1. Early Career Technical Conference 2016
Heat Transfer Modelling and Bandwidth Determination of
SMA Actuators in Robotics Applications
Tyler Ross Lambert
Auburn University
Department of Mechanical Engineering
Austin Gurley and David Beale
1
3. 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 cheap
and robust.
3
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
4. 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 size
What 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
5. 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
6. SMA Phase Transformation Diagram
6
[2]
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
7. Super-Elastic and Shape Memory Effects
7
[2]
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
8. Heat Transfer Analysis
8
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.
9. Heat Transfer Energy Balance
m - mass of the wire
cp - specific heat of the SMA
ΔH - change in energy associated with a phase transformation (“latent heat of transformation”)
ξ – phase fraction (percent martensite)
T - uniform temperature of the wire
t - time
I - current through the wire
R(ξ) - resistance in the wire as a function of its phase fraction
h - convection coefficient between the wire surface and the surrounding fluid
As - surface area of the wire in contact with the surrounding fluid
T∞ - temperature of the ambient fluid surrounding the wire
𝑚𝑚 𝑐𝑐𝑝𝑝
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
+ 𝛥𝛥𝛥𝛥 ̇𝜉𝜉 = 𝐼𝐼2
𝑅𝑅 𝜉𝜉 − ℎ𝐴𝐴𝑠𝑠 𝑇𝑇 − 𝑇𝑇∞
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
10. 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 throughout
the wire ( ̇𝜉𝜉 = 0).
10
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
11. Heat Transfer Simplified Model
𝑚𝑚𝑐𝑐𝑝𝑝
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
= 𝐼𝐼2 𝑅𝑅 − ℎ𝐴𝐴𝑠𝑠 𝑇𝑇 − 𝑇𝑇∞
The thermal behavior of an SMA actuator given these
assumptions is given by the simplified equation:
The closed form solution is then given by:
𝑇𝑇 𝑡𝑡 = 𝑇𝑇∞ +
𝐼𝐼2 𝑅𝑅
ℎ𝐴𝐴𝑠𝑠
+ 𝑇𝑇0 − 𝑇𝑇∞ −
𝐼𝐼2 𝑅𝑅
ℎ𝐴𝐴𝑠𝑠
𝑒𝑒
−
ℎ𝐴𝐴𝑠𝑠
𝑚𝑚𝑐𝑐𝑝𝑝
11
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
12. Heat Transfer Simplified Model
The equation can be further simplified by noting:
𝐴𝐴𝑠𝑠 = 𝜋𝜋𝜋𝜋𝜋𝜋 and 𝑚𝑚 = 𝜌𝜌𝜌𝜌
𝑑𝑑
2
2
𝐿𝐿
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∞ +
𝑰𝑰𝟐𝟐
𝑹𝑹
𝐡𝐡𝝅𝝅𝝅𝝅𝝅𝝅
+ 𝑻𝑻𝟎𝟎 − 𝑻𝑻∞ −
𝑰𝑰𝟐𝟐
𝑹𝑹
𝐡𝐡𝝅𝝅𝝅𝝅𝝅𝝅
𝒆𝒆
−
𝟒𝟒𝟒𝟒
𝝆𝝆𝝆𝝆𝒄𝒄𝒑𝒑
d – wire diameter
L – wire length
𝜌𝜌 – wire density
𝑻𝑻 𝒕𝒕 = 𝑻𝑻∞ + 𝑻𝑻𝟎𝟎 − 𝑻𝑻∞ 𝒆𝒆
−
𝟒𝟒𝟒𝟒
𝝆𝝆𝝆𝝆𝒄𝒄𝒑𝒑
12
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
13. Heat Transfer Coefficient
The heat transfer coefficient, h, is defined as:
ℎ =
𝑘𝑘𝑓𝑓𝑓𝑓 𝑓𝑓𝑓𝑓 𝑓𝑓 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 fluid
Nu𝐷𝐷– surface averaged Nusselt number
d – wire diameter
13
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
14. 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 +
0.62Re𝐷𝐷
1/2
Pr1/3
1+
0.4
Pr
2/3 1/4 1 +
Re 𝐷𝐷
282000
5/8 4/5
Natural Convection
Horizontal Cylinder
Nu𝐷𝐷 = 0.6 +
0.387Ra1/6
1+
0.559
Pr
9/16 8/27
2
Vertical Cylinder 𝑑𝑑 >
35𝐿𝐿
(
Ra
Pr
)1/4
Nu𝐷𝐷 = 0.825 +
0.387Ra1/6
1+
0.492
Pr
9/16 8/27
2
𝐏𝐏𝐏𝐏 - Prandtl Number
𝐑𝐑𝐑𝐑 – Rayleigh Number
𝑹𝑹𝑹𝑹𝑫𝑫 - Reynolds Number
14
[3]
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
16. 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 𝑇𝑇, 𝑇𝑇∞ , 𝑑𝑑 = 65.5𝑒𝑒−
𝑑𝑑
4(𝑇𝑇− 𝑇𝑇∞)
1
6
W
m2K
d – wire diameter in mm
16
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
18. Cooling/Heating Bandwidth
The cooling/heating bandwidth of the system reflects how fast the
system input (either the ambient temperature or electrical power)
can be cycled before the ability of the wire to cool itself is
impeded.
