Early Career Technical Conference 2016

Heat Transfer Modelling and Bandwidth Determination of SMA Actuators in Robotics Applications

Tyler Ross LambertAuburn University

Department of Mechanical Engineering

Austin Gurley and David Beale

1

Introduction2

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions150 g mass lifted by SMA actuator (21 ksi of pressure)

Shape Memory Alloy (SMA) Background

• Shape Memory Alloys (SMA) are specially alloyed materials that change crystalline structure when heated and cooled, or when stressed and relaxed, which results in the alloy contracting with large force.

• Why use an SMA actuator?

• SMA wire actuators can be driven via heating through the use of an electric current, eliminating noise during operation.

• SMA wire actuators can act as their own built-in position sensor, drastically reducing costs in robotic designs.

• Nickel Titanium alloy (Nitinol), a common SMA, is relatively cheapand robust.

3

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Objectives of Analysis

Objectives Model SMA bandwidth in terms of wire size Can an SMA actuator move fast enough to

work in your robotic application?

Model SMA efficiency in terms of sizeWhat are the power demands of using an

SMA actuator in your robotic application?

Bandwidth and efficiency are the two main drawbacks with SMA actuators.

4

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Crystalline Phase Changes

Crystalline Phases Martensite Phase

Characterized by colder temperatures and higher stresses Required some deformation from preload to avoid “twinned

Martensite”

Austenite Phase Characterized by higher temperatures and lower stresses

5

[1]

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

SMA Phase Transformation Diagram6

[2]

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Super-Elastic and Shape Memory Effects7

[2]

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Analysis8

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Modelling Why is it important?

Accurate heat transfer model can allow for powerful predictions to be made for several system properties. Thermal time constant/eigenvalue → bandwidth

Input power → efficiency

Temperature Response → rise time

How it was done Energy balance from First Law of Thermodynamics and use

empirical models.

Heat Transfer Energy Balance

m - mass of the wirecp - specific heat of the SMAΔH - change in energy associated with a phase transformation (“latent heat of transformation”)ξ – phase fraction (percent martensite)T - uniform temperature of the wiret - timeI - current through the wireR(ξ) - resistance in the wire as a function of its phase fractionh - convection coefficient between the wire surface and the surrounding fluidAs - surface area of the wire in contact with the surrounding fluidT∞ - temperature of the ambient fluid surrounding the wire

𝑚𝑚 𝑐𝑐𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝛥𝛥𝛥𝛥�̇�𝜉 = 𝐼𝐼2𝑅𝑅 𝜉𝜉 − ℎ𝐴𝐴𝑠𝑠 𝑑𝑑 − 𝑑𝑑∞

The behavior of an SMA actuator driven by an electrical input and cooled via convection is given by:

9

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Analysis Assumptions

Closed form solutions for this equation exist when the following assumptions are made:

The wire has a uniform temperature.

The wire is long enough so that boundary effects can be ignored at anchor points [wire must be greater than 148.8 mm for this assumption to be true (Furst 2012)].

The wire operates safely outside of the transformation bound (so the latent heat of transformation can be neglected).

The crystalline phase fraction is constant throughoutthe wire (�̇�𝜉 = 0).

10

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Simplified Model

𝑚𝑚𝑐𝑐𝑝𝑝𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

= 𝐼𝐼2𝑅𝑅 − ℎ𝐴𝐴𝑠𝑠 𝑑𝑑 − 𝑑𝑑∞

The thermal behavior of an SMA actuator given these assumptions is given by the simplified equation:

The closed form solution is then given by:

𝑑𝑑 𝑑𝑑 = 𝑑𝑑∞ +𝐼𝐼2𝑅𝑅ℎ𝐴𝐴𝑠𝑠

+ 𝑑𝑑0 − 𝑑𝑑∞ −𝐼𝐼2𝑅𝑅ℎ𝐴𝐴𝑠𝑠

𝑒𝑒− ℎ𝐴𝐴𝑠𝑠𝑚𝑚𝑐𝑐𝑝𝑝

11

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Simplified Model

The equation can be further simplified by noting:

𝐴𝐴𝑠𝑠 = 𝜋𝜋𝑑𝑑𝜋𝜋 and 𝑚𝑚 = 𝜌𝜌𝜋𝜋𝑑𝑑2

2

𝜋𝜋

The closed form solution now takes the form:

And for the homogenous case where the wire is not being electrically heated:

This model is only as valuable as the approximation for h.

