2008 International ANSYS Conference · input is smaller than past input. The energy is accumulated...

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2008 International ANSYS Conference

Nonlinear Piezoelectric Analysis with ANSYS-MATLAB/Simulink coupling

Masahiro MatsumotoCybernet Systems Co., Ltd


Agenda

• Introduction• Overview of Analysis• Modal Analysis with ANSYS• Experiment result

– A nonlinear vibration phenomenon• Examination of new analytical technique

– Consideration of “large-deflection effect”– ANSYS-MATLAB/Simulink coupling

• Other cases (hysteresis model)• Conclusion


• Piezoelectric devices are widely used in electronics, automotive, and many other industries in Japan, such as:– Power devices (electro mechanical)

• piezo actuator、ultrasonic motor ,etc.– Sensors (mechanical electro)

• acceleration sensor, gyro sensor ,etc.– Circuits (resonance)

• ceramic resonator, SAW Filter ,etc.

Introduction (Japanese market)


We propose a piezoelectric analysis to a wide customerin Japan by “customizing Workbench”.

Introduction (sales promotion)

Auto MeshCAD Interface

Optimized approach(DesignXplorer）


Introduction (Today’s problem)

• Under increasingly harsh operation environments of piezoelectric devices, The following nonlinear effects are becoming vital factors in piezoelectric analysis.– Large deflection effects– Material nonlinearity– Temperature dependence

• This paper discusses the techniques to achieve efficient nonlinear piezoelectric analysis using “ANSYS-MATLAB/Simulink coupling”.


Overview of Analysis

Spectacle-type Wearable Retinal Imaging Display (RID)

Piezoelectric mirror

RID consists of three modules: a light source module, an optical scanning module, and an eyepiece module. This optical scanning module uses an optical micro electro mechanical system (Piezoelectric mirror).


Overview of Analysis (cont.)

Model 1 Model 2

Mirror & beam (Si)

Upper electrode

Piezoelectric

Under electrode

The difference between “Model1” and “Model2” is only “ the intervals of the beams”.


Modal Analysis (Model1)

Mode Frequency [Hz]

Mode Shape

1 14,701 anti plane

2 29,340 in plane

3 32,868 torsion

Mode3 : Torsion mode

Because torsion vibration is the key operation movement , Mode3 is important.

Frequency of focus


Modal Analysis (Model2)

Mode Frequency [Hz]

Mode Shape

1 9,086 antiplane

2 29,068 inplane

3 31,045 Bending

4 31,142 Torsion

Because torsion vibration is the key operation movement , Mode4 is important.

Mode4 : Torsion mode

Frequency of focus


Experiment result of “torsion mode”

• The resonance frequency of torsion mode is roughly corresponding. • The resonance frequency of “Model2” is different from “Model1” when the frequency rises and falls. • Moreover the “Model2” amplitude drops when the frequency rises. Why?

Model 1 Model 2

freq. risesfreq. falls


Amplitude drops


A nonlinear vibration phenomenon

• When the deflection increase, a nonlinear effect becomes important. It is known that the reaction force works by the third power of the deflection, (The Duffing equation) .

• One of the vibration characteristic is that the reaction force(= Stiffness) increases as the amplitude grows, and the resonance frequency changes depending on the amplitude.

• it is necessary to consider this nonlinear effect in piezoelectric devices (piezo mirror, ultrasonic motor ,etc) with small damping.

x

mkx

kx+βx3

0 x

F

1 DOF Mass-Damper system

Frequency

Ampl

itude

0

Voltage increase

freq. rises

freq. falls


Examination of analytical technique

• Typical problem of piezoelectric device– The damping is very small

→ The Integration Time Step is very small.→ The transient analysis needs very long solution times .

• Breakthrough in the solution1. Solution times is shortened by “Reduction Model”.2. The nonlinear effect (large-deflection effect) is expressed with

the feedback control.

