Home / Examples / Piezoelectric Analysis [Rayleigh] / Example 19: Tensile Force and Resonant Frequency (Harmonic Analysis)

Example 19: Tensile Force and Resonant Frequency (Harmonic Analysis)

General

To analyze the effect of tension, resonant analysis is used in Example 13 while harmonic analysis is used in this example
[Analysis with the initial stress taken into account] is applied. See Analysis with initial stress taken into account for the details.
The frequency response of the displacement is solved.
Analysis 1: Apply the static load to the bar and pull it at both ends.
Analysis 2: Fix the bar at both ends. Vibrate it by applying the voltage. The resultant displacement in the Analysis 1 is used to take into account the hardening effect of tensile force.

Analysis Space (Common to Analyses 1 and 2)

Item	Settings
Analysis Space	3D
Model Unit	mm

Analysis Conditions (Analysis 1)

Item	Settings
Solver	Piezoelectric Analysis [Rayleigh]
Analysis Type	Static Analysis
Options	N/A

Graphical Objects (Analysis 1)

Two reversely polarized piezoelectric bars are stuck together. The ends are set with the [Pressure] boundary condition.
The top and bottom faces are set with the [Electric Potential Specified] Boundary condition.

Body Attributes and Material properties Setting (Common to Analyses 1 and 2)

Body Number/Type	Body Attribute Name	Material Name
0/Solid	upper	000_P-4 *
1/Solid	lower	000_P-4 *

(*) Available from the material DB

The polarizations are set as follows.

Body Attribute Name

Tab

Settings

upper

Direction

Specified by: Vector

Vector: X=Y=0.0, Z=1.0

lower

Direction

Specified by: Vector

Vector: X=Y=0.0, Z=-1.0

Boundary Conditions (Analysis 1)

Boundary Condition Name/Topology	Tab	Boundary Condition Type	Settings
earth/Face	Electric	Electric Wall	Electric Potential Specified: Electric Potential 0 [V]
hot/Face	Electric	Electric Wall	Electric Potential Specified: Electric Potential 10 [V]
f_left/Face	Mechanical	Pressure	-1X109 [Pa]
f_right/Face	Mechanical	Pressure	-1X109 [Pa]

Results (Analysis 1)

The bar is pulled at both ends. The vector diagram of the displacement is shown below.

Analysis Conditions (Analysis 2)

Item	Settings
Solver	Piezoelectric Analysis [Rayleigh]
Analysis Type	Harmonic Analysis
Options	Analysis with the initial stress taken into account Select Initial Stress Select Specify Analysis Model

The harmonic analysis tab is set up as follows.

Tab	Setting Item	Settings
Harmonic Analysis	Sweep Values	Minimum: 35 [kHz] Maximum: 50 [kHz]
	Sweep Type	Select [Linear Step by Division Number]. Division: 50
	Frequency Sweep	Discrete Sweep

Results Import Tab is set as follows.

Tab	Setting Item	Settings
Results Import	Specify Result	For analysis model, select ”Analysis 1_Static load”.

Graphical Objects (Analysis 2)

The analysis model is the same as Analysis 1 (Static Analysis) except that both ends are fixed in Analysis 2.

Body Attributes and Material Properties Setting (Analysis 2)

Same as Analysis 1.

Boundary Conditions (Analysis 2)

Boundary Condition Name/Topology	Tab	Boundary Condition Type	Setting
FIX/Face	Mechanical	Displacement	Select all UX/UY/UZ components. X=Y=Z=0.0 [mm]
earth/Face	Electric	Electric Wall	Electric Potential Specified: Electric Potential 0 [V]
hot/Face	Electric	Electric Wall	Electric Potential Specified: Electric Potential 10 [V]

Results (Analysis 2)

The frequency response of displacement is solved by the analyses with and without the static load.

With Static Load/44.9 kHz Z Displacement/Absolute

The contour above shows the displacement component Z absolute at the coordinates (5,1,0) near the resonant point in the analysis with static load.

The peak shifts to the higher frequency due to the tensile force. This is the same effect as in the example 13 where it shifts from 4070 [Hz] to 4481 [Hz].

Note: The results of this example are obtained using Femtet2022.0.1.86015 (64 bit).

Displacement in the contour above and peak height in the graph (frequency characteristics of displacement-Z component) tend to be affected by meshes.

Please be aware that depending on the version of Femtet, the displacement may be affected significantly.

The peak frequency will remain at similar values.