Non destructive testing of cables using non linear vibrations:

simple o.d.e models, Duffing equation for one d.o.f.

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B. Rousselet^{†,∗},G. Vanderborck^{‡,∗} † UNSA Laboratoire J.A. Dieudonné

U.M.R. C.N.R.S. 6621, Parc Valrose, F 06108 Nice, Cedex 2

∗Email: br@math.unice.fr and vdb@math.unice.fr ‡ Thales Underwater systems, Département acoustique

BP 157 06903 sophia Antipolis Cedex and UNSA Laboratoire J.A. Dieudonné

U.M.R. C.N.R.S. 6621, Parc Valrose, F 06108 Nice, Cédex 2

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Contents

1 Introduction  1
1.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Transverse vibrations: vibrating masses on stretched cables in large displacement  2
2.1 Masses in transverse displacement . . . . . . . . . . . . . 4
2.1.1 Notations and assumptions . . . . . . . . . . . . . 4
3 Transverse vibration: approximate non dimensional equation for a small load  7
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 A possible damage of a cable . . . . . . . . . . . 13
4 Numerical and asymptotic solution of Duffing equation in the frequency domain  14
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.2 Peaks of the exact Fourier transform ofthe approximate solution u_{0} + ϵu_{1} 15
4.3 Data of the bridge . . . . . . . . . . . . . . . . . . . . . . 20

The strong linkage between damage and nonlinear elasticity of materials leads to many researches in the field of nonlinear response of mechanical structures . Due to severe mechanical environment conditions ( Vibration, Shock...), damage may appear in structures. In a previous S & V symposium we have presented some interesting experiments who have shown that non linear acoustic spectroscopy is a very powerful technique in order to detect damage in structures The proposed method consists of the observation of nonlinear responses obtained from an array of sensors when the structure is excited by dual frequency sources : the low frequency source yields the power while the second acts as a probe . The nonlinear response is obtained by spectral analysis of signals coming from sensors , the response amplitude is proportional to the local nonlinearity (failure nonlinear elastic response) . Now we have developed the theoretical and mathematical approach of the problem and we show a good correlation between the modeling and the experiment. We explain also the ideas that are behind the concept of localization of damage. We also show that this technique based on non linear acoustic is a good concept in order to detect buried objects.

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1 Introduction

1.1 Orientation

Although there is an active investigation of experiments for non destructive testing using non linear vibrations, there is a lack of models and numerical solutions to compare with the experimental results. Obviously to derive local models of the damage is a difficult task and solving partial differential equations with these local non linearities is a challenging task.

The importance of the parameters such as the level of the load and of the damagebut also the time interval are discussed below.

2 Transverse vibrations: vibrating masses on stretched cables in large displacement

As a very simplified model of the experiment described in [?], we consider n masses m_{i} attached to horizontal springs (or cables) which are submitted to a large tension T_{0}, at rest, see figure 1; the tension is positive when the cable is in traction which is assumed trough the vibration; the masses are moving (vertically) transversely to the position of the springs at rest; we denote by upper case letters quantities in the rest position and lower case in the current configuration, except for the tension T_{i} in the current configuration.

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Figure 1: Transverse vibrations

2.1 Masses in transverse displacement

2.1.1 Notations and assumptions

In a linear model of transverse vibration, it is usually assumed that the tension is so large at rest that it remains constant during the vibration. Here, due to large displacements the tension does depend on the change of length, we assume a linear elastic relation between tension and change of length. See the figure 1 with two masses and 3 cables.

• L_{i} length at rest; l_{i} length at time t; as the masses are moving perpendicularly to the position of the cables at rest:

l_{i}^{2} = L_{i}^{2} + (y_{i} − y_{i−1})^{2}

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• the change of tension of the linear elastic spring due to the change of of length T_{i} = T_{0} + k_{i}[l_{i}(y) − L_{i}] = T_{0} + k_{i}(\sqrt{L_{i}^{2} + (y_{i} − y_{i−1})^{2}} − L_{i}); this tension is directed along the axis of the spring and positively oriented from m_{i} to m_{i+1}.

