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

2 / 10

where ςT — thermoelastic internal friction; ςV — internal friction in the
material structure; ςS — losses in the surface layer of the material; ςG — gas
friction.
Contributions of these separate processes (their list is unbounded (2))
are different and depend on the properties of the resonator material, its
design, the quality of its surface processing, the vacuum level in the
instrument. The internal friction in a solid body and in its surface layer as
well as gas friction are thoroughly discussed in [3–5]. The thermoelastic
internal friction, depending on the properties of the material and design,
can either be negligibly small or fully limit the resonator’s quality
factor. The detailed calculation of thermoelastic losses permits to assess
accurately enough the potential possibilities of the vibratory gyroscope
design. In special literature there are examples of these calculations made
for micromechanical instruments [6–8]. In wave solid-state gyroscopes
(WSG), axisymmetric thin-walled shells are used as resonators and the
thermoelastic losses in their material can also be significant. The objective
of the present paper is to consider the influence of thermoelastic internal
friction on the characteristics of WSG resonators made of various materials.
Simulation of thermoelastic internal friction in a resonator. Physics
of thermoelastic internal friction was first revealed by Zener [9] who
associated it with the onset of heat flows under deformation of a solid
body. In resonator’s vibrations, the deformations of its parts are opposite
in sign, i.e. in some places the material expands and in other places it
contracts. The body volume change under deformation requires that some
work A should be done, which can be expressed through thermal expansion
coefficient (α) and modulus of elasticity (Е) [10] as follows:
A = 9α2TEV, (3)
where Т — body temperature; V — molar volume of the substance.
It follows from (3) that when a solid body is under deformation (if
α 6= 0 ), the temperature in different parts of the body will depend on
deformation. In its turn, the inequality of these temperatures will result in
the onset of local heat flows increasing the oscillator’s entropy, which is
equivalent to the irreversible transformation of mechanical energy into heat
energy. To assess the thermoelastic losses quantitatively, Zener suggested
simple formulas that provide good enough complience with the experiment
in relation to a range of metals within the bounds of his phenomenological
model of internal friction in a solid body.
In the present paper, to determine thermoelastic losses, the finite-
element simulation of thermoelastic damping according to the second mode
shape in a cylindrical resonator was used (Fig. 1, b , c).
ISSN 0236-3933. HERALD of the BMSTU. Series Instrument Engineering. 2015. No. 2 29