The Constitutive Equations of a Piezoelectric Duobimorph

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The Constitutive Equations of a Piezoelectric
Duo-bimorph

• Pierre De Lit
• CAD/CAM Department
• Université libre de Bruxelles
• ISATP2003, July 10, 2003

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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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The duo-bimorph is a monolithic piezoelectric
actuator
L
upper electrodes
piezoelectricplates
glue
z
upper electrodes
y
polarisation of the piezoelectrics
lower electrodes
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The duo-bimorph yields two DOFs
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The duo-bimorph yields two DOFs
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The duo-bimorph yields two DOFs
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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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The model takes very general situations into
account

• Each electrode can be set at a given potential
• The piezoelectric plates can be different in
  height and material
• The glue layer is taken into account
• The electrodes are not exactly centered

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Internal parameters are expressed as a function
of external parameters
slope
deflexion
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The fundamental equation describes the behaviour
of the piezoelectric ceramic
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Associated equations describe the behaviour of
clamped beams
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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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The equations are combined to yield the behaviour
of the duo-bimorph
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Equations are analytically integrated
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After integration and simplification

• We set

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The curvature of the beam yields the deflexions
and slopes
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Four constitutive coefficients describing the
slope can then be determined
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Four constitutive coefficients describing the
deflexions can then be determined
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The equation describing the deflexion due to an
applied force is established
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Integrating the differential equation yields the
slopes and the deflexions
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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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The centroid along z axis is not at the origin
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The equations describing the duo-bimorph
behaviour are transposed
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The equations must be integrated taking into
account the position of the centroid
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The solving principle stays the same expressions
are just longer
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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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We finally obtain the complete behaviour matrix
of the duo-bimorph
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What will we talk about?

• The duo-bimorph
• Assumptions, parameters and fundamental equations
• Behaviour along y axis
• Behaviour along z axis
• Complete model
• Conclusions

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Let us conclude

• We developed a complete model describing the
  behaviour of the duo-bi
• Former results (Smits, Yao) are retrieved when
  the model is particularised
• identical plates
• one applied voltage
• no glue layer