Mechanical Characterisation of Materials for String Instrument

Abstract

A large variety of materials are used traditionally for constructing string musical instruments in the Western cultural tradition. Since the last decades of the XIX th century it has been recognised that mechanical and acoustical properties of materials are the most relevant for these instruments. Wood is the predominant material. Other materials such as metals, or natural materials—bone, ivory mother of pearl, etc. are also used. The elastic constants of these materials can be determined experimentally with acoustic non-destructive methods such as resonance methods and ultrasonic methods. These constants are required for modal analysis, which allows the study of the vibration of plates or of instruments. In all materials internal friction is produced during the propagation of acoustic waves, reflecting the viscoelastic characteristics of these materials. Some of these materials are highly hygroscopic, such as wood. The properties of wood are dependent on moisture content. Modal analysis can be used to illustrate the influence of very low variations of moisture content on the vibration of plates of string musical instruments.

Appendices

3.1.1 Appendix 1: Relationships Among the Parameters Measured with the Resonance Method on Plates

The theory used here is based on the work of McIntyre and Woodhouse (1988).

It was assumed that the plate studied is a thin, flat orthotropic plate. The bending vibration of such a plate is governed by four elastic constants D ₁ , D ₂ , D ₃ and D ₄ .

The vibration modes of plates give rise to Chladni pattern corresponding to each resonant frequency. The modes of a rectangular plate were described as follows: mode due to twisting (f ₁ ), mode due to bending (f ₂ ) for two cases—cross section, and along the fibres , the ring mode (f ₃ ) and the X-mode (f ₄ ). The plates were supported on small soft foam blocks.

The relationships between the resonance frequency at different modes and the elastic constants of the plates are the following:

$$\begin{aligned} D_{1} & = 0.0789\,\frac{{f_{2 - cross}^{2} \cdot \rho \cdot a^{4} }}{{h^{2} }}; \\ D_{2} & \approx 0.144(f_{3}^{2} - f_{4}^{2} )\rho \,a^{2} b^{2} /h^{2} ; \\ D_{3} & = 0.0789\,\frac{{f_{2 - alongfibres}^{2} \cdot \rho \cdot b^{4} }}{{h^{2} }}; \\ D_{4} & = 0.274\,\frac{{f_{1}^{2} \cdot \rho \cdot a^{2} \cdot b^{2} }}{{h^{2} }} \\ \end{aligned}$$

where h is the thickness of the plate and a and b are the transverse dimensions of the plate (a > b).

3.1.2 Appendix 2: Resonance Method and Relationships Among Elastic Constants Measured on Plates and Wood Technical Constants (Young’s Moduli, Shear Moduli and Poisson’s Ratios)

The relationships among the elastic constants measured on anisotropic plates D₁, D₂, D₃, D₄ and the Young’s moduli E ₁ , E₂, shear modulus G ₁₂ and Poisson’s ratios ν ₁₂ and ν₂₁, are given in what follows as calculated by McIntyre and Woodhouse (1986) for orthotropic plates

$$\begin{aligned} D_{1} & = \frac{1}{12\Delta }\{ \frac{{n^{4} }}{{E_{T} }} + \left( {\frac{1}{{G_{RT} }} - \frac{{2\nu_{RT} }}{{E_{R} }}} \right)n^{2} m^{2} + \frac{{m^{4} }}{{E_{R} }}\} \\ D_{2} & = \frac{1}{6\Delta }\{ \nu_{LR} m^{2} + \nu {}_{LT}n^{2} \} \\ D_{3} & = \frac{1}{{12\Delta E_{L} }} \\ D_{4} & = \frac{{G_{LR} G{}_{LT}}}{{3\left( {m^{2} G_{LT} + n^{2} G_{LR} } \right)}} \\ \end{aligned}$$

where n = sin θ and m = cos θ; θ = 0° when the plate is quarter cut, in LR plane

$$\begin{aligned} \Delta & = (1 - \nu_{LT} \cdot \nu_{TL} )\frac{{n^{4} }}{{E{}_{L}E{}_{T}}} + \left( {1 - \nu_{RL} \nu_{LR} } \right)\frac{{m^{4} }}{{E{}_{R}E_{L} }} + \left( {\frac{1}{{G{}_{Rt}}} - \frac{{2\nu_{RT} \nu {}_{LT}}}{{E_{R} E_{L} }}} \right)n^{2} m^{2} \\ \frac{{\nu_{ET} }}{{E_{R} }} & = \frac{{\nu_{TR} }}{{E_{T} }} \\ \end{aligned}$$

For resonance spruce McIntyre and Woodhouse (1986) reported the following values were calculated D₁ = 1320 MPa; D₂ = 78 MPa; D₃ = 82 MPa; D₄ = 224 MPa.

