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# Non destructive testing using non linear vibroacoustic

Bernard Rousselet, Gérard Vanderborck

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Bernard Rousselet, Gérard Vanderborck. Non destructive testing using non linear vibroacoustic. 7e colloque national en calcul des structures, CSMA, May 2005, Giens, France. hal-01813014

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Submitted on 12 Jun 2018

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Non destructive testing
using non linear vibroacoustic

B. Rousselet*, — G. Vanderborck**

* U.N.S.A. Laboratoire J.A. Dieudonné

U.M.R. C.N.R.S. 6621, Parc Valrose, F 06108 Nice, Cédex 2, email : br@math.unice.fr

** Thales Underwater Systems, Département acoustique
06903 Sophia-Antipolis CEDEX
and Laboratoire J.A. Dieudonné
U.M.R. C.N.R.S. 6621

ABSTRACT. Several recent experimental studies show that it is possible to detect defects in a structure by considering its vibro-acoustic response to an external actuation. On this topic there is a vast literature in applied physics. We recall some papers related to the use of the frequency response for non destructive testing; in particular generation of higher harmonics, cross-modulation of a high frequency by a low frequency: We intend to present simple spring mass models, simple bar and beam models with damage and use **asymptotic expansions and numerical methods** to try to get results which show some similarity with the experiments of [P.D 03]. Asymptotic expansions have been used for at least a century and for example has been used recently for numerical approximation of bifurcation of structures in PotierFerry-Cochelin and coworkers (1993) [E.H 93]. **The key idea is to look at the solution in the frequency domain for the experiments and consequently for the numerics.**

RÉSUMÉ.

KEYWORDS: Non destructive testing, non linear vibroacoustic

MOTS-CLÉS : Essais non destructifs, vibroacoustique non linÃ©aire
# 1. Some previous papers

- In Ekimov-Didenkulov-Kasakov (1999), [A.E 99],

- In Zaitsev-Sas (1999), [V.Z 99],

- Other results may be found in Sedunov-Tsionsky-Donskoy (2002) [D.D 02], Sutin-Donskoy (1998), [A.S 98], Moussatov-Castagnede-Gusev (2002), [GUS 02] ...

- GDR 2501 (Etude de la propagation ultrasonore en milieux inhomogènes en vue du contrôle non destructif)

- In Vanderborck-Lagier-Groby (2003) [P.D 03], "a vibro-acoustic method, based on frequency modulation, is developed in order to detect defects on aluminum and concrete beams.

## Very recent experiments

- have been performed on a real bridge by G. Vanderborck with four prestressed cables: two undamaged cables, a damaged cable and a safe one but damaged at the anchor;

- See Vanderborck-Lagier (2004) [VAN] for a presentation of the results of the experiment with a new post processing graphic presentation of experimental results.

## Difficulties of the experiments:

- non linearities of the shakers (including piezoelectric actuators)

- Natural non linearities: supports, links of complex multi structures as air planes, bridges etc

# 2. Simplest mechanical example

in which we can exhibit intermodulations. We consider a 1 d.o.f example of a spring mass system with a non linear spring.

$$ \ddot{y} = -\omega_0^2 y - \frac{k_3}{m} y^3 + \epsilon \frac{F}{m} \sin(\alpha t) \quad \text{with } \omega_0^2 = \frac{k}{m} \qquad (2.1) $$

$$ \text{with initial conditions } y(0) = \epsilon\eta_1, \quad \dot{y} = \epsilon v_1 \qquad (2.2) $$

We are going to solve this equation **symbolically** with an asymptotic expansion with respect to $\epsilon$: $y = \epsilon y_1 + \epsilon^2 y_2 + \epsilon^3 y_3 + ...$; then **numerically**...

