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Transcript of ECTC Presentation
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
β’ 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.
β’ Why use an SMA actuator?
β’ SMA wire actuators can be driven via heating through the use of an electric current, eliminating noise during operation.
β’ SMA wire actuators can act as their own built-in position sensor, drastically reducing costs in robotic designs.
β’ 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 βtwinned
Martensiteβ
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 β bandwidth
Input power β efficiency
Temperature Response β 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 SMAΞH - change in energy associated with a phase transformation (βlatent heat of transformationβ)ΞΎ β phase fraction (percent martensite)T - uniform temperature of the wiret - timeI - current through the wireR(ΞΎ) - 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 fluidTβ - temperature of the ambient fluid surrounding the wire
ππ ππππππππππππ + π₯π₯π₯π₯οΏ½ΜοΏ½π = πΌπΌ2π π ππ β βπ΄π΄π π ππ β ππβ
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 (οΏ½ΜοΏ½π = 0).
10
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
Heat Transfer Simplified Model
ππππππππππππππ
= πΌπΌ2π π β βπ΄π΄π π ππ β ππβ
The thermal behavior of an SMA actuator given these assumptions is given by the simplified equation:
The closed form solution is then given by:
ππ ππ = ππβ +πΌπΌ2π π βπ΄π΄π π
+ ππ0 β ππβ βπΌπΌ2π π βπ΄π΄π π
ππβ βπ΄π΄π π ππππππ
11
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
Heat Transfer Simplified Model
The equation can be further simplified by noting:
π΄π΄π π = ππππππ and ππ = ππππππ2
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.
π»π» ππ = π»π»β +π°π°πππΉπΉπ‘π‘π π π π π π
+ π»π»ππ β π»π»β βπ°π°πππΉπΉπ‘π‘π π π π π π
ππβ πππππππ π ππππ
d β wire diameterL β wire lengthππ β wire density
π»π» ππ = π»π»β + π»π»ππ β π»π»β ππβ πππππππ π ππππ
12
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
Heat Transfer Coefficient
The heat transfer coefficient, h, is defined as:
β =ππππππππππππ Nuπ·π·
ππ
The thermal conductivity of a fluid is usually tabulated for a given temperature, but the Nusselt number must be found using empirical formulas.
ππfluid β thermal conductivity of the ambient fluidNuπ·π·β surface averaged Nusselt numberd β 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 Reπ·π·Pr β₯ 0.2
Nuπ·π· = 0.3 + 0.62Reπ·π·1/2Pr1/3
1+ 0.4Pr
2/3 1/4 1 + Reπ·π·282000
5/8 4/5
Natural Convection Horizontal Cylinder
Nuπ·π· = 0.6 + 0.387Ra1/6
1+ 0.559Pr
9/16 8/27
2
Vertical Cylinder ππ > 35πΏπΏ
(RaPr)1/4
Nuπ·π· = 0.825 + 0.387Ra1/6
1+ 0.492Pr
9/16 8/27
2
ππππ - Prandtl Numberππππ β Rayleigh NumberπΉπΉπππ«π« - 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:
β ππ, ππβ ,ππ = 65.5ππβππ4(ππβ ππβ)
16
Wm2K
d β 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:
ππβ3 ππππ =1
2ππ ππ = Ξ»ππππππππππππππ =4β
2ππππππππππ 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: Ξ»π‘π‘βπ‘π‘π‘π‘πππ‘π‘ππ β0.0086ππ2
19
Ξ» = 0.0086d-2
Ξ» = 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)
25Β°C, still air
35Β°C, still air
20Β°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
ππππππ
ππππππππππππππππ
πππ΄π΄ππ
πππππππππ΄π΄ππ
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:
Ξ»π‘π‘π‘π‘π‘π‘πππ π πππππ‘π‘ππ =1
2ππππ 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:
πππ‘π‘π‘π‘π‘π‘πππ π πππππ‘π‘ππ =0.0099ππ2
, ππ <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 βAir Temperature βWire Diameter β
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 (πππΏπΏ): 4%
Transformation Contraction Stress (Ξ©): 150 MPa
Latent Heat of Transformation (βπ₯π₯): 24.2 J/g
The work done upon transformation of the actuator is then:ππ = πππππΏπΏΞ©π΄π΄
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:
ππ =πππΏπΏΞ©ππ
4β(ππ β ππβ)ππ + ππππ(ππππβππ + βπ₯π₯)
24
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
SMA Actuator Efficiency Simplified Model
Assume ππ β ππ ππππβππ+ βπ»π»ππππ
= ππππππ2πΏπΏ ππππβππ+ βπ»π»4ππππ
, 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 βWire Diameter β
Input Power βWire Length β
Air Temperature β
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
πππππππππππππππππ π = ππ.ππππ ππππ
28
Introduction
Background
Heat Transfer
Bandwidth
Efficiency
Experiments
Conclusions
Predictions
ππππππππππππππππ =ππ.ππππππππ
(ππ.ππππππ)ππ= ππ.ππππππππ ππππ
ππππππππππππππππππππ βππ.ππππππππ
(ππ.ππππππ)ππ= ππ.ππππππππ ππππ
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. βNickel Titaniumβ. 2016. http://alchetron.com/Nickel-titanium-156127-W
[2] Gurley, Austin. Auburn University. βRobust Self Sensing in NiTi Actuators Using a Dual Measurement Techniqueβ. 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