Example22 Thermoelastic Damping

General

Q factor is calculated by piezo-resonant analysis with thermoelastic damping being taken into account.
As a result, temperature distribution and Q factor are solved.
Approximated frequency is given by resonant analysis with deselecting thermoelastic damping.
The approximated frequency is explained at the end of this page.
Unless specified in the list below, the default conditions will be applied.

Analysis Conditions

Item	Setting
Solver	Piezoelectric Analysis [Rayleigh]
Analysis Space	2D
Analysis Type	Resonant Analysis
Unit	mm
Options	Take thermoelastic damping into account

Resonant Analysis Tab

Tab	Setting Item	Setting
Resonant Analysis	Number of modes	3
Resonant Analysis	Approximated frequency	1.730×10^3[Hz]

Model

Ten rectangles are stacked in the Z direction. In this way, temperature change in the Z direction can be represented precisely with less meshes.

The model is 20mm across and 0.3mm in thickness. The thickness is displayed five times the actual dimension.

Body Attributes and Materials

Body Number/Type	Body Attribute Name	Material Name
n/Solid (0<=n<10)	BEAM	Material_Property_001

Body Attribute

Body Attribute Name

Tab

Analysis Domain

BEAM

Direction

Specified by: Vector

Vector: X=Y=0.0, Z=1.0

Thickness/Width

Thickness of sheet body: 1mm

Material Property

Material Name	Tab	Setting
Material_Property_001	Density	7.8×10^3[Kg/m3]
	Piezoelectricity tab	Piezoelectricity: No Anisotropy: Isotropic Young’s modulus: 2×10^11[N/m2]
	Specific heat	2×10^3[J/kg/deg]
	Thermal conductivity	42[W/m/deg]
	Coefficient of expansion	1.2×10^5[1/deg]

Boundary Condition

Boundary Condition Name/Topology	Tab	Boundary Condition Type	Setting
FIX/Face	Mechanical	Displacement	UX=0, UY=0, UZ=0

Results

Temperature distributions are displayed as below. A vertical direction is extended 5 times by using nonuniform zoom function.

Change thickness between 0.5mm and 0.03mm. Q factor will be given by the following equation with a complex resonant frequency.

Q factor=real part of the complex resonant frequency/imaginary part of the complex resonant frequency/2

Approximated frequency is given by resonant analysis with deselecting thermoelastic damping. The approximated frequency in the thermoelastic damping analysis will be explained later on.

If the thickness is changed, the resonant frequency and Q factor will be changed accordingly. Figure 1 below shows their relationship. The calculated Q factors and theoretical values are well matching.

Thickness of 0.3 is applied in this session’s project file. A variable dz defined in the project is one tenth of the thickness. By changing dz, you can trace the results below.

List 1. Thickness and Q factor

Thickness [m]	Reference Frequency [Hz]	Resonant Frequency Real part [Hz]	Resonant Frequency Imaginary part [Hz]	Q factor
0.5	2.877e3	2.878e3	5.578e-3	2.579e+5
0.3	1.728e3	1.728e3	1.462e-2	5.913e+4
0.1	5.764e2	5.765e2	7.488e-2	3.849e+3
0.07	4.035e2	4.035e2	4.204e-2	4.799e+3
0.05	2.882e2	2.882e2	1.296e-2	1.112e+4
0.03	1.728e2	1.728e2	1.726e-3	5.008e+4

Figure 1. Real part of resonant frequency and Q factor: A comparison between Femtet calculation and theoretical values

Equations of theoretical value

Source: H.Itoh JJAP 50 (2011) 087203

Approximated frequency

The meaning of using approximated frequency in the analysis with thermoelastic damping taken into consideration In the resonant analysis with thermoelastic damping taken into account, the expression (1) below must be solved, where s represents frequency, x represents vector for distributions of displacement and temperature, A, B, and C represent matrices reflecting model shape and material property. Since solving the expression (1) is not easy, an simplified expression (2) is used for analysis, where approximated frequency is used as sref. The approximated frequency, therefore, must be close to the resonant frequency that is to be obtained.

(A+sB+s2C)x=0 (1)

(A+srefB+s2C)x=0 (2)

See other Piezoelectric Analysis examples.