• Denote by θ_{i}, the angle of the i^{th} spring with the horizontal axis; at rest θ_{i} = 0 we have

y_{1} = l_{1}(y)sin(θ_{1}), y_{i}−y_{i−1} = l_{i}(y)sin(θ_{i}) −y_{n−1} = l_{n}(y)sin(θ_{n})

with:

l_{i} = L_{i} (1 + \frac{(y_{i} − y_{i−1})^{2}}{L_{i}^{2}} )^{\frac{1}{2}}

We enforce y_{0} = 0 and y_{n} = 0.

Using Newton law, the equations of dynamics are:

m_{i}y_{i}” = −T_{i}sin(θ_{i}) + T_{i+1}sin(θ_{i+1}) + F_{i} i = 1 … n  (2.1)

where −T_{i}sin(θ_{i}) + T_{i+1}sin(θ_{i+1}) is the vertical component of the force due to cables acting on mass i; we assume no longitudinal movement so that the longitudinal component of the force does not work. The applied load on mass i is denoted by F_{i}.

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3 Transverse vibration: approximate non dimensional equation for a small load

3.1 Introduction

In order to find precisely the appearance and the behavior of the intermodulations, an asymptotic expansion of the solution with respect to a small parameter is usefull; to determine a small parameter, it is convenient to use a non dimensional equation which is now derived.

It is not obvious to prescribe the right data to obtain clear inter-modulation peaks; this is also a real trouble of the experiments!.

So it is very useful to obtain an asymptotic expansion of the solution in order to determine rough windows of the data for which inter-modulation lobes are non negligible and to relate the level of the inter-modulation lobes to the level of the damage; see subsection 4.2

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For one d.o.f., the linearized equation is:

m_{1}ÿ_{1} = −T_{0}(\frac{1}{L_{1}} + \frac{1}{L_{2}} )y_{1} + F̃_{1}cos(α̃t̃)  (3.1)

Approximate equation

m_{1}y_{1}” = −T_{0}(\frac{1}{L_{1}} + \frac{1}{L_{2}} )y_{1} + \frac{λ̃^{2}}{2} (\frac{T_{0} − k_{1}L_{1}}{L_{1}^{3}} + \frac{T_{0} − k_{2}L_{2}}{L_{2}^{3}} ) y_{1}^{3}+

F_{1}cos(α̃t̃) + O(∥y_{1}∥^{3}) (3.2)

Normalization We normalize time t̃

ω^{∗} = (\frac{T_{0}(\frac{1}{L_{1}} + \frac{1}{L_{2}} )}{m})^{\frac{1}{2}} , t̃= \frac{t}{ω^{∗}} , α = \frac{α̃}{ω^{∗}}  (3.3)

and displacement with the total lenth at rest: And the differential equation may be approximated by a so-called Duffing equation:

\frac{d^{2}u}{dt^{2}} = −u + ϵu^{3} + F cos(αt) + …  (3.4)

with the small parameter:

ϵ = \frac{λ̃^{2}L_{tot}^{2}}{2mω^{∗2}} (\frac{T_{0} − k_{1}L_{1}}{L_{1}^{3}} + \frac{T_{0} − k_{2}L_{2}}{L_{2}^{3}} )

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Figure 2: Up: one damaged cable and 2 undamaged cables ; down: 3 undamaged cables

3.1.1 A possible damage of a cable

Here a simple breakage model of several fibers of a cable is addressed, this causes decrease of rigidity k_{1} say for cable 1; internal contacts are not taken into account.

4 Numerical and asymptotic solution of Duffing equation in the frequency domain

4.1 Introduction

One of the main issues of the use of the inter-modulation lobes for non destructive testing is their level relative to the main lobe located at the load frequency α/2π.

The choice of the data of the system in order to be able to observe the inter-modulation lobes is not obvious numerically. The use of the double scale approximation is a valuable tool in order to design a system. However, in the design process, it should be used cautiously because the generated data fall easily out of its domain of validity.

