Home / Examples / Piezoelectric Analysis [Rayleigh] / Example 22: Thermoelastic Damping

Example 22: Thermoelastic Damping

 

 

General

 

 

 

Analysis Conditions

Item

Settings

Solver

Piezoelectric Analysis [Rayleigh]

Analysis Space

2D

Analysis Type

Resonant Analysis

Unit

mm

Options

Take into account thermoelastic damping

 

Resonant Analysis Tab

Tab

Setting Item

Settings

Resonant Analysis

Number of Modes

3

Approximated Frequency

1.730×103 [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 20 mm in the longitudinal direction and 0.3 mm 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: 1 mm

 

Material Property

Material Name

Tab

Settings

Material_Property_001

Density

7.8×103 [kg/m3]

Piezoelectricity

Material Type: Non-Piezoelectric

Anisotropy: Isotropic

Young's Modulus: 2×1011 [N/m2]

Specific Heat

2×103 [J/kg/deg]

Thermal Conductivity

42 [W/m/deg]

Coefficient of Linear Thermal Expansion

1.2×105 [1/deg]

Boundary Condition

Boundary Condition Name/Topology

Tab

Boundary Condition Type

Settings

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 0.5 mm and 0.03 mm to get a complex resonant frequency. Q factor will be given by the following equation with the 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

 

Fig 1. Real Part of Resonant Frequency and Q Factor: Comparison Between Femtet Calculation and Theoretical Values

 

 

 

Equations of theoretical value

 

Reference: H.Itoh, JJAP,50 (2011) 087203

 

 

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)