Thermoelastic losses in structural materials of wave solid-state gyroscope resonators

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Basing on the fact that the connection between displacements and the
rates at mesh points is set by relation [13]
V = λU, (12)
and also using (10) and (11), we obtain the equation for the calculation of
λ eigenvalues
 −K 0 0
G −L 0
0 0 I    Θ
U
V  = λ  H F 0
0 0 M
0 I 0    Θ
U
V  , (13)
where I is a unit matrix.
The eigenvalue is complex value λ = δ + jω that explicitly determines
resonance vibration frequency ω and damping coefficient δ .
Let us write the relation for obtaining resonator’s quality factor Q on
resonance frequency f in the following form:
Q =
Im(λ)
2 Re (λ)
, δ = Re(λ), f =
Im(λ)
2π
. (14)
In addition, to every λn eigenvalue, there corresponds its own harmonic
field of a heat flow, displacements, stresses and deformations. As the
intensities of heat flows depend on both the thickness of resonator’s wall
( b), and vibration frequency (f ), the results of the calculation for various
structural materials are shown in Fig. 2, 3 in the form of dependencies
Q(f, b). Here, it was supposed that Q = ς− 1
T , while the intensities of all
other dissipative processes mentioned in (2), were equal to zero.
Results and discussion. In Fig. 2 the data is shown on the quality
factors of metal resonators made of duraluminium Д16Т and corrosion-
resistant steel. As one can see in the picture, the quality factor of thin-walled
(less than 1 mm) and low-frequency (less than 1 kHz) metal resonators is
always small (about 1000) because of high intensity of the thermoelastic
internal friction, which is also a dominating dissipative process. To get the
quality factor (4 . . . 5) ∙ 104, the wall thickness should be taken equal to
not less than 1 . . . 1 .5 mm, while the working frequency should be not less
than 4. . . 5 kHz.
Rather small values of resonators’ quality factor can be received
using silicon and sapphire as structural materials (Fig. 3). At the vibration
frequency up to 10 kHz of these resonators, thermoelastic losses will limit
their quality factor on the level 106. In contrast to these materials, quartz
glass has a low level of thermoelastic internal friction. At the vibration
frequency about 1 kHz, even for wall thickness 0.5 mm, the quality factor
34 ISSN 0236-3933. HERALD of the BMSTU. Series Instrument Engineering. 2015. No. 2