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4.2 Peaks of the exact Fourier transform of the approximate solution u_{0} + ϵu_{1}

As

F(cos(2πiat)χ_{[0,A]}) = \frac{A}{2}(τ_{a}e^{−iπAν}sinc(πνA)+τ_{−a}e^{−iπAν}sinc(πνA))

with sinus cardinal (or sampling function)

sinc(t) = \frac{sin(t)}{t}

using section ??, the level of the amplitude of the Fourier transform of each term of the expansion u = u_{0} + ϵu_{1} is neglecting (1 + 2.α + ϵ.β_{1})... terms

angular frequency ↦ level of the peak  (4.1)
1 + ϵβ_{1} ↦ \frac{t_{max}}{2} a  (4.2)
α ↦ \frac{t_{max}}{2} (−ϕ + ϵ\frac{3.ϕ.(2.a^{2} + ϕ^{2})}{4(1 − α^{2})})  (4.3)
3.(1 + ϵ.β_{1}) ↦ ϵ\frac{t_{max}}{2} \frac{a^{3}}{32}  (4.4)
(−2 + α − 2.ϵ.β_{1}) ↦ ϵ\frac{t_{max}}{2} \frac{(3.a^{2}.ϕ/4)}{(1 − (−2 + α)^{2})}  (4.5)
(2 + α + 2.ϵ.β_{1}) ↦ ϵ\frac{t_{max}}{2} \frac{(3.a^{2}.ϕ/4)}{(1 − (2 + α)^{2})}  (4.6)
3.α ↦ ϵ\frac{t_{max}}{2} \frac{ϕ^{3}}{(4.(1 − 9.α^{2}))}  (4.7)

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Proposition 4.1. From this table, it is clear that the level of the intermodulation lobes depends linearly on ϵ; the slope itself depends on the value t_{max} of the final time, on the level of ϕ = \frac{−F}{1−α^{2}} related to the applied load , and on the level of the coeeficient a, related to the sum of initial position and of the applied load.

Figure 3: tm=786 dt =0.003,

4.3 Data of the bridge

The following data are those used in the experiments reported in [?].

General data The total lenth of the stressed cable, young modulus and the area of the cross section:

L_{tot} = 165m, E = 2 ∗ 10^{11}N/m^{2}, A = 4 ∗ 10^{−3}m^{2}

linear mass is ρ = 43.8kg/m; the total mass is 3615.5kg; it is taken as the same for the damaged cable.

For the safe (undamaged) cable: The measured tension in the safe

cable:

T_{0} = 2.794 ∗ 10^{6}N,

from the usual stress-strain relation for bars, the relative elongation or strain between the unstressed ans prestressed configurations: S_{stot} = T_{0}/(E ∗ A) = 0.0035;

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The 1 d.o.f. model provides a good approximation of the fundamental frequency .

[The damaged cable] The measured tension is

T_{0dam} = 1.919 ∗ 10^{6}N;

is assumed to be made of a safe part and a damaged one; the strain for the safe part S_{safe} = T_{0dam}/(E ∗ A) = 0.002375; The coefficients ϵ of formula (10) involved in Duffing equation:

ϵ_{safe} = −572.66 ϵ_{dam} = −635.58.

We notice a substantial increase of ϵ due to decrease of rigidity

of one cable; this increase causes an increase of the intermodulation lobes as noted in subsection 4.2

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For a much larger load F = 0.0378982, the intermodulations are quite visible. However, it is acknowledge that it corresponds to an experimental load of

F_{exp} = mL_{tot}ω^{∗2} = 7676009N

To found a shaker wchich can produce such a large load is quite impossible; moreover it should should produce a spectrally pure load, i.e. with only two frequencies and this point obliges to use the shaker with a limited load.

Finally, we show numerical evidence that if we apply the load at a frequency close to the eigenfrequency of the structure, the experimental load of 489N is high enough in order to show intermodulations; the asymptotic expansions provided in this article are not valid with this type of frequency ; it will be considered in a forthcoming paper.

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Figure 4: tm=2621, alpha=1.06, F=0.00006 eps=-635,

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