The elastic constants used for this calculations are given for spruce and maple, in Table 3.15 and Table 3.16.

Table 3.15 Elastic constants for spruce and maple (Hearmon 1948)

Table 3.16 Poisson’s ratios for spruce and maple (Hearmon 1948)

3.1.3 Appendix 3: Ultrasonic Stiffnesses and Wood Technical Constants (Young’s Moduli, Shear Moduli and Poisson’s Ratios)

Ultrasonic stiffnesses measured on orthotropic plates are related to elastic constants by the following relationships (Bodig and Jayne 1982)

$$\begin{aligned} C_{11} & = \frac{{E_{1} E_{2} }}{{E{}_{2} - E_{1} \nu_{21}^{2} }} \\ C_{22} & = \frac{{E_{1} E_{2} }}{{E{}_{1} - E_{2} \nu_{12}^{2} }} \\ C_{12} & = C_{21} = \frac{{E_{1} E_{2} \nu_{21} }}{{E{}_{2} - E_{1} \nu_{21}^{2} }} \\ C_{66} & = G_{12} - {\text{the}}\,{\text{shear}}\,{\text{modulus}}\,{\text{in}}\, 1 2\,{\text{plane}} \\ \end{aligned}$$

Ultrasonic velocity measurements (i.e. in plane 1, 2 with bulk waves v ₁₁ , v ₂₂ , v ₆₆ and with surface waves v _{12 surface} , which propagates at the surface of the material in plane 1,2, propagation direction 1, polarisation 2) are used for calculation of these constants as follows with:

• Longitudinal waves $C_{11} = V_{11}^{2} \cdot \rho$ and $C_{22} = V_{22}^{2} \cdot \rho$ (propagation direction parallel to polarisation direction)

• Shear waves $C_{66} = V_{66}^{2} \cdot \rho$ (propagation and polarisation directions are perpendicular)

• Surface waves $C_{12} = f\left( {\rho ,V_{surfave} ,C{}_{11,}\cos \alpha } \right)$ for surface waves, propagating at the interface wood-air, the polarisation is elliptical in a plane perpendicular to propagation direction—see Rayleigh wave) as described by Bucur and Rocaboy (1988). Knowing the terms C ₁₁ , C ₂₂ , C ₁₂ and C ₆₆ , the calculation of engineering constants is straightforward from the inversion of the matrix [C]⁻¹ = [S] and S₁₁ = 1/E₁, S₁₂ = −ν₁₂/E₁,

Experiments can be performed on different specimens, such as cubes cut at different angles in each anisotropic plane, a sphere or a polyhedron with numerous faces—i.e. dodecahedron, polyhedron with 12 or 24 faces, depending on the skill of the technician cutting such a specimen. Appendix 4

3.1.4 Appendix 4: About Acoustic Wave Propagation Phenomena in Anistropic Solids (Musgrave 1970; Auld 1973; Every and Sachse 2001)

The elastic properties of solids can be defined by the generalised Hooke’s law relating the volume average of stress [Ϭ_ij] to the volume average of the strains [ε_ij] by the elastic constants [C_ijkl] or [C_ij] (where the indices _ijkl are 1, 2, 3, 4 and are the indices notation of Voigt, or we can have only indices _ij in Voigt’s contracted notation) in the form. Therefore Hook’s law can be written such as.

[Ϭ_ij] = [C_ijkl] · [ε _ij] or [ε_kl] = [S_ijkl] · [Ϭ_ij] or in more condensed form

[Ϭ] = [C] · [ε] or [ε] = [S] · [Ϭ] and [C] = [S]⁻¹

[C _ij ] are termed stiffness and [S _ij ] are termed compliances . Experimentally the terms of compliance matrix [S _ij ] are determined from static tests .

The terms of stiffness matrix [C _ij ] could be determined from ultrasonic measurements and in what follows we describe the derivation of analytical expressions for these terms and the phase velocities for the three bulk waves propagating in an anisotropic, homogeneous and linearly elastic solid.

