With reference to Be vs Ti Table , please find
The density of a material, ρ, is defined as the mass per unit volume or ρ = m/V, where m = mass
and V = volume. For a given transducer geometry, a lower-mass diaphragm allows greater
acceleration of the moving system (F=ma), increasing both passband efficiency and highfrequency
extension (Kinsler & Frey; Eargle).
3.2 Young’s Modulus
This is the ratio of uniaxial stress to strain and is measured in units of pressure. It is defined as E
= σ/ε, where σ = stress and ε = strain. A higher Young’s Modulus equates to a stiffer
diaphragm, all other things being equal. A stiffer diaphragm, of course, does not bend as much in
response to an applied force. Thus, at high frequencies, the bending modes are shifted up in
frequency, extending the useful bandwidth of the transducer (see further discussion below).
3.3 Poisson’s Ratio
When a solid material is compressed in one direction, it tends to expand in the other two
directions perpendicular to the direction of compression. This phenomenon is called the
Poisson Effect. The Poisson’s Ratio ν (nu) relates the contraction or transverse strain
(perpendicular to the applied load), to the extension or axial strain (in the direction of the
applied load). Assuming that the material is stretched or compressed along the axial direction:
ν = - dεtransverse / dεaxial
ν is the resulting Poisson's Ratio,
εtransverse is transverse strain (negative for axial tension, positive for axial compression)
εaxial is axial strain (positive for axial tension, negative for axial compression).
The Poisson’s Ratio of beryllium is unusually low. For acoustic applications, a low Poisson’s
Ratio results in reduced coupling of sound waves from one mode of propagation to another.
For example, an axial wave will remain an axial wave transferring less energy to transverse
modes of wave propagation. This ability to keep the different modes of propagation separated
can be of great importance in acoustics, especially at higher frequencies, or in imaging or surface
wave devices (Materion Electrofusion). In other words, a lower Poisson’s Ratio better
preserves the direction of the applied force.
3.4 Speed of Sound
The speed of sound is the rate of travel of a sound wave through an elastic medium. The speed
of sound waves in solids is determined by the material's stiffness and density and may be
described by the equation c = E/ρ . This is generally understood to be the speed of a
longitudinal wave along the x-axis of a long bar (where x >> y or z). For the purposes of this
discussion, we are interested in the bending modes of a thin plate (x and y >> z) where the
displacement of the wave is in the z-axis. The bending stiffness causes the bending wave speed
to be different than c. These relationships are detailed below.
3.5 Tensile Strength
Tensile Strength is the property of a material that measures its ability to withstand tensile stress
without failure. A material with a higher tensile strength allows a thinner dome to maintain
equivalent strength, giving a lower moving mass.
3.6 Bending Modulus (relative to aluminum)
This property describes the stiffness of a thin plate in response to bending forces (similar to
Young’s Modulus, but in two dimensions). Of course, the actual bending stiffness of a dome
shape is highly dependent on the details of that geometry. For this discussion, we separate the
material’s inherent bending stiffness from the geometry’s stiffness by assuming the same thin
plate geometry for all materials. For the sake of clarity in the table, these results have been
shown relative to the Bending Modulus of the aluminum plate.
Table 3: Predicted Bending Mode Frequencies
Freq(kHz) Aluminum Titanium Tin / Aln Beryllium Ben / Aln
1st mode 10.04 9.98 0.99 26.16 2.61
2nd mode 10.44 10.36 0.99 26.64 2.55
3rd mode 10.56 10.46 0.99 26.79 2.54
4th mode 10.62 10.52 0.99 26.90 2.53
5th mode 10.68 10.58 0.99 27.05 2.53
6th mode 10.76 10.66 0.99 27.28 2.54
First, notice that the Finite Element Model confirms the analytical solution above; if geometry is
held constant, beryllium’s bending modes occur at roughly 2.5x higher frequency than those of
the other materials.
Perhaps even more interesting is the acutely non-harmonic relationship of the modes above the
first. Classical plate theory hints at this, but flat plates and membranes do not exhibit such
closely-spaced modes. While beyond the scope of this study, it is certainly worth noting that
the relationship between the natural frequencies is drastically affected by the geometry of the
dome. The modal density above the first bending mode is much higher than that predicted by
classical (flat) plate theory. Essentially, this lends greater importance to the first bending mode
frequency, since adjacent modes set in so quickly and densely above that frequency.
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Transducer design engineer