assessing the anti-oscillator model parameters from ...

7th EUROMECH Solid Mechanics Conference J. Ambrosio et.al. (eds.) Lisbon, Portugal, 7–11 September 2009

ASSESSING THE ANTI-OSCILLATOR MODEL PARAMETERS FROM NUMERICAL OR EXPERIMENTAL DATA M. Massenzio1 , E. Jacquelin1 , A. Bennani1 , S. Ronel1 , T. Tran Dinh1 , J.P. Lain´e2 1 Universit´ e

de Lyon, F-69622, Lyon, France ; INRETS, UMR T9406, LBMC, Bron ; Universit´e Lyon 1, Villeurbanne. e-mail: [email protected] [email protected], {sylvie.ronel, abdelkrim.bennani}@iutb.univ-lyon1.fr 2

Universit´e de Lyon, F-69622, Lyon, France ; CNRS, UMR 5513, LTDS, Ecully; Ecole Centrale de Lyon, Ecully. e-mail: [email protected]

Keywords: Finite Element Model, Frequency Response Function, Impact, Structural behaviour. Abstract. There are typically two different approaches to model the dynamic behaviour of a structure. The finite element method (FEM) is very general and may be used to solve either static, modal analysis, stability or impact problems. Comprehensive results can be obtained with FEM, but the large number of degrees-of-freedom (dof) required leads to a very high numerical cost. Besides, the method provides a fine knowledge of the response but do not underlines the global comprehension of the structural behaviour. An alternative is to use spring-mass model involving one or at least a very few dof. Nevertheless, these models are not always suitable to describe efficiently the dynamic behaviour of the structure. A new approach has been proposed recently to predict the structural response for low velocity impact [2, 8], the so-called anti-oscillator (AO) method. The AOs are linked to the anti-resonant frequencies [3] and to the shapes associated with these frequencies. The AO model allows a fine prediction of the dynamic behaviour with a very few dof and therefore the numerical cost is low. So far, in order to identify the AO parameters, a first model was required. Particularly the AO characteristics are defined from the mass M and stiffness K operators (or matrices) which are commonly derived from a finite element model. This work shows how to derive the AO model parameters experimentally by measuring an apparent mass frequency response function (FRF) [9] so that the first model is avoided. In fact, it is shown that the AO characteristics and the apparent mass FRF are closely tied. Indeed, the AO mass are the residue of the partial fraction decomposition of this FRF, while AO frequencies are the poles. Consequently, the AO characteristics can easily be derived from an experiment which allow the determination of the apparent mass FRF. In the paper, the AOs are presented first. The determination of the AO parameters from an FRF without any model is then undertaken. Finally, some examples are given to show the efficiency and the simplicity of the method. 1

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

1

Introduction

The finite element method [1] is very general and may be used to solve different kind of problems such as static problems, modal analysis, stability analysis... Nevertheless such a modelling may lead to a lot of degrees-of-freedom (dof) whereas, sometimes, few dof are sufficient. A new approach has been proposed recently to transform a finite element model to an efficient model based on the anti-oscillators (AO) [2]. The anti-oscillators are linked to the antiresonant frequencies [3] and to the shapes associated with these frequencies. It was not new to use such an information to update a model [4], [5], [6] and then to improve the knowledge of a system; the antiresonant frequencies have been used for crack detections in beams as well [7]. Nevertheless, these studies did not use the antiresonances to directly model a structure until the work reported by Jacquelin et al. [2] which introduced the notion of anti-oscillators. An application of the AO model had been done by Pashah et al. in [8] to understand and forecast the behaviour of impacted structures. Nevertheless in [2], a first model of the structure is required to identify the AO characteristics. So, this first model must be updated carefully to derive a good AO formulation. This paper shows how to derive experimentally the parameters of this new model by measuring an apparent mass frequency response function (FRF) so that a first model is avoided. In the following, the anti-oscillators are presented first. Particularly the AO characteristics are defined from the mass M and stiffness K operators (or matrices) which are supposed to be known. The determination of the AO parameters from an FRF without any model is then undertaken. Finally, some examples are given to show the efficiency and the simplicity of the method. 2