From generalised Hook’s law we know that

$$\sigma_{ij} = C_{ijkl} \cdot \varepsilon {}_{kl} = C_{ijkl} \frac{{\partial u_{l} }}{{\partial x_{k} }}$$

(3.1)

where $\sigma_{ij}$ are the components of stress tensor, $\varepsilon_{kl}$ the components of strain tensor, C _ijkl are the components of stiffness tensors, u _l the components of the induced displacement by the wave, x the position vector.

If we consider a harmonic plane wave propagating in the direction of the unit propagation vector n, with a displacement vector u of components

$$u_{i} = Up_{i} \exp [i\omega (t - \frac{n.x}{V})]$$

(3.2)

where U is the amplitude of the displacement, p _i are the components of the unit polarisation vector, collinear with the displacement vector, t is the time, V is the phase velocity in the direction n, x the position vector and ω the circular frequency.

For a continuum body, with no forces acting on it, Cauchy’s elastodynamic equations in terms of the components of the displacement vector u can be written as

$$\rho \frac{{\partial^{2} u_{i} }}{{\partial t^{2} }} - C_{ijkl} \frac{{\partial^{2} u_{k} }}{{\partial x_{l} x_{j} }} = 0$$

(3.3)

where ρ is the density of the medium.

By introducing the second order matrix Γ = C _ijkl n _j n _l , called the Kevin-Christoffel matrix, one obtain the Christoffel equations

$$\varGamma_{il} p{}_{k} = \rho V^{2} p_{i}$$

(3.4)

Which can be also written as

$$[\varGamma_{il} - \rho V^{2} \delta_{il} ]p_{l} = 0$$

(3.5)

This equation is an eigenvalue and eigenvector problems (the solutions of this equation). In order to solve the eigenvalue problem, and to determine the elastic constants function of velocity and density, one must impose the cancellation of the determinant of the matrix

$${ \det }[\varGamma_{il} - \rho V^{2} \delta {}_{il}] = 0$$

(3.6)

where δ_il is called Kronecker determinant, which could be 1 or 0, depending on propagation situation. The function δ _il is1 if the variables are equal and 0 otherwise.

The coefficients of the tensor [Γ] are Γ₁₁, Γ₂₂, Γ₃₃, Γ₁₁,, Γ₁₃, Γ₂₃

The eigenvalues and the eigenvectors of Eq. (3.6) can be calculated for specific anisotropic material

$$\left| {\begin{array}{*{20}c} {\Gamma _{{{\text{11}}}} {-}{\uprho} {\text{ V}}^{{\text{2}}} } & {\Gamma _{{{\text{12}}}} } & {\Gamma _{{{\text{13}}}} } \\ {~\Gamma _{{{\text{21}}}} } & {\Gamma _{{{\text{22}}}} {-}{\uprho} {\text{ V}}^{{\text{2}}} } & {\Gamma _{{{\text{23}}}} } \\ {~\Gamma _{{{\text{31}}}} } & {\Gamma _{{{\text{23}}}} } & {\Gamma _{{{\text{33}}}} {-}{\uprho} {\text{ V}}^{{\text{2}}} } \\ \end{array} } \right|\left| {\begin{array}{*{20}c} {p_{1} } \\ {p_{2} } \\ {p_{3} } \\ \end{array} } \right| = 0$$

(3.7)

For an orthotropic solid, with nine terms of stiffness tensor [C] and three symmetry planes we have the components of the tensor Γ expressed such as explained in the following table (See Table 3.17)

Table 3.17 Terms of Kevin-Christoffel matrix as function of stiffnesses for an orthotropic solid

Equation (3.7) is a polynomial in phase velocity squared (V ²). From it the first issue addressed is the determination of the elastic constants [C] of a given material when the phase velocity is known. On the other hand we have a set of simultaneous equations in p _m (p ₁ , p ₂ , p ₃ ), related to the polarisation vector. For a unique solution to those we have to fulfil the conditions of the Eq. (3.8) presented below, for the particular case of propagation in the plane 1, 2. when the propagation is along the symmetry axes

$$\left| {\begin{array}{*{20}c} {\Gamma }_{{ {11}}} {{-}\uprho {\rm V}}^{ {2}} & {0} & {0} \\ { {~0}} & {\Gamma }_{{ {22}}} {{-}\uprho {\rm V}}^{ {2}} & {0} \\ { {~0}} & {0} & {\Gamma }_{{ {33}}} {{-}\uprho {\rm V}}^{ {2}} \\ \end{array} } \right| = 0$$

(3.8)

The solutions are

Γ₁₁ –ρ v² = 0; ρ v² = C ₁₁ corresponding to a longitudinal wave of velocity v ₁₁ along axia1

Γ₂₂ –ρ v² = 0; ρ v² = C₆₆ corresponding to a fast shear wave of velocity v₆₆, for which the wave propagates along axis1 and is polarised along axis 2

Γ₃₃ –ρ v² = 0; ρ v² = C₅₅ corresponds to a slow shear wave of velocity v ₅₅ . For which the wave propagates along axis 1 and is polarised along axis 3

These solutions, in case of the orthotropic solid, enable us to calculate the six diagonal terms of stiffness matrix [C].

