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**Nonlinear Magnetostrictive Transducer: Actuator and Sensor Characterization**

© 2013 COMSOL. All rights reserved.

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Magnetostriction Joule effect – Magnetostrictive materials exhibit free strain (λ) when exposed to magnetic field (H) Villari effect – The magnetization (M) in this type of materials can be changed by applying mechanical stress (σ) in the presence of a bias magnetic field Applications Acoustic transducer: SONAR, Hydrophone, Ultrasonic cleaning, Ultrasonic friction welding, Actuator: Pump, Active valve, Rotary motor Sensor: Vibration sensor, Position sensor, Torque sensor

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**Magnetostrictive Transducer**

Piston Steel housing Drive coil Magnetostrictive rod

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**Description of transducer**

The transducer has a magnetostrictive core Magnetic field is applied to the magnetostrictive core by a current-carrying drive coil around it The transducer has a steel housing which encloses the drive coil and the core The steel housing, piston and core acts as a closed magnetic flux path

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Modeling objectives In this tutorial we will look at how the non-linear bi-directional magneto-mechanical coupling involved in magnetostrictive transduction can be modeled in COMSOL Case studies: Actuator characterization: Transducer displacement as a function of coil current for different preloads on the piston Evaluation of the blocked force: Clamp the piston and evaluate the force experienced by it as a function of coil current Sensor characterization: Change in magnetic flux density in the magnetostrictive material as a function of axial load on the piston for different DC bias coil current

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Model features 2D axial symmetry geometry used to reduce computation time and memory Drive coil is modeled as a homogenized current carrying domain where the individual wires are not resolved The magnetic and structural simulations do not include time-dependent effects (inertial force, eddy current, damping, etc.) A parametric sweep of current in the drive coil and total force applied on the piston is used to represent quasi-static operation of the transducer Effects of both positive (tension) and negative (compression) loads are studied Effect of only positive coil current is studied as magnetostrictive response is symmetric with respect to the direction of magnetic field Non-linear constitutive relation is implemented by using interpolated data of B(σ,H) and λ(σ,H)

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**Coupled constitutive relations**

The magneto-mechanical coupling takes place due to quantum mechanical interaction but can be modeled at macro-level using magnetomechanically coupled constitutive relations Structural constitutive equation Magnetic constitutive equation ε: Total strain σ: Mechanical stress E: Young’s modulus at magnetic saturation λ: Magnetoelastic strain H: Magnetic field B: Magnetic flux density μo: Magnetic permeability of free space M: Magnetization

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References The modeling approach shown here is based on the following references C. Mudivarthi, S. Datta, J. Atulasimha and A. B. Flatau, “A bidirectionally coupled magneto-elastic model and its validation using a Galfenol unimorph sensor,” Smart Materials and Structures, Vol. 17(3) , 2008. F. C. Graham, C. Mudivarthi, S. Datta and A. B. Flatau, “Development and validation of a bidirectionally coupled magnetoelastic actuator model,” Smart Materials and Structures, Vol. 18(10), , 2009. The material data was obtained from the following reference S. Datta, J. Atulasimha and A. B. Flatau, “Figures of merit of magnetostrictive single crystal iron-gallium alloys for actuator and sensor applications,” Journal of Magnetism and Magnetic Materials, Vol. 321, No. 24, pp , 2009.

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**Modeling steps The following slides show the important modeling steps**

More details can be seen in the accompanying model file: nonlinear_magnetostriction_transducer_43b.mph

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**First few steps Physics interfaces: AC/DC > Magnetic Fields (mf)**

Structural Mechanics > Solid Mechanics (solid) Domain ODEs and DAEs (dode) Change the Field name and Dependent variables name to Lambda

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**Global Definitions > Parameters**

We will later use the parameters I0 and Fz in a Parametric Sweep to vary the current in the drive coil and the load on the piston

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**Implementing the constitutive relations**

The non-linear constitutive relations require us to provide information on M(σ,H) and λ(σ,H) as a model input M(σ,H) and λ(σ,H) can be obtained from experimental data or from nonlinear magnetostriction material models for a set of discrete values of (σ,H) We would need to reformat the data to get H(B, σ) and λ(B, σ) to use it in COMSOL This is better suited to work with how the constitutive equations are implemented in COMSOL

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**How should we arrange the data?**

Grid-data format in COMSOL Our data file should be arranged as follow

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**Global Definitions > Functions > Interpolation**

Import the data file that contain H(B,σ) in a grid format

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**Global Definitions > Functions > Interpolation**

Import the data file that contain λ(B,σ) in a grid format

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Create the geometry See model file for dimensional details

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**2D axisymmetric representation of 3D**