The anti-oscillator model

The AO model is briefly described in this section: no proof is given; more information may be found in [2]. When an action effect acts on a structure at a point P in a given direction, the response changes if the load direction or the point location change. Consequently the “best” model of a structure depend on the location and the direction of the action effect. The AO formulation was proposed to consider that remark. Then the AO formulation is attached to a point P and to a given direction, i.e. to a given degree of freedom (dof) i0 of the studied structure. This description is particularly appropriate for impact problems, when the impact location and direction were known [8]. Nevertheless, this approach is more general and may be used with all kind of loads and, above all, it explains how the structure globally behaves when excited along a specific dof. 2.1

Static, constraint and residual modes

The static mode φst is the shape of the structure when a load Fst is applied, such that φst i0 is equal to unity (see Fig. 1). Note that for the statically indeterminate structures φst is a rigid body mode. The constraint modes φi , ωAR i are the eigenmodes of the structure submitted to an extra boundary condition (see Fig. 1): φ i i0 = 0 (1) Then the eigenfrequencies ωAR i are the antiresonant frequencies of the structure [3] which depend on i0 . 2

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

Figure 1: Static and constraint modes of a beam.

The residual mode φ0 is spanned by the static mode and by the constraint vectors and is such that the residual mode is orthogonal to each constraint vector with respect to the mass operator. Then φ0 verifies: φ0 = φst −

+∞ X

ci φi

(2)

i=1

The coefficients ci may easily be determined, owing to the orthogonal property between φ0 and the constraint modes {φi }i=1..+∞ . The static and the residual modes were introduced for two main reasons: • when a structure is loaded by a quasi-static excitation in the direction of the dof i0 , the model must allow for the static stiffness: the AO model allows the determination of such a static stiffness thanks to the static mode; • when a structure is loaded by an excitation, which is dynamic in nature, in the direction of the dof i0 , the inertia effects act against the action effect to prevent the motion of the dof i0 ; so, at least during the very beginning of the impact, the inertia effects act as an extra boundary condition. Therefore the displacement should be well described by the constraint modes. The idea was then to expand the solution in terms of the static and the constraint modes. Unfortunately the static mode can not be orthogonal to the constraint modes. So, the displacement X was expanded in terms of the residual and the constraint modes i.e. in terms of a sum of orthogonal functions with respect to the mass operator: X(t) =

+∞ X

qi (t) φi

(3)

i=0

A discretization was achieved by truncating the expansion (3) at the rank N : d

X(t) ' X (t) =

N X i=0

3

qi (t) φi

(4)

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

2.2

The anti-oscillators

The discretized displacement (4) may also be written as: X d (t) = q0 (t) φ0 + = λ0 (t) ψ0 +

N X

qi (t) φi

i=1 N X

(λi (t) − λ0 (t)) ψi

(5)

i=1

where

λ0 (t) = q0 (t) ψ0 = φst qi (t) (6) ψi = ci φi λi (t) = ci Actually, this change of variable was very interesting because it led to represent any structure by the model depicted in Fig. 2: N single-degree-of-freedom systems (mi , ki )i=1..N lying on a single-degree-of-freedom system (m0 , k0 ). The degrees of freedom were then the N + 1 parameters {λi }i=0..N . The single-degree-of-freedom systems (mi , ki )i=1..N were referred to be “anti-oscillators” [2]. In fact Eq. (5) indicates that λ0 is equal to the dof of interest, i0 due to the definition of the ψi functions.