If the propagation took place out of principal direction, for example in the plane 12, we can write

$$\left| {\begin{array}{*{20}c} {\Gamma }_{{ {11}}} {{-}\uprho {\rm V}}^{ {2}} & {0} & {0} \\ { {{\Gamma }_{{ {21}}}}} & {\Gamma }_{{ {22}}} {{-}\uprho {\rm V}}^{ {2}} & {0} \\ { {~0}} & {0} & {\Gamma }_{{ {33}}} {{-}\uprho {\rm V}}^{ {2}} \\ \end{array} } \right| = 0$$

or

$$\left( {\rho \cdot V^{2} } \right)^{2} - \rho \cdot V^{2} \left( {\varGamma_{11} + \varGamma_{22} } \right) + \varGamma_{11} \cdot \varGamma {}_{22} - \varGamma_{12}^{2} = 0$$

The solution of this equation is for the quasi-longitudinal (QL) or quasi shear waves (QT) are

$$\begin{aligned} 2\uprho{\text{V}}_{\text{QT}}^{2} & = \left( {\varGamma_{ 1 1} + \varGamma_{ 2 2} } \right) + \left[ {\left( {\varGamma_{ 1 1} - \varGamma_{ 2 2} } \right)^{ 2} + 4\varGamma_{ 1 2} } \right]^{ 1/ 2} \\ 2\uprho{\text{V}}_{\text{QT}}^{2} & = \left( {\varGamma_{ 1 1} + \varGamma_{ 2 2} } \right) - \left[ {\left( {\varGamma_{ 1 1} - \varGamma_{ 2 2} } \right)^{ 2} + 4\varGamma_{ 1 2} } \right]^{ 1/ 2} \\ \end{aligned}$$

The coefficient (Γ₁₁ + Γ₂₂) is an invariant, independent of the angle of propagation for both waves QL or QT.

As noted by Bucur (1988, 1990,  2006), the expression of the invariants deduced as the sum of some of the coefficients of Christofell tensor, can be written for the particular case of the plane 12 or LR (Γ₁₁ + Γ₂₂), of the plane 13 or LT (Γ₁₁ + Γ₃₃) and of the plane 23 or RT (Γ₂₂ + Γ₃₃) such as

$$\begin{aligned} I_{12} & = \left( {V_{11}^{2} + V_{22}^{2} + 2V_{66}^{2} } \right)^{{\frac{1}{2}}} \\ I_{13} & = \left( {V_{11}^{2} + V_{33}^{2} + 2V_{55}^{2} } \right)^{{\frac{1}{2}}} \\ I_{23} & = \left( {V_{22}^{2} + V_{33}^{2} + V_{44}^{2} } \right)^{{\frac{1}{2}}} \\ \end{aligned}$$

The advantage of these invariants is the combination of all types of velocities propagating in one anisotropic plane

Combining the values of invariants as a ratio between the invariant in the transversal plane RT and the average invariants in planes including axes L − LR and LT we have

$$I_{ratio} = \frac{{I{}_{23}}}{{\frac{{I_{12} + I_{13} }}{2}}} = \frac{{2I_{23} }}{{I_{12} + I_{13} }}$$

As invariants Every and Sachse (2001) suggested the coefficients noted A, B and C and called principal invariants of the Christoffel tensor (not influenced by the angle of the propagation direction of the velocity vector). These coefficients are related to three velocities of longitudinal waves in axes, such as

$$\begin{aligned} A & = \rho \left( {V_{11}^{2} + V_{22}^{2} + V_{33}^{2} } \right) \\ B & = \rho^{2} \left( {V_{11}^{2} \cdot V_{22}^{2} + V_{22}^{2} \cdot V_{33}^{2} + V_{33}^{2} \cdot V_{11}^{2} } \right) \\ C & = \rho^{3} \cdot V_{11}^{2} \cdot V_{22}^{2} \cdot V_{33}^{2} \\ \end{aligned}$$

We note that in the expression of A, B and C that shear velocities are not included.