Figure 6: Time history of displacement and velocity response together with the non-linear

restoring force function of a damped bilinear single degree of freedom oscillator

Basics of Non-linear Parameter Identification The determination of stiffness and damping parameters of joints is subject to uncertainty, even in the linear case. In case of non-linear modeling, where non-linear stiffness and damping parameters have to be derived, even more uncertainty is posed on such joint parameters. Consequently, the prediction capability of a finite element model is not necessarily improved by taking into account the non-linear joint effects. While the linear joint stiffness parameters may be identified, or respectively, be improved by means of computational model updating procedures using eigenvalue and mode shape residuals (see [7-9]), the uncertainty of the non-linear parameters will still remain. Non-linear parameters in that sense are physical quantities which have a direct influence on the non-linear restoring force function ( , , )Rf u u t and which allow for the determination of equivalent parameters eqk

and eqc , provided the relative displacement amplitude u∆ is known. In this paper, a computational model updating procedure (parameter identification procedure) is presented, which allows for the identification of non-linear joint parameters by minimizing the differences between experimental and analytical non-linear FRFs. The experimental non-linear FRFs have to be obtained from step sine test with harmonic single point excitation (shaker excitation) with the force amplitude maintained at a constant level by means of a feedback loop controller. The analytical non-linear FRFs are essentially fundamental harmonic non-linear FRFs which can be computed from a non-linear frequency response analysis using the Harmonic Balance Method (in this paper, the harmonic balance response software HBResp was used together with system matrices imported from MSC.Nastran). For computational model updating, the classical substructure matrix approach is used for updating the system matrices, see [7]:

[ ] [ ]AM M= , (12a)

[ ] [ ] [ ]A r rr

C C Cα= +∑ , 1, ,r R= … , (12b)

[ ] [ ] [ ]A s ss

K K Kβ= +∑ , 1, ,s S= … . (12c)

Here, [ ]AM and [ ]AK are the mass and stiffness matrix of the underlying linear system which are generally

provided by commercial FE software. The pre-multiplication factors rα and sβ are the so-called correction parameters (updating parameter), which are adjusted during model updating. These model updating parameters are comprised in the parameter vector p :

1 1T

R Sp α α β β= . (13)

The so-called substructure matrices [ ] [ ]Ar

r

CC

α∂

=∂

and [ ] [ ]As

s

KK

β∂

=∂

define the type and the location of the

model error to be corrected during updating. In case of non-linear updating, these substructure matrices equal the

sensitivities of the non-linear element matrices ( )ˆeqK u and ( )ˆeqC u with respect to the updating

parameters p , see equations (10) and (11).

The underlying linear damping matrix [ ]AC is unknown in most cases and is constructed here from the expansion

of modal viscous damping ratios iξ previously identified from a modal survey test. This approach, however, assumes a diagonal modal damping matrix, see [7].

[ ] [ ] [ ]2 Ti iA A i i A

i i

C M X X Mξ ωµ

=

∑ . (14)

In this equation, iω denote the eigenfrequencies, iµ are the modal masses, and iX are the eigenvectors. These quantities are provided by commercial finite element software as a result of a numerical modal analysis of the underlying linear system. In case of large order finite element models, like the one discussed in the following, a transformation of the equation of motion to the modal domain is useful for order reduction of the equation of motion to be solved. In such cases, the construction of the underlying linear damping matrix [ ]AC in physical degrees of freedom is not necessary, since the modal viscous damping ratios can be used directly. However, it has to be noted, that the non-linear stiffness and damping terms of the equation of motion coming from the non-linear joint elements will generally introduce coupling terms in the modal domain equations, i.e. even though the linear modal equations of motion are uncoupled, the contribution of the non-linear joint elements appear as coupling terms. These coupling terms appear as off-diagonal terms in the modal matrices and can be significant in case of large vibration amplitudes. Model updating procedures seek for optimal numerical values for the correction parameters comprised in p which shall minimize the test/analysis deviations. This is achieved by minimizing an objective function which is constructed from a so-called residual term containing the weighted square sum of the test/analysis deviations and a so-called regularization term which plays an important role in case of an ill-conditioned sensitivity matrix:

[ ] ( ) ( )0 0

residual term regularisation term

minTT

pJ r W r p p W p p = + − − → . (15)

In equation (15) vector r is the residual vector containing the test/analysis deviations, [ ]W and pW are

weighting matrices, and 0p is a vector containing the initial values of the updating parameters. Minimizing the residual term of the objective function in equation (15) yields the desired values for the updating parameters, whereas the regularization term is a modification to the objective function which constrains the parameter variation such that the parameter changes are kept moderate, see [9-10].