𝑻𝑻 𝒕𝒕 = 𝑻𝑻∞ +𝑰𝑰𝟐𝟐𝑹𝑹𝐡𝐡𝝅𝝅𝝅𝝅𝝅𝝅

+ 𝑻𝑻𝟎𝟎 − 𝑻𝑻∞ −𝑰𝑰𝟐𝟐𝑹𝑹𝐡𝐡𝝅𝝅𝝅𝝅𝝅𝝅

𝒆𝒆− 𝟒𝟒𝟒𝟒𝝆𝝆𝝅𝝅𝒄𝒄𝒑𝒑

d – wire diameterL – wire length𝜌𝜌 – wire density

𝑻𝑻 𝒕𝒕 = 𝑻𝑻∞ + 𝑻𝑻𝟎𝟎 − 𝑻𝑻∞ 𝒆𝒆− 𝟒𝟒𝟒𝟒𝝆𝝆𝝅𝝅𝒄𝒄𝒑𝒑

12

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Coefficient

The heat transfer coefficient, h, is defined as:

ℎ =𝑘𝑘𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 Nu𝐷𝐷

𝑑𝑑

The thermal conductivity of a fluid is usually tabulated for a given temperature, but the Nusselt number must be found using empirical formulas.

𝑘𝑘fluid – thermal conductivity of the ambient fluidNu𝐷𝐷– surface averaged Nusselt numberd – wire diameter

13

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Empirical Models for Nusselt Number

Empirical Models for Heat Transfer Coefficient Forced Convection

Churchill-Bernstein Relationship Valid for a cylinder in a crossflow where Re𝐷𝐷Pr ≥ 0.2

Nu𝐷𝐷 = 0.3 + 0.62Re𝐷𝐷1/2Pr1/3

1+ 0.4Pr

2/3 1/4 1 + Re𝐷𝐷282000

5/8 4/5

Natural Convection Horizontal Cylinder

Nu𝐷𝐷 = 0.6 + 0.387Ra1/6

1+ 0.559Pr

9/16 8/27

2

Vertical Cylinder 𝑑𝑑 > 35𝐿𝐿

(RaPr)1/4

Nu𝐷𝐷 = 0.825 + 0.387Ra1/6

1+ 0.492Pr

9/16 8/27

2

𝐏𝐏𝐏𝐏 - Prandtl Number𝐑𝐑𝐑𝐑 – Rayleigh Number𝑹𝑹𝒆𝒆𝑫𝑫 - Reynolds Number

14

[3]

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Cooling Response Comparison15

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Forced Convection Natural (Free) Convection

Intuitively, forced convection results a much faster cooling rate.

Heat Transfer Coefficient

The heat transfer coefficient can then be approximated for the wire by substituting in for the wire properties and assuming the wire is cooling via natural convection in still air.

We reduce the model to the following form:

ℎ 𝑑𝑑, 𝑑𝑑∞ ,𝑑𝑑 = 65.5𝑒𝑒−𝑓𝑓4(𝑑𝑑− 𝑑𝑑∞)

16

Wm2K

d – wire diameter in mm

16

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Heat Transfer Coefficient17

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Cooling/Heating Bandwidth

The cooling/heating bandwidth of the system reflects how fast thesystem input (either the ambient temperature or electrical power)can be cycled before the ability of the wire to cool itself isimpeded.

