Nonlinear Magnetostrictive Transducer: Actuator and Sensor Characterization

© 2013 COMSOL. All rights reserved.

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

Magnetostrictive Transducer

Steel housing

Drive coil

Magnetostrictive rod

Piston

• 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

Description of transducer

• 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

Modeling objectives

• 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)

Model features

Coupled constitutive relations

• Structural constitutive equation

ε: Total strainσ: Mechanical stressE: Young’s modulus at magnetic saturationλ: Magnetoelastic strain

• Magnetic constitutive equation

HMHB oo , H: Magnetic fieldB: Magnetic flux densityμo: Magnetic permeability of free spaceM: Magnetization

HE ,

• The magneto-mechanical coupling takes place due to quantum mechanical interaction but can be modeled at macro-level using magnetomechanically coupled constitutive relations

References

The modeling approach shown here is based on the following references1. 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) 035005, 2008.http://iopscience.iop.org/0964-1726/17/3/035005

2. 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), 104013, 2009.http://iopscience.iop.org/0964-1726/18/10/104013

The material data was obtained from the following reference3. 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. 4017-4031, 2009.http://www.sciencedirect.com/science/article/pii/S0304885309008075

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

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

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

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

How should we arrange the data?

• Grid-data format in COMSOL

• Our data file should be arranged as follow

Global Definitions > Functions > Interpolation

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

Global Definitions > Functions > Interpolation

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

Create the geometry

See model file for dimensional details

2D axisymmetric representation of 3D

Steel housing

Drive coil

Magnetostrictive rod

Piston

Air domain to model magnetic flux leakage

Magnetic Fields – solve on all domains

Domains 2, 3, 4

Domain 5

Domain ODEs and DAEs – solve on Domain 3

Check the option to see the Discretization

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)

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

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

iE

,Bi

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

Materials

• Air– 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)

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

Mesh

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

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

Results

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

Structural displacement and Magnetic flux density

Magnetic flux density and Stress

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

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

Study 2 – Transducer blocked force

• Disable Boundary Load 1 in Study 2

• All other setting are similar to Study 1

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

Results – Blocked force

Study 3 – Sensor characterization

• Setting are similar to Study 1

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

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

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!

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