The residual vector r contains the differences between test data comprised in tv and analysis data

comprised in av where the latter is a function of the updating parameters:

( ) t ar v v p= − . (16)

Since the relationship between residual vector and updating parameters is generally non-linear, the minimization of eq. (15) has to be performed iteratively by applying the classical sensitivity approach [7-9, 11-12], where the non-linear relation between the analysis data av and the updating parameters p at a certain linearization

point k is approximated by a Taylor series expansion truncated after the fundamental terms:

[ ] k kr r G p= − ∆ , (17)

where:

kp p p∆ = − (18)

denotes the parameter changes in each iteration, and

( ) k t a kr v v p= − (19)

denotes the residual vector at the linearization point k , and

[ ] k

ak

p p

vG

p=

∂=∂

(20)

is the sensitivity matrix containing the sensitivities of the analytical data with respect to the updating parameters at linearization point k . For linear models, the sensitivities can be calculated analytically, see for example [9] and [13]. In case of non-linear models a finite difference approach is used instead. If the parameter variation is unconstrained, i.e. when there are no specific limits for the updating parameters, the solution of eq. (15) can be expressed by eq. (21) which has to be solved in each iteration step of updating:

[ ] [ ][ ] [ ] [ ] T Tk k p k kG W G W p G W r + ∆ = , (21a)

[ ] [ ][ ] [ ] [ ] 1T T

k k p k kp G W G W G W r−

∆ = + . (21b)

In case of linear FE model updating, the residual vector r mostly contains eigenfrequency and mode shape deviations. In the non-linear model updating procedure presented here, the residual vector comprises the deviations between experimental and analytical non-linear frequency response functions (FRFs). This means that the test data vector tv contains n non-linear FRFs obtained from measurements, ,1( )th Ω to , ( )t nh Ω

(index t for test data):

,1 ,

,1 , ,1 1 ,1 , 1 ,( ) ( ) ( ) ( )T T

t t n

T TTt t t n t t r t n t n s

h h

v h h h h h h = = Ω Ω Ω Ω

, (22)

whereas the analytical data vector av contains n simulated non-linear FRFs ,1( )ah Ω to , ( )a nh Ω (index a

for analytical data):

,1 ,

,1 ,

,1 1 ,1 , 1 ,( ) ( ) ( ) ( )

k k

T Ta a n

k

T TTa a a n

p p p p

a a r a n a n s

h hp p

v h h

h h h h

= =

=

=

= Ω Ω Ω Ω

(23)

As can be seen from equations (22) and (23), the size of the residual vector r depends on the number of FRFs

used for model updating and on the number of frequency points iΩ per FRF. This results in a strongly over determined equation system for the identification of the updating parameters, see eq. (21b). The sensitivity matrix eq. (20) needed for the calculation of the parameter changes contains the partial derivatives of the analytical non-linear FRFs with respect to the updating parameters. This matrix may be subdivided into n p× partitions, where n denotes the number of FRF used for updating, and p denotes the number of updating parameters:

[ ]

11 12 1

21 22 2

1 2

p

pk

n n np

g g g

g g gG

g g g

=

(24)

Each partition ijg is the partial derivative of a certain analytical FRF ,a ih with respect to a certain updating

parameter jp and can be approximated by a finite difference approach. A closed form analytical solution for the sensitivity of non-linear FRFs does not yet exist to the knowledge of the authors. The performance and the capabilities of the presented sensitivity based model updating procedure for the identification of non-linear model parameters by using frequency response residuals is presented in the following.

Application of Non-linear Model Updating to Aero-Engine FE-Model One of the requirements posed on the non-linear updating procedure is its applicability to large order finite element models. In fact, the procedure was developed within the European Research project CERES together with the European aero-engine industry for non-linear joint identification of aero-engine casing structures. In this section, the non-linear updating procedure is checked for this intended application. For the identification of non-linear stiffness and damping parameters of bolted flange joints, step sine tests were conducted on an aero-engine subassembly. Figure 7 shows the experimental set up, where the different engine components of the overall structure can be seen: Fan Casing Stage 1 (FC1), Fan Casing Stage 2 (FC2), Front Bearing Housing (FBH), Intermediate Casing (IMC), Combustion Chamber Outer Casing (CCOC), High Pressure Turbine Casing (TC), and Rear Bearing Support Structure (RBSS, mounted inside the TC). The interfaces between the aero-engine components are bolted flange joints. The non-linear step sine tests were performed in a narrow frequency band around the resonance frequency of a global bending mode. FRFs for several different excitation force levels were obtained (1N, 10N, 20N, 30N, 40N, 50N, 60N, 70N), where the force amplitudes of each test level were maintained constant. Generally it can be concluded that in this case the bending modes of the aero-engine casing modes are influenced by the joint non-linearity, whereas other modes such as ovalizing shell modes remain almost unaffected.