## The linear case

The first term is solution of:

$$ \ddot{y}_1 = -\omega_0^2 y_1 + \frac{F}{m} \sin(\alpha t) \quad \text{with } y(0) = \eta_1, \quad \dot{y} = v_1 \text{ which gives:} \qquad (2.3) $$

$$ \text{with } \phi = \frac{F}{m(\alpha^2 - \omega_0^2)}, \quad y_1 = \eta_1 \cos(\omega_0 t) + \left(\frac{v_1}{\omega_0} + \frac{\alpha}{\omega_0}\right) \sin(\omega_0 t) - \phi \sin(\alpha t) \qquad (2.4) $$

**Remark 1.** If we set $\eta_1 = 0$, $v_1 = 0$, then the term of pulsation $\omega_0$ has magnitude $\frac{\alpha}{\omega_0}$ times the magnitude of the term of pulsation $\alpha$; this is not a good choice for the non linear case in which $\frac{\alpha}{\omega_0}$ is of order 100; it seems good choice $\eta_1 = 0$, $v_1 = \omega_0(-\frac{\alpha}{\omega_0} + 1) \frac{F}{m(\alpha^2 - \omega_0^2)} = (-\alpha + \omega_0)\phi$
Other terms

The term $y_2$ is zero but the **third term** satisfies:

$$ \ddot{y}_3 = -\omega_0^2 y_3 + \frac{k_1}{m} y_1^3 \qquad (2.5) $$

to simplify, we assume $\eta_1 = 0$ and set $\phi = \frac{F}{m(\alpha^2 - \omega_0^2)}$, $\psi = \frac{v_1}{\omega_0} + \frac{\alpha\phi}{\omega_0}$ so that:

$$ y_1^3 = (\psi\sin(\omega_0 t) - \phi\sin(\alpha t))^3 = \qquad (2.6) $$

$$ -\frac{1}{4}\psi^3\sin(3\omega_0 t) + \frac{3}{2}\psi(1/4\psi^2 + \phi^2)\sin(\omega_0 t) \qquad (2.7) $$

$$ -\frac{3}{4}\psi\phi^2\sin((\omega_0 + 2\alpha)t) - \frac{3}{4}\psi\phi^2\sin((\omega_0 - 2\alpha)t) \qquad (2.8) $$

$$ +\frac{3}{4}\psi^2\phi\sin((\alpha + 2\omega_0)t) + \frac{3}{4}\psi^2\phi\sin((\alpha - 2\omega_0)t) + \qquad (2.9) $$

$$ -\frac{3}{2}(\psi^2\phi + \frac{1}{2}\phi^3)\sin(\alpha t) + \frac{1}{4}\phi^3\sin(3\alpha t) \qquad (2.10) $$

If we go on in the expansion, we get terms of pulsation $\alpha + 4\omega_0$, $\alpha + 6\omega_0$ etc

General tendency:

- The pick of $3\omega_0$ is much larger than the pick in $\alpha \pm 2\omega_0$ which are the most natural picks in the experiments;

it is delicate to find datas such that the secondary picks at $\alpha \pm 2\omega_0$

actually appear when the differential equation is solved numerically.

- Question: algorithm and software for detecting the secondary picks?

- then find (by optimization) datas such that the secondary picks are important: criteria for damage.

### 3. Transverse vibrations: vibrating masses on streched cables in large displacement

Work performed with Theissen (doctoral student of U. Muenster); Erasmus students N. Goris and I. Altrogge worked on this topic during their stay in UNSA (2004-2005). We consider n masses attached to horizontal springs (or cables) which are in tension $T_0$, at rest ; the tension is positive when the cable is in traction which is assumed; at rest the mass $m_i$ is submited to the force $T$ the masses are moving (vertically) transversely to the springs; we denote by upper case letters quantities in the rest position and lower case in the current configuration.

#### 3.1. Masses in vertical displacement

Here we assume that the masses can move only vertically.

##### 3.1.1. A possible damage of a cable

is breakage of several fibers, this will cause decrease of rigidity $k_1$ say for cable 1.
<figure><img src="image_2.png" /></figure>

– Let us start with undamaged cables of same rigidity k. If we note $L_0$, the common length of the unstressed cables, and L their common stressed length, their tension is $T_0 = k(L - L_0)$;

– now, after damage, $k_1 < k = k_2$, cable 1 becomes longer and cable 2 shorter, $L_1 > L_2$, the tension goes down to $T_{00} = k_1(L_1 - L_0) = k_2(L_2 - L_0)$;

– note the limit case of cable 1 broken is $k_1 = 0$ so that the cable 2 gets length $L_0$ but the system is no longer working properly!

– Before such a breakdown, if the change of tension is substantial, this causes a substantial change of the fundamental frequency; indeed, this is the routine monitoring of cable bridges!

– The nonlinear vibroacoustic testing aims at monitoring the cables before such a substantial change.