λ (t)

λ (t)

61

6N

m1

mN

k1

λ0 (t) 6

kN

m0

k0

Figure 2: Model of a structure impacted by another structure

In [2] it was shown that the masses represented in Fig. 2 may be expressed as: • for i > 0 : mi = M (φi , φi ) c2i = • for i = 0: m0 = M (φst , φst ) −

M (φst , φi )2 M (φi , φi )

N X M (φst , φi )2 i=1

4

M (φi , φi )

= mst −

(7)

N X i=1

mi

(8)

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

where M is the mass operator (ie the mass matrix for discretized system). Likewise, the stiffnesses {ki }i=0···N can be identified: • i = 0: k0 = K(ψ0 , ψ0 ) = K(φst , φst )

(9)

Then, - for statically determinate structures, k0 is the static stiffness. - for statically indeterminate structures (i.e. φst is a rigid body eigenshape), k0 = 0. • i = 1 · · · N: 2 ki = ωAR i mi

(10)

In the following the anti-oscillators will be sorted on the basis of their frequency so that the frequency increases with the ascending number of anti-oscillators. 3

The AO parameters and the apparent mass FRF

Ewins FRF [9] defined the apparent mass FRF as the inverse inertance. The point apparent mass FRF associated with the dof λ0 will be used in that paper. Considering a harmonic force applied in the direction of the displacement λ0 which is the th i0 dof: fh (t) = Fh ejωt (11) Each component λi (t) of the displacement given by Eq. (5) is harmonic as well and may be written as following: ∀i ≥ 0 λi (t) = Λi ejωt (12) By considering Fig. 2, the dof verify the following equations: • i > 0

¨ i (t) + ki (λi (t) − λ0 (t)) = 0 mi λ

(13)

− ω 2 mi Λi + ki (Λi − Λ0 ) = 0

(14)

Relation (11) leads to:

2 As ωAR i = ki /mi , this equation may be rewritten as:

Λ0 2 1 − ωω2

Λi =

(15)

AR i

• i = 0 ¨ 0 (t) + k0 λ0 (t) − m0 λ

N X

ki (λi (t) − λ0 (t)) = fh (t)

(16)

l=1

Equations (11) and (15) lead to: 2

− ω m0 Λ0 + k0 Λ0 −

N X l=1

5



ki 



1 1 −

ω2

2 ωAR i

− 1 Λ0 = Fh

(17)

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

Then the point dynamic stiffness FRF is: N X Fh − ω 2 mi = −ω 2 m0 + k0 + ω2 Λ0 l=1 1 − ω 2

(18)

AR i

The point apparent mass FRF Mapp , which is the requested FRF for this study, may be deduced: Mapp

N X Fh k0 Fh mi = = m0 − 2 + = ω2 2 Γ0 −ω Λ0 ω l=1 1 − ω 2

(19)

AR i

¨ 0 (t) = −ω 2 Λ0 ejωt = Γ0 ejωt where the acceleration is λ This result shows that it is possible to determine the AO parameters from experiments: • k0 is determined from a static test with a static force applied at P in the direction of the dof i0 (see Fig. 1); alternatively, k0 is the amount of the dynamic stiffness FRF when ω is equal to zero. • mst is determined from the Mapp FRF: mst = m0 +

N X

mi = Mapp (ω = 0) +

l=1

k0 ω2

(20)

Note that for the indeterminate structures, k0 = 0 and the “static” mass is the rigid body mass: mst = m0 +

N X

mi = Mapp (ω = 0)

(21)

l=1

• the antiresonant circular frequency ωAR i are the poles of the Mapp FRF; • mi is identified as the residue associated with the pole ωAR i of the Mapp FRF; • m0 is then deduced as: m0 = mst −

N X

mi

i=1

This method will be applied in the following to derive the AO parameters from experimental FRF. 4

Experimental identification of the AO characteristics

In this section a Pinned-Free beam and a Pinned-Pinned beam are studied. These examples will deal with both determinate and indeterminate structures. The acceleration was measured with a B&K 4393 accelerometer and a Nexus conditioner while the force was measured with a B&K 8200 transducer force associated with a B&K 2626 conditioner. A Siglab analyser was used to obtain the FRF (the dynamic apparent mass). The sampling frequency was 25.6 kHz.