Figure 7: Aero-engine structure with shaker attached supported by bungee cords

Figure 8 shows the distortions of the real part and the imaginary part of the experimental driving point FRFs. It can be seen that the resonance frequency is shifted towards lower frequencies with increasing excitation force levels whereas the damping decreases with increasing excitation force level.

Figure 8: Real- and imaginary part of driving point acceleration FRF for different excitation force levels

The distortion characteristic of the FRF shown in figure 8 can be observed not only at the driving point but at all other measurement positions as well, i.e. the local joint non-linearity produces a global non-linear behavior of the structure. The arrows plotted in figure 8 indicate the direction of the FRF distortion with increasing excitation force level. The response curves of the linear case were measured with identical experimental setup, however, the excitation force was set to a relatively low level which was not maintained constant. Figure 9 shows the finite element model (ca. 90,000 degrees of freedom) of the aero-engine subassembly shown in figure 7 and the measurement positions (1-7) used in the non-linear step sine tests. In addition, the bending mode which was specifically excited in the tests is shown there. The distribution of the measurement positions allow for rough observations of possible changes of that bending mode due to the non-linearity. The visualization of the bending mode shown in figure 9 reveals that the bolted flange joint between the engine components IMC and CCOC is highly loaded whereas the other joints are loaded much less. Thus, it is assumed in the following that the non-linear effects observed in the test data were produced by this interface. Even though each joint is modeled in a flexible way according to figure 2, only the highly loaded IMC-CCOC interface is supplemented with 48 non-linear spring/damper elements equally spaced on the circumference of the joint.

FC1

FC2

FBH

IMC

CCOC

TC

Shaker

Figure 9: FE model and bending mode (ca. 345 Hz) of the aero-engine structure

Since bolted flange joints may have different stiffnesses in tension and compression conditions (assuming sufficiently high loading), a bilinear stiffness was selected as an appropriate model for the 48 non-linear springs. In order to represent the decreasing damping observed in the test data, a softening quadratic damper was selected as the non-linear damping model for the 48 non-linear damper elements. The negative damping value of the softening quadratic damper gradually decreases the initial linear modal damping ratios with increasing vibration levels. The non-linear restoring force functions and their mathematical expression are shown in figure 10.

Figure 10: Non-linear restoring force functions

Once the location and an appropriate model for the non-linearity were selected, the non-linear parameters (in this case 1k , 2k , cu , and nlc ) describing the non-linear restoring force functions of the 48 non-linear spring/damper elements have to be determined. It has to be noted that the modal viscous damping ratios of the mode shapes were identified from modal survey tests and that the underlying linear joint stiffness was identified by applying a linear model updating procedure prior to non-linear modeling. Thus, the number of unknown non-linear updating parameters can be reduced by the parameter 1k which is the underlying linear stiffness of the bilinear springs.

Three non-linear parameters remain to be identified, which are 2T

c nlp k u c= . Model updating procedures can only yield reasonable results if the initial model does not deviate too much from the real structure. This requirement involves the necessity to determine meaningful initial values for the updating parameters. “Trial-and-Error” is always a possibility to find such initial values, however, in order to reproduce the distortion characteristic of the frequency response functions with a non-linear model, a large number of successive non-linear response analyses may be necessary to find appropriate initial parameter values. A more scientific approach for the rough adjustment of the updating parameters is discussed in [14]. This approach is based on the comparison of linearity plots (or modal characterization functions) obtained from test data and from a non-linear model. This procedure was also applied here for the determination of initial parameter

Rf

ucu1k

2kRf

u

( )1

2 1

,,

cR

c c c

k u u uf

k u u k u u u≤

= − + >, 0R nl nlf c u u c= <

1 (Drive Point) 2

3 4

6 5

7 highly loaded bolted

flange joint

values and shall not be discussed in detail. The initial values of the non-linear parameters 0p which were found by the application of the procedure for the rough parameter adjustment present here are shown in Table 1.