## 4. A non linear string model

A model of non linear string has been introduced first by Kirchoff in 1877 and rederived by Carrier in 1945.

$$y_{tt} - T\left(\int_{0}^{t} y_{x}^{2}\right)y_{xx} = f \quad (4.1)$$

Several mathematical studies of this type of equations have been performed recently (Medeiros (1994), Clark- Lima (1997).
<figure><img src="image_3.png" /><figcaption>Two masses on stretched cables (cable 2 damaged) moving freely</figcaption></figure>

## 5. Masses free to move in a plane

Equation of the dynamics

$$m_i x_i'' = -T_i \cos(\theta_i) + T_{i+1} \cos(\theta_{i+1}) + f_i \quad i = 1 \dots n \quad (5.1)$$

$$m_i y_i'' = -T_i \sin(\theta_i) + T_{i+1} \sin(\theta_{i+1}) + g_i \quad i = 1 \dots n \quad (5.2)$$

## 6. Bar models with defects

Bar models with longitudinal waves (dynamical traction and compression) are considered.

$$\rho \frac{\partial^2 u}{\partial t^2} - \frac{\partial n}{\partial x} = f(x, t) \quad (6.1)$$

With a non linear stress-strain law:

$$n = E \left( A \frac{\partial u}{\partial x} + \epsilon \chi_{[a,b]} \left( \frac{\partial u}{\partial x} \right)^3 \right) \quad (6.2)$$

We could as well assume that the applied load is of order epsilon without any assumption on the nonlinearity. Assuming $\epsilon$ to be small an approximate solution is searched for with the following "ansatz": $u = u_0 + \epsilon u_1 + \dots$

Theoretical justification of the expansions: the situation is complex in full generality: non linear hyperbolic equations exhibit a singularity after a finite time! But: the experiments are performed during a short time interval and the Fourier transforms are computed on these time
intervals! Following a suggestion of Guy Metivier we are addressing the problem during a
small initial time interval in which the solution is smooth: plan to use an approximation of
the equation with a fixed point method proposed in Majda. In any case we should smooth the
characteristic function (the material is changing smoothly)!

7. Conclusion

- Some simple models governed by ODE pr PDE show intermodulations;

- But what is the relative level of secondary peaks for a given set of datas deserves inves-
tigations: indeed it is also the difficulty of the real experiments

- Need to include other behaviors: shocks, friction

- Need of more precise models: non linear beams including tractional, flexural, torsional effects

– Mixture of local models for the defect and global models for the undamaged structure to obtain precise results at low computational cost.

8. References

[A.E 99] A.E.EKIMOV I.N.DIDENKULOV V., "Modulation of torsional waves in a rod with a crack", *J.Acoust. Soc. AM.*, vol. 3, num. 106, 1999, p. 1289-1291.

[A.S 98] A. SUTIN D. D., "Nonlinear vibro-acoustic nondestructive testing technique", *Non-destructive characterisation of material, 7 Ed R.E. green*, Plenum press, New York, 1998.

[D.D 02] D. DONSKOY A. EKIMOV N. S. T., "Nonlinear seismo-acoustic land mine detection and discrimination", *Acoust. Soc. Am*, vol. 111, 2002, p. 2705-2714.

[E.H 93] E.H. BOUTYOUR B. COCHELIN N. D. M. P.-F., "Calculs non linéaires par des méthodes asymptotiques-numériques: applications aux structures élastiques", *Colloque national en calcul de structures, 11-14 mai 1993*, Hermes, 1993.

[GUS 02] GUSEV A. M.-B. C.-V., "Frequency up-conversion and frequency down-conversion of acoustic waves in damaged materials", *Physics letter A*, vol. 301, 2002, p. 281-290.

[P.D 03] P. DUFOURCQ JP. GROBY M. L. P. T. E. G. V., "Détection vibro-acoustique non linéaire d' endomagements dans une structure poutre", Communication au Congrès français de mécanique, septembre 2003.

[VAN] VANDERBORCK GERARD L. M., "Application of non-linear ultrasonic spectroscopy to health monitoring and damage detection in structures,. 38p.", *75th Shock and Vibration Symposium, Virginia Beach (VA) USA, du 18/10/2004 au 21/10/2004*.

[V.Z 99] V. ZAITSEV P. S., "Nonlinear vibro-acoustic response of a metal sample with a dis-continuity like defect as related to damage detection problems", *Proceedings of DECT 99, Las Vegas, Nevada*, 1999.