This quantity can be found from the time constant from the original
differential equation:
𝑓𝑓−3 𝑑𝑑𝑑𝑑 =
1
2𝜋𝜋 𝜏𝜏
= λ𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐 𝑐𝑐 =
4ℎ
2𝜋𝜋𝜋𝜋𝜋𝜋𝑐𝑐𝑝𝑝
This metric allows for an estimate of the transformation bandwidth
by comparing how much the thermal response can be attenuated to
the frequency at which the SMA actuator will not undergo a phase
transformation.
18
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
19. Cooling/Heating Bandwidth
For natural convection: λ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 ≈
0.0086
𝑑𝑑2
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
ThermalTransformationBandwidth(Hz)
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
20. 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
𝑇𝑇𝑀𝑀𝑓𝑓
𝜏𝜏𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑐𝑐 𝑐𝑐
𝑇𝑇𝐴𝐴𝑓𝑓
𝑡𝑡 𝑀𝑀𝑓𝑓
𝑡𝑡𝐴𝐴𝑓𝑓
32
21. 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:
λ𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =
1
2𝑡𝑡𝑀𝑀
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
22. 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:
𝜆𝜆𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 =
0.0099
𝑑𝑑2
, 𝑑𝑑 <
35𝐿𝐿
(
Ra
Pr
)1/4
22
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
23. General Rules for SMA Bandwidth
23
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.
24. 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 (∆𝐻𝐻): 24.2 J/g
The work done upon transformation of the actuator is then:
𝑊𝑊 = 𝐿𝐿𝜀𝜀𝐿𝐿Ω𝐴𝐴
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 =
𝜀𝜀𝐿𝐿Ω 𝑑𝑑
4ℎ(𝑇𝑇 − 𝑇𝑇∞)𝑡𝑡 + 𝜌𝜌𝑑𝑑(𝑐𝑐𝑝𝑝∆𝑇𝑇 + ∆𝐻𝐻)
24
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
25. SMA Actuator Efficiency Simplified Model
Assume 𝑡𝑡 ≈
𝑚𝑚 𝑐𝑐𝑝𝑝∆𝑇𝑇+ ∆𝐻𝐻
𝑉𝑉𝑉𝑉
=
𝜌𝜌𝜌𝜌𝑑𝑑2 𝐿𝐿 𝑐𝑐𝑝𝑝∆𝑇𝑇+ ∆𝐻𝐻
4𝑉𝑉𝑉𝑉
, then the efficiency
and transformation time can be found from only known
quantities:
25
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
26. 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 ↘
27. 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
29. 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
30. Demonstrations using SMA Actuators
30
18 DOF Hexapod Robot
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
31. Demonstrations using SMA Actuators
31
Ball-Beam Balancer
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
32. Demonstrations using SMA Actuators
32
Actuated Gimbal for
Solar Panel Alignment
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
Human Hand Replica
Small Bug
33. References
33
[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