6

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

E (GPa) 210

ρ (kg m−3 ) 7800

R (mm) L (mm) 6 500

Table 1: Properties of the P-F beam

4.1

Pinned-Free (P-F) beam

The studied beam was a cylinder (radius R, length L) made of steel. The pinned support is positionned 19 mm from one end; L is the length if the beam and R is the radius. The geometry and material characteristics are listed in Table 1. The OA were associated with a measurement point which was 410 mm from the axis of rotation. The experimental curve is given in Fig. 3: the peaks give the antiresonant frequencies and the zeros give the resonance frequencies. The anti-oscillator masses were identified as described previously. The anti-oscillator characteristics are given in Table 2. 1

10

0

Apparent Mass (kg)

10

−1

10

−2

10

−3

10

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency (Hz) Figure 3: P-F beam apparent Mass FRF modulus: experiment FRF (—–) and identified AO FRF (- - -)

AO number 1 2 mass mi (g) 15 42.9 frequency fAR (Hz) 173 455 mst (g) 185 −1 k0 (kNm ) 0

3 44.6 758

4 16.8 1296

5 7.5 2141

6 1.4 3286

7 1.1 4406

8 0.8 4463

Table 2: AO characteristics of the P-F beam

Equation (19) shows that, in that case, the apparent dynamic mass must have a horizontal asymptote when the frequency tends to zero that gives the “static mass” mst (actually mst is a rigid body mass). Indeed experimentally the horizontal asymptote was observed (see Fig. 4) and mst was identified (see Table 2). A direct evaluation of the rigid body mode led to mst = 186 g which is closed to the value identified and given in Table 2. 7

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

0.5

0.45

Apparent Mass (kg)

0.4

0.35

asymptote

0.3

0.25

0.2

0.15

0.1

0.05

0

10

20

30

40

50

60

70

80

90

100

Frequency (Hz)

Figure 4: P-F beam: zoom of the apparent Mass FRF (real part)

The identified apparent Mass FRF was plotted in Fig. 3. The figure shows that the identified AO parameters allowed good reconstruction of the experimental apparent Mass FRF. 4.2

Pinned-Pinned (P-P) beam

The studied beam was similar to the P-F beam except for the length (see the characteristics listed in Table 3). The OA parameters were associated with a measurement point located at the mid-length. In fact, this beam was not a pure simply supported beam which may be easily seen from the experimental eigenfrequencies. This is essentially due to vertical stiffness induced by the movement of the ball bearing assembly at the ends of the beam (see Fig. 5). Moreover, both bearing assemblies are not fully constrained to avoid the nonlinear effects due to tensile load: the beam can move in a groove made in the bearings. E (GPa) 210

ρ (kg m−3 ) 7800

R (mm) L (mm) 6 423

Table 3: Properties of the P-P beam

Figure 5: The Pinned-Pinned beam

8

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

The experimental curve is given in Fig. 6. The anti-oscillator masses were identified as described previously. The anti-oscillator characteristics are given in Table 4. 2

10

1

Apparent Mass (kg)

10

0

10

−1

10

−2

10

−3

10

−4

10

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency (Hz)

Figure 6: P-P beam apparent Mass FRF modulus: experiment FRF (—–) and identified AO FRF (- - -)

AO number 1 mass mi (g) 8.2 frequency fAR (Hz) 594 mst (g) 180 k0 (kNm−1 ) 800

2 3 2.92 0.76 1537 3308

Table 4: AO characteristics of the P-P beam

Equation (19) shows that, in that case, the apparent dynamic mass tends to infinity when the frequency tends to zero. This fact was experimentally observed as shown in Fig. 6. This makes the identification more difficult. A solution was to multiply Eq. (19) by ω 2 and to obtain the dynamic stiffness FRF: 

Kdyn (ω 2 ) = − k0 + ω 2  m0 +

N X l=1



mi  2 1 − ωω2

(22)

= − k0 + ω 2 mst

(23)

AR i

which had the following asymptote toward zero: Ã 2

Kasym (ω ) = − k0 + ω

2

m0 +

N X

!

mi

l=1

The slope of this asymptote is the static mass, and the absolute value of Kasym (ω 2 = 0) provides the static stiffness. These parameters have been identified from Fig. 7 and are given in Table 4. By considering the static mode φst , mst may be determined: 9

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

mst = µ

where φst (x) = −4 the mid-span section.

x L

¶3

Z L 0

ρ S φ2st (x) dx = 17/35 mbeam = 182 g µ

+ 3

(24)

