PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Selective identification of structural force distribution
Nicolas TOTARO1
1Univ Lyon, INSA-Lyon, Laboratoire Vibrations Acoustique, F-69621 Villeurbanne, FranceABSTRACT
On the one hand, in the field of structural source identification, some methods have been developed during
the last decades like the Force Analysis Technique, the Virtual Fields Method, the 2D spatial Fouriertransform or the structural holography. The proposed method clearly falls in the same framework and is closeto the philosophy behind the Virtual Fields Method.
On the other hand, in the field of acoustic source identification, the inverse Patch Transfer Function has been
recently proposed to reconstruct the acoustic fields on a complex shape source in any (stationary) acoustic
environment. This method is based on the concept of virtual acoustic volume. The method proposed here
tries to adapt this concept to structural identification.
Therefore, the proposed method is based on the use of a virtual testing structure solved by a Finite Element
solver. It allows selecting what kind of distribution force one wants to identify. In the example of a plate withwelded stiffeners, if one is only interested in localizing and quantifying the applied force, the virtual testing
structure has to include the stiffeners. If one is interested in forces due to the welded points, the virtual testing
structure has to exclude the stiffeners.
Keywords: inverse method, vibration, force distribution

1. INTRODUCTION
Based on the same concept of virtual domain as the recently proposed inverse Patch Transfer
Functions method (1) in acoustics, the present method (2) aims at reconstructing the force distribution
applied on a plate-like structure with any shape and any boundary conditions.
As the Force Analysis Technique (3), the Virtual Fields Method (4), the 2D spatial Fouriertransform or the structural holography (5), the proposed method is based on the equation of motion ofa thin plate. The concept of virtual structure introduced here is close to the virtual fields defined in(4). However, as mode shapes of the virtual structure are used to reconstruct the force distribution ina zone of the structure, it is easy to choose what to include in the model and so what to exclude of theforce distribution. The example of a razor-shaped structure with stiffeners excited by a point force isgiven. If the stiffeners are included in the virtual structure, only the point force appear in the identified
force distribution. If the stiffeners are not included then the force distribution exhibits some force
spots at rigid connections between the stiffeners and the structure. The proposed method is thus
considered as a selective structural source identification method.

2. THEORETICAL BACKGROUND
Let us consider a plate-like structure with any shape and boundary condition as shown in Figure 1.
A force distribution ‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻ is applied on the structure generating a displacement field ܹሺ‫ܯ‬ǡ ߱ሻ onthe structure. The displacement field respects the following equation of motion(1)െ݉߱ଶ ܹሺ‫ܯ‬ǡ ߱ሻ ൅ ‫ܦ‬ȟଶ ܹሺ‫ܯ‬ǡ ߱ሻ ൌ ‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻ
where ݉ and ‫ ܦ‬are respectively the surface mass and the bending stiffness of the plate and ߱ is theangular frequency of excitation. In the following, the objective is to identify the force distribution‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻ by measuring the transverse displacement ܹሺ‫ܯ‬ǡ ߱ሻ.

1Nicolas.totaro@insa-lyon.fr6285

Figure 1: Tested plate of unknown shape and boundary conditions excited by a force distribution ‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻ. A contour Ȟdelimits a surface ȳ on this plate.݊ሬԦ and ‫ݏ‬Ԧ are the normal and tangential vectors of contour Ȟ.

A contour Ȟ of arbitrary shape delimiting a zone ȳ on the surface of the structure is defined. As expressedin (6), the Green’s identity applied on the equation of motion (1) can be written asන ‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻߟሺ‫ܯ‬ሻ݀‫ ܯ‬ൌ න ൫െ݉߱ଶ ߟሺ‫ܯ‬ሻ ൅ ‫ܦ‬ȟଶ ߟሺ‫ܯ‬ሻ൯ܹሺ‫ܯ‬ǡ ߱ሻ݀‫ܯ‬ஐஐ൅ න ܹሺܳǡ ߱ሻ࣮൫ߟሺܳሻ൯ െ ߟሺܳሻ࣮൫ܹሺܳǡ ߱ሻ൯݀ܳ(2)୻െන୻߲ܹሺܳǡ ߱ሻ߲ߟሺܳሻࣧ௙ ൫ߟሺܳሻ൯ െࣧ௙ ൫ܹሺܳǡ ߱ሻ൯߲݀ܳ݊
߲݊where ߟሺ‫ܯ‬ሻ (resp. ߟሺܳሻ) is an arbitrary function (continuous and twice differentiable) defined on ȳ (resp.
Ȟ). ࣧ௙ ሺ‫ז‬ሻ and ࣮ሺ‫ז‬ሻ are respectively the bending moment and shear force operators (1).
In Equation (1), the terms ܹሺܳǡ ߱ሻ ,డௐሺொǡఠሻ
డ௡, ࣧ௙ ൫ܹሺܳǡ ߱ሻ൯ and ࣮൫ܹሺܳǡ ߱ሻ൯ are considered asmeasurable quantities. On the contrary, ߟሺܳሻ is testing function that can be chosen arbitrarily. To determine‫ܨ‬ሺ‫ܯ‬ǡ ߱ሻ in Equation (2), ߟሺܳሻ has been chosen as a mode shape of a virtual plate with clamped boundary
conditions on the contour Ȟ. The mode shapes then respect the following equationsെ݉߱௣ଶ ߶௣ ሺ‫ܯ‬ሻ ൅ ‫ܦ‬ȟଶ ߶௣ ሺ‫ܯ‬ሻ ൌ Ͳ   ‫ א ܯ‬ȳ