%)05.0(≈

ANSYS-MATLAB/Simulink coupling


ANSYS-MATLAB/Simulink coupling

• ANSYS data, which “Structural Dynamics Toolbox (MATLAB 3rd Party tool)” degenerate, is built into MATLAB. Solution time has been greatly shortened by MATLAB.

• To consider the nonlinear model (torsion stiffness), the reaction force shown by the previous slides was added as a feedback control. This technique can adequately express the evaluated behavior.

ANSYS databaseANSYS Reduction Model

Nonlinear term

Reduction model by SDT

Excited by alternating voltage

Feedback control by reaction force


Consideration of “large-deflection effect”

y = -5E-05x3 + 0.0116x

2 - 0.0015x

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

Angle(deg)

Reac

tion Force (%)

y = -4E-06x3 + 0.0041x

2 - 0.0019x

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

A ngle (deg)

Rea

ction

Force (%)

Model 1

Model 2Predominant movement of the piezoelectric mirror is a torsion mode. Therefore, the nonlinear stiffness is calculated by the reaction force when the mirror is twisted.

Model1 = 4e-6Model2 = 5e-5

Because the interval of the beams which support the mirror is wide, Model2’s twist stiffness is high. The reaction force is used as the feedback control.

twist


ANSYS-MATLAB/Simulink coupling (Result)

3.18 3.185 3.19 3.195 3.2 3.205 3.21 3.215 3.22x 104

0

5

10

15

20

25

30Tyep 2

Frequency (Hz)

Angl

e (d

eg)


Model 2

3.28 3.285 3.29 3.295 3.3 3.305 3.31 3.315 3.32x 104

0

5

10

15

20

25

30Tyep 2

Frequency (Hz)

Angl

e (d

eg)


Model 1

Good Correlation !!


Piezoelectric optimization analysis considering nonlinear material (hysteresis)


Overview of Analysis (Cantilever)

x

L

Wp（Design parameter）

disturbance

x = Ax + Buy = Cx + Du&

Reduction

Reduction model

state feedback

u

u = -Kx

（ Design parameter ）

Purpose ：To optimize the shape and the control system of vibration controller

Objective variables : Minimization of Vibration/Control energy

Design variables：Structure System : dimensionControl System : weighting matrix

F (control input)

{ }1 2 n

l u

k k kk k ku

=

≤ ≤= −

Κ

Kx

L

Piezo film


hysteresis of piezoelectric actuator (Model)

• Mechanism of hysteresis (assumption )– A part of the inputs energy is accumulated internally as potential energy. – The inputs are memorized. The energy isn’t accumulated when new

input is smaller than past input. The energy is accumulated when new input is larger than past input.

– The potential energy has retentivity. The energy is maintained without applying the opposite input.

– There is a limit in the energy accumulation, and the energy is saturated.

• Base Model

( ) ( ) ( )∫−= dxtxtkxty μAccumulation of energy


Input

Output

hysteresis of piezoelectric actuator (Result)


Search for multipurpose optimization

0 0.5 1 1.5 2-4

-2

0

2

4x 10-3

T im e (s)

Value

W ithout control

0 0.5 1 1.5 2-4

-2

0

2

4x 10-3

T im e (s)

Value

W ith control

LQ regulatorN B I m ethod

ANSYShysteresismodel

0 1 2 3 40

0.5

1

1.5

2

2.5

3

C ost Function1 y'*y

Cost Function2 u'*u

W idth = 0.1W idth = 0.2W idth = 0.3

Control Energy

Vibr

atio

n En

ergy

Performance evaluation by “Pareto-frontier” Comparison of the regulators

saturation of quality


Conclusion

• Japanese market and the marketing method of a piezoelectric device is introduced.

• The techniques to achieve efficient nonlinear piezoelectric analysis using “ANSYS-MATLAB/Simulink coupling” is proposed.

• “ANSYS-MATLAB/Simulink coupling” can efficiently optimize the controller design.

• This method enables us to built the material model of the research stage such as hysteresis into ANSYS !!


Thank you

2008 International ANSYS Conference · input is smaller than past input. The energy is accumulated...

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