Resonant frequencies computation

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which makes methods for an accurate and complete characterization of their vis-coelastic properties of great interest.

In this work, our objective is to describe some adaptations of the conventional RUS method, including the mentioned Bayesian formulation of the inverse prob-lem, for the application to attenuative materials. Section 3.2 describes the direct computation of the natural frequencies of a sample of given elasticity. Section 3.3 presents the samples for which the method is illustrated and the measurement of their spectra. The dedicated signal processing method used to retrieve the resonant frequencies of the overlapped peaks is detailed in section3.4. Inverse estimation of the elastic properties based on a Bayesian framework is presented in section3.5and the estimation of the damping properties in section 3.6. Results are presented and discussed in section 3.7 and section 3.8 respectively. Concluding remarks end the paper in section4.5.

3.2 Resonant frequencies computation

The Rayleigh-Ritz method is an efficient and accurate method to compute the nat-ural vibration frequencies of solids of regular shapes, including rectangular paral-lelepipededs (RP) (Ohno,1976; Maynard, 1992; Migliori et al.,1993; Migliori and Sarrao, 1997) and cylinders (Heyliger et al.,1993;Heyliger and Johnson,2003;So and Leissa,1998). We here recall the basic principles of the method and then give further details for both geometries. We emphasize on the mathematical properties allowing us to increase the efficiency of the computation. Efficiency is important for the probabilistic formulation of the inverse problem (section IV), which implies several thousands of direct computations of the natural frequencies.

The harmonic solutions of the equations of motion for a freely vibrating elastic body are the stationary points of the LagrangianL

L= 1 2



ρω2u2i −Cijklεijεkl

dV, (3.1)

where V and ρ are respectively the specimen’s volume and mass density, Cijkl are the stiffness constants,uiis theith component of the displacement field, andεij is a component of the strain tensor. The displacement field components are expanded as finite sums of known basis functions and the stationary equation∂L= 0is reduced to a generalized eigenvalue problem:

ω2Mα=Kα, (3.2)

whereMandKdenote respectively the mass and stiffness matrices of the vibration problem and α is a vector containing the coefficients of the expansion. For simple solid shapes and well-chosen basis functions, components of the matrices are easily

computed and Eq. (3.2) is numerically solved, giving the natural angular frequencies ωn and the corresponding eigenvectors αn, which represent the displacement pat-terns of the resonant modes. The partial derivatives of a given resonant frequency with respect to any parameter pcan be obtained from


Partial derivatives of the natural frequencies with respect to the stiffness constants are used in the inverse problem.

3.2.1 Rectangular Parallelepiped

The considered RP solid is aligned with a Cartesian coordinate system {x1, x2, x3} and has length Li in the ith direction. Normalized Legendre polynomials of the scaled coordinates are used as basis functions, i.e.

ui= where Pa denotes the normalized Legendre polynomial of order a. Thanks to the orthogonality of the Legendre polynomials over the interval [−1,1], M is the unit matrix and Eq. (3.2) is a standard eigenvalue problem. Many elements of K are equal to zero, and the matrix turns to be block-diagonal. Eq. (3.2) therefore splits in several independent smaller problems (Ohno,1976;Migliori et al.,1993;Migliori and Sarrao, 1997). For an orthotropic elastic symmetry, involving three orthogo-nal symmetry planes, the problem splits in eight smaller ones. Each sub-problem corresponds to vibration modes with different combinations of symmetry or anti-symmetry in the three direction of space. This split not only drastically reduces the computation cost but allows description and labeling of the resonant modes accord-ing to their belongaccord-ing to one of the eight symmetry group. Moreover, the matrixK has a linear dependence to the stiffness constants (Landa et al., 2009). Therefore, in an iterative computation of the resonant frequencies for many different sets of elastic properties, elements ofKneed not to be computed each time but are linearly obtained.

We found, as Migliori and Sarrao (1997), that a maximum polynomial order N = 10gives sufficient accuracy while keeping the computation time small.

3.2.2 Cylinder

Although the case of the cylinder can be treated in a rectangular coordinates system (Visscher et al.,1991), cylindrical coordinates provide improved efficiency and accu-racy (Heyliger and Johnson,2003). The considered cylinder has radiusRand height

3.2. Resonant frequencies computation 41

H (Fig. 3.1). In the following, we assume that the material has an orthotropic (or higher) symmetry in the cylindrical coordinate system, i.e. the stress-strain relation is (in Voigt notation)

Although it is possible to deal with a more general elastic symmetry, this assumption corresponds to many practical cases and allows many simplifications which increase the efficiency of the computation.

We denote byu,vandwthe displacement components in ther,θandzdirections respectively (Fig.3.1). For an isotopic cylinder,So and Leissa(1998) expressed the displacement field as:

u(r, θ, z) =U(r, z) cos(nθ), (3.6a) v(r, θ, z) =V(r, z) sin(nθ), (3.6b) w(r, θ, z) =W(r, z) cos(nθ), (3.6c) for n = 0,1,2, ...,∞. In Eq. (3.6), the θ coordinate is decoupled from the two other coordinates. This is due to the common circular symmetry of the specimen shape and elastic properties (this is discussed in details by Heyliger and Johnson (2003)). For an orthotropic cylinder, the same symmetry exists and we can use the same expressions. Therefore, the vibration modes involve a single azimuthal wave numbern, and modes with different wave numbers are uncoupled. The problem then degenerates into a set of smaller problems (one for eachn), considerably increasing the efficiency of the calculation.

In the Rayleigh-Ritz approximationU,V, andW are expanded as sums of simple polynomial functions inr and z directions, e.g.

U(r, z) =

Using expressions (3.7) and (3.6), the mass and stiffness matrices are computed and Eq. (3.2) is numerically solved for eachn. It should be noted than another set of solutions than (3.6) exists, replacing the cos(nθ) terms by sin(nθ) and conversely (So and Leissa, 1998). For n ≥ 1 it gives the same natural frequencies, but with modal shapes rotated of angle π/2(i.e. modes withn≥1 are doubly degenerated).

Forn= 0the two solutions do not provide the same frequencies and should be both included in the computation. Natural modes of the type(u= 0, v 6= 0, w= 0) are

Figure 3.1: Cylindrical sample geometry in the (r, θ, z) cylindrical coordinate system.

The components of the displacement field are denoted u,v andw in the directionsr, θ andz, respectively. The material is cylindrically orthotropic so that both the shape and the elastic properties are invariant for any rotation around the cylinder axis.

torsional modes while modes of type(u6= 0, v= 0, w6= 0)are axi-symmetric modes.

For each n the problem can be further split in two uncoupled parts, according to whether the displacement is symmetric or antisymmetric with respect to the plane z= 0 (β is even or odd)(So and Leissa,1998).

In practice, we found that, for the study of the∼30 first resonant modes, it was sufficient to compute resonant frequencies up ton= 5and that maximal polynomial orders of 8 in both radial and axial direction (M = N = 8) provided sufficient accuracy.

The present approach extends the work of So and Leissa (1998) on isotropic cylinders to orthotropic cylinders while keeping the calculation efficient. However, it is less general than the approach of Heyliger and Johnson(2003), which includes trigonal elasticity but in consequence does not split the eigenvalue problem and is less efficient.

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