Results of Non-linear Model Updating After the initial numerical values 0p for the updating parameters were found, these values can be improved using the non-linear updating procedure by minimizing the differences between experimental and analytical non-linear FRFs. Therefore, the following three parameters are defined as updating parameters which are allowed to vary during updating, see figure 10:

parameter 1: tension regime stiffness 2k of bilinear spring

parameter 2: pre-load factor cu of bilinear spring (transition point)

parameter 3: damping factor nlc of quadratic damper Figure 11 shows the evolution of these parameters and of the normalized objective function during 10 iteration steps of updating. In this case, the objective function was comprised of non-linear FRFs of two excitation force levels (20N and 70N) at the measurement positions 1, 2, 3, and 4 (see figure 9). The FRFs of the other excitation force levels may be used for the validation of the model updating results. A weak regularization was used which results in a smooth parameter evolution. Table 1 comprises the initial values, the final values, and the changes of the model updating parameters.

Figure 11: Evolution of non-linear updating parameters and normalized objective function

during 10 iteration steps

The identified final values of the non-linear parameters were used to calculate the non-linear FRFs for all excitation force levels for which experimental data is available. A comparison of the analytical non-linear FRFs with the experimental ones can then be used to check the prediction capability of the finite element model with the non-linear joint. The FRF comparisons are comprised in figures 12 to 15 and it can be concluded that the chosen model for the non-linear stiffness and damping properties of the bolted flange joint is well suited to reproduce the distortion characteristic of the experimental non-linear FRFs. This is also true for excitation force levels for which the FRFs were not contained in the objective function. Instead of using the identified parameters for non-linear frequency response analysis it is also possible to use them in the time domain by introducing the corresponding non-linear elements into an underlying linear model for non-linear transient analysis. This was shown in the cantilever beam example of figure 3.

Table 1: Initial values, final values and changes of updating parameters

Parameter Initial Value Final Value Change [%]

1k 52.253 10 Nmm

⋅ 52.253 10 Nmm

⋅ --

2k 47.129 10 Nmm

⋅ 48.134 10 Nmm

⋅ 14.1%+

cu 54.188 10 mm−⋅ 56.436 10 mm−⋅ 53.7%+

nlc 2

25.107 Nsmm

− 2

27.063 Nsmm

− 38.3%+

Figure 12: Comparison of analytical (left) and experimental (right) non-linear FRFs at measurement position 1

Figure 13: Comparison of analytical (left) and experimental (right) non-linear FRFs at measurement position 2

Figure 14: Comparison of analytical (left) and experimental (right) non-linear FRFs at measurement position 3

Figure 15: Comparison of analytical (left) and experimental (right) non-linear FRFs at measurement position 4

Conclusions and Outlook In this paper, the non-linear modeling of bolted flange joints is discussed together with the mathematics involved in the derivation of equivalent stiffness and damping properties using the Harmonic Balance Method. This provides the basis for non-linear frequency response analysis. A procedure is proposed for the determination of meaningful initial parameter values for non-linear spring/damper elements and for the improvement of these parameters by minimizing frequency response deviations. The performance of this procedure was demonstrated on a large order finite element model of an aero-engine structure together with non-linear FRFs obtained from step sine tests with constant excitation force levels. Even though after model updating the magnitudes of the resonance peaks of the analytical non-linear FRFs shown in figures 12 to 15 do not match exactly the experimental non-linear FRFs for all measurement points, the results are nonetheless promising, especially when taking into account that without non-linear modeling a prediction of vibration amplitude dependent changes of FRFs (resonance frequency shift and change of response magnitude) is not possible at all when using standard tools of commercial finite element software. From a physical point of view it is unconfident, that the identified stiffness of the bilinear spring in the compression regime is only 2.77 times higher than the tension regime stiffness. This is inconsistent with the real stiffness properties which may occur at bolted flange joints. One possible explanation could be the range of the excitation force amplitudes used in the non-linear tests. The maximum force amplitude level was simply not high enough to significantly reduce the compressive pre-stress between the flange faces at the bolted flange joint under consideration, which may be the reason for the stiffness reduction being so small. Furthermore, it is interesting to see that there is a significant loss of damping with increasing excitation force levels. No physically meaningful reason for this damping mechanism was found yet such that the polynomial type of damping model can only be considered as an equivalent non-linear damping model. A possible extension of the procedure presented here could be the inclusion of the constant term 0a of the Fourier series in equation (4). Taking into account the uncertainty inherent in the non-linear parameters could also be a meaningful extension, because nominal identical bolted joints will always have slightly different stiffness and damping properties. This could be achieved by using a probabilistic approach or by using an approach based on fuzzy arithmetic.

Acknowledgement The work presented here was performed within the European Research project CERES (Cost-Effective Rotordynamics Engineering Solutions), funded by the European Commission under the Competitive and Sustainable Growth Programme (Framework V, Key Action 4, 1998-2002). The authors would like to thank the Imperial College of Science, Technology, and Medicine, London, and MTU, Munich, for providing test data.

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