¶

x for x ∈ [0, L/2] and φst is symmetric with respect to L

5

0.5

x 10

asymptote

−1

Apparent stiffness (N m )

0

−0.5

−1

−1.5

−2

−2.5

0

1

2

3

4

ω2 (rad2 s−2)

5

6 5

x 10

Figure 7: P-P beam apparent stiffness FRF (real part) and the asymptote (frequency range: [0-120] Hz)

This proves the accuracy of determining mst with the described method. The static stiffness can also be derived from mst and the first eigenfrequency f1 (f1 = 107 Hz): k0 = (2 π f1 )2 mst = 813kNm−1

(25)

The discrepancy with the value given in Table 4 is around 1 %. The identified apparent Mass FRF was plotted in Fig. 6 which shows that the identified AO parameters allowed good reconstruction of the experimental apparent Mass FRF up to 3300 Hz. To decrease the discrepancy beyond 3300 Hz, more AOs are required. They can be identified with the part of the FRF beyond 3300 Hz. 5

Conclusion

The anti-oscillator model defined in a previous paper [2] was associated to a first model of the studied structure. Accordingly the quality of that model depends on another model. A direct identification of the AO parameters was proposed in this paper to avoid a dependence on a model. This method required the experimental determination of a point apparent mass frequency response function which may be carried out easily. This point FRF is measured at the location and the direction associated with the required AO. This FRF is a rational function and the first step to get the AO characteristics is to identify that rational function. A partial fraction decomposition of the point apparent mass FRF is then achieved. The residues of the decomposition are the AO masses and the poles are the antiresonant frequencies. The only difficulties arose to identify the static stiffness and the static mass. For indeterminate structure, the static mass is the y-intercept of the FRF curve and it may be obtain by 10

M. Massenzio, E. Jacquelin, A. Bennani, S. Ronel, T. Tran Dinh, J.P. Lain´e

drawing the asymptote of the FRF when the frequency tends to zero. For determinate structure the apparent mass FRF has a singularity at zero. So, a solution to identify the static stiffness and the static mass is to use the dynamic stiffness FRF which has an asymptote when the frequency tends to zero. The slope of this line is the static mass while the static stiffness is the y-intercept. Consequently, with a basic experiment, it is possible to build easily an AO model. Particularly, it is easy to obtain the first AO characteristics which may play a great role in an impact problem as indicated in [8]. As the AO parameters may be experimentally and numerically obtained, it is then possible to have a new indicator about the accuracy of a model by comparing the AO parameters of both models. This should be explored in the future. REFERENCES [1] T.J.R. Hughes. The Finite Element Method. Dover Publication, 2000. [2] E. Jacquelin, J.P. Lain´e, A. Bennani, and M. Massenzio. A modelling of an impacted structure based on constraint modes. Journal of Sound and Vibration, 301 (3-5):789–802, April 2007. [3] F. Wahl, G. Schmidt, and L. Forrai. On the significance of antiresonance frequencies in experimental structural analysis. Journal of Sound and Vibration, 219 (3):379–394, 1999. [4] D.A. Rade and G. Lallement. A strategy for the enrichment of experimental data as applied to an inverse eigensensitivity-based FE model updating method. Mechanical Systems and Signal Processing, 12 (2):293–307, 1998. [5] J.E. Mottershead. On the zeros of structural frequency response functions and their sensitivities. Mechanical Systems and Signal Processing, 12 (5):591–597, 1998. [6] W. D’Ambrosio and A. Fregolent. The use of antiresonances for robust model updating. Journal of Sound and Vibration, 219 (3):379–394, 1999. [7] M. Dilena and A. Morassi. The use of antiresonances for crack detection in beams. Journal of Sound and Vibration, 276:195–214, 2004. [8] S. Pashah, M. Massenzio, and E. Jacquelin. Structural response of impacted structure described through anti-oscillators. International Journal of Impact Engineering, 35 (6):471– 486, 2008. [9] D.J. Ewins. Modal testing. Research Studies Press Ltd, 2000.

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