Piston Steel housing Drive coil Magnetostrictive rod Air domain to model magnetic flux leakage

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**Magnetic Fields – solve on all domains**

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**Domain ODEs and DAEs – solve on Domain 3**

Check the option to see the Discretization

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Evaluate Lambda Here the variable Lambda at each point in space is evaluated from the interpolation function L_B_Stress using the arguments mf.normB (magnitude of the B-field) and solid.sz (z-component of stress)

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**Solid Mechanics – solve on Domains 2-4**

Domain 3 only Lambda is obtained from the Domain ODEs and DAEs interface Magnetostriction can be assumed to be a volume-preserving inelastic strain, hence its values along the x and y-directions are –Lambda/2

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**Why can’t we directly use the interpolation function to specify λ(B,σ) as an initial strain?**

COMSOL computes the stress (σ) from the following equation Here the initial strain If the initial strain (εi) is directly specified as a function of stress then this creates a circular dependency preventing COMSOL from evaluating both stress and initial strain The variable Lambda in the Domain ODE and DAE interface is introduced to circumvent this problem by creating extra degrees of freedom which are obtained as a function of stress and B-field In the Solid Mechanics interface the initial strain becomes an “indirect” function of stress The problem expressed in this form can now be handled by COMSOL’s nonlinear solver

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**Solid Mechanics – boundary conditions**

Boundary Load is applied on the top surface of the piston (Boundary 10) The same boundary is Fixed for evaluating the transducer blocked force

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**Materials Air Soft Iron (without losses) Magnetostrictive**

Select from Material Browser > Built-in > Air Soft Iron (without losses) Select from Material Browser > ACDC > Soft Iron (without losses) Assign it to Domains 2 and 4 (transducer housing and piston) Magnetostrictive Right-click Materials and select Material Rename it to Magnetostrictive Assign it to Domain 3 (Galfenol rod)

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**Magnetostrictive material properties**

These are typical properties of Galfenol (Fe81Ga19) Note how the function H_B_Stress is called with arguments normB and solid.sz

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Mesh Domains 2, 3 and 4 have a significantly finer mesh to help resolve the material nonlinearities

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**Study 1 – Actuator characterization**

Turn-off parametric solver to improve convergence as Fz is varied from negative to positive values Disable Fixed Constraint 2 in Study 1 Right-click Study 1 and select Show Default Solver prior to computing Increase the number of Newton iterations so that the Nonlinear solver does not stop prematurely

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Results Choose the value of Load and Drive current to see the results for different operating conditions

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**Structural displacement and Magnetic flux density**

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**Magnetic flux density and Stress**

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**Magnetic flux density (mf. normB) vs**

Magnetic flux density (mf.normB) vs. Drive current (I0) at different Preloads (Fz) evaluated at the center of the rod (Point 4) Saturation effect Stress-induced anisotropy

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**Magnetostriction (Lambda) vs**

Magnetostriction (Lambda) vs. Drive current (I0) at different Preloads (Fz) evaluated at the center of the rod (Point 4) Stress-induced magnetoelastic strain (λs) Magnetic moments are randomly oriented in the absence of pre-load Pre-load aligns the magnetic moments along an easy axis in a plane perpendicular to the direction of magnetic field thereby yielding the saturation magnetostriction of (3/2)λs (3/2)λs

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**Study 2 – Transducer blocked force**

Disable Boundary Load 1 in Study 2 All other setting are similar to Study 1

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**Evaluating the Blocked force**

Find the total reaction force on the top surface of the piston that has been fixed (clamped) Create a plot of the data is this Table

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**Results – Blocked force**

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**Study 3 – Sensor characterization**

Setting are similar to Study 1

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**Magnetic flux density (mf. normB) vs**

Magnetic flux density (mf.normB) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4) I0 = 0: No sensing I0 > 0: Higher load is required to overcome the magnetic field-induced anisotropy Bias current can be used to control sensitivity and operating range

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Strain (solid.eZ) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4) Delta E-effect: Apparent change in elastic modulus due to rotation of magnetic moments Stiffness can be tuned with bias current and load Useful for tunable acoustic waveguide

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**Magnetic field (mf. normH) vs**

Magnetic field (mf.normH) vs. Load (Fz) at different bias current (I0) evaluated at the center of the rod (Point 4) Contrary to popular belief, a constant current in the coil does not create constant field inside the magnetostrictive material!

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Summary This tutorial showed how to simulate a complete quasi-static transducer characterization of a magnetostrictive material The model incorporates full nonlinear material properties and bidirectional physics coupling between structural and magnetic inputs and response Key points: How to implement the nonlinear constitutive relations How to use nonlinear material data as a function stress and B-field How to create the appropriate mesh and solver settings How to create the important transducer characterization plots

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