This quantity can be found from the time constant from the originaldifferential equation:

𝑓𝑓−3 𝑓𝑓𝑑𝑑 =1

2𝜋𝜋 𝜏𝜏 = λ𝑐𝑐𝑐𝑐𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐 =4ℎ

2𝜋𝜋𝜌𝜌𝑑𝑑𝑐𝑐𝑝𝑝 This metric allows for an estimate of the transformation bandwidth

by comparing how much the thermal response can be attenuated tothe frequency at which the SMA actuator will not undergo a phasetransformation.

18

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Cooling/Heating Bandwidth

For natural convection: λ𝑡𝑡ℎ𝑡𝑡𝑡𝑡𝑚𝑚𝑡𝑡𝑓𝑓 ≈0.0086𝑑𝑑2

19

λ = 0.0086d-2

λ = 0.0926d-1.562

0

1

2

3

4

5

6

0 0.05 0.1 0.15 0.2 0.25 0.3

Ther

mal

Tra

nsfo

rmat

ion

Band

wid

th (H

z)

Wire Diameter (mm)

Thermal Transformation Bandwidths (Hz)

25°C, still air

35°C, still air

20°C, v = 2 m/s

Ambient Air

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Transformation Bandwidth

The cooling bandwidth underestimates the transformation bandwidth

Neglects heat of transformation

Does not account for the additional thermal signal attenuation the system can handle

20

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

𝑑𝑑𝑀𝑀𝑓𝑓

𝜏𝜏𝑐𝑐𝑐𝑐𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐

𝑑𝑑𝐴𝐴𝑓𝑓

𝑑𝑑𝑀𝑀𝑓𝑓𝑑𝑑𝐴𝐴𝑓𝑓

32

Transformation Bandwidth

More accurate bandwidth determination can be obtained by analyzing the cooling response when the wire is hot and finding the time taken for the wire to reach the Martensitic transformation bound:

λ𝑡𝑡𝑡𝑡𝑡𝑡𝑐𝑐𝑠𝑠𝑓𝑓𝑐𝑐𝑡𝑡𝑚𝑚 =1

2𝑑𝑑𝑀𝑀 This makes several assumptions

The wire undergoes constant external stress Film temperature of surrounding air remains constant at

all points in time Heating time is the same as cooling time

21

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Transformation Bandwidth

The transformation bandwidth was then found for three common cases for several wire diameters.

For most wires, the empirical scheme derived for the horizontal wire suffices for modelling bandwidth:

𝜆𝜆𝑡𝑡𝑡𝑡𝑡𝑡𝑐𝑐𝑠𝑠𝑓𝑓𝑐𝑐𝑡𝑡𝑚𝑚 =0.0099𝑑𝑑2

, 𝑑𝑑 <35𝜋𝜋

(RaPr)1/4

22

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

General Rules for SMA Bandwidth23

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Increase in: Bandwidth

Air Speed ↗Air Temperature ↘Wire Diameter ↘

Bandwidth increases as convective heat transfer increases.

Bandwidth decreases as heating/cooling times increase with larger diameter wires.

SMA Actuator Efficiency Equation

Nitinol wire characteristics Transformation strain with no external stress (𝜀𝜀𝐿𝐿): 4%

Transformation Contraction Stress (Ω): 150 MPa

Latent Heat of Transformation (∆𝛥𝛥): 24.2 J/g

The work done upon transformation of the actuator is then:𝑊𝑊 = 𝜋𝜋𝜀𝜀𝐿𝐿Ω𝐴𝐴

The electrical power required to actuate the Nitinol can be approximated by the power lost to convection plus the latent transformation energy plus the energy required to raise the wire temperature. The wire efficiency can then be calculated as:

𝜂𝜂 =𝜀𝜀𝐿𝐿Ω𝑑𝑑

4ℎ(𝑑𝑑 − 𝑑𝑑∞)𝑑𝑑 + 𝜌𝜌𝑑𝑑(𝑐𝑐𝑝𝑝∆𝑑𝑑 + ∆𝛥𝛥)