(3)߶௣ ሺܳሻ ൌ Ͳ  ܳ ‫ א‬Ȟ(4)߲߶௣ሺܳሻ ൌ Ͳ  ܳ ‫ א‬Ȟ߲݊(5)Considering Equations (3) to (5) in Equation (2), it yields‫ܨ‬௣ ሺ߱ሻ ൌ ݉൫߱௣ଶ െ ߱ଶ ൯ න ߶௣ ሺ‫ܯ‬ሻܹሺ‫ܯ‬ǡ ߱ሻ݀‫ܯ‬ஐ(6)߲ܹሺܳǡ ߱ሻ൅ න ܹሺܳǡ ߱ሻ࣮൫ߟሺܳሻ൯݀ܳ െ නࣧ௙ ൫ߟሺܳሻ൯߲݀ܳ݊
୻୻where ‫ܨ‬௣ ሺ߱ሻ are the modal masses of virtual mode ‫ ݌‬also defined as ۴‫ ܘ‬ൌ ‫ۯ‬۴ where ۴‫ ܘ‬is the vector ofmodal masses, A is the matrix of mode shapes ߶௣ and F is the force distribution on the degrees of freedom6286

in domain ȳ. Therefore, the modal masses can be estimated measuring displacement on surface ȳ andcontour Ȟ and rotationడௐሺொǡఠሻ
డ௡on contour Ȟ . Finally, as ߶௣ ሺ‫ܯ‬ሻ constitute an orthonormal basis offunctions, the force distribution is simply obtained by۴ ൌ ‫ ܂ۯ‬۴‫ܘ‬(7)

where ‫ ்ז‬stands for the transpose of a matrix.

3. NUMERICAL EXAMPLE
3.1 Structure under study
To illustrate the method proposed previously, a razor-shaped structure excited by a point force isinvestigated as presented in Figure 2. This structure is clamped on two edges and is pierced by two
circular holes of same dimensions. Two stiffeners are fixed on the plate by 10 rigid connections forthe longest one and by 6 for the shortest. This structure will be referred in the following as the “testedplate”. The objective is here to reconstruct the force distribution on the grey zone in Figure 2. To
achieve this result, two testing structures will be used. The first one, presented in Figure 2(b) doesn’t
have any stiffeners and is clamped on most of its edges. The second one, presented in Figure 2(c),exhibits the same stiffeners as those of the tested structure.

(a)

(b)

(c)

Figure 2: (a) Tested structure excited by a point force F; (b) virtual structure without stiffeners, clamped on some edges; (c)virtual structure with stiffeners, clamped on some edges

Mode shapes of the two testing plates are extracted with a commercial FE software and Equations (6) and(7) are applied to reconstruct the force distribution.
3.2 Results
The results are presented in Figure 3. Figure 3(a) presents the displacement field “measured” on
the tested structure on the surface ȳ . Figure 3(b) and Figure 3(c) present respectively the force
distribution obtained from mode shapes of the structure without stiffeners and from mode shapes ofthe structure with stiffeners.

6287

(a)

(b)

(c)

Figure 3: (a) displacement field (m) measured (numerical experiment) on the real structure in the zone of the virtual structure
at 2100Hz; (b) force distribution (N/m²) identified using the virtual structure without stiffeners (Figure 2 (b)); (c) force
distribution identified using the virtual structure with stiffeners (Figure 2 (c))

One can see in Figure 3 that some force spots appear in the first force distribution. They are due to the point
force itself but also to the forces implied by the rigid connections of the stiffeners. It is easy to evaluate herethat 4 rigid connections among the 16 generate non negligible force levels. In the second case, only one spot
due to the point force appears. The influence of rigid connections is canceled because they are taken into
account in the mode shapes of the virtual structure. Figure 3 demonstrates that the proposed method can act
as a selective structural source identification method.

4. CONCLUSIONSThe proposed method aims at reconstructing the force distribution applied on a plate-like structure.
This method is based on the concept of virtual structure. The mode shapes of this virtual structurewith clamped boundary conditions on the contour of the tested zone are used to reconstruct the forcedistribution in this zone.
Thanks to the use of this orthonormal basis of functions, no inversion is needed making theproposed method hardly sensitive to background noise.

ACKNOWLEDGEMENTS
This work was performed within the framework of the LABEX CeLyA (ANR -10-LABX-0060) ofUniversité de Lyon, within the program « Investissements d’Avenir » (ANR-16-IDEX-0005) operated

6288

by the French National Research Agency (ANR).

REFERENCES
1. Forget S, Totaro N, Guyader J-L, Schaeffer M. Source fields reconstruction with 3D mapping by meansof the virtual acoustic volume concept. J Sound Vib. 2016;381:48-64.
2. Totaro N. Selective Structural Source identification. J Sound Vib. 2018;420:114-128.
3. Leclère Q, Pézerat C. Vibration source identification using corrected finite difference schemes. J Sound
Vib. 2012;331:1366-1377.
4. Berry A, Robin O, Pierron F. Identification of dynamic loading on a bending plate using the Virtual Fields
Method. J Sound Vib. 2014;333:7151-7164.
5. Zhang Y, Adin Mann III J. Measuring the structural intensity and force distribution in plates. J Acoust Soc
Am. 1996;99:345-353.
6. Chesné S, Pézerat C, Guyader J-L. Identification of plate boundary forces from measured displacements.
J Vib Acoust. 2008;130:041006-1.

6289