24

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

SMA Actuator Efficiency Simplified Model

Assume 𝑑𝑑 ≈ 𝑚𝑚 𝑐𝑐𝑝𝑝∆𝑇𝑇+ ∆𝐻𝐻𝑉𝑉𝑉𝑉

= 𝜌𝜌𝜌𝜌𝑓𝑓2𝐿𝐿 𝑐𝑐𝑝𝑝∆𝑇𝑇+ ∆𝐻𝐻4𝑉𝑉𝑉𝑉

, then the efficiency and transformation time can be found from only known quantities:

25

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

General Rules for SMA Efficiency

Efficiency typically ranges from 1% - 3% for NiTi alloys.

26

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Increase in: Efficiency

Air Speed ↗Wire Diameter ↗

Input Power ↗Wire Length ↘

Air Temperature ↘

Experimental Results

Experimental Setup Testing Apparatus: Single leg of 18 DOF Hexapod Robot

Wire Diameter: 0.125 mm

Wire Length: 60 mm

Heating method: PWM output from microcontroller with sinusoidal sweep of duty cycle

Sensors: Self-Sensing Probe

27

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions Powered Terminal Blocks

Moving Hinge

Self-Sensing Probe

Antagonist Springs

Experimental Results

𝝀𝝀𝒎𝒎𝒆𝒆𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒎𝒆𝒆𝝅𝝅 = 𝟎𝟎.𝟔𝟔𝟔𝟔 𝐇𝐇𝐇𝐇

28

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Predictions

𝝀𝝀𝒕𝒕𝟒𝟒𝒆𝒆𝒎𝒎𝒎𝒎𝒎𝒎𝒕𝒕 =𝟎𝟎.𝟎𝟎𝟎𝟎𝟎𝟎𝟔𝟔

(𝟎𝟎.𝟏𝟏𝟐𝟐𝟔𝟔)𝟐𝟐= 𝟎𝟎.𝟔𝟔𝟔𝟔𝟎𝟎𝟒𝟒 𝐇𝐇𝐇𝐇

𝝀𝝀𝒕𝒕𝒎𝒎𝒎𝒎𝒕𝒕𝒎𝒎𝒕𝒕𝒕𝒕𝒎𝒎𝒎𝒎 ≈𝟎𝟎.𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎

(𝟎𝟎.𝟏𝟏𝟐𝟐𝟔𝟔)𝟐𝟐= 𝟎𝟎.𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔 𝐇𝐇𝐇𝐇

Percent Error: 3%

Conclusions

Bandwidth can be computed to within three percent error for SMA actuators. This information can be used to size up SMA

actuators depending on the needs of the project and can help when designing a controller to control these systems.

The efficiency of an SMA actuator can be modelled and approximated This information helps gauge the power needs to

maintain a system of SMA actuators.

29

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Demonstrations using SMA Actuators30

18 DOF Hexapod Robot

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Demonstrations using SMA Actuators31

Ball-Beam Balancer

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Demonstrations using SMA Actuators32

Actuated Gimbal forSolar Panel Alignment

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Human Hand Replica

Small Bug

References33

[3] The McGraw-Hill Companies, Inc. Heat and Mass Transfer: Fundamentals & Applications Fourth Edition in SI Units Yunus A. Cengel, Afshin J. Ghajar McGraw-Hill, 2011

[1] Alchetron. Alchetron Technologies Pvt. Ltd. “Nickel Titanium”. 2016. http://alchetron.com/Nickel-titanium-156127-W

[2] Gurley, Austin. Auburn University. “Robust Self Sensing in NiTi Actuators Using a Dual Measurement Technique”. SMASIS Conference on Smart Materials, Adaptive Structures and Intelligent Systems. 2016.

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions

Questions and Discussion34

Introduction

Background

Heat Transfer

Bandwidth

Efficiency

Experiments

Conclusions